Mohr–Mascheroni theorem
In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. The result was originally published by Georg Mohr in 1672,[1] but his proof languished in obscurity until 1928.[2][3] The theorem was independently discovered by Lorenzo Mascheroni in 1797.[4]
Proof approach
To prove the theorem, each of the basic constructions of compass and straightedge need to be proven to be doable by compass alone. These are:
- Creating the line through two existing points
- Creating the circle through one point with centre another point
- Creating the point which is the intersection of two existing, non-parallel lines
- Creating the one or two points in the intersection of a line and a circle (if they intersect)
- Creating the one or two points in the intersection of two circles (if they intersect).
Since lines cannot be drawn without a straightedge (1.), a line is considered to be given by two points. 2. and 5. are directly doable with a compass. Thus there need to be constructions only for 3. and 4.[5]
An alternative algebraic approach uses the isomorphism between the Euclidean Plane and . This approach can be used to provide significantly stronger versions of the theorem.[6] It also clearly shows the dependence of the theorem on Archimedes' axiom (which cannot be formulated in a first-order language).
See also
Notes
- ^ Georg Mohr, Euclides Danicus (Amsterdam: Jacob van Velsen, 1672).
- ^ Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift Euclides Danicus, udkommet i Amsterdam i 1672" [Of a memoir Euclides Danicus published by the Danish mathematician Georg Mohr in 1672 in Amsterdam], Matematisk Tidsskrift B , pages 1–7.
- ^ Schogt, J. H. (1938) "Om Georg Mohr's Euclides Danicus," Matematisk Tidsskrift A , pages 34–36.
- ^ Lorenzo Mascheroni, La Geometria del Compasso (Pavia: Pietro Galeazzi, 1797). 1901 edition.
- ^ Norbert Hungerbühler, "A Short Elementary Proof of the Mohr–Mascheroni Theorem," The American Mathematical Monthly, vol. 101, no. 8, p. 784, Oct. 1994.
- ^ Arnon Avron, "On strict strong constructibility with a compass alone", Journal of Geometry (1990) 38: 12.
External links
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