DOCSLIB.ORG

PIERRE VARIGNON and the CALCULUS of MOTION Michael S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PIERRE VARIGNON and the CALCULUS of MOTION Michael S PIERRE VARIGNON AND THE CALCULUS OF MOTION Michael S. Mahoney Princeton University The new currents of scientific thought that reshaped the Académie des Sciences in the 1690s flowed from sources outside France and met in Paris during the period 1689-92. From England came Newton's Principia, familiar in style perhaps to those who knew Huygens' work, but novel in substance surely to those schooled in Cartesian mechanics. From Germany1 by way of the Acta eruditorum came Leibniz' "Tentamen de motuum coelestium causis", familiar in its vortex cosmology but strange in its use of a new "differential calculus" to analyze the motion of bodies. From Basel came Johann Bernoulli, an unknown young mathematician skilled in that calculus and in the even newer and more mysterious "method of integrals", which his brother Jakob and he had recently be developing. Though the last of the new currents to arrive in Paris, the Bernoullis' infinitesimal analysis was the first to influence the direction of mathematics there. It opened the way to Leibniz' recent work on dynamics and thereby provided the tools with which a new generation of Academicians would recast and reorganize Newton's mechanics, integrating it into a program of research that would remain essentially Cartesian for another half-century. Hence, we shall consider the new currents in reverse order. Although Leibniz had worked out the essentials of his new calculus while still in Paris in the mid-1670s,2 he waited almost a decade before publishing an account of it in the May 1684 number of Acta eruditorum. The article, "Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractis, nec irrationales quantitates moratur, & singulare pro illis calculi, genus"3 set forth only the bare algorithms of differentiation, with no explanation or demonstration to speak of and with the sketchiest of illustrative examples. Articles that followed over the next few years shed a bit more light on the new method, especially on the ways in which it set the bases for a new, "transcendent" geometry, but they still offered little help or guidance to the reader who wanted to gain command over the calculus. Indeed, what constituted command remained itself unclear, as Leibniz failed to undertake a systematic exposition of the techniques of the calculus and of the range of its applications. 4 In particular, although he paid practically no attention at all to the inverse operation of integration and to its application to quadrature, rectification, and the inverse problem of tangents. It was instead Jakob Bernoulli and his younger brother Johann who assumed that responsibility in the late 1680s and who rapidly extended the purview of differential calculus to include all known curves and many new ones. 1More precisely, from Rome, where Leibniz was visiting when he first learned of the appearance of the Principia and hastened to put his own, similar thoughts to paper. See most recently Domenico Bertoloni Meli, Equivalence and Priority: The Newton-Leibniz Dispute on the World System (Oxford : Clarendon Press ; New York : Oxford University Press, 1993), together with the earlier studies of E.J. Aiton, "The Celestial Mechanics of Leibniz", Annals of Science 16(1960), 65-82; "The celestial mechanics of Leibniz in the light of Newtonian criticism", ibid. 18(1962), 31-41; and "The celestial mechanics of Leibniz: A new interpretation", ibid. 20(1964), 111-123. 2See J.E. Hofmann, Leibniz in Paris, 1672-1676: His Growth to Mathematical Maturity (London, New York: Cambridge University Press, 1974). 3pp.467-473; LMS.V.220-226 4See M.S. Mahoney, "Infinitesimals and Transcendent Relations", in D.C. Lindberg and R.S. Westfall (eds.), Reappraisals of the Scientific Revolution, Chap.12. ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 2 The notes on the calculus made by the Marquis de l'Hôpital for Nicolas Malebranche in 1689 show by contrast the necessity and the originality of the Bernoullis' efforts.5 L'Hôpital failed to discern in Leibniz' notation the operational nature of d when prefixed to the symbol of a quantity. Even though Leibniz had earlier spoken of dx as "a certain modification" (quaedam modificatio) of x l'Hôpital treated it as simply a variation on the e that Barrow had used to denote an infinitesimal increment.6 The examples l'Hôpital offered Malebranche scarcely went beyond what Fermat and others had been doing fifty years earlier, either in substance or in style. Two years later, when Johann Bernoulli first appeared in Paris, the situation had changed little. L'Hôpital, who counted as the leading mathematician in Paris, had but a tenuous hold on the new infinitesimal methods. Quick to recognize Bernoulli's superior skills upon first meeting him at one of Malebranche's Thursday soirées, l'Hôpital contracted with the young Swiss to learn the calculus. Installed at the Marquis' summer home in Oucques, Bernoulli composed for l'Hôpital a short exposition of the differential calculus and a long treatise on the methods of the integral calculus.7 In a now notorious arrangement, Bernoulli ceded his rights of authorship to l'Hôpital, who published the work on the differential calculus, along with some material on the integral calculus, as his own Analyse des infiniment petits, pour l'intelligence des lignes courbes in 1696. Bernoulli had another eager student in Pierre Varignon, a friend of l'Hôpital and his sometime collaborator. At the time Professor of Mathematics at the Collège Mazarin, Varignon had established a reputation in the subject through his work in statics pursued in the traditional geometrical mode. Sharing in the lectures at Oucques, and more astute and talented as a mathematician than the Marquis, Varignon quickly made the calculus his own and was soon exchanging problems and solutions with Bernoulli, who recognized in him a colleague rather than a student and quasi-patron and who saw to it personally that Varignon received issues of the Acta eruditorum as they appeared, thus overcoming an embargo on trade with France.8 Hence, Varignon was able to make steady progress toward a command of the differential and integral calculus and to direct it to his own interests. In particular, Varignon strode out on his own precisely at the point at which l'Hôpital admitted having reached the limits of his understanding. Toward the end of the lectures on the integral calculus, Bernoulli offered a series of "physico-mechanical" problems, in which the conditions placed on quantities stemmed from the physics of static forces and of bodies moving under constraint. Bernoulli did not go all that far in his presentation, and he offered nothing like a system of analytic mechanics. His purpose was to illustrate applications of the integral calculus. Only two of the problems involved motion, namely the isochrone and the tautochrone, and both of them could be approached by means of Galileo's analysis of uniform motion and his laws of fall, in particular the law that ties the velocity of a falling body to the square root of the distance through which it has fallen from rest. To apply these 5Cf. Vol. 17,2 of Malebranche's Oeuvres|. 6"De geometria recondita et analysi indivisibilium atque infinitorum", Acta eruditorum (1686), pp. 292-300; LMS.V.226-33. 7/*references*/ 8/*documentation*/ ©1985,1994 - DRAFT - Not for quotation or distribution without permission Varignon and the calculus of motion page 3 laws, Bernoulli made an assumption that became fundamental to analytic mechanics: over infinitesimal distances ds traversed in infinitesimal times dt a motion may be considered as uniform, whatever its actual nature in the finite realm.9 That is, by Galileo's law of uniform motion, ds = vdt. This assumption is revealing in two ways of the conceptual structure of the differential calculus as it was originally developed and applied to mechanics. First, it treats the velocity as a primitive, rather than a derived quantity. If, that is, the relation ds = vdt defines anything, it defines the distance traversed by a body moving at a given speed. The relation stands as it is, rather than as a consequence of what has become a definition of velocity, namely v = ds/dt. Second, recall that, when Huygens moved from indivisibles to infinitesimal segments in his demonstration of the tautochrony of the cycloid, he worried about variations in speed over those segments. So he took pains first to try to show that the mean speed and the speeds at either end of the segment were close enough to be treated as equal and then to follow the laborious path of a double reductio ad absurdum. When Bernoulli first came to Paris, Varignon had been engaged in an effort to capture the science of weights and Galileo's laws of motion in algebraic form. It did not take much for Bernoulli to convince him that in the calculus he had a tool better suited to his purpose. Moreover, through Bernoulli's lectures Varignon gained access to the mathematical details of Leibniz' concise and confusing treatment of planetary orbits in his "Tentamen de motuum coelestium causis", hastily written upon learning of the publication of the Principia, which in turn was opened to for Varignon a new reading.10 Following a generalized account of isochronic curves in 1699, Varignon turned in 1700 to a general theory of motion determined by central forces and in a series of memoirs rendered into the language of the calculus the mechanical substance of Book I, Sections 2 - 10, of the Principia. His first memoir, "Manière générale de déterminer les forces, les vitesses, les espaces, & les temps, une seule de 9Cf. Euler, Mechanica (1736), Book I, Prop.3: In motu quantumvis inaequabili minima spatii elementa motu aequabili percurri concipi possunt.
Load more

Recommended publications
  • Calcul Différentiel Et Intégral De Leibniz Et Newton
    Master de Mathématiques – Sorbonne Université (M1) UE 4M039 : Histoire des mathématiques (Alexandre Guilbaud et Laurent Mazliak) Semaine 4 La naissance du calcul différentiel et intégral : un pas décisif vers l’émergence du concept de fonction Le jeudi 15 février 2018 M1 - Histoire des mathématiques 1 Prologue (1/3) – Les académies Jean-Baptiste Colbert présentant les membres de l'Académie royale des sciences à Louis XIV (H. Testelin, d'après une gravure de Lebrun) Le jeudi 15 février 2018 M1 - Histoire des mathématiques 2 Prologue (1/3) – Les académies § Les académies qui naissent au XVIIe siècle signent l'abandon du travail solitaire. § Les savants s'y réunissent de leur plein gré et de façon informelle pour : échanger des idées et réfléchir de façon collective sur un certain nombre de sujets / soumettre leurs travaux à la critique des autres membres / réaliser des observations et les commenter. § Les « académies privées » : ü L'Académie dei Lincei, fondée à Rome en 1603 (Galilée en est membre à partir de 1611) ü L'Académie del Cimento à la brève existence (1657-1667) ü L'académie napolitaine des Investiganti (1663-1670) ü L'Académie par correspondance de Marin Mersenne (1588-1648) met en relation une quarantaine de savants (dont Descartes, Fermat, Galilée, Gassendi, Roberval, Huygens) ü etc. Le jeudi 15 février 2018 M1 - Histoire des mathématiques 3 Prologue (1/3) – Les académies § La Royal Society de Londres : existe depuis 1661, elle ne reçoit aucun financement de la couronne et vit des cotisations de ses membres (qui devinrent fort nombreux). § L'Académie royale des sciences de Paris, créée par Colbert en 1666 : - l'Observatoire de Paris naît en 1667 sous l'impulsion de l'Académie.
    [Show full text]
  • Le Calcul Différentiel Et Intégral Dans L'analyse Démontrée De Charles
    Recherches sur Diderot et sur l'Encyclopédie 38 | avril 2005 La formation de D'Alembert Le calcul différentiel et intégral dans l’Analyse démontrée de Charles René Reyneau Jean-Pierre Lubet Édition électronique URL : https://journals.openedition.org/rde/304 DOI : 10.4000/rde.304 ISSN : 1955-2416 Éditeur Société Diderot Édition imprimée Date de publication : 1 juin 2005 ISBN : 2-9520892-4-8 ISSN : 0769-0886 Référence électronique Jean-Pierre Lubet, « Le calcul différentiel et intégral dans l’Analyse démontrée de Charles René Reyneau », Recherches sur Diderot et sur l'Encyclopédie [En ligne], 38 | avril 2005, mis en ligne le 30 mars 2009, consulté le 30 juillet 2021. URL : http://journals.openedition.org/rde/304 ; DOI : https:// doi.org/10.4000/rde.304 Propriété intellectuelle 151-176 21/06/05 16:07 Page 151 Jean-Pierre LUBET Le calcul différentiel et intégral dans l’Analyse démontrée de Charles René Reyneau Le premier mémoire envoyé par Jean Le Rond D’Alembert à l’Académie royale des sciences avait pour objectif essentiel de rectifier une erreur contenue dans l’Analyse démontrée de Charles René Reyneau. Avec d’autres travaux de D’Alembert sur le calcul intégral, cette question1 a été présentée en détail par Christian Gilain dans un article récent . Nous proposons de revenir ici sur l’ensemble du traité de Reyneau, pour essayer de caractériser les éléments de calcul différentiel et intégral qu’il contient. Publié pour la première fois en 1708, l’ouvrage a été largement diffusé au point qu’une seconde édition, comportant peu de modifications, a été nécessaire en 1736-1738.
    [Show full text]
  • 0.999… = 1 an Infinitesimal Explanation Bryan Dawson
    0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder .
    [Show full text]
  • Understanding Student Use of Differentials in Physics Integration Problems
    PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020108 (2013) Understanding student use of differentials in physics integration problems Dehui Hu and N. Sanjay Rebello* Department of Physics, 116 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2601, USA (Received 22 January 2013; published 26 July 2013) This study focuses on students’ use of the mathematical concept of differentials in physics problem solving. For instance, in electrostatics, students need to set up an integral to find the electric field due to a charged bar, an activity that involves the application of mathematical differentials (e.g., dr, dq). In this paper we aim to explore students’ reasoning about the differential concept in physics problems. We conducted group teaching or learning interviews with 13 engineering students enrolled in a second- semester calculus-based physics course. We amalgamated two frameworks—the resources framework and the conceptual metaphor framework—to analyze students’ reasoning about differential concept. Categorizing the mathematical resources involved in students’ mathematical thinking in physics provides us deeper insights into how students use mathematics in physics. Identifying the conceptual metaphors in students’ discourse illustrates the role of concrete experiential notions in students’ construction of mathematical reasoning. These two frameworks serve different purposes, and we illustrate how they can be pieced together to provide a better understanding of students’ mathematical thinking in physics. DOI: 10.1103/PhysRevSTPER.9.020108 PACS numbers: 01.40.Àd mathematical symbols [3]. For instance, mathematicians I. INTRODUCTION often use x; y asR variables and the integrals are often written Mathematical integration is widely used in many intro- in the form fðxÞdx, whereas in physics the variables ductory and upper-division physics courses.
    [Show full text]
  • Leonhard Euler: His Life, the Man, and His Works∗
    SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Infinitesimal Calculus
    Infinitesimal Calculus Δy ΔxΔy and “cannot stand” Δx • Derivative of the sum/difference of two functions (x + Δx) ± (y + Δy) = (x + y) + Δx + Δy ∴ we have a change of Δx + Δy. • Derivative of the product of two functions (x + Δx)(y + Δy) = xy + Δxy + xΔy + ΔxΔy ∴ we have a change of Δxy + xΔy. • Derivative of the product of three functions (x + Δx)(y + Δy)(z + Δz) = xyz + Δxyz + xΔyz + xyΔz + xΔyΔz + ΔxΔyz + xΔyΔz + ΔxΔyΔ ∴ we have a change of Δxyz + xΔyz + xyΔz. • Derivative of the quotient of three functions x Let u = . Then by the product rule above, yu = x yields y uΔy + yΔu = Δx. Substituting for u its value, we have xΔy Δxy − xΔy + yΔu = Δx. Finding the value of Δu , we have y y2 • Derivative of a power function (and the “chain rule”) Let y = x m . ∴ y = x ⋅ x ⋅ x ⋅...⋅ x (m times). By a generalization of the product rule, Δy = (xm−1Δx)(x m−1Δx)(x m−1Δx)...⋅ (xm −1Δx) m times. ∴ we have Δy = mx m−1Δx. • Derivative of the logarithmic function Let y = xn , n being constant. Then log y = nlog x. Differentiating y = xn , we have dy dy dy y y dy = nxn−1dx, or n = = = , since xn−1 = . Again, whatever n−1 y dx x dx dx x x x the differentials of log x and log y are, we have d(log y) = n ⋅ d(log x), or d(log y) n = . Placing these values of n equal to each other, we obtain d(log x) dy d(log y) y dy = .
    [Show full text]
  • Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
    Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time.
    [Show full text]
  • Arxiv:1404.5658V1 [Math.LO] 22 Apr 2014 03F55
    TOWARD A CLARITY OF THE EXTREME VALUE THEOREM KARIN U. KATZ, MIKHAIL G. KATZ, AND TARAS KUDRYK Abstract. We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementar- ity rather than a rivalry of the approaches. Contents 1. Introduction 2 1.1. A historical re-appraisal 2 1.2. Practice and ontology 3 2. Grades of clarity according to Peirce 4 3. Perceptual continuity 5 4. Constructive clarity 7 5. Counterexample to the existence of a maximum 9 6. Reuniting the antipodes 10 7. Kronecker and constructivism 11 8. Infinitesimal clarity 13 8.1. Nominalistic reconstructions 13 8.2. Klein on rivalry of continua 14 arXiv:1404.5658v1 [math.LO] 22 Apr 2014 8.3. Formalizing Leibniz 15 8.4. Ultrapower 16 9. Hyperreal extreme value theorem 16 10. Approaches and invitations 18 11. Conclusion 19 References 19 2000 Mathematics Subject Classification. Primary 26E35; 00A30, 01A85, 03F55. Key words and phrases. Benacerraf, Bishop, Cauchy, constructive analysis, con- tinuity, extreme value theorem, grades of clarity, hyperreal, infinitesimal, Kaestner, Kronecker, law of excluded middle, ontology, Peirce, principle of unique choice, procedure, trichotomy, uniqueness paradigm. 1 2 KARINU. KATZ, MIKHAILG. KATZ, ANDTARASKUDRYK 1. Introduction German physicist G. Lichtenberg (1742 - 1799) was born less than a decade after the publication of the cleric George Berkeley’s tract The Analyst, and would have certainly been influenced by it or at least aware of it.
    [Show full text]
  • Deux Moments De La Critique Du Calcul Infinitésimal
    Séminaire de philosophie et mathématiques MICHEL BLAY Deux moments de la critique du calcul infinitésimal Séminaire de Philosophie et Mathématiques, 1987, fascicule 12 « Deux moments de la critique du calcul infinitésimal », , p. 1-32 <http://www.numdam.org/item?id=SPHM_1987___12_A1_0> © École normale supérieure – IREM Paris Nord – École centrale des arts et manufactures, 1987, tous droits réservés. L’accès aux archives de la série « Séminaire de philosophie et mathématiques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Deux moments de la critique du calcul infinitésimal: Michel Rolle et George Berkeley par Michel BLAY 1) La critique de Michel Rolle 1-1) La critique des concepts et des principes fondamentaux. 1-2) Le nouveau calcul conduit à l'erreur. 1-2-1) Le cadre général de l'argumentation. 1-2-2) Le privilège accordé à la méthode de Hudde. 1-2-3) La construction des exemples. 2) La critique de George Berkeley 2-1) L'absence de rigueur. 2-2) L'heureuse compensation des erreurs. 2-2-1) Première détermination de la sous-tangente. 2-2-2) Deuxième détermination de la sous-tangente. Deux moments de la critique du calcul infinitésimal: Michel Rolle et George Berkeley C'est en 1684, dans le tout nouveau périodique des Acta Eruditorum, que Leibniz publie son texte fondateur du nouveau calcul différentiel: "Nova Methodus pro Maximis et Minimis, itemque, Tangentibus, quae nec Fractas nec Irrationales Quantitates moratur, et singulare pro illis calculi genus" (1).
    [Show full text]
  • Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Reviewed by Slava Gerovitch
    BOOK REVIEW Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Reviewed by Slava Gerovitch Infinitesimal: How a Dangerous Mathematical Theory ing, from the chaos of imprecise analogies to the order Shaped the Modern World of disciplined reasoning. Yet, as Amir Alexander argues Amir Alexander in his fascinating book Infinitesimal: How a Dangerous Scientific American/Farrar, Straus and Giroux, April 2014 Mathematical Theory Shaped the Modern World, it was US$27.00/US$16.00, 368 pages precisely the champions of offensive infinitesimals who Hardcover ISBN-13: 978-0-37-417681-5 propelled mathematics forward, while the rational critics Paperback ISBN-13: 978-0-37-453499-8 slowed the development of mathematical thought. More- over, the debate over infinitesimals reflected a larger clash From today’s perspective, a mathematical technique that in European culture between religious dogma and intellec- lacks rigor or leads to paradoxes is a contradiction in tual pluralism and between the proponents of traditional terms. It must be expelled from mathematics, lest it dis- order and the defenders of new liberties. credit the profession, sow chaos, and put a large stain on Known since antiquity, the concept of indivisibles gave the shining surface of eternal truth. This is precisely what rise to Zeno’s paradoxes, including the famous “Achilles the leading mathematicians of the most learned Catholic and the Tortoise” conundrum, and was subjected to scath- order, the Jesuits, said about the “method of indivisibles”, ing philosophical critique by Plato and Aristotle. Archime- a dubious procedure of calculating areas and volumes by des used the method of indivisibles with considerable suc- representing plane figures or solids as a composition of in- cess, but even he, once a desired volume was calculated, divisible lines or planes, “infinitesimals”.
    [Show full text]
  • 3.5 Leibniz's Fundamental Theorem of Calculus
    3.5 Leibniz’s Fundamental Theorem of Calculus 133 spherical surface on top of the ice-cream cone. You may use knowledge of the surface area of the entire sphere, which Archimedes had determined. Exercise 3.24: Imagine boring a round hole through the center of a sphere, leaving a spherical ring. Use the result of Exercise 3.23 to find the volume of the ring. Hint: Amazingly, your answer should depend only on the height of the ring, not the size of the original sphere. Also obtain your result directly from Cavalieri’s Principle by comparing the ring with a sphere of diameter the height of the ring. 3.5 Leibniz’s Fundamental Theorem of Calculus Gottfried Wilhelm Leibniz and Isaac Newton were geniuses who lived quite different lives and invented quite different versions of the infinitesimal calculus, each to suit his own interests and purposes. Newton discovered his fundamental ideas in 1664–1666, while a student at Cambridge University. During a good part of these years the Univer- sity was closed due to the plague, and Newton worked at his family home in Woolsthorpe, Lincolnshire. However, his ideas were not published until 1687. Leibniz, in France and Germany, on the other hand, began his own breakthroughs in 1675, publishing in 1684. The importance of publication is illustrated by the fact that scientific communication was still sufficiently uncoordinated that it was possible for the work of Newton and Leibniz to proceed independently for many years without reciprocal knowledge and input. Disputes about the priority of their discoveries raged for centuries, fed by nationalistic tendencies in England and Germany.
    [Show full text]