E L L I P S I S ?

The Colosseum is generally supposed to be elliptical, but the curve of the arena and of the remaining facade, which has now been measured to a high degree of accuracy, does not coincide exactly with a perfect ellipse. It is in fact an ovoid (a polycentric curve, i.e. a curve with more than one centre), but very approximate to an ellipsis.

This matter is still very much debated (see this very interesting forum in the Nexus Network Journal), because it also implies different interpretations of the geometric skill of the ancient Romans. Everybody admits that Romans could draw plans with polycentric curves. Apparently all amphitheatres are built like this. But what about the ellipsis? Did they know it? And if they knew it, how could they construct one and draw it on the ground? So there are quite a few explanations as to how they could have drawn an ellipsis without a compass.

There are also really esoteric explanations of the geometry of the Colosseum, like the triangle with proportions 3:4:5 (which in the case of the Colosseum becomes 6:8:10), or the Ippia trisector ...
Rationally, the solution should be found by measuring the building with a high degree of precision and assessing what is what, but there are in fact many obstacles to it: the original small deviations of stones and brickwork from the plan (that are natural with those materials), the adjustments made during the construction for the wider corridors on both axis, the general deformation of the structures, due to earthquakes, land movements etc., then the ruined state of the stones, etc. So in practice there are small deviations from the ideal curve (be it an ovoid or an ellipsis), and nobody can demonstrate in full the validity of his theory. I bet that archaeologists will debate on this matter in the next century, too (I'm writing in 2004).

Many are sure that the Romans knew how to draw an ellipsis; the Greeks before them had studied it since Archimedes' time and there is evidence that they knew advanced geometry. But probably it was more difficult to draw it on the ground (the sheer size of an amphitheatre makes a precise measuring quite difficult), or maybe it did not make much sense to struggle with a perfect ellipsis when a polycentric curve is much easier to draw, and the final result is in practice no different from an ellipsis. In my modest opinion the polycentric curve makes more sense. First, it has been adopted for amphitheatres all over the Empire. Secondly, it is easy to draw with simple instruments that the gromatici (land surveyors) used every day to measure land and make plans for roads and buildings. Moreover, if a polycentric curve has a minimum number of centres (say 4) it is almost indistinguishable from an ellipsis.

Polycentric curve
According to the famous Colosseum expert,  Ingegner Giuseppe Cozzo, the hypothesis is that the curve of the arena is a polycentric one, i.e. a curve composed of many different curves, whose centres are found by means of a geometric system.

This was the system,  (see it in the image >): AB is the length of the arena, and BI is a constant measure, corresponding to the width of the building. Find the half of AB and call it O. Draw a circle centred in O,  with the radius OB. Divide the segment OB into three equal parts (BC=CD=DO). The line VM starts from the extreme V of the quadrant BOV, intersecting D.

Divide in 2 parts VD and find E. Draw the line EL intersecting C. Then draw the sector FB by centring the circle in C (of radius BC), draw the sector FG by centring the circle in E (of radius EF), and finally the sector GH by centring the circle in V (radius VG). Add the BI constant to the curves, so as to obtain the measures HN, GM, FL that give the perimeter of the building. Repeat this procedure for the other three quadrants of the amphitheatre. Done it ?.

According to Cozzo, the small differences between the ideal curve and the reality of the amphitheatres should account for small errors made originally by the architect, or by errors made by us when measuring dilapidated monuments.

Second theory:
in a more recent study Professor Camillo Trevisan has advanced a different hypothesis on the geometric system(s) used. By more accurate measurements, he has noticed that there are small differences between the hypothetical curve as described by Cozzo and the real measurements of the amphitheatres, and has provided an explanation.
Unfortunately for you it is in Italian language ....