# Division of the circle and compass constructions

It is well known that every construction that can be performed by a compass and a ruler can be also performed by a compass only. This is a good (and difficult) exercise in elementary geometry. My question:

When did mathematicians start investigating this question ?

The literature on the subject that I know is from the 19th and 20th centuries. However, I also know that there was a time when this question had serious practical applications. And this was much earlier (until the second half of the 18th century).

The practical application I am talking about is the following. Since the 16th century, astronomers began to measure angles to high accuracy. For this they used divided circles made of metal (bronze, brass). The circles were divided to fractions of a minute. How did you do this? (Those who never tried may think that this is a trivial task, but it is not). Late 18th century encyclopedias have long articles called "Division of the circle", where they explain in great detail how this was done and the history of the business. There is also literature written by some of these master dividers.

One feature of the task is that one cannot really use a ruler for very high accuracy constructions. One of these masters puts it clearly: "You cannot really find an intersection of two lines with a ruler". They used a compass which is a much more accurate instrument. Even division of a straight ruler was performed with a compass of very large radius.

In the second half of 18th century this noble trade suddenly came to the end: a dividing engine was invented which permitted to divide an instrument circle hundreds times faster than by hand.

Remark. It is another interesting question: to what extent could they to this in antiquity? There is ONE archeological find which shows that this business existed in Hellenistic world: it is the Antikythera mechanism.

EDIT (after the answer of Uri Zarfaty). I learned about this problem from the writings of Bird, a famous instrument maker, a "master divider" as they called him. It is he who explained that "You cannot find the intersection of two lines with a ruler". He meant "this is impossible in practice, with sufficiently high precision". Now we learn that this very same Bird is mentioned in the introduction of Masceroni's paper! So my conjecture is correct! Bird lived long enough to see the invention of the dividing engine which made his noble art obsolete.

• So you are asking for the history of explicit attempts to prove or disprove the theorem "Every straight-edge and compass construction can be accomplished with compass alone" ? Nov 28, 2014 at 22:22
• My main concern is whether there is any connection between this theorem and practical application I mentioned. But the date when mathematicians considered this problem for the first time can help to solve the main question. Nov 29, 2014 at 5:17

The theorem that any compass and straightedge construction can be performed by a compass alone is called the Mohr-Mascheroni Theorem. It was first described by the Danish mathematician Georg Mohr (1640-1697) in 1672. However, Mohr's proof was lost until 1928 and the theorem was independently proved by the Italian mathematician Lorenzo Mascheroni in 1797.

Mascheroni's interest in the problem was in fact inspired by ad hoc compass-only constructions used for astronomical quadrants at the time. Quoting from Dictionary of Scientific Biography:

"In the preface Mascheroni recounts the genesis of his work. He was moved initially by a desire to make a contribution to elementary geometry. It occurred to him that ruler and compass could perhaps be separated, as water can into two gases; but he was assailed by doubts and fears often attendant upon research. He then chanced to reread an article on the way Graham and Bird [who supplied instruments to Maupertuis] had divided their great astronomical quadrant, and he realised that the division had been made by compass alone, although, to be sure, by trial and error. This encouraged him and he continued his work with two purposes in mind: to give a theoretical solution to the problem of constructions with compasses alone and to offer practical constructions that might be of help in making precision instruments. The second concern is shown in the brief solutions of many specific problems and in a chapter on approximate solutions."

This summary notes that the problem of carrying out geometric constructions under additional restrictions was previously also considered by "da Vinci, Dürer, Cardano and Tartaglia, among others". However, I don't know if these too explicitly linked to astronomical research.

• Great!! So my conjecture is correct! And even Bird's name is explicitly mentioned (he was the greatest master of division of arcs for astronomical instruments). I wonder if Mohr's and Masceroni's papers are available. Mar 1, 2015 at 18:25
• Mascherano's La Geometria del Compasso can be read at archive.org/details/lageometriadelc01mascgoog (or bought in the last link in the answer for €1,200). Mar 1, 2015 at 18:29
• Mohr's Euclides Danicus can be read (in Dutch) at books.google.co.uk/books?id=9SRcAAAAcAAJ. Mar 1, 2015 at 18:33
• A more convenient link for browsing the Mascherano may be books.google.com/books?id=Me1EAQAAMAAJ. Mar 1, 2015 at 18:35