178 NEWTON S PRINCIPIA. 



a spheroid, Newton proceeded to calculate its ellipticity. 

 This he does nearly as follows : 



(1.) From Picart s and Cassini s measures of a degree, 

 he finds, supposing the earth spherical, that its radius 

 must be 19,615,800 Paris feet. From some observations 

 on falling bodies at Paris, he calculates that the force of 

 gravity at that place is such, that a body will fall 2174 

 lines in the first second of its descent. Knowing the 

 earth s radius, and its time of rotation, it is easy to cal 

 culate the centrifugal force at the equator ; viz., such that 

 under its action, a body would describe 7.54064 lines in 

 the first second. Since the resolved part of the centri 

 fugal force perpendicular to the earth varies as the square 

 of cosine of the latitude we can calculate the centrifugal 

 force at Paris, and then adding it to the force of gravity, 

 calculate as above, we find the whole undiminished force 

 of gravity at that place to be such, that a body would 

 describe 2177.267 lines in the first second of its descent. 

 The undiminished force of gravity at the equator will differ 

 from this by a very small quantity ; hence rejecting small 

 quantities of the second order, the ratio of centrifugal 

 force at the equator to equatorial gravity is as 1 to 289. 

 This ratio is still in use. 



(2.) If we took a spheroid, whose axes are as 101 to 

 100, by a simple application of Prop. XCL Book I., 

 Newton shows that the force of gravity at the pole is to 

 that at the equator as 501 to 500. Take now two canals, 

 from the surface to the centre; let one meet the surface at 

 the pole, the other at the circumference. That there may 

 be equilibrium the weights of these two canals must be 

 equal. Conceive these divided by transverse parallel 

 equidistant surfaces into parts proportional to the wholes ; 

 the weights of any number of parts in the one leg will 

 be to the weights of the same number of parts in the 



