The Euclidean tools for carrying out geometric constructions were straightedge and compasses. We need to distinguish between straightedge and ruler. A ruler has markings on it which permit its use for transporting lengths from one place to another; moreover, it has two edges permitting certain parallel lines to be drawn. A Straightedge, on the other hand, simply enables us to join two given points by a straight line. Euclidean compasses also are to be distinguished from dividers; compasses may be used only to draw circles with a given centre A which pass through another given point B (i.e., of radius AB). The modem practice of such maneuvers as “with centre A, radius CD (a transported radius), construct a circle …” are not in keeping with Euclid's use of compasses; he presumed that they would collapse when either arm was lifted from the page. It turns out, however, that the difference between collapsing and “divider-type” compasses is only apparent, for we shall soon show that Euclid's compasses can achieve any construction executed by its modem counterpart.

We are chiefly concerned in this essay with two major discoveries regarding the equivalence of instruments. In 1797, the Italian geometer Lorenzo Mascheroni showed that any construction which can be camed out with straightedge and compasses can be carried out with compasses alone! Following the suggestion of J. V. Poncelet, Jakob Steiner proved in 1833 that any construction that can be executed with straightedge and compasses can be carried out with straightedge alone, provided that just one circle and its centre are given.