![]() ![]() Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics. , sets furnished with various structures having no classical analogues. This is the part of Geometry on which the oldest Mathematical Book in existence, namely, Euclid’s Elements, is writ-ten, and is the subject of the present volume. Book 7 deals with elementary number theory: e.g. the right line, and the circle, is the introduction to Geometry, of which it forms an extensive and important department. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. Book 6 applies the theory of proportion to plane geometry, and contains theorems on similar gures. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. " During the seventies of the last century there occurred another scientific revolution. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. It is safe to say that it was a turning point in the history of all mathematics. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself.
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