(Note on collaboration: If you do the homework in pairs or groups, please list the name(s) of your collaborator(s) on what you each turn in.)
Use whatever information you've gleaned from your readings and your notes as the source material for your essay; attached are some additional excerpts that you can refer to as well.
The design of publishing the following treatise, is to introduce the true method of fluxions, most of the books that have hitherto appeared upon that subject having in them little more than the name, the principles upon which they have proceeded being the same with the differential calculus; so that by calling a differential a fluxion, and a second differential a second fluxion, etc., they have so confusedly jumbled the methods together, that people, who have not been thoroughly acquainted with them, have been led into many mistakes: for although the way of investigation in each be the same, and both center in the same conclusions, yet whoever will compare the principles, upon which the methods are founded, will find that they are very different.
The differential method teaches us to consider magnitudes as made up of an infinite number of very small constituent parts put together; whereas the fluxionary method teaches us to consider magnitudes as generated by motion. A line is described, and in describing is generated, not by an apposition of points, or differentials, but by the motion, or flux, of a point; and that velocity with which the generating point moves, when the line begins to arise, or in the first moment of its generation, or formation, is called its fluxion; so that to call a differential a fluxion, or a fluxion a differential, is an abuse of terms, and an imposition upon the reader; for a fluxion has no relation to a differential, nor a differential to a fluxion; a fluxion cannot be compared with a differential, nor a differential with a fluxion, because they are of a different nature.
The fluxion shews us the law and manner of flowing, by which we are taught how to determine the proportion of magnitudes one to another, from the celerities of the motions by which they are generated, which is a pure and abstracted way of reasoning, and agreeable to the method made use of by the ancient geometers; whereas the differential being but an infinitely small part of the magnitude itself, in the forming of magnitudes after this way, we are to conceive them as made up of an infinite number of these small constituent parts, disposed of in such a manner, as to produce a magnitude of a given form; and that these small constituent parts are to each other as the magnitudes of which they are differentials; and consequently, that one infinitely small part, or differential, must be infinitely great, with respect to another infinitely small part, or differential, which must happen when we consider magnitudes formed after the differential method.
These things being well consider'd, it must be confess'd that the way of considering the different degrees of magnitudes, as arising from an increasing series of mutations of velocity necessary to the generation of the quantities to be formed, is much more simple, and less perplexed than the other, and consequently, all the operations that are founded upon the method of fluxions must be much more clear, accurate and convincing than those that are founded upon the differential calculus. In the former method quantities are rejected, because they really vanish; in the latter they are rejected, because they are infinitely small; which cannot but leave the mind in some ambiguity or confusion. And therefore to treat of fluxions upon differential principles, is leading the reader quite wrong, and giving him such false conceptions of things, which he will, if ever, with great difficulty get clear of; for people having been long accustomed to one way of considering things, it is not very easy to get over the prepossessions, and to bring themselves into a quite different way.
What I have here said relating to the differential method, will, I hope, be no ways construed as if I intended to lessen, or depreciate that method; far be it from me. On the contrary, I highly value and esteem it, and always made use of it, 'till I became acquainted with the fluxionary method: but what I mention it here for, is to compare the two methods together, that the excellency of the fluxionary method above the differential, may the more evidently appear; and thence to shew, how wrong it is to make use promiscuously of two methods, so very different in principles, as if they were one and the same method.
Preface. I seek to know what the true spirit of the infinitesimal calculus consists in. The reflections that I set forth on this subject are distributed in three chapters: in the first I expose the general principles of the infinitesimal analysis; in the second I examine how this analysis has been reduced to an algorithm, by the invention of the differential and integral calculus; in the third I compare this analysis with other methods which can take its place, such as the method of exhaustion, that of indeterminates and indivisibles, that of indeterminates, etc.
Chapter 1. I call an infinitely small quantity any quantity which is considered as continually decreasing, such that it can be reduced as small as one wishes, without one's being obliged by that to vary those quantities whose relation one is seeking.
When one wants to find the relation of certain proposed quantities, some constant and some variable, one considers the general system as being arrived at a determined state that one regards as fixed: then one compares this fixed system with other states of the same system, which are considered as continually approaching the first one, so as to differ from it as little as one wishes. These other states of the system themselves are not, strictly speaking, anything but auxiliary systems, which one introduces in order to facilitate the comparison between the parts of the first one. The differences between the quantities that correspond in all these systems can therefore be supposed to be as small as one wishes, without changing anything in the quantities that compose the first one, which are the ones whose relation is sought. These differences are therefore of the nature of the quantities that we call infinitely small. [...]
Leibnitz, who first gave the rules of the infinitesimal calculus, established it on this principle: that one can substitute at will either one for the other of two quantitites which differ only by an infinitely small quantity. This principle had the advantage of extreme simplicity and very easy application. It was adopted as a sort of axiom, and one contented oneself with regarding infinitely small quantities as quantities that are less than all those that can be appreciated or grasped by the imagination. [...]
Chapter 3. There are many other ways of solving problems which are in the province of infinitesimal analysis; and although there is not one of them that seems to unite the same advantages, it is nonetheless interesting to know what the different points of view are by which the principles of this theory can be seen: that is why I propose here to take a look at the diverse methods which resemble it, and can even take its place.
The method of exhaustion. The method of exhaustion was that which the ancients used in their difficult researches, and particularly in the theory of curved lines and surfaces, and in the evaluation of areas and volumes which they enclosed. Since they admitted only perfectly rigorous demonstrations, they did not consider themselves permitted to consider curves as polygons of a large number of sides; but when they wanted to discover the properties of one of them, they regarded it as the fixed limit which the inscribed and circumscribed polygons continually approached as far as one wanted, as long as one increased the number of their sides. [...] But it did not suffice for the geometers to have recognized and, as it were, guessed these properties; it was necessary to verify them in an incontestable way, which is what they did in proving that any supposition contrary to the existence of these same properties necessarily led to some contradiction; which is why they called this type of demonstration reduction to absurdity. [...]
The method of indivisibles. Cavalerius was the precursor of the scientists to whom we owe the infinitesimal analysis; he opened their path by his Geometry of indivisibles.
In the method of indivisibles, one considers lines as composed of points, surfaces as composed of lines, and volumes as composed of surfaces.
These hypotheses are certainly absurd, and one must use them with caution; but they should be regarded as a means of abbreviation, by means of which one quickly and easily obtains (in many cases) what one could discover only by long and tedious proceedings, if strictly following the method of exhaustion. For example, is there a question of showing that two pyramids of the same base and the same height also have the same volume? One considers them both as composed of an infinity of equidistant plane surfaces, which are their elements; and, since these elements are equal each to each, and since their number is the same in each case, one concludes that the volumes of the pyramids which are the sums of these elements are equal to each other. [...]
The method of first and last ratios, or limits. The method of first and last ratios, or limits, also takes its origin from the method of exhaustion; and strictly speaking it is only a development and simplification of that. It is to Newton that we owe this useful refinement, and it is in his book of Principles that one must study it; it suffices for our object here to give a succinct idea of it.
When two quantities whatsoever are supposed continually to approach each other, so that their ratio or quotient differs less and less, and as little as one wishes, from unity; these two quantities are said to have for their last ratio, a ratio of equality.
In general, when one supposes that diverse quantities respectively and simultaneously approach other quantities that are considered fixed, so as to differ from them at the same time as little as one wishes, the ratios that these fixed quantities have between them are the last ratios of those which are supposed to approach each other respectively and simultaneously; and these fixed quantities themselves are called limits or last values of those which thus approach each other.
These last values and last ratios are also called first values and first ratios of the quantities to which they correspond, depending on whether one considers the variables as approaching or receding from the quantities considered fixed, which serve as their limits. [...]
The method of fluxions. Newton considers a curve as engendered by the uniform motion of a point; he decomposes at each instant the constant velocity of this point into two others, one parallel to the axis of abscissas and the other parallel to the axis of ordinates. These velocities are what he calls fluxions of these coordinates, such that the arbitrary velocity of the point that describes the curve is the fluxion of the described arc. [...]
General conclusion. The diverse methods of which we have given an idea in this writing are, strictly speaking, nothing but one and the same method, presented by different points of view. It is always the method of exhaustion of the ancients, more or less simplified, more or less conveniently adapted to the needs of calculation, and finally reduced to a regular algorithm. [...]
But among all these methods which have their common origin in the method of exhaustion of the ancients, which is the one that offers the greatest advantage for common usage? It seems to me that it is generally agreed that it is Leibnizian analysis.
The labors of Descartes, of Pascal, of Fermat, of Huygens, of Barrow, of Roberval, of Wallis, of Newton especially, prove that this great discovery had been approached for a long time, when it was proclaimed by Leibniz; and I think that there is not one of these illustrious geometers who would not have made it, if he had suspected that there was a great discovery to make in this direction. That is to say, there is not one of them who would not have found a way to reduce the method of exhaustion to an algorithm, if he had had the idea of searching for it, and if he had foreseen all the fruitfulness which such a method would someday have. [...]