History of calculus

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History of calculus or infinitesimal calculus, is a history of a mathematical discipline focused on limits, functions, derivatives, integrals, and infinite series. Isaac Newton and Gottfried Leibniz independently invented calculus in the mid-17th century. A rich history and cast of characters participating in the development of calculus both preceded and followed the contributions of these singular individuals.


CONTENT
A - B - C - D - E-G - H-K - L - M-P - Q-S - T-Z
De lineis insecabilibus (ca. 340 B.C.)
Euclid's Elements (ca. 300 B.C.)
Arithmetica Infinitorum (1656)
History and Origin of The Differential Calculus (1714)
A General History of Mathematics (1803)
A Short Account of the History of Mathematics (1888, 1910)
A History of the Study of Mathematics at Cambridge (1889)
A History of Mathematics (1893, 1919)
History of Modern Mathematics (1896)
The Geometrical Lectures of Isaac Barrow (1914)
See also-External links

Quotes[edit]

A[edit]

  • Nothing in Descartes' work led directly to Leibniz's calculus, but Descartes' discoveries in mathematics were certainly forerunners of the calculus.
    We know that in 1661... Newton read books about Descartes' mathematics. ...without Descartes' unification of algebra and geometry it would have been impossible to describe graphs using mathematical equations, and hence, except perhaps as a pure theory, the calculus would be completely devoid of meaning.
  • In Sorbière's day, European thinkers and intellectuals of widely divergent religious and political affiliations campaigned tirelessly to stamp out the doctrine of indivisibles and to eliminate it from philosophical and scientific consideration. In the very years that Hobbes was fighting Wallis over the indivisible line in England, the Society of Jesus was leading its own campaign against the infinitely small in Catholic lands. In France, Hobbes's acquaintance René Descartes, who had initially shown considerable interest in infinitesimals, changed his mind and banned the concept.. Even as late as the 1730s... George Berkeley mocked mathematicians for their use of infinitesimals... Lined up against these naysayers were some of the most prominent mathematicians and philosophers of that era, who championed the use of the infinitesimally small. These included, in addition to Wallis: Galileo and his followers, Bernard Le Bovier de Fontenelle, and Isaac Newton.
    • Amir Alexander, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014)
  • On the one side were ranged the forces of hierarchy and order—Jesuits, Hobbesians, French Royal Courtiers, and High Church Anglicans. They believed in a unified and fixed order in the world, both natural and human, and were fiercely opposed to infinitesimals. On the other side were comparative "liberalizers" such as Galileo, Wallis, and the Newtonians. They believed in a more pluralistic and flexible order, one that might accommodate a range of views and diverse centers of power, and championed infinitesimals and their use in mathematics. The lines were drawn, and a victory for one side or the other would leave its imprint on the world for centuries to come.
    • Amir Alexander, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014)

B[edit]

  • [Joseph-Louis Lagrange's] lectures on differential calculus form the basis of his Theorie des fonctions analytiques which was published in 1797. ...its object is to substitute for the differential calculus a group of theorems based upon the development of algebraic functions in series. A somewhat similar method had been previously used by John Landen in his Residual Analysis... Lagrange believed that he could... get rid of those difficulties, connected with the use of infinitely large and infinitely small quantities, to which philosophers objected in the usual treatment of the differential calculus. ...Another treatise in the same lines was his Leçons sur le calcul des fonctions, issued in 1804. These works may be considered as the starting-point for the researches of Cauchy, Jacobi, and Weierstrass.
  • Nothing is easier... than to fit a deceptively smooth curve to the discontinuities of mathematical invention. Everything then appears as an orderly progression... with Cavalieri, for instance, indistinguishable from Newton in the neighborhood of the calculus, or Lagrange from Fourier in that of trigonometric series, or Bhaskara from Lagrange in the region of Fermat's equation. Professional historians may sometimes be inclined to overemphasize the smoothness of the curve; professional mathematicians, mindful of the dominant part played in geometry by the singularities of curves, attend to the discontinuities. ...That such differences should exist is no disaster. Dissent is good for the souls of all concerned.
    • Eric Temple Bell, The Development of Mathematics (1940)
  • Descartes' method of finding tangents and normals... was not a happy inspiration. It was quickly superseded by that of Fermat as amplified by Newton. Fermat's method amounts to obtaining a tangent as the limiting position of a secant, precisely as is done in the calculus today. ...Fermat's method of tangents is the basis of the claim that he anticipated Newton in the invention of the differential calculus.
  • Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first.
  • The fundamental definitions of the calculus, those of the derivative and integral, are now so clearly stated in textbooks on the subject... that it is easy to forget the difficulty with which these basic concepts have been developed.
    • Carl B. Boyer The History of the Calculus and Its Conceptual Development (1949).
  • The precision of statement and the facility of application which the rules of the calculus early afforded were in a measure responsible for the fact that mathematicians were insensible to the delicate subtleties required in the logical development... They sought to establish calculus in terms of the conceptions found in traditional geometry and algebra which had been developed from spatial intuition.
    • Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1949).
  • Just as the problem of defining instantaneous velocities in terms of the approximation of average velocities was to lead to the definition of the derivative, so that of defining lengths, areas, and volumes of curvilinear configurations was to eventuate in the formation of the definite integral. ...This definition then invokes, apart from the ordinary operations of arithmetic, only the concept of the limit of an infinite sequence of terms, precisely as does that of the derivative. The realization of this fact, however, followed only after many centuries of investigation by mathematicians.
    • Carl B. Boyer, The History of the Calculus and Its Conceptual Development (1949).
  • Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. ...Fermat applied his method ...and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite.
  • The Calculus of Variations owed its origin to the attempt to solve a very interesting and rather narrow class of problems in Maxima and Minima, in which it is required to find the form of a function such that the definite integral of an expression involving that function and its derivative shall be a maximum or a minimum.

C[edit]

  • Methods of drawing tangents were invented by Roberval and Fermat... Descartes gave a third method. Of all the problems which he solved by his geometry, none gave him as great pleasure as his mode of constructing tangents. It is profound but operose, and, on that account, inferior to Fermat's. His solution rests on the method of Indeterminate Coefficients, of which he bears the honour of invention. Indeterminate coefficients were employed by him also in solving bi-quadratic equations.
    • Florian Cajori, A History of Elementary Mathematics (1898, 1901) Set up and electrotyped January, 1894; Reprinted March, 1895; October, 1897; November, 1901.
  • J.M. Child... has made a searching study of Barrow and has arrived at startling conclusions on the historical question relating to the first invention of the calculus. He places his conclusions in italics in the first sentence as follows Isaac Barrow was the first inventor of the Infinitesimal Calculus... Before entering upon an examination of the evidence brought forth by Child it may be of interest to review a similar claim set up for another man as inventor of the calculus... Fermat was declared to be the first inventor of the calculus by Lagrange, Laplace, and apparently also by P. Tannery, than whom no more distinguished mathematical triumvirate can easily be found. ...Dinostratus and Barrow were clever men, but it seems to us that they did not create what by common agreement of mathematicians has been designated by the term differential and integral calculus. Two processes yielding equivalent results are not necessarily the same. It appears to us that what can be said of Barrow is that he worked out a set of geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus. But to say that Barrow invented a differential and integral calculus is to do violence to the habit of mathematical thought and expression of over two centuries. The invention rightly belongs to Newton and Leibniz.
    • Florian Cajori, "Who was the First Inventor of Calculus" The American Mathematical Monthly (1919) Vol.26
  • It is a curious fact in the history of mathematics that discoveries of the greatest importance were made simultaneously by different men of genius. The classical example is the... development of the infinitesimal calculus by Newton and Leibniz. Another case is the development of vector calculus in Grassmann's Ausdehnungslehre and Hamilton's Calculus of Quaternions. In the same way we find analytic geometry simultaneously developed by Fermat and Descartes.
    • Julian Lowell Coolidge, A History of Geometrical Methods (1940). Reference is to Hermann Grassmann's Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (The Theory of Linear Extension, a New Branch of Mathematics).

D[edit]

  • In the method of exhaustion, Archimedes possessed all the elements essential to an infinitesimal analysis. ...the idea of limit as conceived by Archimedes was adequate for the development of the calculus of Newton and Leibnitz and... it remained practically unchanged until the days of Weierstrass and Cantor. ...the principle ...consists in "trapping" the variable magnitude between two others, as between two jaws of a vise. Thus, in the case of the periphery of a circle... Archimedes grips the circumference between two sets of regular polygons of an increasing number of sides... one set is circumscribed... and the other is inscribed. ...By this method he also found the area under a parabolic arch...
  • The statement is so frequently made that the differential calculus deals with continuous magnitude, and yet an explanation of this continuity is nowhere given; even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner. ...It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858.
  • If a cone is cut by surfaces parallel to the base, then how are the sections, equal or unequal? If they are unequal then the cone would have the shape of a staircase; but if they were equal, then all sections will be equal, and the cone will look like a cylinder, made up of equal circles; but this is entirely nonsensical.
  • ...nor have I found occasion to depart from the plan... the rejection of the whole doctrine of series in the establishment of the fundamental parts both of the Differential and Integral Calculus. The method of Lagrange... had taken deep root in elementary works; it was the sacrifice of the clear and indubitable principle of limits to a phantom, the idea that an algebra without limits was purer than one in which that notion was introduced. But, independently of the idea of limits being absolutely necessary even to the proper conception of a convergent series, it must have been obvious enough to Lagrange himself, that all application of the science to concrete magnitude, even in his own system, required the theory of limits.
  • I have throughout introduced the Integral Calculus in connexion with the Differential Calculus. ...Is it always proper to learn every branch of a direct subject before anything connected with the inverse relation is considered? If so why are not multiplication and involution in arithmetic made to follow addition and precede subtraction? The portion of the Integral Calculus, which properly belongs to any given portion of the Differential Calculus increases its power a hundred-fold...
    • Augustus De Morgan, The Differential and Integral Calculus (1836)
  • When... we have a series of values of a quantity which continually diminish, and in such a way, that name any quantity we may, however small, all the values, after a certain value, are severally less than that quantity, then the symbol by which the values are denoted is said to diminish without limit. And if the series of values increase in succession, so that name any quantity we may, however great, all after a certain point will be greater, then the series is said to increase without limit. It is also frequently said, when a quantity diminishes without limit, that it has nothing, zero or 0, for its limit: and that when it increases without limit it has infinity or ∞ or 1⁄0 for its limit.

E-G[edit]

  • Kepler imagined a given geometrical figure to be decomposed into infinitesimal figures, whose areas or volumes he added up in some ad hoc way to obtain the area or volume... Cavalieri proceeded by setting up a one-to-one correspondence between the indivisible elements of two geometrical figures. If corresponding indivisibles of the two figures had a certain (constant) ratio, he concluded that the areas of volumes of one of the figures had the same ratio. Typically, the area or volume of one of the figures was known in advance, so this gave the other. ...
    Kepler thought of a geometrical figure as being composed of indivisibles of the same dimension [as the original figure]... from some process of successive subdivision... However, Cavalieri generally considered a geometrical figure to be composed of an indefinitely large number of indivisibles of lower dimension. ...an area as consisting of ...line segments, and a volume as consisting of... plane sections... Rigor, he wrote in the Exercitationes, is the affair of philosophy rather than mathematics.
    • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)
  • Newton regarded the curve as the locus of the intersection of two moving lines, one vertical and the other horizontal. The and coordinates of the moving point are then functions of the time , specifying the locations of the vertical and horizontal lines... The motion is then the composition of a horizontal motion with velocity vector having length and a vertical motion with velocity vector having length . ...the velocity vector is the parallelogram sum of these ...It follows that the slope of the tangent line to the curve is .
    • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)
  • Shortly after his arrival in Paris in 1672, [ Leibniz ] noticed an interesting fact about the sum of differences of consecutive terms of a sequence of numbers. Given the sequence
    consider the sequence
    of differences . Then
    .
    Thus the sum of the consecutive differences equals the difference of the first and last terms of the original sequence. ...
    His result on sums of differences also suggested... the possibility of summing an infinite series of numbers. ...
    If, in addition,
    [ ]
    • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)
Pascal'sTriangle.jpg
LeibnizHarmonicTriangle.jpg
  • Pascal's aritmentic triangle and Leibniz' harmonic triangle enjoy a certain inverse relationship... These considerations implanted in Leibniz' mind a vivid conception that was to play a dominant role in his development of the calculus—the notion of an inverse relationship between the operation of taking differences and that of forming sums of the elements of a sequence.
    • C. H. Edwards, Jr., The Historical Development of the Calculus (1979)
  • 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 derivative and integral—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 Fundamental Theorem of Calculus 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.
    • Hardy Grant, Israel Kleiner, Turning Points in the History of Mathematics (2016)

H-K[edit]

  • The subject of infinitesimals 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 method of exhaustion, with its double reductio ad absurdum is still necessary.
  • 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 by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of 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 Savilian Professor of Geometry 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
    for 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 according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for the area of a semi-circle of diameter 1 as the definite integral . 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.
    • Christiaan Huygens, Oeuvres complètes Tome XIV. Probabilités. Travaux de mathématiques pures 1655-1666 (1920) p. 337, as quoted by Margaret E. Baron, The Origins of the Infinitesimal Calculus (1969)
  • 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.

L[edit]

  • 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.
    • Joseph Louis Lagrange, "Lecons sur le calcul des fonctions," leçon dix-huitiéme, (Euvres de Lagrange, publiées par J.A. Serret, Tome X, p. 294 as translated by Florian Cajori, "Who was the First Inventor of Calculus" The American Mathematical Monthly (1919) Vol.26
  • 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 , so that , or the sum of the 's... I propose to return to former considerations. Given and its relation to , to find . Now this comes from the contrary calculus, that is to say if . Let us assume that , or as increases, so will diminish the dimensions. But means a sum, and a difference. From the given , we can always find or , or the difference of the '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.
    • Gottfried Wilhelm Leibniz, Response letter to Newton's Epistola prior (Aug. 27, 1676) as quoted by H. W. Turnbull et. al., The Correspondence of Isaac Newton (1960) Vol. 2 (1676-1687) pp. 65-66.
  • 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 center of gravity. 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.
Leibniz's characteristic triangle
from Pascal's "Letters of Dettonville"
  • 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 conoid. 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 fractions
    1/1 + 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 quadratrix), 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.

M-P[edit]

  • 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.
    • Isaac Newton, De methodis serierum et fluxionum, Mathematical Papers, 3, 32-353, as quoted by John Stillwell, Mathematics and Its History (1989)
  • 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.
  • ...The first derivative, the result of the differentiation of with respect to , was written by Leibniz in the form
    ...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 was considered as the ratio of two "infinitely small quanitites", of the infinitesimals and . ...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. ... is the limit of a ratio of to ... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat so as if it were a ratio... and multiply by to achieve the separation of variables. We get

Q-S[edit]

  • 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 Dynamics, 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 fundamental theorem of calculus, 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 was literally a sum of terms , representing infinitesimal areas of height and width . The difference operator yields the last term in the sum, and dividing by the infinitesimal yields . So voila!
  • 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.
    • John Stillwell, "Logic and Philosophy of mathematics in the nineteenth century," Routledge History of Philosophy Volume VII (2013) ed. C. L. Ten, p. 204.
  • The "exhaustion method" (the term "exhaust" appears first in Grégoire de Saint-Vincent, 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.
    • Dirk Jan Struik, A Concise History of Mathematics (1948) with a reference to S. Luria, "Die Infinitesimaltheorie der antiken Atomisten" Quellen und Studien B 2 (1932) pp.106-185.

T-Z[edit]

  • 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.
    • P. Tannery, "Fermat" in La Grande Encyclopédie (Berthelot) as quoted by Florian Cajori, "Who was the First Inventor of Calculus?" The American Mathematical Monthly (Jan. 1919) Vol. 26, No. 1, p. 17.
  • 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.
    • John von Neumann, "The Mathematician," The Works of the Mind (1947) pp. 180-196.
  • The origins of calculus are clearly empirical. Kepler's first attempts at integration were formulated as "dolichometry"—measurement of kegs—that is, volumetry for bodies with curved surfaces. This is... post-Euclidean geometry, and... nonaxiomatic, empirical geometry. Of this, Kepler was fully aware. The main effort and... discoveries, those of Newton and Leibniz, were of an explicitly physical origin. Newton invented the calculus "of fluxions" essentially for the purpose of mechanics—in fact... calculus and mechanics were developed by him more or less together. The first formulations of the calculus were not even mathematically rigorous. An inexact, semiphysical formulation was the only one available for over a hundred and fifty years after Newton! And yet, some of the most important advances of analysis took place during this period... ! Some of the leading mathematical spirits... were clearly not rigorous, like Euler; but others, in the main, were, like Gauss or Jacobi. The development was as confused and ambiguous as can be, and its relation to empiricism was certainly not according to our present (or Euclid's) ideas of abstraction and rigor. Yet... that period produced mathematics as first class as ever existed! And even after the reign of rigor was... re-established with Cauchy, a... relapse into semiphysical methods took place with Riemann.
    • John von Neumann, "The Mathematician," The Works of the Mind (1947) pp. 180-196.
Pascal Quadrature
  • Riemann gave a rigorous definition of the integral by enclosing it between... the "lower sum"... the sum of the areas of the rectangles below the curve, and the "upper sum"... the sum of rectangles of somewhat greater height, which cover the area. The treatise on conoids and spheroids shows that Archimedes was familiar with this method of inclusion and... used it for the determination of volumes. But... one cannot say that he was familiar with the concept of the integral. His integrals always remained tied to a definite geometric interpretation, as volumes or as areas of plane figures. We have no evidence that he understood that one single concept is the foundation of all these geometric interpretations... he bases his rigorous proofs on totally different methods... Nevertheless, his rigorous determination of areas and volumes make Archimedes the precursor of the modern integral calculus.
  • On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect...
    • Robert Woodhouse, A Treatise on Isoperimetrical Problems, and the Calculus of Variations (1810)

De lineis insecabilibus (ca. 340 B.C.)[edit]

Aristotle, or possibly Theophrastus, source, as translated by Harold Henry Joachim
  • Are there indivisible lines? And, generally, is there a simple unit in every class of quanta?
    §1. Some people maintain this thesis on the following grounds:—
    (i) If we recognize the validity of the predicates 'big' and 'great', we must equally recognize the validity of their opposites 'little' and 'small'. Now that which admits practically an infinite number of divisions, is 'big' not 'little' <or 'great' not 'small'>. Hence, the 'little' quantum and the 'small' quantum will clearly admit only a finite number of divisions. But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates 'little' and 'small' apply.
    (ii) Again, if there is an Idea of line, and if the Idea is first of the things called by its name:—then, since the parts are by nature prior to their whole, the Ideal Line must be indivisible. And on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures—and, generalizing, the Ideal Plane and the Ideal Solid—must be without parts: for otherwise it will result that there are elements prior to each of them.
    (iii) Again, if Body consists of elements, and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indivisible: for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought.
    (iv) Again, Zeno's argument proves that there must be simple magnitudes. For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any 'stretch' which is not simple, motion—unless there be simple magnitudes—involves that the moving body touches successively one-by-one an infinite number of points in a finite time: which is impossible.
    But even if the body which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the 'stretch' which it traverses in an equal time: and since the movement of thought is quickest of all movements:—it follows that thought too will come successively into contact with an infinity of objects in a finite time. And since 'thought's coming into contact with objects one-by-one' is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible there must be such a thing as an indivisible line. ...

Euclid's Elements (ca. 300 B.C.)[edit]

Euclid, as translated, unless otherwise indicated, by Thomas Little Heath, The Thirteen Books of Euclid's Elements (1908) Vol. 1 Intro, Bks 1-2, Vol. 2, Bks 3-9, Vol. 3, Bks 10-13, Appndx.
  • Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.
    • Book X. Proposition 1.

Arithmetica Infinitorum (1656)[edit]

John Wallis, translation via Jacqueline A. Stedal, The Arithmetic of Infinitesimals: John Wallis 1656 (2004) unless otherwise indicated
  • You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used...
  • This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.
  • Around 1650 I came across the mathematical writings of Torricelli, where among other things, he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met.

History and Origin of The Differential Calculus (1714)[edit]

Gottfried Wilhelm Leibniz, as translated with critical and historical notes from Historia et Origo Calculi Differentialis by J. M. Child, in The Early Mathematical Manuscripts of Leibniz (1910) p. 22-58, unless otherwise noted.
  • It is an extremely useful thing to have knowledge of the true origins of memorable discoveries, especially those that have been found not by accident but by dint of meditation. ...the art of making discoveries should be extended by considering noteworthy examples of it.
  • Among the most renowned discoveries of the times must be considered that of a new kind of mathematical analysis, known by the name of the differential calculus; and of this... the origin and the method of the discovery are not yet known to the world at large.
  • Its author invented it nearly forty years ago, and nine years later (nearly thirty years ago) published it in a concise form; and from that time it has... been a method of general employment; while many splendid discoveries have been made by its assistance... so that it would seem that a new aspect has been given to mathematical knowledge arising out of its discovery.
  • Now there never existed any uncertainty as to the name of the true inventor, until recently, in 1712, certain upstarts... acted with considerable shrewdness, in that they put off starting the dispute until those who knew the circumstances, Huygens, Wallis, Tschirnhaus, and others, on whose testimony they could have been refuted, were all dead.
    • Child's footnote: This is untrue. ...the attack was first made publicly in 1699... although Huygens had been dead... Tschirnhaus was still alive, and Wallis was appealed to by Leibniz. ...Leibniz did not appeal to Tschirnhaus, through whom it is suggested by [Hermann] Weissenborn that Leibniz may have had information of Newton's discoveries.
  • They have changed the whole point of the issue, for... they have set forth their opinion... as to give a dubious credit to Leibniz, they have said very little about the calculus; instead every other page is made up of what they call infinite series. Such things were first given as discoveries by Nicolaus Mercator of Holstein who obtained them by the process of division, and Newton gave the more general form by extraction of roots [binomial expansion by the interpolation method of Wallis]. This is certainly a useful discovery, for by it arithmetical approximations are reduced to an analytical reckoning; but it has nothing at all to do with the differential calculus. Moreover, even in this they make use of fallacious reasoning; for whenever this rival works out a quadrature by the addition of the parts by which a figure is gradually increased, at once they hail it as the use of the differential calculus... By the selfsame argument, Kepler (in his Stereometria Doliorum), Cavalieri, Fermat, Huygens, and Wallis used the differential calculus; and indeed, of those who dealt with "indivisibles" or the "infinitely small," who did not use it? But Huygens, who as a matter of fact had some knowledge of the method of fluxions as far as they are known and used, had the fairness to acknowledge that a new light was shed upon geometry by this calculus, and that knowledge of things beyond the province of that science was wonderfully advanced by its use.
    • Child's footnotes: We now see what was Leibniz's point; the differential calculus was not the employment of an infinitesimal and a summation of such quantities; it was the use of the idea of these infinitesimals being differences, and the employment of the notation invented by himself, the rules that governed the notation, and the fact that differentiation was the inverse of a summation; and perhaps the greatest point of all was that the work had not to be referred to a diagram. This is on an inestimably higher plane than the mere differentiation of an algebraic expression whose terms are simple powers and roots of the independent variable.
      Why is Barrow omitted from this list? As... in the case of Barrow's omission of all mention of Fermat, was Leibniz afraid to awake afresh the sleeping suggestion as to his indebtedness to Barrow? I have suggested that Leibniz read his Barrow on his journey back from London, and perhaps tiring at having read the Optics first, and then the preliminary five lectures, just glanced at the remainder and missed the main important theorems. I also make another suggestion, namely, that... in his then ignorance of geometry he did not understand Barrow. If this is the case it would have been gall and wormwood for Leibniz to have ever owned to it. Then let us suppose that in 1674 with a fairly competent knowledge of higher geometry he reads Barrow again, skipping the Optics.. and the wearisome preliminary lectures... He notes the theorems as those he has himself already obtained, and the few that are strange to him he translates into his own symbolism. I suggest that this is a feasible supposition which would account for the marks that [Karl Immanuel] Gerhardt states are made in the margin. It would account for the words "in which latter I found the greater part of my theorems anticipated"... it would account for his using Barrow's "differential triangle" instead of his own "characteristic triangle." As Barrow tells his readers in his preface that "what these lectures bring forth, or to what they may lead you may easily learn from the beginnings of each, let us suppose that Leibniz took his advice. What do we find? The first four theorems of Lecture VIII give the geometrical equivalent of the differentiation of a power of a dependent variable; the first five of Lecture IX lead to a proof that [in our modern notation]...; the appendix to this lecture contains the differential triangle, and five examples on the a and e method, fully worked out; the first theorem in Lecture XI has a diagram such that, when that part of it is dissected out... this portion of the figure is a mirror image of the figure drawn by Leibniz when describing the characteristic triangle. ...and the second theorem of Lecture XII is the strangest coincidence of all. ...The sixth theorem of this lecture is the theorem of Gregory which Leibniz also gives later.
      Yet if all this were so, he could still say with perfect truth that, in the matter of the invention of the differential calculus (as he conceived the matter to consist, that is, the differential and integral notations and the method of analysis) he derived no assistance from Barrow. In fact, once he had absorbed his fundamental ideas, Barrow would be less of a help than a hindrance.
  • On his return from England to France in the year 1673... at the instigation of Huygens he began to work at Cartesian analysis (which afore-time had been beyond him), and in order to obtain an insight into the geometry of quadratures he consulted the Synopsis Geometriae of Honoratus Fabri, Gregory St. Vincent, and a little book by Dettonville (i.e., Pascal [letters to M. de Carcavi]). Later on from one example given by Dettonville, a light suddenly burst upon him, which strange to say Pascal himself had not perceived in it. For when he proves the theorem of Archimedes for measuring the surface of a sphere or parts of it, he used a method in which the whole surface of the solid formed by a rotation round any axis can be reduced to an equivalent plane figure. From it our young friend made out for himself the following general theorem.
    Portions of a straight line normal to a curve, intercepted between the curve and an axis, when taken in order and applied at right angles to the axis give rise to a figure equivalent to the moment of the curve about the axis. When he showed this to Huygens the latter praised him highly and confessed to him that by the help of this very theorem he had found the surface of parabolic conoids and others of the same sort, stated without proof many years before in his work on the pendulum clock. Our young friend, stimulated by this and pondering on the fertility of this point of view, since previously he had considered infinitely small things such as the intervals between the ordinates in the method of Cavalieri and such only, studied the triangle... which he called the Characteristic Triangle...
    • Child's footnote: This theorem is given, and proved by the method of indivisibles, as Theorem I of Lecture XII in Barrow's Lectiones Geometricae; and Theorem II is simply a corollary, in which it is remarked: Hence the surfaces of the sphere, both the spheroids, and the conoids receive measurement..."
  • To find the area of a given figure, another figure is sought such that its subnormals are respectively equal to the ordinates of the given figure, and then this second figure is the quadratrix of the given one; and thus from this extremely elegant consideration we obtain the reduction of the areas of surfaces described by rotation to plane quadratures, as well as the rectification of curves; at the same time we can reduce these quadratures of figures to an inverse problem of tangents. From these results, our young friend [Leibniz] wrote down a large collection of theorems (among which in truth there were many that were lacking in elegance) of two kinds. For in some of them only definite magnitudes were dealt with, after the manner not only of Cavalieri, Fermat, Honoratus Fabri, but also of Gregory St. Vincent, Guldinus, and Dettonville; others truly depended on infinitely small magnitudes, and advanced to a much greater extent. But later our young friend did not not trouble to go on with these matters, when he noticed that the same method had been brought into use and perfected by not only Huygens, Wallis, van Huraet, and Neil, but also by James Gregory and Barrow.
    • Child's footnote: "From these results"—which I have suggested he got from Barrow—"our young friend wrote down a large collection of theorems." These theorems Leibniz probably refers to when he says that he found them all to have been anticipated by Barrow, "when his Lectures appeared." I suggest that the "results" were all that he got from Barrow on his first reading, and that the "collection of theorems" were found to have been given in Barrow when Leibniz referred to the book again, after his geometrical knowledge was improved so far that he could appreciate it.
Pascal's (Arithmetic) Triangle
Leibniz Harmonic Triangle
  • Now it is to be shown how, little by little, our friend arrived at the new kind of notation that he called the differential calculus. In the year 1672, while conversing with Huygens on the properties of numbers, the latter propounded to him this problem:
    To find the sum of a decreasing series of fractions, of which the numerators are all unity and the denominators are the triangular numbers ; of which he said that he had found the sum among the contributions of Hudde on the estimation of probability. Leibniz found the sum to be 2, which agreed with that given by Huygens. While doing this he found the sums of a number of arithmetical series of the same kind in which the numbers are any combinatory numbers whatever, and communicated the results to Oldenburg in February 1673... When later he saw the Arithmetical Triangle of Pascal, he formed on the same plan his own Harmonic Triangle... where, if the denominators of any series descending obliquely to infinity or of any parallel finite series, are each divided by the term that corresponds in the first series, the combinatory numbers are produced, namely those that are contained in the arithmetical triangle. Moreover this property is common to either triangle, namely, that the oblique series are the sum- and difference-series of one another. In the Arithmetical Triangle any given series is the sum-series of the series that immediately precedes it, and the difference series of the one that follows it; in the Harmonic Triangle, on the other hand, each series is the sum-series of the series following it, and the difference series of the series that precedes it. From which it follows that
    and so on. ...
    He considered that any term of a series could in most cases be denoted by some general notation, by which it might be referred to some simple series. For instance, if the general term of the series of natural numbers is denoted by x, then the general term of the series of squares would be x2, that of the cubes would be x3, and so on. Any triangular number such as 0, 1, 3, 6, 10, would be...
    any pyramidal number such as 0, 1, 4, 10, 20, etc., would be
    and so on.
    From this it was possible to obtain the difference-series of a given series, and in some cases its sum as well, when it was expressed numerically. For instance, the square is xx the next greater square is xx + 2x + 1 and the difference of these is 2x + 1 i.e., the series of odd numbers... In the same way... 3xx + 3x + 1... is the general term of the difference-series for the series of cubes. Further, if the value of the general term can thus be expressed by means of a variable x so that the variable does not enter into a denominator or an exponent, he perceived that he could always find the sum-series of the given series. ...
    But our friend saw that it was not always easy to proceed in the same way when the variable entered into the denominator, as it was not always possible to find the sum of a numerical series; however, on following up this same analytical method, he found in general... that a sum-series could always be found, or the matter be reduced to finding the sum of a number of fractional terms such as l/x, 1/xx, 1/x3, etc., which... if the number of terms taken is finite, can be summed... but if it is a question of an infinite number of terms, then terms such as 1/x cannot be summed... because the total... is an infinite quantity, but that of an infinite number of terms such as 1/xx, 1/x3, etc., make a finite quantity, which nevertheless could not up till now be summed, except by taking quadratures. So in the year 1682... he noted in the Acta Eruditorum that if the numbers 1.3, 3.5, 5.7, 7.9, 9.11, etc., or 3, 15, 35, 63, 99, etc., are taken, and from them is formed the series of fractions
    ,
    then the sum of this series continued to infinity is nothing else but 1/2; while if every other fraction is left out, 1/3 + 1/35 + 1/99, etc. expresses the magnitude of a semicircle of which the square on the diameter is represented by 1.
    Finally he gave a method for applying the differential calculus to numerical series when the variable entered into the exponent, as in a geometrical progression, where taking any radix the term is , where stands for a natural number. The terms of the differential series will be , or ; and from this it is plain that the differential series of the given geometrical series is also a geometrical series proportional to the given series. Thus the sum of a geometrical series may be obtained.

A General History of Mathematics (1803)[edit]

...:From the Earliest Times, to the Middle of the Eighteenth Century by Charles Bossut, source, as translated by Tr. John Bonnycastle from Essai sur Histoire Générale des Mathématiques (1802) Vol. 1-2
  • The ancients drew tangents to the conic sections, and to the other geometrical curves of their invention, by particular methods, derived in each case from the individual properties of the curve in question. Archimedes determined in a similar manner the tangents of the spiral, a mechanical curve. Among the moderns, des Cartes, Fermat, Roberval, Barrow, Sluze, and others, had invented uniform methods, of more or less simplicity, for drawing tangents to geometrical curves, which was a great step: but it was previously necessary, that the equations of the curves should be freed from radical quantities, if they contained any; and this operation sometimes required immense, if not absolutely impracticable calculations. The tangent of the cycloid, a modern mechanical curve, had been determined only by some artifices founded on it's nature, and from which we could derive no light in other cases. A general method, applicable indifferently to curves of all kinds, geometrical or mechanical, without the necessity of making their radical quantities disappear in any case, remained to be discovered. This sublime discovery, the first step in the method of fluxions, was published by Leibnitz in the Leipsic Transactions for the month of October, 1684. The ever memorable paper that contained it is entitled: 'A New Method for Maxima and Minima, and likewise for Tangents, which is affected neither by Fractions nor irrational Quantities; and a peculiar Kind of Calculus for them.' In this we find the method of differencing all kinds of quantities, rational, fractional, or radical, and the application of these calculi to a very complicated case, which points out the mode for all cases. The author afterward resolves a problem de maximis et minimis, the object of which is to find the path, in which an atom of light must traverse two different mediums, in order to pass from one point to another with most facility. The result of the solution is, that the sines of the angles of incidence and refraction must be to each other in the inverse ratio of the resistances of the two mediums. Lastly he applies his new calculus to a problem, which Beaune had formerly proposed to des Cartes, from whom he obtained only an imperfect solution of it. ...Leibnitz showed in a couple of lines the required curve to be ...the common logarithmic curve.
  • In two small tracts on the quadratures of curves, which appeared in 1685, [Leibniz] published the first ideas of the calculus summatorius, or inverse method of fluxions. These are farther developed in another tract, entitled, 'Of recondite Geometry, and the Analysis of Indivisibles, and Infinites,' published the following year. In this Leibnitz gives the fundamental rule of the integral calculus; and explains in what the problems of the inverse method of tangents consist, which have since been varied in so many ways. ...and he observes generally, that all the problems of quadratures, before given by geometricians, might be resolved without any difficulty by his method.
  • While Leibnitz was in possession of all these treasures, Newton had yet published nothing, from which the world could learn, that he on his part had arrived at similar results. But toward the end of the year 1686, his Philosophiæ naturalis Principia mathematica issued from the press: a vast and profound work ...the key of the most difficult problems resolved in it is the method of fluxions, or analysis of infinites, but exhibited in a form which disguised it, and rendered the author difficult to follow. Accordingly at first it had not all the success it deserved: it was charged with obscurity, with demonstrations derived from sources too remote, and an affected use of the synthetic method of the ancients, while analysis would much better have made known the spirit and progress of the invention. ...mathematicians did Newton the justice to acknowledge, that, at the period when his book was published, he was master of the method of fluxions to a high degree, at least with respect to that part which concerns the quadratures of curves.
  • Two illustrious men, who adopted his method with such ardour, rendered it so completely their own, and made so many elegant applications of it that Leibnitz several times published in the journals, with a disinterestedness worthy of so great a man, that it was as much indebted to them as to himself. ...I am speaking of the two brothers James and John Bernoulli.
  • Every branch of the new geometry proceeded with rapidity. Problems issued from all quarters; and the periodical publications became a kind of learned amphitheatre, in which the greatest geometricians of the time, Huygens, Leibnitz, the Bernoullis, and the marquis de l'Hopital combated with bloodless weapons; the honour of France being ably supported by the marquis for several years.
  • The following problem, proposed by John Bernoulli, in 1693, contributed greatly to the progress of the methods for summing up differences. To find a curve such that the tangents terminating at the axis shall be in a given ratio with the parts of the axis comprised between the curve and these tangents. This was resolved by Huygens, Leibniz, James Bernoulli, and the marquis de l'Hopital.
    On this occasion Huygens passed on the new methods an encomium so much the more honourable, as this great man, having made several sublime discoveries without them, might have been dispensed from proclaiming their advantages. He confessed, that he beheld 'with surprise and admiration the extent and fertility of this art; that, wherever he turned his eyes, it presented new uses to his view; and that it's progress would be as unbounded as it's speculations.' How unfortunate, that science was bereft of him at an age, when with this new instrument he might still have rendered it so many important services!
  • We find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances, which will be read with pleasure. Here for the first time he repels the unjust and repeated attacks of his brother; and exhorts him to moderate his pretensions; to attach less importance to discoveries, which the instrument, with which they were both furnished, rendered easy; and to acknowledge, that, 'as quantities in geometry increase by degrees, so every man, furnished with the same instrument, would find by degrees the same results.' Very modest and remarkable expressions from the pen of one of the greatest geometricians, that ever lived.
  • In 1696 a great number of works appeared which gave a new turn to the analysis of infinites. ...and above all the celebrated work of the marquis de l'Hopital, entitled: 'The Analysis of Infinites, for the understanding of curve Lines,'...
    Such a work had long been a desideratum. 'Hitherto,' says Fontenelle, in his eulogy on the marquis, 'the new geometry had been only a kind of mystery, a cabbalistic science, confined to five or six persons. Frequently solutions were given in the public journals, while the method, by which they had been obtained, was concealed: and even when it was exhibited, it was but a faint gleam of the science breaking out from those clouds, which quickly closed upon it again. The public, or, to speak more properly, the small number of those who aspired to the higher geometry, were struck with useless admiration, by which they were not enlightened; and means were found to obtain their applause, while the information, with which it should have been repaid, was withheld.'
    The work of the marquis de l'Hopital, completely unveiling the science of the differential calculus, was received with universal encomiums, and still retains it's place among the classical works on the subject. But the time was not yet arrived for treating in the same manner the inverse method of fluxions, which is immense in it's detail, and which, notwithstanding the great progress it has made, is still far from being entirely completed. Leibnitz promised a work, which, under the title of Scientia Infiniti, was to comprise both the direct and inverse methods of fluxions: but this, which would have been of great utility at that time, never appeared.
  • The marquis de l'Hopital had given in his work on the analysis of Infinites a very ingenious rule... No person thought proper to dispute his title to this while he lived; but about a month after his death, John Bernoulli, remarking that this rule was incomplete, made a necessary addition to it, and thence took occasion to declare himself it's author. Several of the marquis de l'Hopital's friends complained loudly... Instead of retracting his assertion, John Bernoulli went much farther; and by degrees he claimed as his own every thing of most importance in the Analysis of Infinites. The reader will indulge me in a brief examination of his pretensions.
    In 1692 John Bernoulli came to Paris. He was received with great distinction by the marquis de l'Hopital, who soon after carried him to his country seat at Ourques in Touraine, where they spent four months in studying together the new geometry. Every attention, and every substantial mark of acknowledgment, were lavished on the learned foreigner. Soon after, the marquis de l'Hopital found himself enabled, by persevering and excessive labour which totally ruined his health, to solve the grand problems, that were proposed to each other by the geometricians of the time. From the year 1693 he made one in the lists of mathematical science, in which he distinguished himself till his death. At this period he was ranked among the first geometricians of Europe; and it is particularly to be observed, that John Bernoulli was one of his most zealous panegyrists. Perhaps he was exalted too high during his lifetime: but the accusation brought against him by John Bernoulli after his death forms too weighty a counterpoise, and justice ought to restore the true balance. ...
    The extracts of letters, which John Bernoulli has brought forward, are far from proving what he has asserted. ...It is true we find from them, that John Bernoulli had composed lessons in geometry for the marquis de l'Hopital, but by no means that these lessons were the Analysis of Infinites... We see too in these extracts, that the marquis, while at work on his book, solicited from John Bernoulli, with the confidence of friendship, explanations relative to certain questions, which are treated in it...
    Amid all these uncertainties, it is most equitable and prudent, to adhere to the general declaration made by the marquis in his preface, that he was greatly indebted to John Bernoulli [aux lumiéres de J. B.]; and to presume, that if he had any obligations to him of a particular nature, he would not have ventured to mask them in the expressions of vague and general acknowledgment. If... any one should think proper to credit John Bernoulli on his bare word, when he gives himself out for the author of the Analysis of infinites, the code of morality... will never absolve him, for having disturbed the ashes of a generous benefactor, in order to gratify a paltry love of self, so much the less excusable, as he possessed sufficient scientific wealth besides.
  • Toward the end of the year 1704, Newton gave to the World in one volume his Optics in english, an enumeration of lines of the third order, and a treatise on the quadrature of curves, both in latin. ...the treatise on quadratures, belongs to the new geometry. The particular object of this treatise is the resolution of differential formulæ of the first order, or of a single variable quantity; on which depends the precise, or at least the approximate, quadrature of curves. With great address Newton forms series, by means of which he refers the resolution of certain complicated formulæ to those of more simple ones; and these series, suffering an interruption in certain cases, then give the fluents in finite terms. The development of this theory affords a long chain of very elegant propositions, where among other curious problems we remark the method of resolving rational fractions, which was at that time difficult, particularly when the roots are equal. Such an important and happy beginning makes us regret, that the author has given only the first principles of the analysis of differential equations. It is true he teaches us to take the fluxions, of any given order, of an equation with any given number of variable quantities, which belongs to the differential calculus: but he does not inform us, how to solve the inverse problem; that is to say, he has pointed out no means of resolving differential equations, either immediately, or by the separation of the indeterminate quantities, or by the reduction into series, &c. This theory however had already made very considerable progress in Germany, Holland, and France, as may be concluded from the problems of the catenarian, isochronous, and elastic curves, and particularly by the solution which James Bernoulli had given of the isoperimetrical problem. Newton's opponents have argued from his treatise on quadratures, that, when this work appeared, the author was perfectly acquainted only with that branch of the inverse method of fluxions which relates to quadratures, and not with the resolution of differential equations.
    Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newton's death, Dr. Pemberton gave it to the world in english. In 1740 it was translated into french, and a preface was prefixed to it, in which the merits of Leibnitz are depreciated so excessively, and in such a decided tone as might impose on some readers, if the writer of this preface [Buffon] had not sufficiently blunted his own criticisms, by betraying how little knowledge of the subject he possessed.
  • Nicholas Facio de Duillier... thought proper to say, in a little tract 'on the curve of swiftest descent, and the solid of least resistance,' which appeared in 1699, that Newton was the first inventor of the new calculus... and that he left to others the task of determining what Leibnitz, the second inventor, had borrowed from the english geometrician.
    Leibnitz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the consequence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his own authority; that he could not believe it was with Newton's approbation; that he would not enter into any dispute with that celebrated man, for whom he had the profoundest veneration, as he had shown on all occasions; that when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia, that neither had borrowed any thing from the other; that when he published his differential calculus in 1684, he had been master of it about eight years; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities; but that he could not judge whether this method were the differential calculus, since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method, which had the same advantages; that the first work of an english writer, in which the differential calculus was explained in a positive manner, was the preface to Wallis's Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, &c.
  • In 1708, Keil... renewed the same accusation. ...Keil returned to the charge; and in 1711, in a letter to sir Hans Sloane, secretary to the Royal Society, he was not contented with saying, that Newton was the first inventor; but plainly intimated, that Leibnitz, after having taken his method from Newton's writings, had appropriated it to himself, merely employing a different notation; which was charging him in other words with plagiarism.
    Leibnitz, indignant at such an accusation, complained loudly to the Royal Society; and openly required it to suppress the clamours of an inconsiderate man, who attacked his fame and his honour. The Royal Society appointed a committee, to examine all the writings that related to this question, and in 1712 it published these writings, with the report of the committee, under the following title: Commercium epistolicum de Analysi promota. Without being absolutely affirmative, the conclusion of the report is, that Keil had not calumniated Leibnitz. The work was dispersed over all Europe with profusion.
    Newton was at that time president of the Royal Society, where he enjoyed the highest respect and most ample power...
  • Newton, gifted by nature with superiour intellect, and born at a time when Harriot, Wren, Wallis, Barrow, and others, had already rendered the mathematical sciences flourishing in England, enjoyed likewise the advantage of receiving lessons from Barrow in his early youth at Cambridge. The whole bent of early youth was toward studies of this kind, and the success he obtained was prodigious. ...
    Leibnitz, who was four years younger, found but moderate assistance in his studies in Germany. He formed himself alone. His vast and devouring genius, aided by an extraordinary memory, took in every branch of human knowledge; literature, history, poetry, the law of nations, the mathematical sciences, natural philosophy, &c. This multiplicity of pursuits necessarily checked the rapidity of his progress in each; and accordingly he did not appear as a great mathematician till seven or eight years after Newton.
    Both these great men were in possession of the new analysis long before they made it known to the world. If priority of publication determined priority of discovery, Leibnitz would have completely gained his cause: but this is not sufficient...
  • If Newton first invented the method of fluxions, as is pretended to be proved by his letter of the 10th of december 1672, Leibnitz equally invented it on his part, without borrowing any thing from his rival. These two great men by the strength of their genius arrived at the same discovery through different paths: one, by considering fluxions as the simple relations of quantities, which rise or vanish at the same instant; the other, by reflecting, that, in a series of quantities which increase or decrease, the difference between two consecutive terms may become infinitely small, that is to say, less than any determinable finite magnitude.
    This opinion, at present universally received except in England, was that of Newton himself, when he first published his Principia... At that time the truth was near it's source, and not yet altered by the passions. In vain did Newton afterward change his language, led away by the flattery of his countrymen and disciples; in vain did he pretend, that the glory of a discovery belongs entirely to the first inventor, and that second inventors ought not to be admitted to share it. ...two men, who separately make the same important discovery, have an equal claim to admiration; and... he who first makes it public, has the first claim to the public gratitude.
  • The design of stripping Leibnitz, and making him pass for a plagiary, was carried so far in England, that during the height of the dispute it was said... that the differential calculus of Leibnitz was nothing more than the method of Barrow. What are you thinking of, answered Leibnitz, to bring such a charge against me? ...If the differential calculus were really the method of Barrow (which you well know it is not) who would most deserve to be called a plagiary? Mr. Newton, who was the pupil and friend of Barrow, and had opportunities of gathering from his conversation ideas, which are not in his works? or I, who could be instructed only by his works, and never had any acquaintance with the author?
  • John Bernoulli who... learned the analysis of infinites from the writings of Leibnitz, ... advances not only that the method of fluxions did not precede the differential calculus, but that it might have originated from it; and that Newton had not reduced it to general analytical operations in form of an algorithm, till the differential calculus was already disseminated through all the journals of Holland and Germany.
  • The death of Leibnitz, which happened in 1716, it may be supposed, should have put an end to the dispute: but the english, pursuing even the manes of that great man, published in 1726 an edition of the Principia in which the scholium relating to Leibnitz was omitted. This was confessing his discovery in a very authentic and awkward manner. Must they not be aware, that the chimerical design of annihilating the testimony, which an honourable emulation had formerly rendered to truth, would be ascribed to national prejudice, or to a sentiment even still more unjust?
  • In later times there have been geometricians, who... have objected... that the metaphysics of his method were obscure, or even defective; that there are no quantities infinitely small; and that there remain doubts concerning the accuracy of a method, into which such quantities are introduced. But Leibnitz might answer: ...I have no need of the existence of infinitely small quantities: it is enough for my purpose, as I have said in several of my works, that my differences are less than any finite quantity you please to assign; and that consequently the errour, which may result from my supposition, is less than any determinable errour, which is the same as absolutely nothing. The manner in which Archimedes demonstrates the proportion of the sphere to the cylinder, has a similar principle for it's basis. ...The metaphysics of my calculation, therefore, are perfectly conformable to those of the method of exhaustion of the ancients, the certainty of which has never been questioned by any one.

A Short Account of the History of Mathematics (1888, 1910)[edit]

W. W. Rouse Ball, quotes are taken from the 1910 edition, unless otherwise noted.
  • It would seem from Fermat's correspondence with Descartes as if he had thought out the principles of analytical geometry for himself before reading Descartes' Discours, and had realized that from the equation of a curve (or as he calls it the "specific property") all its properties could be deduced. His extant papers on this subject deal however only with the application of infinitesimals to geometry; it seems probable that these papers are a revision of his original manuscripts (which he destroyed) and were written about 1663, but he was certainly in possession of the general idea of his method for finding maxima and minima as early as 1628 or 1629.
    Kepler had already remarked that the values of a function immediately adjacent to and on either side of a maximum (or minimum) value must be equal. Fermat applied this to a few examples. Thus to find the maximum value of he took a consecutive value of x, namely where is very small, and put . Simplifying and ultimately putting he got . This value of x makes the given expression a maximum. [This] is the principle of Fermat's method, but his analysis is more involved.
  • [Fermat] obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were and ; but there is nothing to indicate that he was aware that the process was general, and though in the course of his work he used the principle, it is probable that he never separated it, so to speak, from the symbols of the particular problem he was considering. The first definite statement of the method was due to Barrow and was published in 1669.
Barrow's Differential Triangle PQR
where QR = e, RP = a
Lectiones opticæ et geometricæ (1669)
  • In 1669 [Isaac Barrow] issued his Lectiones opticæ et geometricæ: this, which is his only important work, was republished with a few minor alterations in 1674. A complete edition of all Barrow's lectures was edited for Trinity College by W. Whewell, Cambridge, 1860.
    It is said in the preface to the Lectiones opticæ et geometricæ that Newton revised and corrected these lectures adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. ...
    The geometrical lectures contain some new ways of determining the areas and tangents of curves. The most celebrated of these is the method given for the determination of tangents to curves. Fermat had observed that the tangent at a point P on a curve was determined if one other point besides P on it [the tangent line] was known; hence if the length of the subtangent MT could be found (thus determining the point T) then the line TP would be the required tangent. Now Barrow remarked that if the abscissa and ordinate at a point Q adjacent to P were drawn he got a small triangle PQR (which he called the differential triangle because, its sides PR and PQ were the differences of the abscissas and ordinates of P and Q) so that
    .
    To find he supposed that x,y were the coordinates of P and those of Q. ...Using the equation of the curve and neglecting the squares and higher powers of e and a as compared with their first powers he obtained The ratio was subsequently (in accordance with a suggestion made by de Sluze) termed the angular coefficient of the tangent at the point.
    Barrow applied this method to the following curves (i) ; (ii) ; (iii) , called la galande; (iv) , the quadratrix; and (v) .
    ...take as an illustration the simpler case of the parabola . Using the notation given above we have for the point P, ; and for the point Q, . Subtracting we get . But if a is an infinitesimal quantity, must be infinitely smaller and may therefore be neglected: hence . Therefore . That is . This is exactly the procedure of the differential calculus, except that we there have a rule by which we can get the ratio or directly without the labour of going through a calculation similar to the above for every separate case.

A History of the Study of Mathematics at Cambridge (1889)[edit]

W. W. Rouse Ball, source
  • The most notable of [Wallis' mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that stood for the reciprocal of and that stood for the root of . He next proceeded to find by the method of indivisibles the area enclosed between the curve , the axis of , and any ordinate ; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio . He apparently assumed that the same result would also be true for the curve , where is any constant. In this result may be any number positive or negative, and he considered in particular the case of the parabola in which , and that of the hyperbola in which : in the latter case his interpretation of the result is incorrect. He then shewed that similar results might be written down for any curve of the form ; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was its area would be He then applied this to the quadrature of the curves , , , , &c. taken between the limits and : and shewed that the areas are respectively
    .
  • [Wallis] next considered curves of the form and established the theorem that the area bounded by the curve, the axis of , and the ordinate is to the area of the rectangle on the same base and of the same altitude as . This is equivalent to finding the value of . He illustrated this by the parabola in which . He stated but did not prove the corresponding result for a curve of the form .
  • As [Wallis] was unacquainted with the binomial theorem he could not effect the quadrature of the circle, whose equation is , since he was unable to expand this in powers of . He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves and , so as an approximation its area might be taken as the geometrical mean between and . This is equivalent to taking or 3.26... as the value of . But, he continued, we have in fact a series and thus the term interpolated between and ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making
    The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis.
  • In 1659 Wallis published a tract on cycloids in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of algebraic curves: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the semicubical parabola . This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified.
  • The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work.
  • [Isaac Barrow's] lectures delivered in 1664, 1665, and 1666, were published in 1683 under the title Lectiones mathematicae: these are mostly on the metaphysical basis for mathematical truths. His lectures for 1667 were published in the same year and suggest the analysis by which Archimedes was led to his chief results.
    In 1669 he issued his Lectiones opticae et geometricae, which is his most important work. ...The geometrical lectures contain some new ways of determining the areas and tangents of curves. The latter is solved by a rule exactly analogous to the procedure of the differential calculus, except that a separate determination of what is really a differential coefficient had to be made for every curve to which it was applied. Thus he took the equation of the curve between the coordinates x and y, gave x a very small decrement e and found the consequent decrement of y, which he represented by a. The limit of the ratio a/e when the squares of a and e were neglected was defined as the angular coefficient of the tangent at the point, and completely determined the tangent there.
  • Barrow's lectures failed to attract any considerable audiences, and on that account he felt conscientious scruples about retaining his chair. Accordingly in 1669 he resigned it to his pupil Newton, whose abilities he had been one of the earliest to detect and encourage.
  • [Isaac Newton's] subsequent mathematical reading as an undergraduate was founded on Kepler's Optics, the works of Vieta, Schooten's Miscellanies, Descartes's Geometry, and Wallis's Arithmetica infinitorum: he also attended Barrow's lectures.
  • [Isaac Newton] took his BA degree in 1664. There is a manuscript of his written in the following year, and dated May 28, 1665, which is the earliest documentary proof of his discovery of fluxions. It was about the same time that he discovered the binomial theorem.
  • On account of the plague the college was sent down in the summer of 1665, and for the next year and a half Newton lived at home. This period was crowded with brilliant discoveries. ...He ...worked out the fluxional calculus tolerably completely: thus in a manuscript dated Nov. 13 of the same year he used fluxions to find the tangent and the radius of curvature at any point on a curve, and in October 1666 he applied them to several problems in the theory of equations. Newton communicated the results to his friends and pupils from and after 1669, but they were not published in print till many years later. ...
    On his return to Cambridge in 1667 Newton was elected to a fellowship, and in 1668 took his M.A. degree. ...
    During the next two years he revised and edited Barrow's Lectures edited and added to Kinckhuysen's Algebra, and by using infinite series greatly extended the power of the method of quadratures given by Wallis. ...
    In 1673 he had written an account of his method of quadrature by means of infinite series in letters to Collins or Oldenburg; and in 1676 in answer to a request from Leibnitz he gave him a very brief account of his method and added the expansions of a binomial (i.e. the binomial theorem) and of sin-1x; from the latter of which he deduced that of sin x. He also added an expression for the rectification of an elliptic arc in an infinite series.
    Leibnitz wrote on Aug. 27, 1676, asking for fuller details, and on Oct. 24 Newton replied in a long but very interesting paper in which he gives an account of the way in which he had been led to some of his results.
    He begins by saying that altogether he had used three methods for expansion in series. His first was arrived at from the study of the method of interpolation by which Wallis had found expressions for the area of the circle and hyperbola. Thus by considering the series of expressions

    he deduced by interpolations the law which connects the successive coefficients in the expansions of


    He then by analogy obtained the expression for the general term in the expansion of a binomial, i.e. the binomial theorem. He says that he proceeded to test this by forming the square of the expansion of which reduced to ; and he proceeded in a similar way with other expansions. He next tested the theorem in the case of by extracting the square root of more arithmetico. He also used the series to determine the areas of the circle and hyperbola in infinite series and found that they were the same as the results he had arrived at by other means.
    Having established this result he then discarded the method of interpolation, and employed his binomial theorem as the most direct method of obtaining the areas and arcs of curves. Newton styled this his second method and it is the basis of his work on analysis by infinite series. He states that he had discovered it before the plague in 1665-66.
    Newton then proceeds to state that he had also a third method; of which (he says) he had about 1669 sent an account to Barrow and Collins, illustrated by applications to areas, rectification, cubature, &c. This was the method of fluxions; but Newton gave no detailed description of it in this letter, probably because he thought that Leibnitz could, if he wished, obtain from Collins the explanation of it alluded to above. Newton added an anagram which described the method but which is unintelligible to any one to whom the key is not given. He gives however some illustrations of its use. The first is on the quadrature of the curves represented by


    which he says can be determined as a sum of terms if be a positive integer, and which he thinks cannot otherwise be effected except by an infinite series. [This is not so, the integration is possible if be an integer.] He also gives a long list of other forms which are immediately integrable, of which the chief are




    where m is a positive integer and n is any number whatever.
    At the end of his letter Newton alludes to the solution of the "inverse problem of tangents," a subject on which Leibnitz had asked for information. He gives formulae for reversing any series, but says that besides these formulae he has two methods for solving such questions which for the present he will not describe except by an anagram...
    He adds in this letter that he is worried by the questions he is asked and the controversies raised about every new matter which he publishes and he regrets that he has allowed his repose to be interrupted by running after shadows; and he implies that for the future he will publish nothing. ...he did refuse to allow any account of his method of fluxions to be published till the year 1693.
  • Leibnitz did not reply to this letter till June 21, 1677. In his answer he explains his method of drawing tangents to curves, which he says proceeds "not by fluxions of lines but by the differences of numbers"; and he introduces his notation of and for the infinitesimal differences between the coordinates of two consecutive points on a curve. He also gives a solution of the problem to find a curve whose subtangent is constant, which shews that he could integrate.
  • The two letters to Wallis in which [Newton] explained his method of fluxions and fluents were written in 1692, and were published in 1693. Towards the close of 1692 and throughout the two following years Newton had a long illness, suffering from insomnia and general nervous irritability. He never quite regained his elasticity of mind, and though after his recovery he shewed the same power in solving any question propounded to him, he ceased thenceforward to do original work on his own initiative, and it was difficult to stir him to activity.
  • In 1704 [Newton] published his Optics, containing an account of his emission theory of light. To this book two appendices were added; one on cubic curves, and the other on the quadrature of curves and his theory of fluxions. Both of these were old manuscripts which had long been known to his friends at Cambridge, but had been previously unpublished.
  • The second appendix to [Newton's] Optics was entitled De quadratura curvarum. Most of it had been communicated to Barrow in 1666, and was probably familiar to Newton's pupils and friends from about 1667 onwards. It consists of two parts.
    The bulk of the first part had been included in the letter to Leibnitz of Oct. 24, 1676. This part contains the earliest use of literal indices, and the first printed statement of the binomial theorem: these are however introduced incidentally. The main object of this part is to give rules for developing a function of a in a series in ascending powers of x; so as to enable mathematicians to effect the quadrature of any curve in which the ordinate y can be expressed as an explicit function of the abscissa x. Wallis had shewn how this quadrature could be found when y was given as a sum of a number of powers of x and Newton here extends this by shewing how any function can be expressed as an infinite series in that way. ...Newton is generally careful to state whether the series are convergent. In this way he effects the quadrature of the curves


    but the results are of course expressed as infinite series. He then proceeds to curves whose ordinate is given as an implicit function of the abscissa; and he gives a method by which y can be expressed as an infinite series in ascending powers of x, but the application of the rule to any curve demands in general such complicated numerical calculations as to render it of little value. He concludes this part by shewing that the rectification of a curve can be effected in a somewhat similar way. His process is equivalent to finding the integral with regard to x of in the form of an infinite series.
    This part should be read in connection with his Analysis by infinite series published in 1711, and his Methodus differentialis published in 1736. Some additional theorems are there given, and in the latter of these works he discusses his method of interpolation. The principle is this. If is a function of x and if when x is successively put equal to a1, a2,... the values of y are known and are b1, b2,.. then a parabola whose equation is can be drawn through the points and the ordinate of this parabola may be taken as an approximation to the ordinate of the curve. The degree of the parabola will of course be one less than the number of given points. Newton points out that in this way the areas of any curves can be approximately determined.
    The second part of this second appendix contains a description of his method of fluxions and is condensed from his manuscript...
  • It is probable that no mathematician has ever equalled Newton in his command of the processes of classical geometry. But his adoption of it for purposes of demonstration appears to have arisen from the fact that the infinitesimal calculus was then unknown to most of his readers, and had he used it to demonstrate results which were in themselves opposed to the prevalent philosophy of the time the controversy would have first turned on the validity of the methods employed. Newton therefore cast the demonstrations of the Principia into a geometrical shape which, if somewhat longer, could at any rate be made intelligible to all mathematical students and of which the methods were above suspicion. ...in Newton's time and for nearly a century afterwards the differential and fluxional calculus were not fully developed and did not possess the same superiority over the method he adopted which they do now. The effect of his confining himself rigorously to classical geometry and elementary algebra, and of his refusal to make any use even of analytical geometry and of trigonometry is that the Principia is written in a language which is archaic (even if not unfamiliar) to us. The subject of optics lends itself more readily to a geometrical treatment, and thus his demonstrations of theorems in that subject are not very different to those still used.
    The adoption of geometrical methods in the Principia for purposes of demonstration does not indicate a preference on Newton's part for geometry over analysis as an instrument of research, for it is now known that Newton used the fluxional calculus in the first instance in finding some of the theorems (especially those towards the end of book I. and in book II.), and then gave geometrical proofs of his results. This translation of numerous theorems of great complexity into the language of the geometry of Archimedes and Apollonius is I suppose one of the most wonderful intellectual feats which was ever performed.
  • The fluxional calculus is one form of the infinitesimal calculus expressed in a certain notation just as the differential calculus is another aspect of the same calculus expressed in a different notation. Newton assumed that all geometrical magnitudes might be conceived as generated by continuous motion: thus a line may be considered as generated by the motion of a point, a surface by that of a line, a solid by that of a surface, a plane angle by the rotation of a line, and so on. The quantity thus generated was defined by him as the fluent or flowing quantity. The velocity of the moving magnitude was defined as the fluxion of the fluent.

A History of Mathematics (1893, 1919)[edit]

Florian Cajori, source of 1919, 2nd edition, revised and enlarged
  • At one time, while purchasing wine, [Johannes Kepler] was struck by the inaccuracy of the ordinary modes of determining the contents of kegs. This led him to the study of the volumes of solids of revolution and to the publication of the Stereometria Doliorum in 1615. In it he deals first with the solids known to Archimedes and then takes up others. Kepler made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science. In comparing rectilinear figures, the method of superposition was employed by the ancients, but in comparing rectilinear and curvilinear figures with each other, this method failed because no addition or subtraction of rectilinear figures could ever produce curvilinear ones. To meet this case, they devised the Method of Exhaustion, which was long and difficult; it was purely synthetical, and in general required that the conclusion should be known at the outset. The new notion of infinity led gradually to the invention of methods immeasurably more powerful. Kepler conceived the circle to be composed of an infinite number of triangles having their common vertices at the centre, and their bases in the circumference; and the sphere to consist of an infinite number of pyramids. He applied conceptions of this kind to the determination of the areas and volumes of figures generated by curves revolving about any line as axis, but succeeded in solving only a few of the simplest out of the 84 problems which he proposed for investigation in his Stereometria.
    Other points of mathematical interest in Kepler's works... [include] a passage from which it has been inferred that Kepler knew the variation of a function near its maximum value to disappear...
    The Stereometria led Cavalieri... to the consideration of of infinitely small quantities.
    • citing (Greeks' conic sections quote) William Whewell, History of the Inductive Sciences 3rd Ed., (1858) Vol. I, p 311.
  • Bonaventura Cavalieri... a pupil of Galileo and professor at Bologna, is celebrated for his Geometria indivisibilibus continuorum nova quadam ratione promota 1635. This work expounds his method of Indivisibles, which occupies an intermediate place between the method of exhaustion of the Greeks and the methods of Newton and Leibniz. Indivisibles were discussed by Aristotle and the scholastic philosophers. They commanded the attention of Galileo. Cavalieri does not define the term. He borrows the concept from the scholastic philosophy of Bradwardine and Thomas Aquinas, in which a point is the indivisible of a line, a line the indivisible of a surface, etc. Each indivisible is capable of generating the next higher continuum by motion; a moving point generates a line, etc. The relative magnitude of two solids or surfaces could then be found simply by the summation of series of planes or lines. For example... he concludes that the pyramid or cone is respectively 1/3 of a prism or cylinder of equal base and altitude... By the Method of Indivisibles, Cavalieri solved the majority of the problems proposed by Kepler. Though expeditious and yielding correct results, Cavalieri's method lacks a scientific foundation. If a line has absolutely no width, then the addition of no number, however great, of lines can ever yield an area; if a plane has no thickness whatever, then even an infinite number of planes cannot form a solid. Though unphilosophical, Cavalieri's method was used for fifty years as a sort of integral calculus. It yielded solutions to some difficult problems. [Paul] Guldin made a severe attack on Cavalieri... [who] published in 1647... a treatise entitled Exercitationes geometriece sex in which he replied to the objections of his opponent and attempted to give a clearer explanation of his method. ...A revised edition of the Geometria appeared in 1653.
Cycloid and its generating circle
  • There is an important curve not known to the ancients which now began to be studied with great zeal. Roberval gave it the name of" trochoid," Pascal the name of "roulette," Galileo the name of "cycloid." The invention of this curve seems to be due to Charles Bouvelles who...in 1501 refers to this curve in connection with the problem of the squaring of the circle. Galileo valued it for the graceful form it would give to arches in architecture. He ascertained its area by weighing paper figures of the cycloid against that of the generating circle and found thereby the first area to be nearly... thrice the latter. A mathematical determination was made by his pupil Evangelista Torricelli... By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle and published his solution. This same quadrature had been effected a few years earlier (about 1636) by Roberval in France, but his solution was not known to the Italians. Vincenzo Viviani... another prominent pupil of Galileo, determined the tangent to the cycloid. This was accomplished in France by Descartes and Fermat.
  • In France, where geometry began to be cultivated with greatest success, Roberval, Fermat, Pascal, employed the Method of Indivisibles and made new improvements in it. Giles Persone de Roberval... claimed for himself the invention of the Method... Roberval and Pascal improved the rational basis of the Method of Indivisibles, by considering an area as made up of an indefinite number of rectangles instead of lines, and a solid as composed of indefinitely small solids instead of surfaces. Roberval applied the method to the finding of areas, volumes, and centres of gravity. He effected the quadrature of a parabola... [and] cycloid. Roberval is best known for his method of drawing tangents, which, however, was invented at the same time if not earlier by Torricelli.
    Torricelli's appeared in 1644 under the title Opera geometrica. Roberval gives the fuller exposition of it.
Tangents to a curve
Tangent as a secant whose two points of intersection with the curve come together to coincide
  • Roberval's method of drawing tangents is allied to Newton's principle of fluxions. Archimedes conceived his spiral to be generated by a double motion. This idea Roberval extended to all curves. Plane curves, as for instance the conic sections, may be generated by a point acted upon by two forces, and are the resultant of two motions. If at any point of the curve the resultant be resolved into its components, then the diagonal of the parallelogram determined by them is the tangent to the curve at that point. The greatest difficulty connected with this ingenious method consisted in resolving the resultant into components having the proper lengths and directions. Roberval did not always succeed in doing this, yet his new idea was a great step in advance. He broke off from the ancient definition of a tangent as a straight line having only one point in common with a curve,—a definition which by the methods then available was not adapted to bring out the properties of tangents to curves of higher degrees, nor even of curves of the second degree and the parts they may be made to play in the generation of the curves. The subject of tangents received special attention also from Fermat, Descartes, and Barrow, and reached its highest development after the invention of the differential calculus. Fermat and Descartes defined tangents as secants whose two points of intersection with the curve coincide. Barrow considered a curve a polygon and called one of its sides produced, a tangent.
  • Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen from the words of Poisson, himself a Frenchman, who rightly says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems."
  • The labors of L. Euler, J. Lagrange, and P. S. Laplace lay in higher analysis, and this they developed to a wonderful degree. By them analysis came to be completely severed from geometry. During the preceding period the effort of mathematicians not only in England, but, to some extent, even on the continent, had been directed toward the solution of problems clothed in geometric garb, and the results of calculation were usually reduced to geometric form. A change now took place. Euler brought about an emancipation of the analytical calculus from geometry and established it as an independent science. Lagrange and Laplace scrupulously adhered to this separation. Building on the broad foundation laid for higher analysis and mechanics by Newton and Leibniz, Euler, with matchless fertility of mind, erected an elaborate structure. There are few great ideas pursued by succeeding analysts which were not suggested by L. Euler, or of which he did not share the honor of invention. With, perhaps, less exuberance of invention, but with more comprehensive genius and profounder reasoning, J. Lagrange developed the infinitesimal calculus and put analytical mechanics into the form in which we now know it. P. S. Laplace applied the calculus and mechanics to the elaboration of the theory of universal gravitation, and thus, largely extending and supplementing the labors of Newton, gave a full analytical discussion of the solar system. ...
    Comparing the growth of analysis at this time with the growth during the time of K. F. Gauss, A. L. Cauchy, and recent mathematicians, we observe an important difference. During the former period we witness mainly a development with reference to form. Placing almost implicit confidence in results of calculation, mathematicians did not always pause to discover rigorous proofs, and were thus led to general propositions, some of which have since been found to be true in only special cases. ...But in recent times there has been added to the dexterity in the formal treatment of problems, a much needed rigor of demonstration. A good example of this increased rigor is seen in the present use of infinite series as compared to that of Euler, and of Lagrange in his earlier works. ...
    The ostracism of geometry, brought about by the master-minds of this period, could not last permanently. Indeed, a new geometric school sprang into existence in France before the close of this period.

History of Modern Mathematics (1896)[edit]

David Eugene Smith
  • There have been four general steps in the development of what we commonly call the calculus... The first is found among the Greeks. In passing from commensurable to incommensurable magnitudes their mathematicians had recourse to the method of exhaustion, whereby, for example, they "exhausted" the area between a circle and an inscribed regular polygon, as in the work of Antiphon (c. 430 B.C.)
    The second general step... taken two thousand years later,... the method of infinitesimals... began to attract attention in the first half of the 17th century, particularly in the works of Kepler (1616) and Cavalieri (1635), and was used to some extent by Newton and Leibniz.
    The third method is that of fluxions and is the one due to Newton (c. 1665). It is this form of the calculus that is usually understood when the invention of the science is referred to him.
    The fourth method, that of limits, is also due to Newton, and is the one now generally followed.
  • The Greeks developed the method of exhaustion about the 5th century B.C.
    Zeno of Elea (c. 450 B.C.) was one of the first to introduce problems that led to a consideration of infinitesimal magnitudes. He argued that motion was impossible, for this reason:
Before a moving body can arrive at its destination it must have arrived at the middle of its path; before getting there it must have accomplished the half of that distance, and so on ad infinitum: in short, every body, in order to move from one place to another, must pass through an infinite number of spaces, which is impossible.
  • Leucippus (c. 440 B.C.) may possibly have been a pupil of Zeno's. Very little is known of his life and we are not at all certain of the time in which he lived, but Diogenes Laertius (2nd century) speaks of him as a teacher of Democritus (c. 400 B.C.). He and Democritus are generally considered as the founders of that atomistic school, which taught that magnitudes are composed of individual elements in finite numbers. It was this philosophy that led Aristotle (c. 430 B.C.) to write a book in indivisible lines.
  • Antiphon (c. 430) is one of the earliest writers whose use of the method of exhaustion is fairly well known to us. In a fragment of Eudemus (c. 335 B.C.)... we have the following description:
Antiphon, having drawn a circle, inscribed in it one of those polygons that can be inscribed: let it be a square. Then he bisected each side of this square, and through the points of section drew straight lines at right angles to them, producing them to meet the circumference; these lines evidently bisect the corresponding segments of the circle. He then joined the new points of section to the ends of the sides of the square, so that four triangles are formed, and the whole inscribed figure became an octagon. And again, in the same way, he bisected each of the sides of the octagon... and thus formed an inscribed figure of sixteen sides. Again, in the same manner... he formed a polygon of twice as many sides; and doing the same again and again, until he had exhausted the surface, he concluded that in this manner a polygon would be inscribed in a circle, the sides of which, on account of their minuteness, would coincide with the circumference of the circle.
We have in this method a crude approach to the integration of the 17th century.
  • Eudoxus of Cnidus (370 B.C.) is probably the one who placed the theory of exhaustion on a scientific basis. ...[In] Book V of Euclid's Elements (the book on proportion)... it is thought that the fundamental principles laid down are his. The fourth definition... is: "Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another," and this includes the relation of a finite magnitude to a magnitude of the same kind which is either infinitely great or infinitely small. ...According to Archimedes, this method had already been applied by Democritus (c. 400 B.C.) to the mensuration of both the cone and the cylinder.
  • It is known that Hippocrates of Chios (c. 460 B.C.) proved that circles are to one another as the squares of their diameters, and it seems probable that he also used the method of exhaustion... Archimedes tells us that the "earlier geometers" had proved that spheres have to one another the triplicate ratio of their diameters, so that the method was probably used by others as well.
  • It is to Archimedes... that we owe the nearest approach to actual integration to be found among the Greeks. His first noteworthy advance... was concerned with his proof that the area of a parabolic segment is four thirds of the triangle with the same base and vertex, or two thirds of the circumscribed parallelogram. This was shown by continually inscribing in each segment between the parabola and the inscribed figure a triangle with the same base and... height as the segment. If A is the area of the original inscribed triangle, the process... leads to the summation of the series
    or...
    so that he really finds the area by integration and recognizes, but does not assert, that
    ,
    this being the earliest example that has come down to us of the summation of an infinite series. ...
    In his treatment of solids bounded by curved surfaces he arrives at conclusions which we should now describe by the following formulas:

    Surface of a sphere,
    .
    Surface of a spherical segment,
    .
    Volume of a segment of a hyperboloid of revolution,
    .
    Volume of a segment of a spheroid,
    .
    Area of a spiral,
    .
    Area of a parabolic segment,
    .
  • Among the more noteworthy attempts at integration in modern times were those of Kepler (1609). In his notable work on planetary motion he asserted that a planet describes equal focal sectors of ellipses in equal times. This... demands some method for finding the areas of such sectors, and the one invented by Kepler was called by him the... "sum of the radii," a rude kind of integration. He also became interested in the problem of gaging, and published a work on this... and on general mensuration as set forth by Archimedes. ...[Kepler's] was a scientific study of of the measurement of solids in general. ...composed "as it were" (veluti) of infinitely many infinitely small cones or infinitely thin disks, the summation of which becomes the problem of later integration.
  • Kepler's attempts at integration... led Cavalieri to develop his method of indivisibles... which may also have been suggested to him by Aristotle's tract De lineis insecabilibus [On indivisible lines]... It may also have been suggested by one of the fragments of Xenocrates (c. 350 B.C.)... who wrote upon indivisible lines. ...
    Cavalieri... seems to have looked upon a solid as made up practically of superposed surfaces, a surface as made up of lines, and a line as made up of points, these component parts being the ultimate possible elements in the decomposition of the magnitude. He then proceeded to find lengths, areas, and volumes of the summation of these "indivisibles," that is, by the summation of an infinite number of infinitesimals.
    Such a conception of magnitude cannot be satisfactory to any scientific mind, but it formed a kind of intuitive step in the development of the method of integration and undoubtedly stimulated men like Leibniz to exert their powers to place the theory upon a scientific foundation. ...
    Cavalieri was able to solve various elementary problems in the mensuration of lengths, areas, and volumes, and also to give a fairly satisfactory proof of the theorem of Pappus with respect to the volume generated by the revolution of a plane figure about an axis.
Note: Pappus. Theorem I. If a plane curve rotate about an external axis lying in its plane, the volume of the ring thus generated is the same as that of a cylinder whose base is the region S enclosed by the curve and whose altitude is the distance through which the centre of gravity of S has travelled:
,
where h denotes the distance of the centre of gravity of S from the axis, and A, the area of S.
( William Fogg Osgood, A First Course in the Differential and Integral Calculus (1907) p. 362. )
Tangent function animation
  • The problem of tangents, the basic principle of the theory of maxima and minima, may be said to go back to Pappus (c. 300). It appears indirectly in the Middle Ages, for Oresme (c. 1360) knew that the point of maximum or minimum ordinate of a curve is the point at which the ordinate is changing most slowly. It was Fermat, however, who first stated substantially the law as we recognize it today, communicating (1638) to Descartes a method which is essentially the same as the one used at present, that of equating [the derivative] to zero. Similar methods were used by René de Sluze (1652) for tangents, and by Hudde (1658) for maxima and minima.
  • The first British publication of great significance bearing upon the calculus is that of John Wallis, issued in 1655. It is entitled Arithmetica Infinitorum, sive Nova Methodus Inquirendi in Curvilineorum Quadraturum, aliaque difficiliora Matheseos Problemata, and is dedicated to Oughtred. By a method similar to Cavalieri the author effects the quadrature of certain surfaces, the cubature of certain solids, and the rectification of certain curves. He speaks of a triangle, for example, "as if" (quasi) made up of an infinite number of parallel lines in arithmetic proportion, of a paraboloid "as if" made up of an infinite number of parallel lines, and of a spiral as an aggregate of an infinite number of arcs of similar sectors, applying to each the theory of the summation of an infinite series. ...he expresses his indebtedness to such writers as Torricelli and Cavalieri. He speaks of the work of such British contemporaries as Seth Ward and Christopher Wren, who were interested in this relatively new method, and, indeed, his dedication to Oughtred is the best contemporary specimen that we have of the history of the movement just before Newton's period of activity. All this, however, was still in the field of integration, the first steps dating... from the time of the Greeks.
Barrow's Differential Triangle,
Lectiones opticae et geometricae
(1670)
  • What is considered by us as the process of differentiation was known to quite an extent to Barrow (1663). In his Lectiones opticae et geometricae he gave a method of tangents in which, in the annexed figure, Q approaches P, as in our present theory, the result being an indefinitely small (indefinite parvum) arc. The triangle PRQ was long known as "Barrow's differential triangle," a name which, however, was not due to him. ...this method, and the figure... must have had a notable influence upon the mathematics of his time.

The Geometrical Lectures of Isaac Barrow (1914)[edit]

James Mark Child, source
  • Isaac Barrow was the first inventor of the Infinitesimal Calculus; Newton got the main idea of it from Barrow by personal communication; and Leibniz also was in some measure indebted to Barrow's work; obtaining confirmation of his own original ideas, and suggestions for their further development, from the copy of Barrow's book that he purchased in 1673.
    The above is the ultimate conclusion that I have arrived at as the result of six months' close study of a single book, my first essay in historical research. By the "Infinitesimal Calculus," I intend "a complete set of standard forms for both the differential and integral sections of the subject, together with rules for their combination, such as for a product, a quotient, or a power of a function; and also a recognition and demonstration of the fact that differentiation and integration are inverse operations."
  • The case of Newton is to my mind clear enough. Barrow was familiar with the paraboliforms, and tangents and areas connected with them, in from 1655 to 1660 at the very latest; hence he could at this time differentiate and integrate by his own method any rational positive power of a variable, and thus also a sum of such powers. He further developed it in the years 1662-3-4, and in the latter year probably had it fairly complete. In this year he communicated to Newton the great secret of his geometrical constructions, as far as it is humanly possible to judge from a collection of tiny scraps of circumstantial evidence; and it was probably this that set Newton to work on an attempt to express everything as a sum of powers of the variable. During the next year Newton began to "reflect on his method of fluxions," and actually did produce his Analysis per Æquations. This, though composed in 1666, was not published until 1711.
  • Leibniz bought a copy of Barrow's work in 1673, and was able "to communicate a candid account of his calculus to Newton" in 1677. In this connection, in the face of Leibniz' persistent denial that he received any assistance whatever from Barrow's book, we must bear well in mind Leibniz' twofold idea of the "calculus":
    (i) the freeing of the matter from geometry,
    (ii) the adoption of a convenient notation.
    Hence, be his denial a mere quibble or a candid statement without any thought of the idea of what the "calculus" really is, it is perfectly certain that on these two points at any rate he derived not the slightest assistance from Barrow's work; for the first of them would be dead against Barrow's practice and instinct, and of the second Barrow had no knowledge whatever. These points have made the calculus the powerful instrument that it is, and for this the world has to thank Leibniz; but their inception does not mean the invention of the infinitesimal calculus. This, the epitome of the work of his predecessors, and its completion by his own discoveries until it formed a perfected method of dealing with the problems of tangents and areas for any curve in general, i.e. in modern phraseology, the differentiation and integration of any function whatever (such as were known in Barrow's time), must be ascribed to Barrow.
  • The beginnings of the Infinitesimal Calculus, in its two main divisions, arose from determinations of areas and volumes, and the finding of tangents to plane curves. The ancients attacked the problems in a strictly geometrical manner, making use of the "method of exhaustions." In modern phraseology, they found "upper and lower limits," as closely equal as possible, between which the quantity to be determined must lie; or, more strictly perhaps, they showed that, if the quantity could be approached from two "sides," on the one side it was always greater than a certain thing, and on the other it was always less; hence it must be finally equal to this thing. This was the method by means of which Archimedes proved most of his discoveries. But there seems to have been some distrust of the method, for we find in many cases that the discoveries are proved by a reductio ad absurdum, such as one is familiar with in Euclid. To Apollonius we are indebted for a great many of the properties, and to Archimedes for the measurement, of the conic sections and the solids formed from them by their rotation about an axis.
  • The first great advance, after the ancients, came in the beginning of the seventeenth century. Galileo (1564-1642) would appear to have led the way, by the introduction of the theory of composition of motions into mechanics; he also was one of the first to use infinitesimals in geometry, and from the fact that he uses what is equivalent to "virtual velocities" it is to be inferred that the idea of time as the independent variable is due to him.
  • Kepler (1571-1630) was the first to introduce the idea of infinity into geometry and to note that the increment of a variable was evanescent for values of the variable in the immediate neighbourhood of a maximum or minimum; in 1613, an abundant vintage drew his attention to the defective methods in use for estimating the... contents of vessels, and his essay on the subject (Nova Stereometria Doliorum [Vinariorum]) entitles him to rank amongst those who made the discovery of the infinitesimal calculus possible.
  • In 1635 Cavalieri published a theory of "indivisibles," in which he considered a line as made up of an infinite number of points, a superficies as composed of a succession of lines, and a solid as a succession of superficies, thus laying the foundation for the "aggregations" of Barrow. Roberval seems to have been the first, or at the least an independent, inventor of the method; but he lost credit for it, because he did not publish it, preferring to keep the method to himself for his own use; this seems to have been quite a usual thing amongst learned men of that time, due perhaps to a certain professional jealousy. The method was severely criticized by contemporaries, especially by Guldin, but Pascal (1623-1662) showed that the method of indivisibles was as rigorous as the method of exhaustions, in fact that they were practically identical. In all probability the progress of mathematical thought is much indebted to this defence by Pascal. Since this method is exactly analogous to the ordinary method of integration, Cavalieri and Roberval have more than a little claim to be regarded as the inventors of at least the one branch of the calculus; if it were not for the fact that they only applied it to special cases, and seem to have been unable to generalize it owing to cumbrous algebraical notation, or to have failed to perceive the inner meaning of the method when concealed under a geometrical form. Pascal himself applied the method with great success, but also to special cases only; such as his work on the cycloid.
  • The next step was of a more analytical nature; by the method of indivisibles, Wallis (1616-1703) reduced the determination of many areas and volumes to the calculation of the value of the series , i.e. the ratio of the mean of all the terms to the last term, for integral values of n; and later he extended his method, by a theory of interpolation, to fractional values of n. Thus the idea of the Integral Calculus was in a fairly advanced stage in the days immediately antecedent to Barrow.
  • What Cavalieri and Roberval did for the integral calculus, Descartes (1596-1650) accomplished for the differential branch by his work on the application of algebra to geometry. Cartesian coordinates made possible the extension of investigations on the drawing of tangents to special curves to the more general problem for curves of any kind. To this must be added the fact that he habitually used the index notation; for this had a very great deal to do with the possibility of Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make any very great progress on his own account in the matter of the drawing of tangents, owing to what I cannot help but call an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—
(1) a secant, of which the points of intersection with the curve became coincident
(2) a prolongation of an element of the curve, which was to be considered as composed of an infinite succession of infinitesimal straight lines;
(3) the direction of the resultant motion, by which the curve might have been described.
Descartes chose the first. ...I cannot see that it would have been possible for a Descartes to miss the differential triangle and all its consequences if he had chosen the second definition. ...
Descartes did not make another choice of definition and use the second one... for in his rule for the tangents to roulettes, he considers a curve as the ultimate form of a polygon. The third definition, if not originally due to Galileo, was a direct consequence of his conception of the composition of motions; this definition was used by Roberval (1602-1675) and applied successfully to a dozen or so of the well-known curves; in it we have the germ of the method of "fluxions." Thus it is seen that Roberval occupies an almost unique position, in that he took a great part in the work preparatory to the invention of both branches of the infinitesimal calculus; a fact that seems to have escaped remark.
  • Fermat (1590-1663) adopted Kepler's notion of the increment of the variable becoming evanescent near a maximum or minimum value, and upon it based his method of drawing tangents. Fermat's method of finding the maximum or minimum value of a function involved the differentiation of any explicit algebraic function, in the form that appears in any beginner's text book of today (though Fermat does not seem to have the "function" idea); that is, the maximum or minimum values of f(x) are the roots of f'(x) = 0, where f'(x) is the limiting value of [f(x+h) - f(x)]/h; only Fermat uses the letter e or E instead of h.
  • Here then we have all the essentials for the calculus; but only for explicit integral algebraic functions, needing the binomial expansion of Newton, or a general method of rationalization which did not impose too great algebraic difficulties, for their further development; also, on the authority of Poisson, Fermat is placed out of court, in that he also only applied his method to certain special cases. Following the lead of Roberval, Newton subsequently used the third definition of a tangent, and the idea of time as the independent variable, although this was only to insure that one at least of his working variables should increase uniformly. This uniform increase of the independent variable would seem to have been usual for mathematicians of the period and to have persisted for some time; for later we find with Leibniz and the Bernoullis that d(dy/dx) = (d2y/dx2)dx. Barrow also used time as the independent variable in order that, like Newton, he might insure that one of his variables, a moving point or line or superficies, should proceed uniformly; ...Barrow... chose his own definition of a tangent, the second of those given above; and to this choice is due in great measure his advance over his predecessors. For his areas and volumes he followed the idea of Cavalieri and Roberval.
  • Thus we see that in the time of Barrow, Newton, and Leibniz the ground had been surveyed, and in many directions levelled; all the material was at hand, and it only wanted the master mind to "finish the job." This was possible in two directions, by geometry or by analysis; each method wanted a master mind of a totally different type, and the men were forthcoming. For geometry, Barrow; for analysis, Newton and Leibniz with his inspiration in the matter of the application of the simple and convenient notation of his calculus of finite differences to infinitesimals and to geometry. With all due honour to these three mathematical giants, however, I venture to assert that their discoveries would have been well-nigh impossible to them if they had lived a hundred years earlier; with the possible exception of Barrow, who, being a geometer, was more dependent on the ancients and less on the moderns of his time than were the two analysts, they would have been sadly hampered but for the preliminary work of Descartes and the others I have mentioned (and some I have not—such as Oughtred), but especially Descartes.

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