By Nicholas Lobachevski, George Bruce Halsted
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Extra info for Geometrical Researches on the Theory of Parallels
5. For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique. The fifth axiom became known as the “parallel postulate,” since it provided a basis for the uniqueness of parallel lines. (It also attracted great interest because it seemed less intuitive or self-evident than the others. ) All five axioms provided the basis for numerous provable statements, or theorems, on which Euclid built his geometry. The following sections briefly explain the most important theorems of Euclidean plane and solid geometry.
Supposing this decorated window to be the canvas, Alberti interpreted the painting-to-be as the projection of the scene in life onto a vertical plane cutting the visual pyramid. A distinctive feature of his system was the “point at infinity” at which parallel lines in the painting appear to converge. Alberti’s procedure, as developed by Piero della Francesca (c. 1410–92) and Albrecht Dürer (1471–1528), was used by many artists who wished to render perspective persuasively. At the same time, cartographers tried various projections of the sphere to accommodate the record of geographical discoveries that began in the mid-15th century with Portuguese exploration of the west coast of Africa.
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. Greek society could support the transformation of geometry from a practical art to a deductive science.
Geometrical Researches on the Theory of Parallels by Nicholas Lobachevski, George Bruce Halsted