Compass equivalence theorem

In a nutshell

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.

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Construction

The following construction and proof of correctness are given by Euclid in his Elements. Although there appear to be several cases in Euclid's treatment, depending upon choices made when interpreting ambiguous instructions, they all lead to the same conclusion, and so, specific choices are given below.

Given points A, B, and C, construct a circle centered at A with radius the length of BC (that is, equivalent to the solid green circle, but centered at A).

Draw a circle centered at A and passing through B and vice versa (the red circles). They will intersect at point D and form the equilateral triangle △ABD.
Extend DB past B and find the intersection of DB and the circle ◯BC, labeled E.
Create a circle centered at D and passing through E (the blue circle).
Extend DA past A and find the intersection of DA and the circle ◯DE, labeled F.
Construct a circle centered at A and passing through F (the dotted green circle)
Because △ADB is an equilateral triangle, DA = DB.
Because E and F are on a circle around D, DE = DF.
Therefore, AF = BE.
Because E is on the circle ◯BC, BE = BC.
Therefore, AF = BC.

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Euclidean spaceFundamental space of geometry Cartesian coordinate systemCoordinate system using perpendicular axes SphereSet of points equidistant from a center DimensionProperty of a mathematical space Topological spaceMathematical space with a notion of closeness ManifoldTopological space that locally resembles Euclidean space

Construction

Alternative construction without straightedge

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