The History of Mathematics: A Brief Course

382 12. MODERN GEOMETRIES ways into another, the various determinations of it form a continuous or discrete manifold. He noted t ...

DIFFERENTIAL GEOMETRY 383 Differential geometry and physics. The work of Grassmann and Riemann was to have a powerful impact o ...

384 12. MODERN GEOMETRIES whether a surface can be synthesized from any six functions regarded as the coef- ficients of these fo ...

TOPOLOGY 385 Jacobi (see Klein, 1926, Vol. 2, p. 190), Beltrami arrived at the operator where, with the notation slightly mode ...

386 12. MODERN GEOMETRIES was available. The word topology first appeared in the title of his 1848 book Vorstu- dien zur Topolog ...

TOPOLOGY 387 in which æ — w^2 — 0, every complex number æ = a + bi has two distinct complex square roots: w = ±(u + IV), where ...

388 12. MODERN GEOMETRIES y a b / L yl á ï Sphere Torus FIGURE 14. Left: The sphere, regarded as a square with edges identified, ...

TOPOLOGY^389 Β Ε D Β' A' A FIGURE 15. Left: the projective plane triangulated and cut open. If two opposite edges with corresp ...

390 12. MODERN GEOMETRIES homology group. Moreover, he noted, while the order in which the cycles in a chain were traversed was ...

TOPOLOGY 391 Compactness. Another basic concept of point-set topology is that of compactness. This concept is needed to make t ...

392 12. MODERN GEOMETRIES including Henri Lebesgue (1875 1941), and is now generally known as the Heine- Borel theorem. The word ...

QUESTIONS AND PROBLEMS^393 The first part of the book is an exposition of abstract set theory as it existed at the time, includi ...

394 12. MODERN GEOMETRIES 12.4. Deduce Brianchon's theorem for a general conic from the special case of a circle. How do you int ...

QUESTIONS AND PROBLEMS 395 (xg, J/9), the ninth point of intersection of two cubic curves through the other eight points, for wh ...

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Part 5. Algebra The History of Mathematics: A Brief Course, Second Edition by Roger Cooke Copyright © 200 5 John Wiley & Son ...

Occasionally, a practical problem arises in which it is necessary to invert a sequence of arithmetic operations. That is, we kno ...

Chapter 13. Problems Leading to Algebra Algebra suffers from a motivational problem. Examples of the useless artificiality of mo ...

400 13. PROBLEMS LEADING TO ALGEBRA notion that multiplication is distributive over addition (another way of saying that proport ...

MESOPOTAMIA 401 2. Mesopotamia If we interpret Mesopotamian algebra in our own terms, we can credit the math- ematicians of th ...