"Whenever a construction is possible by means of compasses and straight edge, more advanced means should not be used."

Pappus of Alexandria

There has been considerable interest during the past 2300 years in comparing different models of geometric computation in terms of their computing power. One of the most well known results is the proof in 1672 due to Mohr that all constructions that can be executed with straight-edge and compass can be carried out with compass alone. The earliest such proof of the equivalence of models of computation is due to Euclid in his second proposition of Book I of the Elements in which he establishes that the collapsing compass is equivalent in power to the modern compass.

A word is in order concerning this terminology - collapsing compass. With the modern compass one can open it to some aperture on the page and then lift the compass and transfer this same aperture to another location on the page. With the collapsing compass this operation of transferring a distance is not allowed. It is called collapsing because it behaves as if when the compass is lifted from the page it folds and the measured distance is forever lost.

In the theory of equivalence of models of computation Euclid's second proposition enjoys a singular place. However, like much of Euclid's work and particularly his constructions involving cases, his second proposition has received a great deal of criticism over the centuries. I became interested in this problem when I discovered that much of this criticism is not warranted and in fact most textbooks of Euclid's Elements contain incorrect proofs of the second proposition. I wrote a paper (see below) where I argue that it is Euclid's early Greek commentators and more recent expositors and translators that are at fault and that Euclid's original algorithm, according to several trustworthy sources such as Gerard of Cremona's Latin translation of a 12th century Arabic manuscript as well as Peyrard's French translation of a pre-Theonian 10th century Greek manuscript, is beyond reproach.

• Godfried T. Toussaint, "A new look at Euclid's second proposition," The Mathematical Intelligencer, vol. 15, No. 3, 1993, pp. 12-23.
• PDF file euclid.pdf

• Godfried T. Toussaint, "Un nuevo vistazo a la segunda proposicion de Euclides," Mathesis, vol. 9, 1993, pp. 265-294.
• In Spanish - PDF file euclid.spanish.pdf

1. Francois Labelle's Tutorial on the Complexity of Ruler and Compass Constructions (with an awsome interactive Java applet for providing your own solutions to problems)
2. GRACE (A graphical ruler and compass editor)
3. GEObect - A software system for doing Euclidean constructions.
4. A collapsing compass problem.
5. Computing tessellations with straight-edge and compass.
6. Geometric Constructions:
1. Geometric Constructions Page
2. Geometric Constructions with a Compass Alone
3. Geometric Constructiuons with Straight Edge and a Circle or Parabola
4. Constructions (Wolfram Research)
5. History of Constructions
7. Trisections:
• Eric's Treasure Trove
• Trisection at the Geometry Forum
• Famous incorrect trisection algorithm
• Trisecting an angle with marked ruler and compass

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