Classical problems   

Even though they contain indications of ancient, practical origins, the Euclidean Elements are a late work where the axiomatic deductive structure is developed completely, even if not with absolute rigor. Their variety shows the legacy of generations of surveyors who, from the procedures learnt from the Egyptians, were able to create a totally new science as a theoretical system much richer than those who pulled ropes or traced circles could have imagined.

Nonetheless, some problems remained persistently beyond the scope of the line and circle, even when geometry was moved to the world of paper, ruler and compasses. Among them are three classic problems: the duplication of the cube, the trisection of the angle and the quadrature of the circle.

According to legend, when Greece was victim of a great epidemic, a delegation went to the oracle of Delo to ask respite. The response of the oracle was that the rage of the Gods would be placated only when the altar dedicated to them, which had the shape of a cube, was be replaced by one of the same shape but twice the size. The messengers, industrious but not very wise, ordered an altar to be built, again in the shape of a cube, with the side twice as long as that of the original. The epidemic didn't stop and the oracle was consulted again. On that occasion it was discovered that the new altar was not twice as big as the former one but eight times bigger, being, in fact, the volume of the cube with side a equal ³the one with side 2a will have volume (2a)³ If, on the other hand, the cube with side a needs to be doubled, it is necessary to build another one with side , so that its volume is .

Leaving aside the legend and taking an exclusively geometrical point of view, the problem consists in, given a segment of length a, drawing a second segment, with length . Now, while it is easy to build a segment with length with ruler and compasses (it is sufficient to build a square of side a, the diagonal of which is long), and then double a square (the square built on the diagonal of the former one is suitable), to double the cube is not as simple. Instead, it was demonstrated, even in relatively recent times, that the problem cannot be solved using exclusively a ruler and compasses. The problem of Delo requires the employment of more complex tools and techniques.

The same occurs with the other problems. For those, the surveyors invent new curves, pushing themselves beyond the limits of lines and circles. Ippia from Elide (V century B.C.) invented the quadratrix, a curve that he used to obtain the quadrature of the circle and the trisection of the angle.

Later, the problem of the duplication of the cube was solved by Archita from Taranto (or maybe by Menecmo from Proconneso, who also lived around the 400 B.C.) through the intersection of two parabolas, and by Diocle (II century B.C.) through a new curve, the cissoid. The method of Archita is very ingenious and easy to describe, using the modern system of Cartesian geometry. Considering two parabolas with equation y² = 2ax e x² = ay. If the point with coordinates (x,y) belongs to both curves, we'll have x⁴ = a²y² =2a³x, from which, excluding the solution x=0, is found x³=2a³, and therefore . In conclusion, by intersecting the two parabolas, a point with the side of the double cube as abscissas can be obtained.

Naturally, we have simplified the description of the process of Archita, who, not knowing cartesian geometry and in particular the use of coordinates, employed purely geometric methods. What is most interesting is that certain curves appear in the demonstration for the first time. These were to be the object of study for many centuries to come: conic sections.