140 PHILOSOPHICAL TRANSACTIONS. [aNNO 1714. 



Analysis, Mr. Newton found out most of the propositions in his Principia Philo- 

 sophiae; but because the ancients, for making things certain, admitted nothing 

 into geometry before it was demonstrated synthetically, he demonstrated the 

 propositions synthetically, that the system of the heavens might be founded on 

 good geometry. And this makes it now difficult for unskilful men to see the 

 Analysis by which those propositions were found out. 



It has been represented that Mr. Newton, in the scholium at the end of his 

 book of quadratures, has put the third, fourth, and fifth terms of a converging 

 series respectively, equal to the second, third, and fourth differences of the 

 first term, and therefore did not then understand the method of second, third, 

 and fourth differences. But, in the first proposition of that book, he showed 

 how to find the first, second, third, and following fluxions in infinitum ; and 

 therefore when he wrote that book, which was before the year 1676, he did 

 understand the method of all the fluxions, and consequently of all the differ- 

 ences. And if he did not understand it when he added that scholium to the end 

 of the book, which was in the year 1704, it must have been because he had 

 then forgot it. And so the question is only whether he had forgot the method 

 of second and third differences before the year 1704. 



In the 10th proposition of the 2d book of his Principia Philosophiae, in 

 describing some of the uses of the terms of a converging series, for solving 

 problems, he tells us, that if the first term of the series represents the ordinate 

 Bc of any curve line acg, fig. 4, and cbdi be a parallelogram infinitely narrow, 

 whose side di cuts the curve in g and its tangent cp in f, the second term of 

 the series will represent the line if, and the third term the line fg. Now the 

 line FG is only half the second difference of the ordinate ; and therefore Mr. 

 Newton, when he wrote his Principia, put the third term of the series equal to 

 half the second difference of the first term, and consequently had not then 

 forgotten the method of second differences. 



In writing that book, he had frequent occasion to consider the increase or 

 decrease of the velocities with which quantities are generated, and he argues 

 rightly about it. That increase or decrease is the second fluxion of the quan- 

 tity ; and therefore he had not then forgotten the method of second fluxions. 



In the year 1692, Mr. Newton, at the request of Dr. Wallis, sent to him a 

 copy of the first proposition of the book of quadratures, with examples of it in 

 first, second, and third fluxions; as may be seen in the second volume of the 

 Doctor's works, p. 391, 392, 393, and 396. And therefore he had not then 

 forgotten the method of second fluxions. 



Nor is it likely, that in the year 1704, when he added the aforeaaid scholium 

 to the end of the book of quadratures, he had forgotten not only the first pro- 



