VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 217 



it can be only from that proposition of Sir Isaac which informs us, that gravity 

 is in a reciprocal ratio of the distances: because he gave us the proposition but 

 the page before, as a method for determining the figure of the earth. But we 

 are not allowed to make use of this proposition in this case, because it has been 

 shown, art, 1 8, that it can take place only on the supposition of a homogeneous 

 spheroid. Therefore, &c. 



26. It will not be very difficult, without any regard had to the foregoing 

 theory, to find the ratio of the axes of a spheroid, which we may suppose to 

 have a nucleus at the centre, of greater density than the rest of the planet; 

 and hence we shall be easily assured of Dr. Gregory's mistake. 



27. Setting aside all attraction of the parts of matter, if the action of gra- 

 vity is directed towards a centre, and is in the reciprocal ratio of the squares of 

 the distances, the ratio of the axes of the spheroid will then be that of 576 to 

 577: and the gravity at the pole is greater than at the equator by the 144th 

 part, or thereabouts. Which may be a confirmation of what is here advanced, 

 especially to such as will not be at the pains of going through the foregoing 

 calculations. For we may consider the spheroid now mentioned, in which gra- 

 vity acts in a reciprocal ratio of the squares of the distances, as composed of 

 matter of such rarity, in respect of that at the centre, that the gravity is pro- 

 duced only by the attraction of the centre or nucleus. 



28. In the foregoing calculations, in order to find the axes of our spheroids, 

 and to know whether their figure makes a sensible approach to that of the 

 conical ellipsis, we have had recourse to this principle, that gravity ought 

 always to act in a direction perpendicular to the surface. Two reasons have 

 prevailed with us to make use of this principle, rather than the other, which 

 consists in the equilibrium of the columns. The first is, because the calcula- 

 tions founded on it are more simple. The second is, that considering the state 

 of the actual solidity of the earth, it should seem as if this principle were the 

 more indispensably necessary. However, because Sir Isaac Newton, and all the 

 other philosophers, who have treated about the figure of the earth, have taken 

 it, as it were, at its first formation, at which time they suppose it to have been 

 fluid: we shall here make the same supposition, and we shall assume no other 

 ratio for that of the two axes, than that of the spheroid which results from a- 

 coincidence of these two principles. 



We shall begin by inquiring what is the whole weight of any column cn, 

 fig. 15. To do this, we must resume the expression of the attraction in any 

 point M of the column cn; then multiply it by r -f- xr, and by the density 

 frP -\- gr9, and afterwards we must find the fluent. Thus we shall have 



VOL. VIII. Pf. 



