Mathematics

doi:10.1017/CHOL9780521572439.014

13 - Mathematics

from Part II - Disciplines

Published online by Cambridge University Press: 28 March 2008

Craig Fraser

Edited by

Roy Porter

Show author details

Roy Porter: Affiliation:
Wellcome Institute for the History of Medicine, University College London

Book contents

Get access

Summary

Considered broadly, mathematical activity in the eighteenth century was characterized by a strong emphasis on analysis and mechanics. The great advances occurred in the development of calculus-related parts of mathematics and in the detailed elaboration of the program of inertial mechanics founded during the Scientific Revolution. There were other mathematical developments of note – in the theory of equations, number theory, probability and statistics, and geometry – but none of them reached anything like the depth and scope attained in analysis and mechanics.

The close relationship between mathematics and mechanics had a basis that extended deep into Enlightenment thought. In the Preliminary Discourse to the famous French Encyclopédie, Jean d’ Alembert distinguished between “pure” mathematics (geometry, arithmetic, algebra, calculus) and “mixed” mathematics (mechanics, geometrical astronomy, optics, art of conjecturing). He classified mathematics more generally as a “science of nature” and separated it from logic, a “science of man.” An internalized and critical spirit of inquiry, associated with the invention of new mathematical structures (for example, non-commutative algebra, non-Euclidean geometry, logic, set theory), represents characteristics of modern mathematics that would emerge only in the next century.

Although there were several notable British mathematicians of the period–Abraham De Moivre, James Stirling, Brook Taylor, and Colin Maclaurin among them – the major lines of mathematical production occurred on the Continent, a trend that intensified as the century developed. Leadership was provided by a relatively small number of energetic figures: Jakob, Johann, and Daniel Bernoulli, Jakob Hermann, Leonhard Euler, Alexis Clairaut, Jean d’Alembert, Johann Heinrich Lambert, Joseph Louis Lagrange, Adrien Marie Legendre, and Pierre Simon Laplace.

Type: Chapter
Information: The Cambridge History of Science , pp. 305 - 327

DOI: https://doi.org/10.1017/CHOL9780521572439.014 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bos, Henk J., “Differentials, Higher-Order Differentials and the Derivative in the Leibnizian Calculus,” Archive for History of Exact Sciences, (1974).CrossRef Google Scholar

Boyer, Carl, A History of the Calculus and Its Conceptual Development (New York: Dover Publications, Inc., 1959; originally published by Hafner Publishing Company in 1949 under the title “The Concepts of the Calculus, A Critical and Historical Discussion of the Derivative and the Integral”).Google Scholar

Demidov, Sergei, “Création et développement de la théorie des équations différentielles aux dérivées partielles dans les travaux de J. d’Alembert,” Revue d’Histoire des Sciences 35/1 (1982).Google Scholar

Engelsman, Steven, “D’Alembert et les Équations aux Dérivées Partielles,” Dix-Huitième Siècle, 16 (1984).CrossRef Google Scholar

Engelsman, Steven B., Families of Curves and the Origins of Partial Differentiation (Amsterdam: North-Holland, 1984).Google Scholar

Euler, , “De infinitis curvis eiusdem generis seu methodus inveniendi aequationes pro infinitis curvis eiusdem generis,” Commentarii Academiae Scientiarum Petropolitanae 7 1734–1735 (1740).Google Scholar

Euler, , “De la controverse entre Mrs. Leibniz et Bernoulli sur les logarithmes des nombres negatifs et imaginaires, Mémoires de l’académie des sciences de Berlin 5 (1749), (1751);Google Scholar

Euler, , Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solution problematis isoperimetrici lattisimo sensu accepti (Lausanne, 1744).Google Scholar

Fisch, Menachem, “‘The Emergency Which Has Arrived’: The Problematic History of Nineteenth-Century British Algebra – a Programmatic Outline,” British Journal for the History of Science, 27 (1994).CrossRef Google Scholar

Frängsmyr, T. et al. (eds.), The Quantifying Spirit in the 18th Century (Berkeley: University of California Press, 1990).Google Scholar

Fraser, Craig, “The Origins of Euler’s Variational Calculus,” Archive for History of Exact Sciences, 47 (1994)CrossRef Google Scholar

FraserCraig, , “D’Alembert’s Principle: The Original Formulation and Application in Jean D’Alembert’s Traité de Dynamique (1743),” Centaurus, 28 (1985).Google Scholar

Fraser, Craig, “The Calculus as Algebraic Analysis: Some Observations on Mathematical Analysis in the 18th Century,” Archive for History of Exact Sciences, 39 (1989).Google Scholar

Fraser, Craig, “Lagrange’s Analytical Mathematics, Its Cartesian Origins and Reception in Comte’s Positive Philosophy,” Studies in the History and Philosophy of Science, 21 (1990).CrossRef Google Scholar

Fraser, Craig, “Joseph Louis Lagrange’s Algebraic Vision of the Calculus,” Historia Mathematica, 14 (1987).CrossRef Google Scholar

Fraser, Craig, “J. L. Lagrange’s Changing Approach to the Foundations of the Calculus of Variations,” Archive for History of Exact Sciences, 32 (1985)CrossRef Google Scholar

Fraser, , “Isoperimetric Problems in the Variational Calculus of Euler and Lagrange,” Historia Mathematica, 19 (1992).CrossRef Google Scholar

Fraser, , “The Background to and Early Emergence of Euler’s Analysis,” in Otte, M. and Panza, M. (eds.), Analysis and Synthesis in Mathematics History and Philosophy, Boston Studies in the Philosophy of Science, vol. 196 (Dordrecht: Kluwer, 1997).Google Scholar

Fraser, , “The Calculus as Algebraic Analysis,” and Marco Panza, “Concept of Function, between Quantity and Form, in the 18th Century,” in Jahnke, H. Niels et al. (eds.), History of Mathematics and Education: Ideas and Experiences (Göttingen: Vandenhoeck & Ruprecht, 1996).Google Scholar

Funkhouser, H. Gray, “Historical Development of the Graphical Representation of Statistical Data,” Osiris, 3 (1937)CrossRef Google Scholar

Gillies, Donald (ed.), Revolutions in Mathematics (New York: Oxford University Press, 1992).Google Scholar

Goldstine, Herman H., A History of the Calculus of Variations from the 17th through the 19th Century (New York: Springer-Verlag, 1980)CrossRef Google Scholar

Grabiner, Judith V., “Is Mathematical Truth Time-Dependent?” American Mathematical Monthly, 81 (1974)Google Scholar

Grabiner, Judith V., The Origins of Cauchy’s Rigorous Calculus (Cambridge, MA: MIT Press, 1981)Google Scholar

Grattan-Guinness, Ivor, The Development of the Foundations of Mathematical Analysis from Euler to Riemann (Cambridge, MA: MIT Press, 1970)Google Scholar

Grimsley, R. G., Jean d’Alembert (1717–1783) (Oxford: Clarendon Press, 1963).Google Scholar

Guicciardini, Niccolò, The Development of Newtonian Calculus in Britain, 1700–1800 (Cambridge University Press, 1989).CrossRef Google Scholar

Hahn, Roger, The Anatomy of a Scientific Institution: The Paris Academy of Science, 1666–1803 (Berkeley: University of California Press, 1971).Google Scholar

Jahnke, H. Niels, Mathematik und Bildung in der Humboldtschen Reform, volume 8 of the series Studien zur Wissenschafts-, Sozial-und Bildungsgeschichte der Mathematik,” eds. Otte, Michael, Schneider, Ivo, and Steiner, Hans-Georg (Göttingen: Vandenhoeck & Ruprecht, 1990).Google Scholar

Katz, Victor J., “Calculus of the Trigonometric Functions,” Historia Mathematica, 14 (1987).CrossRef Google Scholar

Lagrange, , Traité de la résolution des équations numériques de tous les degrés (Paris, 1798).Google Scholar

Langer, Rudolph E., “Fourier Series, The Evolution and Genesis of a Theory,” American Mathematical Monthly, 54, pt. 2 (1947).Google Scholar

McRae, Robert, “Condillac: The Abridgement of All Knowledge in ‘The Same is the Same,’” in The Problem of the Unity of the Sciences: Bacon to Kant (Toronto: University of Toronto Press, 1961).Google Scholar

Novy, L., Origins of Modern Algebra (Leiden: Noordhoff International Publishing, 1973).Google Scholar

Ovaert, J. L., “La thèse de Lagrange et la transformation de l’analyse,” in Houzel, Christian et al. (eds.), Philosophie et Calcul de l’Infini (Paris: Francois Maspero, 1976)Google Scholar

Pycior, Helena, “George Peacock and the British Origins of Symbolical Algebra,” Historia Mathematica, 8 (1981)CrossRef Google Scholar

Richards, Joan L., “The Art and the Science of British Algebra: A Study in the Perception of Mathematical Truth,” Historia Mathematica, 7 (1980)CrossRef Google Scholar

Stigler, Stephen M., The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, MA: Harvard University Press, 1986).Google Scholar

Taton, René, “Inventaire chronologique de l’oeuvre de Lagrange,” Revue d’Histoire des Sciences, 26 (1974).Google Scholar

Tilling, Laura, “Early Experimental Graphs,” British Journal for the History of Science, 8 (1975).CrossRef Google Scholar

Varignon, Pierre, “Nouvelle formation de spirales beaucoup plus différentes entr’elles que tout ce qu’on peut imaginer d’autres courbes quelconques à l’infini; avec les touchantes, les quadratures, les déroulemens, & les longueurs de quelques-unes de ces spirales qu’on donne seulement ici pour éxemples de cette formation générale,” Histoire de l’Académie royale des sciences avec les mémoires de mathématique et de physique tirés des registres de cette Académie 1704 (Paris, 1706).Google Scholar

Youschkevitch, A. P., “The Concept of the Function up to the Middle of the 19th Century,” Archive for History of Exact Sciences, 16 (1976)Google Scholar

Book contents

13 - Mathematics

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive