SINES. 



iht fine of the half arc, or half o/" A B. Find the chord of 

 the arc A B ; for half of this is its fine. Thus fup- 

 pofinc; A G and A D, as in the preceding problem : we 

 (l\all find the fine of half the arc A B, or the fine of 15°, 

 = 2588190. 



3. The fine D G of the arc D F being given, tofnd the fine 

 D E of the double arc D B {fig. 19.) Since the angles at E 

 and G are right angles, and the angle B is common to each 

 triangle B C G and D E B, we fliall have BC : CG :: 

 B D : D E ; wherefore C G being found by the firll pro- 

 blem, and B D being double of D G, D E is found by 

 the rule of proportion. 



4. The fines F G and D E {fg, 20.) of the arcs F A and 

 D A, tuhofe difference D F is not greater than 45 minutes, be- 

 ing given, to fnd any intermediate fine, as I L. To the dif- 

 ference F D of the arcs, whofe fines are given, the difference 

 of the arc A I, whole fine is required, and A F, viz. I F, 

 and the difference of the given fines D H, find a fourth 

 proportional: this, added to the lefs given fine F G, the 

 aggregate will be the fine required. 



5. Tofnd the fine of 45 degrees. Let H I {fg. 4.) be 

 a quadrant of a circle ; then will H C I be a right angle ; 

 confequently the triangle rectangular ; therefore H I - = 

 HC^-1-CI'=2HC'; wherefore, fince H C the whole 

 fine is 1 0000000; if from 2 H C ', 200000COOOOOOOO, 

 be extraftcd the fquare root 14142136; we ftiall have 

 the chord HI, whole half 7071068, is the fine of 45° 

 required. 



6. The fide of a pentagon being given (fee Polygom), the 

 fine of 36^ may he eafily found. For the fine of 36° is equal 



'to half 1 1755706, i. e. = 5877853. 



7. The fine of a minute, or 60", FG {fg. iO.\ being given, 

 to fnd the fine of one or more fronds M N. Since the arcs 

 A M and A F are very fmall, A M F may be taken for a 

 right line, without any fenfible error in the decimal fradtions 

 of the radius, in which tlie fine is expreffed ; that is, the 

 arcs A M and A F may be taken proportional to their 

 chords. Wherefore, fince M N is parallel to F G, we 

 fhall have AF:FG::AM:MN; therefore, A F, 

 F G, and A M, being given, M N is eafily had. 



Sines, to confnift a canon of. The fines of 30°, 15°, 45% 

 and 36^ (which we have already fhewn how to find) be- 

 ing had ; we can thence condriift a canon of all the fines 

 to every minute, or every fecond. For from the fine of 36^ 

 we find thofe of iS"" 9° 4° 30', and 2° 15', by the fecond 

 problem : the fines of 54° 72° 81° 85° 3c', and 87° 45', 

 &c. by the firit problem. 



Again, from the fine of 45-', find the fine of 22" 30' ii" 

 15', Sec. From the fine of 30^, and the fine of 54", find 

 the fine of 12^. From the fine of 12°, find the fines of 

 6^ 3° 1° 30' 45' 78= 84°, &c. From the fine of 15°, find 

 the fine of 7° 30' 3° 45', &c. till you have i 20 fines fuc- 

 ceeding each other orderly, at an interval of 44 minutes. 

 Between thefe, find the intermediate fines by the fourth 

 problem : thus will the canon be complete. 



From the fine of an arc given, to find the tangent 

 and fccant. See Taxoent, Secant, and fequel of this 

 article. 



To find the logarithm of a given fine, called the artificial 

 fine, fee Logarithm. Wolfii Element. Math. vol. i. 

 p. 214, &c. 



There are various other methods of conffrufting the tri- 

 angular or trigonometrical canon. We (hall here fubjoin 

 thp following, as the rcoft fimple and eafy. But it will be 

 neceHary to premife the three iubfcquciit propofitions. 



I . The fine "EV of an arc A E {fg. 21.) being given, to 

 find its crfiie C F, verfed fine A F, tangent A T, cotangent 



I 



D H, fecant C T, and cofecant C H. The cofine C F is 



evidently = v/C E' — E F% and the verfed fine = C E 

 or C A - C F. And the triangles C F E, CAT, and 

 CDH being fimilar, we (hall have CF : FE :: CA : 

 AT ; or the tangent will be a fourth proportional to the 

 cofine, the fine, and radius; and C F : CE (C A) :: CA 

 : C T, whence the fecant is a third proportional to the 

 cofine and radius ; and EF:CF::CD:DH, or the 

 cotangent is a fourth proportional to the fine, cofine, and 

 radius ; and E F : E C (C D) :: C D : D H, or the co- 

 fecant, is a third proportional to the fine and radius. More- 

 over, becaufe A T : A C :: C D (A C) : D H, the redan- 

 gle of the tangent and cotangent is equal to the fquare 

 of the radius, and, therefore, the tangent of half a right 

 angle, being equal to its cotangent, is equal to the radius; 

 and the cotangents of any two different arcs (reprefented 

 by P and Q), are to one another inverfely as their tan- 

 gents ; for tang. P x cotang. P = rad.^ = tang. Q x 

 cotang. O : therefore cotang. P : cotang. Q :: tang. Q : 

 tang. P, or cotang. P : tang. Q :: cotang. Q : tang. P. 



2. If there be three equidifferent arcs A B, AC, AD, 

 {fis- ^'^•) ""-uc fliall have radius to the cofine of their common 

 difference B C, or C D, as the fine CY of the mean, to half 

 ihefum of the fines B E -|- D G of the two extremes ; and as 

 radius to the fine of the common difference, fo is the cofine of the 

 mean to half the difference of the fines of the two extremes. 

 For let BD be drawn, interfering O C in m ; alio draw 

 m n parallel to C F, and B H and m v parallel to A O. 

 The arcs B C and C D being equal, O C is perpendicular 

 to B D, and bifcfts it ; and, therefore, B to or Dm will 

 be the fine of B C or DC, and O m its cofine. More- 

 over mn, being an arithmetical mean between the fines 

 B E, D G, of the two extremes (becaufe Bm = Dm) is, 

 therefore, equal to half their fum, and D v equal to half 

 their difference. But the triangles O C F, O m n, and 

 D V m, being fimilar, we (hall have OC:OOT::CF:m« 



and O C : D m :: F O : D u. Whence mn /Pg. + BE N 

 and D.f5^^a = ^->LlO. 



V 2 y oc ' 



and confequently D G -f B E = 

 2 D «7 X F O 



20m X CF 



and DG 



- BE = 



OC 



OC 



And, moreover, if the mean arc 



A C be 6o^ O F its cofine will be = fine 30° = i chord 

 60° = :§ O C ; confequently D G — B E will, in th'is cafe, 

 be barely = D m, and D G = Dw; 4- B E. Hence it 

 follows: I. That if the fine of the mean of three equi- 

 different arcs (radius being unity) be multiplied by twice 

 the cofine of the common difference, and the fine of either 

 extreme be fubtrafted from the produft, the remainder 

 will be the fine of the other extreme. And, 2. The fine 

 of any arc above 60* is equal to the fine of another arc, 

 as much below 60^, together with the fine of its excefs 

 above 60°. 



3. To fnd the fine of a very fmall arc, e, gr. of \^' . 

 The chords of very fmall arcs being to each other nearly 

 as the arcs thenifelves, we fliall have ^^^th of the femi- 

 periphery : -,,l^th (:: 360 : 384) :: .00818121, the chord 

 of ~^th: .008726624, the chord of ,,hrth of the femi- 

 periphery, or half a degree, wliofe half, or .004363312, i» 

 the fine of 15', very nearly; and, therefore, 15': i' :: 

 .004363312 : .000290888, the fine of the arc of 1' nearly. 



Upoi» 



