Cpesfestirtee age th ghee es Fae MATHEMATICS LIBRARY NOTICE: Return or renew all Library Materials! The Minimum Fee for each Lost Book is $50.00. The person charging this material is responsible for its return to the library from which it was withdrawn on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for discipli- nary action and may result in dismissal from the University. To renew call Telephone Center, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN MAY 09 RECT L161—O-1096 ie) bee ny te ahs ytd Ge Oe ee Ary (fh ' TREE AAP Yale aN WI ee ied) may uk TREATISE ON PLANE AND SPHERICAL Trigonometry. _ By ROBERT WOODHOUSE, A.M. F.R.S. t : LATE FELLOW OF GONVILLE AND CAIUS COLLEGE, AND PLUMIAN PROFESSOR OF ASTRONOMY iN THE UNIVERSITY OF CAMBRIDGE, THE FIFTH EDITION. 933 =SSea.. Corrected, Altered, and Enlayged. \) 5 i 4 \ y CAMBRIDGE: Printed by J. Smith, Printer to the University; FOR J. & J.J. DEIGHTON; AND G. B. WHITTAKER, AVE-MARIA LANE, LONDON, 1827 "1 mae Pe ae ek is At GT eye 4 \ PREFACE TO THE SECOND EDITION. Ir is not easy to adapt a Treatise on Trigonometry to all descriptions of Students; to state, in its beginning, within a small compass, and with their simplest solutions, those Propositions which relate merely to the cases of oblique-angled triangles, and then, on the ground of those propositions and by the method of their solutions, to proceed to investigations of greater intricacy. The Student, if he be supposed to possess a knowledge of the first six Books of Euclid, may thence, by a few easy inferences, and by the aid of some simple constructions, arrive’ most readily at the Trigonometrical solutions of the cases of oblique-angled triangles. If his views extend no farther, he cannot take a better guide than Ludlam or Robert Simson : nor proceed by any easier method than the Geometrical. But few Students are content with such confined views. Trigonometry is now extended far beyond its original object, and to other investigations than those of the relations of the sides and angles of Triangles. The collateral uses of the science have become the most numerous, and-are not the least- important. To the knowledge of many of these, however, the Geometrical H PREFACE. = method is unable to conduct us. At some point or other of our enquiries (we speak of its present and actual state) it must be abandoned, and recourse be had to that which technically is called the Analytical Method. Since this latter is the sole thoroughly efficient method, will it not be better to make it, in a Treatise on Trigonometry, the predominant one, instead of being compelled merely to call in its aid, when the resources of the former are exhausted ? The Author of the present Treatise has endeavoured to con- struct it on such a plan; and, with this view, he has had as little recourse as possible, to Geometrical constructions and the pro- perties of figures. What he thence has borrowed are not so much to be considered as the first steps in his process of demonstration, as the data and ground-work from which the process itself 1s to commence and to be instituted. . By these means the process is made uniform and systematical. But uniformity may be purchased at too dear a rate; and the main purpose of the Work, which is utility, would be sacrificed, if, for the sake of system, the analytical method were reluctantly compelled to submit to modes of proof that are strange to its nature and genius. The specimens of demonstration contained in the following pages must determine whether or not such sacrifice has been made. The great practical use of Trigonometry is the resolution of rectilinear triangles; but, that it is capable of being extended, and to objects, not merely curious, but of real interest, we may learn from the history and actual state of the science. The first considerable extension of Trigonometry, beyond its original object, was made about twenty years after the death of PREFACE. il Newton. It was then, on the ground-work laid down by that great man, that three Mathematicians of the Continent, Clairaut, Da- lembert and Euler, and Thomas Simpson our countryman, began to establish a system of Physical Astronomy more perfect than what its Author had left. With this view, they laid aside the Geo- metrical method which Newton had used, and which they thought incompetent to their purpose, and adopted the Analytical. Pursuing this method, they perceived the formule of Trigonometry to be of continual use and recurrence, and the language, by which the process of demonstration was conducted, to be formed, ina great degree, of symbols aud phrases borrowed from that science. In order, therefore, to render the process precise and expeditious, it became necessary to improve the means and instruments by which it was carried on; and, accordingly, at the time spoken of, the advancement of Trigonometry, the pure and subsidiary science, was contemporaneous with that of Astronomy, the mixed and principal one. This general statement would be confirmed by an examination of the Memoirs and Treatises on Physical Astronomy published about the year 1750. Clairaut and Dalembert in their Lunar Theories embody in those Works, or introduce as prefatory matter, several, now commonly known, Trigonometrical formulze*. In the Volume of Tracts which Thomas Simpson published, the Author evidently * Tt will hardly be believed that theorems, such as are given in pp- 28, 29, &c. were almost unknown. Yet Clairaut, (Mem. Acad. 1745, p- 342, and Theorie de la Lune, edit. 2, p. 9.) alluding to these Theorems, says, ‘ M. Euler est le premier, que je scache, qui ait fait usage de ces Theoremes pour operer sur les sinus et cosinus dangles, sans avoir recours a leurs formes imaginaires.’ lv PREFACE. intended the one which is inserted at p. 76, as preparatory to the succeeding Theory of the Moon; and Euler distinctly states as a reason for cultivating the algorithm of sines, its great utility in the mixed Mathematics. In the arrangement of the Treatise, which the Table of Coutents sufficiently explains, Spherical succeeds to plane Tri- gonometry. Now, the former has not, like the latter, been ex- tended beyond its original purpose. Ithas no collateral and indirect uses; it has not enriched the general language of analysis, by its peculiar phrases. But, notwithstanding this confined range, and apparent simplicity in the object of the science, its propositions are more easily established by the Analytical method than the Geometrical, And, (at least in the opinion of the Author of this Treatise) this would be the case, even if there existed no simi- larity and artificial connexion, between the processes by which the series of formule in the two branches of ‘Trigonometry were respectively established. But, so far from there being no similarity, the corresponding propositions can be deduced by methods so analogous, that to know the one is almost to know the other. This will appear to be the case, if we refer to pages 25 and 142, &c. of this Treatise. We shall there find similar Algebraical derivations of formule from two fundamental expressions for the cosine of an angle. ‘The principle of the derivation, however, is not new; it originated with Euler, who inserted in the Acta Acad. Petrop. for 1779, a Memoir entitled Trigonometria Spherica Universa, ex primis principiis breviter et dilucide derivata. Gua next, in the Memoirs of the Academy of Sciences for 1783, p. 291, deduced, but by awkward and complicated processes, Spherical Trigonometry “ from the Algebraical solution of the simplest of its Problems.” In 1786, Cagnoli, in his excellent Treatise, fine derived without “ similar triangles or complicated figures,” the PREFACE. Vv fundamental expressions for the sine and cosine of the sum of two arcs. And lastly, Lagrange and Legendre, the one in the Journal de L Ecole Polytechnique, the other in his Elemens de Geometrie, have followed and simplified Euler’s method, and instead of three fundamental expressions, have shewn one to be sufficient. ADVERTISEMENT TO THE FIFTH EDITION. ele Iw the present edition, the work, although much enlarged, retains its former plan. Usefulness has been studiously kept in view, in the alterations which have been made. ‘The prin- cipal additions will be found in the Spherical Trigonometry ; which contains an entirely new Chapter, on the late Trigono- metrical Survey. CAMBRIDGE, November 1827, TABLE OF CONTENTS. Ti tO CHap. [. Page DEV ISTOEN TOL CCE Leas ce LOR St ie Caua) Gennes nie Definition of sine; sine of an arc=ssine of supplement ..... CVE CBCRL STUNG stg tnt ne es city ap Oa lane 3 to 6 OUCOMDG Pcl le ey date te See beer NG Tables of arcs having the same sine and cosine..... Re sraratata ke 16 Tangent, eo-tangent, secant, co-secant’.. ....6..e0. e008 4— 16 Radius =r introduced into Trigonometrical Expressions.... 18 Values of sines, cosines, &c. in particular cases ........ 20 Table for converting degrees, &c. in the French division of the circle, into degrees, &c. of the English EH TAYE AN FA cy Si gn A A eel a 24 Cuap. II. Sine, cosine, of an angle of a triangle expressed in terms of Leg SINGS ee CET Ble a a WS Hn wey ween wate 25 — 27 Sines of angles proportional to sides opposite: area of a Toft edhe gan ta OPIN Sg SP AwaEE “ries Rv Say A Tea 28 Expressions for the sine and cosine of the sum of 2 arcs ... 29 — 30 sin. (4 + B) =sin. A.cos. B+ cos. A. sin. B cos. (4 + B)=cos. A.cos. B F sin. A «sin. B sin, (A + B) +sin. (A — B)=2 sin. 4.cos. B sin. (4-+ B) — sin. (4 — B)=2.-cos. 4.sin. B sin. (A + B).sin.(A — B) = sin.* 4d — sin? B cos.(A — B)-+-cos. (4+B) = 2cos. 4.cos. B cos. (A — B) — cos, (A+B) = 2 sin. A.sin. B sin. 4 + sin. B = 2 sin, Et 2 08.5 ; of AB dow B - COS. 29 — 33 cos. 4 + cos. B = 2 cos. ii Table of Contents. | Page cos. B — cos, d = 2 sin. at 8 . sin. as Nts tee 2 Hy toe A+B sin. A-+sin.B sin. A nies sin. B == eTiguy? ers! erceeeeeee oeeeeen tan. 9 : cos. d+ cos.B _ 1 34 cos. B — cos. A tan eG ee 2 2 sin. 4+ sin. B ete A+B cos. A + cos. B = tan. Tiree oh el a, ©) Aiea -e (ei ieee sin. A ohtan Ten cost — ° Oo ee cee ecoestooreesese ees ta. (A: BY oy oe aE enna 37 1 + tan. A tan. B Expression for the tangent of the sum of any number of arcs 39 - _ Table of the different expressions for the sine, cosine, and tangent of an’ are“). yarn ee eee Sets Al, 42 Cuap. III. cos. 2A = 2cos.2? A—1, or = 1—2sin? JA... 44 cos. 3A = 4 cos.3 A — 3 cos. A... . 2. eee eee 46 cos. 4.4 = 8 cos.4A — 8 cos? A+ 1...... eee AZ cos. 5A = 16cos.8 A — 20cos2 4 + 5c0s. A... 48 cos. 6.A = 32cos.° A — 48 cos.4 A + 18 cos.? 4 mL. S. Vile a cde + ee Cennnnem Sie i pan abid. sin: 24 = 2. sin. Acos. Ait... 2. Waren tes etal sin, 3\A) <3 sin. (A) sind Aire ae ke Sele ee sin.4.d = (4sin. d — 8sin3 A)cos. A........- sin. 5 A = 5sin. A — 20sin3 A+ 16sin’ A.. Sums of series of the cosines of multiples ares, of the sines, &c. 52, &c. Waring’s property of chords; Vieta’s .........- PBs a 55 De Moivie's formule «ssl ose4ras eh Por ee ae 59 Cotes’s resolution of a 'Trinomial ........... My, Dave Bod ae 60, &c Expressions for the powers of sine and cosine ........-++-- 62 Tangent of multiple are*..i4s2Ue2s2 30k OR Tenens 164 Table of Contents. ili Cuap. IV. Page Construction of Trigonometrical canon .........0cc0c0e: 67, &c. BUPAIECOYGOULEO.. oie on hives on Sed ln the Pies Nae 68 sites Of all'arcs*commputed 0... de 2% ssa Re eee ON 69, &c. tABGEDES COMPULCd ss , . Siler Om oN cle siadeldid sietaet clare, o Oe 74 POPU OL VErHICAtION ot ecieey a Wietsd vie e's exes yes baci «epost 76 Cuap. V. Solution of right-angled Triangles Bip Pie By ks RIESE VT Stel 81, &c. BE OUMCUE IA Cees as op 2 Ws SSE o's yoyeie a 4 oy SL Te 83, &c. Uses of different Solutions of the same case ........ reece Mas OLE. Trigonometrical: Problems... cic osc c os ois'n e iv oe ba Ras 99, &c. Cuap. VI. Instances of the utility of Trigonometrical Formule ...... 106—120 Cuap. VII. Solution of a cubic equation by the Trigonometrical Tables 121 eect Ma Matai corre ats Cel En ta Wichin'n whe akg 0 ate Wig ee 123 BOUMIGN: Of Ay CRAULAUIC HAL: ait dele s die ah allt ate acel. Bie bya act 124 POAT DIG Goth sae Seance sce SIM ae Tat ete MTS 126 SPHERICAL TRIGONOMETRY. Cuap. VIII. Definitions cotrericcesarelsistezecersitty tel aid eho are’ oer e’ ath lite biatete aN wan ek 128 Section of a sphere by a plane, acircle..............05. 129 One side of a spherical triangle < sum of two others ...... 130 Three sides of a spherical triangle < great circle ......... abid. Method of finding the pole of a great circle ...........5.. 131 Supplemental or polar triangle.................5044. Hit BB Angles at the base of an isosceles spherical triangle, equal 135 Greater side opposite greater angle .........000 eee eee eee 136 Area of spherical triangle and polygon .........+++.- dora loZ,.&e: iv Table of Contents. CuapP. IX. Expression for cosine of angle in spherical triangle ........ 142 TOPASINIG. vf gies SMa as aie oe Semen wd ae chy w alee’ o's oles 144 Sines of angles proportional to sines of opposite sides ...... 146 Expressions for the sines and cosines of half an angle, and of half the sum and difference of two angles.... 146, &c. Cuap. X. Formule of solution for right-angled spherical triangles ... 149 Crctiiar parts... v4. ele eta et syrate open ae 151 Waper's Rules proof-of: 0.0) ..2 samen, Var eer ener 152 Quadrantal triangle ............: Me Ponies re he eee teat als 154 Example of solution of right-angled spherical triangles ..... 155 Aftections of sides and angles: > 7.7 cea © aren 159 CuHap. XI. Oblique spheneal ‘triangles... sw. vetias oe cee an erry ed: N aper’seANGlOSIES .0%.'.in cv ae | cle Men eet tae tna 175 Expressions for the sine, cosine and tangent of half the sum of the angles of a spherical triangle ....... Here: 189 Example of the utility of these formule .............. ae 194 Cuap. XII. General statement of the mode of conducting a Trigonome- trical Survey is. k oc ac eee eee os Ga Methods for computing the mean refraction; the arc con- tained between two stations being we) known 202, &c. The ‘eontained are eompuited) iio Se ial. hk a ee, 2 (200 A heights of stations found from their reciprocal angles of Depression and Elevation ....... Syne ic Shen mQORa sere The corrections for the angles of Depression....... ee 217 Reduction of an angle observed between two objects; to the hormecwa a. 2.4 1" and of like quantities are, usually, ex- : 6 out pressed by means of decimals; thus, te retaining only the two first figures, equals 0°.85; and =th of the circumference would be expressed by 512.25 497.85. But it is occasionally useful (see Astron. edit. 2. p. 779.) to ex- tend the division of the circle beyond that of seconds, and to in- troduce, with their proper symbols, thirds, fourths, &c. In such 6 at: an extension, 5 1” would equal 51” 25.7 and the foregoing arc, the seventh of the circumference, would be expressed by 519 25) AQ 54%, O57, A. ‘The arcs of circles, it has appeared, are proper measures of the angles which they subtend ; if the angles be increased, the arcs are also increased, and in the same ratio; and knowing the value of one, is, in fact, knowing the value of the other. But, in ‘Trigonometry, the values of angles are made to depend on the values of certain right lines, drawn according to certain rules, but not varying, in the mathematical sense, as the angles vary. The lines just alluded to are called sines, tangents, secants, &c. which it now becomes necessary to define. : The Sine of an Arc is a right line drawn, from one extre- mity of an arc, perpendicularly to a diameter passing through the other extremity. The Cosine of an Arc is a right line intercepted between ithe centre of the circle and that point in the diameter (the foot of the sine) at which the sine of the same arc drawn perpendi- cularly to the diameter meets it. A The Versed Sine is a part of the diameter intercepted between the common extremity of the arc and diameter, and the foot of the sine. The Tangent of an Arc is a right line, drawn from one ex- tremity of it and perpendicularly to a diameter passing through it, and terminated by its intersection with another diameter passing through the other extremity of the arc and produced beyond it. The Secant of an Arc is a line intercepted between the centre of the circle and that extremity of the tangent of the same arc which lies without the circle. The Chord of an Arc is the straight line joining its two ex- tremities. . The Complement of an Arc less than a quadrant is its defect from a quadrant. The Co-tangent, Co-secant of an arc are, re- spectively, the tangent, and secant of its complement, and, therefore, may be drawn according to the preceding directions, by considering the complement of the arc to be the arc itself.” If we now illustrate these definitions, and assume, in the annexed diagram, AB to be the arc: then, see p. 3. |. 24. BF is the sine of AB, and F is what we have called the foot of the sine. CF is the cosine of AB (see p. 3. 1. 27.) AF is the versed sine of AB (I. 1.) AT’ is the tangent of AB (1. 4.) CT is the secant of AB. A line joinmg A and B would be the chord of AB: as Bgé, bfb (see fig. p. 7.) are the chords of the arcs BQO, bab’. * 1 ° Besides these, there have, of late years, been introduced terms such as co-versed sine, su-versed sine, su-co-versed sines (see Mendoza’s Tables): and the reason of their introduction is certain facilities af- forded by their use in computation, such as that of clearing the Moon’s distance, &c. 5 The complement of AB is (p. 4. |. 12.) BQ, since 4B+ BQ == quadrant: now BQ being considered as the arc of which the sine, tangent, secant, &c. are required, its sine, cosine, tangent, secant, by the preceding definitions, are (see fig. p. 7.) respec- tively, Bg, Cg, Qt, Ct: and, accordingly, (see pp. 3, 4.) those same lines are respectively, the cosine, sine, co-tangent, co-secant of the arc AB. : The lines that have hitherto been drawn expound the sine, cosme, &c. of an arc AB less than a quadrant: but if we take \972 Ab greater than a quadrant, then, according to the above defini- tions, bf is the sine of the arc AQO. Cf is the cosine, Af is the versed sine, CS the secant, AS is the tangent. In order to determine the co-tangent and co-secant of this arc Ab we must vary and extend the definition of the complement of an arc: now, (see p. 4. 1. 12.) the arc being less than a 6 quadrant, its complement was defined to be its defect from that quantity: but an extended definition which should make the complement of an arc to be the difference between it and a quadrant would suit both arcs greater and less than a quadrant; and, according to such definition, QO (fig. p..5.) would be the complement of Ab, and Qs (AQ being a quadrant) the tangent of Qb would be the co-tangent of Ab. Cs the secant of QO, would be the co-secant of Ab. Let the arc be called A, then when A is less than a quad- rant (Q), A+-(Q— A)=Q, or, A — (A Q) = Q, when A is greater than a quadrant (Q), A -(A—Q)=Q, therefore, by what has preceded, sin. A=cos. (A — Q), and sin. (A — Q)=cos. A, tan. 4=co-tan. (A — Q), and co-tan. A=tan. (A — Q), sec. A =co-sec. (A — Q), and co-sec. A ssec. (A — Q). Let now the arc be greater than two quadrants but less than three: and let 4Qad represent such an arc, then, by the preceding definitions (fig. p. 5.), de is the sine, Ce the cosine, Ae the versed sine, AT the tangent, CT the secant. Lastly, let the arc be greater than three quadrants but less than four, or less than the circumference of the circle; and let AQak represent such an arc, then ke is its sine, Cg the cosine, Ag the versed sine, Am the tangent, Cm _ the secant. 7 ‘We may also use the definitions of p. 3, 4, and draw the sines, cosines, &c. of arcs greater (if we may so call them) than the circumference: for instance, suppose the arc to be equal to the circumference plus the arc AB; then, guided by the defini- tion (see p. 3,) begin from A, one extremity, and go round the circle in the direction AQaB’ till you arrive at B the other extremity of the arc (= AQaA-+AB): from B draw BE perpendicularly to Aa passing through 4, and BF' is the re- quired sine of the arc, CF is the cosine, AF the versed sine, AT the tangent, and CT’ the secant: which are evidently the sine, cosine, Xc. of the arc 4B. Hence, admitting the existence of arcs greater than a circle, and applying to such the original definitions of p. 3, 4, we have these equalities sin, A.B=sin. (circumference + AB), or sin. d=sin. (27 +A), calling A.B, A, and the circumference 27; also cos. A = cos. (27 + A), tan. A = tan. (Q7 + 4), sec. A = sec. (Qa + A), and in a similar manner we might easily obtain more like equalities. The sine, cosine of an arc are thus expressed by means of the sine, cosine, &c. of other arcs: but they may also be 3: expressed in terms of one another: thus, since by seventh Proposition of the first Book of Euclid, CB’ = CF’ + BF’, we have, making r = CB, r’ = cos. A+sin.” A, and, accordingly, sin.” A =r? —cos.? A. Again, since, CIa= CAAT sec’. A= r° +tan’. A, and, accordingly, tan.” A =sec.” A—7’, Again, by the similar triangles CFB, CAT, CR: FB 3: Cao Ar: FB Mote AT= (Wei Ges CA x CR’ sin. A or, tan. A ='7 een be and, by the same similar triangles, the forty- In like manner we may easily deduce from the similar tri- angles CQt, Cg B, the values of Qt and Ct, the co-tangent and ¢o-secant (see p. 6.) of the arc AB, (=A), and, Sie: Or, we may dispense with this second set of similar triangles (CQé and Cg B) and deduce the values of the co-tangent and co-secant from the previous values of the tangent and secant, by means of their definition (see p. 6.). Thus, the co-tangent of an arc is the tangent of the com- plement of that arc. If A be the arc and Qa quadrant, Q— A is the complement: now by I. 19. of p. 8. sin. (Q — A) : cos. (Q— A)’ but, sin. (Q— A) = cos. A, cos. (Q— A) = sin. A; tan. (Q-— A) =r. cos. A sin, A *, cotan. A [the same as tan. (Q—- A] =r. In like manner, r cos. (Q— A) vad sin. A If it were worth the while, it would be easy to express, under different terms, the preceding equalities: for instance, we may express the two latter, after the manner of stating a Theorem, thus: co-sec. A = sec. (Q— A) = —_— — B 10 The radius isa mean proportional to the secant and cosine of an arc, and, also, to the co-secant and sine. The radius is also a mean proportional to the tangent and co-tangent: which may be thus deduced, sin. A cos. A , co-tan. A = 1, ——-; cos. A sin. A tan. A=r. “*. tan..4X co-tan. A = ‘9°, or, tan, A: 7 :: cotan. A, and this same proportion is immediately deducible from the similar triangles CQt, CAT (fig. p. 7.). The right line 4B is (see p. 4.) the chord of the arc AB. If the right line be bisected at » and Cn be drawn perpendicularly to AB, then the pomt m, where Cn produced meets the circle,. bisects the arc AB (Euclid, Book III, Prop. 30.); therefore AB A 3 the arc Am = Bm = 5 (supposing A = the arc AB). But An is by the definition of p. 3. 1. 24, the sine of Am, the chord (An B), therefore, of an arc A is double the sine (A 7) r A ahi e of half the arc (GS) : or, which 1s the same proposition, the sine of an arc (A) is half the chord of twice that arc (2A). Instead of making A = the arc AB, make, for convenience, 2A to represent it, and let v represent mn the versed sine of Am or Bm (= A): then, since (Euclid, Book HI. Prop. 35.), il mn X np = An’, v xX (Qr—v) = sin.” A, or, 2rv = sin. A + vo. But An? + mn” = Am’ (the line 4m), A 2 or, sn. A +0 = (2 sin. ) (see p. 10. Il. 17, &c.); “. (see |, 3.) 2rv = 4. sin.” a but Cn = Cm — mn, or, cos. A =r—v a le =, therefore, 7 — = sin. — T 2 | | | (XS) = : 20 |B ay, NS 1 A eyes -(2 cos. — — r*) T g since cos le + gin? 2 =; Q g If we multiply the two last expressions (Il. 13, 14.) for cos. A, we have cos. A x cos. A, or cos. A = 1 A A A A — 497? a Pin LS Se <)- ee Asin. tea = { 7 COS Ps Sl 3 7 sm 9 cO5 Of. 1 at a A =~ (' — 4. sin.” — cos.” =) : v7" 2 2 A : 2 2 Hence, - sin.” — cos. — =r —cos.* A r 2 2g = Sifts ch, | and consequently, AY A bal 2 2sin. —cos. — = r sin. A. , 2 Zin 12 The supplement of an arc is the difference between it and a semicircle: accordingly, | BQa is the supplement of AB, . ab is the supplement of AQD, , and adis the supplement of AQad. Now (see the definition) BF is the sine of BQa: for BF is = vf 2 drawn from one extremity B of the arc BQa perpendicularly to the diameter a 4 passing through a the other extremity: but, as we have seen (p. 4.), or by the same definition, BF is the sine of AB. Similarly, bf is the sine equally of the arc 6a and of the arc AQDb: ed the sine of ad and a Qd: and, generally, the sine of the supplement of an are is equal the sine of the arc itself. «"z We have already (see pp. 6, 7.) in one or two instances, ex- pressed by means of general symbols certain equalities that subsist between the sines and cosines of arcs, which, though different, may be said to be related. Such modes of expression are, in the actual « » The parts included within ,*, *,* are less essential, or less elementary, than the other parts, and may, in a first or partial perusal, be passed over by the Student. 13 operations, carried on by the aid of the ‘T'rigonometrical Analysis, of frequent and considerable use. It is desirable, therefore, to extend them, which may be done without much difficulty. Thus with regard to the equality which has just been stated: if 27* represent the circumference of a circle, of which the radius is 1, and A be the arc, then sin. A = sin, (7 — A), and since A may represent any arc, if instead of A we substitute Tv . : “ ahs A in the preceding equation, we have T ; T sin. G ~ A) = sin, CG + A), and if, instead of A, we substitute m — am — A, we have n sin. (= 7-4) = sin. (—"7+4) : n n We may also obtain other general expressions; thus, by ex- tending the definition of a sine (see pp. 4, 7.) the subjoined equations are true sin. A = sin. (Qa7 + A) = sin. (4a + A), and generally, sin. A = sin. (2n7 + A), n being a whole number. For like reasons, sin. (r — A) = sin. (Sa7— A) = sin. (5a — A), and generally, | sin. (r — A) = sin. $(Qn+1) r—Ab, n being any number in the progression, 0, 1, 2, 3, &c. Hence, * The numerical value of 7, that of the circumference of a circle of which the diameter is 1, 1s 3.14159, &c.: 27, or 2 ¥ 3.14159, &c. ex- presses, then, the value of the circumference of a circle, of which the radius is 1. 14 since by p.12, sin. A = sin. (3 — A) there results this general equation, : ~ sin. (Qna + A) = sin. {((@n+ 1)7 — A}, or, what amounts to the same thing, if s be the sine of any arc A, it is also the sine of all arcs comprehended under the two formule, (Qna + A), {(n of 1)r—Al}, in which 2 may be any term of the progression, 0, 1, 2, 3, &c. 11. The definition of a cosine being (see p. 7.) like that of a sine, extended to designate the cosines of arcs, that are greater than the circumference, we may, in like manner, obtain general expressions for it. Thus, CF, which is the cosme of AB, may be considered as the cosine of the (circumference + 4B) &c. Hence, as before the following equations will be true; cos. Ad = cos. (27+ A) = cos. (4a +A) = cos. (Qn7+ A), n being any term of the progression, 0, 1, 2, 3,4,5,&c. But, since the same CF is also the cosine of the arc ABaB’, we have CF=cos. (227 —A) = (for reasons just alleged) cos. (4a — 4) = cos. (Qn — A), n being any term of the progression, 1,2, 3, &c. or, CF, generally, = : cos. {(2n + 2)r— At, ‘n being any term of the progression, 0, 1, 2, 3, &c. Hence CF is the cosine of all arcs comprehended within the two formule (Qnmr + A), 4(Qn + 2)r — A. 1b 12. In a similar manner we may investigate the general formule of arcs that have the same negative cosine, Cf=cos.(m ~ A)=cos.(37 — A)=, generally, cos.{(Qn + 1)7 — A}, n being any term of the progression, 0, 1, 2, 3, &c. and since the same Cf is the cosine also of ABbab’ or r+ A Cf = cos. (r+ A) = cos. (37+ A) = cos. {(Qn+1)r+A}. Hence, Cf is the cosine of all arcs comprehended within the two formule, : {Qn+1)r7—Ah, JQ@n+1)7 + 4h. The arcs of which f0’ is the sme, are a + A, 3a + A, and generally {(Qn + 1)47 + A}. The arcs of which FB’ = f6' is the sine, are Qn — A, 47 — A, and generally (Qn + 2a — A. Hence FB’ is the sine of all arcs comprehended within the two formule, {Qn + 1)7 +A} and {Qn+ 2)7— A}, n being in each case any term of the progression, 0, 1, 2, 3, &c. In calculations, where FB, FB’, and other quantities are involved, if s be the symbol for £.B, —s must be the symbol for FB’. For, conceive a line to be drawn a tangent to the circle, at the point opposite to Q in QC produced, and let the distance of any point in the circumference from this line be called z, then FB(s)=z—r, and FB =r—z, or FB = —(z—7). Hence, if in any equation subsisting between trigonometrical lines we wish to pass from the consideration of the point B to that of the point B’, we must in such equation substitute —(z—7r) instead of z—7, or — s, instead of s.* - 13. The preceding results may be conveniently represented in a Table, s and c representing the sine and cosine of an arc A. * This hinges on the general doctrine of negative quantities: the scrupulous Student, who is not satisfied with what is here said, is referred to Carnot’s Geometrie de position, and his subsequent work on ‘the Theory of Transversals, &c. — 16 PPMP —21-+ue)|p —4(e--us) peg |p —4(g--us) pp —2(t ue) |p —“(-+ue)) p+ 4uz ‘e|NULIO,T [e133 } <8 V+46 V—-«6 V-“oOL | p48 PV —“O1 vV+46 || vp~~6 y+t4s p pre | p-4xn || pug | ptug || pug | pees | p—xe ptug ¢ V4 Vg V-49 | p+? vy—-~9 V+4¢ V—-«S¢ p46 j V+4¢§ V—-§ V-*p | Vs es V+46 V—4¢ V +42 I Aas eae V—-4& VF || P-%S V+e ys V =O I—=='s00 | 9—="s00 9="s00 | 9=='809 || s—=ours | s—=ours | sours $= 9UIS u 17 It is easy from this Table and the expressions for tan. A, co-tan, A, sec. A, &c. namely, sin. A cos. A 1 cos. A’ sin. A’ cos. A to determine the values of the tangent, co-tangent, secant, &c. (rad. = 1) Oeste Cr eee tS thus tan. (4+ A) = RESTO] Go ee = tan. A, cos.(397—A) = c co-tan. (Sa7r— A) = StS aed) = aa = —co-tan. A, 77+ A) A sec. (7 7 Sao re pera me os oo SOC A C Hi cos. (7am Adi —c¢ 18. In some of the preceding expressions, a radius =1 has been used, and, solely for the purpose of lessening the number of symbols in the Trigonometrical formule. For, 1, or any power or root of it, used as a multiplier or divisor of an expression, may be expung- . ox : sin. A ed from such expression; thus, instead of —4e77» We may more simply write sin.* 4. Still, however, it is, on many occasions, necessary to use, for the radius, a general symbol such as 7, or, an arithmetical value such as 10,000. For this reason, it is de- sirable to be possessed of some easy and expeditious rule, for converting formule constructed with a radius =1 into other formule that shall have a different radius. Such a rule may be obtained from the following simple considerations : If (see fig. of next page) BF, bf be drawn similarly inclined to CA, then by similar triangles, BF= yo or BF=6f.= (Gf CB=1, Ch=?). ) : é 1 In like manner, CF=Cf. mae AF= af ~. Now, sines, co- sines, tangents, &c. are drawn after the manner that these lines are; if the angle BFC be a right angle, BF is the sine of the angle BCF to radius 1, bf is the sine to radius r; and, CF, Cf, C are the cosines. Hence, in any formula involving sin. 4, cos. A, &c. 4 calculated for a radius = 1, substitute instead of sin, A, cos. A; sin. A cos. A ar aR and the resulting formula will belong to lines r drawn in a circle, of which the radius is 7: for instance, age sin. A re tan. A smd * ¢ nh A= ——, radius=1; .°. = "xX ! cos. A’ i r ? cos. A” sin. A or, tan. A=r. > (radius = 71). cos. A : 1 4 Again, sec. A= (radius = 1), then cos. A | sec. A 1 = ———— = —— _,, or r cos. A cos. A’. r 2 sec. A = , (radius = 7). cos. ; tan. A co-tan. A Again, tan. A.co-tan. 4 =1, then, .——— =], r r “and tan. A.co-tan. A=r’, And if in any formula, any power of sin. A, or of cos. A, such as ° 3 ° ° e sin. A, sin.” A, cos.‘ A, or cos.” A, occurs, the radius being 1, then, . .. sin® A> sin. A: cos.* A cos.” A by substituting 5 | 7 —— the result- gh REDE bg Oh pail: : ing formula will belong to a circle of which the radius is 7. 19 This is the rule; but, since it would leave a Trigonometrical formula with fractional terms (the denomimators being powers of the radius) it may, with advantage, be modified and made more convenient. Thus, suppose a form should occur such as cos. nd =a.cos.” A + b.cos."~* A + &c. to radius 1; then, by what had preceded, the reduced form to a radius 7, is cos.nA a.cos.” A cos.”-? A a be + be. and, cleared of fractions, is r’—1 cos. nA =.a.cos.”.A + b.r*. cos.”— * A + &c. Here, cos. nA is multiplied by 7”~!, cos.”~* A by 7’, &c.; that is, if we choose to call cos.” A a quantity of m dimensions, cos."~" A, a quantity of n—2 dimensions, cos. A Xx sin. 4, a quantity of two dimensions, we may announce the preceding rule under the following simple form : “Multiply each term of a Trigonometrical formula, in which the radius = 1, by such power of r, as shall make it of the same dimensions with the term of the highest dimenstons ; the resulting formula will be true when the radius is = r. Thus, if cos. 3A =4.cos.? 4A — 3.cos. A (rad. = 1), e } Q . . e . since cos.” 4, the term of the highest, is of three dimensions, and cos. 3 A, cos. A, are of one dimension, we have r .cos.3A = 4.cos.. A — 3r’ cos. A. 19. The Trigonometrical symbols, such as sin. 4, tan. 4, &c. that have been obtained, are merely general ones, and, hitherto, no methods have been given of assigning their values in specific values of the angles. The general methods for this purpose will be given in a subsequent part of the Treatise; but, even at this stage, by peculiar artifices, we may, in certain simple cases, assign the arithmetical values of the sines and cosines of angles. For example, 20 If (fig. p. 14.) AB= BQ, that is, if 4B=half a quadrant, or expressed in degrees, if AB or 4 FBC=45°, since 2 BCQ=45", also cos. 45° = sin. 45°, but cos.?A + sin.”A = 1, (1 =radius in this ‘ | 1 case); therefore, 2 (sin. 45°)” = 1, and sin. 45° ane -707 1068, or (see the preceding rule) = 7071.068, to a radius = 10,000. If ACB =< 90° = 30°, since BCB’ =2.ACB = 60°, and since ZCBB’= 2CB'B=60’", the triangle BCB’ is equilateral, and consequently BB’ (the chord of 60°)=the radius CA = 1, and, : 1 ] . : BF=sin. 30° = 5 BB rugs .5, and .°. cos. 30° or sin. 60° | 1 V3 (see Art. 10.)= \/ G rae) bane .8660254, and (see p. 18.)= 660.254 to a radius = 10,000, and 8660254 to a radius = 10,000000. Hence may be proved, what was asserted in p. 3, that the sines of arcs do not vary as the arcs themselves. For, the : ieee Le : : sin. 30° = 5 radius =, .°., s sin. 90°; in other words, the sines are as 1 to 2, whilst the arcs are as 1 to 3. The values of the tangent may be found, in the above cases, sin. A from the expression tan. A = . Thus cos. A tad. = 1 rad*. = 10,000 Pyare aeons 10,000 an. = (jie hee eT — eeeeeesesteeees = cos. 45° ; f 1 tan. 80° = ~ X —~—= .5773503 .,..... = : a aa 77 57 7 a.aUS haa ates ) tan. 60° = ron hae = 1.7320508 ...... = 17320,508 f : fat. QO See al We may here direct the Student’s attention to the superior augmentation of the tangent above that of the sine; for, we have corresponding to * Arcs Ogre 30° 45° 60° 90° Sines 0, 5000, 7071, 8660, —_ 10000, Tangents 0, 5773, 10000, 17320, oO, * We have adhered, in this Chapter, to the antient and common division of the circle. But, in most of the French scientific treatises that have, of late years, been published, the circumference is divided first, into 400 equal parts or degrees, then, each degree into 100 equal parts, or minutes, then, each minute into 100 equal parts or seconds: so that a French degree is less than an English in the proportion of 90 to 100: a French minute less than an English, in the proportion of 90x 60 to 100x100: and a French second less in the proportion of 90 x 60 x 60 to 100 x 100 x 100: hence, if 2 be the number of French : n(10— 1 degrees, the corresponding number of English equals nS 5 aaa or 1— ~ » which last form points to an easy arithmetical operation for finding the number of degrees in the English scale from the number in the French scale: since from the proposed number we must subtract the same, after the decimal point has been moved one place to the left : Examples: What number of degrees, minutes, &¢. in the English scale correspond to 73°, to 71° 15’, and to 26°.0735, in the French scale, "3 71.15 26.0735 7.3 7.115 2.60735 65.7 64.035 23.46615 6 6 - 6 65° 42’ English. 2.10 27 . 9690 6 6 6 58.140 Answer 23° 27’ 58”. 64° 9! 6" 22 In the same manner as we have found, in the preceding cases, the values of the tangents, we may find those of the versed sine, secant, &c. In the next Chapter we will proceed to investigate certain ex- pressions for the sine. and cosine of the sum and difference of two arcs, in terms of the sines and cosines of the simple arcs. Such expressions are, in this science, very important, since from them, by an easy derivation, may be made to flow almost all ‘other Trigonometrical formule. *,* The conversion of English degrees, &c. into French, is to be effected by increasing the number of English pret peraG &c. by one-~ ninth: for lO£E=9F;:*.. PaE+e: for instance, if the English are be | 23° 33! 54!" — 23,565; the equivalent French arc is 23.565 2.6183333 26.183333 that is, 26° 18’ 33” 33’, &c. The operation of reducing French to English degrees may be superseded, and rendered less liable to mistake, by means of the following Table, in which as.it is usual, the reduction is effected simply .by addition. | Example to the Table. Reduce 26°.0735 to English degrees, &c. French. English, By the Tables 20° 01. Oo - +s. 18°. 0’ 0” 6 Dag OF ere eee 5 24 0 GSA) earl matte soy! 0 3 46.8 0) -OU80L ek. 2 O LEE O78 Os Wie AE ee RS ee 0 (02, 1,62 26. 0735 23 27 58.14 the same as before. oceaimeeiredieaee 23° Reduce 2°.7483 to English degrees, &c. French. English. yan ables Sate, 0 pepe onic os ee 1° 48’ 0” Chee hs pita a eee eiake ese 0 37 48 Oita dee ees Sei eiate ee ais 0) 2 OG GeO S02 ote or 0 0 25.92 RE Teds eA Adiga 0 0 .972 24. TABLE For Reducing French Degrees, &c. to English. French Division : 400° in the circle, 100’, in a degree, 100” in a minute. English division : SOLA vie tees OU eens aes . 60". French. English. || French. ‘English. Terraces bP a oe cee bso tO Oa GAD a wccsnuevere tae «ak Fede DER bk s ka ea yd as re Rg ed Or a a a 18 Sconeet ses «a sik eae ee ee BO ees a ats aoe em olen ed Bevis 5 49 00m Cee bed Or Oa Aon cn cee oe ky enon SP OP ORCL ee 4 30 || 50. : 45 Tee peg retg ae 5 24 || 60. che Ged s ewkat FMR sae se late 6 18 || 70. Pee vate Bg waa Mae shes Se wish (gies €: SOQ ch. 72 + ph ea ar bie Lone 8 6 || 90. 81 Hada es Cede ote tee 9 O 4100 90 Minutes. } 1’ sree oO oe at Me twee awe Gene gee oes fet alwit seis) eae sae 1 4.8 || 20. wee es --10 48 Dade «=4 Li 37.2 30 16 12 Ae. oe 9.6 40 21 36 Deen sca ol eten 2 42 50. i “ 27 0 PNY os oar ote e 3 14.4 60 3 32. 24 ff 8 46.8 70. . 37 «648 Deevecccsceceas 4 19.2 BO atc Ge tee! aes ore --43 12 ieee Ue ats oes ty DLO 90... 48 36 LO a sae Sia'élee's sta 68) faced 100 ; 54 0 Seconds. An os oe Core Tee Ook LOS eieale vce aw eth es 3” 24 CHAP. II. Expressions, for the Sines and Cosines of the Angles of a Triangle, in Terms of the Sides: for the Sine and Cosine of the Sum and Difference of two Arcs or Angles, &c. Tue first step in this investigation will be made by the solution of the followmg Problem: Prosiem 1. In an oblique-angled triangle, it is required to express the cosines of the angles in terms of the sides. Let the 3 angles be A, B, C, the opposite sides, a, 0, c. - Let the line between the vertex of the angle C and the point a0 | : Cc epee 3 where a perpendicular from the vertex of 4 on the line a cuts a be p; then, p is called the cosine of C to the radius 6, and (by p. 17.)=6.cos. C, when the radius = 1. Now, by Euclid, Book I, Prop. 12 and 13, c=aa +B — gap =@ + 6° — 2ab.cos. C; gti —¢ consequently, cos. C = er a _ If we investigate cos. B, and cos. A, the process will be exactly similar, and the result similar, that is : atc — 6 Qac D cos. B= Deeg t— a" cos. A — ] Qbe From these expressions, the angles of a triangle may be found when the sides are given. 4) If C be obtuse, then p = 6.cos. (w — C), and ¢ = a> + b? + 2ap =a’ +b? + Q@ab.cos. (x — C) =a° +b? —2ab.cos. C S since cos. (wr — C)=—cos. Cf; a eb ae 2ab is the same, whether C be less or greater than a right angle. consequently, cos. C = , as before, or the expression Proxsiem 2. Let it be required to express the sines of the angles in terms of the sides. By Euclid, Book I. Prop. 47. sin.* A =(rad’.)° — cos.? 4 = (when rad*. = 1) 1 — cos.” A = (1+ cos. 4) (1=—Cos. A); since the difference of the squares of two quantities 1s equal to the product of their sum and difference. Hence we may find the value of sin.” A, by finding, from the preceding Problem, 1 + cos. A, and 1 — cos. A, and, then, by multiplying together those quantities. Now,. 1 — cos. A = 1 ~ pete mega 2 be _ @& — (8 — 2be + &’) aS 2bc¢ = a — (b — cy us abo _@tb—0) (ate—) be , (by 1, 14, &e.) 27 (bo +2%bc+c7)—a’ Q2be ‘ fe (b+c¢c)—a a Qbce _ (atbte)(bte— a) Qbe Hence multiplying, 1 + cos. A, and 1 — cos. A, a aera eas 8 Cant Na Ak*s bec a+b-+e or, since atb—e=9(— is —c), b+e—a=2( asta), &e. sin. A= ee ce) {(AS). ( -be +a). (3** Nn >) CHEE} ee), Gee ). (Gate). (2) - o)}. This is the expression for the sin.” A, formed by means of that for cos. A. But, the expressions for cos. B, cos. C are pre- cisely similar to that for cos. A, and, therefore, the sin.” B and sin.” C formed from them, by the same process, must be expressed by similar fractions; in which fractions, the numerators must, from - the nature of their composition, be the same as the numerator for sin.” A, and the denominators will be, respectively, a°c°, a6’. Let N° represent the numerator, then similarly, *1-+cos. A = ~ (by p. 26.1.13, &c.) sin’ A = raat ° . . N sin. A = —, sn. B=—, sin. C = — be ac ab * 1+ cos. A = ver. sin. (180°— 4); b o qi “. ver. sin, (180°— d) = SOs or, differently expressed, 4bc : (b-++-c)*— a* :: 2 (the diameter) : ver. sin, (180°— 4) which is Halley’s Theorem. Phil. Trans. No. 349. p. 466. Halley calls it * A new Theorem of good use in Trigonometry.” 28 Yeon N bie N Cor. 1. Since sin. B= —, and sin. C= —, we have ac ab sin. B b sin. C co or, if this equality be thrown into a proportion, i OTT pe Sieh ree Ol ret at Sa This relation, however, of the sines of the angles to the sides opposite, may be immediately deduced from (Fig. 3.); for the perpendicular (g) on & from the vertex of A, is the sine of B to the radius c, or g=c sin. B (radius = 1); similarly, ¢ is the sine of C to the radius. 6, or, g=6 sim. C (rad. 1.); .°. since g=4q, c sin. B=6 sin. C. Cor.2. Hence the area of a triangle may be ai in terms of its sides, for, (see Fig. 5.) area =P 8 5 “sin. B= atb+e atb+e at+bte atbte _ VC) Ca) Ca) ae OF, if (ape St iy O = VW{S.(S—a).(S—b).(S —0)}. PRoBLEM. 3. It is required to express the sive of the sum of two arcs, in terms of the sines and cosines of the simple arcs. From the two preceding Problems, ; N N sn. A= —, sm. B= —, be a C- 2 9 1.2 ae: i'd ¢ a-te—b be =n" and, cos. B= i PMP TR OES COS. A= ithe Beane ° a AC pd Cc Hence, oe Nae eo Be een sin, A.cos. B+cos. A.sin. B= ait ML -VeRECroaeCT Qabe 2 Ne? N Q9abc ab 29 But sm. C= ail ab for A+B is supplement to r—(A +.B) as A+ B+C=zx. Hence sin. (A + B)=sin. A.cos. B+cos. 4.sin. B, (1)* , and sin. C= sin. {a—(A + B)} =sin. (A + B), * It has been rightly objected that the above formula is not obtained by direct investigation, but discovered to be true by observation on the values of sin, 4, cos. 4, sin. B, cos. B, and by previous knowledge of their values. If we use a construction, or Geometrical diagram, there is no difficulty in proving, by direct investigation, the above formula. For instance, Mr. Cresswell, in his Treatise on Spherics, has thus simply demonstrated the fundamental formula of p. 27. 1. 24. let AB 2A, BD =| 2B, and draw Bs perpendicularly to 4D, then AD=As+sD; B : but (see p. 10.) d4D=2sin. ae =2sin. (4+ B), As=AB.cos. BAD=AB cos, a (Euclid, Book III. Prop. 20.) ‘: =2 An.cos. B =2 sin. 4. cos. Bb, Ds=BD.cos. BDA=2smn. B.cos. 4; *, sin. (4+ B)=sin. A.cos. B+-cos. A.sin. B, But a solution still more simple, is the following, of which a young Student of Caius College is the Author. ; | The 30 This is the fundamental form, from which almost all other ‘Trigonometrical forms may be deduced. Cor. 1. cos. (A+B) = (p. 6.) sin. (; —(A 4+ B)) e (p. 13.) sin. ¢ +(A+B)); but, sin. C 7 {4 +B)) == sin. ‘(2 + A) + B} : 7 T ae = sin. ¢ + A) .cos. B+ cos. G + A) . sin B, which is derived from the form (1), by substituting, instead of A, Cc +A) But by pp. 6, 13. sin. (: +4) COS. A I and cos, (; +A) = —sin. A. The two arcs are AB, BD, their sum AD, their respective sines are. Ap, Dg, .sssc.s..005, D7 Since 4p, Dg are parallel, the triangle dp D = the triangle Apq; that is, (putting instead of the wholes their parts,) Apn+pnD=Apn-+ Ang; “. prD.— Ang, but dgC+CaD=AqC+CnD; therefore, adding, dgC+ CpD= AnC+ CrnD=ADC; /. Apx Cq+Cpx D¢=ACx Dr, or, sin. ABx cos. DB-+-cos. AB x sin, DB =rad.~ sin. AD. 3] Hence, cos. (A+ B)=cos. A.cos. B—sin A.sin B....(2)*. Cor.2. By page 6, sin. (A — B)=cos. (~ —(A —B)) == 005. iG —4) 4 Bi 2 But, by the formula (2) that has just been established, ew. {(E-4) +3} = cos. G _ A) .cos. B— sin. G ~ A) .sin, B 64 9 =sin. 4.cos. B—cos. A.sin. B, (by p. 6.) Hence, therefore, sin. (A— B) = sin A. cos. B—cos. A.sin. B..........(3). Again, by page 6, cos. (A — B)=sin, G —(A—B)) = sin ‘(2 -A) ~ Bh. * Hence may be derived, and simply, a theorem relative to the cosines of the angles 4, B, C of any rectilinear triangle. In such triangle, | A+ B+ C=180°; . cos. A =cos. {180°—(B-+C)} = —cos. (B+C); similarly, cos. B= ~ — cos. (A+C). But cos. A.cos. B—sin. A.sin. B= cos. (A+B); therefore, by multiplying the three equations, cos. A.cos. B { cos. A.cos. B—sin. A .sin BY =cos, (A+B) . cos. (A+C) . cos. (B+C), and the right-hand side of the equation being a constant quantity, the left is. 32 But, by the formula (1), : T : Tv oT F sin. 1G —_ A) +B} Ze sin. (Z ~ A).cos. B+cos. (; _ A). sin. B =cos, A.cos. B-+sin. A.sin. B (by p. 6.) Hence, therefore, *cos. (A — B) = cos. A.cos. B + sin. A.sin. B.....(4). Cor. 3. Add together the forms (1) and (3), and there results sin. (A+B) + sin. (A— B)=2.sin. A.cos. B....(a). Subtract (3) from (1), and sn, (A + B) — sin. (A — B) = 2.cos. A.sin. Be... 2s (O): Multiply (1) and (3), and, the wehehand side of the equation is = sin. A x cos.” B — cos.’ A x sin’ B= in” A.(1 — sin.” B) — (1 — sin” A). sin” B= sin.” A = sin.” B. Hence, sin. (A + B) x sin. (4 — B) = sin? A — sin.” B....... (c). Add (2) and (4), and cos. (A — B) + cos.(A + B) = 2 cos. A.cos. B......(d). Subtract (2) from (4), and cos. (A — B) — cos.(A + B) = 2s. A.snn. B...... (e). * We may from these formule easily derive expressions for the sine and cosine of 4+ B+ C: thus sin. (4+ B+0) = sin. (A + B). cos. C -+ cos. (4 + B).sin. C , = sin..d.cos. .B.cos. €.-L.cos, 4 «sin, B.cos, C + cos. 4. cos. B,sin. C — sin. d.sin. B.sin. C. 33 If we substitute in the preceding formule (a), (6), (c), Xe. the quantity (2 + 1) B instead of A, we shall have sin. (x + 2) B+ sm. nB = 2 sin. (xn + 1) B. cos. B, sin. (n + 2) B — sn.nB = 2cos.(n + 1) B. sin. B, sin. (xn + 2) B x sin. nB = sin.? (n + 1) B — sin.’ B, cos.2.B + cos.(n + 2) B = 2 cos. (n -+ 1) B. cos. B, cos.2.B — cos.(n + 2) B = 2 sin. (n +1) B.sin. B. Cor. 4. Some of the preceding forms may be differently expressed, for sia ie S | D ¢ making since A= 5 i ; eta B, eet et AB SD z ere Pe oe ae a ee ee ene we have from (a), : ? ar ee S— D sin, S + sin. De 2, sin. ( : ) «cos. ( 5 ), and from (0), sin. S — sin. D = 2 cos. C : =) . sin. (Z & = : or, since S and D are any arcs subject to this condition alone, namely, that S > D, and since, in a series of formule, it 1s con- venient to use the same characters, instead of S and D we may use A and B, and then, sin. A + sin. B = 2 sin. (- = =) . COS. (< z =a) ti bah sn. A — sin. B= ny (€ x =) . Sine (- " =) Nae Ce} P _ By a similar process we may transform (d) and (e) into these, cos. A - cos. B = 2cos. € - =) . COS. (- - =) a ehrecy ye 2 sin. € us =) . sin. (7 e =). (8), E cos. B — cos. A 3t Cow us Dikidel ibe @y and sim A+ sin, BO (- : =) eos. (A ® =) sin. A — sin. B co. CE = at fe - =) tan. () . cot. (- . =) ; an, (A) yO cece e ee ee wel fe Se ey (Sd ‘ Pia) ee eee As © @08. B—cos, A ca ek ne (ee an. : _tan. 3 sine A + sin. B ae € + ‘) vA ’ eee es ts enan bs eee * cos. A + cos. B- ); which, in a particular case, that is, when B= 0, becomes sin. A A ras —cos.” A 1—cos. A A = tan. — . or————————————, OF —enrmenmninrraertin® om Ree 1+cos. 4 2 1+ cos. A 2 1+cos. A From this last expression, we may express cos. A, in terms } A : of the tan. 53 &c.; for, since oA Lie COs A A Ei taattiar: Memeamawarnmme repr fis) 0th 9 1+ cos. 4’ «*« Cor. 6. The forms for the sine and cosine of A+ B being obtained, it is easy to deduce from them, the sine or cosine of any MT ¢ e e arc, suchas —— + A, and nearly with as much convenience as by VL) A reference to the Table that is givenin page 16, thus, by ....(1), - 35 sin. ( + A) = sin. r.cos. A + cos. r.sin. A, but sin. 7 = 0, and cos. 7 = — 1; . sin. (7 + A) = — sin. A. : : 3a SO aT O's Again, sin. (= _ 4) = sin. —— .cos. 4 — cos. —.sin. A; 2) 2 2 : but, sin (= =) sin ( 1) Ss] Oh nt e = e T re = In. ® Cos. oo Cc « e 1 ama ’ is ‘ Ci g £608. 7 ~sIN. © hae 7 ; = — I, since cos. rr O, and cos. 7 = — 1, cos a cos ( +7) oe i in= = 0 —_—= B T La = r) ° AER hse a7.5Sn. = e r) é COS. 7. COS. 5 — sifl.7.sin. — : 37 The sin. (>- A) , therefore, = — cos. A. ‘ 3 S37 ole eae Again, cos. oe - 34) = cos... cos. 3.4 + sin. . sin. 3.A = — sin. 3A, since cos, ohare 0, and sin. Cpe en 1. o7 5 bea O Fee Again, cos. (— + A) = cos. ee cos. A— sin. or sin. A, 1 e (2 ) 2 : e 9 .*. COS. (= +f A) =— sin. A. and, in like manner, other instances may be reduced *,*+. * The above instances are neither intended as specimens of analytical dexterity, nor as mere trials of skill for the student: but they are cases, such as frequently occur in the computations of Practical Astronomy. For 36 We will now proceed to deduce expressions for the tangents of the sums and differences of arcs: and, the first step will be the solution of the following Problem, which, indeed, is little else than a corollary from the preceding results: For instance, (see Astronomy, Ed. 2. Vol. I. pp. 273. &c.). The quan- tities of aberration and nutation, to be applied in order to reduce the apparent right ascensions and north polar distances of stars to their mean, are most conveniently found by formule such as m.sin. (© + A), in which m is a numerical coefficient dependent on the star, © the Sun’s longitude, and A a certain number of degrees, minutes, &c. dependent also on the particular star, to be added to the Sun’s longitude in order to form a sum such as © + 4, technically called the argument. Jor instance, the term representing the aberration in right ascensiori of the pole star for the year 1826, is 44,303 sin. (@ -+ 8° 14° 6’. 44”), and in order to compute the aberration for any particular day, we have merely to substitute for © the Sun’s longitude for that day: for instance, on January 1st, 1826, © the Sun’s longitude = 9° 10° 37' 41” add tothis 8 14 6 44 and the sum is 17 24 44 25 and in order to find the sine of this arc, we must use the table p. 16, or the formula of p. 35; and since 12° is equal to the circumference, the sine of the above arc is the same as the sine of 53 24° 44) 25", the same as the sine of 6 — (5° 15° 35”), the same as the sine of Sra Sod . so that the quantity of aberration in right ascension of Polaris on Jan. 1 1826, is » 4A" .303.sin. (5° 15’ 35”), that 37 ProsiEM 4. It is required to express the tangent of the sum and difference of two arcs in terms of the tangents of the simple arcs. Since (p. 8.) the tangent of an arc is equal to the sine divided by the cosine, if the arc be A + B, sn. (A+B) sin. A.cos. B+ cos. 4.sin. B io acne (Asi, EY cons A cos. Be an An Now, since the object is to obtain an expression involving tan. A, tan. B, we must divide both numerator and denominator of the above fraction by cos. A.cos. B, an operation which will not change its real value; beginning then with the numerator, sin.A.cos. B+cos.A.sin.B_ sin. A | sin. B om Ae = tan. A + tan. cos. A.cos. Bo Eng cehac Recut ee tan. B cos.A.cos. B¥ sin. A.sin. Bo : = sited site BT cae “ cos. A.cos. B a cos.A.cos.B PR Man ay tan. A+tan. B consequently, tan. (A BLE B) = enya ek han ee. (rad. = 1) (9). This formula may be used for determining the tangents of such arcs as 90° + A, 180° + A, &c. exactly, as in Cor. 6. p. 34, we shewed the formule for sin. (4 +.B), &c. might be used in PEs: poi m : determining sin. (Fx + A): for mstance, / that is, AA 303 X.091669 = 47.0612117. Again, on April 16, © = 0° LG Oe add 8 14 6 44 sum 9 O 15 48 and sin. (9° 0° 15’ 48"), = —cos. (0° 15’ 48”) = — .99998, and the aberration is — 44.302. 38 . tan. 90° + tan. A tan A 2 tan.(90°-++- A) Aah at ee = _© Flan AL as La oY as | 1—tan.90..tan.4 1—o.tand — o.tanA : tan. 180°—tan. A O-—tan. A = — ——; again, tan.(180°— 4) =_—_—$___$_$__$_____s = —____—_ tan. 4d ©” ( ) 1+tan.180°.tan.4 1+0.tan.A sin. 45° res Cor. 2. Since tan. 45° = ———-~=1, if we make A= 45°, . cos. 45 there will result from the preceding expression, Ltt tape B tan. (45° + B) = ————... aah etn 1_+ tan. B Cor. 3. Let ¢, ¢’, t”, &c. be the tangents of the arcs A, B, C, &c. then by the formula (9), considering A+B as one arc, tan. (A + B) + tan. C tan. (4 + B+ Q= BRP te Mi zy yy stig ove nceces (q)- tt But, tan. (A+ B) = age? therefore the numerator of the fraction (q) equals tet — tit Ey Re EO and the denominator of the same fraction, equals 1— (tt ett” + #’t’) 1 — ¢?! Hence, tte +t —ttt! 1—(¢t¢ ttt’ +t") If 4 + B+ C = a, (which is the case when A, B, C, are the three angles of a triangle), since tan. 7 = 0, t+ ete — ttt! f+ ! fe yi" aH tt't”’, which is the theorem given in Phil. Trans., 1808, p. 122. tn. (A+ B+ O= = 0, or But the theorem has an origin much more remote; for, the 39 above formula for tan. (A + B-+- C) and similar formule for the tangents of 4 + B + C+ D, &c. were given as far back as the year 1722, by John Bernoulli, and are mserted in the Lezpsic Acts for that year, p. 361, and in the second volume of his Works, at p. 526. The formule for the tangents of the sums of any arcs A, B, C, &c. are symmetrical in their composition, and their law is easily defined: suppose, the symbols S; (ét’), S, (ét't’), &e. be made to represent, respectively, tt et’ + et, fhe ws pane a Nl oe A DR S3(t) — S, (¢7’t’) 1—S8,(¢t’) ’ — SO Sat) 1—S,(¢¢) + S,c¢e'e')’ woe S.(tt't’) + S,(¢t'...t*”) — S,(¢¢) +8, (¢t't't ty re} 5] then, tan. (A + B+ OC) = tan. (A+ B+ C+ D= tan. (4+ B84+C+D+4)= &e. These formule are easily shewn to be true on the principle, that, if the formula for the tangent of arcs be true, the formula for (nz + 1) arcs must be true also: the latter inference being made by means of the form (9), p. 37. 7 If, instead of a radius = 1, we would introduce a radius =r into the preceding formule, we must avail ourselves of the-rule laid down in p. 19. Thus the formula (1), p. 29, becomes’ | r.sin, (4 + B) = sin. A X cos. B + cos. A xX sin. B, _ the formula (c), p. 32, is the same, whether the radius be | or 7. ; A 1 — cos. A : The formule tan.2 — = ——————— to a radius = 1 “4 0) 1 + cos. A ,A r—r’.cos. A becomes tan. S = ————_——_—_— when the radius =7. r-+-cos, A AO The formula (1) becomes sine A + sin B_ Ga cos. A + cos. B ‘ i but, the formula (/), p. 34, remains the same, whether the radius be equal 1 or 7. «*» We will now subjoin a few additional formule for the sines and cosines, &c. of the sums and differences of arcs, the investi- gation of which the Student, by pursuing a track similar to what has been already proceeded on, will easily discover, cos. A + sin. A V2 , cos. A + sin. A V2 : sin. (60° + A) — sin. (60° — A) = sin. A, cos. (60° + 4) + cos. (60° — A) = cos. A, sin. (45° + A) = cos. (45° — A) = : _1+tan.A tan. (45 ns Ape Ere hee FE OD derek tan. (45 A) PTO EU E, — A tan.” A — tan.” B= om (A ee, S Mees a a! lami! A’ x’cos...B sin. ie B).sin.(A + B) g 2 ot. A — cot. A= sin.” A X sin. B M. Cagnoli, in his Trigonometry, has collected into a Table, under one view, and for the purpose of reference, formule similar to the preceding. He has also in another Table (which is subjoined) exhibited the various values for the sine, cosine, and tangent of the angle A. ie & 12. 14, tay 4] Table, Values of sin, A. cos. 4 tan. A. 1 cos. A cot. A” Q, / (1 — cos.7 A). 3. 1 VY (1-b cot.” A)” 4. tan. A Y (1-F tan.” A) ° 5. 2 sin 4 cos 3 +73 2008. 3 re 1—cos. 24 / ( 2 y: 4 A 2 tan. 9 4 1 +tan2 4 9. + oe Resa Bes cot, 3 -+ tan, 3 re sin, (30°-+A)— sin, (30°— A) EE A TV Eee aa 2 sin.? (45° + =) — 1. 2 1] 1—2 sin? (45° Ss ZY. 1—tan.? (45°— S) Nee 1-+-tan.? Gs2y 2 13 tan. (45°42) —tan, (45°-4) (dE ROE LE PC RO Ee? BU tan. (45°43) + tan. (45°-3 ) 14. sin, (60°-+4) — cos. (60°— A), |15. Values of the cos, A. tan. 4° sin. A.cot. A. Y(i—tan? 4). ON lll aka BR /(1-+tan,? A) © cot. A /(1-+-cot.* A) A : cos.? Fo capa sin.? es a pe tA 1—2 sin.? =. ee 2 Bs — ] 5 : VAG 24) Te fath: A ie A ® 2 1-+tan. 3 cot oe tan 2 1 D) sae ° 9 A A’ cot. 3 + tan. 3 1 BRIER I cet , 1-++tan. 4. tan. 3 2 tau. (45°48) toot (42°44) 2 cos. (45° + 2) «os, (45°— <) cos. (60°-+--A)~ cos. (60°— A). F BE, 12, 42 Table. Values of tan. A. jae rey sin. A Y(1—sin? A)” V(1 — cos.” A) cos. A 2 tan. — 2g 1 — tan.? — A COE 2 ‘ A a 2 cot. rs — tan. ms cot. A — 2 cot. 2A. 1—cos. 2A sin. 2A © sin. 2A 1+ cos. 2A’ A (a 2A l1+cos. ZA tan. (4° -}- ne — tan. (45° —_ — AS The investigation of some of these expressions has been already given; and, by pursuing its plan, the Student will, without dif- ficulty, be able to accomplish that of the others. But the Student, whose object is utility, will feel averse from their investigation, should he suspect them to be mere Trigono- metrical curiosities. Such however is not their character; on the contrary, they, in many instances, (we have shewn it in some), materially expedite calculation, and furnish to the general lJan- guage of analysis convenient forms and modes of expression. It is, in accomplishing this latter purpose, that T'rigonometrical formule are chiefly useful: they serve to conduct investigation where the object has no concern whatever with the properties of triangles. Yet, the investigation of the properties of triangles was the object for which Trigonometry was originally invented; and the Student, if he purposes to limit his enquiries to that object alone, need not, in quest of the requisite formule, advance farther than the present Chapter. He may immediately pass on to the fifth Chapter and apply what he has already learned. If, how- ever, his views should extend farther, and he should wish to be possessed of Trigonometry and its formule as instruments of language, he must pursue his researches, become conversant with expressions merely analytical, and, for a time, defer their application. In order that this latter plan may be adopted, we will, in the next. Chapter, continue the deduction of Trigonometrical formule. CHAP. III. On the Sines, Cosines, &c. of multiple Arcs.—Powers of the Sine and Cosine of the simple Arc.—Series of the Cosines of Arcs in Arithmetical Progression.—Vieta's, Waring's, and Cotes's Properties of Curves—De Moivre’s Expression for the Sine and Cosine of a multiple Arc by means of imaginary Symbols. Prosiem 5. Iv is required to express the sine and cosine ef twice an arc, in terms of the sine and cosine of the simple arc. | By form (i), p. 29, sin. (A + B) = sin. A.cos. B+ cos. A.sin. B. Let B=-A; “. sin. (24) =sit. A.coss A+cos. A.sin. A=2 sin. A.cod. A; or, by the Rule of p. 19. to rad*. r, 7 sin.? A =2 sin. A.cos. A.* Again, by the form (2) of p. 31, cos. (A + B) = cos. A.cos. B — sin. A. sin. B. Let A = B; .*. cos. 2A = cos.” A — sin.” A; or, = cos.’ A — (1 — cos.* A)=2.cos. A — 1; or, = 1 — sin” 4 — sinZ2 A= 1 — 2 sin. A. If we employ a radius = 7, then, by the Rule of p. 19, 7.cos. 2A = 2cos. A — 7°, or = r* — Q.sin.” A, (see p. 11.) * This result has been (p. 12.) already obtained, but it is here re- peated, as being the first of a series of formule deduced on the same principle, | AD Cor. |. By transposition, | y — r.cos. 2A, or r(r — cos. 24) = Q sin. A (see p. 11.). Now, r — cos. 2A = ver. sin. 2A, consequently, r.(ver. sin. 2A) = 2 sin.’ A (see p. 11.) which equality, under the form of a proportion, becomes ; : : r ver. sin. 2A : sin. A 2: sin. A: -: iS) > or, expressed in general terms, announces, that the sine of an angle is a@ mean eee tional between the versed sine of twice the same angle and the semi-radius. Again, r + 7.cos.2A = 2 cos.’ A, or 7r(r + cos. 2A) = 2 cos.” A, or (ver. sin. supp’. of 2A) = 2 cos.” A; or, (if we call the ver. sin. of the supplement of an arc, the suversed sine) 7. suversin. (2A) =-2cos.° A, which equality, like the preceding, may be expressed either under the form of a pro- portion, or in general terms. Cor. 2. Hence the sine of 30° = +, radius = 1, for, sin. 60° = sin. (2.30°) = 2 sin, 30°. cos. 30°, (Prob. boy. but, cos. 30° = sin. (90° — 30°) = sin. 60°; : *. sin. 60° = 2 sin. 30°. sin. 60°; V3 or sin. 30° = 4, and consequently sin. 60°, or cos. 30° Te “et Cor. 3. Since, the radius being equal to 1, sin. A + cos. A = 1, and, (p. 11.) 2 sin. 4.cos. A = sin. 24, we have, by the solution of a ses equation such as x ya y saa | : | ry = ie sin. A = VY(1 -+ sin. 2A) + +4 VY — sin. GA), £V(l + sin. 2A) + V(1 — sin. 2A), i cos. A Ww ee 46 in which expressions the upper or lower sign that affects the second term is to be used, accordingly as sin. A is greater or less than cos. A, that is, in arcs not exceeding a quadrant, accordingly as A is greater or less than half a quadrant. ‘These two formule are useful either to compute arithmetically the sines, cosines, &c. of arcs, or to examine their accuracy when computed by other formule; and, performing the latter office, they are called Formulae of Verification. It is easy to perceive their use in computing sines, cosmes, &c.; since, if we take sin. 2A, a known quantity, for instance, the sin. 30° which equals £, we may, by successive substitutions, regularly deduce the sines of 15°, 7° 30’, &c. Thus, sn. 15° = 2 YW + 2) -—E VG — FJ) = .258819, sin. 7° 30 = + (1.258819) — + (.741181) = «1305262, Mines 45 = ee, PrRoBLemM 6. _ It is required to express the sine and cosine of 3 times an arc, 4 times an arc, &c. in terms of the sine and cosine of the simple arc. If we substitute, in the formula for cos. (A + B) (p. 31.), 2A instead of B, we have cos. (A + 2A) = cos. 2A.cos. A — sin. 2A.sin. A; but, by the preceding Problem, sm. 2A = 2 sin. A.cos, A and cos. 2A = 2cos.. A — 1; *, cos. 3A = (2.cos.” A — 1) cos. A — 2cos. A.sin. A = 2.cos.’ A — cos. Ad — 2cos. A(1 — cos.” 4) ? 3 . = 4cos.. A — 3 cos. A, when the radius = 1, 4 cos.” A : } = ——,— — 3.cos. A, when the radius = 7. bY & This form, if we substitute therein, instead of the are 3A, the are e ji oe 3A, gives us AT 3 COs. e ~~ 34) = 4. (cos. 2 but, by Cor. 6, Prob. 3, p. 35. 3 : : cos. lc a 34) = —sin. 3A, cos. (< - A) = sin. A; 214 _ A) CUS. G om A); 9g consequently, ; sin. 3A = 3 sin. A — 4.sin.” 4, when the radius = 1, 4.sin 8 A 8sin. A — : when the radius = r, T By a similar method may the cos. (4A)=cos. (34+ A) or = cos. (2A + 2A), and the cos. (5A) = cos. (44 + A) or= cos. (3A +2A), &c. be deduced. But, the successive formation of the cosines and sines of multiple arcs may, most easily, be effected after the followmg manner: By the form (d), page 32, cos. (A + B)+ cos. (A — B)=2 cos. A. cos. B or cos.(B + A)-+ cos. (B— 4) =2 cos. B. cos. A Let B = nA, then, by transposing cos. (zn + 1)A =2.cos.nA.cos. A — cos.(n — 1) A, and hence from cos. (x — 1) A, and cos. n 4, may be assigned cos. (2 + 1) A: for instance, ifn = 1, cos.(27 — 1)A =cos.0= 1: ; cos, 2 AC B.cos., Alm le se ings (eo ifn 2, cos. 3A=2.cos.2A.cos. A—cos, A=4. cos.’ A—3 cos. A (cM), Liu 3, cos. 4A = 2cos.3A.cos. A — cos.2A = 2(4cos.3 A — 3cos. A) cos. A — (2.cos.” A —1) = 8cos. A—-8cos7*A+l1,, ate PAV CRE And, by a like process, if x = 4, cos. 5A = AS 16 cos. A — 16 cos.” 4 + 2 cos. A — (4cos.2 A — 8 cos. A) = 16 cos.°*A — 20 cos.? 4 +5 cos. A....... yk eS), or, 2cos. 5A = (2cos. AY — 5(2 cos. A) + cos. A. Similarly, if (2 = 5), cos. 6A = 32 cos.° A — 40 an A + 10cos.? A — (8 cos.. A — 8 cos.2 A 4 1) = 32 cos.° A — 48 cos.4 A +18 cos.® A—1....(c%), Hes or, 2.cos. 6A = 2°. cos.6 A — 6.2 cos.4 A te i 2? cos.” A — 2, and the general form is m.(m— 3) 5 (2 cos. Ayn" (e™) 2cos.mA = (2cos. A)"—m(2cos. AY"~* + m .(m— 4) (m—: —5) = 2.3 — (2 cos. Ay" + &e. The formula for the sines of multiple arcs may be deduced from those of the cosines, and, on the same principle as that which has been already used in deducing sin. 3A. By _ substituting, ; , 57 for instance, in the form for cos. 5A, — — 5A instead of 5A, aot 5 cos. (= — 5 A) = 2 Cy ; T x w 16 (cos. = os A) — 20 (cos. = — A) +5. cos. (7 ~ A) : 5 but by Cor. 6. Prob. 3, cos. (= — 5A) = sin. 5A, T cos. (7 _ A) = sin. A} consequently, sin 5A = 16 sin.” A — 20 sin. A + 5 sin. A. we have Or, the sines of multiple arcs may be successively deduced as the cosines have been, on the same principle, and by like formulz; thus, by the form (a), p. 32, A9 sin, (B + A) + sin. (B — A) = 2@sin. B.cos. A Let B = nA, then, by transposing sin. (n + 1)d4=2sin.nA.cos. A — sin, ee 1) A, hence, ifn = 2, sin. 3A = 2.sin.2A.cos. d — smn. J, (sin. 2A =2 sin. A.cos. 4)=4 sin. A. cos.” A—sin. A....(s”) =3sin. A—4sin. A....,. (9%). Similarly, if n = 3, ; sin. 4A =2 cos. A (3 sin. A — 4sin.? A) — sin. 24, = 6 sin. A.cos. A—8 cos. 4. sin.? A — 2 sin. A.cos. A =(4 sin. A — 8 sin.’ A) cos. A, =(8 cos. A — 4cos. A) sin. A.... osc ce ee a(S Ds sin, 5A = Similarly, if m = 4, A'— (3sin..d — 4sin.° A) (4 sin. A — 8 sin.” A) 2 cos.” = 8sin. A — 16sin° A — 8 sin” A + 16sin.° A — (3 sin. A— 4 sin.° A) : =5 sin. d — 20-sin.® 4 + 16 sin A........... or, 2 sin. 5A = 5.2 sin. A — 5.(Q2sin. A)’ + (Q sin. A) sis"); The general expression for sin. mA (m odd) is sin. mA = ae (nm —1) . *—1)(m?—9) . m.sinnA — IS citer kU Ant did Siacoote MULE mah A — &c. Sa 2 2.3.4.5 : sod @ and (m even) is a AAShbs oo Bs Megas) sin? A -- *sin.mA = cos. A |m.sin. A — 0.3 m.(m® — 4) (m® — 16) in.’ A ~ &e.} 2.36465 . The sine and cosine of the multiple arc (mA) have been * For the general demonstration of these forms, see the Appendix G 50. expressed in powers of the sine and cosine of the simple are (A): but, if we express the cosine of the are (4) by a particular binomial, the cosines of the multiple arcs will, m that case admit — of an expression rather remarkable; or, in other words, they may be said to possess a curious property: thus, let ed 2.cos. A= ax +-; I 1\° then, 2cos.2A = 2(2cos. A — 1) = 24; (« +) - it 1 2 =X +-Z- yi By the form of p. 47. 2.cos.3A = 4 cos. 2A.cos. A — 2 cos. A =@+3) Gt) Cro 1 3 =r + =. x? j 1 Generally, if 2.cos. (n—1) A =a") + ea? 1 ; and, 2.cos.nA = 2” + =e then, since (p. 33,) cos. (n + 1) 4 =2.cos.nA.cos. A — cos.(n — 1) 4, 1 1 1 (+5) G+)- Com ] 1 e grt 2cos-(n+ 1) A # — ptt) + Hence, if the form were true for two successive inferior numbers, n—1, and a, it would be true for 2 +1; but, it has been proved to be true in those cases, when n—1=2, and n=3: conse- * This mode of denoting the cosines of multiple arcs, which leads to several curious results, occurs first in De Moivre’s Miscellanea Analytica, pages 8 and 16. 51 quently, it is true for n -~- 1 = 4, and so on successively for all whole superior numbers. : I The above expression holds good also for cos. 2 A, when nis an even number. For, since »>4 co. A+1 cos. — = ————_,, 2 ase) A 1 4.cos. ~ = 2.cos. A +2 eat 24-5 x A 1 ~» &4.COSs. — = Vr +-——. 9) A Pk4/ Now, n+ 1 A n— 1) cos. AR Sot OF. cos. = A x cos. = — cos. A; n 2 +1 o%e ] «COS 5 A=(:? as) Chana), (F: =) _ 4,2 ee Moe teers x? Yale ct | A, therefore, as before, if the expression be true for cos. . ? ® ee . © nN . it is (since it certainly is so for cos. 3 A, n bemg an even n+l number) true, for cos. . A. But, (1. 5,) it is, whenn=2, Cc A ay. true for cos. et therefore, it 1s true for cos. ron therefore for 5A : cea. COR. TTS and, by virtue of these successive inferences, generally true. The above mode of expressing the cosines of multiple arcs is useful on several occasions: for instance, in finding the sum of a series, such as UNIVERSITY OF ILLINOIS “~. LIBRARY, 52 cos. A + cos. 2A + cos. 3A + &c. + cos. nA, for, retaining the former notation, the sum is equal the sum of the two following series, | 1 3 i Chath hs a + &c. + 2”), 1 1 1 1 ~— ] and 3G tatat & +5) 1 yxett! =z x — 1 fice (Coe see ty 2 x—1 Ti) r—1) G ane "4 og - Kes D@*t+))\ 1 ae x" (x — 1) y) TiS A fea eae 0 Ae ed igh oh ee ] 5 ] ei en. ay fh eco, eee A og x Now, aa v( aa Vx at See V« — Vx . Var & eo] tos) Il , nA nA 1 — cos,“ -—— sin. —— 9 2 x \/ Dona RD EE teers peo Mik ance consequently the sum 2 A ERA 1 —-> cos.” — sil, — A 0) 2, eee g a A sin. i ; 9 n+ i of the series is ed alte COS, a+") A. sin, — 2 . . Pa comm hides Since the snnd = yk (« ck =) » we cannot, using : , om x 53 the above notation, directly investigate the sum of a series such as sin. 4 + sin. 2A -+ sin. SA + &e. except by introducing imaginary symbols. The investigation, however, is not difficult, and may conveniently be effected, {although by something like an artifice of calculation) in the fol- lowing manner : In the form (8) of p. 83, if we substitute, instead of Band A, 7 Bi SA A 5A ‘ successively the arcs — and —~, ~—— and —., &c. there will 2 2 2 2 result ) : ‘ cos. — — cos. —-=2.sin. A.sin. _, 2 2 SA 5A cos, ——- — cos. —- = 2sm.2A.sin. —, 2 2 2 BT A By SA cos. —— — cos. —- = 2sin. 3A.sin. —, ees Q 2 Xe. &e. ee A Qn+ 1 Hence, by addition, cos. oe See ota foe 2 sin. (sm. A + sin. 2A + sin. 34 + &c. + sin. nA); consequently, sin. d + sin. 2A + sum. SA + &e. = 2n 1 Bo 4 . n+) cos. — — Cos. salar A sin. ~.A Sin. 2 OS 9 Q ——_ 9 sin. — sin. — 2 Q By a similar artifice may the sum of the series 1 A 1 A 1 A = tan, — + ~ tan. — -+ = tan. — + &c. Q 2 + 4, A T 8 8 a be found. For, if in the expression for tan. (A+B) we make A=B, A 2 tan. A | tan. 2A a apa ane and, therefore (p. 10.) | 1 — tan’ A 1 tan. A cot. 2A = pads AN ae a lng oreo Me 9 tan. A 2tan. A ) = cot. A _ - tan. A (which agrees with the 9th expression for tan. A in the Table, p. 42.). : ; Hence, transposing, and substituting, instead of 2A, 4, suc- cessively the arcs, A and =< and =, &c. there results (by continually dividing each successive formula by 2), 1 A ~ cot. — — cot. A = - tan. —, 2, 2 Lea A I ies ops ct eA = COts-~ — = Col. —~ ==) —~, tan. s—,5 2 4 2 oh 4 1 aed A 1 A ~ cot. — — — cot. — = — tan. —, &e. 8 8 “4 8 8 Hence, by addition 1 A Loe ad —cot.— —cot.A =- tan. — + - tan. + - tan. — + &e. Qn Qn g 2g 7 y 8 + By a similar process, 3 A co-sec. A +co-sec. 2A +co-sec. 4A +&c. = cot. mat cot. 2"—! A. See Cagnoli’s Trigonometry, second edit. p. 122. Z 1 We will now return to the expression 2cos.n A =2"+ ee and shew its use in demonstrating a property of curves given by Waring, in his Proprietates Curvarum, p. 110. The property is this: If in a circle ABCD, &c. the radius of which is 1 , equal arcs 55 _AB, BC, CD, &c. be taken, andif PB be taken = p, p being 6 A P the coefficient of the second term of the quadratic equation 2? — p2 +1 = 0, of which the roots are, a, PB; 1 then, PB =a + P, or =a+-, since af = 1, a i PC. = ath Bor = & Tes 3 3 yc PD = a’ + (’, or =a Ka Now, PB=chord (7 — AB) = 2 sin. G _ fine ah ee — COs. 2 AC similarly, PC = chord(#— AC)= 2 sin. G- Bay eo 2.COS.——, | AD LEAL eh le Seeaiare IS yi ope NUN Aly me a, = 2 cos. -—, PE = &c. ey i> AB and it has been already proved, that if, (putting rou for A), : A 1 CRS stark See gee 7 AB AC ‘ata then 2. cos. 2 (=). or 2 COS. (=) =a + —} 2, 2 v AB AD - 1 aise 2.cos.3 (——), or 2.008. =e? + = ke. 33 Vv 56 ’ al 1 1 hence, PB being x + Atal ia! + , eae “++; PU= a “t bis PD = & + ’, Xe. = Ke. II Vieta, p. 295, Opera Mathematica, Leyden, 1646, expresses the values of these chords, not by the sums of the powers of the roots, but by expressions equivalent to such sums: thus, he puts for PB, 1 N, or N; (N)? he represents by 1 Q, (N)’ by 1C, (NY by 1QC, &c. then iQ—2=PC N?-—-2=PC 1C—3N=PD reise ‘N° —~3N=PD 1QQ-—-4Q+2=PE (¢ moder notation, )N+—4N*+2=PE 1Q@C—5C+5N=PF N°—5N°-+5N=PE, TCC. &c. but, N*— 2, N°—3N, &c. express the sums, of the squares, of the cubes, &c. of the roots of an equation x» — Nx + 1; for, the formula for the sum of the m™” powers is ar GLa ma? Nm 64+ &e. oo) 3 NN" —mN@-? +m ; m— 3 N”-4—m 2 Vieta, therefore, is not to be entirely excluded from the honour due to the invention of the preceding theorem. Vieta calculated by means of the chords of arcs; and, his formule, which we have just given, are, in fact, the same as the expressions for cos. 2.A, cos. 3A, &c. given in pages 46 and 47. Vieta also has, p. 297, given another form, exhibiting the relations between the chords of AB, AC, AD, &c. He puts the chord of AB=1 and the relation of the chord of AC to the chord AC hord of Widohink Die cnora oO AB, N, consequently, N chord-A B ne Serr met: = * See Simpson’s Essays, p. 106. ye 2. sin ie he cite 4 : a am AB sin. 3 seCOSe 3 ; eee te «= cos. aimee i , Bae . AB 2 2.siIn. —— sin. —— sin. —— 2 2 2 He then forms this Table: 1Q—1=chord AD A avers: N?—1=chord AD 1C—2N=chord AE modern ¢ N°—2N=chord AE 1QQ—38Q+1=chord AF | ™%™; | N*— 3 N?4+1=chord AF Kor &c. Now since the chord AD =2 sin. 3 (=), and the chord AE = FMieaty (AB 5 Loa i 2 sin. 4 eae, , &c. and since 2 sin. ae is put = 1, the : é B preceding formule become, if we. put a2 = tar (2 cos. A)? — 1 = 2sin. 3A, (2 cos. A)’ — 2 x 2.cos. A = 2.sin. 4A, &c. which are the same, in fact, as (s!) (s'”) given in peer 49. We may also employ the above mode of expressing the cosines of multiple arcs, in deducing De Moivre’s formula, which is * (cos. A+a/— 1.sin. HER = cos.mA +f — l.sinemA, for since cos A= —(r+-), co ee * Lagrange, p. 116. Calcul des Foncttons, says, that this form is as remarkable for its simplicity and elegance, as it is for its generality and utility: and M. Laplace, in the Legons des Ecoles Normales, considers the invention of this formula to be of equal importance with that of the Binomial Theorem. 6s 58 ai AN ae a — VM ( yy — - r <7 sin. A = (x ~ -) , 5 (: eran S ty) 1 and similarly, 2VW — 1.sin. ma = 2” — —; santas ; 1 hence, cos. A +4/ —1sin. A=z, cos, A— V —1.sin. A ==, 1 —— : ] and, cos. mA+,/ —l.sin.mA =«™,cos.mA— VY —1sin.mA ae: consequently, (cos. A +,/ —1.sin. A)™=cos. mA ay, —1.sin. mA, and (cos. A —/ —1.sin. A)”™=cos. mA—.»A/ —1.sin. mA. If we expand these expressions, and then add them, we shall have e ‘cos. mA = _ mim — 1) sg AE a LINO ie Rae 2.3.4 cos."7* A x sin.* A — &e. If we subtract them cos.”A cos."—*A xsin.” A sin. mA = m.(m—~ 1)(m— 2) ———____---——— ¢€ m.cos.”—'Axsin. A — Ate os.” 3A x sin A + &c. From the above mode of representing the cosines of multiple arcs we may also deduce, and concisely, the formule of Cotes, page 113, &c.* Theor. Log. Pref. in Harmonia Mensurarum, and of De Moivre, Misc. Analyt. p. 16, &c. thus, * The Theorem of Cotes was not announced to the public by its Author, but by the Editor of his Works, Dr. Robert Smith, who informs us, page 113, Preface, that after various conjectures and trials, he ex- tracted it and its meaning from the deceased Author’s loose papers *‘ Re- vocavt tandem ab interitu. Theorema Pulcherrimum,” . M. Lagrange con- jectures, and with probability, that Cotes arrived at his Theorem by the way of Vieta’s Theorems. See page 56. In the expression, 1 2.cos.mA = x” + —, make mA = 0; rE | | LPR then 2” + Lear 2 cos. 0, or, x°™— 2.cos.0.2” + 1= 0, l 0 and x + - = 2.cos. —, or, 2? — 2.cos. —x +1 =0. 1 es ae 1 Now, froma + ~- = 2.cos.—, a” 4-—— = 2 cos. 6 was deduced; x mm x therefore, if x were deduced from the first expression in terms of cos. =, or, in other words, if a were the root of the equa- m | 0 : tion, x — 2cos.— x + 1 = O, that same value of x, or root a, m : : : 1 substituted in the second expression, 2” -+-—— = 2 cos. 8, would make it a true equation, or @ would be a root of the equation geen = 2 cos, 0. a 4-1, 0s Hence, by the doctrine of equations, x — a, is a divisor, both of | 0 8 x = 2.cos. ~ x-+1, and of 2°” — 2 cos.0.2”+1; and, similarly, m : lit x 0 since ~ is the other root of x — 2cos. —x--1, provided a be a m } tr ne. ; 0 one root, x — “18a divisor both of 2° — 2 cos. 2 + 1, and of 2 | 1 x?” —2 cos. 0.x" +1; and consequently, (x — a) (« --) p.0%5 a x? —2.cos. —x + 1 is a divisor of m ge” — 2cos.0.0” + 1. Now, by Table, p. 16, and by the preceding reasoning, it appears that the arcs 60 Qn -0, 47 — 80, &e. (2n-+2) © — 0, Qn+0, 47+ 80, &c. Q2na +80, have the same cosine as the arc @ has: but if, instead of the equation I é x +-~ = 2cos.—, we assume x m 1 Qnr +6 xc +-=2.cos. Gey or=2.cos. (22) or = 2.cos.( ay; ) r Mm | m m =) or = ae a) or = 2cos. (arpa ~— or or= 2.008. ( Me the resulting expressions will be respectively 1 a ae 2.cos.(27 +6) or =2.cos.(44r +6) or = 2 cos. (Qna+O); or = 2.cos, (27 — 0), or = 2.cos. (4m — 9), or = 2. cos. (Qn + 2) 7 — O, which expressions, by what has just appeared, (see Il. 1, 2, &c.) are all of equal value. Hence, of the same expression a™ —2.cos. 0.4" + 1, Qar+6 2 Y 2 x*— 2cos.—xr + 1, x — 2Ccos, x+1, m ) , Ir — : 4a + 0 uw — 2,cos.————_ « + 1, & — 2.cos. ——— 2+ 1, &e. are divisors; in other words, 2°” — 2 cos.0.x2” + 1 may be re presented by a product, of which these latter quantities are the factors; accordingly, xe" —2cos.0.a"%+1= ! 0 Qa +0 | (i= 2.cos.— x 4. 1) x (2° - 9 cing ee ee 1) x m GO 27 = (2° == 2.COSs ———— © + 1) digest m 61 27 If wemake 6 = 0, we have cos.@ = 4, and cos. and 27” — Qa" + 1, or (e&” — 17 = f Qn Aw * @®-2241).(a*- 2008.2 +1) (2°-2.cos.—" +1) “m m &e. and accordingly, 2, . 4, (r—1). (2° 2.cos.— « -+ 1) : (2°=2. cos. 2+ 1). fe. ™ m which is the analytical expression of Cotes’s Theorem. See Harmonia Mensurarum, p. 114, &c. and De Moivre’s. Miscellanea Analytica, p- 17, &c. , : We will now proceed to investigate expressions for the powers of the cosine and sine of an arc, in terms of the cosines and sines of multiple arcs, which expressions are highly useful in all Mathematical mvestigations connected with porate Astro- nomy. ProsBLeM 7. It 1s required to express the powers of the cosine and sine of an arc, in terms involving the cosines and sines of the multiple arc. I 2.cosw A=axr+-; x ey 28 cost Ar= (« = -) = £ 4 2 1 = 2 — g—4 nm ; ye” +n +n. ( — Roache yeaa +—= L faallecting into pairs the terms equidistant from each extremity of the series) * For the expansion of the Binomial, see Wood’s Algebra, page 109, first edition; or Vince’s Fluxions, p. 45; or Woodhouse’s Principles of Analytical Calculation, page 24, &c. 2 62 (4B) ta (ortega) rt Ce ch) 40 .. by page 50, 2”~* cos.” A = cos. nA + n.cos.(n—2) A+ ate 1 cos. (n— 4) A+&c. The number of terms is 7 + 1, therefore, if nm be even, the last term is n n.(n — 1) (n — 2) poate (C —_ out 1) xX.- cos.(n — n) A. 2) ss ). ; n lige see een t 2 ~ Now, n | nm. n.(n—1) (n—2) so (n-24 1) An (2n— 2) (n- 24 1) 1 mr Ore bin’ Gite die {fo eens oe é 2 ) 8 Bat On 2) El VV, BN SS TIN mM Dae ec oT 133.5 tate (n= rye n.(n—1)(n—2).. (; +- 1) 2x2x2, Ke. (105 terms ) x ——— ———— 1.8.5 : 1.2.3, &c.....4u—I) n > 1 Seo eoccee(m— l see ttl) =O" x 1.8 Sereeenln— 1) 1.2.3 a ® e sree consequently, since cos. (n—n) A = 1, the last term = Setar cies Webel: oe a n— 1) Qo” x 1.3.5 ..-.--@—-V A n VO Sawer Q Hence, as instances of the general form, n=2,%.cos.2 A=cos.2A41.. Me ge Fok ee ee n= 3,2". cos.” A=cos.3A+3. bared eee Gn SiR ois Coen = 4,2. cos. A=cos.4A+4.cos.2A4+3. i Ei Cee n= 5,2". cos. A=cos. 5A +5.cos.3A+ 10 ¢ cos. pip ane?) n= 6,2. cos.6 A=cos.6A +6cos.44 +15.cos.2A+10(e””) Xe. &c. 63 In order to obtain a general form for sin.” A, in the expression . “ as for cos.” A substitute instead of A, ieee A, then oT cosit (< - A) = COS. 2 G ~ A) + n.cos.(n— ye ~~ A) re 2.(2— 1) cos. (n — 4) G = A) + Xo. Now, let x be. even, and of the form 2”. p, p being an odd number, or let 2 be, as it is called, pariter par;. nw & yey then on gem lp om = 2™—* 2a, but 2?"—*, p is a whole number, and therefore cos. (275 ip. 27), 1s)! Again, (v = 2)> = (""p— 2)" = (2"—"p = Im, but(2°"~*p — 1)isanodd number, and .°. cos.(2°"—'p —l)r = —1; T no hence, sin cos. C — A) = sin. A, and cos. CB — nA) = nT. | Pong cos. ——~ Cos. nA = cos.nA, (since SN per as 0) , Ke. qr—! sin.” A = namen— i cos. nA — n. cos. (rn — 2)A + Sagi cos. (1-4) A— &c. If n be even, but of the form 2p, or impariter par nw oT 3 == 2p 3 = pw and cos. pr = — |, and consequently, since cos. (pw — nA) = cos. pm.cos.nA = —cos.nA, o°—* sin.” A = — cos.nA + n.cos.(n—2) A — &e. and, in both these cases, the last term is, as before, Lives Gast iors e: (n— 1). 2? 7, n | Oe a ee g ; nT nor Per vd. when 7 is odd, cos. —- = O, and cos. panini: = sin. sin.nA a ‘ 64 ; | ; = + sin.n A, where the upper sign is to be taken, if n be 1, 5,9, &c. Hence, + ol sin? A, = S n(n — 1) 1.2 ‘sinnnA — n.sin. (n — 2) A sin.(n — 4) A —&ec. Hence, as instances of the general form, = 2,2 sin.) A= — tos RATA). 5 dake al Af) n= 3,2". sin’ A= —sin.3A+3.sin. Ai... 2.0222... ee (f®) n= 4,2°.sin.*A=cos.4A—4.c08.2A 43........00660(f) n=5,2'.sin° A=sin.5A—5.sin.3A+10sin.A....... (f”) n=6, 2°. sin.° A = —cos.64+6cos. 4A — 15c0s.2A+10(f) Prosiem 8. It is required to express the tangent of twice, thrice, &c. an arc in terms of the tangent of the simple arc. ‘ By Prob. 4, page 37, tan. A + tan. B tan.(A + B) = Lo aa 1 — tan, A. tan, B’ consequently, al tan. A + tan. A 2 tan. A 1—tan. A.tan. A 1—tan’ A’ tan. 2A + tan. A 1 — tan. 2A.tan. A’ tan.(A + A), or, tan. 2A = Again, tan. (2A + 4) = and the numerator, 2 tan. A 3 tan. A—tan.” A tan. 2A + tan. A = ———,— at tan... = = Ses 1 — tan’. 1 — tan." A the denominator, which 1s 2.tan” A 1 — 3.tan.” A 1—tan. 4. tan. 2A = 1 — ———— = ——___: 1 — tan.” A 1 — tan.” A Hence 3.tan. A — tan. A tan. SA { =tan.(2A + A)} = aa ed by a similar method, sce 4A = 3A-+4, and 5A = 4A+A, > and, 65 4tan. A — 4. tan? A tan. 4A =? 1 — 6.tan.? A + tan.* 4’ 5 tan. A — 10.tan.? A tan.” A tan. 5A = A Soe CE Sie trdatet 1 — 10tan.” A + 5 tan.4 A These expressions for the tangents of multiple arcs may also be derived from those given in p. 39. For instance, since A, B, C, D are equal, ¢, t’, t”, t’” are, and S,(f) = 4¢ ax Sx Ors ttt” = t° vie Ar Salhb ho) 1Xx2x38 / 4x 3 S, (tt) = fe bt q (tt) Feoae put ea tee age consequently, At—4¢° tan.(d +A +A+ A) = tan. 4A = ——; as before, in 1. 1. In like manner, we may deduce tan. 5A from the expression for tan. (4+ B+C+ D+ E) (see p. 39.) For, S; (£) = Oks 5x4x3 S,(t¢’t’) = ————_ # = 10¢° LO 2 xES S. (tft) =e 5x4 S.(t¢) = ——# = 10° s (tt) 1x @ 0 S5X4X3X2 4 1x2@x3sx4 yee ay 5t— 10° + 2° ». tan. 3b, Se a ee ‘ 1 — 10¢ + 50° S(t”) selee Wits as before. - It is not, perhaps, necessary to multiply farther trigonometrical formule ; such as are chiefly useful, and usually occur in inves- i 66 , tigation, have been given; and the Student, who thoroughly apprehends the principle and mode of their deduction, will be able, by his own dexterity, to deduce others. A sufficient number of formule having been given, it may now be thought proper to proceed to their application; and, the first object of their application seems naturally to be that for which the science of Trigonometry was originally invented; namely, the Solution of Rectilinear Triangles. Now, this solution consists of two parts; first, itis necessary to express the relations of the sides and angles of triangles by Trigonometrical symbols ; and, secondly, to afford the means of arithmetically computing, in specific instances, the values of such symbols. For instance, if two sides, a, 6, and an angle A of a rectilinear triangle should be given, the value of the angle B (see p. 28.) would be truly expressed by : ; b sin. B = sin. AX -. a But this is an algebraical value; in order to obtain a practical result, we must be able, when 6 and a are expounded by numbers, and A by degrees, minutes, &c. to express B in degrees, minutes, &c. we must, therefore, possess the means of assigning sin. 4 from a given value of A, and also of assigning B from a re- sulting value of sin. B. These means, in practice, are afforded by Trigonometrical Tables, and their formation, or, what tech- nically is called the construction of the Trigonometrical Canon, is an easy consequence from the preceding results. We will, in the next Chapter, proceed to the construction of this Canon, which may be viewed either as a distinct application of the preceding formule, or as a preparatory step to their appli- cation in the solution of rectilinear triangles. CHAP. IV. On the Construction of the Trigonometrical Canon.—Methods of computing the sine of 1’.—The Sines and Cosines of successvve Arcs.—Formule of Verification, or Methods of examining the Accuracy of Computed Tables. ProsiEM 9. Ir is required to find a numerical value of the sine of 1 minute; the circle being divided into 360 x 60 or 21600 minutes, and its radius being 1. By Prob. 5, cos. A = VW{4(1 + cos. 2A)} (a), and by Cor. 2, Prob. 5, if 2A = 60°, cos.24 = 4 =.5, consequently, cos. 30° = /{1(1 + 4)} = .8660254; substitute this value into the form (a), and we shall have cos. 15°: and, by a repetition of the operation, successively, the cosines 60° 60° 60° 60° of —, —, —s-, —z: 80 that, if c, c4, c”, &c. stand for the successive cosines, the operation must be thus exhibited: 60° 1 i COS. —Fy , OF; CoS. $0) \/ 2 ( ir *)} =c’= .8660254, &c. 0 1 ; : COs. a or, cos. 15°= VC (1 +c!)) =cU= 9659258, &Kc. 60° 0 : ry | cos. as , Or, ‘C0827? 30: & \/ (; (1+ ea) == .99 14449, &c. 60° 1 COS. Fy 1 OF COS. 3°45 = JV (da ve citt)) = ¢7" = 9978589, &c. XC. &e. a j 68 a | 60° rae He we IV V | Xl ek, *COS. Q2” or, cos. 02 44° .3°° .45° = (; Cl -pe yy) 99999990732. From this value of the cosine, the sine = .000255663462. In order to find the sin. 1‘, we must compute on this prin- ciple, namely, that the sines of very small arcs are to one anouier as the arcs themselves+; and then, since the arcs 52”.44.37".45., and 1’ are to another as 00 : NOU 2° 60x 60 9° * The values of oe may be shortly obtained by the following mode of decomposing numbers : 60° 60.60. : 64— 4) (64— 4) (64—4 Se (the units being seconds) Bian Sess) 1S) ote ye (2A Lh fOtes ANE Dh, a eee aw eA lio ema 3 “Alt ee but 2° -+ — = 64”.45", 3.27 4 + = 12”,0.56.15", Pore. subtracting, mI | + If from the two extremities of an arc there be drawn two lines touching the arc and meeting each other, such lines will be equal. By the principle assumed by Archimedes, the arc is < sum of tangents but > = 52", 44'", 3”. 45”. ae 1 d chord joining the two ends of the are ; consequently 3 are < tangent and > 1 : . A.A Wess” 3 chord or > sine of Z$ arc: and therefore tan. 2 —sin,—> — —sin. —: 2 2 2 hence, if, in the instance given in the text, we find the difference be- : ; sin iy tween the tangent and sine of 1’ (computing tan. 1’ from re >) it will be found to be .000290888216— .000290888204 = .000000000012 : consequently, the arc of 1’ differs from its sine by a quantity less than 000000000012: so that, it is plain, the principle of very small arcs varying as their sines, is very little remote from the truth; or, rather, if assumed will entail on the computation a very small numerical error. 69 or, as 3600 : 4096, mall = .000290888204” 3600 7 See This method of computing the sine of 1’, although very operose, is of no great difficulty, since it requires only the knowledge of the simplest arithmetical operations. But, evenif we avail ourselves of the formule and inventions of the Analytic art, the computa- tion of sin. 1’ cannot, in any case, be very expeditiously effected ; more expeditiously, however, than by the preceding method. we have, sin. 1’ = .000255663462 x If we employ the expression for sin. A which was given, in * Since the sines of small arcs may be assumed, with very little error, equal to the arcs themselves, we have, very nearly, 1’ = .000290888204, the radius being 1; let w be the number of minutes of an arc that is equal to the radius, then x x 1, or x x .000290888205=1, and #=3437'.75, nearly = 206265" (=57° 17’ 45”). The arc, therefore, that is equal to the radius contains nearly 206265” (206264'.8), consequently any one of these seconds, or any one part, whether it be an inch, foot, or yard, subtends an angle of 1”, when the radius is either 206265 inches, or 206265 feet, or 206265 yards. A straight line of one foot, therefore, placed perpendicularly to the line of sight, and at the distance of 206265 feet, or of 397.115, subtends, at the eye of the observer, 1”: and the same line of 1 foot subtends at the distance of pe that is, of 1145.5, one minute: and (for it is easy by the common Rule of Three, to multiply these inferences) one foot, at the distance of 1 mile, subtends an angle equal to 206265” canbe a SS 39”.06, nearly, ROMO E eat oie a and at the distance of 22 miles an angle nearly equal to 15”.626. 70 page 45, we may successively deduce the sines of 30°, 15°, 7° 30’, &c. by operations analogous to those already given; thus, since sin A LY (1+sin.24)—4 V—sin. 2A), and sin. 30° = $ A=30° sin. 30° rt. &. See A=15" sin. 15° Lay/(+4)—4 (1-4) =8' =.258819 A=7° 30! | sin. 7° 30’ = £4/C49)—2 4/19) =s"' = 1305262 Miso 407 Sind) bo oe Ly/(4s")-EV/(1as! ‘)= 5" = 0654031 il ole ole 0 and so we may compute on till we obtain the sin. ir The preceding computation was made to begin from A=30", because the sin. 30° is known; but we might have begun from any other arc, the sme of which is known: thus, if we take 2A = 18", the sine of 18°=2 chord 36°, but the chord 36° = the side of a regular decagon inscribed in a circle, = BD (the base of the isosceles triangle described in the 10th Proposition 4th Book V5-1 V5— ya : 1 of Euclid,)* =——>—: hence, sin. 18° she es and sub- stituting this value for sin. 2A in the above form, we have sin. 9° oY aay Gee —4/5)=8 = 156434 sin. 4° 30’= SV (14s) Ly(i—s) =s' = .0784591 sin, 2° 1bie I SV(+s) -£Y(1-8) =5"= .0392598 enc ‘be. Having thus computed the sine of a small arc that is nearly » the method of determining the sin. 1’ is, in principle, pre- i the same asin Prob.9. There 1s, however, a third method, considerably different in its principle, which finds the sine of 1 by the quinquesection and trisection of an arc. It may be thus explained, 1’ By the form (s”), page 49, sin. 5A = 5sin. A — 20 sin A + 16 sin. A. * BAD+-2ABD=180°, but ABD=2BAD; .-. 5 BAD=180°; «. 10 BAD=360° and BAD= 36°: again, by the Prop. 4B. BC = AC? — BD?, consequently, if BD= z AB= Lye Gh —L)= x, and poe Ars 1 A Ud Let 5A = 30°, then sin. 5A=4 and A=6"; let Qsin. A=2, then 1 = 54 — 52° + 2, | by approximation, find the value of x: thus, suppose a to be a near value, and a + 7v to be the true, then 1l=5(a +v)— 5(a@ + 3a'v) + a + 5a*v, neglecting the terms that involve v’, v°®, Kc. consequently, 1—5a+5a—a> Now, since sin.5 A = ~ , assume, as a first approximate value tS 1 = « One. a. aa .2: substitute in the expression for v this value, and find the resulting value of v; it will appear to be = .009, the corrected value of a then, or a--v, is .209; with this, find a new value of v, and another corrected value of a, and repeat the operation till x is found exact to a certain number of decimals, seven for instance; in which case “x = .2090569, and consequently sin. A = .1045285. Having thus obtained the sine of 6°, in the form (s’”) of page 49, that is, in sin. 3A + 3 sin. A — 4 sin.” A, put 3A = 6°, and 2 sin. A = 2, then the equation becomes .2090569 =3x1—2°. Find, as before, by the method and formula of approximation, a value of x, which, to seven places of figures, will be .0697989, conse- quently x or 2 sin. 2° = .0697989, and sin. 2° = .0346995. In order to find sin. 1°, take the form (p. 44,) sin. 2A = 2sin. A .cos. A, then sin.22A = 4.smn.? d—4. sin.’ A: substitute for sin.” 24, or sin. 2”, its value, and by the solution of a quadratic equation find sin? A, and thence, sin. A, or sin. 1°, the value of which to seven places of figures is .0174524. Repeat the operation, and 0 we have sin. —, or sin. 30’, the value of which is .0087265. ' By this method then we have descended from the sin. 30° to 12 the sin.1° and sin, 30’; and, consequently, by like operations, we can descend from sin. 30’ to sin. 1’ and sin. 30”: and by this method, which is however extremely operose, we are able to find the sin. 1’ without a proportion, and, accordingly, to avoid the use of a principle, which some may think doubtful ; which principle is, that the sines of small ares are to one another as the arcs themselves. The above method is, in fact, the same as that which is given, with all its detail, at page 451, &c. in the sixth Volume of the Scriptores Logarithmici, edited by Baron Maseres. It is plain, however, that there is no necessity for beginning the computation from an arc of 30°; we may make it begin from any arc, the sine of which is known: for instance, by the form, page 70, - sin. P=FV(3 + V5) —-3V(5 — V5)=.156434; since, therefore, sin. 9° = sin. (3. 8°) = 3 sin. 3° — 4sin.* 3°, solve, as before, by approximation, the equation 156434 = 347 — 42°, and the result gives the sin. 3°. Again, solve a similar equation by the same mode and formula, and the result gives sin. 1°. And many like methods will suggest themselves to the mind of the intelligent Student. We shall now proceed to the second part of the construction of T'rigonometrical Tables, the object of which will be under- stood from the succeeding Problem. PRoBLEM 10. It is required from the sin. 30” and sin. 1’, to compute the sines of 2, 3, 4, &c. minutes, and also the sines of 1, 2, 3, &c. degrees. By the form (a), page 32, sin. (4 + B) = 2sin. A.cos. B — sin.(4 — B) = 2sin. A (: — 2 sin.” =) — sin. (A — B) = sin. A + jsin. A — sin. (A — B)!— 4sin. A sin.” =. ob CO if B = 1°, then sin. (A + 1°)=sin. d+sin. 4 —sin. (A — 1°) — 4 sin. A sin.® 30! which is Delambre’s formula. Let B=1' and let 4 successively equal 1’, 2’, 3’, &c. sin, 2’ = sin. 1+ (sin. 1’ — sin. 0) — 4 sin. 1’. (sin. 30”)? sin, 3’ = sin. 2’ — (sin. 2’ — sin. 1’) — 4: sin. 2°. (sin. 30”)? sin. 4’ = sin. 3’ + (sin. 3’ — sin. 2’) — 4 sin, 3’. (sin. 30”)? sin, 5° = sin. 4’ + (sin, 4’ — sin. 3’) — 4 sin. 4’, (sin. 30")? &c. = &c. and thus may the sines of all succeeding arcs be computed, by a process not very tedious, since the only part of it at all long is the multiplication of sin. 2’, sin. 3’, &c. by the constant factor (2 sin. 30”), which is the square of the chord of V’. In the above form substitute, instead of B, 1°, and, instead of A, successively 1, 2, 3, &c. degrees; then sin. 2° = sin, 1° ++ (sin. 1° — sin. 0) — 4 sin. 1° (sin. 30”)? sin, 3° = sin, 2° ++ (sin, 2° — sin, 1°) — 4 sin, 2° (sin. 30”), &e. . and so on for the sines of all succeeding arcs. In order to compute the sines of arcs composed of degrees and minutes; arcs, for instance, such as 3° 2’, 3° 3’, substitute for B, 1’, and for A successively 3° 1’, 3° 2, 3° 3, &c. then sin, 3° 2’ = sin. 3° 1’ + (sin. 3° 1’—sin, 3°) — 4 sin. 3° 1’ (sin, 30%)? sin, 3° 3’ = sin, 3° 2' + (sin. 3° 2’—sin. 3° 1’) — 4 sin, 3° 2’ (sin. 30”)? &c. or, if we wish to compute for every ten minutes, put B = 10, and for A write successively A + 10’, A + 207, A + 30, &e. thus, if 4 = 7°, sin. 7° 20’ =sin, 79 10'+-(sin. 7910’—sin. 7°)— 4 sin. 7° 10’. (sin. 5 gin. 7° 30’ =sin. 7° 20'+4(sin. 7° 20’—sin. 7°10')—4 sin. 7° 20’. (sin, 5’), &e. ; By the preceding methods we are enabled regularly to com- pute the sines of all arcs from 1’ or 1” up to 90°; but, when the Kk TA ares exceed 60°, the application of the Trigonometrical formula, page 32, renders the arithmetical computation more simple and concise: thus, since sin. (B + A) = sin.(B — A) + 2.cos. B. sin. A. Let B = 60°, then cos. B = cos. 60° = 4, consequently, sin. (60° + A) = sin. A + sin. (60° — A). (see form 15 of Table, p. 39.) | Hence, instead of the preceding, we may use this latter method, and compute the sines of all arcs exceeding 60°, by the simple addition of the sines of arcs previously computed: for instance, sin. 63° 5’ = sin. 3° 5’ + 56° 55’ and, since sin. 3° 5’ = .0537883, and sin. 56° 55’ = .8378775 the sin. 65° 5’ = .891665. The sine of all arcs from O to 90° being computed, the cosines of all the arcs of the quadrant are known from the equation cos. A = sin. (90°— A); for instance, cos. 63° 15’ 7” =sin. 26° 44’ 53”, cos. 13° 47'=sin. 76° 13’, Ke. The sines and cosines being computed, the tangents may be : sin. 4 computed from this expression, tan. A = , and the co- cos. A cos. A tangents from co-tan. A = — E sin. A When the tangents of arcs up to 45° have been computed, the Trigonometrical formule, previously given, may be conveniently used in computing the tangents of arcs that lie between 45° and 90": pe by Prob. 4, page 37, tan. A+tan. B 1+tan. 4 tan. S Ce 14, REE. Teese reer pMyiaeateats 1—tan. A. tan. B Cobras Me —tan. A hence, as instances, putting A = 1°, 2°, 3°, &e. Td : 1. -+ tan. 1° tan. (45° -+ 1°), or, tan. a0 = print a cpu: 1 — tan. 1 1 tan. 2° tan. (45° + 2°), or, tan. 47° = pai last sa; | — tan. 2 &e. &e. or, we may thus avoid the fractional form, 1+tan. A 1—tan. A tan. (45°-F.A) = —————_ and tan. (45°— A) =—_—___—_:: 1} —tan. A ( ) 1+tan. A’ (1+ tan. A)? —(1 — tan. A)? ih 1 — tan. A i tans AD ee barn tA aia 2 tan. A. 1 — tan.” A- “. tan. (45° + A)—tan. (45°— A) = but, by the form of page 37, tan. 2A = Hence, tan. (45° + A) = @tan. 2A + tan.(45° — A), con- sequently, tan. (45° +- 1°), or, tan. 46° = 2 tan, 2° + tan. 44°, tan, (45° +- 2°), or, tan. 47° = 2 tan. 4° +- tan. 43°, &e. By these formule and methods may the sines, tangents, &c. of arcs be computed. If we attend, however, to the history of the construction of Trigonometrical Tables, we shall find that all Tables have not been computed exactly by the same formule and methods: modern Tables, from the improved state of analytic sckence, having been computed bythe most certain and expeditious methods. In the immense Tables du Cadastre, formed at the expense of the French Government, the sines of arcs are com- puted regularly by successive addition, according to the formule given in page 73; but, m such a construction, an error committed in the sine of an inferior arc would, it is plain, entail errors on the sines of all succeeding arcs. Hence is created the necessity of some check on the computist, and of some independent mode of examining the accuracy of the computation. For this purpose, formule, such as those given in pages 45, 69, derived immediately 76 from established properties, are employed; if the numerical results from these formule agree with the results obtained by the regular process of computation, then, it is almost a certain conclusion that the latter process has been rightly conducted. As there is, in these formula, called Formule of Verification, besides their practical utility, something curious, several are subjoined and proved. fi] sin. 30° = 4; sin. 45° = et sin. 60° = = : 5— f [2] sin, 18° = v z : , by the Note to p. 70, or, it may be independently proved thus : sin, 36° = cos, 54°. or, sin. (2. 18°) =cos. (3. 18°). Let #= cos. 18°; then by form [c’”], page 47, cos. (3 .18°)=4a°— 3.2, and by form, p. 44, “sin. (2. 18°) = 2a4/(1—a*) = -. 4a°—32, or, 24/(01— a") = 427 ~—3. This equation, cleared of radicals and reduced, is 16a — 2007+ 5 = 0; 5 5 cai bette whence 2 2s and .. l— a? = oo and V1 — a, or, 4751 A [3] sin. 9°, or, cos. 81°= 44/(5+43) — 34/(5— 4/5), by the form, of p, 45, on substituting for sin. 18° its value, cos. 9°, or, sin. 81° = 44/(3+4/5)+24/(5 — V5), by the same form. [4] sin. 27°, or, cos. 63° = $4/(544/5)-4 (38-145) cos. 27°, or, sin. 63° = 14/(544/5) +5 (3-145) for, in the preceding proof of the arithmetical value of sin. 18°, sin. 18° = which is also cos. 72°. 5 : 3— 4445 (COs 718")? = EE and (sin. 18°)* espe *. COS, 2) TS GF, 24/5 : cos. 36° = (cos, 18°) — (sin. 18°)* Bean ee ms ee = sin. 5409, since cos. 36° = sin. 54°, substitute this value of the sin. 54° in the forms of page 45, for sin. 2.4, and there will result the above values for the sine and cosine of 27°. Veet) ago = ee woe 4 AFIS 4, : Since, cos. 36° = 77 The subjoined Table contains the sines of arcs from 0° to 90° - that differ by 9°, or in the French division of the circle, that differ by 10 degrees. French Scale. . English Scale. py sin. 10°, or, cos. 90° sin. 99, or, cos. 81°=44/(3+4/5)—44/(5—4/5) sin. 20, or, cos. 80 sin, 18, or, cos. 72 =i (4/5—1) | sin. 30, or, cos, 70 sin. 27, or, cos. 63 =44/(544/5)—44/(3—4/5) ~ sin. 40, or, cos. 60 sin. 36, or, cos. 54 =44/(10— 24/5) sin. 50, or, cos. 50 sin, 54, or, cos. 45 = WF sin. 60, or, cos. 40 sin. 54, or, cos. 36 =2(4/5+1) sin. 70, or, cos. 30 sin. 63, or, cos. 27 =$4/(5-+4/5) +44/(3—4/5) sin. 80, or, cos, 20 sin. 72, or, cos. 18 =44/(10+4+215) *sin. 90, ‘or, cos. 10 sin. 81, or, cos. 9 =214/(34+V5)+44/(5—4/5). But the most general formula of verification is this, which is to be found in Euler’s Analysis Infinitorum, page 201, vol. I. (Lausanne, 1748). [5] sin. A + sin, (360 — A) +-sin. (72° 4+ 4) = sin. (36° + A) + sin. (72° — A). In order to prove this formula, we will use the numerical values of the cosines of 36° and 72°. V5+1 rate sin. (36° + A) — sin. (36°— A) =2.cos. 36°. sin. A= sin. A * There is a large Table of this kind in Cagnoli’s Trigonometry, p. 58, &c. edit. 2. And, it is easy to see how additional formula may be obtained: for instance, sin. 15°=sin. (45—30°) = sin. 45° (cos. 30°— sin, 30°) er paeo iN P Y3+1 Similarly, sin. 75° = And these and like expressions, besides that. utility which is pointed out in the text, may have a farther one, in the theory of Polygonals. 78 and sin.(72°+ A)—sin. (729 — A)=2.cos.72°. sin. A= v ; in. A; subtract the latter equation from the former, and then we have, sin. (36° +.A)-+sin. (72° — A) — sin. (72°-+ A) — sin. (36°— A) =sin. A, which transposed is the equation (5). ‘If in the above equation, we substitute 90°— A instead of A, there will result ae sin, (90°— A) +- sin. (A— 54°) + sin, (18°-} 4) = sin. (540-+ A) — sin. (18°— 4), or sin. (90°— A) = | sin. (54°-+ A) + sin. (54°— A) — sin, (18°-++A) — sin. (18°— 4), which is Legendre’s mode of expressing the equation. But, it is plain from the mode by which the latter has been deduced, there is no real difference between the two formule, and, with regard to their application, it is quite indifferent, whether we adopt Euler’s or Legendre’s. In using these formulz, different values must be substituted. for A, thus: in Euler’s, if A = 9°, then, sin, 9° + sin. 27° + sin, 81° = sin, 45° + sin. 63°; or, sin. 9° + sin. (3.9°) + sin. (9. 9°) = sin. (5. 9°) + sin. (7.9°). [fin Legendre’s formula we make 4 = 81°; then sin. 9°= sin. 45° — sin. 27° — sin. 81° -++ sin. 63°; or sin. 9° +- sin, 27° + sin, 81° — sin, 450 + sin. 63°, the same as before; "ich proves what) we Rae just asserted, Ga GA ae Again, if 4 = 18°, then, by the formula [5] sin, 18°-Fsin, (36°— 18°) + sin. 90°==sin. (36°-+18°) 4 sin. 7 2° yar or, 2 sin, 18° + 1 = 2sin. 54°. If A = 10°, then sin, 10°-+ sin. 26° + sin, 82° = sin. 46° + sin. 62°. w EXAMPLES. Take Sherwin’s Tables, in which the natural sines of arcs are inserted - from these it appears, that sin. 9°= 1564345 sin. 27 = 4539905 sin, 45° = 7071068 : sin. 81 = 9876883 sin. 63 = 8910065 15981133 15981133 sin. 10° = 1736482 ‘ sin, 26 = 4383711 sin. 46° = 7193398 sin. 82 = 9902681 sin, 62 = 8829476 16022874 16022874 Here the numerical results of the two sums exactly agreeing, as they ought to do according to the formula, we may conclude, almost with certainty, that in Sherwin’s Tables the sines of 9, 27,-81, 45, 63, 10, 26, 46, 62 degrees are rightly computed. , These formule are most convenient for practice; butif, from the solution of an equation of three dimensions, as (s’”’) p. 49, or, from that of an equation of five dimensions, as (s”), the value of sin. 3° or of sin. = be computed, such value would become a means of ascertaining the accuracy of Trigonometrical Tables. In 1610, Pitiscus published great Trigonometrical Tables, inserted in his Mathematicus Thesaurus, and, from the account of this Work, given in the Berlin Memoirs of 1786, page 24, it appears, that formule, which, in fact, are formule of verification, were employed by that mathematician: thus, inorder to ascertain whether the chord of 30° had been rightly computed, he sub- stitutes in the equation, *4Ax° — g' = c° (x = chord of 30°, c = chord 60°) * Pitiscus’s notation is like Vieta’s, given in page 54; the form of his equation ie) ‘instead of x the computed value, and he finds the resulting value of ¢ to be 1000; as it ought to be, since the chord 60° = radius: and accordingly he concludes x to have been rightly computed. Again, in order to ascertain whether the chord of 10° be rightly computed, he substitutes its computed value in an equation, such as chord 30°= 3 chord 10° —(chord 10)* [see form (s!”) page 49] and the chord 30° thence computed ought to agree with its value (x) previously ascertained to be right; and he pursues a similar course, in order to verify the computation of the chord of the fifth part of an arc, Previously to quitting this part of our subject, we wish to employ the arithmetical values of the sines of 18° and of 36°, _ which have been just deduced, in proving that which may be announced as a geometrical property. 5+1 5-V5 ai, and .*. sin.” i be 3— V5 the alam = 4 sin.” 18°, see page 76, or (chord 72°)? — 1 = (chord 36°)*, but the chords of 72° and 36° are respectively the sides of an equilateral pentagon and decagon, inscribed in a circle. Hence, the square of the side of an equilateral pentagon inscribed in a circle, is equal to the square of the radius plus the square of the side of an equilateral - decagon inscribed in the same circle. By the form p. 77, cos. 36° = consequently, 4 sin.? 36°— 1= Having now obtained methods of arithmetically computing the sines, cosines, &c. of angles, when the angles are expounded by a specific number of degrees, minutes, &c., we may proceed to apply our formule to express the various relations that subsist between the sides and angles of rectilinear triangles. equation is 4q— 1 6 g=square of chord of twice the arc. Another account of this Work of Pitiscus, and of similar Works, published about the same. time, and now very rare, is given im the 5th Volume of the Memoirs of the Institute. CHAP. V. On the Resolution of the Cases of Rectilinear Triangles—\st, When the Triangles are Right-angled.—2d, When Oblique.— Reasons for introducing different Solutions of the same Case. Examples, &c. Iw a triangle there are 3 sides and 3 angles: any three of these being given, the remaining may be obtained. ‘Thereis one ex- ception to this, which takes place when the three quantities given are the 3 angles. ‘The reason of the exception is this: take any tri- angle, then, externally or internally, other triangles may be formed with sides parallel to the sides of the proposed triangle, which triangles shall have the same angles, but greater sides or less sides: the magnitudes of the sides therefore are independent of the angles, and consequently cannot be determined from them. We will begin with the solutions of the cases of right-angled triangles. 1st Case of right-angled triangles, in which two sides are given. _ Here, besides the right angle C, a, 6, are given, and c, 4, B are required. Solution. | Example. c determined. a= A3; «.a*= 1849 Ist. c=4/(a*-++b*) Euclid 47. Book 1, b= 55; .*. = 3025 A and B determined. Shh Cd lie is and c = 69,81, &c. ad. = = 7 by Cor. 1. to Prob. 2, Computation. sin, BT r the tabular radius = 10 42 seilans 2 LOGS Te cet rare LO but 44+-B = 2 “ or 90° .*. sin. B=cos. A log, 43 = .. ..1.6334685 sin. A 11.0334085 Une Eaeiae aa 3, ad.= 1), oF log. 55=....1.7403627 aa : ce -. log. tan. 4 = 9.8931058 tan. Ar, 5 and expressed in log™. A=38° 1 8" log. tan. A = log. r'4 log. a—log. 6 and B= 51 38 52 cuts B Be ae hah. Again, ‘log. r= 10 BY P log. 55 = 1.7403627 A being determined, c may: for 11.7403627 __ 6. sec. Aa BE log. cos. 38° 1’ 8” =9.8964202 r ~~ cos. A —sdog. € == 18439425 ‘, log. exlog. r--log. b—log. cos, A. . €==269, 81, &¢. as before. L 82 2d Case, in which the hypothenuse and one of the acute angles are given. Here c, B, and C = = 90°, are given, and a, b, A required. Solution. 6 determined. *..d=c.sin. B=c. In logarithms, log. b=log. c+log. sin. B—log. r. A = 90° — or@=Cc. Example. c=361.4, B=Al® 12’ Computation. log. 361 .4=2.5579881 log. sin. 41° 12’=9.8186807 log. b= 2.3766688 * Dee aao. Us sin. 48° 48’ a4 B = 48° 48’; «. —————- = -; . 0 / eens SE e a8 and, consequently, @ may be deter- mined exactly as b has been; or thus, from }, a=/(c?—b*) = 1/{ (cb) (c—4)}. In Log™. log. a = ; { log. (c-+-b) + log. (c—6)}. 3d Case, in which a side and the acute angle (which is not opposite to it) are given. Here, 6, A and C= 90° are given, and a, c, B, are required. Solution. B determined. B = 90°— a determined. 2 : [Cor. 1. Prob. 2.] sin. B hie sin. A +, \sin. 3A a, SGNBUNL UES (orn LOR ak _ otan, 4. es more *. in logarithms, log. a=log. b-+-log. tan. A— 10 c determined. sin. A sine Cc ony sin. C sincB 8.” ek ORIN, since C= 90° and _ B=cos. A b Eee cos, doo cam co a mae In logarithms, log. c= 10-4 log. b—log. cos. A, Example. b=31.76 A=17° 12! 51” + B=72° 47' 9". Computation for a. log. 6 or log. 31.76=1.5018805 log. tan. 17° 12’ 51” =9.4911132 log. a -+- 10=10.9929937 *, a2=9.8399 Computation for c. log. 107° == 10 log. 31.76 = 1.5018805 10-+Llog. 6. =11.5018805 log. cos, 17° 12’ 51” = 9.9800967 log. c = 1.5217838 Ser ki, fs PL 83 | We now proceed to the Cases of Oblique-angled Triangles. First Case, in which two angles and a side opposite to one of the angles | are given. Here, A, B, a, are given, and ), c, C, are required. Solution. Example. C determined. A= A4+-B+-C=180°.:.C =180°—(A+B)|41° 13” 22”, B=71° 19 5”, a= 55 6 determined. i, oe by Cor. 1, Prob. 2. 180°—(1 12° 32! 27”) =67° 27’ 33" Bin a i Oe. hangs sin. B log. a or log. 55 = 1.7403627 Spork fai ety Site A log. sin. B In logarithms, or log, sin, 71° 19’ 5” = 9.9764927 log. b=log. a+log.sin, B—log.sin. 4 log. a + log. sin, B= 11.7 168554 The side c is similarly determined. log. sin. A= 9.8188779 . log. b= 1.8979775 *, b=79.063. Second Case of oblique-angled triangles, in which, two sides and an — angle opposite to one of the sides are given. Here, a, b, B are given, and A, C, c are required. Solution. Example 1. (ambiguous). A determined. a=:178.3, 6=145, B= 41° 10! sin. 4 a ; : a : —————— = -.°.sin. d=sin, B=. Computation for A. sin. Bb b . log. sin. B In logarithms, log. sin. A =log. sin. B+-log. a—log. b This case may be ambiguous, or or log. sin. 41° 10’=9.8183919 log. a or log. 178 . 3 =2.2511513 will admit of two solutions, when 12.0695432 a>b, and B is acute: for, let MN log. 6 or log. 145 =2.1613680 OO =a, MP=b, 2MNP=B, take Mn log, sin. 4 =9.9081752 = MN, then MP ( aor MN, MP would fall beyond WV and n, as MP’ does, and no other line equal to it can be drawn between P’ and N;: in this case, 4 has one value only. If B be obtuse, 4 cannot; therefore here also the case is not ambiguous, A and B being known, C = 180° —(A + B) is known. c determined. pein. sy sin. C_ sin, BOS Cb te Be "sin. B’ or c may directly, that is without the intervention of the process for finding A, be determined from this expres- sion. c=acos. B+4/(b* —a’ sin.* B). 2) Ae BAP) P28" and C= 84 47 38 or A = 125 57 38 and: Cee oe) 22 _ Example 2. (not ambiguous), a= 145, b=178.3, B=41° 10’ log. sin. 41° 10’=9,.8183919 log. 145=2.1613680 11.9797599 logo A7S8.S. ce 22Gb 51S log. sin. A= 9.7286086 wv. A=e32° 21' 54%, In this instance the supplement of A cannot belong to the case proposed. Computation of c in Ist Example. log. 145 =2 1613680 log. sin. 84° 47’ 38” 9.9982047 12.15957 27 log. sin. 41° 10'= 9.8183919 log. c = 2.3411808 0 & 21057 Third Case, in which two sides and the included angle are given, Here, a, 6, C are given, and 4, B, c are required, a sin. A By Cor. Le Prob, 2. — 3 = crc aie a ay. sin, A i teh Bree a—6b sin. A—sin. B a Le ects Me a+b sin. A-+-sin. B- be aun Boo or similarly Example. a= 562, b=320, C=128° 4’. Computation. a—b = 242 a+b = 882 A + B=180°—(128° 4’) = 51° 56° se me Ma roy a log, is, log, tan. 250 58’ = 9.6875402 ) 7? A-B log. 242 = 2.3838154 a—) _ sin, d—sin. B_ 2 12.0713556 g+-besing 4 -alty Bey ce log. 882 = 2.9454686 . A—B y [Lf], page 34, log. tan. = 9.1258870 n, 428) compute .° from this ex- oes mpute 3 rom this ex A —= = — 7° 36! 40" pression, and ey een orien A+B a ——— = 25° 58 the Trigonometrical Tables, and 2 since oot Ab cat oe ae A-+-B= 1800—C we shall have 4 B = 18° 21’ 20° and B, for if ins = s, and scat syed ; * Computation of c then, A=s +d, and B=s—d. a : abo log. 562 = 2.7497363 c determined. ; ° log. sin. 128 A ir ; 896136 sins Ce pe fee or log. sin. 51 50 f.* nity ? sin. 4". sin, 4 ac In logarithms. log. c = 12.0458732 log. sin. 33° 34/ 40'= 9.7427789 log. c = 2.9030943 * C = SOU,0F log. a + log. sin. C — log. sin. 4, or c may directly be found, thus: a’ --b*—c* 2ab o. c=4/(a*-+b?— 2 ab cos. C) which form however is not suited to logarithmic computation. cos. C = The above is a complete solution of the case, in which two sides and the imcluded angle are the quantities given. But, the analytic art is required to furnish, besides merely ade- quate solutions, commodious and concise ones. And, of this latter character are the solutions which have been given of the third case by Dr. Maskelyne, in the Introduction to Taylor's Lo- garithms, and by Legendre, inhis Trigonometry, p. 369, 4th edit. These solutions we now proceed to explain. 86 Let a>6: find in the Tables an angle 0, such that tan. @= ee and from this logarithmic expression : log. tan. 9 = 10 + log.a — log. b.......-+-+ (a) in this case, since a>0, 0 is>454, for when 0=45°, tan. 0=r. a+b b > ! as i peat a Now, since tan. O=r.-, tan. 0+ r=r G + 1)=r tan. 0 — °) but by Prob. 4, pp. 35, 36, &c. tan. (@—45") =r (ae A-B A+B hence, 7 tan. eae =tan. ( 5 ) «tan. (0 = 45°)....(B) consequently, since @ is known from the expression (a), A—B tan. may be computed; and thence, by means of the Trigonometrical Tables, is known, and A and B may be determined as in the former case. Solution of the preceding Example by this method. Computation of @ by the formula [a]. 10 + log. 562 = 12.7497363 log. 320= 2.5051500 od log. tan. @ = 10.2445863 .-. @=60° 20° 35” 6 — 45° = 15° 20’ 35” " by the formula [9] : log. tan. 15° 20’ 35” = 9.4383476 Z , or log. tan. 25° 58’ = 9.6875402 Computation of log. tan, i -, 10 + log. tan. = 19.1258878 87 ae seer? = 7° 36’ 40” and since ae = 25° 58’ A=33° 34’ 40” and B = 18° 21’ 20” the same result as was obtained by the preceding method, and obtained in practice, almost with equal facility, even when it is necessary to take from the Tables the logarithms of a and 6. The demonstration of this method is not more concise than that of the preceding; but, the rule and the connected com- putation are, and, especially, in those cases in which the loga- rithms of a and 6 should happen to be given; for then 0 would ; : A—B immediately be determined from the form (a), and hit from the form (3), so that the whole of the rule would be expressed by the two forms (a), ((). The above method of determining the angles A, B, is the same as that which is contained in the fourth Proposition of Robert Simpson's Trigonometry, p.486, of his Elements of Euclid, 6th edition: and, in substance, is the same as the method given by Dr. Maskelyne, p. 36. Introduction to Taylors Loga- rithms: the sole difference of the two methods is in the ex- pression: instead of the formula, r.tan. ese = tan. bo Dr. Maskelyne directs us to employ ) tan. (@— 45°) C -0 r.tan. (= A= =) = co-tan. ©. tan. (@—45 ) | CHR eB but, since A + B+ C = 180, = = (90 _ Ps ) C A+B - consequently, co-tan. te tan. ( 5 ) ; Two methods of computing the side c, have been already given, one, from A and B previously determined; the other, independent of such determination: the latter method, however, 88 is not adapted to logarithmic computation; but, it may be by the introduction of an angle, called a subsidiary angle, such as @ is in the preceding demonstration, thus: from page 85, c= V(a + 0 — 2ab.cos. C) = V(a — 2ab + BD + 2ab — 2ab.cos. C) Vita — b) + 2ab (1 — cos. C)} = (a — b) J (1 + ay ver. sin. c) 4ab- ver. sin. C | Assume ————- ——————. = tan.” @, in which cas (a— by oe) , 3 4ab ver. sin. C 1 ee FO = tan.7 0 = sec.® 0: ax (a— by 2 Ht " consequently, aie putadh = Ge a ¢ = (a—6). sec. 0, or = cos sec. O aeraerae “br, bila cos. or c aces and in logarithms, log. c=log. (a—b)+log. sec. 9—10, or=log. (ee —b)+10—log. cos. 8. This agrees with Dr. Maskelyne’s determination of c, given in p. 36. of his Introduction to Taylor’s Logarithms: and the sole difference in the process 1s that, instead of ver. sin. C alg Se orun 2 he uses, sin. Fe: which two values, as it appears from p. 45, are equal. Example of the computation of c: a and 6 being 562 and 320, and Os. 128" 4, 2 logy tan. 0@—10=log. 2-Flog. a+-log. b--log. ver. sin, C—2 log. (a—6) Now, log. 2 = 0.3010300 log. 562 = 2.7497363 log. 320 = 2.5051500 log. ver. sin. 128° 4’ = 10.2085966 15.7645129 2 log. 242 = 4.7676308 10.9968821 89 .. log. tan. 6 = 10.4984410 ;’.*. log. sec. 8 = 10.5192823 .*. log. c= 10.5192823 + 2.3838154—10 = 2.9030977 . C = 800.01. If we make the value of c equal to V(a + 2ab +b? — 2ab — 2ab cos. C) we shall have c=(a+6) Ja- a — (1 + cos. C)) 4ab 1 -C and, if we assume 4 BF = Be GSS > ) = sin. 0, there ‘will result, c = (a + 0). cos. 0, 1 + cos. C is the versed sine of the supplement of C, which Mr. Mendoza, in’ his valuable Tables on Nautical Astronomy, calls the *suversed sine; hence the rule for the solution alge- braically expressed is, log. sin. 0 =1{10+log. 2 + log. suver. sin. C-+log. a+log. 6—2log.(a+b)} log. c = log. (a + 6) + log. cos. 8 — 10. Of the preceding solutions of the third Case, one alone, as it has been remarked, is, in strictness, sufficient: the others have been added, for the sake of rendering, in certain cases, the computation more expeditious. And, when‘a specific instance is presented, it will not be difficult to determine which method of solution it is, that ought to be adopted. If, for example, the side opposite the included angle be alone required, ‘we ought to com* pute it by the method of p. 88, I. 14, avoiding, as unnecessary, the calculation of the angles (A, B,) at the base. Fourth Case, in which ‘the three sides are given. Here, a, 6, c, are given, and 4, B, C, are required. * In these Tables the log. suversed sine = log. M 1-4 cos. Th 90 First Solution. By Prob. 2, page 27, sin. 4d = | (ca) 2) (etbte—-5) ett) at+b-+e and, in logarithms, putting acs S, and instead of sin. A, sin. A ? bi log. sn. A — 10 = log. 2 + | i flog. S+log. (S—a) + log.( S—6)+log. (S—c){ —log. 6 —log. ¢ and similarly, log. sin. C is expressed by the same form, sub- stituting in the negative part, mstead of — log. c, —log. a, so is also log. sin. B, substituting instead of — log. b, — log. a. Second Solution. By Prob. 2, page 26. (a+ b—c)(a+c—6) 2be ads Gee ei ee 1— cos. 4d = , Ser | But 1— cos. 4=2. sin.” a” (page 44.) hence, introducing the radius 7, ie WA. 2 _ 6-)(S=0 r be In logarithms, A : 2 log. sin. ne 20 + log. (S — 6) + log. Carel 4 b — log. c, and similarly, OA aNe , @ log. sin. ee 20 -+ log. (S — a) + log. (S — 6) — log. a— log. 6. Third Solution. By Prob. 2, page 26, Be ar gots (a+6+c) (c+b—a) _ eS ay ‘ 2Qbhe bc —A av A B tl e A= ] = ¢° MS — =— ¢ 7 Dhl ——— 7 2 — nn eS ut 1 + cos cos. (7r~— A) = Qsin. Q sin. (C =) Q —2 In logarithms, A 2 log. cos. one 20+ log. S + log. (S—a)—log. b— log. ¢ and similarly, s 3 2 log. cos. See 20 + log. S+-log. (S — c)—log. a—log. 6. Fourth Solution. | feces : poy A A Divide the expression for the sin.” Rs by that for cos.’ , and since the tangent is equal to the radius multiplied into a fraction of which the numerator is the sine and the denominator the SD) S-o er 2 SoS ce ayes ‘ cosine, we have, tan.” In logarithms, | A 2 log. tan. ae 20+log. (S—6)+log. (S~c)—log. S—log. (S—a) and similarly, I C 2 log. tan. ~ = 20+ log. (S— 6) + log. (S—a)—log. S—log. (S—c). 5 Yo 92. Example by the first method of solution. a=33 log.=1.5185139} .. log. a-+-log. b=3.5479235 [C]* b= 42.6 og =1.6204000 log. b4-log. c=3.3585744 [4] c= 53.6 log. = 1.7291648 log. a-+-log. c=3.2476787 [.B] nla =64.6 log.= 1.810325 aa — a=31.6 log. 1.4996871 ee — $= 22 log.= 1.3424227 psa) c=11 log.=1.0413927 2 | 8.937350 | 2.8468675 € =log. area: Cor. 2. Prob, 2. (10 + log. 2)... .10.3010300 , area = 702.858 13.1478975. Hence, (see the formula of solution) if we subtract from this loga- rithm (13.1478975) the values [C], [4], [B], we shall have, re- spectively, the log. sines of the angles C, A and B: 13.1478975 13,1478975 13.1478975 [C] 3.1479235 [A] 3.3585744 [B] 3.2476787 9. 9999740= log. sin. C9.7893231=log. sin. A 9.9002188=log. sin. B ., C=89° 22° 214 A=37° 59! 53” B=520 37' 463 or 89 22 22 B=52 37 462 to 89 22 25 . C=89 22 202 180 0O O In this case, if we had not determined A and B, the value of C (determined from its logarithm of seven places of figures) would have been doubtful to the extent of 4”: consequently, for the finding the exact value of C, in this instance, the first method is not proper. * [C] [4] EB] are merely marks of reference. 93 By the second method ; the same Example. Angle C computed. Angle A computed. log. (S—a)= 1.4996871 log. (S—6) = 1.3424227 log. (S—b)= 1.3424297 log. (S—c) = 1.0413927 (20 added) 228421098 (20 added) 22.3838154. [C] or log. a + log. 6 = 3.1479235 [A] — 313585744 2 log, sin. c= 19.6941863 2 log. sin, : = 19.0252410 En a log. sin. a5 9.8470931 log. sin. a= 9.5126205 7 C = 2 (449 41’ 104) i A es 211 89. SD 56'S") = 89 22 208 = 37 59 52%. By this method the angle C is determined at once with great accuracy. By the third method. Angle C computed. Angle A computed. log. S= 1.8102325 log 8 a 1.8102325 log. (S — c) = 1.0413927 log. (S — a) = 1.4996871 (20 added) 22.8516252 (20 added) 23.3099196 [Cc] 3.1479235 [A] 3.3585744 19.7037017 19.9513452 Cc A log. cos. = 9.8518508,5 log. cos. 5= 9.97 56726 we C= 2 (44° 41’ 10" 5. “A = 2(18° 59! 56”.5) = 89 22 202 = 87 59°63. As C, in these two last methods, is determined to a great degree of exactness, the value of B is not computed from the formula, but it may be had by subtracting 4 +c from 180°. OF By the fourth method, Angle Cc computed. Angle A computed. log. (GS — 5) = 1.3424227 log. (S — 6) = 1.424227 log. (S —@)= 1.4996871 log. (S—c) = 1.0413927 (20 added) 22.8421098 a, (20 added) 22.3838154 (a) Again, Again, log. S = 1.8102325 : log. S = 1.8102325 log. (S —c) = 1.0413927 log. (S — a) = 1.49960871 —--2.8516252 (d) 3.3099196 (0) ”, (c)—(d) = 19.9904846 .. (4)—(b) = 19.0738958 .. log. tan. —_ 9.9952423 .'. log. tan. é = 9.5309479 . C=2 (44° 41’ 10"F ate ee? (LB BO oO cn) = 89 22 202 = 27 59 53. As far as instances prove, any one of these three latter methods may be used for determining the angle C, and angles nearly of the same magnitude ; and, it is of no material consequence which it is that is used. The first method is plainly, from a mere comparison of results, insufficient to give exactly the value of such an angle as C is: and we need not go through the labour of the arithmetical computation in order to ascertain its insufficiency: for, if we perceive that the square of the side, such as c, is nearly equal to the sum of the squares of the other two sides, we shall know that the value of C does not differ much from 90°.* It may now be worth the while to enquire, more minutely, why, since compendium of calculation is a desirable object, several methods of solution have been given. * This 4th case of oblique triangles is commonly (see Robert Simpson’s Euchd, page 488; Ludlam, page 220,) solved by means of this proposition. ‘The sum of the two sides of a triangle is to the base as the difference of the segments of the base is to the difference of the sides; but the demonstration of this proportion, since the case is other- wise more conveniently solved, is purposely omitted, 95 Now, each of the preceding methods is adapted to logarithmic computation, and each, in an analytical point of view, affords a complete solution. One solution, therefore, would have been sufficient, and one alone given, if the same applied, with equal convenience and equal numerical accuracy, to all instances; but the fact is otherwise. If an example were proposed in which the angle dA should be nearly 90°, as C is in the former example: the log. sim. A might be deduced from the first solution; but, to such logarithmic value, there would not, in the Tables, correspond a precise value of the angle A: for instance, if the numerical value of log. sin. A should be 9.9999998, A might equal (by the Tables) either 89° 56’ 19”, or 89° 57’ 8”, or any angle inter- mediate of these two angles. The reason of this is, the very small variation of the sine of an angle nearly equal to go’. And, this small variation is apparent from the mere inspection of the Geometrical diagram, in which two contiguous sines should be drawn to two arcs each nearly equal to a quadrant; or, ana- lytically, it may be thus shewn. Let A be an are nearly = 90°; let it be increased by a small quantity (1” for instance), then by the formula (1), p. 29, making B = 1”, sin. (A + 1”) = sin. A. cos. 1” + cos. A.sin, 1”. Subtract sin. A from each side of the equation, then sin.(4 +1”) — sin. d=sin. A (cos. 1” —1)+cos. A sin. 1”, ; Bh ho 1 ¥ (by p. 44,) =—sin. A. 2s” rai cos. A Site. t/ . . 9 - 1 bd Now, sin. A x @sm. > may, from the smallness of the f factor 2 sin.” mae be neglected; and, accordingly, sin. (A +1”) — sin. A = cos. A. sin. 1”, nearly ; therefore, the difference of two contiguous sines, or what has been called, the variation of the sine, varies nearly as cos. A; and the cos. A is when A =90°, nearly, a very small quantity relatively to its other values, m which A is of a mean value. 96 It must not, however,, be unnoticed, that the want of precision in the determination of the angle is partly owing to the con- struction of the Logarithmic and Trigonometrical Tables: The Tables referred to, and in common use*, are computed to seven places. of figures; but, if we had Tables+ computed to a greater number of places, to double the number, for instance, then the logarithmic sines of all angles between 89° 56° 18”, and 89" 57’ 9”, would not be expressed, as they are in Tables now in use, by the same figures. In such circumstances, we should. obtain conclu- sions very little remote from the. truth; but, then, such Tables would be extremely incommodious for use, and would, in all common cases, giveresults to a degree of accuracy quite superfluous and useless. Moreover, such Tables, even in the extreme cases which we have mentioned, are not essentially necessary: since'their use can be superseded, by abandoning the first method of solution, and recurring either. to. the 2d, 3d, or 4th method. When the angle (4) sought then is nearly = 90°, the. first method must not. be. used, but one of the. latter methods, in which either the sine, cosine or tangent of half the angle is deter- - mined; and, in such an extreme case, it is a matter of indiffer- ence whether, instead of the first method, we substitute the 2d, or 3d, or 4th. But, in other cases, it is. not a: matter of indifference: for since, as it has been shewn, the variation or the increment of the sine is as the cosine, and of the cosine as the sine, these two variations are equal at 45°, but beyond 45°, up to 90°, that of the sine is less, and that of the cosine greater; * Sherwin’s 8vo. Hutton’s 8vo. Taylor’s 4to. + In Vlacq’s Tables, published at Gouda, 1663, we have Arcs. Log, Sines. SOU CHOI LON. bund due Chee ah ie cs SET Ne 9.9999997300 | 2 SR ca OS Siete Ase a PTs PUMA AMERY ACG 8 OEY SH Rie a SANS SR ee 6s, Glos le nae Inte ieee OP Oat 97958 BRAN 5 Wek ls rs VIR puna re tees | ee eer Lear tea pe 8 98154 89 57 O al cURL ES Wallace fore te) poe RN 6 98346 97 and, the contrary happens between 45° and 0; consequently we have this Rule: If the angle sought be < 90°, use the second method; Ifeccccccescsssee es > 90, use the third method. The 4th method may be used, and commodiously, for all values of the angles sought from O up to angles nearly = 180°: ( A baa when, however, the angle (A) is nearly = 180°, tan. a which is nearly tan. 90°, is very large, and its variations, (which are as the square of the secant*) are also very large and irregular. If, therefore, we use Sherwin's Tables, which are computed for every minute only of the quadrant, the logarithms corresponding to the seconds, taken out by proportional parts, will not be exact: for, in working by proportional parts, it is supposed, if the difference between the logarithmic tangents of two arcs differing by 60 seconds be d, that the difference between the logarithmic tangents of the first arc, and of another arc, that differs from it only by 2 seconds is 60% : now, this is not true for arcs nearly equal to 90°; and an example will most simply shew it: by Sherwin’s ‘Tables, * For by the formula, p. 37, tan. (4-1) = tan. A -+- tan. 1 1—tan. A.tan. 1“ Subtract tan. A from each side, and tan, 1” + tan.” 4 tan. 1” tan. (4+1”) — tan. 4d = iA ced) ze ‘1 — tan. A,tan, 1” Y tas ae meee = (tan, 1”-++tan,? 4. var 1”) (1-+tan. A tan. 1%) = tan. 1” (1 + tan.” 4) nearly; OY .. since tan. 1” is an assigned quantity, tan. (A+1"} ~ tan. 4 oc 1-++ tan,” A oc sec,” 4. ‘ N 98 log. tan. 89° 30° = 12.0591416 log. tan. 89 29 = 12.0449004..) 0.02.08: (2) log. diff. corresponding to 60° = 142412 diff, corresponding to 30” = 71206. ..(4) .“s by Rule, log. tan. 89° 29’ 30” ([2] +[4)=1 12,0520210 whereas true log. tan. 89° 29’ 30”, by 'Taylor’s Log. = 12.0519626 Again, | log, tan. 89° 50! = 12.5362727 log. tan. 89 49 = 12.4948797 log. diff. corr’. to 60” = 4.13930 *, diff. corrg’. to 6” = 41393 *, by the Rule log. tan. 89° 49’ 6” = 12.4990190 whereas the true log. tan. 89° 49’ 6”, by Taylor’s Log™*. = 12.498845 In these instances, the log. tangent, determined by the pro- portional parts, is too large, which it plainly must be; for, the logarithmic increment of the tangent increasing as the arc does, that is, the increment during the last 30” being greater than the increment during the: first 30°, if we take half the whole increment for the increment due to the first 30”, or one- tenth of the whole increment, for the increment due to the first 6”, we plainly take quantities too large. The same reason would, it is true, hold against calculating logarithmic tangents of any arcs by proportional parts, if the values of logarithmic tangents were exactly put down in Tables; but, (we speak of the Tables in ordinary use) the values are expressed by seven places only of figures; and, as far as seven places, the irregularities in the suc- cessive differences of the logarithmic tangents of arcs that are of some mean value, between O and 90°, do not appear; thus, by Sherwin’s Tables, log. tan. Aa 30’ = 9.9924197 - log. tan.°44 29 = 9.99 21670 log. diff. as Heer to 60” = 2527 *, diff. corresponding to 30 = 12635 .. by the Rule log. tan..44° 29’ 30” = 9.992290335 and the true log. tan. by Taylor’s Tables = 1.99292934. i 99 It appears then, from the assigned reason, and by the instances given, that an angle nearly 90° cannot exactly be found from its logarithmic tangent. The determination of the angle by means of proportional parts will be wrong in seconds by Sherwin’s Tables; and will be wrong in the parts of seconds by Taylor’s Tables. From the whole of what has been said then, it appears that in computing the values of angles, two inconveniences may occur, either when the successive logarithmic numbers are too nearly alike, as in the case of sines of angles nearly YO°, or too widely different, as in the case of the tangents of angles nearly equal to 90°. It is the busmess of the Analyst to provide formule, by which these inconveniences may be remedied or avoided; and hence have arisen the different methods for attaining, apparently, the same end. Before we entirely relinquish this digression, we wish to observe, that, although the log. sine or log. tangent of the angle A may be determined exactly either by the first or the fourth method, yet, if it should be very small, its value cannot, with sufficient exactness, be determined by the ‘Tables in common use. For, very small angles cannot be exactly found from their loga- rithmic sines and tangents; not exactly in seconds, by Sherwin’s Tables, nor exactly, in parts of seconds, by Taylor’s Tables; and therefore, as great exactness may be required, and is commonly required, in those cases, in which a very small angle is to be de- termined, the ‘Tables are not to be used. ‘They are to be super- seded by a peculiar computation, of which, without demonstration, Dr. Maskelyne has given the rule in his Introduction to 'Taylor’s Logarithms, p. 17 and 22. This rule aud similar rules will be stated and demonstrated in a subsequent part of this Work, when the analytical series for the sine and tangent of an arc are deduced. To the several cases of the solution of oblique triangles, examples have been given, but, merely arithmetical examples; it may be proper therefore, to subjoin a feigned case of practice and observation, in order to shew, more plainly, the use and ap- plication of the formulz of solution. An observer at 4 wishes to determine his distance from two inaccessible objects B, C, and also the distance BC, of the same objects. 100 ry. ' 7 , ‘ Phe observer takes a new station D, and measures the distance AD; suppose it to equal 1763 yards: at A and D, by means of proper instruments, he makes the following observations : ? BAC = 45° 1' 3" BDC = 36 15" 8” oF ik ‘Caes 30° 0’ a) at tt Cua SRE eae wp consequently, CDA = 69° 22’ 45”, ACD = 180° — (ADC + CAD) =80°37'13" BAD=75° 1! 5", ABD=180°—-(BAD+ BDA) =71°51'15" AB determined. c By ist Case of oblique triangles, p. 83 log. 1768 = 3.2462523 AB sin. ADB log. sin. 33° 7’ 40” = 9.7375966 ide log. AB = log. AD wn log. sin. log. sin, v1 51’ 15’— 9.9778456 fet im ae pee gr log. AB = 3.0060033 2) AB = 1014, AC determined. By ist Case of oblique triangles, log. 1763 = 3.2462523 AC _ sin. ADC- log. sin, 69° 22'45"= 9.9712441 AD ~ sin. ACD’ 13.2174964 Met Bh om ci ie + log. sin. log. sin. 80° 37’ 13” = 9.9941543 ADC — log. sin. log. AC = 3.2233421 «. AC = 1672.4. 10L CB determined, By the formula of computation | 10-Llog. 2 = 10.3010300 given in page log. 1672.4= ' 3.2233421 tan.*0 = an Qa _ ver. sin, C log. 1014 = 3.0060033 eee. (des Can log. versin. 45° 1’ 3”= 9.4670294 and, log. tan. 6 25.9974048 eer deere: Va 8 F : 2 log. 658.4 5.63697 96 +log. versin. C— 2 log.(a— b) —————— i , 20,3604252 x (a—6) . log. tan. 0 = 10.1801466 He and log. cos. 06 = 9.7412271 [a] log. c=10--log. (a—b)—log. cos. @ | -** Since here, a= AC=1672.4, b= AB=1014 10-log, 658.4== 12.8184898 [b] C= BAC=465° 1'3", a~b=658.4 | 18. ¢ = 3.0772627 . [6 ]=[a] and c = 1194.7. - The last part of this Example (the determination of CB) belongs to the third case of oblique-angled triangles, in which two sides and the included angle are given; but, since the angles at the base, (the angles ABC, ACB,) are not required, the method of solution given (see p. 84,) was not adopted: BC, indeed, may be computed, and, according to the common practice of calculators, would be computed, by first determming the angles ABC, ACB from the form ABC— ACB AC—AB ABC + ACBy | tan. (——) = an. (=) | 2 AC+AB g in. BAC and then BC from this expression, BC=AB. ee but it is plain, that the computation, without being at all more exact, would be longer than that by which BC has been already determined. The last part of the Example will also serve to illustrate the use of that solution of the third case, which was given in page 87: for, the logarithms of 4C and AB being determined in the previous part, we have immediately log. tan. 9 = 10+(3.2233421 — 3. 0060033) = = 10. 2173388 ; whence @ = 58° 46 23”, and 0 — 45° = 13° 46! 98". consequently, since log. tan. 13° 46’ 23.8 = 9.3893876, 102 log. tan. coe or co-tan. 22° 30’ 31.5 = 10.3835881 ; *. log. tan. oo = 9.7729757 and peti as 30° 39’ 48” but cael a = 67° 29’ 28.5 consequently, ABC = 98° 9’ 16".5 ACB = 36° 49' 40".5 The process had been somewhat more tedious if we- had found these angles by the formula given in page 84, in the solution of the third case; for then we must have computed AC — AB, AC + AB, and have taken out their logarithms. This instance, and the remark on it, have been introduced to shew, not that the common and general solution is insufficient, but that other solutions may conveniently, that is, with some gain of expedition, be imtroduced. The Student, however, who shall peruse this Treatise in order to be initiated into Trigo- nometry, is advised, in the first perusal, to attend solely to the general solutions, and to postpone to a time of leisure and of ac- quired knowledge, the consideration of the methods that are either more expeditious, or are adapted to particular exigencies*. * With the view of rendering every thing as easy as possible to the Student, separate investigations of the cases of right-angled triangles, have been made to precede those of oblique-angled triangles; but, con- sidered generally, the former cases are really included in the latter, and their solutions comprehended within the general solutions ; we will shew this in two instances: suppose in the third case of oblique triangles, that the included angle is a right angle, then, by,the solution, ALB aan) 5) ae age tan. (AE 2) = tan (= a) gps but A+B+C=180 and C= 90 Ade B : : ee Pe y == 45 43 consequently, 103 We will terminate this Chapter with one or two Problems, the solutions of which may be deduced, almost immediately, from the preceding formule. 1. In a triangle of which the sides are a, b,c, and the angles opposite, A, B, C, it is required to determine C, when a, 6, and A— B are given. . By the formula of p. 85, tan. —) = tan. cake Se tan. i wee ) x = (see p. 9,) cot e x (see 110,)-—— sat — £0, Ae —— (. a : 2 — FE “E * a+b’ tan. — 2g A—B 7>t@—-b ab , consequently, tan. = tan. 45°, PET ah » since tan. 45° = 1 but 4 = 90°— A—B 4 a A— ok = 45°—B; «, tan. ( B ) = tan. (45°—B)=(by Prob. 4, —tan. B Con 2s pe 81.) oF Bi a—b 1—tan. B b = ——___. DB — 5 ion, i hence —— ey fale ad tania and tan ; the same solution, in fact, as was given, page 81, in the solution of the first case of right-angled triangles. Suppose next, in the same instance, we employ the form used in ‘the fourh case of oblique-angled triangles, thus : b Db ae 1+-cos. A = (AS x we =), bot an hs roles ct : bc 2 - Ail 2 28 hy Nee Mh Aa mh since c? = a” + 3’: 4 2 b3-+-bc Alo b hence, 1-++ cos. A = =- + 1 and cos. 4 = b] be ah BY oa the same result as that which was obtained, page 81: so that it is ‘plain the variety of cases might have been diminished, but not without a considerable loss of simplicity and facility. 104 a—b A-— Bb x cot. . a+b 2. Given, A, B, C, and a+0; required a and 6. *. tan. —- = 2g E e A—B If the upper sign be used, then a—b = (a+b) x tan, e x tan. 3 : C A—B If the lower sign be used, then, a-+ 6 =(a — 0). cot. x cot. —> and in each case, by adding and subtracting, we obtain Re and. _ 20, and thence a and 6. 3. Given A, a and b + c; required 6, c, &e. | By the third formula of solution, p. 91, eth se A ESOT) RO Ee re ea 2 2". COs. be Hence dc is known, let.it = p, and let b+ ¢ = s3 then b-++ c= s : be = p; .*. (by the solution of a quadratic equation) tat Ge) eV Geo 4. Given a + 6+ ¢, the sum of the sides, the area of the triangle, and the angle A; it 1s required to find the side a (see Newton, Arith. Univ. Prob. 8.) BY Prob..2,. p. 27, 1.2. : . (64+c¢) — a? Hp (a+b6+c—a)—a’* 1+cos. A = os 2bc Qbec Again, the area TS = sin. Ax “ (a+b+cyY—2a(at+b+c) 4K ak 1+ cos. A. pater oeiaiude (sid 1+ cos. A = x sin. A; “a=h(atbte) - 105 : 2K = = (a + b +. c) “+ reer xX cot.—3 A in. A A | since (Cor. 5. p- 34.) AF pear =tan.—- = ae cot. a The formule of Trigonometry have now been applied to the resolution of rectilinear triangles; the original object, for which the science was invented. And, it is to be observed, such application is the most easy, and is of very extensive practical utility. In the next Chapter we will continue to apply, still farther, the preceding formule. The first instances will shew their utility in expediting Arithmetical computation: the latter, selected from Works and Writings on Physical Astronomy, will shew their utility in subjects of great importance and of arduous investigation. In this last application, the original object of the science, the Properties of Triangles, seems entirely to be lost sight of, and the Trigonometrical analysis is peculiarly and almost solely useful, because it confers precision and power on mathe- matical language. * — CHAP. VI. Instances of the Utility of Trigonometrical Formule. i. _ Iv is required to compute the logarithm of a + 5b, atbaa(it"). ‘ a mtd b Assume, for the upper sign, ~ = tan,? 9, for the lower, - = sin.? 6 a * Votes: a then log. (a + 6) = . log. a + log. (1 + tan.”@) = log. a + 2 log. sec. 0 — 20, ana log. (a — 6) = log. a + log. (1 — sin.” 0) = log. a + 2 log. cos. 0 — 26, - In the application of this method, a and 6 are not supposed to be numbers: for, then, the simple way would be to add them and to take the logarithm of the sum; but a and 6 are com- pounded quantities, formed of the sines and cosines of angles : thus, in finding the Moon’s distance from a star, versin. dist. = versin. (d’ — d) -++ cos. d. cos. d’. versin. A (d.d’ being the declinations, and A the difference of the right ascensions) a is = versin. (d — d) and 6 = cos. d.cos. d versin. A, and tan.” @ is assumed = eee. renee = cos. d.cos. d’. sin.” — cos. d.cos. d versin. 4 2 versin. (d’ — d) » or, ,d—d P] Q sin. and then, by the process above stated, the form is adapted to logarithmic computation. 107 M. Sejour solves this imstance somewhat differently (see his fraité Analyt. tom. I. p. 103.) Thus, leta + 6 = y, then, a sin. 45°46 cos. 45° = y.sin. 45° (1). Sits ro | Assume tan. z =~; ...——— = ~-, and at 42) COB: Ze as : asin. z — 6 cos.z =O : (2). Eliminate a from the equations (1) and (2), and | 5 (sin. z cos. 45° + cos. z . sit. 45°) = y sin. 45° sin. z, or &. sin. (45° 4+ z) = y sin. ss sin. 2, 6. sin. (45) z) sin. 45°. sin. 2 and .*. ¥, ora + b= Again, in order to compute the logarithm of a + 6 + c, since a+ 6 +c = y +, make tan, go 218 : On. * : c¢ sin, 45°. sin. z =e a _ then “as before, ~ b” sin. (45°+ 2)? sins (45° + 2’) fou An a eatin 45°. sin. 2° and so on fora + 6.+c+d, &e. 2. It is required to compute the sine of an angle; for instance, the sine of 3°. Sin. 3°= sin. (2°-+ 1°) = sin. 2° cos. 1° + cos. 2° sin. 1°, sin. 1° cos. a cos. 1° sin. 2° 0 , tan. 1 = sin. 2° cos. (i+ yi tan. 2° sin, 2° cos. 1° (: + ee tan. 1° Assume atan. @ = tan. 2°? and we shall niver in a logar priate form, 108 log. sin. 3° = log. sin. 2°+ log. cos. 1°+ 2 log. sec. 6 — 30. 3. Let it be required to compute a quantity P, such, that —- (1 +e) +e’) (1+e”) x &c.; the law of the formation i e', e', e 5 &c. being , 1-VU-e) mle VW(1 — e?) ne 1-V(1—e”) STEVE Oe THe Se ae ee The computation is conveniently effected thus, put ¢ = sin. 0; 1— cos. 0 = (pa Jrguqtiy ae Eo Tis ' Q . V(1—e*) =cos. 0, and e = 0 6 and, 1 + e = 1+ tan.” 5 = sec.” 5 Again, put e’ = sin. 0"; : 1—cos. 0 : ; oe = = tan.’ — and 1 + e” = sec.* — 1+ cos. 0 2 Again, put e’ = sin. 0”; gain, p vf Ld 1 —cos. 0” g uw Q == ‘tan, a and 1 +e = sec. = &ec. arty +cos. 0” Ud co] ee 4 “n OS Gua Hence P = sec.” on sec. — z . sec.?— . &e. and log. P = (supplying the tabular radius) " 0 6! 0 2 (log. sec. = 5 + log. sec. — o TueE, sec. oe Kc. ) _ 2(10 + 10 + 10 + &c.) 4, * Required the integrals of the differential expressions, d@.sin.@.cos.@; dO.cos.20; d@.sin20; d@cos.? 6; dé d@.cos. 0. cos. nO; ap * This, in other words and symbols, is to require the fluents of the Auxionary expressions 0. sin. @. cos. 0; 6 cos.” 0, &c. 109 In order to integrate these expressions, it Is necessary to premise (see Simpson’s Fluxions, Vol. I. p. 165.) @ being any arc, that do = d(sin. 0) 60) ile (d cos. @) pees d (tan. 0) Foo 8 < sin. 0” sec.” 8 : Ist. d0.cos. 4. sin. 0. By the formula of p. 44, cos. 8.sin. 0 = = + sin, 20; -, fd0.cos. @.sin. 6 = 5fd0.sin 96.= = /'2d0. sin. 20 cos. 20 +c (c¢ = correction). oe I pea ad. fd@.cos.? 6 By the formula of p. 44, Os Us = (1 + cos. 20); Hay hE oonf 2+} 5 J 40. COS. 26 7 ee | wre =f ++ 20 .cos. 28 = : + sins 26 +c. ) sd. fdé. sin.’ 0. By the formula of p. 44, 7 I sins.@ = 5 (1 — cos. 26); if dd jsine @ =/ <2 - = fd .cos. 20 110 Paes ee ta ; — 4 S240 . cos. 20 l roe sin. 20 Co 4 3 | o ~ Q 4th. [d@. cos. 9. By the formula of p. 62, , 1 BS cos.2 9 = Z cos. 30 + A cos. @; 1 3 “ f dé cos.” 9 = 7/48 . cos. 30 + °7i/ a0. Coste ie —_ f'3d0 . cos. 3 ean 3.4 - 4 = Sica, (ty + = sin. 9 + c. 3.4 4 Sth. fd0.cos. 6 .cos.n8. By the formula of p. 47, 1. 16, cos. 9. cos. nd=- $cos. (n— 1) 0 + cos. (x + 1) 9}; , 2 ; 1 s . [dO «cos. 9.cos.nd == fd0. cos.(n—1)0 + gJ 48. cos. (nt 1)6 le- —s PAE ec 1)0 1 “F 2.(n+1) f(n+1)dé@.cos. (n+1)0= sin. (2 —1)0 + sin. (2+1)0 +. 1 2(n+1) d@ + cos.4+@- a Sp. 2.(n—1) 6th. By p.8, —; a sec.1 @ = (sec.? 0)’ = (1 + tan.? 0)’; ill oJ SHS a0. 01+ tan. 6) = fd (1 +tan. 6) x (1+ tan: 0) cos = ffd(tan. 6) (1 + tan.” 0)} = tan. O + : tan.” 0 + ¢. l—e COS.¢§ l-é& in terms involving the cosines of multiples of the arc s; see Mayer’s Theory of the Moon, p. 14. 1 re a 15.1 6. [f-= , it is required to express — , —, &c. y yoy 1 1 — €.cos.s Sie oa 1 + e? —e€.cos.s, y —e rejecting* the terms involving e°, e, &c. “Saad + e)'— 2.11 + &)e.cos.s + €% cos. 5, hee 1 cos. 2 = 1426-26. 008.5 +t? + ‘| 2 2 5é e" = Met aeee cos. s+ 5 cos. 2s. Again, ] : ; — = (1 +e)’ — 3e.(1 +e’) cos.s + 3e°(1 +’) cos.’ y g fl cos. 2 1 + 36° — 3Se.cos.s + 3e€ 3 + Z co) Oe Se 1 +-— — Se.cos.s + — cos. Qs. 2 2. And generally 1 — = (1+e?)”— n(l teyrle, cos. s+n A tele, cos. ¢ y" * The rejection of the terms involving ¢, «ef, &c. is not, as it is plain, essential to the illustration of the use of Trigonometrical formula, but, we have given the instance as Mayer has, and as all similar instances in Physical Astronomy will be circumstanced, in which e de- notes the eccentricity of an orbit. 112 il | ; na—1 1 cos, 2 l+ne—ne.cos.stn. 3 é {; ed 2, 2 n.n+s3 m.n—l == 5) Tarren eee. tae 29. In this instance, it is plain, if all terms involving higher powers of ¢ than that of the square be rejected, that no higher multiple of s than 2s can be introduced. ‘The quantity ¢ is the mean anomaly, and y is the radius vector. 6. Ifs = V(f?—2fr.cos. t +7°), it is required to express 1 ; : : : 3 by a series of terms involving the cosines of the multiples of the arc ¢. (See Mem. Acad. des Sciences, 1754, p. 538. Clairaut sur l’orbite apparent du Soleil). Let 2fr.cos.t — r° =a, then s= W(f’— a), and by the binomial theorem, 2 3 4 ae ee Beg Fn But, a =r Ha Bie cos. ¢+4f°r. oo.) t, (by Prob. 6,) = r tate cos.t + 2f" 7? (1 + cos. 2t) = r+ Of? r— afr.cos.t + 2f? 1°. cos. Qt. + &c. Again, a= — 7h + 6fr° + C08. t = 12f? r*. cos.°t + 8f° 1°. cos. t = —r — 6f' rt + (6fr? + 6f? 7°) cos. t — 6f* r*. cos. 2t + 2f?r*. cos. 3¢t. (Prob. 7, [e””’ ]) Sith : : 1 . Substituting these values in the series for -;, there results § , Le Lara) sr2aon Pea aot ti hirer es Crore & Winiaiit 4 f 64 f 3r Ad r? . oy —— } cos? +: Gat af) 137° 4. 1059" 35r + (GE er) cos, 2¢ +> —-= Is In this instance, 7 is the radius of the Earth’s orbit, and f that : : ae. : of Jupiter’s, and since ~, is a small fraction (=.19245) the terms 5 6 : Fee ase - yotang a » Pp , Xc. are rejected, (see Phys. Astron. p. 283.) rp Sas The quantity —; is a factor ina term dependent, in the theory s of a planet disturbed in its orbit, on the disturbing force ; and, the object of the above resolution is to resolve the term into a series such as A® +. A®, cos. ¢ + A®. cos. 2t + &c. a: ee which is easily effected when ~, is a small fraction, and may be, ey ; Lees ; but not without artifice, when — is nearly equal 1, that is, when the radii of the disturbing and disturbed body are nearly equal. (See Phil. Trans. 1804. pp. 265, &c. Mem. Acad. 1764, p. 545, and Phys. Astron. Chap. XVIII.) 7. If the terms of the series A® + A. cos.t + A. cos. 2¢ + &e. be multiplied by cos. mé, it is required so to transform the terms that the series shall preserve its original form. By the formula (d) of p. 32, 1 cos. nt .cos. mt = —.{cos.(m—n)t + cos.(m + n)t! 2 Hence, cos, ¢.cos, mt = {cos. (m—1) t-++ cos. (m+ 1) tt cos. 2f.cos.mt = Xe. which values being substituted in Rd me POT me {cos. (m—2)t + cos. (m+ 2) ¢} A®, cos. mt + A™ cos. t. cos. mt -+- A®. cos, 2t. cos. mt + &e, Pp 114 and the terms properly arranged, what is required will be done. (See: Laplace, Mem. Acad. 1785, pp. 54, &e. and Mecanique Celeste, p. 263.) 8. Required the sin. ¢, cos. ¢, sin. Qt, &c. Qe : when ¢ = no — — (1 — n) sin. mv; m Linea e : : the coefficient — (1 — 2) being a very small quantity. m (See Acad. des Sciences, 1745, p. 539: also for similar instances, Acad. des Sciences, 1754, p. 348, and Clairaut’s Theory of the Moon, p. 20, and Phys. Astron. p. 140, &c. By the formula (1), page 29, i : Qe i sin. = sin. nv.cos. ;— (1 —n) sin. mv m vot Qe : — cOs.nv.sin. J— (1 —7) sin. mv¢ . m : Ze ‘ Ayet: Now the quantity* oF (1 — 7) sin. mv, by the hypothesis, is very small; therefore its cosine is nearly = 1, and its sine is nearly the arc which it is supposed to represent: consequently, ? ; 2e ) sin. # = sin. mv — — (1—7) sin. mv. cos, nv, Mm a7 (by (b) p. 32,1] = sin. nv — = (1 —n)}sin. (n +m) v— sin, (n—m)v} Again, Qe : i cos. = cos.nv + —(1—n) sin. mv. sin. nv , m - [ by (e) p. 32, | = Cos. nv +=(1 —n) {cos.(n—m) v—cos. (n-+m) vt | * As in the former instance, (see Note, p.111,) so in this, the iam. By ; ; ; smallness of the quantity me (i1—7) is, in no wise, essential to the illus- tration of the use of the Trigonometrical formula. 115 and, by a like process, may sin. @¢, cos. 2¢, &c. be deduced. 9. Required the value of * sin. of Q cos. v.dv—cos, vfQ sin. 0.dv, when Q is represented by a series of terms a.cos.nv + 6.cos.mv + &c. (See Clairaut, dead. des Sciences, 1745, p.341, also his Theorie de la Lune, edit.2. p.9. Lalande, Acad. des Sciences, 1760, p. 313. Laplace, Mec. Celeste, p. 241. Thomas Simpson’s Tracts, pp. 92, &c, Cousin's Physical Astronomy, pp.23,&c. Vince's Astronomy, Vol. I. pp. 168, &c. and Woodhouse’s Phys. Astron. pp. 99, &c. If we substitute the first term a.cos. nv of the series in the. expression, then sin. of Q cos.0.dv = a.sin. of cos.v.cos.nv.dv J [ form (d) p. 31, |=a.sin. v au icos. (n—1)0+cos.(n+1)v} dv sin. (n—1)v ~~ sin.(n+1)¢ n—- 1 n+ 1 where C, the correction, will = 0, if the integral = O when v=0. (p. 109,) = a. sin. v { a ch Again, cos. Uf Q sin. v.dv = a.cos.vf sin. v. cos. nv. dv, (form [| p- 32,)=a .cos.vi f{ sin. (n-- 1) v—sin. (n— 1)} dv (p: 106,) = 4.05.» fe AVP _ sos O02 . ch 2(n—1) 2(n+1) in which, according to the preceding hypothesis of the correction, ] 1 O + C, a 2.(n—1) ys 2.(n+1) n— 1 and C’ =— * dv is the differential of » answering to v the fluxion of v.. 116 Hence, combining the two parts of the expression, sin. vf Q. cos. 20.dv — Cos. ofQ sin. 2v0.dv = ey ‘sin. (n— 1)v.sin. v — cos. (n— 1)”. cos. vt a | aE 2.(n+1) }cos. (n+ 1)v.cos. v + sin. (n+ 1) v sin. v} a.Ccos. v pany = [by the forms (2) and (4) pp. 31, 32. | @.COS.NV | G.COS.NV | a.COS.U __ Wa olan) oo adh Tt: a.COS.¥ COS. ND fe ant If, instead of a. cos. nv for Q, we had substituted 6. cos. mv, the resulting value of the integral would have been b. cos. v b. cos. mv m —1 m —3 Hence, the whole integral, when Q is represented by the series a.cos.nv+b6.cos.mv + &e. is at # a r COs. Vv n— | m a.cos. nv b. cos. mv a ae ese ay ries m— 13 CF aa ; 10, Expand (see Ex. 5.) —;s into a series of cosines of arcs, u when 1 A u =—(1 + e cos.cmv) a 1 mu = - (1 +e cos. cv) a “as = y.sin. (gv — 6) ee, ty being very small quantities, (see Phys. Astron. p. 247). 117 1 ist, nearly, u? = ~=(1 + Se’ cos. cmv) a — = a*(1 — 4e cos. €v); wu ; BG Kees '.—y = (1 — 4e. cos. cu+3e cos. ¢mv) u a but s = ¥ sin. (gv — 0); sin. (gv — @) — 2e.sin. (gv + cv — @) i at — 2e.sin. (gu —co — 0) Syne Se seri) , Pek | + Zz sin (gv + cmv — 8) ao : + gt; sin. (gv — cmv — 9) 11. Required the value (I) of fdz.cos. mz x cos. nz x cos.pz. (See Simpson's Tracts, p. 89.) 1 €O8.mz X cos. NZX cOs.pz=— Scos.(m—n) z+cos.(m+n) z} cos. pz cos. (m~- n—p)z +cos. (m—n+p)z} oe Md aa : {cos. (m-+n — p)z-cos. (m-+-n-+p) zt consequently the integral (1) equals sin. (m—n—p)z . sn.(m—n-+p)z 4.(m—n—p) 4.(m—n-+p) sin. (m+ n— p)z sin. (m-+n-+p) z 4 (m+n — p) 4(m+n-+p) 12. Let it be required to compound Ci sin, (20 — 20) + sin. (20° — 20) — sin. Qu — Qv’) into one term, the product of three sines or cosines (see Phys. Astron. pp. 441, 442.) 1s By forms (5), (6), p. 33, F=Q sin. {(v — 0) +(e —6)t fcos. ~—v')} —sin. (20—2v) =2sin. }{(v—6) + @—6)} —sin. (W—v’)} x cos. @—V) = 4 cos. (0 — 9) sin. (v —8) cos. (v— 7). This process is, in its nature, the reverse of that in the pre- ceding example. 13. It is required to resolve(7” — 2r7" cos. wtr’)—”, (F) into a series such as A + B.cosw + C.cos.2w + D.cos. 3m + &e. (see Phys. Astron. p. 260: also Lacroiz, tom. II. p. 132, Laplace, tom. I. pp. 271, &c.) Make 2cos.8w =2@+-, then 1 L 1 : ”—Qrr cos.w - rs r—r'r (« 4 -) + 7" x | r (r —r2) (7 - ~) W Su by 40 a =o) (; eter lhe nam Yio “.0-3.2) Now, aad Botte ext ganmecrnn ED m(m + 1) (m+ 2) Mr ee Pe ee (3) Fete r Key Mr Vo mm. ma 1 (:-5-) i Eis ea rane das Oh eta (- yi MEE ; eat 5 La PERS m(m + 1) (m + 2) + {wou | 0: — + &c. * See Examples 6 and 7. i Te ee eee Pe 119 es Le (ee () m. ei cael re) + &e.! Weitere 2 (mr. m mt gr Ego UP Nr he ae (2) ot ty, m+ 2 () soso teres 3 (- &e. + 1.2 3 J vk \ 2 7m.m+l1 rr m m+l m+2 ia see re eee D = &e. 14. Inthe Lunar Theory, one of the equations (that of the evection) is represented by —@r .sin. (2X — 2), 7 being supposed to be of a mean constant value: if r should vary and proportionally to 1—Sc.cos. Y, what is the new equation that would thence arise? (See Vince’s Astronomy, vol. II. p. 53.) Let m be ihe mean value of 7, then the SELON a Sin 8c. cos. Y) sin. (2 X — 2) = —2Qm sin. (2X — Z)+6me cos. Y sin. (2 X— 3.) Therefore, since the first term denotes the mean value, the second, or, Gmc cos. Y sin. (2X — %), denotes the variation from that mean value, or the uew equation. Now, 6mc.cos. Y.sin. (2X — Z) = (by form [a] p. 32.) 8mc jsm. 2X —-S+Y)+sn.@X—Z— V)} which is the form of the new equation, in which, the angles QX—2+Y, 2X —Z > Y, are technically denominated the arguments, and 3mc (= 2 1”) is the coefficient. The Student, perhaps, may now be inclined to believe that the formule demonstrated in the preceding pages, are not entirely without their use, nor invented and shewn as mere specimens of analytical dexterity. The instances, indeed, have been, almost : : 120 all, taken from Tracts on Physical Astronomy; and it is, there- fore, on the assumption of the utility of that science, that the Trigonometrical analysis has been affirmed to be useful. We will now proceed to apply the formule of ‘Trigonometry to the resolution of certain numerical equations: an application less extensive than the preceding, and of more doubtful utility. CHAP. VII. —— On the Solution of certain Numerical Equations by means of Trigonometrical Formule and Tables. Iw this application of the Trigonometrical Analysis, the utility, whatever it be, relates to the expedition and convenience of the resolution of the equations, and not to any thing novel or curious in its principle and method; moreover, the expedition and conciseness of the resolution depends not on any essential and real abridg- ment, but on that sort and kind of abridgment which registered computations or tables afford: for instance, we shewed (page 71.) that a cubic equation, such as sin. 34 = 3sin. A — 4sin.’ A when solved by approximation, might be used for the computation of sines, or the construction of Tables. Reversely, the Tables constructed may be used for the resolution of similar cubic equa- tions. Again, we shewed (page 70) that an equation of 5 dimen- sions, such as 3 5 Q2sin.5A, ors =5e —5e +4 solved by approximation, might be made subservient to the con- struction of Tables. Reversely, Tables constructed either by the approximate solution of equations, or by other methods of ap- proximation, may be employed in the numerical resolution of similar equations. De Moivre solved equations of the third and fifth degree by the cosines of the third part and fifth of an arc; and Vieta divided an arc into three and five equal parts, by equa- tions of the third and fifth degree. ‘This is sufficient, perhaps, to explain the real principle of the solution of equations by Trigo- nometrical Tables. . The convenience or expedition of the method, as we have already said, is of the same nature as the expedition of computation by logarithms. If we do not avail ourselves of com- puted Tables, the whole process of the solution of a cubic equa- tion, for instance, will be tedious; we must employ some method Q 122 of approximation, Now, if Trigonometrical Tables are employed, the process is short and easy, but only so, because the most labo- rious part of it is already done for us. In these cases, we avail ourselves of the registered computations of preceding mathe- maticians. Solution of a Cubic Equation. It may be proper, however, to shew, more in detail, the method of solving equations by the aid of Trigonometrical Tables. If we take the equations (c’”) and (s’”) pp. 47, 49, supply a radius 7, and put c and s for cos. 3A, and sin. 3 A, respectively, and w for cos. A and sin. A, then y 2 4 3r° a 3 cr = 44° — 37%x, or wv — —a@ —- — =O... (1) 4 4 r? sr and sr’ = 39x — 42°, SLE Winelnernat i eeptrer ( ere oe eu Hence, c being given, find in the Tables the arc 3A cor- responding to it, and, from the same Tables, take out the cosine of A; this latter value is the root of the equation (1). If s be given, find 3A corresponding to the sine s, and then take from the tables the sine of 4, which is the root of the equation (2). But, cubic equations have 3 roots; now, by the Table of p. 16, or by Cor. 6. page 35, if c = cos. 3A, then also c = cos. (27 —3A)=cos. (27 + 3A)=cos, (427 —3A), Ke. hence, Hi erield « Q°97 Qo substituting instead of the arc 4, the arcs, Sune A, Tas + A, 477 — ~— A, &c. any and all, of the followmg equations, are true, 5 orts 4, (om 4) ~3 t*s COS <= — A).o.ee (a) cra. (cos. == + A) —3.9°. cos, (= “+ A). 6646) cr = 4. (cos. = - A) — 3.97. cos. (= — A). hart tee Xe. 125 4, Qe rp — - = re eee But, cos. (<7 — A) = cos. for - (74 4)P = 20 2% i 5s cos. 27. cos. (art Ay = Cos. Ge +A) (since sin. 27 = 0, and cos. 27 = 1). Hence, the equation (d) is precisely the same as the equation (0). Again, if we take the equation that would follow the equation(d), : 4 47 the cosine in it would be cos, ee + A); but, cos. — A) = COS, {or _ (= o 4)} = cos. 27 . cos: (= _ A) = 20 ’ yea COs. (= = A) . Hence this equation is precisely the same as the equation (a); and, in like manner, succeeding equations would recur, so that, essentially, there are only 3 different equations (1), (a), (6), and hence, in the equation cr = 42° — 3r°x, or, 2 — —zx — Fat ‘ Qa x may, either, = cos. A, or, = cos. (=- A) » Ose 2%, Pe cos. (> ar A) » ¢_being the cosine of 3 A. The same reasoning applies to the equation (2), and similar conclusions will follow. - Example. Let #—147 #«—285.5 = 0; compare this with Ae an cr? ree pane y 25.0 . i — = 147, and r=14: andc = ° - and the tabular cosine = 285.5 | 10000 x 49 x14 = 416.808, &c. Now the arc, corresponding to this cosine, or . the are 3.4, is 65° 24’; 124 +, A= 21° 48’ nat. cos. = 9284858 .-. to rad. 14 = 12.9988012, first root. Again, 120°— 21°, 48’ = 98°. 12’, nat. cos. = —.1426289; *, to rad. 14= —1.9968046, second root. Again, 120°-+-21° 48/ = 1419. 48’, nat. cos. = —.7858569 ; » -, to rad. 14 == = 11,.0019966, third root, and the sum of these two last roots, that is, of the second and third roots, exactly equals the first root, which ought to be the case, since the coefficient of the second term in the proposed equation is = 0. But, cubic equations of any form may be solved by substi- tuting ‘T'rigonometrical lines, sines, or tangents, instead of the quan- tities under the radical sine in Cardan’s form: as Dr. Maskelyne has done in page 57 of his Introduction to Taylor’s Logarithms. Not only cubic, but quadratic equations may.be solved by the aid of Trigonometrical Tables, and conveniently so, ere the cocipichta! are expressed by many figures. Solution of a Quadratic Equation. | Let x? + pr — ¢-=0, P ~ 4g theno= —2 + iVG +9) =-2f 3 VOD}. 4 Assume “ = tan.” 6, then, = — - (1 + sec. 8). Now by Prob. 3. Cor. 5, p. 34. 1 — tan.” cos. 9 = 1 + tan. ~1+tan. we ar) ©nlD HID oie . sec. 0 = 1 = tan. ee 125 2 tan. = ‘ 3 1—tan.” - 1 — tan.’— But by Prob. 8, p. 64, 2 tan. 3 tan. 9 = ————_—_; 1— tan.” - 2 *, l—sec. 6 = — tan, 0. tan. ae tan. @ and 1 + sec.@ = = tan. 0. cot. 3° tan. - 2 c nly ds Hence, one root (a) = a tan. 0. tan. Bs 0 the other (3) = — ; tan. @. cot. a And the whole formula of solution, in logarithms, is log. tan. 9 = 7 (20 + log. 4-+-log. ¢g— 2 log. p), 0 and log. aslog. 2 Ae log. tan. 0 -+ log. tan. 3 20; 0 and for 2d root, log. B = log. © + log. tan. 8+ log. cot. : — 20. The solution may be exhibited under a different form, thus: r=hhis V+} --2 1 0 0 sie a Cone — ) rg ede Cae aT) = (with the lower sign) — - Vqx cot. = 38 and e) = (with the upper sign) Vg .tan. a 126 Example. Let x* + 13.56” — 72.31 = 0. Computation of 6. AO is Bete ate beers ait a oes ss gins e 84) mit A LOR Sota te alge es tewe ars os4 5 «6 90 = .3010300 ZlOg. TQS icc cecececcccescocecerescerees = .9295992 11.2306292 IOP AS. AO eee eee eae oe So ante = 1.1322597 log. tan. Oc cecees cesses cacsescerceseds = 10.0983695 ot Oe Le, eh. 2 and 5 = 25 .43.1 Computation of positive root. Computation of negative root. log. 6.78== —.8312297 log. 6.78= —«.8312297 - log. tan. 51° 26/2"= 10,0983695 | log. tan. 51°.26'.2" = 10.0983695 log. tan.25.43 1 = 9.6827151 | log. cot. 25 .43.1 = 10,3172849 20.6123143 21.2468841 *. log. = 6123143 - log. (—a#) = = 12468841 and #, ora = 4.09557 and —w, or—8 = 17.6556 Hence the two roots are 4.09557 and — 17.6556, and the sum of these two roots is — 13.56, the coefficient of the second term of the equation, as it ought to be. The equation that has been solved is x°-+px—g =O; if it had been xv — px — g=0, the two roots would have been aS +h th-JSO+D} that is, the two roots of the former equation taken negatively. If the equation to be solved be x°— px + g = 0, then AG : 4 ; aie oe {1 + VAG _ Ey assume ial = sin.’ 0 2 p coe | cena age Ms Pay pon , and « == (1 + cos. 0) =p. cos: Be or = p sin.” 5. 2 127 Hence the rule of computation, logarithmically expressed, is log. sin. 0 = § (20 + 2 log. 2 + log. g — 2 log. p) and 0 4.8 log. « =log. p +2 log. cos, "s — 20, or = log. p+2 log. sin. ou 20. If the equation be x°-++-pxr+g=0, its roots are the roots of the preceding equation taken negatively. The preceding solutions are, in fact, the same as those given by Dr. Maskelyne, at page 56 in the Pe aa to Taylor’s Logarithms. If, in mathematical researches, equations, like those that have been given of the second and third degree, presented themselves to be solved, their solution would be conveniently effected by the preceding methods, and by the aid of the Trigonometrical Tables; but, the truth 1s, in the application of Mathematics to Physics, the solution of equations is an operation that very rarely is requisite, and consequently the preceding application of Trigo- nometrical formule is to be considered as a matter rather of curiosity than of utility. SPHERICAL TRIGONOMETRY. CHAP. VIII. Definitions. 1. A spHERE is a solid terminated by a curve surface, of which all the points are equally distant from an interior point, called the centre of the sphere. _ The surface of a sphere may be conceived to be generated by the revolution of a semi-circle round its diameter. 2. Every section of a sphere, made by a plane, (so it will be demonstrated) isa circle. A great circle is that, the plane of which passes through the centre; a small circle, that, the plane of which does not pass through the centre. 3. The pole of a circle of asphere, is a point in the surface, equally distant from every point of the circumference of the circle. 4, A plane is said to be a tangent of a sphere, when it has one point only common with the sphere. 5. CE, and, FH + HG > FG; “. a fortiort, AC + CE + EF + FG + AG < Qz. Prop. IV. If CD be drawn from the centre C, perpendicularly to the plane of the circle ANMB, then, D is the pole of the circle ANMB and of all small circles, such as anmé, that are parallel to it. For, since DC is perpendicular to the plane of ANMB, it is perpendicular to all lines in it, as CN, CM, &c. (See Euclid, Book XI. Def. 3.). Hence, DCA, DCN, DCM, each = 90°; consequently, since DC is common, and CA, CN, CM, are equal, the hypothenuses, which are the chords of the arcs DA, DN, DM, are equal; .. the arcs DA, DN, DM are equal to one another and to 90°, and .*. D, (by Def. 3.) is the pole of ANMB. Again, in the small circleanmb, ca, cn, cm, are equal, as are the angles Dca, Den, Dem; .*., as before, the chords of Da Dn, Dm, are equal, and the arcs Da, Dn, Dm; .. D is equally distant from every point of the circumference of anm6, and therefore is its pole. 132 Cor. 1. By Definition 6, the spherical angle AMD is equal to the inclination of the planes 4MBC, DCM, and therefore is a right angle. Cor. 2. Hence, to find the pole of a great circle, draw, from the point M, a great circle perpendicular to AM, and take MD = 90°; then D is the pole: and, reversely, if D be the pole, DMN, DN are right angles, and DN, DM are quadrants. Cor. 3. Hence, to describe a great circle, of which D is the pole, take DN, DM, each=90°; then let a plane pass through N, M, and C, and its intersection with the surface of the glk is the circle required. Cor. 4. Since NDM may be of any magnitude from O to 180°, and since the angles DNM, DMN, are, each =90°, the sum of the three angles of a triangle, as DNM, may be any angle between 180° and 360°, which are the two limits. Cor. 5. "The radius of a small circle anmb is Cn, which is the sin. Dn, or, cos. Nn; and, if the great circle ANMB be divided into any number of equal parts, each equal to NM, the small circle anm6 will be divided into the same number of equal parts, each part being equal to nm; but, the magnitude of nm will be to the magnitude of NM, as the Gineurntoraten of anmb to that of ANMB, consequently, as the radius cz to the radius CN; or, as sin. Dnto sin. DN; as sm. Dn to radius; or, as cos. Nn to radius. Pror. V. If a plane T'D/ is perpendicular to CD, it is a tangent to the sphere. For, take any point T, ies DT, then CDT is a right angle; *., CT is greater than CD; .. T is without the surface, and since this! is true of every point in the plane, except the point D, the plane T'D/ (by Def. 4.) is a tangent to the sphere. Proe. VI. If DT, Dt be drawn, in the planes DCA, DCN respectively, tangents to the arcs Da, Dn, at the point D, the angle 7’D¢ is equal to the angle ADN, both of which angles are measured by the arc AN, ° 133 For, DT, Dt are perpendicular to DC, which is the inter- section of the planes DCA, DCN; .. by Euclid, Book XI. Def. 6. T’Dé measures the inclination of the planes, and there- fore is equal to the angle ADN; but T-D¢, the inclination of the planes, = ACN, of which AN is the measure; .*. AN is the - measure of T’D¢ and of ADN. Cor. If two arcs of great circles intersect each other, their vertical angles are equal. Prop. VII. If from the points A, B, C, of the triangle 4 BC, as poles, the arcs EF, FD, DE, be described, forming a triangle DEF; then, reciprocally, the points D, E, F are the poles of the arcs BC, AC, AB. Since A is the pole of EF, the arc of a great circle drawn from A, to any point in EF, and therefore to a point as E, is a quad- rant; similarly, since C is the pole of DE, the arc of a great circle from C, to any point in DE, such as E, is a quadrant. Hence, E is distant from two points A, C, in an arc AC, by the ‘quantity of a quadrant; .°. by Cor. 2, Prop. 4, E is the pole of AC; and similarly, F'is the pole of AB, and D of BC. Prop. VIII. The former construction remaining, the measures of the angles at A, B, C, are the supplements of the sides opposite, that is, of EF, DF, DE; and, reciprocally, the mea- sures of the angles at D, E, F, are the supplements of abe sides BC, AC, AB. 134 For, the measure of the angle at 4 (see Prop, 6.) is GH. Now, GH= EH — EG= EH — (FE-—FG)= EH + FG— FE=90° + 90°— FE= 180°— FE = (p. 12,) the supplement of FE, 2 Again, . the measureoftheangleat B,or KI= FK — FI= FK—(DF-—DI) = FK+DI— DF=90°+90°— DF = 180°— DF =supp’. DF; and similarly, LM, the measure of the angle at C, is the supple- ment of DE. Secondly, the measure of the angle at D, or MI= MC+ CI= MC + IB— BC =90°+90° —CB=180°— CB = supp*. CB; and similarly, LH, KG, the measures of the angles at E and F, are the supplements of AC and AB. From its properties, the triangle DEF has been called, by English Geometers, the Supplemental Triangle; and from the mode of its description, by the French, the Polar Triangle. Cor. 1. Ifany angle as B, =90°, then DF = 180°—90° = 90°, or is a quadrant: and if B and C each=90°, DF, DE, each is equal to a quadrant. i Cor. 2. If (Cor. 4. Prop. 4.), the angles at B and C each = 90°, and the angle at A is nearly 180°=180°—2, x being a very small angle, then the side of the supplemental triangle opposite to A 1s equal to x; and the sum of the three sides of the supplemental triangle = semicircle + x, = 180° + w. ~ Prop. IX. The sum of the three angles of a spherical triangle is > 2 right angles < 6 right angles. For, the angles of the triangle + sides of the supplemental triangle = 180° + 180° + 180° =6 x 90°; .°. since the sides of the sup- plemental triangle must have some magnitnde, the angles of the triangle must be less than 6 x 90°: again, by Prop. 3, p. 130, the sides of any triangle, and .*. the sides of the supplemental triangle, are less than 4.x 90°, = (let us suppose,) 4x 90—2; consequently, the angles of the triangle = 6 x 90°—(4.90° —2) =2 x 90° +r. -135 Cor. 1. Hence a spherical triangle may have two or three. right angles, two or three obtuse angles. Cor. 2. If the angles at B and C are right angles, 4B, AC are quadrants, and A is the pole of BC: if also the angle at 4 is. a right angle, the triangle ABC coincides with the supplemental or polar triangle, and the triangle ABC is contained eight times in the surface of the sphere. Prop. X. The angles at the base of an isosceles spherical triangle are equal. Inthe triangle ACB, let AC = BC, draw the tangents AS, BS, . which are equal and which cut their secant OS in the common point S. Draw also from A and B two tangents AT, BT, which, by Euclid, Book III. Prop. 37. are equal. Hence, in the triangles SAT, SBI'; SA, AT, ST, are re- spectively equalto Sb, BT, ST; .. by Euclid, Book I. Prop. 8, the angle SAT'=the angle SBT, and .”. by Prop. 6, page 132, the spherical angle at A = the spherical angle at 8 To provethe reverse proposition, that is, to prove if the anglesare equal, the sides are equal; take the supplemental triangle; then 136 since its sides opposite to the angles at A and B are supplemental of equal angles, the two sides are equal, and the supplemental triangle is isosceles; .*. by this proposition, the angles at the base are equal; .*. their supplements which are the sides AC, BC, are also equal. / Prop. XI. In a spherical triangle the greater side is opposite to the greater angle. In the triangle ACB, let 2 ACB be greater than ABC*; make BCa=CBa; ... Ca=aB (Prop. 10.); but 4a +aC is greater than AC; .. Aa + aB, or AB is greater than AC, by Prop. 2, page 130. Prop. XII. The surface of the sphere included between the arcs DN, DM, 1s proportional to the angle NDWM or the arc NM. - See Fig. 9. For, if the circumference ANMB be divided into equal parts as NM, and great circles be drawn from D through the points of division as N, M, the portions of the surface, such as NDM, will be all similar and equal; hence, if 4M contains NM, p times, or, if AM=px NM, the surface ADM will=p x NDM. Cor. When DM coincides with DB, the angle ADB and its measure AM B=180°: hence, if S=the whole surface of a sphere, and if a=the angle NDM, or the arc NM, the surface @ S NDM = i Xs or, since S = area of 4 great circles of a sphere, (see Simpson’s Fluxions, page 189, vol. I. or Vince’s, page 89,) = 47 (7 = 3.14159, &c.) the surface NDM =a a . The surface may be differently expressed, thus: a is the cir- cumference of a circle the radius of which is 4; .*. 2a represents * See Fig. p. 130. 137 the circumference of a circle, such as ANMB, the radius of which is supposed = 1, and as we represent arcs of great circles by the. number of degrees which they contain, 27 = 360°, and = =7Ts= a 180°: hence, the surface NDM=180°. ae a’. Prop. XIII. The measure of the surface of a spherical tri- angle is the difference between the sum of its three angles and two right angles. Let the triangle be A BC, a, 6, c, representing the magnitudes of the angles at A, B, C; let P=surface BCmB, Q=mCnm, R=ACnrA; produce the arcs Cm, Cn till they meet at ¢ (which will be on the hemisphere opposite to that represented by ABmnA) then, each of the angles at C and e, equals the. angles of the planes in which the arcs Cme, Cne, lie; .*. the angles at C and e are equal. Again, the semicircles AC m, CA BCn, Cne are equal; or, AC + Cm=Cm +me, and .. AC=me, and BC=ne: and the triangle men =the triangle ABC; let x =its area, then, by Cor. to Prop. 12., 3 S 138 S ca re P= - ,— Lares nee | S oc S 7 so hy Sas P Ra— + oe tage and, ++ P+Q+ 2 S. 6. ia aml order, consequently, by addition, S S satb+c Q2xe+(¢4+P4+Q+R) or @xr + a =-.(—); S iI Ss + o a o | poe ie 2) oe) .° by the Cor’. to the last Proposition. This result is the same as that given by Caswell, in Wallis’s Works, vol. II. page 875, who attributes the Theorem to Albert Girard*. In order to obtain a more commodious form for compu- tation, let r be the radius of the sphere, 7 =3.14159, the cir- cumference of a circle in parts of the radius when the ‘radius is 5, then since S the surface of a sphere is equal to four times the area of a great circle of that sphere = Agr ; 9 Tr COP 2 enna 180° {4+ B+C— 180°. @ Now 7, toa radius 4, is the circumference containing 360°: to a radius 1, therefore, it expresses, in terms of the radius, the * This expression for the value of the area was a merely speculative truth (see a subsequent note), and continued barren for more than 150 — years, till 1787, when General Roy employed it, in correcting the spherical angles of observation made in the great Trigonometrical Survey, Phil. Trans. vol. VIII, year 1790, page 163. See also Mem. Acad. Paris, 1787, page 358, and Mem. Inst. vol, VI, p. 511. 139 value 180°, or of 180 x 60 x 60 (=648000). Hence the value of 1’, or, which is nearly equal to 1”, of sin. 1”, in parts of the Be Oo Winn Thater on SEG DRT Wt Mees Ai eer ee Pet ereror area a 648000’ ~ 648000” y be thus expressed : v= rsin. 1” {A+ B+C—180°}, radius, must equal which is an expression for the area in terms of a great circle of the sphere, when the spherical excess (the excess of the 3 angles above two right angles) is expressed in seconds; for instance, if A = 121° 36° 20 B= 42 15 14 Gore 1 S415 Ss A+B+€-— 180° = 18" 6 36” = 65196", now log. sin. 1” .eseeeees+ 4.6855749 idehO51NG TM Ee 4,8142210 ae LOO(O) cs ce 2 9.497959 ; the area of the spherical triangle = r° X .316079, the whole surface of the sphere being r’ ¥& 12.56636. In an ensuing part we will shew how to express the area of a spherical triangle in terms of its sides, and also explain the use of the latter expression. Cor. 1. Since by Prob. 9, page 134, the limits of a+6--c¢ are 180° and 540°, the area of the triangle ABC may be equal to any number of degrees between O and 360°. Cor. 2. If each of the angles a, 0, c, =90°, the area of the triangle ABC = 270°— 180°= 90°, which is 4th of the whole surface, since 47 =8 (90°): this agrees with Cor. 2. Prop. 9. 140 S ; Cor. 3. The area of DMN (see Fig. 9.), = , x a (if e be its angle), =e; if we conceive DN, DM, to be continued till they meet in the opposite pole, and name the space, included be- tween the great circle, a Lune, then the lune=2. DNM=2e: but the area of 4BC=a+6+c-— 180°: equate this with the area of the lune, and which is the value of the angle NDM, when the area of the lune equals the area of the triangle. Prov. XIV. If x be the number of the sides of a spherical polygon, its surfuce equals the sum of its angles, minus the product - of two right angles multiplied by n— 2. Let the polygon be AGFECA: divide it by the means of the arcs of great circles into triangles, then area AGF = 2 AGF+ 4GFA+ 4 GAF— 180° area AFC = 4 FAC + 4 AFC + 4 ACF — 180° area CFE = £CEF + 2 CFE + £4 FCE — 180° *. area of polygon = 4 AGF + (4 FAC + 4 GAF) + (4 GFA + 4 AFC + 24 CFE) + 2 CEF + (2 FCE+ Z ACF) — 3 x 180° = ZAGF+ 4 GAC + 4 GFE'+ 4 CEF + z ECA — (5 — 2) 180°. oe This demonstration proves the Proposition to be true for a polygon of five sides, and a similar demonstration will prove it true for a polygon of x sides: for, it is plain, if, instead of AC, we introduce Ca, Aa, that is, if we introduce an additional side, we introduce an additional triangle, and consequently we must introduce an additional 180° to be subtracted, that is, the negative part will become — (5 — 2) 180° -+ 180°} or —(6—2) 180°. 141 The preceding Propositions belong, more properly, to Sphe- rical Geometry than to Spherical Trigonometry; they have, how- ever, been here inserted, because they exhibit certain properties of spherical triangles rather curious, and very easy of demon- stration; so that, if not of essential use im assigning the relations between the angles and sides of spherical triangles, (which it 1s the special object of Trigonometry to assign,) they will not materially divert or impair the Student’s attention. In the next Chapter, we will proceed to deduce those formule, by which the relations between the sides and the angles of sphe- sical triangles are expressed. CHAP. IX. On the Expressions for the Cosine and Sine of the Angle of a Sphe- rical Triangle in terms of the Sines and Cosines of the Sides. We will begin this Chapter by establishing, in the fol- lowing Problem, the fundamental formula, from which all the methods and forms of solution will be deduced. It corresponds to the fundamental one of Plane Trigonometry, inserted in page 25; and the Student who understands, in principle, the use made of the latter formula, possesses, in fact, the clue to the sub- sequent demonstrations of Spherical Trigonometry. Prop. XV. PROBLEM. : It is required to express the cosine of the angle of a spherical triangle in terms of the sines and cosines of the sides. Let the triangle be mtn, let the sides be a, 6, c, and the opposite angles A, B, C; conceive O to be the centre of the sphere, — 143 and draw the two tangents ¢Q, ¢P, at the point ¢, to the arcs tm,tn; then 42 QtP = £4 A: by Prop.6; and the angle at O is measured by mn, or a; and, by the definition of the secant, OQ is sec. 6 OP is sec. c. The principle of the demonstration that follows, is, to obtain two values of QP, one from the triangle POQ, the other from PQ, and then to compare them: now by Euclid, Prop. 183, Book II, or Prob. 1 of this Treatise : in APOQ, PQ’ =sec.’ b+ sec.” c — 2 sec. b. sec.ccos. a (rad. 1.) in O PtQ, PQ? =tan.’ b +tan.? c— 2 tan. db. tan. ¢.cos. A. Subtract the lower expression from the upper, then, since sec.” 6—tan.’ 6 = (rad.)” = 1, we have O=1+1+2 tan. b.tan.c.cos. A—2sec. b.sec.c.cos.a and thence, cos.a.sec.6.sec.e—1 |. cos.ad—cos. 0.cos. € cos. A = FO tan. 6. tan. ¢ cos. 6 cos. c.tan. 6. tan. c : fk cos. a—cos. 6. cos. ¢ = (since cos. 0. tan. b= sin. 6) ———_-—-—_ pe eteetet GQ) sin. b e sin. Cc and similarly, since the process for finding cos. B, cos. ce will be exactly the same, changing a for 5, 6 for a, &c. the result must be similar for cos. B &c., that 1s, cos. b— cos. a. cos. € cos. B = —_—_-__—____—__- ........ ... w (0) sin. a. sin. € cos. Cc — cos. a.cos. 0 cos. C = ————_——_—__.. . ..... . . (e). sin. a.sin. 6 Cor. 1. Hence, since cos. c=sin. a.sin. 0. cos. C+ cos. a.cos. 6 by substituting this value of cos.¢ in the expression (a), we have ° ° 2 cos. a— cos. 0. sin. 6. sin. a. cos. C—cos. a. cos. cos. A = ; - sin. 6. sin. c ; | ete se fae But, cos. a—cos. a.cos. b= cos. a (1—cos.” 6) = cos. a.sin.” b; » 144 eins Heid - cos. a sin. 6— cos. 6.sin. 6. cos. C. sin. a@ .. coe. A = as sin. 0. sin. ¢ ~ cos. a.sin. b—cos. 6.sin. a.cos. C sin. C ae | cos. b. sin. a— cos. a. sin. b.cos. C SSUUARLY, (COS. 5 ere terres sin. C (cos. a.sin. 6 + cos. b.sin. a) (1 — cos. C} *, cos. A+cos. B = : sin. ¢ Rear G, sin. (a + 6) 2 sin.” e : 3 sin. € sin. (b— a). 2 cos.” 10 and cos. d= cos. B = ; sin. ¢ Prop. XVI. Peoniwat It is required to express the sine of the angle of a spherical triangle in terms dependent on its sides. me cos. a— cos. b.cos.c¢ By the last Proposition, cos. A . II sin. 6. sin. e -“. l+cos. A = cos. a—(cos. b.cos. c—sin. b.sin.€) _ cos. a—cos.(b +c) sin. 6.sin. ¢€ sin. 6. sin. c “by the form (2), p. 31: and by the form (8), page 33, b E b+c-—an cos. a— cos. (6-+c)=2 sin. (5) sin. (——) A Hence, g cj b ‘ b — an 1 +008. A = x sin, (SFEES) sin, (CFE=2) sin. 6 sin. c 2 fa) 2 (3*) : (Crees ) = ——-—— _ * sn. (| —————— J. Sin. (| ——_———_ — a sin. 6 sin. ¢ 9 2 145 yA 1 . fatbt+ey . satbte and cos.” — = —————- sin. ( ) sin. —_ a) : 2 sin. 6 sin. 2 Sunilarly, | cos. 6. cos. c-+sin. 6 sin. e— cos. a 1—cos. A = sin. 6 sin. ¢ = NU altar aot [by the form (4), p. 32. ] sin. O.siIn. C es ery is (F**) sin (F= -) by form (8), p-83; sin. 6 sin. c 2 . atb+c i atb+ec = X S10, (Gearsa _ b) sin, (—— -<), sin. 6 sin. c 2g Q A and consequently ae A 1 . satb+e . satbte sim. TES tea) SORTER ER ET ee ( —b) - SIN. (——-- c) ° 2, Q 2 sin. 0. sin. c If we multiply together 1-+cos. 4, and 1—cos. A, the pro- duct of which is sin.” 4, and substitute S instead of - He ater ; we shall have sin.* A= A aaa bai sin.” 6. sin.? c x sin. S.sin. (S — a) sin. (S— 6) sin. (S —c), and consequently, sin. A = 2 sin. 6. sin. ¢ Y isin. S.sin. (S—a) sin. (S— bd) sin. (S —c)}*: Cor. 1. If we wish to compute sin. B, we must begin cos. 6b— cos. a. cos. € i from cos. B = ———-_-__—_———_, and proceed exactly in | sin, @ . sin. c * The Student, for his own convenience, is desired to compare these expressions and the manner of deducing them, with the corresponding ones in Plane Trigonometry. T 146 the steps of the former process: the result will be a fraction, the numerator of which is the numerator of the above fraction for the sin. 4, and the denominator will be sin. a sin. c; call the common numerator N, then ; | N : N. ; N sin, A = ——-—,, sin. B= ———_—, sn. C = ——_-_-. sin. 6. sin. ¢ sin. @.Sin.c sin. a.sin. 6 sin. 4 N sin. @.8in. C Cor. @. Hence — hf a inte so RE Me ea ON © = sin. B sin. Ob. sin. € N sin. a sin. A sin. a > an 2 Coed UAT IES sin. 6 sin. C sin. C -or (if these equations be expressed, after the manner of expres- sing a proposition) The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. Ri fi dee ; The expressions for sin. pe ole &c. enable us easily fel : Ark B A+B to deduce expressions for sin. (——) » COS. ( 5 Dic Thus, since Cy: | \/ sin. (S — 6) sin. sin. (S — 6) sin. (S —c)} sn. = = 2 sin. 6. sin. ¢ EO OUSIn eb Maina eee = ofA eu Oe a S.sin. oe B =V sin. b. sin. ¢ ioe (S — a) sin. ae sin. @. sin. C jae S. sin. (S— 2} sin. a. sin. ¢ C EST (S — a) sin. (S — 2} sin. oma: = ; sin. a.sin. 0 2 iain aN yates hee) s. sin. (S—o) sin, a@.sin. 6 eins Se a 147 fey A+B B 7 oes cd ) =sin. 4 cos. 2 al cos. sin. = bl Ae .S.sin.(S— Oy {sin. (S—b)+ sin.(S—a)}, sin. a.sin. b.sin.*c but the first factor = and by formula (5), p. 33. : : Ss so 3: a-b sin. (S—b) + sin. (S — a) = 2.sin.—. cos. : 2, 2 cos. — A+B 2 Laie: a—b sn. -—-——- = .2sin. — . cos. : 2, c g 2 C cos. er 2 a—b prea! » COS. Cc a cos. — and in like manner, ” cos. — . A-B 2 . a—b sin. —= ® sin. Saar ete 2 2g cos. — Again, A+B A B a et. erent cOSs. == COS. =. COS. Soe Silke es SIN. 2 2 2 2 If J (= (S—a) sin. (S — a. eo Siku (S =o sin. a. sin. 6 * sin. Cc eis 1 C 2 sin. oh cos. = (a +o sin. —. — . 2 sin, € =e cos. + (a+), cos. = A-B sin. 4 (a+ 5). and cos. cos. — A+B Hence may be derived tan. ( ) ; Right-angled spherical triangles may be considered as par- ticular cases of oblique. The solutions of the latter, then, would necessarily include those of the former; and, accordingly, if we wished to generalise as much as we could generalise, it would not be requisite to consider separately the former. Since, how- ever, one main object of this Work is to render investigation as simple and as easy as it is possible to the Student, we will not avail ourselves of this abridgment and seek to be compendious, but proceed, in the ensuing Chapter, to treat distinctly of the cases of right-angled spherical triangles. CHAP. X. Formule of Solution for Right-angled Spherical Triangles.— Affections of Sides and Angles——Circular Parts—Naper’s Rules.—Quadrantal Triangles. — Examples. Prope. XVII. ProsieMm. Ir is required to investigate formule of solution for all the cases of right-angled spherical triangles *. These are to be derived from the fundamental expressions (a), (b), (c), given in page 143, in which if C be the right-angle and = 90°; cos. C = 0; | cs hence, cos. ¢ — cos. a.cos. 0 : , , and .*. cos. c= cos. a.cos.6 (1) cos. C, or O= . . sin. a.sin. 6 ih ,cos.6 = ; substitute these values hence, cos,.¢ = cos. 6 COS. @ respectively, for cos.@ and cos. 6, in the expression for cos. A, cos, B, and we have cos. da—cos. 6.cos. € cos. 4 = : 3 sin. 6. sin. c 1 cos. Cc ————— — cos. 6.¢cos.c} = sin. O.sin. ¢ \cos. 6 cos. c (l—cos.” 6) cos, c.sin.” } cos. 6.sin.6.sin.c cos. 6. sin. 6. sin. € * * Spherical triangles that have one right angle only, are the subject of investigation: and spherical triangles that have two right angles, and three right angles, are excluded, 150 cos. S$. sin. 6 : me COMME tan, Di) sie Sy peak eels a eee cos. 6.sin.c Similarly, ee oR COs. C (1 = cos.! a) _ 608. Beale Oo cot ee (3) COS. da.siN.a@.sin.¢ cos. a.sin.¢ Now in (1) a, 6, ¢ are involved, and, two of these bemg given, the third may be found: in (2) A, c, 6 are involved: and if we choose to represent, symbolically, by (A, c, 6) the form m which three quantities as A, c, 6, are involved, then, similarly, the other several forms, that can arise by combining angles and sides, may be thus represented : bey a7b) 5 1 CAL'c, By CB) ca) ts) CB, 0)” Cages {(A, a,b) (B,a, b)}; {(A, B,a) (A, B, by}; (A; De, which are in number 10, as they ought to be; the combination ra se A ; Le Oates } ; in 5 things, 3 and 3 together, being ig 7 = 10: those combi- nations that are similar, such as the second and third, are included within brackets. The forms (1), (2), (3), have been already deduced, and the re- a _ sin. A maining ones must be deduced from them, and from the form- sin. @ Ny ald Ren Tangs by the common process of substitution and elimination, sin. Cy aD) a that 16,0 COs. 0 == COS. COSy Ocoee lee (1) (A, c, b) cos: A= cot. ¢\'tam D Ade Oey ae) (B, c, a) cosic B= 'cotic ata, Wa he ee (3) sin.C sin. sin. 6 sin. B= -—— , or sin. 6=sin. c. sin. B (4) sin. ¢ (B, c, 6) Ale sin. B in. b ‘ b i is Cor. to Prop. 16, for = — ; ..sincesin. C= 1. (A, c, a) is exactly similar; cau sin. a .*. sin. B =~ sin. € ; OF sin. a@=sin.c.sin. A (5) 15] cos. € (A, a, 6): cos. A=cot. c.tan. 6 by (2), but cot. ¢ = oe 31. Ss] ——— by (5), p. 144 sg DY (5); ps 144. and cos.c=cos.d.cos. 6 by (1), and sin. ¢ = _ sin cos. a.cos. 6.sin. A .*. cos, A =tan. b. 3 sin. @ : ss COL. A, = CObe a. SIRO. (6) (B, a, b) is similar, and os COL. — COLO ein Eeen ao (7) cos. A cos. ¢ (A, B, a): divide (2) by (4), then fins BR eos De haan from (1) or, cos, A=cos.a.sin. B.........(8) (A, B, 6) is similar, and cos. B=cos. b. sin. A......... (9) (A, B,c): multiply together (6) and (7) | and cot. A.cot. B=cos. 6. cos. = COS. C,...00.0+55- (10) Thus are all the forms easily deduced from (1), (2), (3). Indeed, the quantities that are to be combined, indicate plainly the proper process of elimination; but these forms, although very simple and fitted for logarithmic computation, are not easily remembered, and therefore an artificial memory has been supplied to the Student and Computist, by rules, known by the title of Naper’s Rules for Circular Parts; and im the whole compass of mathematical science there cannot be found, perhaps, rules which more completely attain that, which is the proper object of rules, namely, facility and brevity of computation. The rules and their description are as follow ; Description of Circular Parts. The right angle 1s not considered; the complements of the two other angles, the complement of the hypothenuse, the two sides, making in all five quantities, are called by Naper, circular parts. Any one of these circular parts may be called a middle part (M), and then, the two circular parts immediately adjacent -to the right and left of M are called adjacent paris; the other two remaining circular parts, each separated from M the middle part, 152 by an adjacent part, are called opposite parts, or oppostie extremes : thus, let the side a be M, then, comp. B, or 90°— B, and 6 are adjacent parts, and, comp. ¢, or 90°—c, and 90°— A, are opposite parts. If 90° — 4 be M, 90° — c and 0 are adjacent parts; 90°— B and a, opposite parts. This necessary explanation being premised, we come to Naper’s Rules. 1,. The rectangle of the sin. M and radius =rectangle of the tangents of adjacent parts. 2. The rectangle of the sin. M and radius = rectangle of the cosines of the opposite parts. ; There is no separate and independent proof of ihawe rules; but the rules will be manifestly just, if it can be shewn that they com- prehend every one of the ten results, (1), (2), (3), &c. given, in pages 150, 151; for, those results solve every case in right- angled spherical ‘aatigles. , 7 ‘ ee ae OD a ee ee, ee ee a q ‘D ‘sjieg oyisoddg q — 0.7 = (06 I= ,06(q — 96) Uri (7 —,06) ‘Ue = (9 — 906) "uIs “10 £9°809 = FF +409 ' ¥ ‘309 @ — 06|"-*" (7 — 06) 809° g:s00 = (gq — ,06) "UIs “10 ‘Poms’ g*sOd = F°sod V —,06)"*** (gq —,08) ‘s09'n soo =(P7 — 0G) “UIs ‘10 ‘gq suis'D°seo = Psd D sertesseesce@ ney (Gg — 06) Ue} = 7 ‘UIs “10 “D *UIS* g°j09 = g ‘309 q os eetese seen trpy (¥ —,06) Ue} = g ‘UIs ‘10 ‘9 ‘UIS* D°309 = Pi "joo D ee + (Fe =,06).£809_ (3 — 06) ‘sOO== Duis “10 “FY ‘UIS* 9. "UIs = @ ‘UIS q "*' 5° *(7 = OG) ‘809 (9 — 06) S09 = @ ‘UIs ‘10 “qPeuIs* 9suIs= g ‘US D9 — 06 |g — 06)"**** p-uey (9 — 06) "ue = (q — (06) "UIs ‘10 ‘7 ‘ae}* 9°09 = q'so9 g 2 — 06 Va (00 Ae SOR q *0b} (9 — 06) "UIe} —_ (V — ,06) "UIs f10 9 ue} ° 9 409 = V "S09 ‘syleq juaoelpy De COG See Sti ee "*@ 809° D ‘S09 = (9 — OG) “UIs “10 9 ‘809° D ‘sod = 9°S09 ‘W a ao FH ON OS Newt re Nee Gee’ 154 This is a complete proof of the truth of the rules, and, as we have already said; the only kind of proof which the rules admit of; but, after the proof, the rules ought to be used, and the formule having performed the service of proving the rules, are then superseded. |The rules ought to be used also, not only in the immediate solution of right-angled spherical triangles, but in de- ducing, where they can be so made subservient, the properties of oblique triangles. We here allude to those properties announced in the Propositions 24, 25, 26, 27 of Robert Simpson’s Trigonometry. These immediately appear on the application of Napers Rules, and their deduction is so obvious, that it is, practically, agaimst the interest of the Student to make them the ‘subject of three or four formal Propositions; since it is not worth the while to burthen the memory with the terms and enunciation of a Pro- position, for the sake of formally making one or two steps in the process of deduction. The forms ‘for the solution of right-angled triangles have been deduced from the general expressions for the cos. A, cos. B, cos. C, in which, as in @ particular ‘case, C =90°. The same general expressions may also be used for any other case, in which, to any one or more of the sides or ‘angles particular values should be assigned: for instance, ‘the side c may be a quadrant=90°, in which case the triangle has been called a quadrantal triangle; in this case cos. C— cos. a. cos. 6 cos.a.cos.6. . cos, C = = — ————_ (since cos. c = 0) sin. a. sin.'b sin.a~ sin. b> and .*. sin. (90° —C)= — Cot. a. cot. b = — ‘tan. (90° = a) tan. (90° — 0)... ete) : » . cos. b— cos. a. Cos. c cos. 6 Again, cos, B & == Ss te sin. a@.sin.'c sin. a@ ‘ cos.a@—cos.6.cos.c cos. a cos. A = sin. b.sin.c sin. b° From these ‘three equations, ‘and the additional one, viz. sin. Ad sinwa ome =e eB aie zy mdy be deducéd all the cases in quadrantal, as they have already been, in right-angled spherical triangles. \ 1Dd, If, in the above expression (a), we call 90’—C the middle part, and 90°—a, 90°—0 which are the complements of the sides, the adjacent parts, the above result might be comprehended under a Rule like Naper’s; but, should such a case occur, the surest method, and, perhaps, on the whole, the most expeditious for the Student and Computist, would be to take the supplemental or polar triangle, and to solve it by Naper’s Rule; for, the angle in the supplemental triangle opposite to c = 180° — c= 180° —90*, is a right angle. Thus, in the case adduced, the angles of the supplemental triangle are 180° — a, 180° — d, and the hypothenuse is 180° —C; therefore by Naper's first Rule, 1 x sin: (C — 90°) = tan. (a — 90°). tan. (b — 90°) the same result as (a). Examples of the solution of right-angled spherical triangles. Example }, A = 23° 27' 42” Fi Given Required b=10 39 40 BR Oo. c determined. 90°— A middle part, 90°—c, 6, adjacent parts ; Peat? + COS. mn COG, ey LAO. Gt sa + ooo [ist rule] .°- log. cot.c = log. r--log. cos. d—log. tan. 6 log. r= 10; SAG TT ae MOS NCON, SCOURS CE hc er Ad a coia's 0 eS bee dieige = 19.9625240 ia rtatty 10. SOMMERS ss). o's ob wrest = 9.2747329 LOgCOSAB OT GIT 8 a St == 10.6877911 . Cx dP 35! 40”, 156 a determined. 6 middle part, 90°— A, a, adjacent parts ; .r.sin. b = cot. A. tan. a [1st rule, p. 152,] 00 tan. @ = log. r+ a sin, 6 — log. cot. A 10 4 log. sin. 10° 39’ 40”.. AM Pena O. 2O7 27 09 vlogs cot 23 27/42: Sat 10.3624932 f. LOGS COIL ME sie anes Seas eS SOR Ce * a= 4° 35 26" B determined. 90°— B= M, 90°— 4, b ute ri 47 COS SHY. A/S COS OP ee, ee. oe . [2d rule] *, log. cos, B-+-log. r=log. sin. wag cos. ig “ hick. BINDS 27 AZ Segre acetone = 9.6000308 log; cos. 10 339 (AO Mae ete ot pres = 9.9924380 “LORS CORSE he EONS ope Pee es ae = 19.5924688 f. eO0 OB al" The above is the solution of a Problem in Astronomy, in which from the obliquity 4, and the right ascension 6, as given quantities, the longitude c of the Sun, and the declination a are required, Example 2. Given 3 actor’ Required ; 6 c= 71 . 39.37 B 6 determined. 90—c = M, a, b, opposite parts. 2 7. COS. C = cos. a.cos. b........ at AAD ned ae 2d rule; *. log. cos. b= 10 + log. cos. c — log. cos. a, 10 = log. cos. 719 39’ 37"... 6. cee eee eee ee = 19.4978286 log. cos. 27 sO) Bee oe" « pete eceee eee 9.9467376 MOR COGSD: sys we've (2 REE «eo» == 9.5510910 and b = 69° 9 48" 157 B determined. 90—B=M, 90°—c, a, adjacent parts; Br CO mEL Ea COU I UIT os. 2's Ye ae: ete by 1st rule log. cos. B + 10 = log. cot, c + log. tan. a, LOG COUN I ho aie eo: s sic o cee ele geen ss = 9.5204674 Op RAINES fe) Oss eis cla ces Gs Bares vie = 9.7220085 BOinbestOC CUS ART oes co's sloci ole a ys sities Se = 19.2424759 Se Oo SOMA ‘This is the solution of an Astronomical Problem, in which, from the latitude of a place = 90°— 27° 48’ = 62° 12’, and the latitude of the Sun at six o’clock = 90°—71° 39’ 37”“=18° 20/ 23”, it is required to find the Sun’s declination, which, by the result, would be 90° — 69° 9 48”= 20° 50’ 12”; the coe B= 79° 56 11’, in the same Problem, is the Sun’s azimuth. Example 3. C not a right angle, but c, the side opposite to it, = 90°. Given $0 = re x : ; Required i di C determined. By the expression [a], p. 143, cos. C—CcOos.a.cos, Bb cos. C = —— sin. @.sin. 0 cos.a@.cos.bo... = — ——_—_—_..._-,,, Since cos. c= 0 sin. a. sin. 6 =— cot. a.cot.b; . in logarithms, log. cos. C = ie cot. a ++ log. cot. 6 — 10. Now = 10 4- lopPeptia2o57 Gee eels wee et = .1882850 LO GEO AROS ses ate ease my aa 6376106 Wega; Cannes. PO ESAS TS ok Hata = 0.8258956 *, C= 180°— (47° 57’ 16”) = 132° 2’ 44”, The supplement of 479 57’ 16” is taken, since from the expression cos. C =— cot. a.cot. b cos. C is negative, .. C> 90°. ee fav = 158 B determined. By the expression LO], page 143, cos.b6 — cos. @.CO0S. cos. B = : sin. @.sin, Cc cos; Oe = ——-,, since cos. c = 0; sin. a .. log. cos. B = 10 + log. cos, 6 — log. sin. a HLO gt Jos cos, 0G> S20 ey St Ts ie Seer lne = 19.6001181 lpg sip. (3257 (Ope Are woods veto ae = 9.7355441 LOPS COB: Era e s tr cele NUE « Ee so 4 = 9,8645740 «. B = 42° 56° 12”. Here, the Problem is most simply resolved from the founda- mental expressions for the cosines of the angles: and there is no need that we recur to the supplemental triangle: if however we do, then since its two angles are 180° — 7, 180°— 6, and the adjacent side = 180°— C, by Naper’s first Rule, 7 sin. (C —90°) = tan. (a—90°) tan. (6 -90°) or — r. cos. C = cot. a. cot. b, as before; and similarly for B. The above is the solution of an astronomical Problem, in which, from the latitude = 90°— 32°57’ 6” = 57° 2 54”, and the Sun’s declination = 90°— 66° 32’ = 23° 98’, it is required to find the time at which the Sun rises. In the solution of plane triangles, one of the cases, see p. 84, is ambiguous: in spherical right-angled triangles, there are three ambiguous cases: and these are when the quantities, a,c, A are to be found from 6 and B given: now, by Naper’s Rules, a,c, A are given by these three equations; r.sin. a = tan. 6. cot. B r.sin. 6 = sin. c. sin. B; r.cos. B= sin. A. cos. b; but after that sin. a, sin. c, sim. 4, have been deduced from these equations, there is nothing to determine us, whether we ought to take a,c, A, or 180°— a, 180°— c¢, 180°— A; for, the sines of the 3 latter quantities are the same as the sines of the 3 159 former: and it is easily shewn, that there are two right-angled spherical triangles, which have an angle and ‘side opposite the same in both, but in which the remaining sides, and the remaining angle of the one, are respectively the supplements of the remain- ing sides and the remaining angle of the other. The other cases are not ambiguous, and Naper’s Rules, with an attention to the signs of the quantities involved, will enable us to remove the ambiguity which some of these cases appear to have: ‘thus, if 6 be the middle part, 90° —c, and 90° — B, the opposite parts, then sin. 6 = sin.c.sm. B: if 6 be required from this equation, will it not be doubtful, whether b or 180° = 6, [since sin. b =sin. (180°—) | ought to be taken? The ambiguity is removed by this property, that, if B be > or < 90°, 6 is > or < 90°: for by Naper’s Ist Rule, sin. a = cot. B.tan..b. Now, sin. a is positive whena is between Oand 180°, and if B be > 90°, cot. B is negative ; and consequently, sin. tan. 6 = oF , is negative, andb> 90°. If B< 90°, cot. B is cot. ue . sil, a), ie positive; .". tan. 6 = Rs positive: and b < 90°. , c Similarly, make 90°— B= M, and 90°—c, and a adjacent parts, then, cos. B = cot. c.tan. a; .’. if c be sought, tan. a cos. Bo cos. B are both negative; .°. tan. c is positive, and consequently, c is< 90°: if a>90°, and B<90°, or, if a< go°, and tan B>90°, then cos tan sic ic If a be > 90°, and B > 90°, then tan. a and e is negative; .". tan. cis negative; .*. c>90°. These considerations are so simple and so easily made, that it is, perhaps, better to let the Student endeavour to avail him- self of similar ones, than to burthen his memory with the terms and results of formal Propositions: for it must be noticed that, .in order to prevent the ambiguity of solutions in right-angled ‘triangles, terms have been invented and propositions framed relative to the affections of the sides and angles: sides and their opposite angles being said to have the same affection, when each 160° is less or greater than 90°: see Simpson’s Euclid, Prop. 13, p. 500, 8vo. edit. 1781. It has been already proved, that sides and the opposite angles are each greater or less than 90°, that is, have the same affection: again, since by the form (1), p. 149, cos. c = cos. a.cos. b, if a and b be both > 90°, cos. a and cos. 6 are both negative ; .. their product, which = cos. ¢, is positive, and ¢ < 90°; .°. if both be < 90°, cos.c is positive, and c < 90°: if a be > 90”, and b < 90", or if a< 90°, and 6 > 90°, the product cos. a. cos. b is negative; .‘. cos.c is negative; ...c is >90°. This may be easily translated into the terms and language which Robert Simpson uses in his Trigonometry. See Prop. 14, Spherical Trigonometry at the end of Euclid’s Elements, p. 500. The several cases of right-angled spherical triangles being now solved, we will proceed in the next Chapter to the pie) of oblique-angled triangles. CHAP. XI. Equations exhibiting the Relations of the Sides and Angles of Oblique-angled Spherical Triangles.—Formule of Solution deduced from such Equations.—LExamples, &c. Iw the cases of oblique-angled spherical triangles six quan- tities are concerned, a, b,c, A, B, C: and the general problem requires us to determine three of the six by means of the three others. We must have equations then between four of these quantities combined all possible ways; but the number. of the combinations of six quantities, taken four and four, 6.5.4.3 12.3.4 (abcA), (abc B), (abcC) (ABCa), (ABCb), (ABCc) (aCbA), (aCbB), (aBcA), (aBcC), (bAcB), (bAcC) (aAbB), (aCcA), (6C cB). . Now, the number of combinations essentially different 1s the number of the horizontal rows, or four: for instance, the com- binations of the first row depend on three similar equations : equals , or 15. These combinations are cos.a— cos. 6.cos.c cos, A = (1) sin. O.sin. c cos. b— cos. a. COS. ¢ cos. B = ——__—_— (2) sin. @. sin. c cos. c— cos. a.cos. 6 Ae eect rasmus cap Ere panS (3). sin. a. sin. 6 The combinations of the fourth row depend on three similar equations, sin.ad sin. A sin.a = sins A sin.b sn. B sin.6b ~=sin. B’ sin. cc 3 sin. ©’ sinwc sin. C’ and similarly for the remaining two rows: hence the solution of all xX 162 the cases of oblique-angled triangles is reduced, in fact, to four equations, and these four equations must be deduced, as the equa- tions in p. 150 were, by the ordinary processes of substitntion and elimination. We will now proceed to deduce these four equations. First equation belonging to the form (abcA), cos. a— cos. 6. cos. c - COs: AS - : sin. 6. sin. ¢ Second equation belonging to (4 BC a). In order to obtain this, eliminate cos. 6, cos. c, sin. 6, sin. ¢ from the equations (1), (2), (3): or, more simply to obtain it, take the supplemental triangle; then, if a’, 6’, c’ are its sides, A’, B’, C, its angles, we have, by the form (1), / é fs cos. a@ — cos. 6 cos. ¢ / COS, ta oS; : sin, 6. sin. c but cos. A’ =cos. (180°~a) = — cos. a cos. @ = cos. (180°— A) = —cos. A cos. b' =cos. (180 — B)=—cos. B cos. c’ =cos. (180° — C) = —cos. C sin. b’ =sin. (180°— B)=, sin. B sin. c == sin. (180°—C) = sin. C; — cos. A—cos. B.cos. C consequently, —cos, a = Bane es ; sine B. sin. C cos. A+cos. B.cos. ¢ Pag hi Sok ot YR ie tea ES : sin. B.sin. C Third equation belonging to (aC 6A). In order to obtain this, substitute in the equation (1), instead of cos. c, its value derived from the equation (3): and instead C in. a . sin. A A ] Bad, lite Cae then, cos. A. sin. b Perens | sin. / : ; sin. of sin. c, substitute — sin 168 cos. a—cos. 6 (cos. C.sin. a. sin. b+ cos, a. cos. b) = cos, a sin.2 b—cos. C. sin. asin. 6. cos. 6. _ Hence, after dividing each side of the equation by sin. a x sin. 6, there results cot. A..sin. C= cot. a. sin. b—cos. C. cos. b. Fourth equation belonging to (a Ab B). This equation is deduced in Cor. 2. to Prop. 16, where it is sina _ sin. A These four equations analytically resolve the Problem; or, ‘by means of them, any three quantities being given, the fourth may be found. But it 1s plain, from their mspection, that they do not afford convenient solutions, since none of them are under a form adapted to logarithmic computation; and even, if, in order to find one of the quantities involved in the equation, we were to express the equation under a form adapted to a logarithmic com- putation, such modified form would be useless, except in the case for which it was contrived: that is, would be useless, if one of the quantities, by which the required quantity was expressed, should itself be required to be computed, the previously required quantity becoming, in this second case, one of the given quan- tities. For instance, in the combination (abcA) if A be the quantity sought, we have, by p. 144, z 1 + cos. A, or, 2 cos. ao —p, jsin. (45 sin. (Cas _— a)} whence, by a logarithmic computation, cos. A and 4 can be found; but, from such form, if A were given, and a required, a could not be immediately and conveniently found: and, on this account, something more is required of the analyst, than mere quations that exhibit the possibility of solutions: he ought to furnish formule, from which, the quantity, whatever it be, side er angle, may by a direct, certain, and commodious process, be found. Formula then, as it will easily be seen, by which one 164_ quantity may be computed from others that are given, must be more in number than the equations which merely exhibit the re- lations of the quantities. This does not take place in right-angled spherical triangles; in which the formule of solution need not exceed in number the equations. For instance, the equation cos. c= cos. a.cos. 6 represents the relation between c, a, 6; and from the same equation may be found, by a like logarithmic com- putation, either c from a and 6; or a from 6 and c; or 6 froma andc; the same may be said of the second equation, that is, of cos. A=cot.c.tan.6: either A, or c, or b, may, from the equation as it stands, be found with equal facility. | The solution of oblique-angled spherical triangles will be found, by what follows, to require six cases; and in using the foregoing forms of combination such as (abcA), the quantity sought will be placed last. Cases of Oblique-angled Spherical Triangles. Prop. 18. Case Ist, (abcA). Sa i ‘ . ° The sides a, b, c, are given, and the angle is required. first Method of Solution. at+tb+e AS It, 2. V§sin. S. sin. (S — a). sin. (S — 6) sin. (S—c)} sin. 6. sin. c and, log. sn. A=F $20+ 2 log. 2+log, sin. S+-log. sin. (S—a) + log. sin. (S—) + log. sin. (S — c)t — log. sin. b— log. sin. c. By Prob. 16, page 144, if § = sin, A = In order to find log. sin: B, subtract, in the above form, instead of log. sin. 6 and log. sin. c, log. sin. a and log. sin. ¢; and, to find log. sin. C, subtract, instead of log. sin, b, and log. sin. 6, log. sm. a, and log. sin. db. 165 Second Method. By Proposition 16, page 144, sin. S.sin. (S—a A sin. S.sin. (S—a) Len) jy a Oa cos sins S. sin. (S =a) sin. 6. sin. c 9) sin. 6. sin. c and log. Soe 4{20+log. sin. S + log. sin. (S — a) — log. sin. b—log, sin. c}. Third Method. By the same Proposition, in the same page, 1—cos. A= sin. (S — 6) sin. (S— oe ~ tite Al silt CO 7— Gain) ae Ge sin. &.sin. c Bhi he Owe sin. Ob. sin. c rel and log. sin. — = 2 = $20+log. sin. (S — b) + log. sin. (S —c) — log. sin. 6 — log. sin. ct Fourth Method. Divide the third equation by the 2d, and iter. ae fe 2 / sin. (S— db) sin. (S — c) or tan. i Ae Gee an sin. S.sin. (S — a) cos. ry A and, log. tan ee or +520 +log. sin.(S—6) +log. sin. (S—c)—log. sin. S—log. sin. (S—a) ] since sin. b = log. sin. b= 20 — log. cosec. b; cosec. 6’ *, instead of subtracting log. sin. 6, &c. in the above forms, we nity (which with certain Tables is a convenient operation)! add log. cosec. b ~ 20, &c. 166 Example. a= 50° 54! 32” Numerator computed. 637 47.18: ¢=74 51-50 20 +2 log. 2=20.6020599 163 33 40 S= att BUR. SOs SO Wet. ys sin, = 9.9955158 Soret a BY BSI Rl RI, sin. = 9.7102163 Simoes ey A648 Bis Pig emai sin. = 9.8417102 Weer yy yas st Ay 5 Oy nee Fe ee sin. = 9.0807189 2 | 59.2302211 29.6151105 [a] B determined. A determined. log. sin. 50° 54’ 32”= 9.8899424 log. sin. 37° 47/ 18” =9.7872806 log. sin. 74 51 50 = 9.9846660 log. sin. 74 51 50 =9.9846660 [c] 19.8746084 19.7719466 [a]j—[c]= 9.740502] [a]—[b]=9.8431639 t Bodo 22 aot “, 4=44° 10! 40" C determined. AGEING OO. SABO noe ste Mace ree es = 9,.8899424 HOS 7 271) By Pec ts pices inate «bie nna = 9.7872806 [m]=19.6772230 [aJ—[m]= 9.9378875 . C= 180°— {60° 4’ 54”) eait1O 65°) .0; C is greater than 90°, since Cos. C—cOs. a. cos. 6 cos. C = : , : sin. @.sin. 6 which is negative, since cos. c < cos. a cos. b. The sum of the three angles, or A+B4C = 197° 98° 31”, 167 Solution of the former Example by the Second Method. 20 = 20 nistas S =81° 46’ 50"...............8im, = 99955158 S=ax30 52 18 ........ ii, et sin. s= 9.7102163 39.7057321 [d] b= 37° 47' 18"...... sin. = 9.7872806 me te LEN ln ig se ace 6 sin. = 9.9846660 19.771 9466 Jeon [oF [d)—[b] =19.9337855 4 ((¢]—[6]) = 9.9668927 = log. cos. 4; > = 22° 5! 20”, By the Third Method. 20 as 20:35 S— b= 453° 5932" ek: ...sin. = 9.8417102 Deep eS) (OS. ee sin. = 9.0807189 SS. 922420 wet est | [b]=19.7719466 Cf]—[b]=19.1504825 E(LfI-[OD) = 9.5752412 = log, sin. 4 a “35 22° 5’ 20", and 4 =44° 10’ 40”, as before. _ By the Fourth Method. 20 = 20 log. sin. (S— 6) + log. sin. (S—c), by last, or [f ]=38.9224291 log. sin. S+log. sin. (S—a), by 2d solution, or [d_] =39.7057321 : 19,2166970 .. log. tan, “= 9.6083485. ° oe == 22° 5' 20”, and A = 44° 10’ 40”, as before. 168 The solution of this Example becomes the solution of an Astrono-_ mical Problem, when from the co-latitude of the place 6, or 37° 47’ 18”, the co-declination of the Sun c, or 74° 51’ 50”, and the zenith distance a, or 50° 54! 32”, the time from noon or the angle 4 is required; the angle C is the azimuth, Any one of the four preceding methods may be used, but not, in point of brevity, with equal advantage: if one angle only be required, the shortest solution is plainly by means of one of the three last formule: but, if all the angles of the spherical triangle be required, the first method is as short, and quite as convenient, as the three last methods*. Any one of these four methods may be used, but not, with regard to numerical accuracy, in all cases, with equal advantage. If the angle sought, A for instance, should happen to be nearly 90°, then the first method is to be superseded by one of the three latter, and this for reasons precisely the same, as those which have been stated in page 95, to which the Reader is referred. Cass 2d, (ABCa). The angles A, B, C, are given, and the side a required. First Method of Solution. Let the sides and angles_of the supplemental or polar tri~ angle be a’, 0’, c; A’, B’, C’: then, by the last Case, V §sin. S.sin. (S —a'). sin. (S — b’)sin. (S— c’)} ° ! ano = 2 : re Bes sin. 6 sin. ¢c * In the Logarithmical Arithmetike, published in 1631, the first case is solved by the second method, and in Vlacq’s Tables, published at Gouda in 1633, the year in which the Trigonometria Britannica was published, the solution is by the third method: the rules of solution,. then, 200 years ago, were as plain and precise as they now are: yet, in the mode of proof we have gained something, which is certainly more plain and direct than Vlacq’s. ~ 169 eo i oe eed e Pa oe ore = £8044 180°— B + 180°—C) = 270° — st = (270° — S’), if es = S; hence, sin. S= —cos. 8’, Again, S — a’ =270°— S’ —(180°— A) = 90° —(.8’ — A) - sin. (S —a’) =cos. (S’— A); also, S—b’ =90°—(S’ — B); .*. sin. (S—b’)=cos. (S’ ~ B); and similarly, sin. (S —c’) = cos. (S’ — C). © Again, sin. b’ = sin. (180° — B)=sin. B; sin. c =sin. C, and sin. A’ =sin. a. Hence, Vv} —cos. 8’. cos. (S’— A).cos.(S’— B). cos. (S’—C)} sin. B. sin. C sin.a==2 and in logarithms, log. sin. a=4 (2log. 2 + 20 + log. cos. S’ +- log. cos. (S’ — A) +log. cos. (S’ — B)+log. cos. (S’ — C)} — (log. sin. B+ log. sin. C) and sin. 6, sin. c, are represented by fractions that have the same numerators as sin. a, and denominators, which are equal to sin. A.sin. C,, and sin. A.sin. B, respectively. The cos. S’ under the vinculum is affected with the negative sign. Now, by Prop. 9, page 134, 4 + B+ C >180°, and A+B+C 2 < 540°; .°. > 90°, and < 270°; .*. cos. S’ is nega- . poe one Py ’ B C— A tive, or—cos, S is positive. Again, S’— A= ee but, by Prop. 2, page 130, b' + ¢ >a’; .*. 180°— B+180°— C>180° — 4; B+C—A .. B+C—A < 180°, and =o < 90°; .°. cos. (S’— A) is positive; so are, by similar proofs, cos. (S’ — B) and cos. (S’— C€), y's 170 Hence, in the foregoing expression for sin. a, the quantity under the radical sign is really a positive quantity. As the expression for sin. 4 has, by the aid of the supple- mental triangle, been employed to represent a side in terms of the A angles, so, in like manner, may the expressions for cos. ae sin. oe A Salome, thus fe the expression, page 164, ROA Os ses col fy CRUE EE S.sin. (Sera) sin. 6. sin. ¢ ’ 180° —a ea Rts FINCE | COBs ea = C Oa = sil. >» we shall have a Second Method of Solution, And, a may be found from the expression, ONG \/ — cos. S’. cos. (S’ — A) sin.- = an 2 sin. B. sin. C and similarly, a Third Method, And, a may be found from the expression, a vi (S’ — B). cos. (S — C) cos.- = SEER 2 sin, B. sin. C and similarly, a Fourth Method, And, a may be found from-the expression, Hides. V/( — cos. S’.cos. (S’ — A) "9 cos, (S’ — B). cos. (S’ — C) 171 Example by the First Method. a determined. Dee naAr 10) AOR. ss sin, = 9.8431624....-[1] Bie oo 22 AB eee ve ae sin, =9.7405025.....[ 2] Cret10 055. OF eo. kates sin, =9.9378874.... [3] 197 28 31 wo. S's 98 44 15.5.-..2.00% cos, =9.1815867 S—A = 54 33 35.5......06. cos. == 9.7633172 otis OB DLT. c's s'e ales & cos. = 9.6200732 S’— C= —(21 19 50.5).......- cos. = 9.9696235 B02 10S Na dats oes eth sows «4 24 200020099 2)59.1366605 ... 29.5683302 [2]+[3]=19.6783899 log. sin. a = 9.8899403 and a =50° 54’ 30”,8. b determined. LCA e eeee e ABS = 29.5683302 DSSS Ge Aaba eel euro geile oa = 19.7810498 ODS AULA PalU sat a MUNN atte PA, 3 = 9.7872804 ATE Fes abi PN olla Se RR = 37° 47’ 18”, c determined. TED Weiser PII aera aieiearae = 29.5683302 WE rl 24) BAS is Oe ae Stage = 19,5836649 OH Ses PR SNP ey SS pe ER ROR ORY te = 9.9846653 and ¢ fii Series euiehe aie dae fale setae = 7A SULAO, By the Second Method. Pi Seu AP ornrace Bice eet or me = 20 = 98° 44/ 15", Deiat vera cs ee cos. = 9.1815867 S'— Fa EE a fiat be ga nia mma cos. = 9.7633172 38.9449039 [2J+[3], or log. sin, B+-log. sin. C..... =19.6783899 2)19.2665140 -. log, sin, 3= 9.6332570 need == 25° 27" 152, te) and a= 50 54 304. bet | 172 By the Third Method. log. Baca ieee See Rises Ur alas a =20 S’— B= 65° 91305 ssc cccicweeees cos. = 9.6200732 S'— C= —=21 10 50.5 oes. e ee. 008. = 9.9696235 39.5896967 RS p neal Repco coke Sc iancen eankac a 19.6783899 | 2)19.9113068 .. log. cos. == 9,9556534 m= 25° 27' 15"2, and a= 50 54 304. By the Fourth Method. ; LOS Tae ea pee eee hee eer Speyeinas = 20 log. cos. S' ++ log. cos. (S’— A) Pe ge ee ae = 18.9449039 log. cos, (S’— B)--log. cos. (S’—C)...... = 19.5896967 2)19.3552072 log. tan. 5 = 9.6776036 * ee z= 25° 27! 15", wl and a= 50 54. 302. CaseE 3d. (aC6A). a, b, two sides and C the included angle are given, and the angle A is required. By Cor. 2. Prop. 16, page 146, sn. M4 sine a, | sm.A+sin. B sin.at+sin.b sin. B sin. 6’ sin. B” ; sin. b ; sin.(a+6) . ,C ._———_—— sin.” —; sin. c 3? > By Cor, to Prop. 15. cos. A+-cos. B= 173 . sin. A-+sin. B sin. a+sin. B sin. c.sin. B 3 I ey el eer cos. A+cos. B eemtta +b) sin? C oa b sin. A + sin. B but ———_—___-_—_- cos. A +cos. B = tan. (=) rae 37, and sin. c.sin. B ow) Be C ; —_—_—_——— = sin. C= 2 sin. a COS. =>. p- 42, and in. b m7 + =a x sin. a-tsin. } es Cc : @ sin. (- = >) . COS, (- a - 2 sin, (=) . COS. (- = 2) C Si Ce wie ( epP TTL [ by form (5), page 32] 2.sin. (=) . COS. (=) os. ( —— and .*, tan. =) SS er ee es 8 COL. se tere wile (1) 2 ey? 2 Similarly, sin. A — sin. B sin. a —sin. b : = sin. C mites Beco 2 sin. (a+). sin.® 5 2.,s1n. (a3). cos. (=) CO 2.2.smn. ‘ex °) «co Tempe or, tan, a) a Trae ae aie 2, am (**) | 2 * The above is a simple deduction of the forms (1), (2), (which, in substance, are Naper’s Analogies), and a compendious one, supposing the formulz for sin. aes cos. feet , not to have been previously de- duced: see p. 146. WE: heesd A+ By. : Hence, since by the formule (1), tan. (—) is determined, A+B becomes known, let it = S: and, by the formula (2), tan. ‘e - pose it equal to D, then, since ; B becomes known; sup- BNG : A— ) is determined, and A+B ae Q A-B_ D: (4 by addition, A= S+ D, by subtraction, B= S— D. Example. A--determined. - C = 36° 45’ 28” : = 18° 22’ 44"... .. cot. = 10.4785395 m —b a = 84 14 29 ~ = 20 0 22....cos.= 9.9729690 20.4515085 b b= 44 13 45 _ = 64 14 7....cos.= 9.6381663 .. log. tan. — = 10.8133422 A+B st = =81° 15' 44" 41 » . 4A — B determined. C " 3= 187 22" 4a" oo . cot. =10,4785395 — B a ? a 20 O 22....sin.= 9.5341789 20.0127184 a-+-b ne : =a C4 147 gin a 9.9545255 10,0581929 A—B 37 = 48° 49! 38”. 175 A and B determined. c determined. — — 81°15 44.41 * gin, 369 45’ 28" = 9.7770158 cic ndpe AS AQ 38 | sin. 44 13 45 = 9.8435629 - 4 =130° 5 22".41 19.6205787 B= 32 26 6.41 sin, 32 26 6 = 9.7294422 . log. sin. c == 9.8911365 2. Ca 51° 6’ 12” =) and tan. (—) » expand- ed into a proportion, are called, from their Inventor, Naper’s Analogies. The angles A and B being determined by the above forms, the side c may be determined, as it has been in the foregoing sin.c __ sin. b sin, Cin. B | desirable, as in the corresponding case of rectilinear triangles (see page 85), to determine c immediately without the interven- ing process of finding the angles A and B: and, in fact, many Problems in Astronomy* require, from the data of two sides and the included angle, solely the determination of the opposite side c. ! A The expressions for tan. ( us Example, from the expression. : but it may be Determination of the side c. Second Method. 3 cos. c— cos. a. cos. b since COs, C= = page 143; sin. a. sin. 6 .°. COS. c = cos. a.cos. 6 +sin. a. sin. 0. cos. C; but cos. C=1—ver. sin. C, (ver. sin. stands for versed sine); * For instance, in finding the Moon’s distance from a Star; in de- ducing the altitude of a Star from the latitude, declination and hour- angle (two Problems useful in determining the Longitude) ; in deducing, in the case of an occultation, “the Moon’s distance from a Star; in deter- mining the altitude of the nonagesimal (see Astronomy, p. 364.): in de- termining the latitude from two altitudes and the time between, (see Astronomy, p, 422.), &e. 176 “. COS. = Cos. a.cos. b+sin. a.sm. b— sin. @. sin. b.ver. sin. C = cos. (a—b)— sin. a. sin. 6. ver. sin. C; 4 ° of ° ° ° e + 1—cOS. ¢, ors 2 sin. 3 = Ver. sin. (a—6b)+sin. a.sin. b. ver. sin. C 4% sin. a. sin. &. ver. sin. C = ver. sin. (a ra b) (: aE ee) ver. sin. (a—6) sin. a. sin. 6. ver. sin. C Assumeé®* tan.” 0 = ———_@_—______—__ ver. sin. (a— 6) : which in logarithms is 2 log. tan. @ = log. sin. a +log. sin. 6-+log. ver. sin. C—log. ver. sin. (a — 6) (p) then 2 sin.” — = ver. sin. (a— 6). sec.” 0, and c 3 log. 2+-2 log. sin. 7 = log. ver. sin. (a—b)+2 log. sec. 9-10 (q). Former Example. ¢ computed independently of A and B by the 2d Method. Determination of the subsidiary angle @ by the form (p) @ ex 84° 1A) 20" ek el ea xe sin. = 9.9978028 Thee 7 TN Ws al: a Dg a a De ata A sin. = 9.8435629 C = 36 45 28 .....2000- gras ver.sin. = 9.2984762 29.1398419 a—b =40 O 44............52. Ver Sin, = 9.3693878 ESM GN ERTOU Ss ecegee el scu hss seen: =19.7704541 and log. tan. @...... pom canirdae eat fare = 9.8852270 * This is the instance to which we alluded in speaking, page 106, of the use of Trigonometrical formule in computing log. (a+). ie Determination of c by the form (q¢), page 176. D log. SeenGe ques Wk eee viele o o's 8 = 20.2012488 logs ver. sin. 40° 0! 44 oo... ee eee = 9.3693878 _ 29.5706366 jae fateh a = 10.3010300 2 2 log. sit. 5. ree (0 EST G 5606000 and log. ASS on sh ieee 9.6348033 ag = 25° 39° 5A, and c¢ = 51 6 11.5, nearly. This is, perhaps, the most commodious form for computing c; for, when we use it, we need not consider whether the fraction sin. a.sin. b.ver. sin. C . , d — ——— 3 > or < ], since tan. 0 admits of all ver. sin. (a — 6) degrees of magnitude. It 1s easy, however, to give another formula of computation, thus: Third Method of computing c. cos. c=cos. a.cos. b+sin. a sin. b. cos. C : ; C =cos. a.cos. 6+sin. a.sin. Db. (2 cos.” pelo 1) S . * 2 C = cos. (a+6) + sm. a.sin. b. 2 cos.” 2? tg € C a3: 1—@. sin." = = cos. (a+6)+sin.a.sin.b.2 cos. —; 9 a+b 2 and .°. 2 sin.7- = 2.sin. — 2sin.a.sin. 0.cos.—. rls : C Let sin. a. sin. b. cos.” aie sin.” M; | C sk | and .". log. sm. M= (2 log. cos. x + log. sin. a+log. sin. b—20) b sha sin.” M a, o then sin.” Cc Be ~ = sin. 2 * Z 178 = sin. (5 +5 + M).sin. (; +5- M) by the form (c), p. 32; and in logarithms, log. sin. = = 1 Nog sin. G +-~ = +M) + log. sin. (; +2-m)} fet This is the kind of form which Laplace has employed in his Mecanique Celeste, Livre 2, page 227, (see also the use made of this form, Astronomy, vol-I. p. 159.) Former Example by the third Method. rj” = 18° 22' 4.4.2 . cos. =19.9545254 a=84 14 29....sin.= 9.9978028 b=44 13 45....sin.= 9.8435629 ron) 6 UR sir i fete 39.7958911 M=52 14 23...... «. 9.8979455 =log. sin. M=log, sin, 52° 14/ 93” _ a+b Tula + M=116° 28' 30”,,.sin.= 9.9518856 — —~ M= 11 59 44,,..sin.= 9.3177204 19.2696060 0.6348038 = log, sin. 33 = 25° 33! 5°28 ° ° = 51 611.33, nearly. Fourth Method. cos. c=cos. a.cos. b-+-sin. a. sin. 6. cos. C 1—ver. sin. c= cos. a.cos. b6+sin. a.sin. b—sin. a.sin. b.ver. sin. C;. .’. ver. sin. c= 1 —cos. (a—6)+sin. a.sin. b. ver. sin, C = ver. sin. (a — 6) + sin. a. sin. OD. ver. sin. C, “which formula, translated mto words, becomes the precept given in Sherwin’s ‘l'able, page 44, (edit. 1771) for finding the side opposite 179 to the included angle. The author gives in his Work a Table of natural versed sines, which are plainly necessary in his mode of computation. If we use one of the preceding formule for computing c, we may, if we please, determine A and B without the aid of Naper’s Analogies, and by these expressions, : sin. C . sin. a é sin. C. sin. sin, A = ———-——_; Te Bi nee sin. € 7 sin. C Case IV. (Ac Ba).* A, B, two angles, and c the adjacent side are given, and the side a is required. The solution will be deduced from the former, by the aid of the supplemental triangle; 4’, B’, C’, a’, 0’, c’ being its angles and sides. tan. (F2) =n (SEO) = == tan, (480°— a2) b = —tan. (= a Ak B b—a a—b tan. ae = fan, ( ) = —~ tan. ( ) 2 Q 2 sin. (x 2 = sin. (180°— at”) = sin. (=) [gee Cor. 6, Prob. 3.] | sin. ( —"") = Sit. =) = — sin. (—); * This combination is in the third row (see p. 161,) and therefore not essentially different from the first of that row, namely, (a Cb A) which has been already considered. In fact, the two equations involving — the four quantities are precisely similar: but, from what was said in p. 163, the same formula of solution cannot suit each case, since, in the former, an angle is the quantity required which, in the latter, becomes one of the quantities given. 180 eS) Sire ein Ee, : A+B, _ A+B —Q ". generally, sin. ( Again, a +. b’ 2 = Cos, (180° aa cos. [ see Cor. 6, Prob. 3. | cos. Ee» Be (Ce ) 5 oF = cos. ES Ud lastly, cot C cot (90' =) tans Wiese Be = e ars Se aay = an. ee y» 2 9 2 Having now transformed all the terms in (1), (2), (see p. 173), into different expressions, if we substitute the transformed terms in (1), (2), there results and tan. (a) = in (44) tN. = sees ee 2 (4); Ay, 3 may be derived. These equations (3), (4), expanded into a proportion, are called, [as the former similar ones (1), (2) have been, (see p. ae J Naper’s Analogies. 181 Example. €= 51°. 6’ 19” A=130 5 22 FeO? 200 0 A—B ae See 8 49 38....C0S, = 9.8184449...... sin. = 9.8766379 += 25 33 6....tan. = 9.6795032......tan.= 9.6795032 19.4979481- 19.5561411 A+B ; : PaNe bar bap 81 15 44....cos.= 9.1815936...... sin. = 9.9949301 a+b a—b tan. - = 10.3163545} tan. ——= 9,5012110 b —b and —64° 14’ 6”2| and ——=20° 0° 29"3 a—b - S201) 0 225 Brea nape Medes 6b =14 13 43%, ‘ 104 C determined. BLL GU pete 2 iereie coer Mer ined Cen ..ae 9.8836842 SE Oa rE, Oot Qtecrty atti Cen eal = 9.8911327 19.7748199 Bie A LA 20. shoe te, = 9.9978028 POR SINS Coke totes we ee as ate oases, So os cet gered OME Tak oe LOpeeeBOn 45°) 28,-. a and 6 being determined, C may be determined from this ok ee . sin. ¢ a : -expression, sin. C=sin. A x — » as it was in the preceding sin. a ’ Example, or, without the intervening process of finding a and }, by the following method. — 182 Determination of the angele C ’ cos. ¢ = — In the supplemental triangle, by the original form (c), p. 143 sin. a. sin. 0. cos. C’+ cos. a. cos. b or cos. (180°—C) = sin. (180°—.4). sin. (180°— B) . cos. (180°—c) +- cos. (180°— A) cos. (180° — B) or — cos. C= —sin. A.sin. B.cos.c-+-cos. A.cos. B : ? — sin. A.sin. B (1 — ver. sin. c) + cos. A.cos. B and .°. 1—cos. C, or, ver. sin. C= 1+ cos. A.cos. B—sin. A.sm. B+ ver. sin.c.sin. 4.sin. B = 1+ cos.(A + B) + ver. sin. c.sin. A.sin. B sie & .. ver. sin. C, or 2 sin. ye ver. sin.c.sin. A sins B= A+B ( ver. sin. c.sin. 4.sin. B cos SO ) A+B ) ‘ 2 cos.” (>) 9g ver. sin. c.sin. A.sin. B Assume tan.”@ = AnR , or, in logarithms, 2 cos.” (——) 2 log. tan, 9 = A 5 (log. ver. sin. c+-log. sin. A +log, sin. B — 2 log. cos (=) + 10—log, 2). hy & then, ver. sin. C, or, 2 sin.” a> 2.cos. ( A+B A =) sec. 0; hk in logarithms, log. sin. —- = log. cos + log. sec.9 ~ 10. ~ 183 C, in the former Example, found independently of a, d. 10—log. 2=9.6989690 c= 51° 6°12"... .*. ver. sin. =9.5706390* MTOR Soe ee. - SIN. = 9.8836842 Ste OOO eek caves ears sin: = 9.7294.422 38.8827344 A+B a= S115 toe See } cos. = 18.3031872....cos.= 9.1815936 20.5195472 “. log. tan. @= 10.2597736... sec. 0=10.3171290 . log. sin. “ = 9.4987226 ae 5 == 18° 29! A3"S, and C = 36° 45’ 27”, nearly, as before. If we express 1 + cos. (4 + Bb), the versed sine of the supple- ment of A + B, by suver. sm.(4 +B) we may employ this form for computing C ver. sin. C = suver. sin. (4+ .B) + ver. sin. c.sin. A. sin. B = suver. sin. (A+B). sec.” 0, putting ver. sin. C ‘ suver. sin. (A -+ B) tan. 9 = sin. A.sin. B Case V. (aAOdB). Two sides a, 6, and an angle A opposite to one given, the angle B, and the remaining angle and side are required. sin. A.sin. b By Cor. 2, page 146, sn. B = ‘ Re sin. a 7 . . c 3 ae C * Since ver. sin.c=2 sin.” ee log. ver, sin. c—log. 2=2 log. sin. « — 10, .*. log, tan, 0=5 (2log. sin. Hos. sin. A+-log. sin. B— 2log. cos, ath) which form is rather more conyenient than the one used in the compvu- tation. 184 and in order to find C, take the first of Naper’s Analogies, p. 173, : cos. 4 (a+b) ee L(aepBy 3 a then cot. : = tan. 2 (A+ B) C and log. cot. — = 2 log. tan. 1 (4 + B) + log. cos. ¥(a + b)—log. cos. 5 (a — 8). C being found, c may be had from the expression n. C ; » or directly thus from the third of Naper’s sin. A Analogies, p. 180, sin. C=SIN. a. cos. s (4+ B) c 1 aca ir ERS SES Dai Ce Uo, ELT ort aa cos. + (A — B) Example. B computed. AP Lae wer Nie ek oa oe Sate eno eee sin. == 2.7390354 POL AD LOU Oras ok ce so clette cee at sin. = 9.97346063 19.7125017 NOP NAS ila Coen Meal, sin.= 9.9934638 COREE AS RL AR Bre SS 1 OS SR ne sin. = 9.7190379 re) and B = 31° 34 37".71.. C coraputed. b We % + Pi Baht be MR AS LS es RE dg cos. = 9.4093099 ca SS eee a eee a ae nee aera tan.= 9.8027553 19.2120652 ones Nae on MEO ORG s cleo t Paice bale e's bse es cos.= 9.9983741 cot. f 9.013691 = 80° 42’ 38" ? and © =161 25 173, 185 c computed from C. Siecle COI ee bc osc bebesd = 9,5032532 BEEING MaRS reer ents ce 9 dv ele ee = 9.9934038 19.4967170 pints d517] Serene ee ars he PERE = 9,7390354 TOR USING, Wem Pmene eee Ear om has sr uy a = 9.7576816 he x= 145° 5) 2". t ¢ computed independently of the value of C. f J b OG halal oir Ee RS See eae eee tan. = 10.5758962 eles BS LOL ROT ae ae te mean cos. = 9.9264417 20,5023379 A—B SEM Om LOIN oe. ofa PEC ee eee. cos. = 9.9999536 ~. tan, = = 10.5023843 “3 = 79° 39' 30”, andc=145 5 Iz. Case VI. (AaB6). Two angles 4, B and a side (a) opposite to one of them, are given, the other side b, and, besides, the remaining side c and the angle C, are required. | b is determined from this expression, sin. b=sin. a Aa sin. B sin. A’ C from the first of Naper’s Analogies (1), p. 173, and ¢ from the third (3), p. 173, as in the preceding case. ay 186 6 computed. a = 89° 16’ 53".5......06 ives. 3 agin ,8in, = 9.9999658 Bt 4B O95 ve Oe. oe rah sin. = 9.8751256 | 19.8750914 ~ BREN Ge in) a OL TA | ee ah sim. = 9.9747475 log. sin. b = 9.9003439 we b = 52° 39' 4.5. The sin. b=sin.(180°—6), but b cannot = 127° 20’ 56”, for since A> B.a must be > 0. c computed from the form (6). e = k 7007 OCU eeleatae dest aicen tan. = 10.4622011 Se ee = 59 37 30 .eeeee So ereee nae cos. = 9.7038563 20.1660574 ee PE. BON oss eee aes cos. = 9.9919097 log. tan. 5 = 10.1741477 “5 = 56° 11’ 29”.33, *andc=112 22 58.6. * This last Example is taken from the T’rigonometry, of M. Legendre, who has, however, found c and C by a process different from the above. Subjoined are the data and results in French measure (/’) and reduced by the Rule, page 21, to English (£). 187 , C computed from the form (a). ot? = WOO ISO VOOee tenis saciid COs. = 9.5133811 Oe a 59 37 30 A ee *.... tan, = 10.2320208 19.7454019 ve aie De aes ee eons cals w seete cos. = 9.9774230 cot. rae 9.7679781 fe, 3 = 59° 39’ 30", and C=119 15 O. gd O0220 01775 ( 2 ) B= 54° 0' (F) A=78° 50° (F) 9 92 017 | 4 4, 7 85 89 28 135 AS 6 70 65 6 6 6 16.8918 36 | 39 O 6 . B=48° 36 (F) . A=70° 39’ (£) 53.508 ; ==. 80° 10° 35'.5 (2) c= 1249 86'99".3, b= 58° 50’ 14” (F), C= 132° 30! 12.48 69.93 5 85 O14 13 25 112 38 29.37 52 65 126 119 25 6 6 6 22.97622 39.0756 15.0 6 6 Css 110) 15° GE) 58.5732 4,536 Ce Lie QOL NB OM Bly. <*> “0 2590 307405.) and these quantities (c, 6, C,) agree with those determined in the text. The above reductions may, more easily, be performed by the aid of the Table inserted at the end of Chap. J, p. 24. 188 In Case 5th, the ae C has been determined by means of B previously determined, and by the aid of one of Naper’s Analogies; and this method, on the grounds of facility and certainty, is, perhaps, the most convenient: still, analytically considered, the determination of C does not require the previous determination of B; for, by the third equation, p. 163. cot. d.sin. C=cot.a.sin. 6—cos. C.cos. b, in which A, a, 6, C are alone involved. But, this form is not adapted to logarithmic computation; in order to adapt it, we must introduce what has been called a subsidiary angle: thus, if we take @ such, that tan. 9=cos. b.tan. A (c); then, sin. C. cos. 6 tan. @ + cos. 6.cos. C=cot.a.sin. 6; or, cos. 6.(sin. C. cos. 8 + cos. C. sin. 0)=cot. ad.sin. 6 .sin. 8; cot. a.sin.@. sin. 6 or, sin. (C +0) = = tan. 6.sin. @.cot. a...(d) cos. 6 Hence, by deducing the logarithm of C +0, we shall know C, since 9 is determined by this form, log. tan. @=log. cos. 6 + log. tan. A — 10, and by a similar method, that is, by the assumption of a sub- sidiary angle, may c be determined solely from A, a, 6. It is sufficient, however, to have noted these methods, for, the com- putist 1s not recommended to avail himself of them; the preced- ing ones, those by which the Examples have been numerically solved, being fully adequate to the purpose of solution. In Case 6th, C and c may be also solved by the introduction of a subsidiary angle; and its introduction, in these cases, corresponds to the Geometrical resolution of the oblique-angled into two right- angled triangles: thus, in the last case, conceive a perpendicular (p) the arc of a great circle, to be drawn from the angle C on the base c, and let the angle contained between this perpendicular and the 189 side 6 be supposed equal to (90° — 9) ; then, by Naper’s first Rule, 1 x cos. b = cot. (90° — 0). cot. A, and tan, = cos. b.tan. A, which agrees with the assumption (c), p- 188. | Again, by Naper’s first Rule, sin. {90° — (90° — 6)} or, sin. 0=tan. p.cot. 6, and .*. tan. p = sin. @. tan. db. And finally, by Naper’s first Rule, Cos. {C— (90° — 0)} = tan. p.cot. a, or 3 *, sin. (C +6) = sin. Q. tan. 6. cot. a, which agrees with the result (d) in the preceding page. We have deduced in pp. 147, &c, expressions for the sine and cosine of arcs, such as : and it is worth the while to extend the deduction to the sine and cosine of half the sum of- the angles of a spherical triangle, since the resulting formule are, in certain geodetical operations, capable of an useful appli- cation. A, B, C, being the 3 angles of a spherical triangle, it is re- : A+ B+C. quired to find cos. Bei , AA. B+C A+B C Bade Bice cos. ————_—— = cos. . cos. — — sin. . sin. ~, 2 2 2 he SiN. + COS. COB Tae Le a I sae I Mirae L see p. 147, = a meee 2 et) : cos. (a+b) cos. = cos. — sin. C ( ay ae _) a : -. (sin. -. sin. - 2 Cc 2. cos. — 2 sin. S.sin, (S—a) sin. (S—)d) sin. (S—c) $ see p- 145, =— — Lote ningun, (S—a)sin: (577 O) Staged a b Cc 2.COS. ~» COS. ~. COS. — 2 0, 2 which is a formule demonstrated by Cagnoli, p. 329. of his “Trigonometry, and by Delambre, Astron. Vol. I, p. 232. A Bae OC. sin. LM ERT Ric A+B+C_ . A+B C ABB Fe C€ sin. - cos. — -++ cos. ———. sin. — 2 2 2 aE te 3 sin. cos.* ~ sin.” — 2 : ‘ cose. (4 0) 7 5 cos. $(a +b) C cos. = COS. = 2 g “(p. 147.) = = ficos.L(a- b) +L cos. 4(a+6)} + {3 c0s. 3(a—b)—3 008. 5 (a +6)} ———, 2 2 oo cos. c— cos. a.cos. 6 Cc a b c cos. — 4..COS. ~ . COS. ~ .COS. = 2 2, 2 2 ee ey aC atl 20 4.cos. —.cos. ~+2cos. <—1—(2¢0s.°4—1) (2c0s.7=—1) 2, 2 2 2 2 . — —— a b Cc 4.COS. ~ . COS, ~. COS. — oO , 2 oO aod ea 19] 9 @ Pp on cos.” = +cos. - + cos. = — 1 2g 2 g — a £ a b c . COS. = . COS. ~ . COS. = COHCE 1 COS. 3 * The ‘values of cos. (Gay » sin. (AEE , are by the above formule expressed in terms of the sides, a,b,c of a spherical triangle. We have the means, therefore, of computing the sum of the 3 angles (an useful operation, as we shall hereafter see) from the 3 sides. But it may, in some cases, be convenient to deduce such sum from other data, from, for instance, two sides a, b, and the included angle C: which may be thus effected : Cae : cos.” ‘s sin.” = sin, (44 Bee) _ = sare cos, 4(a—b) + = cos. 3 (a-+) cos. 3 cos. 3 substitute instead of sin.” = 1 — cos.” 2 , and develope cos. $ (a+b), then Me Gee Oh auch se aul b C: sin, ea eA } cos. a Cos. 5 — sin. 3 sin. 5(1-2 cos.7 =) 2 cos ¢ °Q a b SEK ee D COS. = COS. = sin, = sin. — cos. C a 74 IE 2 g oe cos. 3 B4C sin. C b but cos FETC _- SEE (sin 7 Sin 3 ; m Goes i a cos. C foe ~ SIN. = A+B+C COS. 5 COS + sin 3 seas G 2 )=- b A aris Lies sin, C, sin. = sin. - 2 b 1-ptan. 5 = , tan. earls C Geis tan. F tan, 3 a $ln. C 192 an expression of great simplicity, and easy to be remembered, but not capable of being adapted, (as far as the author of the present treatise knows), to logarithmic computation. As far, therefore, as we have gone, we ought, should it be necessary to compute the sum of the 3 angles of a spherical triangle from its 3 sides, to use the former formula, p. 190, in- stead of the latter. But it so happens, Gf such an expression can be admitted, in speaking of the modifications of analytical language), that we may, from the latter, deduce a formula for the computation of : A+ B+C A + B-+C still more commodious than that for cos. eS : Thus, He A+ B+C 2 1 — sin. a b c a C 1+2 cos. 5° C08: 5 - COs. 5 — cos.” wh cos.* = — cos.” = pp 2 2 2 a b, c : 2 .cCoS. ~ .COS. —'. COSe — 9, 9 2 the numerator if a Be ateNa 9 0 ae 2 0 be =1-— (cos. -—cos. = . cos. = } —cos.” ——cos.” —-+ cos.’ — . cos.’ — 2 2, 2 2g Qg 2 2 a 2 sin.? - (cos. = p =) .sinn = — .= — COS. ~ . cos. = Q 2 g 2 = sin.” = jsin. \ { aber ae a ie b C sin. ~ sin. ~ — cos. ~ COs. — COs, — 2 Q - Q 2 i = {cos ei cos eae 3 s 7 —, "3 OS, b eS — COS. } At Dla NIN ean ey ite a b Cc . sin, - COS. ~ — COS. — COS. ~ ro) 2 2) Qh? ge . atb+te . b+te—a . ate—b. atb—e ore § SIN, eee. SIN, OO SOOO 4 4, 4, = 2”, sin. 2S .sin. + (S—a) sin. (8 —4) sin. 4(S—c). Hence, since LjA+ BC 5, AR OR 1 — sin, ———~———- = | — cos. (90 Se : A+B+C fhe 2 OGL. ae Ae = Q sin. (4 4 dh we have sin. (45° — art’) sin. 5S. sin. -(S—a). sin. £(S—b).sin. 4 (S—c) a b c COS. =. COS. ~ . COS. = i 2g, 2 29 from which expression, suited to logarithmic computation, A+B+C ee deduced. But it is plain that the computation of A+ B+C from this expression, is quite as long as from the preceding one of p. 190, each requiring the taking out of seven logarithms. If, however, we divide this last formula, (I. 8.) by the preceding one, we shall have, 45° , and, of course, A-+-B+C may be directly since cos. atte ame HTN (90° ‘ak A i =) ; = sin, (45° _ Aas a an =) ae (45° 2 ae) Bes 194 ie (a0 - tert A+ B+) or 4 tan. (45° _ i A+B+C 4 2.cos. 45° — 2.sin. +S. sin. $(S—a). sin. $(S — 6) sin. 5 (S— ¢) 7 {sin. S. sin. (S—a).sin. (S—6) sin.(S—c)} * bat 2 BS sin. = S aif os 4 sin. S V2.sin. +S. cos. 5 8 2, cos. = tan. — S —— po sual and so on. 2 and since tan. (45°— X) = — tan. (X — 45°), we have tan Cae —_ 45°) 4 = V{tan.1S.tan. $ (S$ — a) tan. 5 (S— 6) tan. $(S —c)}. By taking out, therefore, four logarithmic tangents, we obtain at A+ B+C once the tangent of er aa 45°, and thence immediately we obtain A+ B+ C. The above formula for computing A + B+ C, or the sum of the three angles of a spherical triangle, from the three sides, are not formule of mere curiosity, but applicable to practical pur- poses. The last formula, for instance, enables us immediately — to compute the excess of the sum of the three angles of a sphe- rical above two right angles. Let ¢ be that excess, then SOM Mis ity i aby oo ; since e = A+ B+C-— 180°, ii 4 4 — 45°: € Ed ; *. tan. PORT, if, as is generally the case, ¢ be very small, will equal the right hand side of the preceding equation, and this 195 excess, as we shall hereafter see, it is necessary, or, more pro- perly, convenient tocompute. Again, in measuring the surfaces, or spherical areas included within the intersecting lines of the survey, we cannot compute more commodiously, r’.sin. 1". {4+ B+C}, which (see 188.) is the area of a spherical triangle, than by A+B-+-C A and, as an instance, we will take that to which Delambre, in the ist Vol. p. 235. of his Astronomy, has applied Cagnoli’s Theorem, and his own Series d= 76, 35° 36" Gee bo" 10 5G c=40 9O 10 computing — 45°, from the formula of p. 194.* Ss log. tangents. @ al 41 34........ . .9,9497516 Sha) on AD eee ees ss 8.7733683 Cr iia LO) 1 om Le ne nel eg 9.4745269 PA S10). = 81 AV 80 cae. 9.5996367 2) 37.7972838 (deducting 20) - $.8986417=tan. 4° 31’ 39”. 4 AtB+C Hence, <2 — 45°= 4° 31’ 39”, and 4+ B+C — 180 is 6 36, il or the excess of the 3 angles of the spherical triangle over 2 right angles is 18° 6’ 36”. This, therefore, is not an instance that occurred in the geodetical operation. The area of the above triangle = (see p. 139.) * In the next Chapter we shall see why, in practice, it is convenient to compute from this formula. 3 196 gin, 1” (A+ B+ C— 180°) = 7°. sin. 1".x 65196", log. Sin ss. ss 6 4. 0850740 log. 65196 . ... 444 +4.8142210 COS LOO Se ck vee 9.4997959 the area, therefore, equals to r* x .516079, the whole surface of the sphere being 7° x 12.56636. A great variety of instances to the preceding methods might easily be collected from Plane Astronomy. It is not, however, necessary to give any; since, amongst other purposes, the present ‘Treatise is meant to be merely preparatory and subservient to the study of the latter science, and to be intelligible to the Student who may -happen to be unacquainted with its technical terms and language. Astronomical Examples, stated and numerically resolved, would, indeed, be useful tothe Student. One part of their utility would be, to communicate the art of translating Astronomical conditions into bare Mathematical conditions; it is not, however, the special business of a Trigonometrical ‘Treatise to teach such art. Another part of their utility would consist in teaching the method of trans- forming general symbolical results and formule into numerical values; but, of this method sufficient specimens, it is hoped, have been given in the preceding pages. Still, however, it is desirable to apply and illustrate the pre- ceding formule; and, it happens fortunately, we can effect this without introducing either the principles or the terms of a new science. ‘The accounts of those Trigonometrical Surveys, by which the figure and dimensions of the Earth have been attempted to be determined, will furnish us with very interesting instances of exemplification. In the next Chapter we will turn our attention to this point. We shall there perceive how results may be obtained by the direct application of the preceding methods of solution; and, besides, for what reasons and by what means, those methods, in certain circumstances, are either modified, or completely superseded by methods of approximation. CHAP. XII. pa i Object of the Trigonometrical Surveys.—Conditions for de- termining the Nature of the Line to be measured and com- puted.— Apparent Depression of one Station seen from another by reason of the Earth’s Convexity.—Elevation by Refraction.—The mean Terrestrial Refraction an Aliquot Part of the Arc contained between two Stations observed re- ciprocadly, the one from the other.—Determination of the Heights of Stations from their reciprocal Angles of Depression and Elevation.— Reduction of the Angle observed between two Objects to an horizontal Angle.—The three Reduced Angles of Observation the Angles of a Spherical Triangle —Their Sum ought to exceed 180°.—The excess in Practice always very small; Theorem for computing it—JIts real Use— Llegendre’s Theorem for adapting the Small Spherical Tri- angles of Geodetical Operations to the Rules of Plane Tri- gonometry.—Reduction of Spherical Triangles to Triangles formed by the Chords. Ir is proposed, in the present Chapter, to give some account of a Trigonometricul Survey; to describe first its object, and the general mode of conducting its operations, then to shew the kind of aid it derives from Trigonometry: and for what purposes it applies formule long known and established in that science, or requires the aid of new ones to be derived from it. Beyond this connexion of the practical operation of the survey, and the scientific theorems,of ‘Trigonometry, it is not intended to proceed. The description of the znstrumental means of conducting it, ingenious and interesting as they are, will not be attempted. The nature of the present Work does not de- mand such a description, and an useful description, one suffi- ciently full and exact, would add, preposterously, to its bulk. 198 The object of the survey is to measure the distance DM Ma between two stations D and M, situated in the same meridian, or two stations that have the same longitude, (see Astronomy, Chap. xlii.) This distance DM, if the Earth be considered to be a spheroid of small eccentricity, is nearly a circular arc. Suppose it to be determined, and to equal D, then the latitudes (L, 2, (see Astronomy, Chap. xli.) of D and M are to be ‘found; lastly, from this proportion, D Pee BPS Anas nio o, L-l:1 2 Dae we obtain, in terms of D, (in feet, if D be expressed in feet,) » . the value of one degree. This general statement, like all other general statements, includes many subjects of consideration. —_- In the French survey the line DM extends over the whole of France, from Dunkirk to a station near Barcelona. .The — 199 inequalities of ground, therefore, were there no other obstacle, would prevent the determination of the length of DM by direct measurement. DM, therefore, cannot be determined by mea- suring its parts Da,ab, &c. But Da, ab, &c. are to be computed. Da, for instance, can be computed from DC and the angles DCW, CDa. The angles can be observed, but DC must be measured, by direct means, or must be computed from some other known or measured line. It may be com- puted, for instance, from DJA, a measured line, and the observed angles of the triangle DAC. Some such line, sooner or later, must be measured, and, then, for distinction’s sake, it is called a Base. Suppose now DC to be known, or to be the unit upon which the whole succeeding series of triangles is to depend, and the observer to be at D. In that station, (Dunkirk, in the French survey) he sees no fixed and distinct object in the direc- tion of the meridian DM, but, to the right he sees the tower of Watten (W), and, to the left, Cassel. He observes at D the angle WDC, and, since he knows the direction of the meridian, he observes also the angle WDm. Next, at the stations W, C, he observes, in the first instance, the angles DWC, DCW. In the triangle DWC, then, one side DC is known, and the three angles: consequently, sin. DOW sn. WDC DW=DC.— aaa and WC = DC. re may be computed. In proceeding towards the south, the observer at W and C observes the angles which F, (Fiefs) subtends, that is, the angles FWC, FCW, and thence computes WF, CF in terms of WC already known in terms of DC. After this manner, observing stations more and more to the south, the operation is carried on to the extreme southern station: suppose that station to be F (for the observations, cal- culations are all of the same kind, whatever be the number of a 200 triangles intervening between the extreme stations), draw Fm perpendicular to DM, and Dm is the length of the meri- dional line that is to be valued. Now Dm=Da+ab+bm. Da is known from DW, and the angles DWC, WDa. From the same data, Wa, Ca, and the angle DaW are known. ab is known from Ca, and the angles Cab, WCE. From the same data, Cb and the angle Cha (= Fbm) are known, Fb=CF-—Cb, and bm=Fb.cos. Cha. The above is a general statement of the principle and mode . of proceeding: but, like the brief title of a very long chapter, it affords us a very incomplete notion of what is to succeed. The measuring of the distance DM between the pomts D and //, must mean a measuring according to certain rules and conventions. The distance cannot be merely made up of parts as Da, ab, bm, &c. these parts being determined from DW, WF, and the angles of the respective triangles WDa, FWa, &c. and lying in sae planes of those triangles: because, in such a case, the distance DM would be formed of lines Da, ab, &c. lying in different planes irregularly inclined to each other accord- ing to the unevenness of the country in which the stations D, W. C, &c. shall be situated. Let us refer to the first triangle WDC, in which D, W, C represent Dunkirk, Watten and Cassel. The first place being situated in Downs near to the sea is dower, that is, nearer to the centre of the Earth, than Watten or Cassel. Is it possible from observations to find, instead of Wand C, two other points W’, C’, the projections of W and C on the Earth’s surface, situated at the same distance from the Earth’s centre as D is? If we could do that for the triangle DWC, we could do the like for the other triangles WCF, and from the original series of triangles find another series of imaginary or computed tri- angles, the angles of which should be situated on the surface of a sphere of which the distance of D from the centre of the Earth is the radius. 20] If we could effect this plan, the distance DM would be systematically measured, and, for equal differences of latitude, would be the same in England as in France, and the inequalities of surface, although they might effect the difficulties of local practice, would have no influence on the result. We have supposed Dunkirk, or D, to be the original point of levelling. But, (for in these matters the greatest nicety is affected), this supposition is not sufficiently precise. We must go a step farther, and determine the height of D above the level surface of the sea, above (that the whole matter may rest upon a natural and determinable basis) the mean height of the sea, or the height which is the mean of the greatest and least tides. This operation is to be effected by the usual means of levelling practised in land surveying, that is, by determining a series of successive elevations on the slope ground that separates the sea and the first land station. This last method, (more exact however than any other) would be inconvenient if it were applied to determine the elevations of W and C above D, the stations being separated from each other by several miles. Another method is to be resorted to, which, in the general statement, may be described as consisting in determining the angular elevations of D and W, as observed respectively from W and D. : If the Earth were a plane, and D and W were equally elevated above it, D and W, viewed from each other, would appear equally distant from their respective zeniths, and, were there no refraction, 90 degrees distant. If D and W be on the surface of a sphere, and equally above it, the depression of D seen from W, would equal the depression of W seen from D; the depressions, (the zenith distances being greater than 90°,) arismg from the Earth’s convexity. These depressions, we may now remark, are greater than what are actually observed by reason of the elevations of observed objects from refraction. If objects, then, were equally distant from the Earth’s surface, they would, viewed respectively from each other, appear equally depressed: but if W, the tower of Watten, should be higher C0 202 than D, (or Dunkirk) D, observed from W, would appear more depressed than W observed from D. ‘The difference, then, of the actual heights of Wand D will depend partly on the difference of the observed or instrumental depressions of W and D, and partly on the actual distance of W and D: for it is _ plain, if W, retaming still the same actual*height above the Earth’s surface, were removed farther from D, that the apparent depression of D below the horizon of W would increase. Let A, B be two objects or stations, C the centre of the Earth, CB>CA, take CQ=CA; jom 4B, AE. If a per- pendicular line to CZ at the point A, passes above B, then since such perpendicular line is A’s horizon, B viewed from A would appear depressed. If the line passed beneath B, or between C and B, then B viewed from A would appear ele- vated, setting refraction aside. Let VBA the zenith distance of A viewed from B=A’, Cc 2AB the zenith distance of B viewed from A =A, If the depressions be called 0’, 6, then A’ = 90° + 8, A = 90° + 3. ti 203 Now A+A’=(C+ 4 BAC) +(C+ 4 ABO (C= ¢ BCA) =C+(z BAC +C+2ZABO) =C + 180", and consequently 6 +0 = C. If B should be elevated so that the perpendicular to CZ from A falls below B, (see p. 202. 1. 12.), then A=90°—6=90°—e, (e denoting the elevation), and 6 —e=C. In order to determine approximately the difference of heights, supposing there were no refraction, we have sin. BAQ BQ=AQ Pad RO ae AQ sin. BAQ, nearly, but BAQ=ZAQ—2AB =180-4{AQC+QAC}—A C = 180 +> — 5 {AQC+QAC+C}-—A 0 C Pilg 01 S00 +3 og 1180 }— (90+) | =—— 0, 2 But the depression 0 is not that depression which is actually or instrumentally observed. For, by the effect of refraction, an object B seen from A is elevated to the point 6, for instance. The zenith distance observed is ZAB=ZAB-— BAb. Hf, therefore, we continue to represent the zenith distances that are observed by A, A’, we must add to them, in the pre- ceding equations, the refractions (p, p,) due to those zenith distances. Hence, (see !. 1, &c.) 204 — A+p+'+p =C-+ 180°, | and 6+0+p+tp =C, C and the value of BAQ will equal nh od — p. These refractions p, p', which take place near the Earth’s surface, are, for distinction’s sake, called terrestrial refractions, not to be derived from those formule which are used in Astro- nomy, but by peculiar methods. In the above equation, suppose the objects A, B to be equally elevated above the Earth’s surface : then d=0, and p=9’; *,23+2p=C, C—268 rae and p= Hence, subtract from the angle C, (which is formed by lines drawn from the two objects to the centre of the Earth), the sum of the depressions, and half the difference is the refraction. Suppose the objects to be at different distances from the Earth’s surface, then ptp = C-(6+0), if we suppose (which in these cases is no improbable suppostaenD p to equal ep , then _ C-(6+0) i g Hence, General Roy’s. Rule, (see Phil. Trans. 1790. pp- 242, &c. and Trigonometrical Survey, Vol. I. p. 175.) Subtract the sum of the two depressions from the contained arc, and half t the remainder is the mean refraction. 205 If one of the objects, instead of being depressed, 1s elevated, then, ! | (0 +p')-—(e—p)=C, (p being supposed = p’); : ee 8 ity OSs nae Subtract the depression from the sum of the contained arc and elevation, and half the remainder is the mean refraction: which is General Roy's second rule, (see Trigonometrical Survey, Vol. I. p. 176.). The depressions 6, 0’ can be observed: the only part, then, of the preceding rule that requires explanation, is that which re- spects the determination of the contained arc. By the process described at the beginning of this Chapter, the distance A.B between two stations A, B may be determined, and thence, by an easy reduction, the chord AEH. This, in all instances that occur in a trigonometrical survey, is, from the smallness of the angle C, very nearly equal to the arc that subtends C: call it F, and the Earth's radius r, 7 being = 3.14159, then F: wr :: C : 180°; sets 180° F180 x 3600" rT Tw r Tv Ae F r.sin. 1” For instance, if we take General Roy’s instance, (Vol. I. p. 176. Trigonometrical Survey), in which Feet. F, the distance between Tenterden steeple and Allington Knol= 61777 SHG te ks SPREE pia r, the Earth’s radius be assumed..... = 20970255 206 we have log........ 61777...4.7908268 log. 20970255. .7.3216037 log. sin. 1” 4.6855759 2.0071786. : .2.0071786 2.7830482...... N°. = 607.64; therefore the angle C=10! 7”.64. We may now illustrate the formula for refraction, (see p. 204.) Let the depression df Tenterden viewed from Allington be 4' 1”.4, rh 4 Oa A Allington ...... from Tenterden 3. 16.6, — f ‘wit 9 PP Sip pyee eaeh a UE LER ee EE Fy Eh Vc and, consequently a, LV 24h8 24".8 at , 2 Co EON TGA ai or the mean refraction is about sth of the contained arc. , hearly, This is one result, and it is plain that every reciprocal ob- servation will give a similar one, that 1s, will give the mean refraction some aliquot part of the contained arc. ‘The results, however, differ considerably from each other. That which has been just obtained, makes the mean refraction only a little less than one seventh of the contained arc, whereas the mean result, (the mean of several hundred PLAS U 1S more nearly one twelfth. * The observed depression of D seen from W, and of W seen from D, (see fig. of p. 202.) with the computed distance between D and WW, enables us, as we have seen, to find the mean refraction. Every like observation, during the survey, furnishes data for a like result. The surveys, therefore, of England and France enable us to determine the mean quantity of terrestrial refraction from several hundred observations. But the mitisey of this mean quantity is, in particular instances, only useful 907 The mean quantity of refraction bemg found from many experiments, may be applied to determine the relative heights} of objects in default of reciprocal observations. ‘Thus, in the survey carried on by General Roy, the height of St. Ann’s Hill was found, from that of Hampton Poor House, by taking ath of the contained arc for the effect of refraction. This height, however, was afterwards found to be too great, and as a proof of the great uncertainty in these matters arising from the va- riableness of refraction, St. Ann's Hill, which, viewed from Hampton Poor House in 1787, was elevated 17° 39”, was, in 1792, elevated only 8’ 11”. In order to determine the difference of the altitudes of two objects A, B, from their respective depressions and distances, we must find BQ from AB, or AQ and the angle BAQ; and since BQ, AQ are very small compared with the Earth's radius, it will, in most cases, be sufficiently accurate to find BQ, by finding the subtense of the angle BAQ at the distance AQ. This, (see pp. 176, &c. of the Trigonometrical Survey) is General Roy's method, but it is easy to attain greater accuracy, by finding the values of the angles BAQ, ABQ. Thus Both Depressions. One an Elevation. VBA=A'+p=90°+0 +p =90° +0 +p, ZAB=A +p=90°+0 +p or =90°—e +9, : : 1 useful to a certain extent. If the mean quantity, (o f) should be nae and in a particular instance the value of a should appear to be - we ought to suspect some error to have occurred in the observation, or the ob- servation to have been made under such peculiar physical circumstances, as to require repetition, That, indeed, ought to be practised on every occa- sion, in order to get rid of partial errors, or, more properly, of errors that arise from some unknown cause. Col]. Mudge gives us, in p. 352. Vol. I. and p. 182. Vol. II. of the Tregonometrical Survey, remarkable instances of the variableness of refraction. 208 BAQ=ZAQ—ZAB, BAQ=VBA—VQA; P dear Oo +e AG a) ai Coma - OL ere ABQ=180°—V BA, ABQ=cAB-C; RODS ABs eC Nels Og s GR Ee ier ORAL Fe CG EG ed es a ee Q 2 Q Q° OMe C or = 90 — segeet hea g, +9) BAQ B sin. Benes 1S be es sin. ABQ EO Eto, a pte oon) eA se LE pe ieee Cc Hees ens Bt a cos. (5 + There are very few cases im which we may not, without impair- ing the exactness of the practical result, reject the denominator. ord Since cos. es + — g oO ze é C Oo = COS. —. COS, — sin.— .sin. , g Q 2 we have, by dividing the numerator and denominator of BQ by cos. L(@- 6), the following value of BQ: BQ= 40. gE __ O08. 1 — tan, 2" tan. L@FS} a ne Ve oe dbs Es + tan. & stam, 40 FO + Ket. 209 Since BQ= AQ. tan. £(0'—0), very nearly, e B we have tan. 10’ $0) = 0° Pe BQ Vi ) Ce td Fete ema te Me ; 2 dZ@Fd= AO ssa? Very neatly; BQ consequently, BAQ = Akane Ki . From the formula, then, in its original state, we derive BQ from the subtended angle BAQ, and now the angle BAQ from its subtense BQ. The operation of finding the angle that BQ, or Bb, subtends at a distance AB, when, as is almost always the case in practice, AB is nearly perpendicular to CV, is, indeed, so simple, that we would not have introduced it here, except for the purpose of noticing the circumstances that render it ne- cessary to be performed every time the depression of A below B, or of B below A. is to be computed. The matter is easily explained. In reciprocal observations B is supposed to be observed from 4, and A from B. Now it will happen that it is not convenient to observe the very point B, which is to be the station of the observer when 4 is observed, but some other point b either above or below B. For instance, it is convenient to observe J, the top of a steeple or tower which may be an incom- modious station for the observers instrument when J is observed. .The same will hold good for A.. The place where the axis or the centre of the instrument is, may afford no distinct mark to the observer at B, but he looks at some other point a, above or below A. But the angles on which the preceding’ calculations are founded, are ZAB, VBA.. We must, therefore, reduce the observed angles (2Ab, VBa), by adding to them, or sub- tracting from them, the angles BA 6, A Ba: or the angles which the nina differences of beet Bb, Aa, subtend at the distance AB. General Roys instances will illustrate the preceding formula, D vp 210 At Allington Knoll the top of the staff on Tenterden steeple was de- pressed 3’ 51” by observation; and the top of the staff was 3.1 feet higher than the axis of the instrument when it was at that station. The distance of the stations was 61777 feet. * Again, on Tenterden steeple the ground at Allington Knoll was de- pressed 3’ 35”, but the axis of the instrument, when at Allington Knoll, was 34 feet above the ground. Suppose B to represent Allington, and 4 Tenterden, then 8 = 3’ 51”, Aa= 3.1, =5 35 Bb = 5.5; 3.1 ee ABa —s 61777 . sin. 1’ 5.5 BA bee 61777 .sin. 1” | log. 61777 = 4.7908268 log. 3.1== .4913617 log. sin, 1” = 4.6855749 (D) 9.4764017 9.4764017 (D) (N°=10.35) 1.0149600 — -7403627 (D) 9.4764017 # (No. = 18.36) 1.2639610 Hence 8’, corrected, =: 3’ 51”-+4-10".35 = 4’ 1”.35 8, corrected, = 3 35 —18 .36 =3 16 .64 ate A or Oe sg the hie mix, Cale. 6 eas os 4 oe mona eT * The values of 4Ba, BA6, and of like angles, may be as aly deduced thus : By the note to p. 69, 1 foot subtends 1” at 206265 feet; SP Pe i 5".5 at 206265 feet; therefore at the distance of 6177 7 feet, 5.3 feet subtends an angle 206265 =5".5 x 61777, os 1 8.36. 21) and BQ =61777 x tan. 22".35..... log. 61777 = 4.7908268 log. tans 227.25... 0.06095 ++ 60347542 (No. = 6.69238).......+2.4. 0,8255810 The place, therefore, of the axis at Allington Knoll, is higher than its place when on Tenterden steeple by 6 feet, 8 inches. The mean terrestrial refraction (p), which is represented by C—(6+40') 2 a r) is, in this instance, , that is, 1’ 25”,8, since C, (see 10’ 7”.64—'7' 17”.99 2 p. 206.) is 10’ 7”.64. Second Example. At Allington Knoll the ground at High Nook was depressed MO EST evs bs (0’). At High Nook the ground at Allington Knoll was elevated A RB its (€). _ The height of the axis of the instrument above the ground at each of the stations, was 54 feet (~dH=5.5—dH'=5.5). The distance AQ was 23186 feet. ; ; 5.5 Hence the correction of 6 = 231865 1” ee 5.5 of 23186.sin. 1”? Woe 5.3 Os ae Oe e log. 23186. .4.3642258 log. sin. 1” 4.6855749 | D.OAGSO0T I orl es Ce a onosooe (No=49.04).......... 1,6905620 Hence, the corrected value of & = 46° 53”~49"= 45’ 54” of e= 42 34-449 —43 23 ie ana gran s 238 ik bo and BQ= AQ. tan. §(8’-e) = 23186. tan. (44’ 38”.5), 212 IOS POLO Meeks) ee sgt fet 4.3652258 Dee tATL) Ba SS Os Wetniualy «> ss Soela per lik ¢ 8.1134909 SOL ie oie teis cy: die slater +++ 2.4787 167 Hence, Allington Knoll is 301.1 feet above High Nook, which, added to 27.6 feet, the height of the axis at High Nook above low water gives 328.7 feet, the height of Allington Knoll above low water. To find the contained arc and the refraction, we have op: CAI BG ete te on! ae Ba 43652258 (see p. 205.) log. sin. 1’--log. 7 ..........2.0071786 (N°, 228.5).........2.3580472 Pd Pe ane I 1S dp = = OH A 2 9 2 ee SS PB Of gi 1 . d Lights he Ly 4h an Cc 998.5 ; near y 6 This result is different from the preceding one, (p. 207.) and the quantity of refraction is so variable even under circumstances apparently the same*, that it is not safe to rely for its determi- nation on a single observation. The reciprocal depressions of A and B, if observed at different times, are worth scarcely any thing, since, in the interval of the observations, the state of the air, with regard to temperature, weight and other circumstances, may have changed. One source of uncertainty, therefore, may be got rid of, if two observers, at the same hours, should at A and B observe the depressions of B and J, noting at the same time the barometer and thermometer, wind, &c. as Col. Mudge caused to be practised. ! But there are cases in determining the refraction, when we * See Trigonometrical Survey, Vol. II. pp. 181, &c. See also the Tables of Terrestrial Refraction, deduced from observations made during the Surveys of General Roy, Colonel Mudge, &c. Trigonometrical Survey, pp. 179, 349, 383. Vol. II. pp. 176, &c. 213 may dispense with the contemporaneous observation of the reci- procal angles of depression and elevation. For instance, when from one station we observe the depression of another, the dist- ance of the stations and their respective heights above low water, (or above the height of the mean tide) being previously known. Thus, by levelling, (a much more certain mode than by calcu- lation from observed angles of depression) the station on Dover Castle was found to be 469 feet above low water (spring tides). The top of the balustrade of Calais steeple, (the pot observed) 1404 feet, and the distance of Calais from Dover 137458 feet. If B and A, therefore, represent the above points of Dover and Calais, we have . BQ = 469 — 140.5 = 328.5, BQ now, BAQ = AQ cane? nearly, AQ and C = Perey cana near 7 AGC. sin, 1/7 log. 137455...... 5.1371606 log. 528.511. 535 2.5165354 log. sin. 1”....... 4,6855749 CS) earpiece ae 9.8227355 SHIN (5) ve sae sens 9.8227355 (No. =494.18).. 2.6937999 git) eras 0.4515857 (DY Aes FON 0.4515857 log. G’s rad. ... 7.3216037 (NOl=2 1349) 0. Fe 3.1299820 we have, therefore, 2 BAQ = 8’ 14".18...... 8' 147.18 ZC = 22 20.) wala 14.5 sh ”. 2 BAQ+ Botte estes 19 29.23 but 4 VBA= 4 BAQ+ 2 BQA C = 4 BAQ+90° +7, the depression of A, therefore, which, were there no refraction, would be equal to 4 VBA—90°, would be 214 £4 BAQ+ 5 Takes Shs = 19’ 28".23 but the depression observed was.+.eee-++- 17 59 and the difference, or effect of refraction... 1 29.23 Pe VPRO" B8 1 and Fue maennn Se nearly, and any similar observation made at Dover will give the actual refraction at the time of the observation. Under circumstances like the preceding, and with a similar result, is the case in which the horizon of the sea can be seen from a station. Thus, in the station near Paddlesworth, the depression of the horizon of the sea was observed to be 26’ 27”. Now the height of the station (BQ) is 642 feet; and since C BA, a tangent to the sea at A is perpendicular to the radius CA; “. VBA (=90° + depression) =C+ 4 BAC=C+90°; therefore the depression = C, (were there no refraction) ; 215 AC 209070255 | but cos. C = Cp 20070255 +642 9999008861", * In the common Tables of logarithmic cosines, &c. the same seven places of figures represent the logarithmic cosines of arcs from 27’ 49” to 27' 35". By such Tables, therefore, and using the above formula for cos. C, the determination of C would be uncertain to the amount of six seconds. By transforming, however, the formula, we may get rid of this uncertainty, and use the common Tables. Thus, (see p. 42.) C 1—cos.€ 2 ~ 1+cos. C’ AC , but cos. C = TB AK tan.” (making / to represent BQ) ; J 1—cos. Ch h : : 1-+-cos.C 89 2r-+h on C ee aLVe Vx fi- ; -. tane= = = * +&e, ° 5 2Qr Gy Qr° =) Now t= 642....... log. = 2.8075350 r= 20970255....log. = 7.3216037 Q log. 0.3010300 76226337 5.1849015 _ If we stop at the first term, we have , 2 log. tan. — 10 = 5.1849013 ¢ and log. tan. 5= 7.5924506 and < = 13/ 27” C = 26 54. If we take account of the second term, or suppose tan. S=a/p —— ' en or Ar 4’ we 216 the natural cosine of 26° 54” the dip, but the apparent dip was 26° 27”; .. 27” is the quantity by which the horizon was elevated by refraction. In the preceding instances, the elevations, compared with the Earth’s radius, are so small, that little more is required to be done than to find the value of a line which, at a given dist- ance, subtends a small angle, or to find a small angle subtended by a small line, (see p. 207.). General Roy, therefore, in his computations used no exact formule for finding the height of a station from observed angles of depression, or for finding from a station of known height how much below its horizon, other stations appeared to be depressed. The exact formule of com- putation, then, with which the foreign Treatises abound, are, in almost all cases that occur in a Trigonometrical Survey, formule of curiosity. They are tools finer than the work to be done with them requires. Thus, in the Example of p. 211. the value of BQ computed from AQ. tan. 3 (6’— 0) was found 6.692 feet, the exact formula is . BQ=AQ pail Ae ae C —o cos. (- a ——-) 2 2 but ©... 5 9”.82 AQ ho, 61777»... 47908268 TOSS 8ine 52s x's As be yee —6.0347542 aah 99-95 -8255810 PUTT oO: QO LT Sadak leh ee a odes aes cos. = 9.9999995 (No. 6.69239)...... 0.8255815 we shall have log. tan. = 7.5924489, C 5368 d == 13’ 96” ——— erent 20" raaR? so that it is, in practice, quite useless to go beyond the first term. 217 which is a result, in a practical point of view, the same as the former which was fgets For it would be absurd to be th of a foot, especially m cases in scrupulous about the aad which, from the uncertainty of the angles of depression, the probable error in the height will exceed 5 feet. Remarks, similar to those that have been just made, may be applied to the methods of computing the small corrections of the angles of depressions. Thus, in p. 202, B being the point observed from A, but 6 being the place, above B, where the centre or axis of the instrument was placed, the correction of the angle 2AB is BAb. That was found, (see p. 210.) by simply finding the angle which a line Bb subtends at the distance AB. This, Ainge is not strictly accurate, smce Bb is not equally inclined to AB and Ad. Let us deduce a formula that shall represent more accurately the angle BAD. Let the angles BAb, ADB be represented by A, C, the sides Bb, AB, Ab by a,c, b, then b? + Zz g Tae. 6 cos. C = ————_—_,, 2ab b° +c? —a’? cos. A = ————_——_ Q2bec ; _ (a? —c’) ie Qbe _ +8 —2ab.cos. C gh Qbc b—acos. C ’ c a, but sin. A = ~ sin. C; € a sin, C .. tan. A b—a.cos. € Er 218 7 aD — sin. C Hg ta 1 — = cos. C b n.sin. C —er q ae en I—ncos. C b Take the differential, or fluxion of this expression, and dA __ n.cos. C—n dC 1—2n.cos. C+n*" In order to expand 1 EES SE A Ee A oe N—1 Piro 7 kat Clnd a Qn cos. C+n*)~", compare it with (r?—2rr’ +7°)~™, (see p- 118.) / ——- — then “Fr = 15°7 =n om = 1, ; 1 and the first coefficient = Ll+n’-+n'+&c. = area) the 26 eu ® A ee By 5 (1-+n* +n‘ + &c.) = = 23 —n 2 raat oR 4 8 Pag coeanGe esi rnias No La lu be lig eseeeere — eee 1—n &e. Hence, dA .cos. C—n? Tie Sa {1+2n.cos. C+2n’.cos. 2C + &c.}, let 2n™ cos.mC+2n™+! cos. (m+1)C+2n™**. cos. (m+2)C, be three consecutive terms, then since, generally, 2cos. C.cos. pC =cos. (p—1)C + cos. (p +1) C, from the multiplication of 2.cos. C—n’ into the three above- 4 219 mentioned terms, there will be produced three other terms, and only three, involving cos. (m-+1) A, which will be © n™*1 cos. (m+1)C, —2n™*, cos. (m+1)C, n™t*.cos.(m+1)C, the sum of which =n™*! cos. (m+ 1) C —n™**. cos. (m +1) C = n+! (1—n"). cos. (m+1) C. Hence, since the constant part n° —n” disappears, dA ; qc = 7. COS. C+n°.cos.2C +n. cos.3C+&e. sine C On? sin. 2C — nr’? sin. 3C and A =2.——— + — .——— + —- sin. 1 2 sm.il 3 sin. 1 To apply this to the instance of p.210, which has been already solved, we have Bi at Me eee Bape ion = 90° 3’ ”, n= aS 61777” C=90°-}+-depression = 90° 3’ 51”; 4 _*' 5.5 © cos. 3’ 51” 6.00 NA sine 40" A ie aia Giyr7) an et Now 10g. (5:5. .2... 5. .7403627 log. Ol7 7a. 47908268 5.5 \3 5.9495359 (a). .log. (a7) 1.8990718 (c) log. cos. 3’ 51”...9.9999997 ... log. sin. 7°42”.. 7.3502165 lop. sin. 1’... 4. A OSSS74 9 sie 5's sin. 2”... 4.9866049 5.3144248 (6) 2.3636116 (d) (a)-+-(0)......- 1.2639607 (c)-+(d)... 4.2626834 (10 borrowed) ; .. the first term = 18”.364, the second = 00000183, the second term, therefore, may be neglected, and the first gives the same result as was obtained in p. 210. As far then as practical utility is concerned, it is quite un- necessary to apply the preceding formula. It will suit, how- ever, other purposes. 220 In what has preceded, we have shewn the use of what are called the reciprocal angles of depression and elevation in de- termining the absolute heights of stations, and, which is a matter of more importance, the quantities of terrestrial refraction. We shall now be enabled to investigate other points. If the places on the Earth’s surface, which are alternately the observed objects, and the stations of observation, were on a spherical surface, the apparent depression of one station, seen from another, or the dip, setting aside the refraction, would be half the arc contained between them. But this can scarcely ever happen. The objects are almost always at different heights above the level of the sea. The observed angle, therefore, be- tween two objects, is different from that which they would sub- tend, if they were in the horizon of the observer. But to this latter angle it is necessary to reduce the observed angle, in order that the angles and sides of the triangles to be computed may be those of spherical triangles. Let O be the station of the observer, A, B, the two objects, then the angle subtended by them at O, is AOB, which AB measures; but, if Za, 2b are each=90°, ab, and not AB, measures the angle a 26, which is the angle required. What to be done, then, is, from the observed angle AOB, and the observed 221 zenith distances ZA, ZB, to find the angle a <8, or, which is the same thing, to find the difference of the angles AOB, aZé. By p. 145, sin. .sin. 2A.sin. 2B = sin. $(AB+ Z2B—2ZA).sin. 3 (AB+ ZA— ZB). Let AB=a, Aa=H, Bb=h, and angle AS B=A, then sin? .cos. H.cos. h = sin. 3 (a+H—h).sin. $@+h— A), _ from which A may be computed. Let us take M. Delambre’s instance given in the Ist Vol. of the Base Metrique, Observed Angle. Zenith Distances. @= 51° 9! 29.744 Aubassin..... 91° 32' 45", H=1° 32’ 45” Phas A So Aag Bastide....... QUT 1 Ode bee gu LO A= 1 710 Log. Sines. 4 (a+ H—h)... 2.08. Zh LALO S OP ee ore 9.6385995 £ (a+ H—h)......0e- QS hips 2 sy «ees os 9.6318474 (20 added)... 39.2704469 (a) Log. Cosines. ) 5 hee eh ey | iA i: Da aR a A 9.9998419 Ee bP AV ART MAB Gi Stn a) 9.9999171 19.9997590 (6) (a)—(B)..,... 19.2706879 (25 SU BOR) cass Ree ai 9.6353439.5 A “63 = 35 35 85, and 4 = 51 10 17.16, the reduced angle. 222 There are no Examples of this kind in the Volumes of the English ‘Trigonometrical Survey, because the instrument used by Roy, Mudge, &c. (a Theodolite) gives, by means of its azimuth circle, the horizontal angle between the objects observed. The French observers used a repeating circle, which gave solely the. angle contained hetween the objects, which angle would almost always lie in a plane inclined to the horizontal plane of the observer. The reduction, therefore, of the angle to its hori- zontal angle, was an operation to be performed at every obser- vation. It became, therefore, desirable to abridge the operation, which was effected by means of Tables, giving a small correc- tion to the observed angle: the Tables themselves being con- structed from the following approximate formula. Let x be the correction, then cos. (a +.x).cos. H. cos, h= cos. a—sin. H.sin. h, when A is small, 2 sn. h=h, cosh=1 — oe nearly, sihee—wv,* cos. cel — —= 1, if x be still more minute than /; *. (cos. a— «x sin. a).(2 oa =) (: ~ —) =cos.a— Ah, Hh—< cos. a. 4) sin. a and + = neglecting the terms that involve xH*, xh*, Hh? x may be thus differently expressed, Hh=z {(H+h)?—(H—h)’}, 5 (A? +) =F f(H+hyY + (H—hyt; H+h)-(H—-h ee ( i ) ( ) —LS(H +h +(H—2} 222 sin. a sin. a 223 > CHARA) legos sa “a ae hy? t+ cos. a4 4, sin. @ sin. a H+h H—h a ers ae ~_— -_-C> ig — if x be expressed in seconds. If this approximate formula be applied to the foregoing Example, we have H= 1° 32’ 45" A= 1 7.10 = 119° ARS ope Si APOE EE Sloe nee 73620300 | 5=2 PRY We PY MR ys Win GU log. tan. .... 9.6800387 log. sin. 1”... 4,.6855749 - (1st term 53’.412)...... 1.7276436 ae er Oo BD Age icwens: ZO0 eo ne eo: 2 log. 3.20. 25,77 01568 >= 25. SA AA LQ Peo acute ghiall. log. cot.... 10.3199613 log. sin. 1”... 4,0855749 (2d term 5”.966)...... 0.7756930 Hence the correction = 53”.412 — 5.966 = 47”.44, and the reduced angle is 51° 9! 20".74 +. 47.44, that is, 51 10 17.18, which differs only %th of a second from the result obtained by the direct process, (see p. 221.). By this formula, then, or by the Tables constructed from it, may the angles observed between two objects, at differed heights above the observer’s horizon, be reduced to horizontal angles. The latter angles are immediately given, or rather in- strumentally given by such a Theodolite as Ramsden constructed for the English Survey, and which, for distinction’s sake, was called the Great Theodolite. The French observers made use 224 of a repeating circle of small dimensions, and easily portable, and by the means of a great number of series of observations, they hoped to get rid of the inaccuracies of individual observa- tions made with it. Formule of computation, like the pre- ceding, are essentially necessary to the use of such an instru- ment. By what has preceded, then, it is plain that, either directly by instruments, or by the intervention of computation, we can obtain angles subtended by objects such as they would subtend if projected on a sphere. And by such means, and by com- puting the arcs belonging to the measured and computed chords, the series of triangles DWC, WCF, &c. would become a series of spherical triangles; and it is now to be considered whether the operation is to be carried on by solving such triangles by the rules of Spherical Trigonometry, or whether we should endeavour to abridge the processes of solution by approximate methods. It is to be observed, that the triangles of which we have spoken, although strictly spherical, are very small spherical triangles differing very little from rectilinear ones. Thus, in p- 205, the distance between Allmgton and Tenterden, (which we may consider as an arc between those places) was 61777 feet, which is only 10’ 7”.64, and the distance between Dover and Calais, a considerable one, (137455 feet), 1s represented by an arc of 22’ 20”; and, if we wanted a practical proof of the small difference between the triangles of the Survey and rectilinear triangles, we should find one in the observed angles, or rather the reduced observed angles of those triangles. Thus, in some of the first triangles mentioned by General Roy, (see Phil. Trans. 1790, pp. 172, &c. also Trigonometrical Survey, Vol. I. pp- 139, &c.) Observed Angles. Observed Angles. Hanger Hill Tower .,.42° 1’ 32” | Hundred Acres ....53° 58’ 35”.75 Hampton Poor House.67 55 39 | Hanger Hill.......68 24 44 King’s Arbour ....... 70 148 | St. Ann’s Hill ....57 36 39.5 179 59 59 179 59 59 .25 225 also in the French. Survey, (Base du System Metrique, PP- 513,. 535.) Observed Angles. Observed Angles. Dunkirk,.,....42° 6° 97.34 Violam. »5 092° 10,434.31 ‘Watten.......74 28 44.88 Aubassin.,...83 15 22.17 CPE elit 63 2h7°5 .78 Bastide.......45 34 29 57 — 180 O O 180 0 37.05 In the two, first instances the sum of the three angles is less than 180°, in the third equal, and in the fourth it only exceeds it by 3”. Theoretically, however, we know that the sum ought always to exceed 180°. The above circumstances, therefore, must arise, not solely from the errors of observations, which were made with the greatest care, but from the spherical excess being nearly of the same magnitude as those errors, or, in other terms, from the angles of the spherical triangle, differing little more than a second, or parts of a second, from rectilinear angles. That there ought to be a spherical excess, or that the sum of the three angles of a spherical triangle exceeds 180°, is plain from the expressions of p. 194. Thus tan. 4 (5-90 } =V ftan. £ -§ . tan. £(S —a) tan. 4(S— b) tan. 4 L(S—c)t, atb-+c Bers a positive quantity; .*. tan. +(S—90°) must be positive, or S must be greater than 90°, or A+ B+C(=28) greater than 180°. but since § = , the quantity under the radical sign 1s The spherical excess, then, exists in theory, but, as far as the preceding cases prove, is not discernible in practice. The fourth case, indeed, shews an apparent excess of 3”, but the first case shews a defect, and each may be attributed to errors of observation: for, with the instruments used inthe Sur- vey, an error of one or two or more seconds might easily occur. Can the knowledge, then, of a theoretical truth be made, in Ip 226 cases like those we are treating of, subservient to practicaf utility? The use that General Roy made of it, (the only use that has since been made of it,) was to correct his angles of observation. Thus, in the first triangle, the defect from 180° of the sum of the three angles was 1”: but there ought to have been an excess above 180° of .23. The observations, then, altogether were wrong, by 1''.23. If each observation were supposed to have been made with equal care, then the obvious mode of correcting each would be by adding to it one-third of 1.23: and this was, in fact, done, but with no certainty of procuring an exact result, as it is plain from the principle of correcting the angles. As some check and means of correcting the angles of observation was, however, thus obtained, it became necessary to compute the spherical excess. In order to compute it, we have two convenient expressions : one requiring the knowledge of the three sides of the spherical triangle, the other that of two sides, and an included angle. Thus, by the first, (see p. 194.) tan. $(S—90°) = V{ tan. $ S.tan.5(S—a) tan. £(S—d) tan.5(S—c)}. Take the first triangle mentioned in p. 224, in degrees, (see Rule, p. 205.) Feet. AT oe aes EP DI ADALS Wide 8 Oe" 269".54 BM ay kOe SBA bbe ies 378.3 PRE ae ue ily Ak PS ie RRA 373 bash bbe: Du Sale as, ae 1020.84 s Log. Tangents. = or eae ee TOaU Ge it Ae ge 255.21 ....-. 7.0924566 BC Size ai casio (bees tid dg clsich 3 bedi 120.43 ...... 6.7661978 BSD) iit stds at aes BRN wake aE 66.06 .....+» 6.5055106 E(S=0) exjdniess dean ween dee! Anes OSFiulshiat 9 6.5218219 , 2)26.8859869 13.4429934 therefore log. tan, (S—90°) = 3.4429934, 22% but the Tables do not contain the tangents of so small an arc which is less than 1”. We must have recourse, then, to Mas- kelyne’s Rule, (see Taylor’s Logarithms, Introd. p. 22, also the Appendix to this Work): accordingly i 2 3.4429934 5.314425 1 8.7574185 No. = .0572; S : e On 90° = 0".1144, and S— 180°, the spherical excess, =0’.228. This is a very small quantity, and obtained with considerable trouble, because the tangents of such arcs as 4 15".21, 2’ 0.43 (=255".21, 1207.43) can only be found from the Tables by proportion. It would be convenient, then, to modify the expression for tan. £(S—90"), and to render it more easy of application. Now in cases like the above, the lines a, 6, ¢ compared with the Earth’s radius, are very small: the lines or arcs, then, re- atb+c atb+c ats aa? a A &c. are also very small, and their tangents are nearly equal to them. Hence, presented by ’ a S S—a 1 prs Belg Fee Yer ee ONE LYE tan. 3S =; tan. 2(S a)=4. eC Ges f(t Iles — 1 BRNO ERR .. tan, £(S—90) = Vs pn SE OE and since tan. 2 (S—90°), which is very small,=4(S—9Q0°) sin. 1”, we have .(S — 90°). 7%, sin. 1” = VjiS.5(S—a)$(S—b)F(S—oc)}; (A+ B+C—180°).r*.sin. 1" = V{S.(S—a).(S—b)(S—c)}. Let, therefore, the excess be denominated e, and the quantity on 228 the night hand side of the equation, and we have, since, | log. r°.4.44. . 14.6432074 log, Sif. Ley hase 4.6855749 9.3287823 log. e= log. # — 9.3287823 * This, it is plain, is a much shorter method of computing e, than the former one of p. 226, and, although an approximate method, quite exact enough for the occasion: to solve the former in- stance by it, we have CER SU Habe ny das Car 27404:.7 Dre RIG th Re CL 38461.12 Gis ce oS ak eR sikh OS 37922.57 *. S=at+b+c = 103788.39 . Logarithm. — Ss oe 3 ELM tae eh $1894.196. 2004. eel 4.71512 Ss | eae $e wees » 24489,.495.......26. 4.38898 S *® mic Dri maladie i Rn cir ein eo 4.12817 ae Gadd 6% Hieis 13971626 -<...% Sees 4.14525 2)17.37752 8.68876 9.32878 (No. TPA FOr reee AG 9.35998 e == 0’.229, as before. ne een See ne env a * Tf we take log. r = 7.3205995 2 log. r 14.6411990 4.08557 49 9.3267739 log. e= log. « — 9.326774, which is General Roy’s Rule, see p, 138. Trigonometrical Survey, Vol. I. ; 229 The quantity W{.S.(S—a) (S—b) (S ~c)} is, (see p. 28,) the area of a rectilinear triangle, of which the sides are a, b, c. The logarithm of the spherical excess, therefore, 1s to be had by subtracting 9.32878 from the logarithm of the area of the tri- angle considering it as a rectilinear triangle, of which the sides are a, 6, c. What is gained in facility of computation by this last step is this, that in many cases it may be more convenient to compute the area from other data, than those of the three sides: from, for instance, two sides and the included angle. Thus the distance from Violan to Aubassin is 18283 toises, Haat. rove ol te. to Bastide.. . 25423 and the observed distance from Violan, of Aubassin, and Bastide 51° 10’ 11”; therefore since a toise =6.396 feet, we have 18283 x 25423 x (6.396) (x) the area = san. 1 TO” Fi" 2 Logarithms, BOTOOS Hes ee es 4.262047 Qe Sis pak th 4.405226 (6.39007 22 2. 25 16817 BIN ele On Lore cara 9.891541 10.170631 re are Ps Pie .301030 9.869601 : 9.32878 (BoA) ias Ais. a his 0.54082 e = 3" 47. This spherical excess is above the average quantity; but there is still a larger in the 23d triangle of the Trigonometrical Survey, (see Phil. Trans. 1808. p. 428, and Trig. Survey, Vol. I. p- 181.) ) 230 Dunnose .... 55° 43' 7” Butser Hill .. 67 12 22 Dunnose from Dean Hill ...48 4 32.25 Butser Hill. .140580.4 Dean Hill...183496.2 Logs. sages tg alidereg eect 10.027648 -70290.2. ..- 4.846894 9.326774: sin. 55° 49! 77.6... oe O917128 (5.022) 700874 20.027648 Hence the spherical excess is very nearly 5”. By the above method and Rule, (for so it may be called), General Roy computed the spherical excess: and its application is not tedious. M. Delambre, however, thought it worth the while to render the computation of the spherical excess still more commodious, by the means of Tables, the-principle of the construction of which, may be understood from the formula of page 191. e being the spherical excess, A+B+C- 180° = A+B+C € Rue Ta ae A+B+C l si ( et Ere «” tan. — consequently, (see p. 191.) a Bes tan. —.tan.-~.sin. C 2 2 = [2 RE SION RY ETDS. a b 1 + tan. ¢ . tan. 5° oO & compare this with the formula of p- 219, and we have € a b sin. C a 6 sin. 2C > = tan. =. tan. - .———;, — tans tan.” » ——— + &e. 2 9 2 sin. 1 Q 2 sin. 2 b which series must, from the smallness of tan. 5 . tan. - 3? quickly converge. : 231 If we apply this formula to the instance of p. 229, we have In Feet. log. 5 Mase ee Riley Waste log. heer eee 4.91010 Log, 7;. Sil 1 vy ee = BOTT Ge veces vitae teas wee Un LT, (57S. 1) os ceers 2.75974 (799.7) 20+ +02 2.90293 log. tan. 5 wes CY Solel}. paces oy oe cecn 2s) (saab oe log. tan. 3 god > CUS LOINT Yisyis om ole este Sages pie DOORL 102, SIs viens oD Wick COND LEE ae PAS Ne Sa! SS gs 4.92495 log. sin. 1”,..... 4.68557 (No. 1.7354)..... «23938 € was 3 = 1.7354, and ¢€ = 3.47. The *first term, then, gives 3.47 for the spherical excess, * We may, in cases like the present, without any loss of practical accuracy, avoid the trouble of taking out the log. tangents of arcs, such as 9’ 35”.1, 13’ 19".7, by assuming, instead of tan. = ; tan, ; 5 : =: In such an assumption, we have ; a b log, Q ee eered 4.76692 log. 2° e@eese 4.91010 log. G)’s rad. 7.32160 7.32160 7.44532 7-58850 7.58850 log. sin. 51° 10’ 11”...9.89154 492536 | 4.68557 O17 97 ues bu i 23979 , € = 3".474, 232 which is to be held as the same result as that obtamed by the exact formula of p. 231. And if we investigate the value of ithe second term 6 sin. 2C = tan. wal tan ~~ A ud 2 sin. 2 5 it will be found 1 Sh 0.000012, a value altogether insignificant. The smallness of this last value arises from that of tan © ROM Ae the product of: tan. < . tan? f ; would give a 2 g 2 2 quantity still less: but the arc differs from its tangent by terms that involve the cube and higher powers of the tangent. Con- sequently, with less error than arises from rejecting the term 2b sin. 2C . tan. ca Chora aT a tan.? - ; 2 2 sin. 2 € we may compute s from € a 6b-sm. C we eb Se AEE Se Q Qr Qr sin)” ab sin, C | or ¢ = — .—— Or sin, 1”? y being the Earth’s radius, which agrees with the result of p. 228, 3 ) ab. and with General Roy’s Rule, since eae sin. C expresses the area of a rectilinear triangle, of which a, b, are the two sides, and C the included angle. The approximate method, (and General Roy’s is no other) has now been derived from those exact formule which express, in terms of the sides of a spherical triangle, or in terms of two sides and the included angle, the excess of the sum of the three angles above 180°, whatever be the magnitudes of the sides. 233 There is no case in a Trigonometrical Survey, that requires the exact formule: the approximate formule are always sufficient. It is rather curious to observe the kind of consequence that arises from the application of these approximate formule to the cases that occur in a Trigonometrical Survey. By theory it 1s certain, that the sum of the three angles of a spherical triangle is greater than two right angles. But this is rarely verified by practice. It frequently happens, and (see p. 224.) one or two instances have been given of the fact, that the sum is less than two right angles. ‘These errors arise from the imperfection of that part of the practice which depends on the observation of angles. But the other part of the practice, that which gives the — values of the measured sides of triangles, or the values of com- puted sides from measured lines, supplies the corrections of such errors. The tangent of one-fourth of the spherical excess is equal to VY {tan. 48 .tan. + (S$ —a) tan. 1(S$—b) . tan. i(S—of, which is always a positive quantity: so are b sin. C tan. -.tan.-.— 7) g 2 .sin. 1 a0 EE fats C, r # € ; the approximate values of 3) and ¢, and from which expressions, either the one or the other, the excess may be always computed with an exactness more than is required. One point, however, remains to be cleared up. The errors in the observed angles A, B, C are ‘such, that the spherical excess can never be known from their sum. But A, B,C are used in determining a, b, c, and must impart to them, in degree, at least, some of their errors, and, of course, must impair the exactness of e, the excess computed from a, b,c, or from a, b, and the included angle. This must be admitted: the real. point Ge 234 of enquiry, however, is, what error will be produced in the spherical excess, by the probably greatest error that will occur in observing the angles. In the preceding Example, let the error in observing the angle be 1’: or let C (instead of 51° 10’ 11”) be 51° 9! 11”. then log. tan. 7 eee: 7.44524 b log. tan. 5 HELA PRIA 7.58817 log. sin. 51° 9! 11”... ... 9.89144 compl’. log. sin. 1”. .....+ 5.31443 23928 .. ..(No. = 1.735), * consequently, « = 3”.47, in the former case € was = 3’.4708, * This may be solved as one of the cases of ‘Errores in mixta mathesi.’ Thus b cos. C.d€ ° tan. red RR IT Hee See SITS A Sea Of b e tan. ~ .COS. CMe emer" 4 : sin. 1” OTR WIS PWIA b . tan. 5 C08. C. 60, ; log. tan. Gites eet seers 7.44524 b log. CAs Gece cece cece cereeees 7.58817 oor Sime il Oud 7h re std ce w's based el ew aun 9.79728 BOS DO ec ciaiy be ee a eietn'e ie pa a errs 22) 1.77815 (.000406) ...... 6.60884 é € “5 —d(§)= 1.7354 — 0004 = 1.735 the result in the text, 239° so that the difference of the two results 1s, practically, insensible, whilst an error of 1’ in the observation is, probably, greater than. will occur, when we consider that what is called an observation, is the mean of several. We are thus enabled to know, we may say, with certainty, the excess of the sum of the three angles of every spherical triangle, (such as the process of triangulation presents) above 180°; and, consequently, to know the sum, or the result, of the errors committed in observing the three angles. ‘The next step is to correct those errors; and, if it should be believed, that each observation was made, under nearly the same circumstances, and with equal care, the simple and obvious mode of distributing the correction, would be to assign one-third of it, to each ob- servation. ‘Thus, in the Example of p. 225, for which the spherical excess (¢€) has been computed, and found equal to (see p. 229.) 3”.47, the sum of three angles was 180° 0’ 3”.05, which is too small, since the sum ought to have been 180° 0’ 37.47; therefore the sum of the errors is equal to 3.47 — 3.05 = .42, one-third of which is .14; the observed and corrected angles, therefore, will be Observed Angles. One-third of Error. | Corrected Angles. Violan ass 3 DIS LOVEL TSI + 14 51° 10° 11.45 Aubassin .... 83 15 22.17 + .14 PRS He wes Bastide...... 45 34 29, 57 +. .14 45 34 29.71 180 0 3.05 180!) 05%) Sea? The angles of observation, being thus corrected by means of the computed spherical excess, are fit for calculation: but the: calculation will be that of the sides and angles of a spherical triangle, more tedious than that if the triangle were rectilinear. If we diminish the above corrected spherical angles, each, by one-third of the spherical excess, the resulting angles must, as it is plain, be those of a rectilinear triangle: thus the spherical excess being 3°.47, one-third is 1°.1566, &c.; therefore, we have 236 Corrected Spherical Angles. | One-third of Excess. Rectilinear Angles. .- 51° 10/ 11".45 — 1.566, &c. 51° 10/ 10.2933, &e. Baio. 22.31 — 1.566 83 15 21. 1533 45 34 209.71 — 1.566 45 34 28. 5533 1s0 0 O The question, now, is, what use can be made of such a re- duced triangle, since we are ignorant of the values of its sides. It is worthy of notice, that the mathematicians who were em- ployed in the early geodetical operations, having obtained, by the process just described, reduced rectilinear triangles, considered their sides to be the same as the sides of the original spherical triangles. M. Delambre says they were led to this inference, a just one as it has since been proved to be, by a kind of instinct. Without calling in, however, the mterference of such an expla- nation, it is probable, that the mathematicians of whom we have spoken, were persuaded from instances, or examples, of the justness of their proceeding. The subject, indeed, had no diffi- culty. It was only necessary to take an extreme case, (one In which the sides of the triangle were large) to solve it by the — rules of spherical triangles, and to compare the result with that obtained from the rectilinear triangle, the angles of which were reduced, (see p. 235. |. 30.) from those of the former, and the sides of which were assumed equal to the sides of the spherical tri- angle. The matter, however, did not rest here: for M. Le- gendre, in 1787, shewed that the assumed process was a just one, or that ifs errors, in practice, were of no account. ‘The following is the demonstration of what is called | Legendre’s Theorem : _ Let A be the spherical angle to which the arc a is opposite: 6, c are the two remaining arcs or sides. Let A’ be the angle of x a rectilinear triangle of which the sides, equal to the sides of the spherical triangle, are a, 6,c. Let A==A’ +a: then, r being the radius of the Earth, ; r r ri. cos. (A’ +2) (=cos. A) = ———_______—__ iy sin. — sin. — r r i the numerator of this expression (F’) 2 40 = ] — —~ + ———,, — &e. Sue ols ay b° b ; e cf i TI xe.t 1 — ee ( Breas ant whorl nidist wi onside c.} Pos ala —Ob ec —b— c ar oR OTe saa moe ee e—a (6? +c) - 2a’ 46° ¢° my Or 2.3 .4r* 243.474 ie b? +c? — a* ‘a (+e —a*). (6 +e +a) 4b ¢° Kah 27° 2.3.47) 8.3.47"? neglecting the terms: that have in their denominators higher powers of r than the fourth. The denomimator = sin. - . sin. — r r b° c . das 8 sg & ; ve ae 2.3059" c} bc {. b? + ¢° =h- i. r Hence, the expression, or fraction Ff, = r’ fi Bey b?+-c’—a? Oe ere a’) (b +c +a") 46° ¢’ be eee Oke PX sik 2.3.47" at casual (rejecting, as above) 238 BP+e—a b+ce—-a ; 4bc Boy fof. CA eae pase AEA SS EE Be 2c°—(b* a bo quest oe? > Se hataiearaiee Shar M2 SABRC. Te NO ed a eine o +te—a (6+ce—a’yP be i 2be 2.3.4.b¢.r 2.37r° ue B--ec® — a" A bc {(Ateue) i} ne 2bc 2.3.77 — 2be : Q 2 bit es ~ = cos. A’: 2bec therefore making cos. x=1, and sin. y=2, we have X : beta cos. A’—sin. A’.x=cos. A’ — aiGe sin.” A’; .3r 1 bc.sin. A’ 3° Or c= € . : re i being the spherical excess. The Theorem, therefore, although not mathematically true, is true, as far as regards all practical purposes, since the denomi- nators of the rejected terms involve r* and higher powers of 1, and, if retained, would affect only the 5th or 6th decimal figure of the result obtained without them. The spherical excess, therefore, which before enabled us to correct the errors of the observed angles, and to prepare the | triangle for solution as a spherical triangle, may now be em- ployed to reduce the spherical, to a rectilimear triangle. But these are matters of convenience, not of essential im- portance. ‘There are other methods, quite as simple and obvious, as the two we have discoursed on, for mathematically conduct- ing the process of triangulation. For instance, instead of solving the spherical triangles by the rules of Spherical Trigonometry, or, of reducing them to certain imaginary rectilinear triangles, we 239 may make the rectilinear triangles, the sides of which are the chords of the spherical, the subjects of investigation. The only point of difficulty in this mode, is the determination of the angles of the chords; for the things given, or determined by observa- tion, are the spherical angles. It is necessary, therefore, by some formula, or correction, to deduce the former from the latter angles. This technically is called The reduction of the angles of a spherical triangle, to the angles formed by the chords. In this, as in the preceding instances, it is supposed that the sides, arcs or chords, of the triangle, are very small relatively to the radius of the Earth. Let C be the spherical angle, C — the angle formed by the chords, a, G, y, then, . 9g A 2 gO nae ANC wes. sin.” ; + sin.” 5 = sin.” 5 cos.(C— x) = Maat. ds Sigs cauy 0. gp RARE 4.2.sin. =. sin. — 2, 2, If we divide this fraction by 2, the numerator of the result- ing fraction will be Noah Ch ore D bree’ 2 sin. ~ + 2sin.2- — 2 sin.’ - 2 2 a ale c =2.sin.” bray 2 sin.” ant Fees ] (: an 2 sin,” -) fe ig Dies 53 2 . 9 ey ey ( Ss =9 sin. 5 + 2 sin." 5 — 1+sin.a.sin.0.cos. C-+cos. a. cos. 6 Deane 4b me UL Met 4 ‘ig hye! = 2 sin.” eine sin.” 5 Lt € — 2 sin.” =) (-2 sin? =) +sin. a.sin. b.cos.C : Bio 6b write a ; b b = 4 ° sin.” P) e sin.” 2 + 4 - Sin, 3 e cos. 3 e SIN. 3 + COSe 3 e COS. 4 bs 240 e ¢ J J a 2 ; * a 7 6 divide this by the denominator 4. sin, gtine'gs and gy iw D a b cos. (C — x) =sin. =. sin. — + cos. —.cos. —. cos, C, 2, 2 g g This, in a merely mathematical point of view, is a complete solu- tion, but, practically, a most incommodious one. ‘The reason of which is, that in almost every case that occurs in practice, the angle of the chords differs very little from the corresponding spherical angle; rarely, by two seconds, most frequently, by a fraction of a second. But our best Trigonometrical Tables, (those of Taylor) do not go beyond seconds: consequently, were we to use the above formula, it would be necessary in every case, (for we cannot suppose, even in extreme cases, the difference x to be an exact number of seconds) to interpolate between the logarithmic numbers; which is a troublesome ope- ration. In order to remove this impediment, instead of finding the value of cos.(C— x), it is expedient to find a formula for ‘the difference, or the correction x: for, this quantity being very small, we may, instead of interpolating between logarithmic sines and tangents of seven places of figures, neglect the three or’ four last figures, without. practical error. Expand cos.(C—.), and it becomes cos. C. cos. r-+sin. C. sin. x . HH eMiviuale, bY re = (1 ~ 2sin.?=) cos. C+2.sin. =. cos. —.sin. C; 2 2 2 ) WH ne o ° 1 Cee {sin. C e COS. = = COS. C . SIN. = wee p) uae 241 —b b dili-bcos.C }. {sin. (C — 3)}. If, therefore, x being very small, we seek for a first value* by . a+b ae = sin.) ae: { 1 — cos. Cc} — sin.” But (e) =@. sin. to1& assuming sin. C — Sar su C, we have 24+b 1-—cos. C . ,a4—b6 1+cos. C _——-——_— — sin. ——— 4 sin. C 4 sin. C s xv ° 2.sin.- = sin. 2 * In order to use a formula of approximation with confidence, we ought to know the errors, or limits of errors, committed by rejecting quantities. In the case before us, the true value of « must be derived from a quadratic equation: thus, make the right hand of the equation of p. 240, 1.22=2 sin. C, and cot. C=—a, then AT tae sae a fats asin.’ 5 -f sin. 5 COS. 5 = ,; and «@ tan,” 5 + tan. 5 = 6.sec.”2=G-+-/ tan.” = y V11+48 .(e—£)} Expand the right hand side of the equation, and tan. 5 = B—(a—) B?+2(a—f)?—5.(a—fBYBt+&e. = B—af?+(2 a’+-1) 63+ &e. the other terms involving (4, 85, &c. Now is very small, and, accordingly, all the terms to the right of 8 are much smaller, and so small as to be safely rejected. To be assured of this latter part, take an extreme case, and the value of the term involving 7 will convince you, that the rejecting of terms involving higher powers of 8, can induce no practical error. Hu a | . 8 O) Cc « Q a = b g Cc = sin, .tan. — — sin. COU @ 4 2, i a), CORNEA oo or, since x is very small, sm,~ = -~.sin. 1, Q g e “ee e Q a « Q a Suna? b e.sin. 1 =sin. a — sin, cot. 3° from this approximate value of 2, (which, however, is, almost always, sufficient), find 2 (=p), and for a nearer value compute C . ,a-b sini* = gee . tan. — — sin.” cot. — . & 4 2g 4 Qg 2 sin. 5 J : 1 ° C ae r) sin ( 4 Take the instance of p. 230. | Observed Angles. Distances or Values of Sides. Dean’ Hill (4). 2054 2) ARO AP SRNO5OMP sais. 83 4 140580.4 Butser Hill (B) ....... 76 12 22 Bi Luis Reet YOS49GH Dunnose (C) ...... Se oon aan cre ae wo. 156122.1 Correction of the angle C. Log. Log. ae eS ty pa 4.90859 uh Bly ee 4.03056 Gp’s radius ............ 7.32060 7.3206 6.70996 * log. Sch in seconds.... 7.08799 38300920 : 3.41992 G a, AR 5.17598 cot...... 10.27691 4 —_—__—_ 0 FY! QQ" 3.69683 TaD ADT OE GLO Bo ok idee 9.72309 4.68557 4.89907 LOS SIN ee aN 4.68557 Cees: COURS. (UGBAG) iL 0s) os 8 .21350 | .. correction = 1.6349 —.1026 = 1.5323. * In this computation, from the smallness of a, 6, in~minutes, se- conds, &c., the arcs have been assumed equal to the sines. If we com- pute exactly according to the form, the result will be 1.534. 243 Correction of the angle A. Log. PES .. 84904.57 2... 4.92893 Set ... 6843.52 . CPS lade woe ve tents 7.32060 | 7.60833 5.21666 tan. 24° 2' 16” ...... 9.64935 Cot, 24.7 2 1G ho, 4.86601 LEAD OET EAD AMAR 4.68557 (1.5151) .. .18044 (.04948) ..... -*» correction = 1.5151 —.0495 = 1.4656. Correction of the angle B. ik Log. “te ... 74175.62 ... 4.87026 <= 1. 3885.42 .. @)’s rad. ...-... 7.92060 7.54966 5.09932 tan, 38° 6°11” ...... 9.89442 COU SBE Gul biwe eos ! 4.99374 TERE ss shea «sss - 4.68557 CRED PSR se 0.30817 (.0090718).... * +. correction = 2.0332— .0091 = 2.024. Log. .. 3.83528 7.32060 6.51468 3.02936 10.35065 3.38001 4.68557 8.69444 Log: . 3.58944 7.32000 6.26884 2.53768 10.10558 2.64326 4.68557 7.95769 * I'he second nearer values, (see p. 242. 1.6.) of these corrections, will be : sin, 55° 43’ 7” sin. 55 43 0.24’ sin. 48° 4' 32” sin. 48 4 31.27’ bin. 76°. 1.2.22" sin. 70 1220 ’ 1 bs HS Vhs 1% 1.4656 x 2.024 x the results of which, as far as four places of figures are concerned, are the same as before. 244 The spherical angles, their corrections, or differences, (as General Roy calls them) and the angles of the chords, will stand thus Observed Angles. Corrections. Angles of Chords. Dean Hill .. =... 48° 4) 39.25 —1 1 oekOO 30".784 Butser Bill’) oe. 76 12 22 — 2 .024 19.976 Dunnose ........ 55 43° 7 phy Weta s: 5.468 180 0 1.25 — 5 022 56.228 The sum of the angles of the chords is 179° 59° 56”.228, but it ought to be 180°: consequently, the corrections being supposed to be right, the spherical angles were erroneously observed, and the amount of the errors must be the defect of the sum of the angles from 180°. The amount of errors, there- fore, must be 3°.772, which added to 1.25, the excess of the sum of the observed angles above 180°, makes 5,022, the sum of the corrections, as it ought to be. This sum of the corrections, or of the differences of the spherical angles and the angles of the chords, must always, as it is plain, be equal to the excess of the sum of the spherical angles above 180°*, and, therefore, the sum of the corrections * It would be quite absurd to seek for a proof of this: but, as a matter of curiosity, and merely as such, it may not be amiss, in the subordinate station of a note, to shew that the sum of the analytical formula, repre- senting the reductions to the chords, is equal to the spherical excess. Reduction (R) = (S52. tan. =* (= y. cot. C 2 2 tan.” pe io 1 b 2 “{@Q+0y} (as ete i ee 2 245 will always answer the same end as the spherical excess. The Temaneee a) HAa) Sick ata 1 Bs a Seah A 0 7 ae ge = san (248 (a* 0°) .cos. Ct, similarly, for B, reduction (R’) = Fe }2ac—(a’+4c*).cos. Bh, . fortmhics eo (ROS saq 12bc—(b?-4c7) .cos. A}, a’ -- b?— c” 2a 4: a” +c? — b* Boe Qac” 62+ c2— a 3 Chr 2bc c oe R+R+R'= ; 4 ah? — (a* +b") (a?-+-5*—~c*) ee ee 4 2 2 2 az 2 i Dust z 16a6.sin, C 1 ygtoiit teens ae) (a*--e a2 $400 (VEC) (Ofc! a") 1 2 a2b?-+-2 a2%¢27-4+-2 b*c? } = 'sab.sin. C ; — (at+-bt-+c4) 1 faery} ~ §a6.sin, 4a? b? Bs SeeeaneN but, cos, C = : 3" cos. sin, B = ~ sin. C, c = Sab.-sin. C A a”b? ab ae 1—cos, C 2 sin: Cs -( ) Ga Ses sins C: v0 | which, see p. 232. is the value of the spherical excess. 246 latter, in the above instance, (see p. 230.) was found equal to 5.022, which is the sum of the corrections. ‘There is, therefore, no essential use in the theorem for the spherical excess. It is, however, applied for the obvious purpose of checking the com- putation of the reduction to the chords. If, now, we employ the defect of the sum of the angles formed by the chords from 180°, in the same manner as the spherical excess was used, (see p. 235.) in correcting the angles of observations, that is, by adding to each one-third of 3” las they will stand thus, Corrected Corrections Angles Observed Angles. | 4d. of 3”.772 Spherical “for of. ngles. Chords. Chords. 48° 4) 327.25 +1°.257 3S 1007 — 1.466 32”.041 76 12 22 +1 .257 23 .257 —2 .024 21.233 $5 AS 7 41 .257 § 257 —1 .532 6 .725 5 O02 59 .999 After these corrections, the excess of the sum of the observed angles is 5°.021, and the sum of the angles of the chords is 179° 59" 59°.999, &c. or 180°. We will now take the first triangle in General Roy’s Survey. Feet. Hanger Hill Tower): (4) 5. ...42% 2°32"... - (a) 274042 Hampton Poor House (B).... 67 55 39...... (6) 38460.4 King’s} Arbour .4./.. °(C) .0..°70) UT) 48. ian oe) 3 7921.9 1 hie ia Pe 27404.2 esi 38460.4 a+b .... 65804.6 ° ogs. da Logs. ae Ma 1G4GG,15 «1. 421058 . chee ov Ganat 6 eat p's rad. ....-- 7-32000 7.32060 6.89598 6.12094 | 3.79196 2.24188 ) tan, 35° O' 54.... 9.84547 cot, ..... 10.15453 | 3.63743 ~ -2,39641 SINS Maen ce ie 4.68557 4.08557 (0895)... 005. 8.95186 - (0051). (0987.7 1084 orr. 0895 | 0051 0844. Ostia FLTAOS = AU Fo -- 37921.9 65326.1 wae 169025 J... 247 Logs. A, 21301 7.32060 6.89241 3.78489 (; yh _ tan. 33° 37’ 49”... 9.82839 3.61321 ; 4.68557 b ..:.... 38460 Goes. 37921.9 _ b-bc .... 76381.9 as Sp ignbO TAD An co 32 6.96146 8.92764 Logs. 4.28206 7.32060 3.92292 htan. Ole 110". 3.50757 (.06638) «0. ..06 9.58464 4.68557 8.82200 fai Logs. «> 2629.4 20, 3 41986 7.32060 6.09926 2.19852 Cot, -. 1. 10,1716) 2.37018 4.68557 aa ene 00484 ... 7.08455 Corr. .0846 .004:8 0798 b— Logs. RE IST haba 2.19872 7.32060 4.80812 9.61624 cot. .... 10.41545 0.03169 4.68557 ' {0000232}.< ait. 5.34612 Corr, 0664. 248 Hence, the following Table : Corrected Observed Angles. Corrections. Ss ious Angles of Chords. ngles. . AO? 2’. 32" — .0844 32”.41 42° 2’ 32", 33: 67 55 39 — .0798 39 .41 67 55 39 .33 70 1 48 — .0664 48 .41 70 1 48 .34 179 59 59 — .2306 0 .23 180 0 0O* sum of errors ... 1.25 oS pee 41. * The above Example has been introduced for more than one pur- pose. The sum of corrections, or the sum of the differences of the sphe- rical angles, and the angles of the chords, is 0”.23, which, as it ought to be, is the spherical excess, found either by General Roy’s Rule, or by the formule of p. 250, but General Roy makes the spherical excess .29. In the next place, the angles of the chords, deduced by subtracting the computed corrections from the corrected spherical angles, are quite different from General Roy’s angles corrected for calculation, (see Trigonometrical Survey, p. 139.) They only agree in one point: in each their sum is, as it ought to be, 180°. We may understand some- thing of the principle which guided the author in correcting his angles, (for he himself partly explains it) but we find little or no trace of the detail. In p. 141, Trigonometrical Survey, it is said, ‘ As that part of the Earth’s surface, to which the operation is confined, has been con- sidered as a plane, it is evident, that the mode of correcting the angles for computation must, in some degree, have been arbitrary; and, therefore, it follows, that in reducing the observed angles to those of plane triangles, each angle may be varied to certain limits; and, consequently, the opposite sides may be varied to certain limits also; but it is evident, that the means of the extreme results, obtained in this manner, must be very near the truth, and perhaps will be considered more accurate than the distances deduced from a single correction of the same angles. Accord- ingly, if we vary the angles, (in reducing them to 180°), from Hounslow Heath, to the XIIIth triangle, so as to produce the greatest and least lengths of the opposite sides, we shall have 141746 feet, nearly, for the mean. 249 The above are instances from the Trigonometrical Survey of England; we will now take one from the French Survey, (Base du Systeme Metrique, Tom. I. p. 535.), in which the spherical excess is considerable, although less than in the former instance. Toises. Feet. Violan ,.. (C) ... 51° 10’ 11.31 @.% 19922... 12786112 Aubassin. .(B) .,. 83 15 22 .17 b ... 25423 .., 162605.508 Bastide .. (A)... 45 34 29 .57 a... 18264 ... 116816,544 Correction or reduction of the angle C. : Log. Log. a sean. OUBNS rie 4.84419 me a. 11447 .. 4.05869 @’s rad. ...... 7.32060 7.32060 7.52359 6.73809 5.04718 3.47618 tan. 25355 ...... 9.68015 EOE ink tas 10.31984 | 5.72733 3.79602 gS RS eat 4.68557 4.08557 ChIOEP eA .04176 (.12896) ..... + 9.11045 Hence the reduction = 1.101 —.129=.97. mean distance of Hollingborn Hill from Fairlight Down, which, however, is only 14 feet more than the distance in the XIIIth triangle.’ In the former editions of this work, the author having, in the above example, wrongly transcribed, from the Philosophical Transactions, the distances of the stations, and, accordingly, wrongly computed the sphe- rical excess, attributed to General Roy’s calculation of that quantity, an error which did not belong to it. The real error consists in the quantity being put down equal to .29, whereas, by two different methods, (see | pp. 227. 248.) it is found to be .23, Ir 250 Reduction of the angle B. a--e ‘ eee OLOSO bs.06', 7.46515 4.93030 tan, 419 37’ 41”... (1.561) ... Log. 4.78575 7.32060 9.94876 4.87906 4.68557 0.19349 Log: cma... 2651 .. 3.42341 7.32060 - 6.10281 2.20562 COte Sele 10.05124 2.25686 (.003726) ... Hence the reduction = 1.561 — .0037= 1.557. Reduction of the angle A. oserah 12000 sn ee b+c. A, 4.86037 7.32060 7-93977 5.07954 tan eo Ar toe (1.0407) ... 9.62335 4.70289 4.68557 0.01732 AA, eae 8796 eee 4.68557 7.57129 3.94428 7-32060 6.62368 3.24736 » COL ates 10.37665 3.62401 4.68557 (.08678) ... 8.93844 Hence the reduction = 1.0407 —,0868=.9539. . Sum of reductions. 954 1.557 970 3.48 This result agrees with that which is specially denominated the spherical excess, and which was deduced in p. 229, 5 251 We may now arrange the results as M. Delambre has done, pp. 533. Tom. I. Systeme du Base Metrique. nee Mean | i Chords, | Angles. Violan.... LE bee Key ge ees y | — 07 51° 10’11".45 | 10.48 | 10”.29 Aubassin ., 33.15/22 17 — .56 93 419122.0 7 20 .75 | 21 .15 Observed Angles. | Reductions. | Spherical Angles. Bastide.... | 45 34 29.571 — .95 | 45 34 29 .72* | 28.77 | 98 .56 180 \0:.3.05,| .~3.48 .|1180. 0 3.48 O04 66 Sum of errors = .43. The first column contains the observed angles: the second their reductions to the angles of the chords, or the differences between the angles of the arcs and chords; the sum of the num- bers in the second must be the excess of the sum of the angles of the true spherical triangle above 180°: but the excess of the sum of the observed angles is only 3.05: therefore, the sum of the errors of observations is 3.48—3.05, or .43. Add 4 of this sum of errors, or .14 to the numbers in the first column, and you form the third column which contains the corrected, and, probably, true, spherical angles. Diminish each of these corrected sphe- rical angles by its corresponding reduction contained in the second column, and you form the fourth column, the sum of which ought to be 180°. Lastly, if you diminish the corrected observed angles by 4 of the spherical excess, or, which ought to be the same, by 4 of the sum of the reductions, you have Legendre’s reduced rectilinear angles, which Delambre calls angles moyens. We may see distinctly, in the above Table, the three modes of solving the triangles of a Survey. We may. take from the third column the corrected spherical angles, and solve the triangle by the rules of spherical triangles: the resulting sides will be arcs of great circles; or we may take the angles of the fourth column, and solve the rectilinear triangle of the chords; the re- * Tn order to avoid the unnecessary putting down of decimal figures, the last decimal numbers in column 3) are put down .72, instead of .71, &e. 252 sulting sides, as well as the given, will be the chords of arcs of great circles: or, lastly, we may take the mean angles of the fifth column, and solve the triangle as a rectilinear triangle: the resulting sides, as well as the given, will be equal, in length, to the arcs or sides of the corrected spherical triangle. M. Delambre and his associates computed, by all three methods, the series of triangles from Dunkirk to Barcelona. The conductors of the Trigonometrical Survey of Great Britain, ‘computed by means of the triangles formed by the chords, but to check the reductions, they computed the spherical excess by General Roy’s Rule. We are very far, although much has been said, from having exhausted the subjects of a Trigonometrical Survey. As yet, little has been determined concerning the azimuths of stations, their distances from the principal meridian, their bearings with that meridian, and their latitudes and longitudes, as resulting from the observed angles and computed sides of the triangles of the Survey. If DaM be the direction of the meridian, the azimuth of W, (Watten) seen from D, is the angle WDa. Now the 258 direction of the meridian, or the point directly south of D being determined by previous observations, this azimuthal angle WDa is easily found. If, for instance, the instrument used were a theodolite, like that of the English Survey, furnished with a graduated horizontal arc, the azimuth of W would be simply the difference of two readings, the first, when the telescope is di- rected to W, the second, to a mark south of D. But at other stations not provided with their south or north marks, other methods must be resorted to, and the most simple is that in — which the pole star is observed at its greatest elongation. Let G be the observer’s station, P the pole, p, p’ the points of the pole star’s greatest elongations, N an object to be observed from G. The telescope is directed to N and to p; then the difference of the readings off on the graduated horizontal arc, is the measure of the angle pGN: next the telescope is directed to p’, the greatest western elongation of the pole star: the difference of the arcs or readings off on the azimuthal circle of the instrument of the pole star thus observed and of N, measures the arc p'GN: and pGN+pGN | PGN+PGp+PGN-—PGp 2 2 = PGN the azimuth of N with respect of G. 254 In the figure of p. 252, the angle WDa determined by the above or a similar method, is the azimuth of W_ with respect to D, which being known together with DWC. determined from observation and DW, enables us to solve the triangle DWa. The angle Cab, which equals DaW being then known, Ca being determined from the solution of the triangle DCa and WCF being known from observation, the triangle Cab. can be solved, and ab determined:* also Cb and the angle Cha= Fbm by the same process, and CF being known by the solution of CWF, Fb=CF— Fob is known, and thence bm=Fb.cos. Fbm: so that Dm= Da+ab+bm; - the arc intercepted between the latitudes of D and F, (F being near to the meridional line DM) would thus become known. This is an illustration of the use of azimuths. Picard, and the mathematicians who made the first surveys, continually employed them in projecting their computed oblique distances on the meridian. ‘Thus, through the point F conceive a parallel to DM to be drawn from W, and also a perpendicular be drawn to such parallel, then the projection of WF would be WF multi- plied into the cosine of the angle made by WF and the parallel: and a series of projections of such lines extending from D to the other extremity of the meridional arc would constitute the length of the arc between the two extremities: and if the lines such as WF were so selected as to be near to the meridian, and slightly inclined to it or its parallel, the result would be tolerably exact. We will presently poimt out the cause of its want of exactness, and the means of correcting it. It was by drawing parallels, such as have been described, that General Roy, and those that succeeded him m conducting the Survey of England, determined the bearings, &c. of places from the meridian of Greenwich. * This is of the nature of a general statement: a few pages farther, and we will speak more in detail. 255° Thus let G be Greenwich, GM the meridian, PG perpen- dicular to it: S, Severn-droog Castle, W, Wrotham Hill: draw SB parallel to the meridian, SA, WM parallel to PG, the perpendicular. By observation* the angle GSW = 152° 28' 56” GSA=90°—azimuth AGS, (=73° 49’ 34”); ». GSB = 106 10 26 The angle BSI, or the bearing of Wrotham Hill from the parallel SB south-eastward By the solution of one of the principal triangles, _ GS =14610.3: and the azimuth AGS = 73° 40’ 34”, ”. AG =14610.3 x cos. 73° 49’ 34” and AS = 14610.3:«: sin. 73. 49 34 log. 14610.3...... 4.1646391 | 4,1646391 cos. 73° 49’ 34”... . 9.4449085 7 sin. 9.9824615 (4069.56) .. 3.6095476 (4031.6). . 4.1471006 “. AS =14031.6, AG = 4069.56. Again, by the solution of another triangle, SW= 79960.6, and since BSW= 46° 18’ 30” * The angle GSW was not determined by immediate observation, but by combining other direct angles of observation. 256 log. 79960.6 eee 4,.9028760 4.9028760 cos. 46° 18’ 30”... 9.8393380 sin. 9.8591789 55234.9 ... 4.7422140 578168 ... 4.7620549 .. BW=57816.8 BS = 55234.9 but AS = 14031.6 AG= 4069.6 . MW=71848.4 GM = 59304.5 The distances, therefore, of Wrotham Hill from Greenwich meridian, and its perpendicular arc, in whole numbers are 71849 feet, and 59305 feet; and so for other stations. In order to obtain the direct bearing of Wrotham and its distance from Greenwich, we have tan. MGW=r. seal and r.GW=MG. sec. MGW, MG log. r ++e+-lO log. MW. . 4.8564207 log. MG. . 4.7730913 4.7730913 10.0833294 = tan. 50° 27’ 48” sec. 10.1961525 (93613) . . 4.9692438 The bearing, therefore, is 50° 27’ 48”, and the distance 936158 feet. By these methods, both the English and foreign mathema- ticlans determined by a series of successive additions the per- pendicular distances of the stations from their assumed meridians of Greenwich and Dunkirk. ‘To express the distances alge- braically, let 0, 0’, &c. be the distances of Severn-droog from Greenwich, of Wrotham from Severn-droog, &c., 2 the azimuth of Severn-droog on the horizon of Greenwich, 2’ that of Wro-. tham on Severn-droog, &c.; then the respective perpendicular distances from the meridian of Greenwich will be : 257 © sin. 2, 6 sin. +0 sin. 2, O sin. +0 sin. 2 +0" sin. 2”, &c. The sum of the distances on the parallels from the perpen- dicular to the meridian at G will be Ocos. 2+0 cos. 2 +0” mae 2’ +&c. and we will now consider what is the error. There can be no mathematical error, if we attend merely to the diagram that we have used: for SB must be equal to AM: but the case is different, if, as the fact is, SW is an arc, and § is situated in a great circle representing its meridian. Let BC represent the arc (SW), extending from one station A to another, MER the meridian of Greenwich, MBF that of B, draw CR, BE perpendicular to MER, and BD perpendicular to DR, then BD, in the process described in p. 255, 1s assumed equal to ER. Produce RD, EB till they meet in A: then the difference between ER, BD is, in principle, the same as the angular difference between two objects above the horizon, and those objects reduced to the horizon, see pp. 220, &c. In this case H,h (BE, DR,) are equal, therefore the error, (what in pp. 220, &c. was called the reduction) is equal, (see p. 223.) to Kx 208 2 ri 18 “a. tan.?-7. tan. = sin. 1”, or — an Ss tan. ? in. 1’ a beng BD. Let H= 1°, a= 30 then, log. tan.? H = 2 log. tan. 1° = 6.48384 tan. 15’ = 7.63982 ar. comp. log..sin. 1 = 5.31443 (0.2742) .... 9.43809 I} but an angle of 1” at the Earth’s centre is subtended, nearly, by 100 feet*, on the surface, consequently, the above error would be about 27 feet. There would be more than this error, if we projected an arc at Dunkirk on the meridian of Greenwich, since it is distant from that meridian by more than 1°: but such a case would not occur in measuring an arc of a meridian, since the obvious policy in measuring it by the projections of sides of triangles, or dist- ances slightly inclined to the meridian, would be to select sides as near as possible to it. The error, as it is plain from its expression, varies chiefly from the variation of H. If H should be 30’, the error would be one-fourth of the former result, and would be equal to 0.0685, or about 6 feet. The diagram of p. 257, which has been used for computing the error in assuming BD=ER, will also serve to find the true value of LR; .. ER is the measure of the angle (A) BAC. In the triangle BAC we have, (see p. 163. 1.5.) ee ee ee ee eee * Log. 1” = 46855749 log. @)’s rad, in feet ... 7.3216037 (101.6) ... 20071786 259 cot. A. sin. ABC=cot. BC.sin. AB— cos. ABC. cos. AB, ABC=90° —- CBD=90° —- CBF — FBD=90° —(2 +2), BC=6, sm. AB=sin. (90° — BE) =sin. (90° —y)=Ccos. ¥; cos.(2+2) one e A = Ne oP Sue YA eR ED, RS ESE TS ce } cot. 0. COs. y—su.(S+2).sin. ¥ _ tan. d.sec. y. cos. (2 +2) J—tan.d.tan.y.sin. (2 +a)) If we expand the denominator as far as the second term, rejecting the following terms that involve tan.” 0. tan.?y, &c. tan. A=tan. d.sec. y.cos.(2+2) + tan.’ 6. sec. y. tan. y. sin. (+2) .cos. (2 +2). If we reject this second term, and assume, instead of the tangents of A and 0, the arcs A, 0, and make sec. y= 1, (for y or BE is always small), we have A=0.cos.(2+x)=0.cos. CBD, which, in fact, is the method formerly used, (see p. 255, &c.) and involves several sources of inaccuracy, viz. the assuming arcs equal to their tangents, y=0, and the rejection of the second term tan.” 0, sec. y. tan. ¥ Ot (2+2) cos.(2+2). M. Delambre, who, for many reasons, is a most excellent authority im these matters, says, that if the exact formula, (p. 259.) be used, that then the method of estimating the are of the meridian from the sum of such arcs as E.R is by far the most convenient method. He thus adapts the formule to logarithmic computation . log. tan. A =log. tan. d-+ log. sec. y + log. cos. (2 +2) —log. {1 —tan. 6. tan. y. sin. (2+ 2)} =log. tan. d + log. sec. y + log. cos. (2+ x) — K .tan, 0 tan, y. sin. (2 +2), 260 K being the modulus, and equal to .43429448. Now 4A, 0, y are very small, we have Ae A? tan. d= A + sake nearly, = 4. (: 4. G2 A *, log. tan. 4 = log. log. — log. tan. .4 =log. A +2 log pa G2 similarly for tan. 0. Again, cos.y = 1 — a T — | oo Ney, y (ee y\ y ? .. log. sec. y= — log. cos. y = 3 log 2 e ‘sin. y Hence, log. A =log. d+ log. cos. (2 +2) — Koy.sin.(2+ x). sin.” 1”, (K modulus) 1 fe all + 3 log. —2— ain, A) gin. 8 eo sin Dione 261 For the application of this formula, M. Delambre has eon A lox; ——. sin. A’ °8 sin. 0” structed tables which give the values of log. &eo. * As it is necessary from computed chords and sines to deduce the corresponding arcs, such expressions as those in the text, or equivalent ones, or tables deduced from them, are required in every step of the Survey. They effect no more than the rules which Maskelyne has given in the Introduction (pp. 21, 22.) to Taylor’s Logarithms. These rules are intended to deduce from the tangent, or sine, of a small arc, the arc itself, and vice-versd. Thus A A”. A’\ -= tn. daA+S=4.(14+5)=4.01-5) 3: .. log. tan. A =log. A+log. sin, 1” — = log. cos. A, a or log. 4 =log. tan. d— 4.6855749 + 3 log. cos. A Og | =log, tan. 4+4-5.3144251 — ae arith. comp. cos. 4—10, which is Maskelyne’s rule: and which gives exactly the same result as the expression loo. A=log. tan. A+3. x 5.3144251—2 (log. A— log. sin. d)— 10, If we take Maskelyne’s Example, p. 22. of his Introduction ‘to Taylor’s Tables, the two processes will stand thus: given A tan, J =7.5228031. Maskelyne’s. Delambre’s, 7.5228031 .. A=11' 27”, nearly. 75228031 5.3144251 15.9432753 12.8372282 23.46607 84 16 ... % arith. comp. cos. 4 log. 4 =2.8369567 2.8372266 ... (687".427) sin. A 7.5225308 wafers LLNQ7CA27. 5.3144259 2 10.6288518 2.837 2266 Delambre’s 262 The above method of M. Delambre relates principally to the measuring the arc of the meridian. ‘That was the main, or ostensible, object of the French Survey: but surveys have ] Delambre’s formula, however, was intended for another purpose than the mere Computation of 4. In like manner, in order to derive the arc from the sine sib. A= A +e A, 4.1 ay = A G aa) =A. (cos. 4); is he A-+log. sin. 1” (4.6855749) =log. sin. A~4log. cos, A, or log. d=log. sin. d+ 5.3144251-+4-4 arith. comp. cos. 4, which is Maskelyne’s Rule. In order to complete this subject, into the discussion of which we have been drawn by the matter of the text, we will subjoin the rules for the reverse operation, namely, that of finding the tangent and sine of a small are from the arc ttself. Rule for finding the sine of a small arc from the arc itself. To the logarithm of the arc reduced to seconds with the decimal annexed, add the constant 4.6855749 (log. sin. 1"), and from the sum subtract one-third of the arithmetical complement of the logarithmic cosine. The remainder will be the logarithmic sine of the given arc. Rule for the tangent. To the sum of the logarithmic ave and of 4.6855749 add two-thirds of the arithmetical complement of the logarithmic cosine, thé sumis the logarithmic tangent of the given arc. Proof of the ist Rule. | Res sin. A= A > f.. (4 being small) = 4 7@ ry, 3) = A, G- =) 263 been undertaken for other objects: for instance, the special object of that of 1792, was to join the Observatories of Paris and Greenwich, or to determine their differences in longitude and latitude. Part of the method used by General Roy* and others may be explained from what has preceded, by adding together the distances from the perpendicular to Greenwich, such as GA, SB, &c. the distance GM was determined. By the same process MMW was determined. ) GM is not the difference of latitude and longitude of Greenwich and Dunkirk, (supposing W to represent the latter place), although it serve to find out the difference. Let P be the pole, D Dunkirk, DA a parallel of latitude, then the difference of latitude is GR-+-RA, or GA is the length between the latitude of G, and a place on the meridian, having the same latitude as D. = (expressing A in terms of the radius) -sin. 1”. 4 (cos, Ay =sin.1".4. (== ==) 10” i .. log. sin. d = 4,.6855749 + log. A 4. I (log. cos, A— 10) = 4.6855749 + log. 4 — arith. comp. cos. A. Proof of the 2d Rule. tan. A=A $fe4(i } =4(i -=)-: Ne .*. on the same principles as before = 4 (cos. A) 73; . log. tan. d = 4.6855749. * In speaking of General Roy, we have, perhaps, in some places, attributed what did not belong to him. His name has been used to de- nominate generally the conductors of the Trigonometrical Survey of England. 264 The difference of the longitudes of D and of G is measured by the angle GPD, and, on the hypothesis of the Earth being a P G R D A sphere, these differences of latitude and longitude may be thus computed. The meridian is supposed to pass through G the principal place of observation, PD is a meridian passing through D, GD is the arc of a great circle passing through G and D: RGD is the observed azimuth (2). D being a station so near to G as to be observed from it, the distance DG compared with PG, the co-latitude of G must, in all cases, be very small: for instance, if GD should be equal to 60000 feet, or about 103 miles, PG (supposing it to be equal to 38° 98’ 40") would exceed GD in the proportion of about 230 to 1. Let DL be the latitude of G: dl the difference oe the latitude of D and G: then | PD=90° —(L+dL), and what is now required to be done is, to find, by approximation, dL. from the solution of the spherical triangle PGD, in which, PG (=90°—L), GD(=8), and the angle PGD=(180° — 2) 265 are given. The formula expressing the relation between these quantities and PD 1s cos. PD =cos. 2 PGDsin. PG.sin.GD +cos. PG.cos. GD, or sin.(L+dL) = —cos. 2.cos. L.sin.6 +sin. L cos. 0; but sin. (Z+dL)=sin. L cos.dL+cos. L. sin. dL dL dL =sin. L (:- —2 sin.” <=) +2 cos. Li. 80s SCOSse. or 2 Transpose sin. L to the right hand side of the oat oD and then divide each term by cos. DL, and q | d 2 sin. ne sec. L (cos. L.cos. gi sin. L. sin. ) 2 a if =tan. L (cos.6 —1)— cos. 2. sin. 0 = (2 sin tan, L-+-cos, Zsin. : = fH, cos. (4 + Hence 2 sin. = —____——— = E, cos. i since dL is, by hypothesis, small, find, as a first value, dL from the equation Sd; i, 2.sn.—— = E, or dL = ——; = «, oO sin. l then the second or nearer value of d IL will be cos. L L €. cos. ——-— , and _ so on. ie 266 As an example to this formula, let us apply it to find the difference of latitudes of Greenwich and Dunkirk, in which instance, however, the azimuth Z is not determined, at once, by observation (the places being too distant for such an opera- tion), but by computation from the distances of stations, inter- vening between Greenwich and Dunkirk, from the meridian and the perpendicular to the meridian of the former place. (See p: 256, &c.) In this case, (see Trigonometrical Survey, Vol. I. p. 168.) G R BD A Feet. RD= 547058 - RG = 152556; “. GD = 567931 ,'... (6) £4 RGD=74 25’ 5”. The log. of the chord GD in seconds of the Earth’s cir- cumference = 3.7471170 log. of the arc GD (see p. 266, &c.).... 3.7471302 267 E computed. log. 8... 3.7471302 — 3.7471302 log.2... «30103 4.6855749 log. § ... 3.4461002 log. sin.8 ... 8.4327051 sin. 1”... 4.6855749 an. ee oeloloeas sins . 6.2633502 tan. L ... 10.0990491 (No. = .00023035) ... 6.3623993 again, cos. Z -+- 9.4291323 sin. d ... 8.4327051 (.007275) ... 7.8618374 *, E = .007275-+4-2 x .00023035) = .0077357, hee E = 7.8884996 ar. comp. sin. 1”... 5.3144251 | log.e ... 3.2029247 *, —e€ = 1595".6 = 26’ 35”.6 € SPP US CO AE Wo7177.8 9 7 (lat. Greenwich) 51° 28’ 40° log. = 9.7943612 es ... 51 18 2.2 log. = 9.7964635 .0021023 loge, eh en eee ce 3.2029247 (1587.89) ... 3.2008224 *, 2d value = 1587.89 = 26’ 27.89. The difference, therefore, of the latitudes. of Greenwich and Dunkirk, thus estimated, is 26’ 27”.8. The difference of the latitudes of the above places is, according to the Trigonometrical Survey, Vol. I. p. 163, 51° 28" 40" — 51° 2! 11".4= 26! 28/6, according to Delambre, Arc du Meridian, Tom: II. p. 295, 51° 28’ 40’— 51° 2 9".7 = 26’ 31".3, 268 but neither with the one or the other can it be expected, that the result just computed should agree; since it was obtained on _the hypothesis of the Earth being a sphere. The spheroidical form of the Earth will alter the value of 6, which we have con- verted into seconds, by dividing the arc subtended between G and D by @’s radius x sin. 1”, whereas it equals - .(i—tLe?®. sin.? L). We will now, on principles like the preceding, deduce an approximate formula for determining the longitude of a station. Let P represent the angle GPD, which is the difference of the longitudes of G and D: then sin. PGD $10. FP = sin. GD. pT sin. (180° — 2) (90° —- L=—dL) sin. 0. sin. Z cos. (L+dL) sin. oO. = sin. O.SIN. : (1+ tan. L.dL), nearly, cos. L but, see p. 265. Sat therefore if we substitute this value of di in the former ex- pression for sin. P, and neglect the terms involving 0°, : sin.O sin. 2 ~~ sin? 0. sin. S cos. Z tan. L sine P = ————_—— = cos. L cos. L : O.sin. & Bie GITh. < CORsme EATS Lament ape or P = ————— = 0. in. 1”. cos, ) cos. L 269 In order to exemplify this formula, let G denote Greenwich, D, Dunkirk: then, see p. 266. 0 .-- 3.7471302 3? .,. 7.4942604 sin. Z ... 9.9837378 9.9837378 arith. comp. cos. L .. 10.2056388 cos. Z ... 9.4291323 (8639.8) tala as 3.9365068 tan. E ... 10.0990491 78.957 arc. comp. cos.2.. 10.2056388 8560.843 L.sin.1” .., 4.6855749 Hence the longitude of Dunkirk is ... (78.957) ... 1.8973933 8560".843 or 2° 22’ 40”.843. In the triangle PGD, PD the co-latitude of D and the angle P, the difference of the longitudes of G and D have been approximately determined. We will now find the angles PGD, P DG, and then shew the use that can be made of such a result. By Naper’s analogies, (see p. 173.) P cos.4(PD— “le (EC REEDC i) P cos. 4(2D— PG) "2° cos. £(PD+PG): Let PGD=180° — 2, PDG=130°—2Z, OM iat then tan. (HE) tan. (180° Rls ate g / tan. {90° — CS - go") PRESTR UVES Maboe a tan. (4 _ 90”) Again, let PD=90°—L, or=90°—L' + dL, PG=90— L’, or=90°- L—dL; PD—-PG L-tL dL fs a o = +.—-, 2 2g 2 ED REG:: ‘ L+L’ Pi es dL Tae OO 3 ,or=90— L— oe Hence, aL Z Pp sin (2 aE — 0 ow tan. ( ; 90”) tan a TL SOS iis In order to reduce this farther, we have, (sce p. 268.) ; sin. O.sin. Z sin, P= cos. but (see p. 42.) P sin. P sin. P sin. P LADS meen a ot ae = Cae eee 1 P i ge : - TiSOE: 2.cos.7 — 2.(1=sin : -) 2 a Yd ° a amy » (teint = + Kc. ). Now the first term in the expression (see p. 268.) for P in- volves 6, the second 0°; if, therefore, we reject the terms in- volving 6°, and higher powers of 6, we shall have tan aa = $sin. P, nearly: accordingly dL Z42' 4 sin. P. sin. (1+> Hy aaa aye an ( 5 90 aT F cos. —— 7 ‘ and since P is very small, and consequently — 90°, we have 271 2+ 2! ‘i Oo. sin at (L+> Oey ~ 2. cos. ETE ———4;— : cos. —— dL yea sin. (L4 > d 2Z= 0 ca, an 180°— tre a aE cos. —— 2 In order to exemplify this formula, which is M. Delambre’s, we will take his instance, (p.14. Tom. III. Base Metrique.) D represents Dunkirk, W .ooee--. Watten, = eee . + Gravelines, WM is a parallel to the meridian Db, of Dunkirk 212 By observation, (see p. 253.) the azimuth of Gravelines on the horizon of Rae = 90°21! 15" Or the angle GIVI eyes tel was oegs vas sels us GWD, the angle subtended by D and Gat W........=45 33 44.65 ate MW De «o's vee 2325) 12 20:05 If the diagram represented rectilinear angles on a plane surface, MWD would be equal to WDa, which would then be the azimuth of Watten on the horizon of Dunkirk: But the angles GWD, GWM are spherical angles, and WDa is to be com- puted as a spherical angle. The numbers, therefore, that represent 6, 2, L, will be as follow : Logarithms, the arc DW’, or 8, in seconds, is 822.43 ... 2.91510 Z = 180°—25° 12' 29".65; .. sin. Z... 9.62932 ate RE ee hg ge ee arith, comp. ... 0.20147 ihe = = 50 49 38-+6' 16” sin. ... 9.89008 a cpt OA Ue cos. ... 9.999999 + 2.63597 The number answering to this logarithm is 432.48 =7' 12”.48 ; ”, Z'=180° —(180°—25° 12’ 29".65) 4 7’ 12.48 =25° 19! 32”.13, which is the value of the spherical angle WDa, or that, of the azimuth of Watten on the horizon of Dunkirk. A similar process enables us to compute the azimuths at other stations, and, as the azimuths may be observed, the com- parisons of the computed and observed azimuths will serve as a — check to the operations and calculations. * [’ the latitude of Watten may be computed from the latitude of Dunkirk (Z), by the method of p. 266. + This value might have been omitted. * 213 The computed azimuths, such as have been just deduced, will also serve for the solution of what are the secondary tri- angles. Thus, the triangles WDa, Cab, &c. in which the points a, 6, &c. made by the intersections of the oblique sides with the meridian, and technically called Nodes, are secondary triangles, and serve, amongst other purposes, as verifications of the results obtained by the primary or principal triangles, such as DWC, WCF, &c. To examine the matter more nearly, we have, in the triangle DWa, DW determined from observation, and the measured base, (see p. 272.) its logarithmic sine being 4.11647.23980. The angle DWa, or DWC determined by observation, and = 74° 28' 45'.28 The angle WDa computed, (see p. 272.) =25 19 42 .13 From these data, two angles DWa, WDa, and the inter- vening side DW, we have, by Naper’s Analogies, (p. 180.) the angle DaW = 80° 11’ 33.27, so that the three angles of the spherical triangle D/Va are Dito re: 74° 28' 45”.28 DW GaP EN: 25.19° 42 13 BP EPO 80’ 11°33°.27 180 .0 .0.68 accordingly the excess of their sum above 180°, or what tech- nically is called the spherical excess”, is .68. * This spherical excess, as we have seen in pp. 195, 236, 244, is derivable from other sources than what was first thought to be its natural one, the area of the spherical triangle, It has happened here, as it has happened in many other cases. “« In a progressive state of the sciences we are enabled, or obliged to Mu ) take 274 In order to find Wa, Da, we have iS sin. WDa sin. Wa=ssin. WD. and log. sin. Wa=log. sin. WD......... 4.11647 .23980 : +log.sin. WDa ....... 9.63124.63656 + ar. com. log. WaD ... 0.00639.36863 log. sin. Wa, in toises*, ... ... 3.75411,24499 AP ei sin. DWa ; Again, sin. Da = sin. WD, annals i Wée. Sins ae log sn WD oe. 4.11647.23980 + log.sin. DWa ......... 9.98386.68657 +-ar.com. log. sin. DaW.. 0.00639.36863 log. sin. Da in totses ...... 4,10673.29500 add 4 ar. com. log.cos. (see p. 262.) ...... oy L1OS6 lopiare Da.) 208 4.10073.40586 — therefore the arc Da = 12785.98086. take different views of the derivation (we mean the scientific and philo- sophical derivation) of formule and theorems. Their genealogy seems continually changing. In the subject, for instance, on which we are speaking, the theorem for computing the spherical excess seems natu- rally to be derived from that by which the area of a spherical triangle is computed, and such was its historical derivation. But view the con- nexion of theorems as it is given in pp. 193, &c., and, we shall find, the theorem for computing a spherical area, is no necessary link between the formula for the sines and cosines of the angles of a spherical tri- angle, and that by which the spherical excess is computed. This last theorem is, (see p. 193.) an, (LBC 4 =4/{ tan. ZS. tan. 5 (S—a) tan. $(S—6) tan. § (S—c)}, derived from cos. 4, cos. B, &e. cos. mabe , &c. without the slightest aid from Albert Girard’s Theorem. It so happens that the spherical excess, and the area of a spherical Triangle may be computed by the aid of the same formula: and such, under this point of view, is the sole relationship which the two theorems bear to each other. They may be consanguineous, but the one does not precede the other, as ancestor pre- cedes descendant. * A toise =6,39406 feet, 275 in order to solve the triangle Ca, we have Ca=CW— Wa‘ Cab= DaW =80° 11' 33”.27 wCb, (= DCF)=79 48 35 .35 DCF being the angle between Dunkirk and Fiennes (I), observed at Cassel. This triangle being solved, like the former, gives the angle Bbha=19° 50’ 51.85, so that the sum of the three angles=180° 0’ 0.47, and the spherical excess 18 .47. By the same solution log. sin. ab = 4.07464.17054 add 4 arith. comp. log. cos. ab . 09563 log.arcab.... START SRET? *, ab = 11875.2470 but Da = 12785.98086 Db = 24661.22786 toises. This is the value of the arc Db of the meridian from the addition of the two arcs Da, ab: but it is plam that Db may be computed directly and independently of the former solution, from the triangle DCd, in which DC is known from the solu- tion of the principal triangle WDC: the angle DCb or DCF from observation, and the angle CDOd as being the difference of the observed angle WDC, and the angle WDa, the azimuth of Walton on the horizon of Dunkirk. These three angles will be DCb ... 143° 13’ 41.52 CDb... 16 46 27 .59 deduced from the two meet es Ye DAG Uw AO, SOC SL, Se angles and DC The log. sin. D6, (deduced as before, 1. 16.) ..... 4.39201.05977 diff. of sine and arc ...... 41244. log. arc Db ...... 4.39201.47291 » Db = 24661.2293 toises, from former solution Db = 24661.2279 0014 276 The difference of the two results is .0014 toise, or, since a toise = 6.3946 feet, the difference is one-tenth of an inch. The coincidence of these two results verifies the accuracy of the processes by which they were obtained. By the method just described, Delambre doubly CoH UNES the length of the arc between Dunkirk and Barcelona. The angles DaW, Cba, &c. are the azimuths of the stations, D, C, &c. on the horizons of the points of intersection of the oblique sides WC, CF, &c. with the meridian. But it rarely happens in Surveys, (it did not once happen in the French Survey conducted by Mechain and Delambre) that the stations of observation are on the principal meridian. But, as we have already explained, this circumstance is no obstacle to the measuring of the whole arc: since, if F’, for instance, should be the last rer we have =6 F . cos. 'Cbha, bF=CF—C8, and ae determined as above. What has been described in p. 273, &c. is one of the moby which Delambre used in computing the arc of the meridian. The triangles DaW, Cab, &c. are solved as spherical triangles. The results of the solutions give the snes of Da, ab, &c. whence, by a small correction, (the sines of small arcs beimg nearly equal to the arcs) the arcs Da, ab, &c. are obtained, Legendre’s method, (see p. 236.) is somewhat different. It is this : The spherical excess being obtained, one-third of it is sub- tracted from each angle, and the triangle is solved as a rectilinear one, and the deduced sides are from his theorem, (see p- 236.) the spherical arcs of the triangle. But simple as this process is, there is. some inconvenience belonging to it. The spherical angle DaW, in Delambre’s method, is equal to the vertical spherical angle Ca 6, and, as such, is subservient to the solution of the triangle Cab: but this equality does not subsist in Legendre’s method, because in that, the angle DaW, prepared for solution, is the observed or spherical angle Da W diminished by 4 of the spherical excess, which is the area of DaW: 277 similarly, the angle Cab, prepared for solution, is the spherical angle diminished by $d of the area Cab, which latter area will, B A a3 probably in every instance, be unequal to the former area DaW, For instance, the spherical excess in the triangle DaW, (see p. 373.) is 0”.68: one-third is 0”.227: the reduced angle DaW, therefore, in Legendre’s method, is 80° 11' 33”.043. The spherical excess in the triangle Cab is 0.47: one-third is 0".157; therefore the reduced angle Cad, instead of being 80° 11’ 33”.043 is 80° 11’ 33”.113. The principle, therefore, of Legendre’s method, in this circumstance, impedes, and ren- ders less simple, the process of computation. The method of computing the arc of the meridian, by the values of Da, ab, &c. (see p. 274, &c.) is called that of oblique- angled triangles. It was employed, both on the common, and on Legendre’s principle, by Mechain and Delambre; but the latter mathematician, (whose authority, on this subject, for several reasons, is great,) thmks the method much less expeditious than what he calls the method of perpendtculars, and which has been 2718 explained in p. 259. To this method or process there must, as it is plain, be joined another for computing the triangles formed by the stations: and he prefers, for this purpose, the method of chords: that is, having reduced the observed angles to horizontal angles, (see p. 221.) he again reduces the angles to the angles formed by the chords, (p. 239.) and solves the resulting recti- linear triangle. The arcs corresponding to the deduced sides, or chords, are found by means of tables. The very near agreement of the values of the arc of the meridian obtained by different methods, (see p. 276.) establishes the accuracy of the observations and computations, by which such values were obtained: but it does not establish the value to be a true one: for all three methods set out from the same value of the base. This base is the unit on which all the computed results depend. In the measurement of this base, therefore, the greatest nicety is required. The measurement of the base on Hounslow Heath took nearly five months: it was conducted by many able men of science; with the assistance of the greatest artist of his day: and what will serve as a kind of practical proof of its difficulty, is the description of its occupy- ing nearly 90 quarto pages in the Account of the Trigonometrical Survey. This part of the Survey, as every other, gave rise to many ingenious contrivances, and valuable experiments, and enriched both art and science. It was a fortunate circumstance, that a plain, like that of Hounslow Heath, could be found near to the metropolis. The levelness of this plain is such, that the ascent from the south-east to the north-west is only a little more than one foot in a thou- sand, in the distance of five miles. The base measured in 1784 by glass rods reduced to the level of the sea, and at the temper- ature of 62°, was found to be Feet. 27404.0157, 219 and the same base, measured in 1791 by means of an hundred feet steel chain, was found to be Feet. 27404.2449, so that the difference was 0.2312 foot, or about 2% inch. But, in an enterprise mm which it was attempted to measure the length of a large kingdom, within two or three feet, it was necessary to resort to every means of examining the accuracy of the original measurements, and the subsequent observations, No single measure could, for this end, be so well devised as that of measuring, at a considerable distance from the original base, a second base, and of comparing it so measured with its value deduced from the former, by the computation of intervening tri- angles. This step would do more than the double measure- ment of the original base: since, if the computed value of the base of verification agreed with the measured, there would arise a strong presumption, both that the original base had been truly measured, and the subsequent processes, of observation and calculation, rightly conducted. It would be much against probability, that a compensation of errors should have caused such an agreement. The jirst base of verification in the English Survey was measured on Romney Marsh; but little reliance was placed on the result. Indeed it is plain from the Account of the Trigono- metrical Survey, that far less pais were bestowed upon it, than upon the Hounslow Heath base. ‘The measured length of the Romney Marsh base was Feet. Inches. 28535 8.128 by computation 28533 3.6 the computed base, therefore, was about 28 inches short of the measured; a discordance not to be endured. Instead of measuring again the base of Romney Marsh, the conductors of the English Survey sought for a base of verification 280 i more remote from the original one, and in 1794, measured one on Salisbury Plain: its length was found to be Feet. 36574.4; and, by direct computation from Hounslow to Salisbury, the com- puted base was found to be Feet. 36574.3, or, probably, 36574.7, differing not more than 34 inches from: the measured value. The French, also, had their base of verification: and M. Delambre makes a comparison on this head between the respective accuracies of the English and French measurements.. The balance, (we need net wonder at it,) is, according to Delambre, all on the side of the French. But the French mathematician, no doubt from carelessness, which really seems to have been habitual to him, assumes that the English considered the base of verification on Romney Marsh to have been accu- rately measured: whereas it is plain, from the description of that measurement, that the contrary was the fact. ‘To conduct and superintend it, like the base on Hounslow Heath, there was no gathering together of London artists, of men of real science, and of philosophical diletanti regaled at Spring Grove; only two officers of artillery and their men were employed for the mea- surement. In p. 103, of the Trigonometrical Survey, it is expressly said, that the apparatus was defective, and the weather tempestuous: and again, p. 143, “we should have computed the distances in the vicinity, and to the eastward of Romney Marsh, from the base of verification only, but there are reasons to suppose that it was not so accurately measured as the other on Hounslow Heath.”’ It was. useless after this to recompute, on principles more exact than those of General Roy, the base of Romney Marsh from the base of Hounslow Heath. If the former had been inaccurately measured, no computation could make it right. Yet M. Delambre remarks, and rightly, that General Roy 281 arbitrarily corrected, (see p. 248.) his angles of observation, and computed his triangles, as if they were situated on a plane, recomputes the series of triangles from Hounslow to Romney Marsh on correct principles, that is, by reducing the observed angles to the angles formed by the chords, &c. and finds, in- stead of 28 inches, only 6 inches difference between the mea- sured base on Romney Marsh and the computed. But this labour, it is clear, was all thrown away; as long as there existed an uncertainty respecting the actual measurement, mere com- putations were out of the question. The fair way of judging of the respective accuracies of the English and French measurements, would have been to have taken the base of verification on Salisbury Plain, which the English themselves asserted to have been accurately measured. It is plain from the account given of it, that great pains were taken with it. Ramsden assisted and directed; and although the base, being on a sloping ground, required the measured hy- pothenuses to be reduced, yet the error of the measurement did not, on that account, probably, exceed 3 inches. The hypothenuses were first reduced to the level of the horizon, and, being at unequal heights, again to the horizon of Beacon Hill, the highest point of the slope. But Beacon Hill being 690 feet above the level of the sea, the measured base after the above two reductions would be too great, in the pro- portion of the Earth’s radius + 690 feet to the Farth’s radius, if the base be reduced to the level.of the sea: or in the propor- tion of the Earth’s radius + 588 feet to the Earth’s radius, if the base be reduced to the mean height of King’s Arbour, (118 feet,) and of Hampton Poor House (186 feet). To com- pute this reduction, let FR, the Earth’s radius, be 3481794 fathom, dR 98 fathom, then the reduced base is equal to 36575.401 . | a A but a , 1 ] but ————_ = ————_ = -_— Ty "Ra R ak Ree ew Nw 282 log. 36575.401 .... 4.56317 lOg 368 Veterans += +. . 1.99123 ar. com. 34817904 .... 3.45807 (1.029) .... 0.01247 Therefore, deducting 1.03 foot from the former value, we have the length of the base 36574.4 feet, nearly. The reduction, in this instance, was intended to bring the bases on Hounslow Heath, and on Salisbury Plain to the same level, for the purposes of verifying the measurements and opera- tions. By like reductions, the computed arcs were reduced to the level cf the sea. It has been already said, that the subject of the Trigonome- trical Survey has been selected, as affording ample matter for the illustration of the theorems and formule of Trigonometry; of this sufficient instances have been given. But it is far from being pretended to explain in the present Chapter the subject completely. Much that is necessary for the full explanation has been omitted, as being besides the plan and purpose of this treatise: for instance, the description of the construction and use of instruments, many of which were specially constructed for the occasion, and of experiments on temperature, the expansion of metals, &c. But this is not all. Even the mathematical part is ncomplete. The various computations have been made on the hypothesis of the Earth being spherical, which it is not, although nearly so. If the Earth’s form differed much from that of a sphere, all that has been imvestigated concerning the lengths of arcs, the azimuths, the latitudes and longitudes of stations would be nugatory. In the spherical triangles formed by the stations, the normals from the stations are supposed to meet at the Earth’s stations. But if the Earth be an ellipsoid, which it probably is, or, which it certainly is, some figure different from a sphere, then not only will the three normals not meet in the same point, but probably no two will intersect the axis in the same point. The spheroidical horizontal angles on the sur- 283 face, therefore, will be different from the spherical, of which we have treated. ‘Their excess above two right angles may not be the area of the included triangle: in short till the errors (for errors they must always be) are investigated, which the hypothesis of. the Earth’s spherical figure introduces, the results from that hypothesis cannot practically be adopted. But even in searching for the limits of the magnitudes of the above errors, we cannot proceed upon the surest grounds. ‘The Earth’s figure, certainly not spherical, is not certainly spheroid- ical: yet on the hypothesis of its being spheroidical, we are obliged, or rather inclined, to proceed to investigate how much we have neglected on the simple spherical hypothesis. In order to institute an investigation, we must take some regular figure, and we ought to take that regular figure, if there be any such, that most probably is the true one. Hence, it becomes neces- sary to Investigate the properties of an ellipsoid of small eccen- tricity, (if an ellipsoid, certainly one under that condition) to deduce the values of the normals at the several points of the surface, the values, in degrees, of the arcs subtending points on the surface, the differences of the angles of the spheroidal and corresponding spherical triangle. This investigation is some- what tedious and embarrassing, although its results with regard to the magnitudes of the errors introduced by the spherical hypo- thesis are not important, or important only, as shewing that the many results obtained under that hypothesis are, very nearly, true, and may be retained. It is sufficient for most of the objects of a Trigonometrical Survey to know, that the corrections due to the spheroidical form of the Earth are of little value: for instance, in forming maps of counties, or what are called Ordnance Maps, it is sufficient to suppose the Earth to be a sphere. A few inches of difference in the distances of places 15 miles asunder, can be of no consequence. But there are other objects, in which it is requisite to know the values of the corrections, in order that they may be used in calculation. For instance, in determining, from the results of 284 the Survey, the latitudes and longitudes of the stations. What was done in pp. 265, &c., produced only near results; im order to produce nearer, it is necessary to assume a spheroid of a certain eccentricity, and on such assumption to compute the corrections for the spheroidal arcs, &c. The few preceding paragraphs, are intended to state what mathematically remains to be done on the subject of the pre- sent Chapter. ‘That subject was chosen as affording the amplest illustration of the theorems, or formule, of Trigonometry. But it is not intended to pursue the subject farther, since it would lead us into investigations, in their kind and extent, not exactly suited to the nature and design of the present treatise. CHAP. XIII. On the Relations between the corresponding Variations of the Angles and Sides and Triangles ; and, on the Means of select- ing, in the application of Trigonometrical Formule, the Con- ditions that are most favourable to accuracy of result. Tue preceding Chapter contains some illustration of the use of Trigonometrical formule. These formule are applied to certain data or conditions furnished by observation. Now, the Mathematical process is sure and infallible; but all instrumental observation, in a greater or less degree, is liable to error. The practical result then cannot be perfectly exact: but it will not, necessarily, Be inexact to the full extent of the error of the ob- servation. That error, according to the conditions of the case, will be variously modified by the Mathematical process. If it changes it magnitude by changing the conditions, it will be least when the conditions are of certain values. Hence, if it should happen, that we are able to vary the conditions, it would un- doubtedly be expedient to assign to them such a magnitude, that the errors of observations should least vitiate the results: that, in the words of a Mathematical statement, the error of the result should be least with a given error of observation. These remarks stand in need of some illustration. The height of a tower may be determined by observing the angle which its summit makes with the horizon, and by measuring the horizontal difference between its base and the station of the observer. Now, in observing the angle, a certain error may be committed: but the error of the result (that which 1s Mathema- tically obtained) will vary as the distance between the tower and the observer is varied. If, therefore, we have it in our power, to regulate that condition, that is, if we can observe the height of the tower at what distance we please from its base, we plainly ought to select that which renders -least inaccurate the result. 2386 Again, in Astronomy, the time is determined from an observed altitude of the Sun or of a Star, from the declination and the latitude of the place. It 1s not then a question of mere curiosity to determine in what position, or part of the heavens, the Sun or Star ought to be observed, in order that the instrumental error, supposed to be of a certain magnitude, may least vitiate the determination of the time. The determination of the /east errors is only one branch of the general Problem, which assigns, in its solution, the relations between the corresponding errors in the data and results; that is, between the given errors in one or more of the conditions of the Problem, and the consequent errors in the results. Thus, the right ascension and declination of the Sun are computed in the Nautical Almanack from the longitude furnished by the Solar Tables and from the obliquity of the ecliptic. Now, the determination of this latter condition 1s subject to some error. If we assign a value to that error, we may theu investigate the corresponding errors in the right ascension and declination, and, im the result of such investigation, we should necessarily include the cases, in which the original error would least affect the values of the right ascension and declination. The errors, that hitherto have been spoken of, are, mathe- matically viewed, small variations or increments in the angles and sides of rectilmear and spherical triangles. Hence, an investi- gation of their corresponding values will comprehend a great variety of Problems that occur in Astronomy. Jor instance, it would assign the effects of parallax, refraction, aberration, pre- cession, &c. in declination, right ascension, &c. since the effects of these inequalities, always very small, may be represented by very small portions of the arcs or circles along which those effects originally take place. It is not here intended to extend this enquiry beyond tri- angles; but, there are a great variety of Problems belonging to other figures and other subjects of investigation that might have been included under the class of Evrores in mixtd Mathesi. This was the title which Roger Cotes gave to his Tract* on * Zstimatio errorum in mixta Mathesi per Variationes partium Trianguli plani et Spherici, 287 this subject; and Lacaille*, in treating of the same subject, properly describes the object of Cotes’s Tract to be the deter- mination of the limits of enevitable errors in the practice of Geo- metry and Astronomy. We purpose to treat this, as we have treated all the preceding subjects, analytically. Suppose the relation between an angle A and a side b, to be expressed by this equation, Sige ==). tat. 0 then, if A should be increased by AA, whilst 6 was increased by Ab (AA, Ab, representing the entire differences or in- crements of A and 0), the equation belonging to the changed tri- angle would be sin. (A+ AA)=m.tan.(6+A5), and the corresponding errors of A and 6, or AA, Ab, would be to be determined from this equation, which is the difference of the two former, namely, from sin. (A+ AA)—sin. A=m.$tan. (6+ Ab) —tan, bt. If we expand} sin. (4+ AA), the left-hand side of the equation will become d.sin. A d’.sin. A 5 endl te ee 2d t (NAY Pac. Now, in most of the cases that come under this enquiry, AA, whether it represents the quantity of precession, or of parallax, or of aberration, &c. is always a very small quantity: so small, that without vitiating the result, we may reject all terms involy- ing (A 4); (A A)’, &c.; in which case, the preceding quantity would become d.sin. A ar mAs * «Le but de l’Auteur est de determiner les limites des erreurs in- evitables dans la pratique de la Geometrie et de l’Astronomie.’ Acad. des Sciences 1741, p. 240. + See Principles of Analytical Calculation, pp. 72, 73. 288 In like manner, if the right-hand side of the equation be evolved and the terms that involve (A 6)?, (Ab)’, &c. be rejected, it will be reduced to d.tan. b 7 Ab. Hence AA, Ad, are to be determined by this equation, Go site Ay ay ota O ll 1b sayih If AA, Ab, should not be very small, or if considerable accu- racy were required, the terms involving (A.4)’, (Ab)” may be retained, in which case the equation will be d.sin, A d*® .sin. A ; 1. tan. b *, tan. ( an iNet an (Ay and for deducing A A in terms of Ad, or AOD in terms of A A, the d.sin. A d.tan.b GA alae (see Principles of Anal. Calc. p. 74,) are the differential co- efficients of sin. A, tan. b, and are respectively equal to cos. A, sec’ b. solution of a quadratic would be requisite. We have taken a particular form; but, if we assume a general one, the method will be the same, and the formula of solution similar. For instance, let X denote any function of A, and Y of b, and let the equation be A = mY then the equation for determining A A, Ad, will be dX dX —_———- a eee 2 — qa At Ted ge 4” Fes oes wiser GY F ) m{o Ab + 1.2.dp A> Re 289 and if AA, AB, are very small, dX dY 7° O4 = m—_. Ab, nearly ; dX Noe id Y; or, qq 54 => * 7p De nearly. And, in like manner, if V should be a function of C, and U of a, &c. and the finite equation of relation should be X+n.V+K&e.=mY+pU +&e. n,m, &c. being constant quantities, the equation of relation be- tween AA, Aa, &c., these quantities being very small, would be nearly dX dV adY dU pera LN a = mre : TA A+n TC AC=m Tb Ab+p Ja Aa In order to facilitate the solutions of the following cases, we will prefix the values of the differential coefficients of sin. x, COS 8 O00. 2-310. & d.cos. x . d.tan. x 2 + = COS. 7, ———_———__ = — sin. ©, ———*—* = sec. &£ dx ; ax der d.sec.x tan.x d.co-sec. 2 co-tan.x d.co-tan.x 3 Te! Pes oe > FS ,, = —€0-8€0.° 2. dr cos. 2 ad.x sin. # (Le EXAMPLE 1. In a right-angled triangle, of which one side 1s /, the other a, and the angle opposite h, 0, it is required to find the error or variation in h, from a given error in 0, (See Cotes’s Est’. Errorum in mexta Mathesi, p. 20.) Here, h=a.tan. @; .°. pity enced = d@ eS a. A@.sec.@ = pas a hAO ! tan. 6 h.A@ h.A@ _ ah. d0 tan. 9.cos.2@ — sin. 8. cos. 0 sin. 20’ eo d.sin, x. SOME Ue OCR , in fluxionary notation is sin, —.—. av ‘ v Oo 290 consequently, if A@ be given, Ah will be least when sin. 26 is the greatest, that is, when 0 = 45°, and consequently, when Fs Hence, if h represent the height of a tower, and A@ be the error of observation, it will be most advantageous to observe the angular height of the tower at ‘a distance about equal to its height *. EXAMPLE 2. In a right-angled spherical triangle, where C is the nght angle, and 4 is invariable, it is required to find the corresponding variations of the hypothenuse ¢ and the side 6. By Naper’s first Rule, p. 152, making the complement of A the middle part, and the radius equal 1, | 1X cos. A=tan. 6. cot. ¢; .. (see p. 201, 1. 12,) O=Ab. sec.2 b. cot. e— Ac.tan. b. co-sec.’ c; Ab tan. b co-sec.” C e Q D4 Te Reale CF Ac sec. 5 cot. c in. &. cos. 6 SID gag, COs aD sin. C. COS.C sin. 26 sin. 2c (see p. 11.) EXAMPLE 3. Let now c be invariable, and let it be required to find the ratio between the variations of the sides a, and 6. Make the complement of c the middle part, then, by Naper’s second Rule, p. 152, Ix) COSs ci COS. 0 COS.ui- * ¢ Commodissum erit ad eam distantiam (AC) observationem instituere ut angulus (ACB) sit graduum 45 quamproxime. Cotes’s Est’. Errorum, p. 20. 291 sra( ps. 201 seh 2) = — Aa.sin.a.cos. b—Ab.sin. 6. cos. a; Aa j cos. a °°. —_=>- sin. 6 x — = — tan. 6 x co-tan. a. Ab cos. 6 sin. a EXAMPLE 4. In an oblique-angled spherical triangle (SZP), if one side (PS), vary, it is required to find the corresponding vanation in one of the angles (SPZ). (See Lacaille, Mem. Acad. 1741, p. 242.) Let the angles SPZ2, SZP, 2SP be A, C, B respectively, and a, c, b, the opposite sides; then, see p. 143. of this Work, cos. A.sin. b. sin. c=cos. a — cos. b.cos.c; are by p- 201, —AA sin. A.sin. 6. sin.c+Ac. cos.c.cos. A.sin. b= Ac.cos.6. sin. ¢; Ac i ? A =——_—_——__——_ (cos. A .cos. c. sin. b—cos. 6.sin. c) sin. A.sinu. b.sin. c = Ac (co-tan. A.cot.c—co-tan. 6. co-sec. A). If Z be the zenith, P the pole, and S the Sun, then the above solution will, in the method of finding the time by equal altitudes, assign the correction of the time (A A) which is due, by reason of the variation or error (Ac) inthe co-declination. (See Astron. vol. I.) EXAMPLE 5. In the preceding triangle (SZP) if SZ (=a) vary, it is re- quired to find the corresponding variation in the angle SPZ (=A). See Est’. Errorum, &c. p. 21.) By p. 143. of this Work, cos. d—cos. 0 . cos. ¢ cos. A = = sin. 0. sin. ¢ 292 *. by p. 201, } sin. a —AA.sin. A= —Aa. sin. Ob. sin. € ‘ p sin. a But, sin. A=sin. C xX ——; sin. C AA i uN sin. 6. sin. c Hence, since 6 is supposed to be constant, and C to be variable, A A is least, (Aa being given,) when sin. C is the greatest, that is, when C is a right angle. If S, 2, &c. designate, what they were made to do, in the latter part of the preceding Example, then this solution determines the error in the time (A A) consequent on a given error in the ob- served altitude (90° — a), when from such altitude and the known latitude of the place, it is proposed to find the time; and, the solution also determines that the error in the time will be the least when C (= SZP) is 90°, that is, when the Star S is on the prime vertical. (See Astron. vol. I.) By similar processes we might find (as Lacaille has done, Mem. Acad. 1741, p. 248.) the effects produced in the right ascensions and declinations of stars, by the precession of the equinoxes. But this and like Problems require no new or peculiar principle for their solution, the first step of which (the essential one in this class of Problems) is to be made, as in the foregoing cases, by taking the differential or fluxion of each side of the equation (see p. 201, |. 12). The other steps necessary to produce results of a certain form, must vary with the conditions of the case, and consequently cannot be anticipated and prescribed by any fixed rules. Here it is intended to terminate what specially belongs to Trigonometry. In the course of the Treatise, considerable aid has been drawn from certain auxiliary branches of science : for instance, in almost every example, the processes and formule of logarithms have been introduced. Logarithms, it is true, have 4 293 neither a more intimate nor a more natural connection with Trigo- nometry, than with many other branches of science. There is no eminent reason, then, why the properties of the former should be discussed in a Treatise on the latter science. Still, since it is usual to treat together of the one and the other, the custom is here not departed from. And, accordingly, for the purpose of investigating the properties of logarithms, and for the discussion of some other subjects connected with the preceding matter, the following Appendix is now added. APPENDIX. Tr we look to those branches of Science that are mathema- tically treated of, such as Dynamics, Astronomy, &c. we meet continually with instances, in which it is necessary to multiply numbers together, to divide one number by another, and to extract the roots of numbers. The common rules of arithmetic are adequate to these operations: but the operations themselves, especially if the numbers consist, as is generally the case, of several places of figures, are very tedious. The conditions of the problems that are really presented to us in Natural Philosophy for solution, are rarely expressed by small integer numbers. They are most frequently the results of methods of approximation; and, as such, are necessarily expressed either by decimals solely, or conjointly by integers and decimals. For example, according to the method of determining the eccentricity (e) of a planet’s orbit (see Astronomy, vol. I. pp. 473, &c.) the eccentricity can never be exactly expressed: it1s merely the result of a method of approximation: it can only, therefore, be nearly expressed: very nearly by five decimal places, more nearly by six, still more so by seven, andso on. ‘The case is the same with other quantities: so that when we are obliged numerically to expand (which in practice we are always obliged to do) our formule, or the results of our mathematical processes, we have to multiply, divide, or extract the roots of such quantities as 1.016814, .983185, &c.(1 +e, 1—e); which operations, indeed, not difficult, are yet tedious; and, if of frequent recurrence, very embarrassing to the computist. There is, besides, this circumstance to be noted in these simple operations of divisions, extractions, &c. namely, that the operations 295 performed in any particular case cease, when the case is resolved, to be of farther use. The extractions of the square roots of an hundred numbers do not aid us in determining, with any increase of facility, the root of the hundredth and one number. Previous operations become not subservient to the abridgment of similar subsequent ones. The labours of preceding mathematicians are, in these cases, of no use to those that come after them. These inconveniences, (such as have been described) could not but be felt by the early Analysts: and, as it is natural, having once possessed themselves of sure methods of calculation, they began to seek after expeditious ones. After many trials and immense labour they discovered such, or rather invented such, by means of Logarithms. These have had various definitions assigned to them, and have been computed by a great variety of methods. They have, with no great propriety of language, been styled Artificial Numbers. They have no more title to that denomination than the square or cube roots of the numbers 2, 3, 5, &c. have. If 10°=3, wis the loga- rithm of 3, and is some number between O and 1, and must be expressed, for the practical purposes of computation, by some vulgar or decimal fraction. But a) 2 NS &c. are in the same predicament. There is no number that exactly expounds / 2: Its value (if we may so express ourselves) is between 1 and 2, but not capable of being exactly assigned: it is greater than a but less 3 141 142 than 3; greater than 75 but less than 75, &c. &c.; and these limits between which the value of / 2 is always placed, may be found either by the common rules for the extraction of roots, or by a series of tentative methods. The case is nearly the same with the equation 107=3. A series of limits between which x is, successively, still more and more narrowly placed, may be found by trial and the simplest operations: the value of x is between + ve 3 au 5: it is less than 4, hae than §: less than 1, greater than 76: less than +, greater than 5; or less than .5, but greater than -46875: and, in this way, we may make approaches to the loga- rithm of 3, with as much certainty as towards the square or cube root of 2, 3, or of any other number which is not a complete power. ‘The results are no more artificial in one case than in the other. | 296 It is true that the direct process for approaching to the value of x in an equation such as 10°=8, or 10°=2, &c. is not so simple nor so easily practised as the ordinary processes or rules for the extraction of roots. But if we examine the matter on those grounds on which all analytical calculation rests, there is no es- sential difference between the two processes. ‘They are in the same line of consecutive deductions: one nearer, indeed, to the common source than the other. We have chosen to consider logarithms as the values of x in the equation 10°=N, when N is represented by the several numbers fromO to 1000 and upwards. This, however, is not the form under which logarithms were originally exhibited, or need necessarily to be exhibited. It is the form rather to which (after many trials) as essentially embodying their properties, they have been reduced by analytic art. By such reduction all numbers are made equal to, or feigned to be equal to certam powers of 10, and the indices of those powers’are the logarithms of the numbers. But it is plain if we may assume such an equation as or 10°°N=N, We may also suppose 2°=N, or 3°=N, &c.; that is, there may be several systems of logarithms: alike in their general pro- perties, but differimg from each other by reason of their bases, which are the technical denominations of the numbers 2, 3, &c. in the equations 2*= N, 3°=N, &c. The general formula for the value of x in the equation N= a" cannot be obtained by any ordinary or simple processes. But there are particular cases in which, without any trouble, we may assign the values of x: for instance, if 2 should be the base, then since = 4 2% =8, 2? = 16, 2 = 32, &e. 2, 3, 4, 5, &c. would be the logarithms of 4, 8, 16, 32, &c. If 3 should be the base, then since Gy eet. Ceo ray = Bana, 9917 : 2,3,4,&c, in such a system would be the logarithms of 9,27, 81, &c. In like manner, in the common or Briggs’s System of Logarithms, in which 10 is the base, 2, 3, 4, 5, &c. are the logarithms of 100, 1000, 10000, 100000, &c. | } There is no need of calculation, not even of the slightest, in these simple instances: and, if we selected others, we might still, by very-simple, although tedious, processes, deduce or approximate to (and almost all the values are approximate ones) the values of the logarithms of numbers. For instance, if the base should be 2, then the logarithm of a number intermediate to 4(=°) and 8 (=2°) must be some number (using that term in its general meaning) between 2 and 3. Suppose the number to be 6=2": then x is intermediate to 2 and 3: if the arithmetical mean of 5 — 2, and 3, namely x, be used to represent it, then, since 2? = x/ 32 which is less than “36 (= 6) 35 or 22, or 2.5 1s too small an index. If we assume r= 2s, or 5, then since Q> = eT ima * *-+- &e letir= 1; and tp =1; ae a will have a peculiar value, and be =1+1+——+- aa &tc. = 2.7182818284, &c. call this e, then (e— 5. —F ie —1°+4¢e- 1 — &c. (which is the value of p in this case)=1. ; 1 In the expression for a® put x = -, then P = 1 1 a® =1+ 1+ = + &c.=e, and-~ = e?; PE a but if e~? = QR le , then, by the form (J), page 302, (- -1)- ‘Gouna ine @adyows =) -Fe=1P +EE- Ike. or, since the denominator = 1, p= Now, since a— 1 then Ce fraction would be respectively, ak rae! log. N = log.(N — 1) + log. (1 + ni =) = Lo (ieolda dd aig ati ty ac Roupar log. a ae cea 2 (Nae see (N= 1 kc.) by (L), page 302; and thus, the logarithms may be computed by series that converge with sufficient rapidity. For instance, 1 ] if N=13; log. 13=log. 12+-—- mire = ay Pid ena oP 3— &e. ) ] «, PEN celts BPE Grapes roa 2. ait e ee) if N= 23; log. 23 =log. 22 + —- pee zaa tse) 1 if N=29; log. 29 = log. 28 +— ACS 2.28" at ae) 308 In these expressions, the logarithms of 12, 16, 22, 28, are known from the logarithms of their factors, see p. 307: and when N is a prime number, N— 1 can be always resolved into factors. There are, however, besides the preceding, various other artifices and methods for computing logarithms. * But, as it has been in substance remarked before, the art of computing logarithms, and dexterity m that art, would, by them- selves, be of no use in expediting calculation: if, for instance, we had to multiply 31.523 by 17.81, and to divide the product by 5.4312, it would be a most long method of performing the ope- ration, to investigate the logarithms of these numbers. It is the circumstance of registering computed logarithms in Tables, and, by the art of printing, of multiplymg such Tables, that enables us to compute quickly. The calculation of logarithms is exceedingly operose; but one man calculates for thousands, and the results of tedious operations are made subservient to the abridgment of similar ones. By the methods already described, the logarithms of all numbers from 1 to 100000, are computed and registered in Tables. ‘Those in common use contain the logarithms of numbers, according to Briggs’s System, in which the base (a) is 10. Naper’s, or the Hyperbolic logarithms are so seldom required in numerical cal- culation, that it is more convenient to deduce them from Briggs’s, by multiplying the latter into the number 2.30258509299, &c. than to search for them in separate Tables t. But Naper’s System, 1 in which, (e— 1) —4 (e — 1°+4 (e— 1° — &c.=1 (e=base) is, apparently, so ‘ae simple, that there must exist some substantial reason for the adoption of Briggs’s. Now, in this latter system, the logarithm of 10 is 1, the loga- rithms of 100, or 10°, of 1000, or 10°, &c. are 2,3, &c. re- spectively; consequently, the logarithm (Z) of a number N being * See Principles of Anal. Calc. pages 142 to 183: Phil. Trans. 1806, p. 327: Bertrand, p. 421 to 676. + Thomas Simpson, has given a short Table of Hyperbolic Loga- rithms at the end of his Fluxions: in Callet’s and Hutton’s Logarithms there is a Table, of a single page, for SO ee common into Hyper- bolic Logarithms, 309 known, the logarithms of all numbers corresponding to N x 10”, or om can be expressed by an alteration in L of the simplest kind. Thus, if the logarithm of 2.7341 be .4368144, the logarithms of the numbers 27.341, 273.41, 2734.1,27341, 273410, are 1.4368144, 2.4368144, 3.4368144, 4.4368144, 5.4368144; that is, these latter logarithms are formed from the first by merely prefixing to the decimal, 1, 2, 3, 4, 5, which are called characteristics, and which characteristics are always numbers one less than the number of the figures of the integers in the numbers whose logarithms are required: the reason is this, 27.341 =10 x 2.7341; .*. log. 27.341=log. 10+log. 2.7341 = 1.4368144 2734.1 =1000 x 2.7341; .. log. 2734.1 slog. 1000+-log. 2.7341=3,4308144 and generally, log. 10" x N = log. 10” + log. N=m-++ L and, similarly, it is plain, that the logarithms of 9.7341 °° 2.7841" 2.7341 (2.7841 10 eee OO nat 000 4705.10.00? 27341, 027341, 0027341, -00027341,_ must be the logarithm of 2.7341, or .4368144, subtracting, re- spectively, the numbers 1, 2, 3, 4, which subtraction, it is usual thus to indicate: 1.4368144, 2.4968144, 3.4368144, 4.436814. The logarithm of a number (N), then, being inserted in the Tables, it is needless to insert the logarithms of those numbers that can be formed by multiplying or dividing N by 10 and powers of 10. that 1s, of Hence, we are enabled to contract the size of logarithmic Tables: and this advantage is peculiarly connected with the decimal system of notation. If there had been, im common use, scales of notation, the rodts* of which were 9, or 7, or 3: then the most convenient systems of logarithms would have been those, * The root or radix of a scale is that number according to the powers of which any digit, as it is moved more and more towards the left, increases in value: in our scale the root is 10: thus 723=7 x 10* 42x 10+3, 310 the bases of which, are 9, 7,3, respectively. For, in such cases, after having computed the logarithm of any number N, we could immediately, by means of the characteristics, assign the logarithm N : of any number represented by 9” x N, or on (root =9,) which numbers would, analogously to the present method, be denoted by merely altermg the place of the point or comma that separates integers from fractions. The root, then, in the scale of notation ought to determine the choice of the base in a system of logarithms. We may construct logarithms with a base = 3, and then, having computed the logarithm L of a number N, the logarithms of all numbers corresponding to N 2 IN, and di would be m + L, and — m + JL, and, there- fore, could be assigned by merely prefixing the proper charac- teristics; but then, in order to know the numbers corresponding N to 3” x N and ga we must multiply and divide N by 3, and the powers of 3. We cannot multiply and divide by simply altermg the place of the point or comma that separates integers from decimals: so that, in fact, not knowing, by inspection, such numbers as 3 N, 9N, 27 N, and 3” x N, we should be obliged to insert the logarithms of a// numbers in the Tables. A single instance will elucidate this statement: with a base = 3, the logarithm of 2.7341 equals .915519, then the logarithms of the numbers 8.2023, 24.6069, 73.8207, 221.4621 are 1.915529, 2.915519, 3.915519, 4.915519; for, the numbers 8.2023, 24.6069, &c. are produced by multi- plying 2.7341 by 3, 3°, 3°, 3’, respectively; they are known, however, only by actual multiplication, and consequently it would not be sufficient to insert in Tables of logarithms constructed to a base = 3, the logarithm of 2.7341 only; but, those of 8.2023, 24.6069, 73.8207, 221.4621, &c. must be also inserted: and it is plain, that the logarithms of 27.341, 273.41, 2734.1, 27341, .027341 must be also inserted. If these latter logarithms are not inserted, the computist would be obliged to undergo the labour 311 of forming them, by adding to the logarithm of 2.7341, re- spectively, the logarithms of 10, 100, 1000, &c. Se Saki toa base = 3. This is not the sole principal inconvenience that would arise from using a system of logarithms with a base not equal to 10, We might indeed, as it has been explained, by slight Arithmetical operations, directly find the logarithms of numbers from Tables of no greater extent than those which are in use; but, the reverse operation of finding the number from the logarithm, could not at all conveniently or briefly be performed: for, the logarithm proposed might be nearly equal to a logarithm which the Tables did not contain. These considerations will, perhaps, be suf- ficient to shew the very great improvement that necessarily ensued on Briggs’s alteration of the logarithmic base. ‘The real value of that alteration does not seem to have been duly appre- ciated by writers on this subject. For the description and use of Tables, in which the computed logarithms of numbers are recorded, the Reader is referred to the volumes of the ‘Tables themselves: and, as nothing seems wanting to the plainness and precision of the rules therein delivered, it would be a needless accumulation of matter to insert them here. The principle however of the construction of certain small Tables for proportional parts, that are nearest the margin of every page, requires explanation. The use of these Tables is to find the logarithms. of numbers, consisting of more than five places. See Sherwin, p. 6, Hutton, p. 128, first edition. Let the number composed of the first five figures or digits of the number N be 7; therefore, the number next to n, or which differs from 7 by 1, ism-+1; let x be the digit, which placed after the digits composing 7, shall make it N, then N=10n-+2, and . N=log. = : 1 a log. N=log. (10x + x) =log ve + Tah = log. 10n + log. € Ss aes ron 2 ta ome iar, 18 ae *(10n)° + ke. 312 xr 2 x =log. 10n-+- ———, if the t rs og dp SET eS Ua ies (ony count of their minuteness, neglected. &c. are, on ac- tron : n Again, log. 1 1 : = log. (: ae -) = a (neglecting 1 ee ‘cider ae dale pn = log. (x + 1) — log.n, and av x p Xx 107 “fe 10 jlog. (n-+1 — log. n)t, and from this formula the small Tables of the proportional parts may be computed: for instance, Let N = 678323, then » = 67832, and x + 1 = 67833, and log. 67833 — log. 67832 = 8314410 — 8314346 = 64, : 3 and since r=3, F; Slog. (2 + 1) —log.nt = Feiss 64 = 192, (or taking the nearest whole number) = 19: and by putting for z, 1, 2, 3, &c. we may form the small Table which is in the page containing the number 6783, &c. thus: Proportional part Proportional Part. in the nearest integers. ig he! | 6.4 6 2 12.8 tS 5S 19.2 19 4 25.6 26 3 32 32 6 38.4 38 7 44.8 45 8 51.2 ol 9 57.6 58 See Sherwin’s Tables, page 6, and at number 6783, and also Hutton’s, page 128. The above proof establishes the truth of the precept for finding the logarithms of numbers consisting of more than 5 places 313 of figures: the other precept* which directs us to find the number corresponding to a logarithm not found exactly in Tables, may be thus proved. Let L be the proposed logarithm, N the number: / the tabular logarithm next less; / the tabular logarithm next greater; n, n, their corresponding numbers. Let x be the difference of N and n, or let N=n-+2x; then x © _ log. N=log. (n+27)=log. n (: -+- =) =log. n+ log. G +=) L or, L=1+ fee aan “.c=np(L—2). Again, r- 1 ] log.2 =log. (n+ 1) =log. n (: -- -) = log. n +log. (1 +-) 1 1 L— ’=(4+—. », f—/=— See = or +o i—l ia consequently x a, and N a-r=n + ae from which expression, the precept (Sher- win, p. 8. Hutton, p. 130.) and the small Tables are derived: for instance, let L=.4414728, then (see the Tables), = 4414595, n = 27635 l' = 4414752, n’ = 27636 133 1 1330 “, L—l=133, l =157, and 1B7 «10 x 157 1 1256+ 74 74 1 7400 Sa ER CR Rg — =.8 + — x — =.8 10. 187 8+ 570 hoo © ison 1 6280 + 1120 100 ~ 1570 being the two figures given according to the Rule and Table, 8 corresponding to 125.6, or 126 the nearest integer, and 4 to 62.8, or 63 the nearest integer. = 6 + 04+ &.=.84-+ &c. 8 and 4 * Hutton, p. 130, first edition: Sherwin, p. 8, fifth edition. Rr 314 It may now be worth the while to illustrate, by a few more instances, the uses of logarithms; and this will be done chiefly with a view of relieving the Student from any embarrassment which the negative index or characteristic (see p. 309.) as it 1s called, may occasion. 7 | In the common system of logarithms in which the base is 10, 1 is the logarithm of 10, and O the logarithm of 1: consequently, every number that can be assigned between 10 and 1 must have for its logarithm a proper fraction, or (since a fraction may always: be decimally expressed) a decimal fraction. The logarithms, therefore, of 2, 3, 4.56, 6.9345, &c. must be such decimals as 3010300, _—-.47'712.125, .6589648, 8410152, &e. which, as it has been already argued (see pp. 295, &c.) are not to be called artzficzal aap but are FPS numbers such as make good the equations, 10° -3010300 = 9 : 11 0 Yar medanad =—3 103 ts 4.56, 10S hae. 3 6.9345 E The Logarithmic Tables contain, in fact, the logarithms only of those numbers which are contained between 1 and 10; and, from these registered logarithms, those of other numbers, less than 1, and greater than 10, are to be derived by means of the properties of logarithms. Thus, in the following extract from Sherwin’s Tables : Numbers. Logarithms. 1255 093 6437 56 9896 57 099 3353 58 6806 : 59 100 0257 the logarithms, with a Re cum point prefixed to their first figure, are respectively the true or real logarithms of 1.255, 1.256, 1.257, 1.258, 1,259, — ‘and since log. 12.55 = log.(1.255 X 10) = log. 1.255 +log. 10 = log. 1.255+1, 315 and since, similarly, 7 | log. 125.5 =log. 1.255 = log. 1.255 +2, &c. we find, by these properties of logarithms, that the logarithms of 12.55, 125.5, 1255, &c. ought to be expressed by 1.0986437, 20986437, 3.0986437 : and similarly, we may, from the logarithms of 1.256, 1.257, &c. immediately assign, by prefixing the proper indices or charac- teristics, the logarithms of 12.56, 125.6, 12560, &c. 12.57, &c. 1257000, &c. But if .0986437 be, as it is, the real logarithm of 1.255, the ri j 1.255 real logarithm of .1255 (since 01255 = —-) must equal 0986437 — 1, or —.9013562, ; : . 1.255 and the real logarithm of .01255 (since 01255 = ——) must equal .0986437 — 2, or — 1.9013562. These negative quantities, then, are the real logarithms of the above decimal numbers, that is, the equations, 1 iain ah 0018502 Arte palo 1255 = 10 > or = 109018862 — 1], 1 .01955 = 10 oh ce on 1919013568 1 001255 = 107 79019562) oy = Jo? SE » &ec. = &e. are, within certain limits of exactness, true equations. Now, although, by means of the registered logarithms, the logarithms of decimal numbers may always be assigned~by the preceding method, yet they are not immediately assigned: there intervenes, as an operation, the subtraction of the logarithm taken out of the Tables, either from 1, or 2, or 8,&c. Inorder to get rid 316 of this subtraction, the Authors of the Rules or Precepts for the use of Logarithmic Tables have devised a notation for negative logarithms (which the logarithms of all proper fractions are) by which the number or the series of figures assigned by the Tables for the logarithm of any number may be retained. They have chosen — to represent the logarithms of 11255, 01255, 001255, &c. nelther by .0986437—1, .0986437—2, &c. nor by the results — .9013562, — 1.9013562, but by i : eae ® 1.0986437, 2.0986437, Sc. and they apply a similar conventional notation to designate the logarithms of all other decimal numbers. The advantage of this notation is obvious: the same set of figures or of cyphers, which the Tables assign to a number, are to be forthwith used, what- ever that number be, whether an integer or a decimal fraction: thus, if, in the logarithmic Tables, 8785218, under the column marked log., stands opposite to the number 756, then, by the properties of logarithms (see pp. 298, &c.) and the peculiar notation, Numbers. Logarithms. 75600 4.8785218 7560 - 3.8785218 756 2.8785218 7.56 1.8785218 756 0.8785218 0756 “1,8785218 00756 28785218 .00756 "3,8785218 The first 5 logarithms are real numbers in which the figures and cyphers have, according to their order or arrangement, that significancy which they have in all ordinary arithmetical opera- tions: the 3 last logarithms might be called Artificial Numbers, since their significancy cannot be inferred from analogy, but is altogether arbitrary or conventional, 317 Now, this being the case, we cannot, relatively to these latter logarithms, establish any rules for operating on them, that is, any rules for adding to them, or for multiplying and dividing them, except by reference to what they are made to stand for. In so doing we make a recurrence toa kind of first principles. In order then to add the logarithms, 1.4329698 and 2.6901961, (which it is necessary to do in finding, by means of the Logarithmic Tables, the product of .271 and .049 of which the above quan- tities are respectively the logarithms) we must substitute the quantities they*stand for: thus (a) 1.4329693 = .4329693 — J (b) 2.6901961 = .6901961 — 2 1.1231654 — 3. Now 1.1231654 means 1 + .1231654; ; *. 1,1231654 — 3 equals .1231654 — 2, which, according to the peculiar notation, may be thus written - 2,1931654: which is the result that would be obtained by adding (a) and (6), and causing the unit carried over by the addition of 4 and 6 to destroy 1. Generally, n431 = 4381 — 27 m.752 = 6752 — m 7. 2+m+1.183 = 1.185 —n—m = .183 —(n—l)—m; but .183 —(n—1)—m may be written n—1 -+- m. 183; oe “. inn -+m-+1.183 the 1 is to be incorporated with the ” or m, and written thus, ey +m. 183, orm —1+7.183. Suppose it necessary to subtract the preceding logarithms (which it would be were it required to divide, by means of loga~ rithmic Tables .271 by .049.) 318 ~ then Bice (c) 1. 4329693 etande for .4329693 — 1, and (d) 2.6901961 6901961 — 2; by subtraction, the right result is — 1 ++ .7227732 + 1, or .7227732 which is the same result as will follow by subtracting (d) from (c) in the common way and by considering — 1 and 1 as the same. In order to procure instances for the multiplication and division of such indices or characteristics, as 2, 3, &c., suppose it were required to find, by the aid of the Logarithmic Tables, the value of (.0756)”. Now, by the properties of logarithms (see pp. 298, &c.) log. (.0756)° = 3 log. 0756 = 3. x 2.8785218 Now 2.8785218 stands for .8785218 — 2 (multiply by, 3) ene prarecit 2.6355654 — 6 but 2.6355654—6, 1s the same as 2+.6355654—6, which equals .6355654— 4, which, according to the peculiar notation of logarithms (see p. 228,) may be thus noted 4,6355654: the same result as will be obtained by multiplying 2. 2.8785218 by 3, considering 6(=3% 2) + 2 to be the same as 4. Since .6355654 1s the logarithm of 4.3208, 4.6355654 [which is the logarithm of (.0756)’], is the logarithm of .00043208: con- sequently, (.0756) = .00043208. When the index or characteristic is to be divided (which happens in finding the root of numbers by the method of loga- rithms) the operation is less direct: suppose it were required to find the value of (.0756)*. ] ioe Now log. (.0756)? = : log. 0756 = = (2.8785218). (eae 319 a But, 2.8785218 = .8785218 — 2 = 1 + 9785218 —2 —1 = 1+ .8785218 — 3 or = 1.8785218 — 3 Lo a) “. = (2,8785218) = = (1.8785218 — 8) = .6261739 — 1 or 1.6261739. But it is evident we may at once obtain this result by changing Q, into 3, then by dividing by 3, and, in the division immediately succeeding to that of the index, carrying 1, as a quantity borrowed, to the next figure. If the value of (.0756)5_ had been required, then, since it would be necessary to divide 2.8785218 by 5, we must make 2, 2, 5, and carry, as quantity borrowed, 3 to the next figure: for 2.8785218 = .8785218 — 2 = 3.8785218 — 5 1 — i * 2 @. 8785218) = ae + 3.87875218) = 1.7757043. ) _A few more instances, involving such characteristics as Q, ry &c. are subjoined. Required the sum of 3.6989700, 7,3467875, 1.4771213, 5.4313638, by separate additions, "73467875 3.6989700 54313638 1.4771213 12.7781513 5.1760913 12.7781513 7.9542426 320 or by one operation, | 768467875 5.4313638 3.6989700 1.4771213 79542426 Instances of Subtraction. Ist, 7.9788107 (=log. 00000095238) subtrahend 3.1549020 ( = log. 0014285) ———— 4.8239087 _(=log. 00066666). 2d, 2.2218487 (=log. 016666) subtrahend 4.6989700 (=log. .50000) 7.5228787 (=log. 00000033333) In the first of these instances the quotient (.00066666) arising from dividing .00000095238 by .0014285, is found by the method of logarithms: and in the second the quotient (.00000033333) arising from dividing .016666 by 50000. Instances of Multiplication and Division. 3 Find the values of (.05)° of (.0000625)#, and of (075218)5. Ist multiplicand 2.6989700 (the log. of .05) multiplier 5 '7.4948500 (the log. of .0000003125) -". (.05)” = .0000003125 ad, 4 | 5.7958828, the dividend (=log. 0000625) 2.9489707 =log. .088914; -". (.0000625)! = 088914. 3rd, 1.8763253 (= log. .075218) 2 5 | 1.7526506 (see pp. 230, 231.) 119505301 (= log. .89234); *. (075218)® = .89234. 321 The logarithms of decimal fractions (see pp. 315, &c.) are truly and properly expressed by negative quantities: but since (see pp- 316.) they are not commodiously so expressed, a peculiar notation with negative zndices or characteristics has been invented. Their meaning is to be derived not from analogy, but from the terms of that prescription that assigns them their meaning. We must refer, as we have seen (see pp. 229, Xc.), to the same source for establishing the truth of rules for operating on such indices. But there is another contrivance for designating the logarithms of fractions, in which no negative indices are employed. This consists in borrowing 10, or 100, or 1000, &c. and by prefixing, to the decimal part of the logarithm, the difference between 10, or 100, or &c., and 1, 2, &c. Thus, instead of “1.8763253, 7.4948500, &ce. the numbers 9.8763253, 3.4948500, &c. or 99.8763253, 93.4948500, &c. are written: but then, in these cases, to prevent ambiguities, or to derive rules for operating on these aréificzal logarithms, it must be noted or understood that 10, 100, &c. is borrowed. For, .4948500, 1.4948500, really representing the logarithms of 3.125, 31.25, the logarithm 3.4948500 would naturally and analogously be that of 3125; but it is made, according to the notation we are now de- scribing, to represent the logarithm of .0000003125. There will exist, therefore, in this, and in similar cases, anoccasion of ambiguity which cannot occur in the notation with negative characteristics. In order to establish any rules relatively to the last-mentioned method of noting the logarithms of fractions, we must, as in the case of negative characteristics, refer to the real quantities they are made torepresent. For instance, 3.4945800 (which, without any convention expressed or implied, would designate the loga- rithm of 3125) is made to represent the logarithm :.0000003125, it stands for 3.4945800 — 10 (= —6.5054199): and this mul- tiplied by 4 equals . 4x 3.4945800 — 40 = 13,9783200 — 40, Ss 322 which may be thus expressed, 13.9783200........40 being borrowed, or 3.9783200..++....30 being borrowed; whence we derive a rule for multiplication; which is, to multiply the logarithm (expressed by borrowing 10) by the multiplier (m for instance) and to reject the 10’s from the characteristic: the number then borrowed is m x 10 — the 10’s rejected. Thus, if 5.9635786 stands for .9635786 — 5 6 or .9635786 — (10 — 5) 35.7814716 represents 6 times the logarithm of een Again, if 5.9635786 stands for .9635786— —(10— —5), six times the logarithm equals 35.7814716, in which 60 is borrowed, or rejecune 30, 5.7814716 may represent it, 30 (= 60 — 30) being borrowed. In order to divide 9.7526506 by 5 (to take such an instance) 9.7526506 stands for .7526506 — 1, or 9.7526506 — 10; .. 49.7526506 is equivalent to 49.7526506 — 50; / ] ad ; (49.7526506) equivalent to Ee (49.7526506) — or 9.9505301 to 9.9505301 — 10; but the left-hand set of figures is made to stand for the right: therefore we may divide a logarithm (expressed by the borrowing of 10) by changing the characteristic (c) into m—1 x 10 +c, if mbe the divisor, and then by dividing by m: the quotient is the real quotient, 10 being supposed to be borrowed, or is the real quotient — 10; thus if 3.4948500 standing for 3.4948500—10, be to be divided by 5, add 40 (= 5 — 1 x 10) to the characteristic ERE it 43; then 1 5 (43.4948500) = 8.6989700. There is then no difficulty in finding rules for operating with 323 logarithms thus noted. But since (as we have already explained the matter in pp. 323, &c.) the logarithms, expressed by 10, or 100, &c. being supposed to be borrowed, have a meaning different from their usual or natural import, there exists a cause of ambiguity and some danger of confusion, ‘The logarithms expressed by means of negative characteristics are free from these objections: they have indeed, like the others, a conventional meaning, but they have only one meaning. The rules for operating with them are distct: and the only ec oaacy against them is, their typo- graphical uncouthness. | ; __ We will pass on to other investigations more nearly allied to the subject of the Treatise, than what has just preceded. In page 48, we gave, after several instances, the general form for cos. mA in terms of the powers of cos. A, but with- out demonstration. This deficiency will be now supplied. 1 1 If 2cos. A=x + - =p, then 2cos.mA =a" + —,. x Assume 1 1\” IN? iNet IN +—=(«+-) +8.(2+-) +B(2+-) +C.(¢+-) i L i x wy 1 m—2n +- 4 1 m—2n +2 , 1 m~2n +N,(«+-) +N(e+-) +N’.(«+-) H nh Lt Expand the terms on the right-hand side of the equation, then that there may be an identical equation, or, that 2” + = may be equal to 2” + = , the following equations must take place. (1) (m-+S) p™-*=0, yet + (m—2).8+ Bi pm-*=o, m—l1 m—2 (m—2) (m— 4) pita Amn. => et a Sten) B+ hp 0, + &e. 324A and similarly the coefficient of p”~ *” will be (2) m.(m—1) (m—2).... (m—n+ 1) Pig Qi Soe. te (m—2).(m—3).. rei (M=M) tee aa Ce art ©e¢e@e m—I 4 (m—4).(mM—5).» 00% EY / ome | medal | B eee ee eee eeeeece n—2Q + &c. + (m—2@n+2).N +N, which must also be equal to nothing; from the equation (1), we have S=—m an m .(m—3) ise Cae m .(m—4).(m—5) Lots 3 therefore, 7f N followed this law, we should have N= m.(m—n).(m—n — 1) .e0e (m— 2n+3) re Loe Soc ees tid) : Hence, the equation (2), written over again, after substituting the above values of A, B, C, &c., and making each term to have the common denominator 1.2.3 ....(m—1).m, becomes O.= m.(m— 1) (m— 2) weeeee (m—n+1) 1, OD Mee acl gla othe anne m.(m — 2) (m— 3) wee. (mM— 2) Artec) ty plcatalad’s hiece ete X sess (3) 325 4, m+ (m—4) (m— 5) wees (M—n—1) n.(n—1) LS he eietc ot akeinen « 7 1.2 — &c. Doel) ied acide Oa 2 8) ae OR aaa IG otetcts sale eel caehetonc n + Nj. Now the sum of all the coefficients preceding N’ a m.(m—n— 1) (M—N—2@) wevese (m—2n-+1)- TRC OR Mi ee ee z For, ot Ben at head) eee n.(n—1) (n—2 gm gm 9 Peele — ee Xe. = be BY Ty cute ie i eich oi —-) z in which the upper or lower sign is to be used accordingly as m is even or odd. Take the differential of this equation, divide by dz, and eaten RAUL TMT. Gs (m—1) 2~? — (m—2)*n2™~ > + &c.—(m—n) = (m—1) mo, (: *) + neg, € ~ -) Repeat this process till the differential of the original equation r—1 ; et Lie be taken n—1 times, then the index of 1 — =~ im the last term | z will be n—(n— 1)=1, and all preceding terms produced by the . e ° e 2 ] e process of differentiation will involve powers of 1 — - higher z than the first, therefore 326 $(m—1).(m— 2) eee ra es (m—n+1)}2"—" a 4 (m— 2) (m—S)... ee (m—n).nt ym—n—} MANOA or ite +{im—3)(m—4) er eelals (m—n—1). et — &ec. i {(m—n).(m—n— 1) States (m—2n + 2) nt} gm nth + {(m—n—1)(m—n—2).... (m—2n+1)} 27” nn al we -) +a = 1)" +a04V (1 —< -) P, Q, V, &c. involving powers of z. Let z=1, then the right-hand side of the equation = 0, and if the: coefficients of the left-hand side be multiplied each by m IT ie he anion ees n’ the resulting terms will, excepting the last which is (m—-n—1)(m—n—2).... (m—2n+1) ‘ 1.92 oeooneeseesane y be precisely the same as the terms of the equation (3), except- ing the last N’; the two last terms, therefore, of the respective equations are equal, that 1s, Ne m.(m—n—1)(m—n —2).. + (m— 2n +1) MeO ois ele tae coeeee N This formula, therefore, expresses the law of the series for LA rine | he a” + —, since it has been deduced on the supposition that © N, the coefficient of the 2 term, and the coefficients of the preceding terms, are formed according to that law: which they evidently are, since by making n=2, 3, 4, &c. we have fag eee) Cie Mm .(m—4).(m— 5) 1.2.3 “ ne m.(m— 5) (m—6).(m— 7) Leo 34 f &c. and the above kind of inference is generally expressed by saying that if the law be true for the n™ term, it can be proved to be true for the (n +1)". The relation between the n™ and (n-+ 1)™ term is this ; (m — n)n DN oor NG : (m—2n+ 2) (m — 2n + 1) Let m=2n, then ee rah) ies sieges Bissly vis Pe2723 evecesosveee ve Mm Ld Soret mio GC 1) m* (3 ne ) m.(m* — 27) Mee om eS A eat similarly, N = m*. (m? — 2°) (m? — 4”) Heine 3.4.5.6.2° ; &C. Hence, if we revert the order of the series, and begin from N'p™-*" or N'p® or N’, we have 328 1 a” -+- — = 2cos.mA=+ > xr = aA Mh Dien: (mi 12") pe : 2.9? 2.3.4 2 3 me. m — 97) .(m* — 47 ( De £ seus a DO) Bete core O the upper sign taking place, if m be 4, 8, 12, &c. or of the form 4S, the lower sign taking place, if m be 2, 6, 10, &c. or of the form 48+ 2. Again, if m’ be odd, or m=2n +1, Nae m .(m* — 1) (m? — 9) N= ‘ BB SAB MOF 7 Xe. 5 *,co. mA= ln I = sige ie PMc AaG Pe 2, ees 9, ae A m.(m* —1). (m? se = + &e. OSS CAS where the upper sign takes place, if m be 1, 5, 9, &c., or be of the form 4S +1: the lower, if m be 3, 7, 11, &c., or be of the form 4S-+3. The two last formule for cos.mA are derived from the original one of 1. 1, which, as we will now shew, is the source of other formule. For the sake of introducing symbols in some degree sig- nificant of what they are intended to represent, let c ae s be the cosine and sine of an arc A: then c=cos.A=p, and the three preceding series will be 329 cos.m A= ((ed™—m(edn—? FA) (9, yn 4 Be.) (a) or, (m odd) | mM. ao) a 2 m (m?—1)(m? Or, =+ (me - Bere a) 3 x m ,, mm —4) S| pete wie Tamrato sane ang — Ke)... (c). Since the expression (4) is true whatever be the arc, let the : 7 T : arc, instead of A, be Sn A, then cos. ( —_ 4) = sin. A=s,and Tv (Ne F : COs. m G _ A) =c0s. CG _ mA) = + sin.mA, (+1f m be of the form 4s + 1, —if of the form 4s + 3:) hence, in all cases, (m odd) | m(m’—1) , . m.(m>~ 1)(m*—9) 2 Se ae — &c. (d) 1 S03 {COR Saris sin.mA=ms — Take the differential or fluxion of (a), and d(cos.mA)= —sin.mA.m.dA, and 2”.d(c”) = Q” x mc™—*. dc=—2". mce™—!5. dA, &e *. dividing by md_A, we have, since s enters each term, sm. mA = (m—3)(m —4) S ((20"-"— (m—2) (20)" "+ 3 (2cy"->— Ke.) (e). If we perform the same operations on (6) and (c), we shall have (m being odd) sin. mA = ! eer a Tab bale es(QQ-7 ep ce 9) 4 80) verse and (m being even) sin.m A= m (m — 4) m (m* — 4) (m” — 16) £8 (me— eee ee gee - &e.) (8) dhe 330 If we perform a like operation on the equation (d), then, since d(sin.mA)=mcos.mA.dA (p.110.), we shall have " one | sg ae Pike cos. mA=c ¢ _ ——— $* pein? Ke). watt). R ‘ , = Tv If in the equation (c) we substitute, instead of A, ea ‘A. aia . = COS. G o_ 4) = sin. A=s, and cos.m G = A) = cos. Ge _ mA) = +cos.mA then (+, if mis of the form 4s, —, if m is of the form 4s +2). Hence, in both cases, m? (m* — 4) an! 2 m cos. mA=1 — — 3 + Ov 3 A ma: Ce is ea stete ts _ Take the differential or fluxion of this equation (7), and divide by md A, then we have (m being even) sin. m A ) m (m° — 4) 3 (m* — 4)(m?— 16) ° ) =o (ms ULE gam Bt mee (ads These formulz for the sine and cosine of the multiple arc, (ten in number,) require not, as it has appeared, separate demon- strations, since the nine latter are derived from the first (a). If in the formule (a), (2), (e), (k), we substitute for m, 2, 3, 4, 5, there will result, as el instances, the forms designated by (cl), (c™), (c7”), (c” ), (s""7), (s7”), &c. in pages 47, 48, 49; and if in (f), (A), (2), we pxpgung m by different numbers, we shall have m=3 sn. 3A=—s(1—4c’) m=5 sin. 5Az= a Se rom \t) 3 Xe. 331 m=3 cos.34 = — c(1—4s°) Fe. i m=5 cos.5A= c(l— ley i) rp) &e. m=4 cos.4A= 1-8s + 8s? ? f , mG cos. GA=1—18s° + 48st— 3255 Ah &ec. which particular forms were not deduced in the above-mentioned pages. Series for the Sine and Cosine. By the form (d), p. 329, m.(m°>—1) 5, m.(m*>—1)(m’—9) 3° 1.2.3 Za diet.O — &e. sin. mA=ms— Let A be very small, and m very large, and such, that md=z, hy Rs eee x y : i then s=sin. A=sin. — = —, nearly; and m’—1, m—9, &c. mm m™ 2 ° =m’, m*, &c. nearly; .°. sin. mA, or, cies cr mM. m = (- = Oe al mem .m Cy ie pee hi —~}) —&c. sin. v=m. Pag > | eae iat (v) ms ee SAPS OF Te e@e¢e 6 OB Bee UV). PUTEE Gg tO ig Mig Dae Bitte SUE We shall have the same result if we take the series (k) in e ° xv oO which m is even; for c=cos. A =cos.—=1 nearly, and m —4, m m — 16, &c. =m’, m’, &e. If in the series (A), or (2), we make the same substitutions as we have already made, we shall have x x cos. mAs Or cos fs ee a Re. Ga) “Niet Wo io 874 (u) de Instead of computing sin. 1’ by the methods given in pp. 70 and 72, calculators, in the construction of Trigonometrical ‘Tables, have employed the preceding series (v) for the sin. 2, and under this form, oa? + &c. 3 Tw 5 * gd ao nr Or eee Se Al he pare: ee substituting —. a instead of x: they have also, availing themselves 7 of previous computations*, taken 7 = 3.14159 26535 89793, and accordingly have been able to represent the above series, and the series (u) for the cosine, with numerical coefficients, after the following manner: * Dr. Horsley, in his Elementary Treatise on Mechanics, p. 153, says, that this is ‘‘taking things in a preposterous order;” and, un- doubtedly, it would be so in a Treatise intended specially to explain the principles of the construction of the Trigonometrical Canon, but not in a Treatise giving rules for practically constructing it with as miuch ease and conciseness as_ possible. 333 é m Oa m odin * Sin, 7 90° = cos. 5 90° = 1.57079 63267 948966 —...[1] | 1.00000 00000 000000 i 3 m? — 0.64596 40975 062463 — [2] | 7 1:23370 05501 361698 4 + +++17969 26262 461670, ...[3] | ++0-25366 95079 010480 = 7 6 —..+,.468 17541 353187 —...[4] | — ...2086 34807 633530 ~ 9 m8 tesseeeee6 O4411 847874 — “b+ «+91 92602 748394 — mt gr? — vapedeasus ODUGS t92002 — 05000002 52020 423731 ee n n m3 gm}? See @eevce 569 217292 has oa erveeesse ee 4710 874779 “12 n n 15 | 14 mets Sibsesesss60 6 O88035 0 BAF) OSES S003 lees ns nit ip. yn}? mi wt ASTOR PR ONC oR, 7 BN es, ciate Neier 656596 =e m'9 5994, ms cas ct eRE PTCTEE TOO REL YC hs: no “pe ee @eoee ones e toe 9 nis 2° +.. 54 eT 3 a4 : 7 oy 86 . From this series not only the sin. 1, but the sine of any arc m I : may be computed ; for instance, let — = —, then, computing n 10 | exactly as far as seven places. * Euler, Introd. ad Anal. Inf. p. 99. Callet’s Log. 27, 28. since a i. [1 ]=.157079632 5 since —=.00001, [3]=.000000796f The fourth term [4] and the re- at ee, maining terms produce no 157080428 significant figures in the 9th 3 since ~ =.001, [2] =.000645964 8th, &c. places. “. sin. 9° =.156434464 or, in nearest numbers, as far as 7 places, sin. 9° = .1564345., See p. 79, 1. 6. : age Ae : From the series for sin. —90°, we may, by assigning different nt m Ate values to — , deduce as many formule of verification as we please ; n for instance, suppose we wish to know whether sin. 20°, computed according to the methods of pages 73, 74, be rightly computed; make oy G0 == 203 ..". eG: which value is accordingly to be m.. : substituted for — im the several terms of the preceding series n _ om for sin. — x 90°. n The sines of arcs, deduced by the preceding series and the formule of pages 73, 74, will be expressed in parts of the radius, and be, what are called, natural sines; but, computation is usually conducted by means of logarithmic sines, which latter may, by the aid of the common logarithmic Tables, (if the log. sines are required to seven places only,) be computed by taking the loga- rithms of the numbers that express the natural smes; and, in order to avoid the inconvenience of negative logarithms, (for if the radius = 1, the sines are all fractions and the logarithms consequently negative) the Trigonometrical Tables are con- structed to a radius = 10'°, the logarithm of which = 10: so that, imstead of 1.6006997, the logarithm of .3987491, which is the natural sine of 23° 30! to a radius 1, 10 +-(1.6006997) or 9.6006997 is made to denote the logarithm. 339 But it is not absolutely necessary to compute the logarithmic from the natural sines; and, indeed, if the latter consist of more than 8 places, their logarithms cannot, from the Tables in common use, be obtained: on this account it becomes necessary to shew, by what means independent of Logarithmic Tables, or re- quiring the aid only of the Tables that are in ordinary use, logarithmic sines may be computed to any degree of exactness. 3 5 | x B the fi '. in, t= re — Bah gk yee y the form (v), sin . 268 OS h4es ; 2 4 =2(1 eeu a epsteiteds Ma Ma &e.). Oa 9.3.4.5 Now, for the purpose of finding the composition of 2 1— F + &c. put sin. xz = 0, then (Table, p. 16. making A=0) x may be 0, or a, or 27, or 3a, &c. and x may also be either — w, or — 27, or — 3m, &c. Hence, viewing the above series as an equation, O, 7, 27, 37, &c. are its d 1 ; roots, or if we put — for xv, and then reduce the equation, so that the term of the highest dimensions (y” for instance) stands Pisane ] I 1 1 first, -,—, —, &e.— -, —-—, — —, &c. are roots of 9 a’ On’ 3a wT Qa’ 3a’ Eis 7 Pea aah + &c.and consequently, for reasons like those stated in 1 1 1 1 qu p- 59, y— ay ee Key + Das ee &Xc. are divisors of the equation: hence, “O-D GHD) O-#) G+) or y” (: TE Sy > + &e. yar > C+) &C. or (dividing s y”), 1 ] ee eee peace? etoile a 1 aoTey & ° eh, ane ¢ =) ( a a . and .°.. Hence, sin. v=a@ | ( mn om up or uttng x = —.- > \P 6 n 2 Cage) Cae _ mn. ar and consequently, log. sin. Bio n p14 i 3|3 m e ] ye —} &Cs S ( 10n a By a like decomposition we shall have m 1 Oe oe 4 n | da g Rolls f3 eke m o1 m m and log. nas an log. (; sey + log. (: = 5) + &c¢. &c.* O85 ne mM * If in the expression for sin. — n " } . (N—-m x Se (ge tho m 7 then, since sin. .~} = sin. (~ — ——) = cos,—.-=, we have n 24 2 nm 2 cos. BF A (AE) ) OE) CRA ac. Itwe nm 2 n 4A m equate this expression for cos. ae 3 with the former one, and divide each 1 bythe commonfactors, there results Pract eo Oe ahs ol 3.3.5.5.7.1, XC. 2° 2,2,4.4.6.6, &e. 2.2.4.4.60.6, Oe : . whence 3 a ee which is Wallis’s expression; and many other curious results, which are not, however, the proper objects of this Treatise, might be obtained. 337 The preceding series may be differently expressed: since* i ( = log. ( —_;— log. 4 An? =log. (Qn-+m) (Qn — m) — log. 2n = log. (2n +m) +log. (2n—m) — 2log. 2—2 log.n; .. log. sin. - .90°= log. + log.m — 3log.n — 3 log. 2 xy log. (2n-+m) + log. (Qn—m) Q 2 ee) + &c. + log. ( and by steps exactly similar we may obtain log. cos. . . 90° =log. (n —m) +log. (n-+m) — 2 log. n g.(1- : =) + Ke. + log. (-- and since, (p. 302), 2 2 * Lor. € _ a); or log. (1 ~~ saa) is not expanded, as the : : m” : similar expressions log, (1 _ ye &c. are, and for this reason; if 42 n* expanded, it would increase the coefficient of nie ga? now = 1 ————- = 00000009, &c. or the significant figures would come in th 1048570 9 "8 Fr eae i 1 1 420 ~ 1099511627776 and the significant figures do not come in till the fourteenth place: if, eighth place, whereas — .00000000000009, &c. 2 therefore, log. G ae =) had been expanded, or the powers of — 1 yy retained in the computation, we must have computed a greater number of terms, (see succeeding series, p. 333,) in order to have had the series exact to fifteen places of decimals. Uv , m° lem m* m® log. CF) oat + saa t ae} n 2.4'n* § 3.4°n8 2 2 4 6 1 § m m m log. (1 PD er lent * 2. Gn! sore + 8} 2 zg 4 6 m 14m m m ale 870° p Sn? ng 2.8'nt * 3.8 ‘i 2 4 : : mm : If we sum the coefficients of —-, —,, &c. taking the columns n° n vertically, that is, if we find the arithmetical value of H 1 1] m . 4342944, &c. zB +- e + ge a e.} (coefficient of =) 1 lop! :. i $.4342944, &C. ia + 6 + 3 + we} (coeficient of —) : Le 1 1 V : m° 74342044, &c. x +- 68 th 3 + &c. (coefficient of “) » and add 10 the log. tabular radius, we shall have two Tables resembling those given in page 333. 339 5 BRI ey! ie log. sin. aa 90° = log. m +- log. (2n—m) +log. (2n+m)—3 log. [a]+9.59405 98857 02190 gn” [1] — 0.07002 28266 05901 * : 4 [2] —....111 72664 41661 ~ 6 [3] —......3 92291 46453 — aeoeed¢e m8 »»17292 70798 a mio + »-843 62986 >i0 miz a: 48715 — log. cos. * 90° = nN log. (n— 7m) log. (n--m)— 2 log. + 10 gm — ...0.10149 48593 41892 — Bg Oe eae: 318 72940 65451 —] m® - + +204 +420 94858 00017 — § i ouie ..1 68483 48597 — 7S. m0 —seee sees 14801 93986 —| m2 see ee 2000 .. 1365 02272—7, | m4 — PP yer A. | 81715 m6 sr NEL gee are 12 61471 — mmm oe oes 82e@ ees aS f 24507 —, m*0 “*eessa se ee serene 12486 — me R®ecosa cece esn 2eserver see +1258 3 “=eeet oe Sal elerala’e be chase Ae Oe From these series, may logarithmic sines and cosines, inde- pendently of the values of the natural sines, be computed to 15 places ; and, this inconvenience is avoided; if the natural sines had been taken, consisting of more than 7 places, no Tables in common use would give their logarithms. The logarithms thdeed of the numbers m, 2n—m, 2n-+-m, &c. are supposed to be taken to 15 340 places, and these can be had, since the numbers will not consist m of more than 6 figures: for — cannot exceed 3*; ; therefore, since n n = 90.60.60 = 324000, n Ts m, 2n + m, &c. cannot exceed 1000000. As an instance to the preceding formula, suppose the loga- rithmic sine of 9° to be required: here m=1, n= 10. .*. log. m or log. 1=0 log. (2n—m) or log. 19..... Er A laine = 1.27875 36009 hop. (2 2-1) OF 1Oe. ea one eye Means set =1.382221 92947 3 [OY ses cte cary: ob ee oe) =O Osis Osea 12.19503 27813 [d] Teh P 1. Oe ee es ae fae 00070 02282 BO Eilstate: oho alpdalys’ o's Seite GMa nels ta ee eee 1117 oS MOUS 1 Oi ee ih eee ds sce eee seh men mee Se 3.00070 03399 [e] ». log. sin. 9°, that is, [d]—[e] ... 0.23... =9.19433 24414 This is the log. sin. 9° to 10 places: and the decimal part is the logarithm of 15643446 the natural sine of 9°, found, p. 76, &c. We will now ps some other instances, and find the loga- rithmic sines of 1”, 45°, 2' 3”, and of 1! 3”. * If — ~ > 3 the series for the cosine would be used for computing the sine, since sin. (45°-+- 4)=cos. (45° — A): it is obvious, that the logarithms of m, n+ m, &e. may be dispensed with entirely, by ex- ee me Stig panding log. (1 — a} ; but then, to attain the same exactness, we must make the series consist of more terms, It is also plain, that instead of fifteen places in the numerical coefficients of the series, any number may be used, See Callet’s Logarithms, p. 48. 34] 4 Logarithmic sine of a, | 90° x 60x60=1; +. m=1, n=324000. ; Numbers. Logarithms. en ata isia tate tate Pili fe ci sie ie ote 4 0 Dm We adens O41 990 a0 a veld eyes 5.81157 43357 Qn-+-M ...0.6- GABO0T wid es os ale = 5.81157 56761 Raheee in) aaa Boe 9.59405 98857 21.21720 98975 Ee Poe eee ANS Behe FOP DUO ues esere: chad own 16,53163 50306 5.68557 48669. The other terms [1], [2], &c. by reason of the large divisors ?, n°, &c. produce no effect. on the result obtained; therefore, as far as ten places of decimal figures log. sin.1” = 5.6855748669 Logarithmic Sine of 45°. m 79 Sap Ss Ms hy fb oe 2 Numbers. Logarithms. Tb: ool artiege etenats eos 1 OR ERR Ny A i 0. Di ME eas cee eT GES SEN, Ro SE eae 0.47712 12547 — QN-M w2. ceoses Se A a 69897 00043 [Gi faec ire orm olets n ths cee einem etn. 2 9.59405 98857 [m] .... 10.77015 11447 “ [LJ = 7 - (07002 28266 ) ....... 01750 57066.5 1 [2] = 1662 (LiL COMA a yy, wc eat 6 98290.3 | 1 [3] = 64 sg 4 Sao bap rete str a 6129.5 1 Lad oe se fe ( Ly a3) Be) Vokes ans 67.5 | 1 . [5] = aes 843.6) eeee eee 8 01757 61554.6 BS Loge ae PO egy Te ee COR eS -90308 99871.1 ; 92066 61425.7 but La owe waren Meter yok De te bs 10.77015 11447 value of log. sin. 45° to 10 places ...., 9.84948 50021 342 This instance has been selected, not that the method of solving itis the most simple, (for the instance is a particular one), but as one which shews the great convenience of the series. Logarithmic Sine of 2 3”. ba m 123. 41 7X 324000 = 2 3"= ae NAN 5 es re oat) os 108000" Bninhers Logarithms. NB \s\e's wale ed's Deke tie a Ih cccoms bale ». 1.61278 38567 Qh see UB ested alos 216959 vs dees wo 5.33437 33078 QZN4HM wercccess QIGOAL ...ceeceee- 5.93453 41787 [a] Soe ates ¢ aa es ee ey ae Pea ac 9.59405 98857 [9]... 21.87575 12289 To find the term marked [1], we have log. m .......«. 1.61278 38567 LOR. vesences es WO0Gt2 a7 005 6.57936 01012 ogee) wniwowean ase. aes 3.15872 02024 m* log. ... 0.07002 282 ... 8.84523 95006 2.00395 97030 ... No. 0000000101 [1] eee eeeee @eneeeeeesee 00000 Oo101 BDO MEAL Cy nar ie oe. « 15.10027 12665 15.10027 12766 DUE OT ee ener cts. SLny OT ote coU 6.77547 99523 ... the log. sine of 2’ 3”. Logarithmic Sine of \' 3”. m Hi As ad -63 7 . m ~ 324000 ~ 324000 ~ 36000* ** m=, 7=36000. Numbers. Logarithms. | AY COIR Bag Pe Mees hk 0.84509 80400 Baers Ginpoupweee TIONS wees. etd NR 720) 09718 Dal micpeagy ltl. 720074 os. us nen 4.85737 47175 1 Rea nes isu ae AU 9.59405 98857 [7] «1. 20.15382 29145 | 343 A gelin si1OD) 1850 oa) he oe ce winnie s+ ° 84509 80400 Per Gk nee? A ee 4.55630 25008 lop. ee MOTB 25 78 6.28879 55392 n log. TE Se Nac WR 2.57759 10784 n log ss ve. 07002282, &c. ... 8.84523 95969 1.42283 06833 ro NDE oss Pe ea con't ofS .00000 00026 9 (in Cat eI bocce omer cic le 13.66890 75024 Ey 7 hee Ey Se ..-- 13.66890 75050 . [g]—[Lp]=6.48491 540985 The above instances shew, with what facility the logarithmic sines of arcs may be computed to 10 and 15 places of decimals. In Taylor’s Tables, the places of decimals are only 7, which, for all common purposes, are sufficient. It is convenient, how- ever, especially in finding the logarithmic sines of very small arcs, to have a larger number of figures than seven: and these sines we may compute by the above formula, if we possess tables that will give us the logarithms of numbers, for expressing the loga- rithms of 2n + m, 3n to a number of places beyond 7. The Books* for such purpose are, however, rare. The Trigonometrical Tables called Taylor’s, give the loga- rithmic sines, cosines, &c. of arcs to every second of a quadrant, and express them by seven places of figures. Sherwin’s and Hutton’s express by the same number of figures, the logarithmic sines, &c. to every minute of the quadrant. The two latter and like works are sufficient for almost all calculations, and their size, (which is no immaterial point), renders them manageable ; and, under certain rules, they may be used, but not very safely, to find the sines, cosines, &c. of arcs containing seconds. For instance, to find the sines of 44° 30° 30”, 44° 30’ 10’, we have * Logarithmicall Arithmetike, 1631. Trigonometria Artificialis, 1631. 344 Ares. Log. Sines. Diff. of Logarithm. Pee Bio" ow bein «. 9.8457903 72 Ars 10 Ra a ek picccies CARE ua aes This difference is for 60”, £ of which is ... -00006425 Bek | cc hampering 0000214 Hence log. sin. 44° 30’ 30” is 9.8450618 , gio a aia _ .9.84572605 log. sin. 44 30 10 ... 9.8 nei J eet Senge 9.8456832 and both these sines are right, by adding what are called proportional parts. But the logarithmic sine of 1° 0’ 30”, cannot be so found: for WO ae inca ti scal state ',.. 8.2418553 LORE SI die Pocea tote ee fe §.2490332 16.4908885 half of which is 8.24544425, which is not the logarithmic sine of 1° 0’ 30”, the. real value being 8.2454590.737, or, to the nearest seven figures, 8.2454591. The reason of this is obvious: the sines of small ares varying rapidly, and those of large ones slowly; still the caution that has been given must be attended to, of watching those cases, in which we have to find the logarithmic sines’of arcs, intermediate to those inserted in the Tables, by the above method of pro- portional parts, or that method which supposes the whole dif- ference between the logarithmic sines of arcs differing from each other by 1’, to be 60 times the difference between any contiguous two of the 60 logarithmic sines of the intermediate arcs, differ- ing by! The like is true with Taylors Tables, which give the loga- rithmic sines, &c. of arcs to every second. The logarithmic sine of 1”.75 cannot by proportion be found from them, and, therefore, Dr. Maskelyne has given, in his. Introduction to the Tables, a rule for findmg such sines; which rule, with other similar rulés, is inserted at page 261. of this Work. cS ee “ a a hes a eat ear e ii A ¥ ‘ De i Lit ae rf at) i) Ball ‘ re wi Mr et ea ae do . x “ , n ' ‘i . ‘ f - : mM bat ‘ : pit *e i dim sib ua x es 4 j wale ; : “ ar J 7 7 : ' at ‘ e ‘ . " Fieger > tu " . af £ + ‘ . 4 ” tg 59 ” a — te Nee saa aaryi rere