112 Of the Figure of the EartL 



first of these, viz. the magnitude of the earth, on the 

 supposition of its entire sphericity, or globular figure, is 

 easily determined. It is only requisite that the whole, 

 or some given part of one of its great circles, be ascer- 

 tained according to known measures. With this view, 

 the arch of the meridian has been selected, as best adapt- 

 ed to celestial observations. This work, for nautical and 

 astronomical purposes, has been performed long since 

 by Picard, Norwood, and others. The more general 

 question of the earth's figure, which necessarily involves 

 that of its magnitude, is of a different nature ; and though 

 not difficult to those who are well versed in the higher 

 geometry, is considerably remote from ordinary inves- 

 tigations. Its analysis affords an illustrious instance of 

 the utility of those abstract mathematical speculations, 

 which we have partly derived from the Greeks ; but for 

 which we are chiefly indebted to the moderns, viz. Des 

 Cartes, Huygens, Clairaut, the Bernouillis, D'Alem- 

 bert, Euler and Newton. 



The question may be propounded in general terms, 

 thus : To detennine in any curve ^ but more particularly 

 in the conic sections, the dimensions of that curve ; or the 

 principal lines which regulate it, the diameter of the Os- 

 culatory c^cle, in two or more points of the curve being 

 given. 



The Osculatory circle, or circle of curvature of any 

 curve, is that which not only touches the curve in a point, 

 but so nearly coincides with it, that no other circle can 

 be drawn between them. The curvature of the curve, 

 and circle, in that point, is therefore considered as the 

 same. As this curvature, however, in all curves, the 

 the circle only excepted, is perpetually varying ; it can 

 be considered the same no where but in the very point 

 of Osculation, or very near it. The measure then of a 

 small portion of the curve at or near this point, may be 

 obtained from the corresponding portion of the circle, 

 and vice versa, that of the circle from a portion of the 

 curve. 



The osculatory circle of any two points of the meridi- 

 an of the earth, be the curve of any kind whatever, may 

 be found by the mensuration of a small portion of it, at 

 those two points corresponding to any small arc, or am- 



