Imatges de pÓgina
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world, is not in my power to account for nor do I believe any one can : but what I

is one of those surprizing things, and I lamented not a little the lofs of such a master. Still however I continued to study by many written rules he had given me, and to this day, mathematics are the greateft pleasure of my


As to our method, my master, in the first place, made me perfectly understand arithmetic, and then geometry and algebra, in all their parts and improvements, the methods of feries, doctrine of proportions, nature of logarithms, mechanics, and laws of motion : from thence we proceeded to the


doctrine of fluxions, and at last looked into the Differential Calculus. In this true way my excellent mafter led me, and in the same difficult path every one must go, who intends to learn Fluxions. It would be but lost labour for any person to attempt them, who was unacquainted with these Precognita.

When we turned to fluxions, the first thing my master did, was to instruct me in the arithmetic of exponents, the nature of powers, and the manner of their generation. We went next to the doctrine of infinite series, and then, to the manner of generating mathematical quantities. This generation of quantities was my first step into fluxions, and my master fo amply explained the nature of them, in this operation, that I was able to form a just idea of a first fluxion, though thought by many to be incomprehensible. We proceeded from thence to the notation and algorithm of first fluxions ; to the finding second, third, & c. fuxions ; the finding fluxions of exponential quantities ; and the fluents from given fluxions ; to their uses in drawing tangents to curves; in finding the areas of spaces; the valves of surfaces; and the contents of solids ; their percusion, oscillation, and centers of gravity. All these things my master so happily explained to my understanding, that I was able to work with ease, and found no more difficulty in conceiving an adequate notion of a nascent or evanescent quantity, than in forming a true idea of a mathematical point. In short, by the time I had studied fluxions two years, I not only understood their fundamental principles and operations, and could investigate, and give the solution of the most general and useful problems in the mathematics; but likewise, folve several problems that occur in the phænomena of nature.



Here Maria stopped, and as soon as astonishment would permit me to speak, I proposed to her several difficult questions, I had heard, but was not then able to answer. I requested her, in the first place, to inform me, how the time of a body's descending through any arch of a cycloid was found: and if ten hundred weight avoirdupoise, hanging on a bar of steel perfectly elastic, and supported at both ends, will just break the bar, what must be the weight of a globe, falling perpendicular 185 feet on the middle of the bar, to have the same effect ? ---My next questions were, how long, and how far, ought a given globe to descend by its comparative weight in a medium of a given density, but without resistance, to acquire the greatest velocity it is capable of in descending with the same weight, and in the same medium, with resistance? And how are we to find the value of a solid formed by the rotation of this curvilinear space, ACD about the axis AV, the general equation, expresling the

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How is the center of gravity to be found of the space inclosed by an hyperbola, and its asymptete? And how are we to find the center of oscillation of a sphere revolving about the line P AM, a tangent, to the generating circle FAH, in the point A, as an axis ?- These questions Maria anfwered with a celerity and elegance that again amazed me, and convinced me that, notwithstanding the Right Rev. metaphysical disputant, Dr. Berkley, late bishop of Cloyne in Ireland, could not understand the doctrine of fuxions, and therefore did all he could to disgrace them, and the few mathematicians who have studied magnitudes as generated by motion ; yet, the doctrine, as delivered by the divine Newton, may be clearly conceived, and distinctly comprehended; that the principles upon which it is founded, are true, and the demonstrations of its rules conclusive. No opposition can hurt it.

When I observed, that fome learned men will not allow that a velocity which continues for no time at all, can possibly describe any space at all: its effect, they say, is absolutely nothing, and instead of fatisfying reason with truth and precision, the human faculties are quite confounded, lost, and bewildered in fluxions. A velocity or fluxion is at best we do not know what ; whether something or nothing: and how


can the mind lay hold on, or form any accurate abstract idea of such a subtile, feeting thing?

Disputants (Maria answered) may perplex with deep speculations, and confound with mysterious disquisitions, but the method of Auxions has no dependance on such things. The operation is not what any single abstract velocity can generate or describe of itself, but what a continual and successively variable velocity can produce in the whole: And certainly, a variable cause may produce a variable effect, as well as a permanent cause a permanent and constant effect. The difference can only be, that the continual variation of the effect must be propor. tional to the continual variation of the cause. The method of

fluxions therefore is true, whether we can or cannot conceive the nature and manner of several things relating to them, though we had no ideas of perpetually arising increments, and magnitudes in nalcent or evanescent states. The knowledge of such things is not essential to Auxions. All they propose is, to determine the velocity or flowing wherewith a generated quantity increases, and to sum up all that has been generated or described by the continually variable fluxion, On these two bafes Auxions stand.

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