Differential and integral calculus
A general algorithm for tangents, velocities and areas: differentiation and integration as mutually inverse operations on the equations of curves.
The English physicist Isaac Newton built the method of fluxions in 1665-1666, but his manuscripts (the 1666 tract, De analysi of 1669) circulated in copies and were printed only in the 18th century. The German Gottfried Leibniz independently arrived at his own calculus in 1675 and printed it in 1684 in Acta Eruditorum.
The priority dispute reached a Royal Society committee (1712) whose report Newton wrote himself; historians today regard the two discoveries as independent.
Why this date
the axis uses the first publication: Leibniz's Nova methodus pro maximis et minimis in Acta Eruditorum, 1684. Newton worked out the method of fluxions in 1665-1666 (October 1666 tract, De analysi 1669, De methodis 1671), but these manuscripts were printed only in 1711 and 1736
What it grew from
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Analytic geometry
prerequisite for · Confidence: high
The calculus of Newton and Leibniz is an algorithm on the equations of curves; Newton taught himself mathematics from van Schooten's Latin edition of Descartes's Geometrie, Leibniz from a reading list by Huygens that included Descartes, and historians call analytic geometry the essential framework of the calculus.
Dispute: Domski (SEP, §3.3) stresses that Descartes excluded infinitesimals and 'mechanical' curves from his program: "Descartes eliminated from his geometrical program the very elements of mathematical and geometrical reasoning that made it possible for Newton and Leibniz to develop the calculus". What Descartes supplied, therefore, was the algebraic language of curves, not the idea of infinitesimals, which came from Cavalieri, Fermat, Wallis and Barrow.
Link sources
- Smith G. «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026, quotation assembled from fragments
By early 1664 he had also begun teaching himself mathematics, taking notes on works by Oughtred, Viète, Wallis, and Descartes ... the latter via van Schooten's Latin translation, with commentary, of the Géométrie.
- «The Development of Analytic Geometry», Science and Its Times (Gale), via Encyclopedia.com (2001) Verified 20 Sep 2026
provided the essential framework for the calculus of Isaac Newton and Gottfried Leibniz
- Cambridge University Library, exhibition «Footprints of the Lion», section «Calculation» Verified 20 Sep 2026
The young Newton owed his greatest intellectual debt to the French mathematician and natural philosopher, René Descartes.
- O'Connor J. J., Robertson E. F. «Gottfried Wilhelm von Leibniz», MacTutor History of Mathematics (1998) Verified 20 Sep 2026
Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint-Vincent, Descartes and Sluze.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.
- Smith G. «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026, quotation assembled from fragments
What grew out of it
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The language of analysis: the function as the basic object
improved · Confidence: high
The calculus of Newton and Leibniz existed as a set of devices, and it was the Introductio of 1748 that assembled it into a system in which the function became the explicit and central object.
Link sources
- Carl B. Boyer, The Foremost Textbook of Modern Times, 1950, American Mathematical Monthly 58 (4), 223-226 (1951) Verified 20 Sep 2026
yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate
- Israel Kleiner, Evolution of the Function Concept: A Brief Survey, The College Mathematics Journal 20 (4), 282-300 (1989) Verified 20 Sep 2026
Euler's Introductio was the first work in which the concept of function plays an explicit and central role
- Introduction to the Analysis of Infinities, Encyclopaedia Britannica (2024) Verified 24 Sep 2026
he made the notion of function the central organizing concept of analysis
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Carl B. Boyer, The Foremost Textbook of Modern Times, 1950, American Mathematical Monthly 58 (4), 223-226 (1951) Verified 20 Sep 2026
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The wave equation and the vibrating string controversy
influenced · Confidence: medium
The wave equation is written in partial derivatives, that is, it grew out of the differential calculus extended to functions of several variables: that apparatus itself took shape after 1684, and clear examples of such equations are known from 1719 onwards.
Link sources
- Jesper Lützen, Euler's Vision of a General Partial Differential Calculus for a Generalized Kind of Function, Mathematics Magazine 56 (5), Sherlock Holmes in Babylon, MAA, 354-360 (1983) Verified 20 Sep 2026
the type of functions which could be allowed in analysis, particularly in the new partial differential calculus
- Florian Cajori, The Early History of Partial Differential Equations and of Partial Differentiation and Integration, American Mathematical Monthly 35 (9), 459-467 (1928) Verified 20 Sep 2026
Partial differential equations stand out clearly in six examples on trajectories published in 1719 by Nicolaus Bernoulli
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Jesper Lützen, Euler's Vision of a General Partial Differential Calculus for a Generalized Kind of Function, Mathematics Magazine 56 (5), Sherlock Holmes in Babylon, MAA, 354-360 (1983) Verified 20 Sep 2026
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The analytical theory of heat and Fourier series
influenced · Confidence: medium
Fourier worked with the means of the analysis that grew out of the calculus of Newton and Leibniz and directly continued the work of Bernoulli, Euler and Lagrange on trigonometric series, although the heat conduction equation required the later apparatus of partial differential equations.
Link sources
- T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999)
Fourier was clearly aware of the earlier work of Bernoulli, Euler, and Lagrange relating to solutions in the form of trigonometric series. He was also aware that Euler, D'Alembert and Lagrange viewed trigonometric series with great suspicion.
- Joseph Fourier, MacTutor History of Mathematics Archive, University of St Andrews (1997) Verified 20 Sep 2026
The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- T. N. Narasimhan, Fourier's heat conduction equation: history, influence, and connections, Reviews of Geophysics 37 (1), 151-172 (1999)
Links under review
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Mathematical Principles of Natural Philosophy
influenced · Confidence: medium
The link between Newton's early calculus and the Principia is disputed: Whiteside found no drafts supporting Newton's 1715 claim that the propositions were found by analysis; the sources confirm only the presence of "moments" and series.
The Principia took over the notions of limit and of "moments" from Newton's early work on the calculus, but the apparatus of the calculus itself is not used in the book: it is written in the language of the geometry of limit ratios, not in the algorithm of fluxions.
Dispute: Newton himself in 1715 (the anonymous Account in the dispute with Leibniz) claimed to have found the propositions of the Principia by the new analysis and only then rewritten them geometrically. Whiteside, editor of Newton's mathematical papers, found no draft supporting this; Guicciardini (1989, 1999) and the Stanford Encyclopedia describe the mathematics of the Principia as a geometry of ultimate ratios without the algorithm of fluxions. On the other hand, Section I of Book I (the lemmas on first and last ratios) and Lemma II of Book II on moments are calculus in geometric dress, and in that sense the book would have been different without Newton's work of 1665-1671. A further difficulty: the calculus is counted from Leibniz's publication of 1684, which Newton did not use when writing the Principia, so the connection makes sense only through Newton's own unpublished fluxions.
Link sources
- Guicciardini N. The Development of Newtonian Calculus in Britain, 1700-1800. Cambridge University Press, p. 1 (1989) Verified 5 Oct 2026
In the Principia there was no proper algorithm of fluxions, even though series and some algebraical manipulations with 'moments' occurred, especially in the second book.
- Smith G. E., «Newton's Philosophiae Naturalis Principia Mathematica», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026
But the geometric mathematics used in the Principia offered no way of representing second derivatives.
- Newton I. «An Account of the Book entituled Commercium Epistolicum», Philosophical Transactions 29 (1715), The Renaissance Mathematicus, «Newton Lied!» (1715) Verified 20 Sep 2026
By the help of the new Analysis Mr. Newton found out most of the Prepositions in his Principia Philosophiae: but because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically, that the System of the Heavens might be founded upon good Geometry.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Guicciardini N. The Development of Newtonian Calculus in Britain, 1700-1800. Cambridge University Press, p. 1 (1989) Verified 5 Oct 2026
Node sources
- Smith G. «Isaac Newton», Stanford Encyclopedia of Philosophy (2007) Verified 20 Sep 2026
having extended his earlier examination of cutting-edge problems into the discovery of the calculus, as presented in his tract of October 1666
- O'Connor J. J., Robertson E. F. «Gottfried Wilhelm von Leibniz», MacTutor History of Mathematics (1998) Verified 20 Sep 2026, quotation assembled from fragments
In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... in Acta Eruditorum.
- «Mathematics: The calculus», Encyclopaedia Britannica (2024) Verified 24 Sep 2026
Although a bitter dispute over priority developed later between followers of the two men, it is now clear that they each arrived at the calculus independently.
- O'Connor J. J., Robertson E. F. «Isaac Newton», MacTutor History of Mathematics (2000) Verified 20 Sep 2026
Newton's De Methodis Serierum et Fluxionum was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong. "Assembled from fragments" means the words do not stand together on the page: we found each fragment separately and joined them with an ellipsis.