Calculus

"In the first two thirds of the seventeenth century mathematicians solved calculus-type problems, but they lacked a general framework in which to place them. This was provided by Newton and Leibniz. Specifically, they a. invented the general concepts of and —though not in the form we see them today... b. recognized differentiation and integration an inverse operations. Although several mathematicians... noted the relation... in specific cases... the clear and explicit recognition, in its complete generality, of... the belongs to Newton and Leibniz. c. devised a notation and developed algorithms to make calculus a powerful computational instrument. d. extended the range and applicability of the methods... While in the past those methods were applied mainly to polynomials, often of low degree, they were now applicable to "all" functions, algebraic and transcendental."

"The subject of s was forced upon the Greek mathematicians so soon as they came to close grips with the problem of the quadrature of the circle. Antiphon the Sophist was the first to [inscribe] a series of successive regular polygons in a circle, each of which had double as many sides as the preceding, and he asserted that, by continuing this process, we should at length exhaust the circle: [according to Simplicius, on Aristotle, Physics] 'he thought that in this way the area of the circle would sometime be used up and a polygon would be inscribed in the circle the sides of which on account of their smallness would coincide with the circumference.' Aristotle roundly said that this was a fallacy... Antiphon's argument.. as early as the time of Antiphon himself (a contemporary of Socrates) had been subjected to a destructive criticism expressed with unsurpassable piquancy and force. No wonder that the subsequent course of Greek geometry was profoundly affected by the arguments of Zeno on motion. Aristotle... called them 'fallacies', without being able to refute them. The mathematicians, however, knew better, and, realizing that Zeno's arguments were fatal to infinitesimals, they saw that they could only avoid the difficulties connected with them by once for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or diminishing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please. If they used infinitesimals at all, it was only as a tentative means of discovering propositions; they proved them afterwards by rigorous geometrical methods. An illustration of this is furnished by the Method of Archimedes. ...Archimedes finds (a) the areas of curves, and (b) the volumes of solids, by treating them respectively as the sums of an infinite number (a) of parallel lines, i.e. infinitely narrow strips, and (b) of parallel planes, i.e. infinitely thin laminae; but he plainly declares that this method is only useful for discovering results and does not furnish a proof of them, but that to establish them scientifically a geometrical proof by the , with its double ' is still necessary."

"The history of modern mathematics is to an astonishing degree the history of the calculus. This calculus was the first great achievement of mathematics since the Greeks and it dominated mathematical exploration for centuries. The questions it answered and... raised lay at the heart of man's understanding of not only geometry and number, but also space and time and mathematical truth. It began with the surprising unification of two rather different geometrical problems, and almost immediately its ideas bore fruit in dozens of seemingly unrelated areas. The methods it developed gave the physical sciences an impetus without parallel in history, for through them natural science was born, and without them physics could not have progressed much further than the mystical vortices of Descartes."

"In the beginning there were two calculi, the differential and the integral. The first had been developed to determine the slopes of tangents to... curves, the second to determine... areas... bounded by curves. Algebra, geometry, and trigonometry were simply insufficient to solve general problems of this sort, and prior to the late seventeenth century mathematicians could at best handle only special cases."

"The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating \pi by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of \pi to any assigned degree of approximation. The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression\frac{\pi}{2} = \frac {2}{1}\cdot\frac {2}{3}\cdot\frac {4}{3}\cdot\frac {4}{5}\cdot\frac {6}{5}\cdot\frac {6}{7}\cdot\frac {8}{7}\cdot\frac {8}{9}\cdotsfor \pi as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value \frac{\pi}{2} according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for \frac{\pi}{8} the area of a semi-circle of diameter 1 as the definite integral \int\limits_{0}^{1}\sqrt{x-x^2}dx. The expression has the advantage over that of Vieta that the operations required are all rational ones."

"As to Cavalierian methods: one deceives oneself if one accepts their use as a demonstration, but they are useful as a means of discovery preceding a demonstration. ...Nevertheless, that which comes first and which matters most is the way in which the discovery has been made. It is this knowledge which gives most satisfaction and which one requires from the discoverers. It seems, therefore, preferable to supply the idea through which the result first came to light and through which it will be most readily understood. We will thereby save ourselves much labour and writing and the others the reading; it is necessary to bear in mind that mathematicians will never have enough time to read all the discoveries in Geometry (a quantity which is increasing day to day and seems likely in this scientific age to develop to enormous proportions) if they continue to be presented in a rigorous form, according to the manner of the ancients."

"This history of the development of calculus is significant because it illustrates the way in which mathematics progresses. Ideas are first grasped intuitively and extensively explored before they become fully clarified and precisely formulated even in the minds of the best mathematicians. Gradually the ideas are refined and given polish and rigor which one encounters in textbook presentations. In the instance of the calculus, mathematicians recognized the crudeness of their ideas and some even doubted the soundness of the concepts. Yet they not only applied them to physical problems, but used the calculus to evolve new branches of mathematics... They had the confidence to proceed so far along uncertain ground because their methods yielded correct results. Indeed, it is fortunate that mathematics and physics were so intimately related in the seventeenth and eighteenth centuries—so much so that they were hardly distinguishable—for the physical strength supported the weak logic of mathematics. Of course, mathematicians were selling their birthright, the surety of the results obtained by strict deductive reasoning from sound foundations, for the sake of scientific progress, but it is understandable that the mathematicians succumbed to the lure."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"That method [of infinitesimals] has the great inconvenience of considering quantities in the state in which they cease, so to speak, to be quantities; for though we can always well conceive the ratio of two quantities, as long as they remain finite, that ratio offers the to mind no clear and precise idea, as soon as its terms become, the one and the other, nothing at the same time."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be useful to write \int for\, omn., so that \int l = omn. l, or the sum of the l's... I propose to return to former considerations. Given l and its relation to x, to find \int l. Now this comes from the contrary calculus, that is to say if \int l = ya. Let us assume that l = ya/d, or as \int increases, so d will diminish the dimensions. But \int means a sum, and d a difference. From the given y, we can always find ya/d or l, or the difference of the y's. Hence one equation may be changed into the other..."

"My method is but a corollary of a general theory of transformations, by the help of which any given figure whatever, by whatever equation it may be accurately stated, is reduced to another analytically equivalent figure... Furthermore, the general method of transformations itself seems to me proper to be counted among the most powerful methods of analysis, for not merely does it serve for infinite series and approximations, but also for geometric solutions and endless other things that are scarcely manageable otherwise... The basis of the transformation is this: that a given figure, with innumerable lines [ordinates] drawn in any way (provided they are drawn according to some rule or law), may be resolved into parts, and that the parts—or others equal to them—when reassembled in another position or another form compose another figure, equivalent to the former or of the same area even if the shape is quite different; whence in many ways the quadratures can be attained... These steps are such that they occur at once to anyone who proceeds methodically under the guidance of Nature herself; and they contain the true method of indivisibles as most generally conceived and, as far as I know, not hitherto expounded with sufficient generality. For not merely parallel and convergent straight lines, but any other lines also, straight or curved, that are constructed by a general law can be applied to the resolution; but he who has grasped the universality of the method will judge how great and how abstruse are the results that can thence be obtained: For it is certain that all squarings hitherto known, whether absolute or hypothetical, are but limited specimens of this."

"The prime occasion from which arose my discovery of the method of the Characteristic Triangle, and other things of the same sort, happened at a time when I had studied geometry for not more than six months. Huygens, as soon as he had published his book on the pendulum, gave me a copy of it; and at that time I was quite ignorant of Cartesian algebra and also of the method of indivisibles, indeed I did not know the correct definition of the . For, when by chance I spoke of it to Huygens, I let him know that I thought that a straight line drawn through the center of gravity always cut a figure into two equal parts... Huygens laughed when he heard this, and told me that nothing was further from the truth. So I, excited by this stimulus, began to apply myself to the study of the more intricate geometry, although... I had not at that time really studied the Elements. But I found in practice that one could get on without a knowledge of the Elements, if only one was master of a few propositions. Huygens, who thought me a better geometer than I was, gave me to read the letters of Pascal, published under the name of Dettonville; and from these I gathered the method of indivisibles and centers of gravity, that is to say the well-known methods of Cavalieri and Guldinus."

"When M. Huygens lent me the "Letters of Dettonville" (or Pascal), I examined by chance his demonstration of the measurement of the spherical surface, and in it I found an idea that the author had altogether missed... Huygens was surprised when I told him of this theorem, and confessed to me that it was the very same as he had made use of for the surface of the parabolic . Now, as that made me aware of the use of what I call the "characteristic triangle" CFG, formed from the elements of the coordinates and the curve, I thus found as it were in the twinkling of an eyelid nearly all the theorems that I afterward found in the works of Barrow and Gregory. Up to that time, I was not sufficiently versed in the calculus [analytic geometry] of Descartes, and as yet did not make use of equations to express the nature of curved lines; but, on the advice of Huygens, I set to work at it, and I was far from sorry that I did so: for it gave me the means almost immediately of finding my differential calculus. This was as follows. I had for some time previously taken a pleasure in finding the sums of series of numbers, and for this I had made use of the well-known theorem, that, in a series decreasing to infinity, the first term is equal to the sum of all the differences. From this I had obtained what I call the "harmonic triangle," as opposed to the "arithmetical triangle" of Pascal; for M. Pascal had shown how one might obtain the sums of the figurate numbers, which arise when finding sums and sums of sums of the natural scale of arithmetical numbers. I on the other hand found that the fractions having figurate numbers for their denominators are the differences and the differences of the differences, etc., of the natural harmonic scale (that is, the fractions 1/1, 1/2, 1/3, 1/4, etc.), and that thus one could give the sums of the series of figurate fractions1/1 + 1/3 + 1/6 + 1/10 + etc, 1/1 + 1/4 + 1/10 + 1/20 + etc. Recognizing from this the great utility of differences and seeing that by the calculus of M. Descartes the ordinates of the curve could be expressed numerically, I saw that to find quadratures or the sums of the ordinates was the same thing as to find an ordinate (that of the ), of which the difference is proportional to the given ordinate. I also recognized almost immediately that to find tangents is nothing else but to find differences (differentier), and that to find quadratures is nothing else but to find sums, provided that one supposes that the differences are incomparably small. I saw also that of necessity the differential magnitudes could be freed from (se trouvent hors de) the fraction and the root-symbol (vinculum), and that thus tangents could be found without getting into difficulties over (se mettre en peine) irrationals and fractions. And there you have the story of the origin of my method."

"In the famous dispute regarding the invention of the infinitesimal calculus, while not denying... the priority of Newton... some... regard Leibnitz's introduction of the integral symbol \int as alone a sufficient substantiation of his claims to originality and independence, so far as the power of the new science was concerned."

"Many of the greatest discoveries of science, — for example, those of Galileo, Huygens, and Newton,—were made without the mechanism which afterwards becomes so indispensable for their development and application. Galileo's reasoning anent the summation of the impulses imparted to a falling stone is virtual integration; and Newton's mechanical discoveries were made by the man who invented, but evidently did not use to that end, the doctrine of s."

"Since the operations of computing in numbers and with variables are closely similar... I am amazed that it occurred to no one (if you except N. Mercator with his quadrature of the hyperbola) to fit the doctrine recently established for decimal numbers in similar fashion to variables, especially since the way is then open to more striking consequences. For since this doctrine in species has the same relationship to Algebra that the doctrine in decimal numbers has to common Arithmetic, its operations of Addition, Subtraction, Multiplication, Division and Root extraction may be easily learnt from the latter's."

"In a correspondence in which I was engaged with the very learned geometrician Mr. Leibnitz ten years ago, having informed him, that I was acquainted with a method of determining the maxima and minima, drawing tangents, and doing other similar things, which succeeded equally in rational equations and radical quantities, and having concealed this method by transposing the letters of the words, which signified: an equation containing any number of flowing quantities being given, to find the fluxions, and inversely: that celebrated gentleman answered, that he had found a similar method; and this, which he communicated to me, differed from mine only in the enunciation and notation, and in the idea of the generation of quantities."

"\frac {dy}{dx} = \frac {\omega^2x}{g}...The first derivative, the result of the differentiation of y with respect to x, was written by Leibniz in the form \frac {dy}{dx}...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative \frac {dy}{dx} was considered as the ratio of two "infinitely small quanitites", of the infinitesimals dy and dx. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...\frac {dy}{dx} is the limit of a ratio of dy to dx... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat \frac {dy}{dx} so as if it were a ratio... and multiply by dx to achieve the separation of variables. We get {dy} = \frac {\omega^2x}{g}xdx"

"The philosophical theory of the Calculus has been, ever since the subject was invented, in a somewhat disgraceful condition. Leibniz himself—who, one would have supposed, should have been competent to give a correct account of his own invention—had ideas, upon this topic which can only be described as extremely crude. He appears to have held that, if metaphysical subtleties are left aside, the Calculus is only approximate, but is justified practically by the fact that the errors to which it gives rise are less than those of observation. When he was thinking of , his belief in the actual infinitesimal hindered him from discovering that the Calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead. And in his mathematical expositions of the subject, he avoided giving careful proofs, contenting himself with the enumeration of rules. At other times, it is true, he definitely rejects infinitesimals as philosophically valid; but he failed to show how, without the use of infinitesimals, the results obtained by means of the Calculus could yet be exact, and not approximate. In this respect, Newton is preferable to Leibniz: his Lemmas give the true foundation of the Calculus in the doctrine of limits, and, assuming the continuity of space and time in Cantor's sense, they give valid proofs of its rules so far as spatio-temporal magnitudes are concerned. But Newton was, of course, entirely ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word fluxion, and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. Whether Leibniz avoided this error, seems highly doubtful; it is at any rate certain that, in his first published account of the Calculus, he defined the differential coefficient by means of the tangent to a curve. And by his emphasis on the infinitesimal, he gave a wrong direction to speculation as to the Calculus, which misled all mathematicians before Weierstrass (with the exception, perhaps, of De Morgan), and all philosophers down to the present day. It is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosopher..."

"In connection with the study of curves Fermat proceeded to apply the idea of infinitesimals to the questions of quadrature and of maxima and minima as well as to the drawing of tangents. In this he seems to have anticipated the work of Cavalieri, but the date of his discovery is unknown."

"Leibniz's thirtieth year and his last in the City of Light was his annus mirabulus. ...The year of miracles began in late August 1675 with the arrival of Walther Ehrenfried von Tschirnhaus. ...The two young Germans became instant best friends, achieving a degree of intimacy rarely matched in the course of Leibniz's life. ... In the Hôtel des Romains, the two expatriots promptly engaged in mathematical parleys. ...the papers preserved in Leibniz's files are crisscrossed with the scribbled handwriting of both men. It was around this time that Leibniz passed the threshold of the calculus. In a note from October 29, 1675, two months after Tschirnhaus's arrival, Leibniz for the first time used the symbol ∫ to stand for integration, replacing the earlier "omn" (for "omnes" [all]). Two weeks later, on November 11, he used dx for the first time to represent the "differential of x." Leibniz now believed himself to be in sole possession of the general method we call calculus. At some point he shuffled his new equations over to Tschirnhaus ...[who] dismissed it all as mere playing with symbols."

"[Leibniz] introduced the sign, ∫, in his De geometria... and proved the , that integration is the inverse of differentiation. The result was known to Newton and even, in geometric form, to Newton's teacher Barrow, but it became more transparent in Leibniz's formalism. For Leibniz, ∫ meant "sum," and \int f(x) dx was literally a sum of terms f(x) dx, representing infinitesimal areas of height f(x) and width dx. The difference operator d yields the last term f(x) dx in the sum, and dividing by the infinitesimal dx yields f(x). So voila!\frac{d}{dx}\int f(x) dx = f(x)"

"Algebra made an enormous difference to geometry. Whereas Archimedes had to make an ingenious new approach to each new figure... calculus dealt with a great variety of figures in the same way, via their equations. That was the whole point. Calculus was a method of calculating results, rather than proving them. If pressed, mathematicians could justify their calculations by the method of exhaustion, but it seemed impractical if not unnecessary... Huygens was probably the only major mathematician who stuck to the 'methods of the ancients.' The methods of calculus were so much more powerful and efficient that rigour became secondary. ...By the middle of the eighteenth century, calculus had solved almost all the problems of classical geometry, and new ones the ancients had not dreamed of. It had also revealed the secrets of the heavens, explaining the motions of the moons and planets with uncanny precision."

"The "exhaustion method" (the term "exhaust" appears first in , 1647) was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... by reducing problems... to... formal logic only. ...This indirect method... the standard Greek and Renaissance mode of strict proof in area and volume computation was quite rigorous, ...It had the disadvantage that the result... must be known in advance, so that the mathematician finds it first by another less rigorous and more tentative method. ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...several mathematicians before Newton, notably Kepler, used essentially the same conceptions... our modern limit conceptions have made it possible to build this... into a theory as rigorous as... "exhaustion"... The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The calculus was the first achievement of modern mathematics, and it is difficult to overestimate its importance. ...it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking."

"Differential geometry originally sneaked into theoretical physics through Einstein's theory of general relativity."

"現代幾何学においても、特性類の重要性は減ずるどころかますます大きくなっている。しかも単なるコホモロジー類としてだけでではなく、それを表わす微分形式自 身の詳細な解析がなされるなど、より深い役割を果たすようになってきている。たとえば、de RhamコホモロジーとRiemann計量の関わりを記述する理論である調和積 分論を、ずっと大きな枠組みのなかで一般化するという、壮大な試みが進行中である。そしてこれらの新しい展開のなかで、微分形式は、たとえてみれば生物にとっ ての水や空気のような役割を果たしているといっても過言ではないだろう。"

"Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language —the language of vectors —to describe those quantities. This language is also used in engineering, the other sciences, and even in common speech."

"Fermat applied his method of tangents to many difficult problems. The method has the form of the now-standard method of differential calculus, though it begs entirely the difficult theory of limits."

"One may regard Fermat as the first inventor of the new calculus. In his method De maximis et minimis he equates the quantity of which one seeks the maximum or the minimum to the expression of the same quantity in which the unknown is increased by the indeterminate quantity. In this equation he causes the radicals and fractions, if any such there be, to disappear and after having crossed out the terms common to the two numbers, he divides all others by the indeterminate quantity which occurs in them as a factor; then he takes this quantity zero and he has an equation which serves to determine the unknown sought. ...It is easy to see at first glance that the rule of the differential calculus which consists in equating to zero the differential of the expression of which one seeks a maximum or a minimum, obtained by letting the unknown of that expression vary, gives the same result, because it is the same fundamentally and the terms one neglects as infinitely small in the differential calculus are those which are suppressed as zeroes in the procedure of Fermat. His method of tangents depends on the same principle. In the equation involving the abscissa and ordinate which he calls the specific property of the curve, he augments or diminishes the abscissa by an indeterminate quantity and he regards the new ordinate as belonging both to the curve and to the tangent; this furnishes him with an equation which he treats as that for a case of a maximum or a minimum. ...Here again one sees the analogy of the method of Fermat with that of the differential calculus; for, the indeterminate quantity by which one augments the abscissa x corresponds to its differential dx, and the quantity ye/t, which is the corresponding augmentation [Footnote: Fermat lets e be the increment of x, and t the subtangent for the point x,y on the curve.] of y, corresponds to the differential dy. It is also remarkable that in the paper which contains the discovery of the differential calculus, printed in the Leipsic Acts of the month of October, 1684, under the title Nova methodus pro maximis et minimis etc., Leibnitz calls dy a line which is to the arbitrary increment dx as the ordinate y is to the subtangent; this brings his analysis and that of Fermat nearer together. One sees therefore that the latter has opened the quarry by an idea that is very original, but somewhat obscure, which consists in introducing in the equation an indeterminate which should be zero by the nature of the question, but which is not made to vanish until after the entire equation has been divided by that same quantity. This idea has become the germ of new calculi which have caused geometry and mechanics to make such progress, but one may say that it has brought also the obscurity of the principles of these calculi. And now that one has a quite clear idea of these principles, one sees that the indeterminate quantity which Fermat added to the unknown simply serves to form the derived function which must be zero in the case of a maximum or minimum, and which serves in general to determine the position of tangents of curves. But the geometers contemporary with Fermat did not seize the spirit of this new kind of calculus; they did not regard it but a special artifice, applicable simply to certain cases and subject to many difficulties, ...moreover, this invention which appeared a little before the Géométrie of Descartes remained sterile during nearly forty years. ...Finally Barrow contrived to substitute for the quantities which were supposed to be zero according to Fermat quantities that were real but infinitely small, and he published in 1674 his method of tangents, which is nothing but a construction of the method of Fermat by means of the infinitely small triangle, formed by the increments of the abscissa e, the ordinate ey/t, and of the infinitely small arc of the curve regarded as a polygon. This contributed to the creation of the system of infinitesimals and of the differential calculus."

"This great geometrician expresses by the character E the increment of the abscissa; and considering only the first power of this increment, he determines exactly as we do by differential calculus the subtangents of the curves, their points of inflection, the maxima and minima of their ordinates, and in general those of rational functions. We see likewise by his beautiful solution of the problem of the refraction of light inserted in the Collection of the Letters of Descartes that he knows how to extend his methods to irrational functions in freeing them from irrationalities by the elevation of the roots to powers. Fermat should be regarded, then, as the true discoverer of Differential Calculus. Newton has since rendered this calculus more analytical in his Method of Fluxions, and simplified and generalized the processes by his beautiful theorem of the binomial. Finally, about the same time Leibnitz has enriched differential calculus by a notation which, by indicating the passage from the finite to the infinitely small, adds to the advantage of expressing the general results of calculus, that of giving the first approximate values of the differences and of the sums of the quantities; this notation is adapted of itself to the calculus of partial differentials."

"It will be sufficient if, when we speak of infinitely great (or more strictly unlimited), or of infinitely small quantities (i.e., the very least of those within our knowledge) it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please, or as small as you please, so that the error that any one may assign may be less than a certain assigned quantity. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be less, it follows that the error is absolutely nothing; an almost exactly similar kind of argument is used in different places by Euclid, Theodosius and others; and this seemed to them to be a wonderful thing, although it could not be denied that it was perfectly true that, from the very thing that was assumed as an error, it could be inferred that the error was non-existent. Thus by infinitely great and infinitely small, we understand something indefinitely great, or something indefinitely small, so that each conducts itself as a sort of class, and not merely as the last thing of a class. If any one wishes to understand these as the ultimate things, or as truly infinite, it can be done, and that too without falling back upon a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, ay, even though he think that such things are utterly impossible; it will be sufficient simply to make use of them as a tool that has advantages for the purpose of the calculation, just as the algebraists retain imaginary roots with great profit. For they contain a handy means of reckoning, as can manifestly be verified in every case in a rigorous manner by the method already stated. But it seems right to show this a little more clearly, in order that it may be confirmed that the algorithm, as it is called, of our differential calculus, set forth by me in the year 1684, is quite reasonable."

"To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus."

"In the beginning of the year 1665 I found the method of approximating series and the rule for reducing any dignity [power] of any binomial to such a series [ i.e. the binomial theorem ]. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions [i.e. the elements of differential calculus], and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions [i.e. integral calculus], and in the same year I began to think of gravity extending to the orb of the moon … and having thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them to answer pretty nearly. All this was in the two years of 1665 and 1666, for in those years I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since."

"[S]cientific physics dates its existence from the discovery of the differential calculus. Only when it was learned how to follow continuously the course of natural events, attempts, to construct by means of abstract conceptions the connection between phenomena, met with success. To do this two things are necessary: First, simple fundamental concepts with which to construct; second, some method by which to deduce, from the simple fundamental laws of the construction which relate to instants of time and points in space, laws for finite intervals and distances, which alone are accessible to observation..."

"Fermat is... honored with the invention of the differential calculus on account of his method of maxima and minima and of tangents, which, of the prior processes, is in reality the nearest to the algorithm of Leibniz; one could with equal justice, attribute to him the invention of the integral calculus; his treatise De æquationum localium transmutatione, etc., gives indeed the method of integration by parts as well as rules of integration, except the general powers of variables, their sines and powers thereof. However, it must be remarked that one does not find in his writings a single word on the main point, the relation between the two branches of the infinitesimal calculus."

"The calculus of utility aims at supplying the ordinary wants of man at the least cost of labour."

"In this work I have attempted to treat economy as a calculus of pleasure and pain, and have sketched out, almost irrespective of previous opinions, the form which the science, as it seems to me, must ultimately take."

"In the history of mathematics and science, few conflicts have attained the notoriety of the Newton/Leibniz dispute. … A carefully reconstructed chronology reveals that Newton had formulated the essentials of his calculus by 1666, years before Leibniz had attained the mathematical knowledge necessary to develop his own point of view on the calculus. … There is much that can never be known about such a feud. This feud is peculiar in that it erupted late, and was both sparked and carried on to a large degree by the followers of the men involved. There are scientific reasons for it (the divergences in their interpretation of "the calculus" itself; personal reasons (a history of suspicion, not only between the two principles but between each of them and other rivals; nationalism, never a negligible factor; and the bitterness associated with disputes on related matters, notably the ongoing rivalry between the Newtonian and Cartesian theories of gravity. At a personal level, Newton's pride, suspicious character, and reluctance to publish collided with Leibniz' naive optimism, arrogance, and his belief in "systems" as more valuable than inspiration, in a long-delayed but virulent explosion."

"The theory of probabilities is basically only common sense reduced to a calculus. It makes one estimate accurately what right-minded people feel by a sort of instinct, often without being able to give a reason for it."

"Among all of the mathematical disciplines the theory of differential equations is the most important … It furnishes the explanation of all those elementary manifestations of nature which involve time."

"Who has not been amazed to learn that the function y = ex, like a phoenix rising from its own ashes, is its own derivative?"

"In England, where it originated, the calculus fared less well. ...by siding completely with Newton in the priority dispute, they cut themselves off from developments on the Continent. They stubbornly stuck to Newton's dot notation of fluxions, failing to see the advantages of Leibniz's differential notation. As a result, over the next hundred years, while mathematics fluorished in Europe as never before, England did not produce a single first-rate mathematician. When the period of stagnation finally ended around 1830, it was not in analysis but in algebra that the new generation of English mathematicians made their greatest mark."

"Accordingly, we find Euler and Alembert devoting their talent and their patience to the establishment of the laws of rotation of the solid bodies. Lagrange has incorporated his own analysis of the problem with his general treatment of mechanics, and since his time M. Poinsôt has brought the subject under the power of a more searching analysis than that of the calculus, in which ideas take the place of symbols, and intelligent propositions supersede equations."

"This is a tricky domain because, unlike simple arithmetic, to solve a calculus problem - and in particular to perform integration - you have to be smart about which integration technique should be used: integration by partial fractions, integration by parts, and so on."

"The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.""

"In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Bionomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the Theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon & (having found out how to estimate the force with wch [a] globe revolving within a sphere presses the surface of the sphere) from Keplers rule of the periodic times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces wch keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about wch they revolve: & thereby compared the force requisite to keep the Moon in her Orb with the force of gravity at the surface of the earth, & found them answer pretty nearly."