SINES. 



given eircle, *■ = ad, or AD any abfcifs or arc, andj- = de, 

 Its ordinate, which will be either the fine D E = j, cofine 

 C E = <r, verfed line A E = i", tangent A F = /, cotan- 

 gent G H = T, fecant C F = /, or cofecant C H = j-, 

 according to the nature of the particular conllruftion. 

 Now, from the nature of the circle are obtained thefe fol- 

 lowing general equations, exprefling the relations between 

 the fluxions and the circular arc, and its fine, cofine, &c. 



r^tt 



— -; the corredl fluent of which is ^ r' . hyp. log. 



•llAll = r' . hyp. log. :;Ltl+-^ = r^ . hyp. log. 



r' / — r' T r'f _ ~ r'' <^ 



r' + t- r» + T^ f^f{f-r-) a- ^ {cr^ - r') 



and thefe alfo exprefs the relation between the abfcifs and 

 ordinatcs of the curves in queftion, each in the order in 

 which it ftands ; where x is the common abfcifs to all of 

 them, and the refpeilive ordinates are /, c, v, t, t,/, and o- ; 

 and hence the area of any of thefe curves may be found as 

 foUowi. 



I. In the Figure of the Sines, (Jig. II.) Here x = ad, 



and s = the ordinate de, and the equation of the curve, as r / [f- r^)' 



— ; and hence the figure of the tangents may be ufcd for 



the conftruftion of the logarithmic fecants ; a property firft 

 pointed out by Gregory at the end of his " Exercit. Geo- 

 metricje." 



When a d becomes the quadrant af, t being then infinite, 

 this becomes infinite for the area afg : and the fame for 

 the figure of cotangents, beginning at /, inftead of a. 

 See A- 15- 



5. Figure of the Secants, (fg. 16.) Here .r =. ad, and 

 _/" =: d e ; and the equation of the curves is x = 



— , and the fluxion of the area is fx = 





the fluent of which is r'' . hyp. log. 



ab 



ove, IS jr = 



area, or /at = 



V (r- - '1 



Hence the fluxion of the /+_^/l ll!) for the general area ade; which is 



, — ■ — , the correft fluent of which is 



V {'■ - ' ) 



r'^ + r ^ (r^ — s^) = r' — re ■= rv =: the reftangle 

 of radius and verfed fine minus or pius, according as s is 

 increafing or decreafing ; which is a general expreflion for 

 the area a de in the figure of fines. When j =: o, as at a, 

 or i, this expreifion becomes o, or 2 r^ ; that is, o at a, 

 and 2 r' at i ; that is, the whole area acb =■ 2 r\ 



2. In the Figure of the Coftnes, {fig. 1 2.) Here x — ad, and 

 c ■=. de; and the equation of the curve is x ; 



^(r~n' 



hence the fluxion of the area ^=. c x 



; and 



the correA fluent of this ■=z r ^ (r^ — ^ ) = s r, the 

 reftangle of the radius and fine for the general area a dec. 

 When s =■ r, or c = o, this becomes r ' = area afc, whofe 

 abfcifs af is equal to a quadrant of the circumference ; 

 the fame as the figure of the fines upon an equal abfcifs. 



3. In the Figure of the verfed Sines, {fig. 13.) Here 

 X =■ ad, and -v =1 de, and the equation of the curve is ;«■ = 



; hence the fluxion of the area is v x := 

 rv j^/ V 



^/ {zrv — I)') 

 r w 



, ,. — , , : and the fluent of this 



V (-2 r 1) — 1;^) ^ (2 r — t») 



isrx — r s := r (x — s) ■= the general area in the figure 

 of the verfed fines. When a </ is a quadrant, this becomes 



—~ — r' — r' = the area afg ; and when a 7 is a femi- 

 2 



circle it becomes 3.i4l6r', the whole area of the figure 



of verfed fines. 



4. In the Figure of theTangents, (fig. 14.) Here x =:ad, 



and t =. d e ; and the equation of the curve becomes x = 



r'' t 

 ; ; and hence the fluxion of the area \t t x =■ 



infinite when a d becomes equal to a quadrant. The fame 

 procefs anfwers for the cofecants, beginning at/ inftead of 

 a. See fig. 17. 



Hence the meridional parts in Mercator's chart may be 

 calculated for any latitude A D or a ^. For the meridional 

 parts : arc of latitude :: the fum of the fecants : the fum 

 of as many radii, or :: area ade : ad x rad. ac ; or A D 

 X A C in fig. 10. 



Sines, Line of, a line on the feftor, Gunter's fcale, &c. 

 the defcription and ufe of which fee under the articles 

 Sector, and Gunter'j Sca/e. 



In eltimacing the quantity of fines, &c. we afluiiie radius 

 for unity ; and determine the quantity of the fines, tan- 

 gents, and fecants, in fraftions thereof. From Ptolemy's 

 Almageil we learn, that the ancients divided the radius into 

 60 parts ; which they called degrees, and thence deter- 

 mined the chords in minutes, feconds, and thirds ; that is, 

 in fexagefimal fraftions of the radius, which they likewife 

 ufed in the refohuion of triangles. 



The fi.Tcs, or half chords, for aught that appears, were 

 firft ufed by the Saracens. 



Regiomontanus, at firft, witli the ancients, divided the 

 radius into fixty degrees, and determined the fines of the 

 fevcTal degrees in decimal fraftions of it. But he after- 

 wards found it would be more commodious to alTumc radius 

 for I ; and thus introduced the prefent method into trigo- 

 nometry. 



In the common tables of fines and tangents, the radius is 

 conceived divided into looooooo parts, beyond wliich we 

 never go, in determining the quantity of the fines and tan- 

 gents. Hence, as the fide of an hexagon lubtends the 

 fixth part of a circle, and is ecjual to radius, the fine of 30" 

 is half the radius, or joooooo. 



1. The fine A D heing given (fig. /[.),to find the fine-complement. 

 From the fquare of the radius AC hiljtraft the Iquare of 

 the fine A D ; the remainder will be the fquare of the fine- 

 complement DC or AG: whence the iquare root being 

 extrafted, we have the fine-complement. E.gr. Suppofing 

 AC 1 0000000, and AD 500000O1 AG will be found 

 86602 J4, the fine of 60°. 



2. Thefime AD of the arc A E iting^iven (fig, 1 8.), to find 



tit 



