The four-bar linkage defines a complete quadrilateral—a figure with extensive elementary projective properties. The undertaking is to search out and discover evidences or indicia of projective interrelationships in the geometry of the four-bar, such as concurrencies of lines, collineations of points, constancies of cross-ratios, harmonic divisions, tangencies of lines and curves, and others, toward the practical application of such properties in four-bar motion. Several of these have indeed been discovered and proved, each worth-while in its own right and each expressible as an instantaneous projective relationship between members of a simply-defined family of linkages. The most startling and applicable of the properties is the collineation of the centers of the inflection circles of the connecting-rod motions of the defined family.

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