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There are many proofs known, but one of the most attractive is due to the nineteenth century Belgian mathematician Germinal Dandelin. It has the virtue of being perhaps the most geometric in flavour. This proof has much in common with the best known proof of Desargues' Theorem, in that it can be interpreted in terms of a three-dimensional configuration. Both Desargues' Theorem and Pascal's Theorem assert that three points defined as the intersections of certain pairs of lines in a configuration are collinear. One very natural way to see that three points in a plane are collinear is to find a plane other than the given one containing the three given points, which must all therefore lie on the intersection of two planes. For Desargues' Theorem, this plane is simple to construct, for Pascal's not quite so simple.
The three-dimensional proof of Pascal's Theorem is now well known, but seems to have originated with Dandelin.
H. S. M. Coxeter, The Real Projective Plane, Cambridge Press, 1955. [p. 103]
D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, 1965. [pp. 102-106]
G. P. Dandelin, MISSING, .
We include below a translation of this paper into English, as well as a set of pictures to illustrate its argument. Although the basic argument is relatively simple and geometrical in anture, it is not so easy to illustrate it convincingly. It is interesting in this regard to examine critically the figures in Bachmann and Hilbert & Cohn-Vossen accompanying their proofs. Bachmann's figures [on p. 253 and p. 254] are schematic in nature, and while the figures in Hilbert & Cohn-Vossen [pp. 104 & 105] are somewhat more realistic, they do not make the argument as transparent as one might wish.
An English translation of Dandelin's paper
A gallery of pictures to accompany Dandelin's paper (work of Tzu-Pei Chen)