326 Prof. De Morgan on the Early History 



ton has had the goodness to communicate in manuscript. This 

 assertion corroborates itself, for Craig does use the binomial 

 theorem, which at the time he was writing had not been in print : 

 Wallis^s Algebra was not published till the veiy end of 1684. 

 His account of the second difficulty is as follows : — 



'' The second difficulty is when the value of the ordinate has 

 irrational terms [asymmetris] ; for analysts well know that it is 

 very laborious to free an equation from irrationality of more 

 than four terms. But a most excellent remedy has been applied 

 by the celebrated geometer G. G. Leibnitz in his new method of 

 finding tangents published in the Acta Eruditorum of last year : 

 in which an easy method is shown of finding tangents without 

 removing the irrational terms, be they ever so nmch involved in 

 the equation." Craig then proceeds to use the differential cal- 

 culus under the symbols of Leibnitz, but with some elementary 

 mistakes. We see here the singidar indifi'erence which Newton 

 at that time, and long afterwards, showed towards his own cal- 

 culus. It appears that when he communicated to Craig, for 

 help in the quadratm-e of curves, his binomial theorem, at the 

 very period when Leibnitz had just announced the differential 

 calculus, he never gave a hint that he himself had had long pos- 

 session of a similar method, and had exchanged communications 

 with Leibnitz on the subject eight years before. 



The second of Craig's separate tracts is Tractatus Mathema- 

 ticus de Figurarum Curvilinearum Quadraturis et Locis Geometricis, 

 London, 1693, 4to (pp. iv + 76 and plate) . That this was the tract 

 which Newton examined before it was printed, I infer as follows. 

 In the preface of 1718, before mentioned, Craig says that New- 

 ton proposed two curves, of which he gives the equations, as 

 examples in corroboration of Craig's objections against D. T. 

 (Tschirnhauss). Now the attack upon D. T. is at the end of 

 this second tract, and the curves specified are the first two ex- 

 amples at the beginning. Moreover, in the first pamphlet Craig 

 was no deeper in the differential calculus than to imagine that 

 Tdy = Q,dx always gives P?/ = Qa', which we may undertake to 

 say Newton could not have passed without detection, if he had 

 seen the manuscript, even though he had only given it a glance. 



It is also to be noticed that in the second tract the name of 

 Newton docs not occur once, though it is full of the differential 

 calculus, and Leibnitz, Sluse, Barrow, Gregory, &c. are frequently 

 mentioned. This, under all the circumstances, we may suspect 

 was Newton's own doing. And I am strongly inclined to think 

 that it was this very tract of Craig's which immediately sug- 

 gested to Newton the progress which the views of Leibnitz were 

 making, and induced him to forward to Wallis the extracts from 

 the Quadr. Curv. which I have already mentioned. 





