Relating to the FIGURE of the EARTH. •y 



he treated as fmall quantities, or mere fluxions of the earth's 

 circumference, the calculation mufl be made by rules quite dif- 

 ferent from thofe that have been hitherto employed. Thefe 

 new rules are deduced from the following analyfis. 



4. Let the ellipfis ADBE (fig. 1. PI. I.) reprefent a meridian 

 paffing through the poles D and E, and cutting the equator in 

 A and B. Let C be the centre of the earth, AC, the radius of 

 the equator, r= a, and DC, half the polar axis, := b. Let FG be 

 any very fmall arch of the meridian, having its centre of cur- 

 vature in H ; join HF, HG cutting AC in K and L. Let <p 

 be the meafure of the latitude of F, or the meafure of the angle 

 AKF, exprefTed, not in degrees and minutes, but in decimals 

 of the radius 1 ; then the excefs of the angle ALG above 

 AKF, that is, the angle LHK or GHF will be zz <p, and there- 

 fore FG = <p X FG. Alfo, if the elliptic arch AF = z t FG = 



z— cpx FH. 



But FH, or the radius of curvature at F, is =r 



2 £» 



2 7 



a 



r ss a 2 b 2 {a 2 — a 2 fin 2 p -J- b z fin 2 <p) ~" *, as is 



demonftrated in the conic fections. Therefore, if c be the 

 compreflion at the poles, or the excefs of a above b y 

 b 2 — a 2 — 2ac -f c% or becaufe c is fmall in companion of #, if 

 we reject its powers higher than the firft, b 2 zza 2 — 2ac, and 



FH = a 3 (a — 2c) {a 2 — a 2 fin 2 $ + a 1 fin 2 <p — 2ac fin 2 <p) ""^ = 

 a* (a — 2c) [a 2 — 2ac€m 2 <p)~~^. 



But (a* — 2ac{m 2 <p)—^ tza~^ 3 (1 — ~€m 2 <p) "~* s 

 a ~~ 3 (1 + ~ fin 2 <p) nearly, rejecting, as before, the terms that 



involved, &c. Hence FH = (a — - 2c) ( 1 + ^ fin 2 <p) = 

 a — 2c -f 2,0 fin 2 <p. 



Now, 



