132 PHILOSOPHICAL TRANSACTIONS. [aNN0 1714. 



like that which ought to be made by the tangent, the ordinate, and the sub- 

 tangent : so that by one simple analogy, this last method saves all the calcula- 

 tion which was requisite, either in the method of Descartes, or in this same 

 method before. Mr. Barrow stopped not here ; he invented also a sort of cal- 

 culation proper for this method. But it was necessary in this, as well as in that 

 of Descartes, to take away fractions and radicals, for making it useful. On the 

 defect of this calculus, that of the celebrated Mr. Leibnitz was introduced, and 

 this learned geometrician began where Mr. Barrow and others left off. This his 

 calculus led into regions hitherto unknown, and there made discoveries which 

 astonished the most able mathematicians of Europe, &c." Thus far the Mar- 

 quis. He had not seen Mr. Newton's Analysis, nor his Letters of Dec. 10, 

 1672, of June 13, 1676, and Oct. 24, 1676: so that, not knowing that Mr. 

 Newton had done all this, and signified it to Mr. Leibnitz, he reckoned that 

 Mr. Leibnitz began where Mr. Barrow left off, and by teaching how to apply 

 Mr. Barrow's method without sticking at fractions and surds, had enlarged the 

 method wonderfully. And Mr. James Bernoulli, in the Acta Eruditorum of 

 January 169I, p. 14, writes thus : "Whoever understands Dr. Barrow's cal- 

 culus (which he sketched out in his Geometrical Lectures, and of which all the 

 propositions, there contained, are specimens) can scarcely be ignorant of that 

 other, invented by Mr. Leibnitz, since it is founded on the former, and differs 

 not from it, unless perhaps in the notation of differentials, and some compendia 

 in the operation." 



Now Dr. Barrow, in his Method of Tangents, draws two ordinates in- 

 definitely near each other, and puts the letter a for the difference of the ordi- 

 nates, and the letter e for the difference of the abscissas : and for drawing the 

 tangent gives these three rules: 1. "In computing, says he, I cast away all the 

 terms in which the power of a or e is found, or in which they are multiplied 

 into themselves : for these terms will become inconsiderable. 2. After con- 

 stituting the equation, I cast away all the terms, consisting of symbols that de- 

 note known or determinate quantities, or in which either a or e is not found : 

 for these terms, being always brought to one side of the equation, will be equal 

 to nothing. 3. I substitute the ordinate for a, and the subtangent for e\ hence 

 at length the quantity of the subtangent will be known." Thus far Dr. 

 Barrow. 



And Mr. Leibnitz, in his letter of June 21, 1677, above-mentioned, where- 

 in he first began to propose his differential method, has followed this method of 

 tangents exactly, excepting that he has changed the letters a and e of Dr. 

 Barrow, into dx and dy. For in the example which he there gives, he draws 

 two parallel lines, and sets all the terms below the under line, in which dx and 



