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Mohr-Mascheroni Theorem

The Mohr-Mascheroni theorem says

Everything you can construct with a straightedge and a compass you can do with the compass alone.

This surprising result  has interesting history. The result proved by the Italian mathematician Lorenzo Mascheroni [1] in 1797, and for more that two centuries it was credited to Mascheroni only.  In 1928 the Danish mathematician J. Hjelmslev [2] discovered in an old book shop in Copenhagen the book Euclides Danicus (The Danish Euclid)  by G. Mohr   and published in 1672 in Amsterdam 1 . To his great surprise he found  the Mascheroni result in the first part of the book. The book was immediately reprinted in facsimile.

The proofs given by Mohr and Mascheroni were complicated. A proof using inversions was found by A.Adler in 1890. Simpler proof can also be found in [3] , [4] , [5]  

Motivated by Mascheroni’s result J.Steiner proved that given a fixed circle and its center, all the constructions in the plane can be carried out by the straightedge alone. [6]  

For instance to find the mid-point of a given arc typeset structure by two endpoints typeset structure, typeset structure  using a straightedge and compass is an easy exercise:

  1. construct the axis typeset structure of the segment typeset structure using two circles centered at typeset structure and typeset structure  with equal radii,
  2. the line typeset structure crosses the given arc typeset structure at the center typeset structure of the arc typeset structure.

[Graphics:HTMLFiles/MohrMascheroniTheorem_12.gif]

Without loss of generality ( cf. [3] , p.26,   [4] ) we can suppose that we know the center typeset structure of circle containing the given arc typeset structure of a circle with center typeset structure( [3] ,p.11,  [4] ):

  1. draw circles at typeset structure, typeset structure with radii typeset structure, and the circle at typeset structure with radius typeset structure; they intersect in points typeset structure, typeset structure;
  2. draw circle at typeset structure with radius typeset structure, and the circle at typeset structure with radius typeset structure; they intersect in typeset structure and typeset structure,
  3. draw circles at typeset structure and typeset structure with radii typeset structure;  they intersect in typeset structure and typeset structure, one of them halves the arc typeset structure.

[Graphics:HTMLFiles/MohrMascheroniTheorem_38.gif]

Notes

1 The book was published simultaneously in a Danish and a Dutch edition, but not in Latin as it  was usual in that time.

References

[1]  Mascheroni, L. (1980). Geometrie du compas. Monom: Coubron.

[2]  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, 1-7.

[3]  Kostovskij, A. (1984). Geometric Constructions Using Compasses Only (2nd ed.) (Russian). Moscow: Nauka Publisher.

[4]  Kostovskii, A. (1961). Geometric Constructions Using Comapasses Only.  Blaisdell Publication Company.

[5]  Eves, H. (1972). A Survey of Geometry. Allyn and Bacon.

[6]  Rademacher, H., & Toeplitz, O. (1933). Von Zahlen und Figuren (2nd ed.). Berlin: Springer Verlag.

Cite this web-page as:

Štefan Porubský: Mohr-Mascheroni Theorem.

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