from the measurement of an arc of the meridian , &c. 511 
Let ADBE be a meridian of the earth, where A is at the 
equator, and D at the pole. 
Suppose F to be any point on 
that meridian, and FH the ra- 
dius of curvature of the ellipse 
at that point. Put A C =a ; 
DC = b ; c being the centre of 
the ellipse ; and let A be equal 
the angle AKF, the latitude 
of F ; or let it be the measure of the arc of latitude a K f, 
to radius unity, or #K : that is, the measure of the angle aKf 
in parts of the radius aK, or unity. Let GF be an inde- 
finitely small part of the ellipse. Then if AF =s z, GH == z 
the fluxion of the arc AF of the ellipse; and if GHbe drawn, 
then the angle GHF = < g Kf = A or fg the fluxion of the 
arc of latitude af to radius i. Hence as 1 : A: : FH : z — A.FH. 
But the radius of curvature FH = a 2 (a? — a 2 . Sin . 2 A -f- 
b z Sin. A) - *. Let e be the ellipticify, or a — b ; then b = a — r, 
and b 2 = c? — 2 ae very nearly, since & is very small. Hence 
FH = a 3 (a — ae) . ( a 2 — 2 ae. Sin . 2 A) But (a* — 
— 2 ae. Sin. 2 A)“"" = (a*)-"”*. (1 — Sin. a A) “~ 2 = 
= a ~~ \ (1 + Sin . 2 A) very nearly, by rejecting all the 
terms involving e~ and its higher powers. Hence FH = 
= a? (a — ae). a~ .(3 Sin. 2 A) == a~~ se + $e. Sin. 2 A, 
which substituted for FH, we get z = A{a—ae -j- g<?. Sin. 2 A]= 
= A (a — ae) + A(se. Sin . 2 A). But Sin . 2 A = — 
and therefore z — A ( a — ae) + f e A— \e A. Cos. 2 A ; whose 
fluent is z==(a— ^e)A-\-\e. Sin. 2A=aA—e A.Sin.aA) 
which requires no correction. And this is the measure of an 
