The History of Mathematics: A Brief Course

342 11. POST-EUCLIDEAN GEOMETRY to become negative, which was surely impossible. Unfortunately, the possibility of repeated doub ...

NON-EUCLIDEAN GEOMETRY 343 (1784-1846) in 1829 that he had been thinking about the foundations of geome- try off and on for ne ...

344 11. POST-EUCLIDEAN GEOMETRY Lobachevskii's geometry. Lobachevskii connected the parts of a hyperbolic triangle through his f ...

NON-EUCLIDEAN GEOMETRY^345 Thus, ironically, the acceptance of the logical consistency of hyperbolic geom- etry was accompanie ...

346 11. POST-EUCLIDEAN GEOMETRY pseudonym Lobachevskii, since "it is hardly likely that two or even three people knowing nothing ...

NON-EUCLIDEAN GEOMETRY 347 FIGURE 12. Projection of the Lobachevskii-Bolyai plane onto the interior of a Euclidean disk. FIGUR ...

348 11. POST-EUCLIDEAN GEOMETRY three-dimensional Euclidean space that was a perfect model for the Lobachevskii- Bolyai plane, i ...

QUESTIONS AND PROBLEMS 349 FIGURE 14. Greek use of a fundamental inequality. Left: from Euclid's Optics. Right: from Ptolemy's ...

350 11. POST-EUCLIDEAN GEOMETRY that is, the intersections of the sphere with planes passing through its center. Let one "line" ...

Chapter 12. Modern Geometries In geometry, as in number theory, the seventeenth century represents a break with the past. The tw ...

352 12. MODERN GEOMETRIES to look at the Æ as a variable and the A and Å as constants, exactly the reverse of what Fermat intend ...

ANALYTIC AND ALGEBRAIC GEOMETRY 353 AC Ε G FIGURE 1. Descartes' linkage for generating curves. The curve x4n = a^2 (x^2 + y^2 ...

354 12. MODERN GEOMETRIES FIGURE 2. Left: AB = 1, so that BE = BCBD. Right: FG = 1, so that GI = y/GH. the problem that Pappus r ...

ANALYTIC AND ALGEBRAIC GEOMETRY 355 FIGURE 3. Descartes' analysis of the ç-line locus problem. general: "the most brilliant vi ...

356 12. MODERN GEOMETRIES cubic curves generally intersect in nine points, so that some sets of nine points do not determine the ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 357 A circle seen in perspective is an ellipse. C both between A and D and C between  and ...

358 12. MODERN GEOMETRIES FIGURE 4. Central projections preserve tangency, midpoints, and (usually) parallelism. treatise on ari ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 359 FIGURE 5. Diirer's paper model of a truncated icosahedron. In those days before photog ...

360 12. MODERN GEOMETRIES Two modern applications of the truncated icosahedron: a mole- cule of buckminsterfullerene ("buckyball ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 361 One of Diirer's devices for producing an accurate painting. The artist's assistant at ...