new Divifien of the Quadrant. 33 
48. From the fines and cofines alfo, the tangents, cotan- 
pents, fecants, and cofecants, may be made by thefe known 
proportions, wz. as _ 
Ks come 2 tadius 3: fine < 2% tangent, 
2. fine _- :; radius :: cofine. : cotangent, 
a cole, of radius: =: radius. : fecant, 
4. fine | FaAdduS 2: Lagius -.- Colecant, 
5. radius : fine .:: fecant : tangent, 
6. radius : cofine :: cofecant : cotangent, 
7. tangent: radius :: radius : cotangent. 
Wherefore, the reciprocal of the cofine will be the fecant; the 
reciprocal of the fine, the cofecant; the quotient of the fine by 
the cofine, the tangent; and the quotient of the cofine by the 
fine, the cotangent; or the produé of the fine and fecant will 
be the tangent, and the produé of the cofine and cofecant; 
the cotangent ; Ory, laftly, the reciprocal of the tangent is the 
cotangent ; proper regard being had to the number of decimals, 
on account of our radius being 100000 inftead of 1 only. 
_ And thefe are to be ufed when the application happens to 
be eafier than the general feries, and eafier than a. propor- 
tion from RHETICUS’s canon. 
But there are other particular theorems, which, by a little 
addrefs, may be rendered more coueLio than he of the 
former: thus, 
19. Tn any two ares this is a general proportion, 
As the difference of their fines : | 
to the fum of their fines :: : | 
fo tangent of half the difference’of the arcs : 
to tangent of half their fum. 
So that by nie continually the arcs, having the common’. 
difference 2, the third term of this proportion will be 1, and: 
the fourth term will be found by dividing the fum of the fines 
VoL. LXXIV. 7 ae by 
