# Mathematics - Unknown Origins, On Greek Mathematics, Traditions Elsewhere, The Wakening Europe From The Twelfth Century

This article comprises a compact survey of the development of mathematics from ancient times until the early twentieth century. The treatment is broadly chronological, and most of it is concerned with Europe.

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It seems unavoidable that mathematical thinking played a role in human theorizing from the start of the race, and in various ways. Arithmetic (as the later branch of mathematics became known) would have been one of them, motivated initially by forming integers in connection with counting. But other branches surely include geometry, linked to the appreciation of line, surface, and space; trigonomet…

The Elements comprised thirteen Books: Books 7–9 dealt with arithmetic, and the others presented basic plane (Books 1–6) and solid (Books 11–13) geometry of rectilinear and circular figures. The extraordinary Book 10 explored properties of ratios of smaller to longer lines, akin to a theory of irrational numbers but again not to be so identified. A notable feature is that Eucl…

Mathematics developed well from antiquity also in the Far East, with distinct traditions in India, China, Japan, Korea, and Vietnam. Arithmetic, geometry, and mechanics were again
prominent; special features include a powerful Chinese method equivalent to solving a system of linear equations, a pretty theory of touching circles in Japanese "temple geometry," and pioneering work on n…

From the decline of the Roman Empire (including Greece) Euclid was quiescent mathematically, though the Carolingian kingdom inspired some work, at least in education. The revival dates from around the late twelfth century, when universities also began to be formed. The major source for mathematics was Latin translations of Greek and Arabic writings (and re-editions of Roman writers, especially Boe…

By the mid-seventeenth century, science had become professionalized enough for some national societies to be instituted, especially the Royal Society of London and the Paris Académie des Sciences. At that time two major mathematicians emerged: Isaac Newton (1642–1727) in Cambridge and Gottfried Wilhelm von Leibniz (1646–1716) in Hanover. Each man invented a version of the differ…

However, Newton's theory was not alone in mechanics. Leibniz and others developed an alternative approach, partly inspired by Descartes, in which the "living forces" (roughly, kinetic energy) of bodies were related to their positions. Gradually this became a theory of living forces converted into "work" (a later term), specified as (force x traversed distance). E…

Laplace published his large book in a new professional and economic situation for science. After the Revolution of 1789 in France, higher education and its institutions there were reformed, with a special emphasis upon engineering. In particular, a new school was created, the École Polytechnique (1794), with leading figures as professors (such as Lagrange) and as examiners (Laplace), and with…

By the 1840s Britain and the Italian and German states were producing quality mathematicians to complement and even rival the French, and new posts were available in universities and engineering colleges everywhere. Among the Germans, two figures stand out. From around 1860 Karl Weierstrass (1815–1897) gave lecture courses on many aspects of real-and complex-variable analysis and parts of m…

A new leader emerged: the German David Hilbert (1862–1943). Work on abstract algebras and the foundations of geometry led him to emphasize the importance of axiomatizing mathematical theories (including the axioms of Euclidean geometry that Euclid had not noticed) and to study their foundations metamathematically. But his mathematical knowledge was vast enough for him to propose twenty-thre…

The amount of mathematical activity has usually increased steadily or even exponentially, and the growth from the mid-twentieth century has been particularly great. For example, the German reviewing journal Zentralblatt Math published at the beginning of the twenty-first century a six-hundred-page quarto volume every two weeks, using a classification of mathematics into sixty-three numbered sectio…

Bottazzini, Umberto. Il flauto di Hilbert. Turin: UTET, 1990. Cajori, Florian. A History of Mathematical Notations. 2 vols. La Salle, Ill., and Chicago: Open Court, 1928–1929. Cantor, Moritz. Vorlesungen über Geschichte der Mathematik. 4 vols. Leipzig, Germany: Teubner, 1899–1908. Classic source of the history of mathematics to 1799. Chabert, Jean-Luc, et al., eds. A History of …

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