THE ASTRONOMICAL VIEW OF NATURE. 321 



scientific research are involved and opened out by this 

 statement. 



First, There is the purely theoretical task of defining 

 clearly what is meant by the different words which are 

 used, and which in the formula are expressed in algebraic 

 symbols. What is the definition of force, what of mass, 

 what of distance ? The ' Principia ' give Newton's defi- 

 nitions. 1 



Second, The definitions must be given in such a way 

 that they express definite measurable quantities ; and in 

 order to verify and apply the formula, methods must be 

 devised for measuring these quantities as they occur in 

 nature, and these measurements must be actually carried 

 out. 2 



13. 



Lines of 

 thought 

 emanating 

 from it. 



1 It will be readily admitted that 

 the definition of force as measured 

 by change of motion, and the defi- 

 nition of mass as the quantity of 

 matter, are definitions involving 

 some difficulty. As to distance, 

 it may be thought that this is a 

 purely mathematical, not a physi- 

 cal quantity. So it would be if 

 physical bodies were mathematical 

 points, such as the planets in a 

 first approximation may be con- 

 sidered to be. But in comparing 

 the attraction of the earth upon 

 a body at its surface with that on 

 the moon, the dimensions of the 

 earth could not be neglected, and 

 the problem presented itself how 

 the quantities of mass and distance, 

 in the case of the earth and the 

 body on its surface, had to be de- 

 fined. It appears from a statement 

 by Prof. Glaisher (see Rouse Ball, 

 ' History of Mathematics,' p. 297, 

 (.-.) that the publication of the 

 ' Principia,' containing the gravita- 

 tion formula, was delayed, because 

 Newton found it difficult to prove 

 that in a sphere the different parts 



VOL. I. 



with their different distances from 

 any point need not be considered 

 separately, but that a quantity 

 equal to the whole mass situated 

 at the centre of the sphere may 

 be substituted. Laplace showed a 

 century later that this property of 

 the sphere exists only for one de- 

 creasing function of the distance 

 viz., that of the inverse duplicate 

 ratio. It exists likewise for that 

 function which increases in propor- 

 tion to the distance, but for none 

 other (see 'Principia,' 1st ed., pp. 

 198, 200 ; 'Mdcanique celeste,' 1st 

 ed., vol. i. p. 143). Hitherto the 

 delay in publishing the ' Principia ' 

 was (see Brewster, 'Life of New- 

 ton,' vol. i. p. 290) always attrib- 

 uted to the erroneous figure of the 

 moon's distance from the earth, 

 with which Newton had been 

 reckoning, and which did not sat- 

 isfy the gravitation formula. 



^ Up to the beginning of this 

 century the merit of carrying out 

 accurate measurements of astrono- 

 mical constants is about equally 

 divided between France and Eng- 



