106

represented in magnitude by cg,tg , and zg,respectively, and that the directions inwhich they act, are respectively parallel tothe three sides y, z, and x, of the trianglefrF, it follows, by the laws of statics, thateg, tg, and zg, and consequently c, t, and z,must be to each other as y, z, and x :whence we have i - and - all equal to each

c 2 t Z > x

other.

Again, the indefinitely small arc F f,being considered as a straight line, inter-secting the two parallel lines hf, HF, wehave (by Euc. i. 29.) the angle rf F equalto the exterior angle CFH = ?, and (byEuc. i. 47.) Ff *== Fr* -j- rf* or z* = x*-| -y* : and since r-F is the tangent, and f Fthe secant of the angle rfF to the radius rf,we have, by putting radius =zl, y,x, and z,to each other respectively, as 1, tang.?, andsec.?. And since it has been shown, thaty, x, and z are to each other as c, z, and t;it follows that c, z, and t must be to eachother as 1, tang. ?, and sec.?, or that c,

and —— must all be equal to each other.

sec.? 1