of the Angle fubt ended by Two QbjeEts , &c. 421 
From thefe geometrical proportions, having given any arc and 
the variation of its fine or cofine, the cotemporary variation of 
the arc may be eftimated by computation in general for any 
part of the quadrant. Let the fine of any arc be r, the 
cofine — r, the chord of the arc’s variation — x, the given varia- 
tion of the cofine = d , or the given variation of the fine = by 
radius = 1 ; then if the cofine of the arc increafes by the dif- 
ference d, the chord of the cotemporary decreafe of the arc* or 
— x~ 2 S Z — 2 dc =+= \j 2s 1 - 2 dc — 4 d z 
and if the fine of the given arc increafes by the difference b 
4 * X = sf 2C Z - 26s V 2 C 2 — 2 bs — \b % ^ 
which are the mathematically true values of the chord FB, 
and will approximate to the magnitude of the arc FB as that 
arc is continually diminifhed. The following expreflions for 
the chord of the variation x are more compendious, and will 
be fufficiently near the truth when FB is very fmall. 
In thefe four expreflions it muft be obferved, that the fine and 
cofine are fuppofed to vary by increafe : fhould the variation 
be a decrement, the fign of x and of b or d muft be changed. 
26. Let the quantities py s , m, n , vary by fmall in- 
crements/*, s, my n , refpeftively, then to obtain the cotempo- 
rary variation of cof. ED, becaufe (art. 2 1 .) 
Co f. ED = 1—8 s z p z x 1 — s z p z — m z + m n -f m z p z — 2 p z m z n z 4- 
m z n z s z p z z±z 2 m z np x/i — p z x \/ 1 ~n z x \/ 1 — s z 9 
by taking the fluxion of the equation we have 
Vox,. LXXL Kkk 
Cofi 
