46 



Bernouilli, nothing would have been easier than to have 

 made tiie necessary correction; and it Avould not have been 

 requisite for Newton to have invented that solution which he 

 has given in the second and third editions of the Principia, 

 and which is much more intricate than the former would 

 have been when so corrected. 



Lagrange, in the above-cited work, has given Newton's 

 solution somewhat simplified, deducing the same conclusion 

 as Newton, but does not attempt to point out the precise 

 error. He then gives a second solution, which he considers 

 as proceeding upon the same principles as the solution of 

 Newton, and by which he obtains the same result. He points 

 out the precise error of his second solution, and concludes, 

 that the error in the solution of Newton is of the same kind. 



Now it may be remarked, that the process of Newton is 

 entirely different in all its steps from that of Lagrange. 

 Therefore they have no common error: but as they give the 

 same result, the error in each must admit of being traced to 

 a common source. What that common source is, M. La- 

 grange has not shewn. And it still appeared hn object of 

 some importance, to enquire into the precise error of the 

 Newtonian solution. The conclusion, deduced by that solu- 

 tion, is confessedly wrong; and, therefore, eu'or must exist in 

 some of its steps, and be assignable without reference to any 

 other solution. To point out that error, is the principal ob- 

 ject of this paper. It will be found to have originated from 

 an erroneous application of the method of prime and ultimate 

 ratios. It will also appear, that, had not this error occurred, 



the 



