Did Euclid consider circle segments as another magnitude?

Question

_{[I adapted the question to reflect what I've learned from Alexandre's answer: that Euclid never talks of lengths and areas but only of line segments and figures (like squares). The question itself remains unchanged.]}

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For Euclid, magnitudes were things that can be compared, be equal, smaller or greater than each other, that can be added and subtracted, and that can have proportions. (Next to these, there were numbers, another type of thing with the same properties.)

There are four magnitudes Euclid explicitly and mainly deals with:

• lengths of straight line segments

• angles between straight line segments

• areas of plane figures (polygons and circles)

• volumes of solid figures (polyhedra and spheres)

Only lengths of straight line segments can be multiplied (giving areas and volumes figures), as well as numbers (giving numbers).

He also deals with lengths of circle segments, at least with the length of the whole circle (circumference).

But does he anywhere talk explicitly about the length of arbitrary circle segments and compares, adds, or substracts them?

Furthermore: Was he aware that an angle is "proportional" to the length of a corresponding circle segment and did make use of this (which would not have meant for him, that the two were the same)?

Answer 1

Magnitudes could be compared and have proportions, added or subtracted, but only to like magnitudes. For instances, adding segments meant roughly concatenating them (same with areas), not relating them to some other type of thing, like number, and then adding those. Lengths could not be compared or related to areas, for example. As for the lengths of circle segments, to Euclid "circumference" applies not just to the whole circle but to circle arcs as well, see e.g. Book VI, Proposition 33:

"Angles in equal circles have the same ratio as the circumferences on which they stand whether they stand at the centers or at the circumferences."

Whether circumferences could be related to line segments, the problem of rectification, is of a kind with the problem of quadrature, relating curved areas to rectiliniar ones, and can only be done by using what is now called "method of exhaustion". Euclid does not take this up in the Elements, that circumferences are as their diameters is enough for his purposes.

Answer 2

Contrary to what you say, Euclid never assigns any numbers to geometric figures. He never speaks of "length", or "angle measure" (he does not use degrees). It is more difficult with areas and volumes, which Euclid really compares but without assigning any numbers to them. From Euclid's point of view, two polygons have the same area if one can be dissected into parts and then these parts be rearranged to form the other polygon. This does now work with a circle or 3-d polytops, but the area of the circle and the volume of a polytop is never really defined in Euclid.

All these things are very well explained in the book of R. Hartshorne, Companion to Euclid, which on my opinion is the best exposition of Euclid from the modern point of view.