stoutfellow

Eudoxus of Cnidus is not one of the better-known mathematicians of the classical era. Euclid and Archimedes left behind extensive writings which have survived to the present; Eudoxus did not. Pythagoras gathered a school of ardent, even fanatical, disciples; Eudoxus did not. Thales was named one of the Seven Wise Men of Greece; Eudoxus came too late for that. But, in my estimation, Eudoxus ranks second among the mathematicians of that era, behind only Archimedes (who has no peers, then or ever). In addition to several major contributions to astronomy, laying the groundwork for Ptolemy, he came up with two crucial mathematical ideas, without which the work of Euclid and Archimedes might not have been possible. One of those, which resolved the problem left by the failure of the Pythagorean approach, is the subject of this post.

Two of the three key ideas of the Pythagorean program, isometry and dissection, remained unexceptionable. What was needed was a way of talking about the proportion between two magnitudes that did not rely on the existence of a common measure - a theory of proportionality. It had proven impossible to represent the proportion between two magnitudes as a proportion between two numbers, but might it still be possible to compare proportions between two pairs of magnitudes - to say that A is to B as C is to D?

The first step towards such a theory was the following realization. Suppose that twice magnitude A is greater than three times magnitude B; then, though we can't identify the proportion A:B exactly, we can at least say that it is greater than 3/2. If 5A is less than 3B, then A:B is less than 5/3. For any pair of numbers m,n, we can compare mA to nB, and determine whether A:B is greater than, equal to, or less than n/m; and by so doing, we can get finer and finer approximations to its value. We may not, however, be able to find the value itself.

Eudoxus's achievement was to point out that we don't need the value itself - not, at least, in numerical terms. Suppose that A,B are two magnitudes of the same kind, and that C,D are two other magnitudes of the same kind (not necessarily of the same kind as A,B). Then, for each pair of numbers m,n, we can ask two questions:

Is mA greater than, equal to, or less than nB?
Is mC greater than, equal to, or less than nD?

If both questions always have the same answer, regardless of the choice of m,n, is it not clear that the proportions A:B and C:D must be equal?

To illustrate the use of Eudoxus's theory of proportionality, let's look again at the proposition, If two rectangles have the same height, their areas are in proportion to their bases. Let A,B be the bases of the rectangles and C,D their areas. If we lay m copies of the first rectangle side by side, and likewise n copies of the second, we obtain two larger rectangles, with the same height, with bases mA and nB and areas mC and nD, and it is easy to see that the order relation (greater than, equal to, less than) that holds between the bases also holds between the areas. The proposition then follows from Eudoxus's theory.

Once we have this proposition, the rest of the program follows: right triangles, parallelograms, general triangles, general polygons. (Areas which are not polygons, such as circles, remain out of reach, but Eudoxus came up with a way of dealing with them as well. More on that later.)

The Pythagorean approach attempted to link arithmetic to geometry, with arithmetic holding the dominant position. Eudoxus's methods put geometry in at least an independent position; the actual result was the subordination of arithmetic to geometry, a subordination that was to last for millennia. It proved remarkably easy to subsume arithmetic into geometry, but there were significant consequences. That, however, is a story for another day.

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