244 
OF FLAN L TRIflOKOMETHY. 
But there is another method of constructing the trig 
onometrical canon; which, though less direct, is more 
geometrical; and that is by determining the sines and 
tangents of different arches, one from another, as in 
the ensuing propositions. 
PROPOSITION II. 
The sine o f an arch being given : to find its co-sinc, 
tangent, co-tangent, secant, and co-sccant. 
Let AE he the proposed arch, F.F its sine, CF its 
co-sine, AT its tangent, DH its co-tangent, Cl' its 
secant, and CH its co-secant: then, (by Euc. 47. l.) we 
jsliall have CF — \/CE 2 —EF*; from whence the 
co-sine will lie known; and 
then, byreason of the similar 
triangles, CFE, CAT, and 
CDH, it will be, 
1. CF : FE :: CA : AT; 
whence the tangent is known. 
2. CF : CE :: CA : CT; 
whence the secant is known. 
3. EF : CF :: CD : DH; 
whence the co-tangent is known. 
: CH; whence the co-secant 
is also known. 
/ 
c 
A 
4. 
FF : CE 
:: CD 
Hence it appears, 
1. That, the tangent is a fourth proportional to the 
co-sine, the sine, and the radius. 
2. That the secant is a third proportional to the 
co-sine, and the radius. 
3. That the co-tangent is a fourth proportional to the 
sine, the co-sine, and the radius. 
4. That the co-secant is a third proportional to the 
sine, and the radius. 
.5. And that the rectangle of the tangent and cq- 
tangent is equal to the'square of the radius.