Customer Review

January 18, 2007
Desargues's fundamental invariant is that of six points being in involution. Like the cross-ratio, it has harmonic division as a special case. C and D divide AB harmonically when AC/CB=AD/BD. When D is at infinity, C must be the midpoint of AB and projective invariance can then be used to find any fourth harmonic point as follows. Suppose we have ACB on a line and we are looking for the missing D. Draw any line Acb through A such that Ac=cb. Draw cC and bB, and make their intersection O the eye point for a projection. The fourth harmonic point of Acb is at infinity, so its image---the intersection with ACB of the parallel of Acb through O---will be the sought fourth harmonic point D. This is one example of how Desargues' projective ideas provided simpler constructions for classical problems, which is what gave his work credibility. Naturally, such simplifications occur especially in the theory of conic sections. Consider for example his constructions of the polar of a point with respect to a conic. Draw an X through the point. It will meet the conic in four points. Draw the line through the two points on the left and the line through the two points on the right. These two lines will meet in a point A. Also draw the line through the two upper points and the line through the two lower points. These two lines will meet in a point B. AB is the sought polar. Desargues's theorem of triangles in perspective is not in here (although some people mistakenly believe it is; e.g., Burton, The History of Mathematics, p. 378). It appears that this theorem was never published by Desargues himself (see chapter VIII, which includes a translation of the first known publication of the result, by Bosse).
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