SINES. 



«' la'' IT a^ 62a' , , 



6. tan. a =a + — + ^^ + - — + -5— + &c- 

 3 15 315 2835 



^. a = 



I 



+ 



I 



cot. a 3 cot.' a 5 cot. ^ a 7 cot.' a 



+ &c. 



I a a' 2 a 



8. cot. a — ■ — 



a 



3 4; 945 4725 



I 1 3 



- &c. 



&c. 



^' " ^° fee. a 2 . 3 fee. ' a 2 • 4 • 5 ^c. "• a 



rfec.a - 1 fee. ' a -13 (fec.^a " , 3 -5 (fec .'a - 1) , 



5 



_ i^_ i_ I • 3 I • 3 • 5 



~ eofec. a 2-3 cofec a 2.4.5 cofec.^a 2.4.6. 7 cofec.'a 



I a 7a' 31a' 127a' 



12. cofec. a = — + -V- + -^ + -^ + ;^ — 5 f" &c. 



a 6 360 15120 604800 



+ &c. 



-— — r verf. a 



verf. a 



^ + &c. \ 



2^ . 4 . 6 . 7 5 



,- -— r verl. a I . 3 verl. a i • 3 • S ^^''f- '' 



13. a = a ^/2verf.aJ. + _- + -^TT^rTi + -^T^.-^T 



I f a' a< a' - 1 



14. locr. fin. a = log. a — i-;. J + — + —. + &c. 5- 



^ ^ ^ M (.2. 33'. 3. 5 3-5-7 J 



I fa" a'* a' „ 1 



I f . locr. cof- a = log. I —^Ti \ — + + -r — — + &c. \ 



16. log. tan. a - log. " + ^ j^ + 



2 • 3-5 



02 a" ,7 

 + &c. [ 



* ■ 5 -7 S 



In all thefe feries, as alfo throughout the entire article, 

 we have fuppofed radius = i ; if in any cafe the radius r 

 be required, it may be introduced by giving it fuch a 

 power as will render the quantities on both fides of the 

 equation of the fame dimenfion with regard to the powers of 

 its faftors. 



Sines, Cofints, IsSc. Figures of, are figures made by con- 

 ceiving the circumference of a circle extended out in a right 

 line, upon every point of which are erefted perpendicular 

 ordinates, equal to the fines, cofines, &c. of the correfpond- 

 ing arcs, and drawing a curve line through the extremities 

 of all the ordinates, which is then the figure of the fines, 

 cofines, &e. 



It appears that thefe figures took their rife from the eir- 

 tumftance of the extenfion of the meridian line by Edward 

 Wright, who computed that line by collefting the fuc- 

 ceffive fums of the feeants, which is the fame thing as the 

 area of the figures of the feeants. This being made up of 

 all the ordinates or feeants by the conftruAion of the figure, 

 and in imitation of this, the figures of the other lines have 

 been invented. By means of the figures of the feeants, 

 James Gregory (hewed how the logarithmic tangents may 

 be eonltrufted in his " Exercitations Geometricae," 410. 

 1668. 



ConflruSion of the Figures of the Sines, Cofines, l^c. — Let 

 A D B, &e. {Plate I. fg. 10.) be the circle, A D an arc, 

 D E its fine, C E its eofine, A E the verfed fine, A F the 

 tangent, G H the cotangent, C F the feeant, and C H the 

 cofecant. 



Drawalineaa {fig. ll.),equalto the whole circumference 

 A D G B A of the circle, upon which lay off alfo the 

 length of feveral arcs, ae of every ten degrees, from o at a, 



to 360° at the other end at a ; upon thefe points rai(e per- 

 pendicular ordinates upwards or downwards, according as 

 the fine, eofine. Sec. is affirmative or negative in that part 

 of the circle ; laftly, upon thefe fet off the length of the 

 fines, cofines, &c. correfponding to the arcs at thofe 

 points, or circumference a a, drawing a curve line through 

 the extremities of all thefe ordinates, which will be the 

 figure of the fines, cofines, verfed fines, tangents, cotangents, 

 feeants, and eofecants, as m Jigs. 11, 12, 13, 14, 15, 16, 

 1 7, where it may be obferved, that the following curves are 

 the fame, namely, thofe of the fines and cofines, tangents 

 and cotangents, and thofe of the feeants and eofecants, only 

 fome of their parts a little differently placed. 



It may be known when any of thefe lines, vi%. the fines, 

 cofines, &c. are pofitive or negative ; that is, to be fet up- 

 wards or downwards ; by obferving the following general 

 rules for thofe lines, in the lit, 2d, 3d, and 4th quadrants 

 of the circle. 



fin the lit and 2d 

 \ in the 2d and 4th 

 fin the ift and 4th 

 1^ in the 2d and 3d 



{in the i(t and 3d 

 in the 2d and 4th 

 J in the i ft and 3d 

 ^ in the 2d and 4th 

 fin the ift and 4th 

 J^in the 2d and 3d 



and all the verfed fines are affirmative. 



To find the Equation and Area of each 

 Draw an ordinate, de, putting r = the 



The fines 

 The cofines 

 The tangents 

 The cotangents 

 The feeants 



are affirmative, 

 negative, 

 aflirmative, 

 negative, 

 affirmative, 

 negative, 

 affirmative, 

 negative, 

 affirmative, 

 negative. 



of thoft Curves. — 

 radius A C of the 



given 



