116 Of the Figure of the Earth. 



taken too near each other. It will be necessary, how- 

 ever, to have recourse to an analytical process, when the 

 measures of a degree are not at E and P, in order to in- 

 vestigate the relation of the principal lines involved in 

 the general expression, gi^'en b}' mathematicians for the 

 radius of carvature. For it is to be observed, that as in ' 

 a given ellipsis, the radius of curvature, or the diameter 

 of the osculator}^ circle, is determined from the principal 

 lines, viz. diameters, ordinates, &c. ; so also, when the 

 radius of curvature is given, the principal lines, which 

 regulate the figure, may be ascertained : this will be ex- 

 emplified in the following 



PROBLEM. 



The measure of a degree^ in two known latitudes of the 

 meridia?!, being given, to determine from thence, thg 

 figure of the earth. 



From physical and other principles, it is known tliat 

 the earth, if not a sphere, must be a solid generated by 

 the revolution of an ellipsis about one or other of its 

 ^xes. Let therefore the ellipsis* PEPE, be a meridian 

 of the earth, without knowing which axis is the longest. 

 A one latitude and B another, where the measures of a 

 degree are known, AD, BF, perpendiculars to the tan- 

 gents, at the points A and B, and cutting the diameter 

 EE, in D and F. GD, IF subnormals to the same. 



A degree being known at the points A and B, the ra- 

 dius of curvature is likewise known, for each respective- 

 ly, which let be represented by R, and r, and put the ra- 

 tio of CP to CE, that of j& to q. From the nature of the 



ellipsis, we have AD =^- vj'"— CG^-f-iL' CG* ; AG*= 

 ^' ^-_CG% and GD^=^^CG^ also BF =^' 



V^<Icr- + ^CI\BP=f^ ^'^^^CF, and IF*=^^ CP. 



But the radius of curvature at A, acco rding to the dete r- 

 mination of mathematicians, is R= -o'^— CG*+^— CG"" '2' 



and in like manner, at B, it is r=- cr* — CP + ■— Cl^r ' 



p * 5' ' 



If we put the sine of the angle ADG, or of the lati- 



* See Fig. 4, plate I . 



