VOL. XLVIII.] PHILOSOPHICAL TRANSACTIONS. • 309 



matter that is taken away, has, in both cases, the same ratio to the matter that 

 is left; and its position, with respect to that which is left, is in both cases nearly 

 the same ; and therefore the successive attractions will be nearly in continued 

 proportion, g:g — d :g — d ~ S -^. Or multiplying and rejecting c/' as incon- 

 siderable, Gd = G^, and d = S. 



Thus, if the attractions of the sphere Apsa, and of the prolate spheroid, at 

 its pole A, be 126 and 125 respectively; the attraction of the intermediate oblate 

 spheroid at its equator will be 125-1^: and how nearly this approaches to the truth, 

 may be seen from an exact computation of those attractions. For if the axes of 

 the generating ellipse be JOl and 100, and the attractive force at the surface of 

 the sphere 12(3; the attraction at the pole of the prolate spheroid will be 1 24.9838^ 

 and that at the equator of the oblate 125.5077; which exceeds the arithmetical 

 mean between the two former, only by .0O68 ; that is, by about -rr-r-nr part of 

 the attraction of the oblate spheroid at the equator. 



This reasoning is more shortly expressed in the Princip. lib. iii, prop, ig, as 

 follows. " Gravitas in loco a in sphaeroidem, convolutione ellipseos ApBg circa 

 axem ab descriptam, est ad gravitatem in eodem loco a in sphaeram centro c radio 

 AC descriptam, ut 125 ad 126. Est autem gravitas in loco a in terrain media 

 proportionalis inter gravitates in dictam sphaeroidem et sphaeram ; propterea quod 

 sphaera, diminuendo diametrum pa in ratione 101 ad 100, vertitur in figuram 

 terrae; et haec figura, diminuendo in eadem ratione diametrum tertiam, quae 

 diametris ap, pa perpendicularis est, vertitur in dictam sphaeroidem ; et gravitas 

 in A, in utroque casu, diminuitur in eadem ratione quam proxime." 



In which the expression " eadem ratione" occurring a second time has misled 

 F. Frisi and others, to think, that this last ratio is also that of the axes, or of 

 101 to 100; wheyeas the identity of ratios here asserted, is to be referred only 

 to the words " utroque casu;" the ratio itself being not that of the axes, or of 

 mton; but the half of that ratio (whatever it is found to be by prop. Ql, lib. i) 

 which the attraction of the sphere has to the polar attraction of the inscribed 

 spheroid. 



This inadvertence, however, of his own, Frisi charges on Sir Isaac Newton ; 

 and files it up, as the 6th of the errors, which he says have been discovered in 

 the Principia. ..." Ita dum stabilitae in IQ lib. 3 proposition! terrestrium axium 

 proportionis fulcimentum et patrocinium quaerimus, aliud in propositione eadem 

 sophisma sese offert, quod eorum, quae in principiis mathematicis Newtoni 

 nacta (i. e. detecta) sunt hactenus, sextum est, &c." But we may take it off 

 the file again; and for the present leave the other 5, till they are considered of 

 at more leisure. 



In his 10th and last chapter, our author sums up the evidence, and finds 

 that all the good observations that have been made, as well by pendulums as by 



