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# A treatise on the analytical geometry by Casey J.

By Casey J.

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Additional info for A treatise on the analytical geometry

Sample text

Desargues’s theorem allows their interchange. So, as Steiner showed, does Pascal’s theorem that the three points of intersection of the opposite sides of a hexagon inscribed in a conic lie on a line. Thus, the lines joining the opposite vertices of a hexagon circumscribed about a conic meet in a point. Poncelet’s followers realized that they were hampering themselves, and disguising the true fundamentality of projective geometry, by retaining the concept of length and congruence in their formulations, since projections do not usually preserve them.

Books XI–XIII deal with solids: XI contains theorems about the intersection of planes and of lines and planes and theorems about the volumes of parallelepipeds (solids with parallel parallelograms as opposite faces); XII applies the method of exhaustion introduced by Eudoxus to the volumes of solid figures, including the sphere; XIII, a three-dimensional analogue to Book IV, describes the Platonic solids. Among the jewels in Book 31 7 The Britannica Guide to Geometry 7 XII is a proof of the recipe used by the Egyptians for the volume of a pyramid.

Transformation French Circles Desargues was a member of intersecting circles of 17thcentury French mathematicians worthy of Plato’s Academy of the 4th century BCE or Baghdad’s House of Wisdom of the 9th century CE. They included René Descartes (1596–1650) and Pierre de Fermat (1601–65), inventors of analytic geometry; Gilles Personne de Roberval (1602–75), a pioneer in the development of the calculus; and Blaise Pascal (1623–62), a contributor to the calculus and an exponent of the principles set forth by Desargues.