Only this kind of analysis is capable of giving a true insight into the properties of curves. For curves are merely polygons with an infinite number of sides, and curves differ from one another only because these infinitely small sides form different angles with one another. Only by using these methods of analysis of infinitely small quantities can we determine the positions of these sides and thus find the properties of the curve they form: properties such a the directions of tangents and normals to the curve, its points of inflection, turning points, how it reflects or refracts rays, etc.
It has long been realised that polygons inscribed within a curve or circumscribed about it become identical with the curve as the number of their sides is increased to infinity. But there the matter rested until the invention of the type of analysis we are about to describe at last showed the scope and implications of such an idea.
The work done in this field by ancient scholars, particularly Archimedes, is certainly worthy of admiration. But they only considered a very few curves and those only cursorily. We find no more than a succession of special cases, in no particular order: and they provide no indication of any general and consistent method of procedure. We cannot reasonably blame the ancient scholars for this: it required a genius of the first order to find a way through so many difficulties and do the very first work in this hitherto completely unknown field. They did not go far, and they proceeded by roundabout ways, but, whatever Vieta says, they did not lose their way: and the more difficult and thorny the paths they trod the more we must admire these ancient mathematicians for having succeeded so well. To put it briefly: it does not appear that the Ancients could, at the time, have done better than they did. They did what our mathematicians would have done in their place, and if they were in ours it is likely that they would do as we do now. All this is a consequence of the unchanging nature of the human mind and the fact that discoveries can only be made in an orderly succession.
It is therefore not surprising that the Ancients did not get any further: but it is a matter of great astonishment that great men, some assuredly as great as the Ancients, should have failed so long to make any further progress, confining themselves to merely reading ancient authors and writing commentaries on their work, turning the knowledge they acquired to no use other than continuing to read, without daring to commit any such crime as to think for themselves and look beyond what Ancients had discovered. Thus it was that though there were many scholars who wrote a great deal, so that the number of books multiplied, yet for all this activity there was no progress made: all the work of several centuries merely served to supply the world with respectful commentaries and repeated translations, often of quite uninteresting works.
Such was the state of mathematics, and, above all, of philosophy, until the time of M Descartes, whose genius and self-confidence led him to abandon this study of ancient authorities and turn instead to reason, the authority to which these same Ancients had appealed. His boldness was considered to be merely a revolt but it led to a number of new and useful insights in physics and geometry. Then it was that men opened their eyes and began to think.
In mathematics, which is what concerns us here, M Descartes began where the ancients had left off, by solving a problem which Pappus said that no-one had been able to solve. It is well-known that he made great advances in analysis and in geometry, and that techniques he evolved from combining the two make it possible to solve many problems which had previously been completely intractable. But since he was mainly concerned with the solution of equations he was interested in curves only as a way to finding roots. He was therefore completely satisfied with using ordinary analysis, and did in fact use this new technique successfully in constructing tangents, and was so pleased with his method of solving this problem that he said it was the most useful and general problem he knew, or ever wanted to know, in all geometry.
M Descartes' Geometry made it fashionable to solve geometrical problems by means of equations, and opened up many possibilities of obtaining such solutions. Geometers applied themselves to the task and soon made new discoveries. Indeed, more and better new results are still being obtained. M Pascal was concerned with quite different matters: he studied curves as curves, and also as polygons. He found the lengths of some of them, the areas they enclosed, the volumes swept out by these areas, the centres of gravity of these areas and volumes etc etc. By considering only elements, that is infinitely small portions of the curve, he discovered methods which were of general application, and which, moreover, we must find the more surprising in that he seems to have arrived at them merely by the power of his imaginative grasp and not by means of analysis.
Soon after M Descartes had published his method for finding tangents, M Fermat discovered a different method, which M Descartes himself finally admitted was in many ways better than his own. At the time, however, this method was not as simple as M Barrow has since made it, by having paid closer attention to the properties of polygons, which naturally suggest that one consider the small triangles each made up of a part of the curve cut off between two infinitely close ordinates, the difference between these ordinates and the difference between the corresponding abscissae. This triangle is similar to that formed by the tangent, the ordinate and the subtangent, so that this method of finding the tangent uses straightforward similarity instead of the calculations which were required by M Descartes's method, and which this new method previously required. Barrow's work did not stop there: he also invented a kind of calculus based on this method, but this calculus, like that of Descartes, could only be used once all fractions and roots had been removed.
Barrow's calculus was replaced by that of M Leibniz, an accomplished geometer who started his own work where Barrow and others had ended theirs. His calculus led him into domains hitherto unknown and the discoveries he made amazed the most brilliant mathematicians of Europe. The Bernoullis were the first to recognise the elegance of Leibniz's method, and they in turn developed his calculus to a degree which enabled them to solve problems which had previously seemed too difficult to attempt.
This calculus is of immense scope: it can be used for the curves which occur in mechanics, transcendental curves such as the catenary, as well as for purely geometrical curves, squares or other roots do not cause any difficulty (and may even be an advantage), any number of variables may be considered, and it is equally easy to compare infinitely small quantities of any type. An endless number of interesting results can be obtained concerning tangents (including tangent curves) concerning problems connected with maxima and minima, points of inflection and turning points, evolutes, caustics derived by reflection or refraction, and so on, as we shall see in the work that follows.
I shall divide the work into ten sections. The first describes the principles of the calculus of differences [the differential calculus]. The second describes how it is used to find the tangent to any curve whatever the number of variables in the equation of the curve: though M Craig did not believe this method could be used for the transcendental curves which occur in mechanics. The third shows how the calculus is used in problems connected with maxima and minima. The fourth shows how it is used to find points of inflection and turning points. The fifth describes its use in finding M Huygen's evolutes for all kinds of curves. The sixth and seventh sections show how it is used to find the caustic curves discovered by the distinguished scholar M Tschirnhaus, both the type formed by reflection and that formed by refraction. The method is again applicable to all kinds of curves. The eighth section describes how the calculus is used to find the curves which touch an infinite number of given straight lines or curves. The ninth consists of solutions to various problems arising out of the earlier work. The tenth section describes a new way of using the differential calculus for geometrical curves: from which we can derive the method used by M Descartes and M Hudde, which is applicable only to this kind of curve. ...
I had intended to include an additional section which was to have described the marvellous use to which the calculus may be put in physics, what accuracy can thereby be obtained, and to show how useful the calculus would be in mechanics. Illness, however, prevented me. The public will, nevertheless, not lose this, since the work will eventually be published with additional material that has been accumulated in the meantime.
All this is only the first part of M Leibniz's work on calculus, which consists of working down from integral quantities to consider the infinitely small differences between them and comparing these infinitely small differences with each other, whatever their type: this part is called Differential Calculus. The other part of M Leibniz's work is called the Integral Calculus, and consists of working up from these infinitely small quantities to the quantities of totals of which they are the differences: that is, it consists of finding their sums. I had intended to describe this also. But M Leibniz wrote to me to say that he himself was engaged upon describing the integral calculus in a treatise he calls De Scientia infiniti, and I did not wish to deprive the public of such a work, which will deal with all the most interesting consequences of this inverse method of tangents, showing how it can be used to find the lengths of curves, to find the area they enclose, to find the volumes and surfaces of their solids of revolution, to find centres of gravity etc. I have said as much as this only because M Leibniz wrote and asked me to do so, and I myself think it necessary to prepare people's minds so that later they will be in a better position to understand all the results that are eventually obtained.
Finally, I am greatly indebted to the Bernoullis, particularly the younger Bernoulli [Johann Bernoulli], who is at present a professor at Groningen [Johann Bernoulli took up the appintment at Groningen in September 1695]. I have made free use of their work as well as that of M Leibniz. They may take the credit for as much of this work as they please, and I am quite content with what little they leave to me.
M Leibniz himself acknowledges his debt to M Newton, who, as it appears in his excellent work Philosophiae naturalis principis mathematica of 1687, had already invented a technque very like that of the differential calculus, which he uses throughout his book. But M Leibniz's use of the characteristic makes his calculus much simpler and quicker, and sometimes also proves very helpful.
As the last pages of this book were being printed, I came across M Nieuwentiit's book. Its title, Analysis infinitorum, excited my interest, but when I read it through I found that it was very different from the present work: not only does the author not use M Leibniz's characteristics but he also completely rejects second, third and further differences. Since he rejects what I have made the basis of most of my work I should feel obliged to reply to his objections, to show that they are unfounded, except that M Leibniz has himself already made a more than adequate reply in the Acta of Leipzig. Moreover, the two postulates or suppositions which I make at the beginning of this treatise, and which alone form the basis of what follows, seem to me to be so evidently true that no serious reader can reject them. I could, in fact, have proved them, in the manner of the Ancients, if I had not preferred to deal briefly with what was already well-known and enter into details only where the material itself was new.
Although this is the end of the Preface, it would be unfair to leave readers with de L'Hôpital's comments that "no serious reader can reject" his two postulates without giving these postulates which, as he says, come right at the beginning of the first chapter:
Postulate I: Any two quantities may be replaced by one another if they differ from each other by no more than an infinitely small amount. ...
Postulate II: We may consider a curve as an assemblage of an infinite number of straight lines each infinitely short, or (equivalently) as a polygon with an infinite number of sides, each infinitely small, which, by the angles they make with one another, determine the shape of the curve. ...
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