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I'm curious to learn more about the history of the introduction of the concept of the point at infinity into mathematics. The sum of my knowledge of the historical aspect is from this paragraph (which may be enough to answer the question but I'd like to know if anyone has anything to add, especially if there is anything prior to Kepler and Desargues):

More than two hundred years before Poncelet, the important concept of a point at infinity occurred independently to the German astronomer Johann Kepler (1571-1630) and the French architect Girard Desargues (1593-1661). Kepler (in his Paralipomena in Vitellionem, 1604) declared that a parabola has two foci, one of which is infinitely distant in both of two opposite directions, and that any point on the curve is joined to this "blind focus" by a line parallel to the axis. Desargues (in his Brouillon project..., 1639) declared that parallel lines "sont entre elles d'une mesme ordonnance dont le but est a distance infinie." (That is, parallel lines have a common end at an infinite distance.) And again, "Quand en un plan, aucun des points d'une droit n'y est a distance finie, cette droit y est a distance finie." (When no point of a line as at a finite distance, the line itself is at an infinite distance). The groundwork was thus laid for Poncelet to derive projective space from ordinary space by postulating a common "line at infinity" for all the planes parallel to a given plane...the emancipation of the subject was carried out by another German, K.G.C. von Staudt (1798-1867), when Felix Klein provided an algebraic foundation for projective geometry in terms of "homogeneous coordinates," which had been discovered independently by K.W. Feuerbach and A.F. Möbius in 1827.

quoted from "Projective Geometry", Coxeter, 1964

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  • $\begingroup$ what a great writer coxeter is. $\endgroup$ – abel Mar 14 '15 at 14:11
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Kepler's ideas about infinity have their source in the work of Nicholas of Cusa (whom Kepler greatly admired), particularly his "bridge of continuity" (a way of "connecting" the finite and the infinite). This ultimately inspired Leibniz to formulate his Law of Continuity. This is discussed further in this article.

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