A History of Mathematics- From Mesopotamia to Modernity

190 A History ofMathematics A B l
a l b Fig. 1The figure for postulate 5. modern mathematics was perhaps a handicap to evaluati ...

Geometries andSpace 191 Right angle Obtuse angle Acute angle Fig. 2Saccheri’s three hypotheses. HRA corresponds to Euclid’s geom ...

192 A History ofMathematics was achieved in stages through the later nineteenth century by Beltrami, Klein, and Poincaré among o ...

Geometries andSpace 193 We have given the traditional outline of the story, and it is easy to criticize. An up-to-date, serious ...

194 A History ofMathematics 2. First problem: the postulate Let us return to Euclid’s postulate 5.^3 It is worth noting that pos ...

Geometries andSpace 195 geometers are, or should be studying ‘forms’ rather than things in the world makes more sense than appea ...

196 A History ofMathematics while Proclus agreed that to prove the postulate one would need extra definitions and theorems, he d ...

Geometries andSpace 197 3. Space and infinity As lines (so loves) oblique may well Themselves in every corner greet: But ours so ...

198 A History ofMathematics Fig. 6Classical descriptive geometry as it is still practised today. The three projections are unite ...

Geometries andSpace 199 XY D B A C Fig. 7Traditional perspective generates the ‘ideal line’ at infinity XY of projective geometr ...

200 A History ofMathematics coincidence, it has always been dismissed as the HAA has not, as clearly contrary to what is obvious ...

Geometries andSpace 201 A D B G Fig. 8Lambert’s quadrilateral. was 80◦: then one would have the absolute measure of a Paris foot ...

202 A History ofMathematics This means that you are adopting what Saccheri called the hypothesis of the acute angle. The simple ...

Geometries andSpace 203 are congruent, of which more later. Staying for the moment with ‘internal’ factors, we can see an obviou ...

204 A History ofMathematics geometry addressed—what sort of a world do we live in?—was not considered. In this sense, the new ge ...

Geometries andSpace 205 p l July l
January b g a Fig. 10The parallax of a star.pis the width of the Earth’s orbit, and the line ...

206 A History ofMathematics I discovered [about 1890] that in addition to Euclidean geometry there were various non-Euclidean ge ...

Geometries andSpace 207 a strong belief that the way to make mathematics ‘safe’ was through the construction of axiom systems. I ...

208 A History ofMathematics BC E A F D G Fig. 11The picture for Euclid I.16. B C D E A F Fig. 12A ‘large’ triangle on a sphere, ...

Geometries andSpace 209 C A B R R b O c a Fig. 13Solving a spherical triangle. Appendix B. The formulae of spherical and hyperbo ...