Compass equivalence theorem

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]

Construction

The following construction and proof of correctness are given by Euclid in his Elements.^[2] 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,^[1] 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, $AF = BC$ .

Alternative construction without straightedge

It is possible to prove compass equivalence without the use of the straightedge. This justifies the use of "fixed compass" moves (constructing a circle of a given radius at a different location) in proofs of the Mohr–Mascheroni theorem, which states that any construction possible with straightedge and compass can be accomplished with compass alone.

Given points $A$ , $B$ , and $C$ , construct a circle centered at $A$ with the radius $BC$ , using only a collapsing compass and no straightedge.

Draw a circle centered at $A$ and passing through $B$ and vice versa (the blue circles). They will intersect at points $D$ and $D'$ .
Draw circles through $C$ with centers at $D$ and $D'$ (the red circles). Label their other intersection $E$ .
Draw a circle (the green circle) with center $A$ passing through $E$ . This is the required circle.^[3]^[4]

There are several proofs of the correctness of this construction and it is often left as an exercise for the reader.^[3]^[4] Here is a modern one using transformations.

The line $DD'$ is the perpendicular bisector of $AB$ . Thus $A$ is the reflection of $B$ through line $DD'$ .
By construction, $E$ is the reflection of $C$ through line $DD'$ .
Since reflection is an isometry, it follows that $AE = BC$ as desired.

References

^ ^a ^b 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.
^ Heath, Thomas L. (1956) [1925]. The Thirteen Books of Euclid's Elements (2nd ed.). New York: Dover Publications. p. 244. ISBN 0-486-60088-2.
^ ^a ^b Eves, Howard (1963), A survey of Geometry (Vol. I), Allyn Bacon, p. 185
^ ^a ^b Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 212, ISBN 0-534-35188-3

[Toussaint-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.

[2] Heath, Thomas L. (1956) [1925]. The Thirteen Books of Euclid's Elements (2nd ed.). New York: Dover Publications. p. 244. ISBN 0-486-60088-2.

[Eves-3] Eves, Howard (1963), A survey of Geometry (Vol. I), Allyn Bacon, p. 185

[Smart-4] Smart, James R. (1997), Modern Geometries (5th ed.), Brooks/Cole, p. 212, ISBN 0-534-35188-3

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