In geometry, the compass equivalence theorem is an important statement in compass and straightedge constructions. The tool advocated by Plato in these constructions is a divider or collapsing compass, that is, a compass that "collapses" whenever it is lifted from a page, so that it may not be directly used to transfer distances. The modern compass with its fixable aperture can be used to transfer distances directly and so appears to be a more powerful instrument. However, the compass equivalence theorem states that any construction via a "modern compass" may be attained with a collapsing compass. This can be shown by establishing that with a collapsing compass, given a circle in the plane, it is possible to construct another circle of equal radius, centered at any given point on the plane. This theorem is Proposition II of Book I of Euclid's Elements. The proof of this theorem has had a chequered history.[1]
- ^ Toussaint, Godfried T. (January 1993). "A New Look at Euclid's Second Proposition" (PDF). The Mathematical Intelligencer. 15 (3). Springer US: 12–24. doi:10.1007/bf03024252. eISSN 1866-7414. ISSN 0343-6993. S2CID 26811463.