Alan J. CainSpencer expressed enthusiasm for mathematics, and was proud enough of a rather slender theorem he proved in 1840 to reprint the paper as an appendix to his autobiography. (The result had actually been known since the mid-18th century.)
But his geometrical knowledge was actually quite limited, and the statement he gave of Monge’s theorem was carelessly imprecise.
Further, Gaspard Monge’s (1746–1818) original proof is simple and explains the theorem:
Imagine the circles are great circles of three spheres. Then there are two planes that are tangent with all three spheres. (See attached image.) These planes are also in contact with the cones defined by each pair of spheres, and the tangent lines are where these cones pass through the original plane. Thus both these planes contain the apices of the three cones, which are the intersections of pairs of the tangents. That is, the intersection points lie in the intersection of the two planes, which is a straight line.
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