The History of Mathematics: A Brief Course

362 12. MODERN GEOMETRIES D A C Â FIGURE 6. Menelaus' theorem for a plane triangle. Although Desargues' terminology is very diff ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 363 FIGURE 8. Desargues' theorem for two triangles in the same plane. {A", C, Â}, {B", A, ...

364 12. MODERN GEOMETRIES Having received a copy of this work from Marin Mersenne, Descartes took the word draft literally and r ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 365 Q FIGURE 9. Pascal's theorem (third lemma), the two directions can be described as M " ...

366 12. MODERN GEOMETRIES FIGURE 10. Newton's degree-preserving projection. FIGURE 11. Brianchon's theorem for a circle. of oppo ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 367 and declared a military secret. He wrote a book on descriptive geometry and one on the ...

368 12. MODERN GEOMETRIES building more and more intricate figures by allowing simpler ones to combine ge- ometrically was novel ...

PROJECTIVE AND DESCRIPTIVE GEOMETRY 369 Mobius is best remembered for two concepts, the Mobius transformation, and the Mobius ...

370 12. MODERN GEOMETRIES to choose ì so that there will be a seventh common zero. With that choice of ì, the polynomial t(x) mu ...

DIFFERENTIAL GEOMETRY 371 2.11. Arthur Cayley. Homogeneous coordinates provided important invariants and covariants^19 in proj ...

372 12 MODERN GEOMETRIES FIGL TB2JL Huygens' cycloidal pendulum, from his Horologium oscillatoriurn. © Stock Montage, Inc. requi ...

Diirer's mechanical drawing methods—does not really work any better than the standard methods.^21 Referring to Fig. 12, in which ...

374 12. MODERN GEOMETRIES a much more complicated problem. It was, however, once again a cycloid. This evolute made it possible ...

DIFFERENTIAL GEOMETRY^375 In the infinitely small parts of a curve it is possible to consider not only the direction or inclin ...

376 12. MODERN GEOMETRIES functions (t,u) t-* (x(t, u),y(t, u), z(t, u)). Quantities such as curvature and area are then express ...

DIFFERENTIAL GEOMETRY 377 in geometric astronomy. This unit sphere is used in mapping a curved surface by taking the normal li ...

378 12. MODERN GEOMETRIES Adpdq, where Ä = sjEG - F^2. Gauss denoted the analogous expression for the mapping (p,q) ^ (X(p,q),Y( ...

DIFFERENTIAL GEOMETRY 379 // a curved surface is mapped on any other surface, the measure of curvature at each point remains u ...

380 L2. MODERN GEOMETRIES of the two principal curvatures derived by Euler would be the same for any two mutually perpendicular ...

DIFFERENTIAL GEOMETRY 381 when applied to vectors in physics. Grassmann remarked that parentheses have no effect on the outer ...