Since the beginning of mankind, we have tried to figure out the mathematical problems of the world around us, starting with counting, adding and multiplying. People started looking for mathematical solutions in nature. Everything had to be solved and calculated. If they could not calculate it, it had to be proven ‘unsolvable’. The tools that were to be used to solve these problems consisted of an ‘unmarked ruler’ and a ‘compass’. For a problem to be solved it had to be ‘exact’, which meant that it could not be an approximation and that it had to be applicable in all possible cases.

The old Greeks were very interested in proving all kinds of mathematical constructions and formulas. However, they encountered three big problems, which they could not explain. These problems were those of ‘squaring the circle’, ‘doubling the cube’ and ‘trisecting an angle’. All through the history of mathematics, these problems caused headaches for professional and amateur mathematicians. Although all these magnificent brains were working on the solution, the problems stayed unexplained and eventually were found really unsolvable. Nonetheless, many people are still in search for an exact answer.

## The Three Classical Greek Mathematical Problems

### A Short History of Mathematics

Finds from ancient history tell us that mankind has been involved in some kind of mathematics for a long time. Archaeologists found counting-sticks, which are dated back to Prehistoric times. Even 70.000 years ago there was a sense of form and measure. The Babylonians (1800 – 1600 BC) were the first society that used mathematics in agriculture, trade, astronomy and navigation, and they already new some kind of Pythagoras’ proposition. Even a clay tablet was found which gives an approximation of Ö2. The Babylonians also ‘invented’ the sexadecimal system. This gave us for instance, the 60 seconds in a minute and 360 degrees of a circle (O’Connor and Robertson).

### Babylonian Clay Tablet

Also mathematicians in Egypt were not sitting still. We can see this on the Rhind papyrus, discovered in 1858, which dates back to 2000 – 2550 BC. This papyrus shows us an instruction book for algebra and geometry. It also gives an approximation of ‘p’, an attempt to square a circle and the use of some kind of cotangent. Similar findings are also discovered in Russia, Germany, India and China (O’Connor and Robertson).

Euclides of Alexandria (300 BC) wrote, in his famous book “The Elements”, about the system of geometry. Many propositions and theories had been discovered already, Eyclides made them into a logical system. He also described the properties of geometric shapes and natural numbers (Wilson). Pappus of Alexandria (290 – 350), who wrote the famous “Collection”, described also three types of methods used by the Greek mathematicians. The first one is a ‘plane’ solution; problems can only be solved with an unmarked ruler and a compass. The second one is a ‘solid’ solution; problems can be solved with the use of one or more sections of a cone. The last method is ‘linear’; problems can be solved with the use of curves and more complex motions. All the classical Greek problems could not be solved using a plane method (O’Connor and Robertson).

## The Three Classical Greek Mathematical Problems

### Squaring a Circle (or Quadrature)

This problem is the most famous one of the three. It is the attempt to make a circle with the same area as the area of a given square. This is a problem that concerns the use and determination of ‘p’. Basically it meant that a line had to be constructed with the length of ‘p’. The problem was first formulated by the mathematicians of Greece like Anaxagoras and Archimedes (Bold 1-17).

Some Greek researchers tried to invent other means in finding an answer. Hippocrates used ‘lunes’ (half moon shapes) and Hippias invented a ‘quadratrix (Wilson)’. All of these were fine attempts, but of course, neither of them used only a ruler and compass. Therefore, the solution was not found.

Although the problem was eventually ‘solved’ as being unsolvable, it did not stop the search for an answer. People who tried, and still try, to solve this problem were mockingly called ‘circle-squarers’, which meant that they attempted the impossible.

## The Three Classical Greek Mathematical Problems

### Doubling a Cube

The context of this problem is said to come from a Greek legend. In Greece, a terrible epidemic plague was killing many people at that time, about 430 BC. The Greeks were desperate and went to see an Apollo oracle in search for help. The gods told them that the cubic altar, in the oracle temple, was too small and that they wanted it doubled in size. The Greeks thought that was easy, so they doubled all the sides. Coming back to the oracle, the gods were very angry; the altar was not doubled this way, but it was eight times the original size (Bold 29-31; Wells 33).

Many attempts were made to solve the problem. The mathematicians even made some completely new theories, constructions and inventions. They might have not being able to solve this problem, but the new theories did solve a lot of other problems and conducted new theories for all mathematics.

## The Three Classical Greek Mathematical Problems

### Trisecting an Angle

The last of the three mathematical problems the Greeks encountered was the trisection of an angle. This meant that they had to try to part a random angle up in three lines or sectors just by using a ruler and compass. This problem is different from the other two because of two reasons. The first is that this problem has no real relation with the other two problems, and the second is that it is a problem of a different order. It means that this problem is solvable in some cases, with some angles, but not with all angles. It is, for instance, possible to trisect an angle of 90° with a ruler and compass. The problem is that the ‘general rule’, which the mathematicians are always trying to find, is not applicable. If at least one angle is not trisectable, the problem stays unsolved (Bold 33-37).

## Trisecting an Angle

Trisecting an angle of 90°. Archimedes’ solution for trisecting an angle, using a spiral. The Three Classical Greek Mathematical Problems:

## Conclusion

The answers, the proofs of impossibility, to all of the three classical problems were given in the 1880s. In 1837, Pierre Wantzel (1814 – 1848), proved that ‘trisecting’ and ‘doubling’ were impossible, with ruler and compass. The idea that ‘p’ is a transcendental number is made by Carl Louis Ferdinand von Lindemann (1852 – 1939) in 1882. This makes the ‘squaring’ unsolvable because you can not find an exact answer (Wilson).

Many smart people tried to solve these problems. They even ‘cheated’ and bended the mathematical rules, that had to be applied. Because many Greek mathematicians did not restrict themselves in using the ‘plane’ methods, they developed a great variety of methods, various curves and new constructions. The attempts, therefore, have not been worthless. They were extremely important in the development of geometry, algebra and more mathematical systems.

### Works Cited

- Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, 1982.
- Wilson, R. “Squaring the Circle and Other Impossibilities” Gresham College Website. 16 January 2008. 25 February 2009. <http://www.gresham.ac.uk/event.asp?PageId=45&EventId=624>
- O’Connor, John J., and Edmund F. Robertson. The MacTutor History of Mathematics archive. School of Mathematics and Statistics, University of St. Andrews, Scotland. April 1999. 25 February 2009. <http://www-history.mcs.st-andrews. ac.uk/history/index.html>
- Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 33-34, 1986.