By Eisenhart L. P.
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Non-Euclidean Geometries The Enlightenment was not so preoccupied with analysis as to completely ignore the problem of Euclid’s fifth postulate. In 1733 Girolamo Saccheri (1667–1733), a Jesuit professor of mathematics at the University of Pavia, Italy, substantially advanced the age-old discussion by setting forth the alternatives in great clarity and detail before declaring that he had “cleared Euclid of every defect” (Euclides ab Omni Naevo Vindicatus, 1733). ” Saccheri took up the quadrilateral of Omar Khayyam (1048–1131), who started with two parallel lines AB and DC, formed the sides by drawing lines AD and BC perpendicular to AB, and then considered three hypotheses for the internal angles at C and D: to be right, obtuse, or acute.
Their compulsion and the multiplication of theorems it produced fit perfectly with the endless questioning of Socrates and the uncompromising logic of Aristotle. Perhaps the origin, and certainly the exercise, of the peculiarly Greek method of mathematical proof should be sought in the same social setting that gave rise to the practice of philosophy—that is, the Greek polis. There citizens learned the skills of a governing class, and the wealthier among them enjoyed the leisure to engage their minds as they pleased, however useless the result, while slaves attended to the necessities of life.
The resultant area is then given by (a + ε)(b − ε). Fermat observed what Kepler had perceived earlier in investigating the most useful shapes for wine casks, that near its maximum (or minimum) a quantity scarcely changes as the variables on which it depends alter slightly. On this principle, Fermat equated the areas ab and (a + ε) (b − ε) to obtain the stationary values: ab = ab − εa + εb − ε2. By canceling the common term ab, dividing by ε, and then setting ε at zero, Fermat had his well-known answer, a = b.
Affine Geometries of Paths Possessing an Invariant Integral by Eisenhart L. P.