gures, the editors of the Acts faid Leibnitz was the first inventor of the differential calculus, and Newton had fubftituted fluxions for differences, just as Honoratus Faber, in his Synopfis Geometrica, had fubftituted a progreffion of motion for Cavallerius's method of indivifibles; that is, Leibnitz was the first inventor of the method, Newton had received it from him (from his Elements of the Diffe rential Calculus), and had fubftituted fluxions for differences; but the way of investigation in each is the fame, and both center in the fame conclufions.

This excited Mr. Keil to reply; and he made it appear very plain from Sir Ifaac's letters, published by Dr. Wallis, that he (Newton) was the firft inventor of the algorith, or practical rules of fluxions; and Leibnitz did no more than publish the same, with an alteration of the name, and manner of notation. This however did not filence Leibnitz, nor fatisfy the foreigners who admired him. He abufed Dr. Keil, and appealed to the Royal Society against him; that they would be pleased to restrain the Doc tor's vain babblings and unjust calumniations, and report their judgment as he thought they ought to do, that is, in his favour. But this was not in the power of the Society, if they did juftice; for it appeared quite clear to a committee of the members, appointed

to

to examine the original letters, and other papers, relating to the matter, which were left by Mr. Oldenburgh and Mr. J. Collins, that Sir Ifaac Newton was the first inventor of fluxions; and accordingly they published their opinion. This determines the affair. When this is the cafe, it is fenfeless for any foreigner to fay Leibnitz was the author of fluxions. To the divine Newton belongs this greatest work of genius, and the nobleft thought that ever entered the human mind.

It must be fo (Maria replied): As the cafe is ftated, Sir Ifaac Newton was most certainly the inventor of the method of fluxions: And fuppofing Leibnitz had been able to discover and work the differential calculus, without the lights he received from Newton, it would not from thence follow, that he understood the true method of fluxions: for, though a differential has been, and to this day is, by many, called a fluxion, and a fluxion a differential, yet it is an abuse of terms. A fluxion has no relation to a differential, nor a differential to a fluxion. The principles upon which the methods are founded fhew them to be very different; notwithstanding the way of investigation in each be the fame, and that both center in the fame conclufions: nor can the differential method perform what the fluxionary method

method can. The excellency of the fluxionary method is far above the differential.

This remark on the two methods furprized me very much, and especially as it was made by a young lady. I had not then a notion of the difference, and had been taught by my mafter to proceed on the principles of the Differential Calculus. This made me request an explication of the matter, and Maria went on in the following

manner.

Magnitudes, as made up of an infinite number of very fmall conftituent parts put together, are the work of the Differential Calculus; but by the fluxionary method, we are taught to confider magnitudes as generated by motion. A described line in this way, is not generated by an appofition of points, or differentials, but by the motion or flux of a point; and the velocity of the generating point in the first moment of its formation, or generation, is called its fluxion. In forming magnitudes after the differential way, we conceive them as made up of an infinite number of small conftituent parts, fo difpofed as to produce a magnitude of a given form; that these parts are to each other as the magnitudes of which they are differentials; and that one infinitely fmall part, or differential, must be infinitely great, with respect to an

other

other differential, or infinitely small part: but by fluxion, or the law of flowing, we determine the proportion of magnitudes one to another, from the celerities of the motions by which they are generated. This most certainly is the pureft abftracted way of reafoning. Our confidering the different degrees of magnitude, as arifing from an increafing feries of mutations of velocity, is much more fimple, and lefs perplexed than the other way; and the operations founded on fluxions, must be much more clear, accurate, and convincing, than thofe that are, founded on the Differential Calculus. There is a great difference in operations, when quantities are rejected, because they really vanish; -and when they are rejected, because they are infinitely fmall: the latter method, which is the differential, muft leave the mind in ambiguity and confufion, and cannot in many cafes come up to the truth. It is a very great error then to call differentials, fluxions, and quite wrong to begin with the differential method, in order to learn the law or manner of flowing.

With amazement I heard this difcourfe, and requested to know by what master, and what method, fhe obtained these notions; for they were far beyond every thing on the fubject that I had ever met with. What the faid concerning the nature and idea of flux

An account of Martin

ions, I thought juft and beautiful, and I be lieve it was in her power, to fhew the bafes on which they are erected.

My mafter, Sir, (Maria anfwered) was a poor traveller, a Scotchman, one Martin Murdoch, Murdoch. who came by accident to my father's house, to afk relief, when I was about fifteen years old. He told us, he was the fon of one of the minifters of Scotland, and came from the remoteft part of the Highlands: that his father taught him mathematics, and left him, at his death, a little ftock on a small farm; but misfortunes and accidents obliged him in a fhort time to break up houfe, and he was going to London, to try if he could get any thing there, by teaching arithmetic of every kind. My father, who was a hospitable man, invited him to stay with us a few days, and the parfon of our parish foon found, that he had not only a very extraordinary understanding, but was particularly excellent at figures, and the other branches of the mathematics. My father upon this agreed with him to be my preceptor for five years, and during four years and nine months of that time, he took the greatest pains to make me as perfect as he could in arithmetic, trigonometry, geometry, algebra, and fluxions. As I delighted in the ftudy above all things, I was a great proficient for fo few years, and had Murdoch been longer with me, I should