# 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 '14 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 '14 at 5:17