Painting -Centers of Similitude (La Hire)

Two circles or other similar figures can be placed such that a line drawn from some fixed point to a point of one of them passes through a point on the other, such that the ratio of the distances from the fixed point to the two points is always the same. The fixed point is called the center of similitude. The circles shown in this painting have two centers of similitude, one between the circles and one to the right (the center of similitude between the circles is shown). Crockett Johnson apparently based his painting on a diagram from the book College Geometry by Nathan Altshiller Court (1964 printing). This diagram is annotated in his copy of the book. In the figure, the larger circle has center A, the smaller circle has center B, and the centers of similitude are the points S and S'. S is called the external center of similitude and S' is the internal center of similitude. The painting suggests several properties of centers of similitude. For example, lines joining corresponding endpoints of parallel diameters of the two circles, such as TT' in the figure, would meet at the external center of similitude. Lines joining opposite endpoints meet at the internal center of similitude.

This painting emphasizes the presence of the two circles and line segments relating to centers of similitude, but not the centers themselves. Indeed, the painting is too narrow to include the external center of similitude.

Some properties of centers of similitude were known to the Greeks. More extensive theorems were developed by the mathematician Gaspard Monge (1746–1818). It is not entirely clear why Crockett Johnson associated the painting with the artist and mathematician Phillipe de la Hire (1640–1718). A bibliographic note in the relevant section of Court reads: LHr., p. 42, rem. 8. However, Court was referring to an 1809 book by Simon A. J. LHuiler on the elements of analytic geometry.

This oil painting on masonite is #14 in Crockett Johnson's series. It was completed in 1966 and is signed: CJ66.

References: R. J. Archibald, "Centers of Similitude of Circles," American Mathematical Monthly, 22, #1 (1915), pp. 6–12; unpublished notes of J. B. Stroud.

Currently not on view