310 PHILOSOPHICAL TRANSACTIONS. [aNNO 1753. 



actual mensuration, concur with the theory, in making the ratio of the earth's 

 axis and equatorial diameter to be as 22g to 230. This is indeed a sufficient 

 confirmation of the theory of gravitation : but it must be observed, that the co- 

 incidence is not perhaps quite so perfect as Frisi imagines. That ratio corres- 

 ponds well enough to the exactness to which the first elements of the calculus 

 can be obtained ; the length of a second pendulum, and that of the earth's 

 equatorial diameter, from which the centrifugal force (-g-^) is deduced. But, if 

 we suppose that force to be accurately f-i-<r> and compute more rigorously, we 

 shall find the ratio in question to be very nearly that of 225 to 226 ; agreeing 

 still with the observations as well as can be desired ; and showing, at the same 

 time, the inimitable art of Sir Isaac Newton in the contrivance and use of ap- 

 proximations ; seeing the strictest calculation raises the equator not the third 

 part of a mean geographical mile above what he had found by his method. 



I sent, says Mr. Short, Frisi's book to my learned friend the Rev. Mr. Mur- 

 dock, who has fully considered the question concerning the figure of the earth ; 

 and who, after having perused the book, and discovered the above mistake of 

 Frisi, sent me the above theorem, and its demonstration. He likewise sent me 

 the following theorems, which, he says, he had communicated to M. de Bre- 

 mond, in the year 1740, when he was translating his treatise on sailing : but 

 M. de Bremond dying soon after, those who had the care of publishing the 

 translation, printed it incorrectly in several places ; particularly the theorems for 

 the prolate spheroid : on which account, he says, if they are thought worth 

 preserving, they may be inserted in the Phil. Trans. 



Postscript. — Theorems for computing the ratio of the attractive force of a 

 spheroid, at its pole or equator, to that of the inscribed sphere. 



2. In a prolate spheriod, the ratio is, 



?n 



Polel-;;^,— ^ +___^3X/:. 



m' 



1 . In an oblate spheroid, the ratio is, 



1 m^ 



Equator j^;^r—[)i X A - ^^— y : ^ Equa. ^tzTT " ^;;rr)i X ^ 



In which m : 1, as the greater axis of the generati ng ellip se is to the lesser, 



A is a circular arc, to the radius 1, whose tangent isV^m^— 1, or its reciprocal 



if ^2 _ 1 c- 1 . And / is the natural logarithm of -, s being the sine of the arc, 



whose co-sine is - X V'm^ — 1, and v the versed sine of the same arc. 



m 



Note, The first two theorems, by substituting l for '^ ni^ — 1, coincide with 

 those of Mr. Maclaurin for the oblate spheroid, in his dissertation on the tides. 



HI. Concerning the Year of the Eclipse foretold by Thales. By the Rev. Mr. 

 George Costard, Fellow of fVadham Coll. Oxford, p. 17. 

 Riccioli supposes that the eclipse foretold by Thales happened the year a. c. 585 ; 



