mired him. He abufed Dr. Keil, and appealed to the Royal Society against him; that they would be pleased to restrain the Doctor'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 examine the original letters, and other papers, relating to the matter, which were left by Mr. Oldenburgh and Mr. J. Collins; that Sir Isaac Newton was the first inventor of fluxions; and accordingly they published their opinion. This determines the affair. When this is the cafe, it is fenfelefs 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 fluxi

ons:

on.

ons: 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 fluxiThe 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 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 fmall constituent parts put very 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

C 5

way,

way, is not generated by an appofition of points, or differentials, but by the notion 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 another differential, or infinitely fall 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 moft certainly is the pureft abstracted 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, muft be much more clear, accurate, and convincing, than those 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

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 discourse, and requested to know by what master, and what method, she 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 fluxions, I thought juft and beautiful, and I believe it was in her power, to fhew the bafes on which they are erected.

My mafter, Sir, (Maria answered) was a poor traveller, a Scotchman, one Martin Murdoch, who came

An account of

Martin Mur

doch.

by accident to my father's house, to ask relief, when I was about fifteen years old. He told us, he was the son of one of the ministers 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 stock on a small farm;

but misfortunes and accidents obliged him in a fhort time to break up house, 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 an hofpitable 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 study above all things, I was a great proficient for fo few years, and had Murdoch been longer with me, I fhould have been well acquainted with the whole glorious ftructure: but towards the end of the

fifth year, this poor Archimedes was unfortunately drowned, in croffing one of our rivers, in the winter time, and went in that uncomfortable way, in the thirty-fixth year of his year, to the enjoyment of that feli city and glory, which God has prepared for a virtuous life and honeft heart. Why fuch men, as the poor and admirable Murdoch, have often fuch hard measure in this: world,