Mathematics begins with counting. It is unreasonable, however, to assume that early counting was mathematics. We can say that mathematics began only when some record of counting was preserved and, therefore, some representation of numbers occurred.
In Babylonia, mathematics developed from 2000 B.C. onward. Previously, a system of signed numbers with ordinal numbers evolved over a long period with a numerical base of 60. It allowed for the representation of arbitrarily large numbers and fractions, and so proved to be the basis for a more powerful development of mathematics.
The Babylonian basis of mathematics was inherited by the Greeks and the independent development of the Greeks began about 450 B.C. Zeno’s paradoxes of Eleas led to the atomic theory of Democritus. A more precise formulation of the concepts led to the realization that rational numbers were not sufficient to measure all lengths. A geometrical formulation of irrational numbers emerged. Research in the field led to a form of integration.
The theory of conic sections shows a high point in Apollonius’ pure mathematical research. Further mathematical discoveries were driven by astronomy, such as the study of trigonometry.
Great advances in mathematics in Europe began again in the early 16th century with Pacioli, followed by Cardan, Tartaglia, and Ferrari with algebraic solutions of cubic and quartic equations. Copernicus and Galileo revolutionized the application of mathematics to the study of the universe.
Progress in algebra had a great psychological effect and enthusiasm for mathematical research, particularly research in algebra, spread from Italy to Stevin in Belgium and Viets in France.
In the seventeenth century Napier, Briggs, and others greatly expanded the possibilities of mathematics as a computational science by discovering logarithms. Cavalieri made progress in computation with his infinitesimal methods, and Descartes added the power of algebraic methods to geometry.
The most important mathematician of the 18th century was Euler, who, in addition to his work in a wide range of mathematical fields, was to invent two new branches, namely, the calculus of variations and differential geometry. Euler was also instrumental in advancing the research in number theory begun so effectively by Fermat.
The 19th century was marked by rapid progress. Fourier’s work on heat was of fundamental importance. In the field of geometry, Plücker produced seminal works on analytic geometry and Steiner on synthetic geometry.
Non-Euclidean geometry, developed by Lobachevsky and Bolyai, led to the characterization of Riemann geometry. Gauss, considered one of the greatest mathematicians of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionize this subject. He also made major contributions to astronomy and magnetism.