94 NEWTON S PRINCIPIA. 



ing in his approximation neglected some quantities as 

 extremely minute, and supposing they could not affect the 

 result, whereas one of the quantities upon which the 

 result mainly depends, having a small numerator, is nearly 

 doubled by the introduction of the quantities omitted. 

 Upon again going through the investigation without those 

 omissions, this great geometrician had the satisfaction of 

 finding that the result made the motion of the moon s 

 apsides agree with the fact ; and both Euler and D Alem- 

 bert now found that in their solutions they had, by a 

 singular coincidence, fallen into the same error. Laplace 

 has since in his great work* given a complete investi 

 gation of the problem, and the results to which he is 

 conducted by the theory are also most satisfactory. He 

 finds the amount to differ only one four hundred and 

 forty-fourth part from that given by observation, which, 

 reduced to our sexagesimal degrees, is only a difference 

 of 24&quot; 12 &quot; from the observed amount. His solution 

 in the case of the nodes does not come so near the 

 observation ; it is only correct within the 350th part ; 

 and yet the success of the theory in the case of the nodes 

 was always reckoned its great victory in the hands of its 

 author, while the case of the apogee cast some doubt 

 upon it. Laplace made a discovery in the course of this 

 inquiry of a similar variation in the apogeal movement, 

 and that it becomes slower at the rate of 15&quot; in 100 

 years, which the recent observations confirm. 



It was certainly impossible for the Newtonian theory 

 to obtain a more brilliant triumph. f But it deserves to 



* Mec. Gel. liv. vii. s. 16. 



f For Clairaut s papers, see Mem. de 1 Acad. des Sciences, 1745 and 

 1748. But there is an admirable paper of the same illustrious mathema 

 tician on the motions of the orbits in the Mem. for 1754. The first cited 

 volume contains both Clairaut and D Alembert s famous investigation 

 of the problem of the three bodies, to which reference is made in the text 



