VOL. XXIX.] PHILOSOPHICAL TRANSACTIONS. J 33 



dy are (severally or jointly) of more than one dimension, and all the terms 

 above the upper line, in which dx and dy are wanting, and for the reasons 

 given by Dr. Barrow, makes all these terms vanish. And by the terms in 

 which dx and dy are of only one dimension, and which he sets betv/een the two 

 lines, he determines the proportion of the subtangent to the ordinate. Well 

 therefore did the Marquis de I'Hospital observe, that where Dr. Barrow left 

 off, Mr. Leibmtz began : for their methods of tangents are exactly the same. 



But Mr. Leibnitz adds this improvement of the method, that the conclusion 

 of this calculus is coincident with the rule of Slusius_, and shows how that rule 

 presently occurs to any one who understands this method. For Mr. Newton 

 had represented in his letters, that this rule was a corollary of his general 

 method. 



And whereas Mr. Newton had said that his method in drawing of tangents, 

 and determining maxima and minima, &c. proceeded without sticking at surds j 

 Mr. Leibnitz, in the next place, shows how this method of tangents may be 

 improved so as not to stick at surds or fractions, and then adds : " I suppose 

 that what Mr. Newton would conceal, about the method of drawing tangents, 

 does not differ from this. And what confirms me in this is, that he adds, that 

 from the same foundation quadratures may also be rendered more easy; for 

 every figure is always quadrable, when the ordinate drawn into that differential 

 of the absciss, becomes the differential of any quantity. By which words, 

 compared with the preceding calculation, it is manifest that Mr. Leibnitz, at 

 this time, understood that Mr. Newton had a method which would do all these 

 things, and had been examining whether Dr. Barrow's differential method of 

 tangents might not be extended to the same performances. 



In November l684, Mr. Leibnitz published the Elements of this Differential 

 Method, in the Acta Eruditorum, and illustrated it with examples, of drawing 

 tangents and determining maxima and minima, and then added : and these in- 

 deed are the rudiments of a certain kind of sublimer geometry, which extends 

 even to the most curious and difficult problems of mixed mathematics, and 

 which without the differential calculus, or some such method, are not easily to 

 be attempted. The words some such methody plainly relate to Mr. Newton's 

 method: and the whole paragraph contains nothing more than what Mr. 

 Newton had affirmed of his general method, in his letters of 1672 and 1 676. 

 And in the Acta Eruditorum of June 1686, p. 297, Mr. Leibnitz added : 

 " I choose rather to make use of dx, and the like, than of letters for them, 

 because dx is a certain kind of modification of x itself, &c." He knew very 

 well that in this method he might have used letters with Dr. Barrow, but he 



