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mired him. He abused 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 justice ; 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. 7. 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 case, it is senfeless for any foreigner to say Leibnitz was the author of Auxions. To the divine Newton belongs this greatest work of genius, and the nobleft thought that ever entered the human mind.
It must be so (Maria replied :) As the case is stated, Sir Isaac Newton was most certainly the inventor of the method of fluxions : And supposing 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 : 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 fluxi
The principles upon which the methods are founded shew them to be very different; notwithstanding the way of investigation in each be the same, and that both center in the fame conclusions: 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 master 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
Magnitudes, as made up of an infinite number of very small constituent parts put together, are the work of the Differential Calculus ; but by the fluxionary method, we are taught to consider magnitudes as geneFated by motion. A described line in this
way, is not generated by an apposition 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 Auxion. In forming magnitudes after the differential way, we conceive them as made up of an infinite number of small constituent parts, so disposed 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 small part, or differential, must be infinitely great, with respect to another differential, or infinitely small part: but by fiuxion, 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 purest abstracted way of reasoning. Our considering the different degrees of magnitude, as arising from an increasing feries of mutations of velocity, is much more simple, and less perplexed than the other way; and the operations founded on fluxions, must 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 are rejected, because they are infinitely small: the latter method, which is the differential, must leave the mind in ambiguity and confusion, and cannot in many cases 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 subject that I had ever met with. What she said concerning the nature and idea of fluxions, I thought just and beautiful, and I believe it was in her power, to shew the bases on which they are erected.
My master, Sir, (Maria answered) was a poor tra
An account of
Martin Murs veller, a Scotchman, one
doch. Martin Murdoch, who came 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 remotest 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 short 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 k.nd. My father, who was an hospitable man, invited him to stay with us a few days, and the parson 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 Auxions. As I delighted in the study above all things, I was a great proficient for so few years, and had Murdoch been longer with me, I should have been well acquainted with the whole glorious structure: but towards the end of the fifth year, this poor Archimedes was unfortunately drowned, in crossing one of our rivers, in the winter time, and went in that uncomfortable way, in the thirty-sixth year of his year, to the enjoyment of that felicity and glory, which God has prepared for a virtuous life and honest heart. Why such men, as the poor and admirable Murdocb, have often such hard measure in this