Abstract
The purpose of this chapter is to give a brief outline of Newton’s methods for “squaring” a curve, which in Leibnizian terms one would call “integrations.” These methods are rarely considered by scholars, even by Newton scholars, with the exception of those, who like George—the dedicatee of this volume—are familiar with the “technical” Newton. My purpose here is not to address the specialists in the history of seventeenth-century mathematics, but rather to offer a reader-friendly primer in Newton’s “quadrature” techniques. I will not help the reader by adapting the notation to our standards: I will strictly adhere to Newton’s notation. But I will try to make things as simple as possible by choosing the most elementary examples. It is often stated that Newton’s mathematical methods were “geometrical.” But once Newton associated an equation to a curve, he could proceed by relying mostly upon symbolic manipulation. This algebraic aspect of Newton’s mathematics should be better known. I hope that the readers non familiar with this side of Newtonian mathematics will derive some instruction, and even some intellectual pleasure, by scratching the surface of the treasure trove of Newton’s algebraic quadrature techniques.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
On Barrow’s influence on Newton, see Feingold’s chapter “Isaac Barrow: Divine, Scholar, Mathematician” in Feingold (1990, 1–104).
- 2.
See the term “inventio æquationis” in Newton (1967, VII, 306).
- 3.
“areae valor” see, e.g., Fig. 10.3.
- 4.
“Præterea quo fluentes quantitates a se invicem clarius distinguantur, Fluxionem quæ in Numeratore Rationis disponitur, sive Antecedentem Rationis haud impropriè Relatam Quantitatem nominare possum, et alteram ad quam referetur, Correlatam; ut et fluentes Quantitates ijsdem respectivè nominibus insignire. Et quo sequentia promptiùs intelligantur, possis imaginari Correlatam Quantitatem esse Tempus vel potiùs aliam quamvis æquabiliter fluentem quantitatem qua Tempus exponitur et mensuratur, et alteram sive Relatam Quantitatem esse spatium quod mobile utcunque acceleratum vel retardatum in illo tempore transigit.” Cambridge University Library, MS Add. 3960.14: pag. 26. In Newton 1967, III, p. 88 and 91.
- 5.
I follow here (with some liberty) the proof in Newton’s De analysi (Royal Society, MS LXXXI, no. 2). For a detailed commentary, wholly respectful of Newton’s notation, see (Guicciardini 2009, 105).
- 6.
“arearum fluxiones erunt ut ordinatae.” See the quotations and discussion of De quadratura in (Guicciardini 2009, 204–6).
- 7.
It might help the reader to note that the rectangle \(B\beta HK=\dot {z}o\) is equivalent to Leibniz’a dA. These “equivalences” raise a whole series of issues concerning the comparison between the Leibnizian and the Newtonian algorithms I cannot broach here. See, e.g. Bertoloni Meli (1993).
- 8.
On the origins of the inverse method of tangents in Descartes, see Scriba (1961).
- 9.
“Quinimò si in æquatione quantitates involvantur quæ nullâ ratione geometricâ determinari et exprimi possunt, quales sunt areæ vel longitudines curvarum: tamen relationes fluxionum haud secus investigantur” (Cambridge University Library, MS Add. MS. 3960.14, pag. 20\()=\) “To be sure, even if quantities be involved in an equation which cannot be determined and expressed by any geometrical technique, such as the areas and lengths of curves, the relations of the fluxions are still to be investigated the same way” (Newton 1967, III, 79).
- 10.
Many propositions in the Principia indeed open with the clause “granting a method for squaring curvilinear figures.” In these propositions the problem at hand is reduced to the squaring of a plane curve that represents the “correlation” (let us use this term in order to avoid “functional dependance”) between continuous magnitudes, such as the distance and the speed of a falling body. For a well-known example, see (Newton 1999, 530).
- 11.
Translation by D. T. Whiteside.
- 12.
The translation from Latin is mine.
- 13.
A recent paper devoted to Newton’s quadratures is Malet (2017).
- 14.
The translation from the French is mine.
- 15.
Newton’s terminology varied. Curves “may sometime be squared by means of finite equations also, or at least compared with other curves (such as conics) whose area may after a fashion, be accepted as known” (Newton 1967, III, 237). Translation by D. T. Whiteside.
- 16.
Newton might have continued to add and tweak the De methodis until the middle of the 1670s. The manuscript (missing the first folio) is Cambridge University Library, MS Add. 3960.14. It is edited and translated in Newton (1967–81, III, 32–328).
- 17.
In Leibnizian notation: \(\tau =\int ydz=(1/\eta )\int vdx=(1/\eta ) s\).
- 18.
In the second Tabula of De quadratura, this corresponds to the first case of the fourth Forma. This quadrature is often employed by Newton in the Principia. Newton discussed the use of this and similar integrations in the Principia with David Gregory and Roger Cotes. At several points, he even planned to add a treatise on quadratures as an appendix to the Principia, in order to allow expert readers to see how the quadratures used in the main text could be achieved. See, Guicciardini (2016).
- 19.
Newton did not use the modern symbol for the absolute value \(|\frac {1}{2}xv - s|\) but rather one that he found in Barrow’s works. Newton wrote \(\div \) for “the Difference of two Quantities, when it is uncertain whether the latter should be subtracted from the former, or the former from the latter.”
- 20.
To recapitulate. The first case of the seventh order translated into Leibnizian notation is as follows. For \(\eta =2\), Newton evaluates the integral \(\int \delta /(z\sqrt {e+fz^2})\,dz\) (\(\delta \), e, f constants). By substitution of variables \(z=x^{-1}\), he reduces it to the conic area \(s=\int vdx= \int \sqrt {f+ex^2}\,dx\). Namely,
$$\displaystyle \begin{aligned} \tau= \int \frac{\delta}{z\sqrt{fz^2+e}}\,dz=\frac{2\delta}{f}\left|\frac{1}{2}xv-s\right|+C=\frac{2\delta}{f}\left|\frac{1}{2}x\sqrt{f+ex^2}-\int \sqrt{f+ex^2}\,dx\right|+C. \end{aligned}$$ - 21.
To help the reader’s understanding, Newton considers equations of the following form: \(f(x , \dot {x}, \dot {y})=0\). In the equation (10.18), indeed, two fluxions \(\dot {x}\) and \(\dot {y}\) as well as a fluent x occur.
- 22.
Newton (1967), III, 89–91.
- 23.
- 24.
This should be read as a tendency, not as a rule. It is true that Cotes and Maclaurin, for example, contributed to techniques of integration in closed form. Of course, series solutions were studied also in the Bernoulli’s school.
- 25.
The classic paper on the different approach to the inverse method in Newton’s and Leibniz’s works is Scriba (1964).
- 26.
Discussed in Giusti (2007, 45–6).
- 27.
In propositions I.10, I.11–13, I.41 (Cor. 3). For a limited number of other exponents n, closed solutions in terms of elliptic functions (not available to Newton’s contemporaries) are possible.
- 28.
References
Bertoloni Meli, Domenico. 1993. Equivalence and Priority: Newton versus Leibniz. Oxford: Clarendon Press.
Bos, Henk. 1974. Differentials, Higher Order Differentials and the Derivative in the Leibnizian Calculus. Archive for History of Exact Sciences 14(1): 1–90.
Feingold, Mordechai (ed.). 1990. Before Newton: The Life and Times of Isaac Barrow. Cambridge: Cambridge University Press.
Giusti, Enrico. 2007. Piccola Storia del Calcolo Infinitesimale dall’Antichità al Novecento. Roma: Istituti Editoriali e Poligrafici Internazionali.
Greenberg, John L. 1995. The Problem of the Earth’s Shape from Newton to Clairaut: The Rise of Mathematical Science in Eighteenth-Century Paris and the Fall of “Normal Science”. New York: Cambridge University Press.
Guicciardini, Niccolò. 2009. Isaac Newton on Mathematical Certainty and Method. Cambridge (Mass): MIT Press.
Guicciardini, Niccolò. 2016. Lost in translation? Reading Newton on inverse-cube trajectories. Archive for History of Exact Sciences 70(2): 205–241.
Jean-Étienne Montucla. 1799–1802. Histoire des Mathématiques. Revised by Joseph Jérôme L. de Lalande, 4 vols. Paris: Henri Agasse.
Jesseph, Douglas M. 1999. Squaring the Circle: The War Between Hobbes and Wallis. Chicago: University of Chicago Press.
L’Hospital, Guillaume F. A. 1694. Solutio problematis geometrici. Acta Eruditorum (May 1694): 193–194.
Malet, Antoni. 1996. From Indivisibles to Infinitesimals: Studies on Seventeenth-Century Mathematizations of Infinitely Small Quantities. Bellaterra: Universitat Autònoma de Barcelona.
Malet, Antoni. 2017. Newton’s Mathematics. In The Oxford Handbook of Newton, ed. Eric Schliesser and Chris Smeenk (online edn, Oxford Academic, 6 Feb. 2017).
Newton, Isaac. 1704. Opticks: Or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light, Also Two Treatises of the Species and Magnitude of Curvilinear Figures. London: for S. Smith and B. Walford, p. 183.
Newton, Isaac. 1711. Analysis per Quantitatum Series, Fluxiones, ac Differentias: Cum Enumeratione Linearum Tertii Ordinis. Edited by William Jones. London: Pearson.
Newton, Isaac. 1736. The Method of Fluxions and Infinite Series. London: H. Woodfall for J. Nourse.
Newton, Isaac. 1908. Newtons Abhandlung über die Quadratur der Kurven. (1704). Aus dem lateinischen übers. und hrsg. von Dr. Gerhard Kowalewski, Leipzig, Ostwalds Klassiker der exakten Wissenschaften: W. Engelmann.
Newton, Isaac. 1961. The Correspondence of Isaac Newton, Volume III (1688–1694). Edited by Herbert W. Turnbull, Cambridge: Cambridge University Press.
Newton, Isaac. 1964–1967. The Mathematical Works of Isaac Newton. Edited by Derek T. Whiteside. 2 vols. New York: Johnson Reprint Corp.
Newton, Isaac. 1967–1981. The Mathematical Papers of Isaac Newton. Edited by Derek T. Whiteside. 8 vols. Cambridge: Cambridge University Press.
Newton, Isaac. 1999. The Principia: Mathematical Principles of Natural Philosophy … Preceded by a Guide to Newton’s Principia by I. Bernard Cohen. Translated by I. Bernard Cohen and Anne Whitman, assisted by Julia Budenz. Berkeley: University of California Press.
Roche, John. 1998. The Mathematics of Measurement: A Critical History. London: The Athlone Press.
Scriba, Christoph J. 1961. Zur Lösung des 2. Debeauneschen Problems durch Descartes: Ein Abschnitt aus der Frühgeschichte der inversen Tangentenaufgaben. Archive for History of Exact Sciences 1(4): 406–419.
Scriba, Christoph J. 1964. The Inverse Method of Tangents: A Dialogue between Leibniz and Newton (1675–1677). Archive for History of Exact Sciences 2(2): 113–137.
Venuti, Lawrence. 1995. The Translator’s Invisibility: A History of Translation. London and New York: Routledge.
Acknowledgements
I would like to thank Marius Stan for his correspondence and competent editing. Toni Malet for sharing many insights on Newtonian mathematics with me. This research was funded by the Department of Philosophy “Piero Martinetti” of the University of Milan under the Project “Departments of Excellence 2018–2022” awarded by the Ministry of Education, University and Research (MIUR).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Guicciardini, N. (2023). Newton on Quadratures: A Brief Outline. In: Stan, M., Smeenk, C. (eds) Theory, Evidence, Data: Themes from George E. Smith. Boston Studies in the Philosophy and History of Science, vol 343. Springer, Cham. https://doi.org/10.1007/978-3-031-41041-3_10
Download citation
DOI: https://doi.org/10.1007/978-3-031-41041-3_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-41040-6
Online ISBN: 978-3-031-41041-3
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)