"Lecture" for Week 4

Later Greek mathematics

Euclid and Lakatos

When we were using Katz's book, what I found most interesting in his chapter on the Greeks was an implied criticism of the traditional way of writing mathematics. He writes (pp. 36, 38, 49):

"Euclid's Elements ... to the modern reader is incredibly dull. There are no examples; there is no motivation; there are no witty remarks; there is no calculation. There are simply definitions, axioms, theorems, and proofs.... If one reads Book I from the beginning, one never has any idea what will come next. It is only when one gets to the end of the book, where Euclid proves the Pythagorean theorem, that one realizes that Book I's basic purpose is to lead to the proof of that result.... As usual, Euclid does not show how he arrived at the [pentagon] construction ..."

It seems like Prof. Katz has been reading one of my favorite books, "Proofs and Refutations: The Logic of Mathematical Discovery", by I. Lakatos (Cambridge UP, 1976). (That would be a good choice for a book review, come to think of it.) Lakatos sees Euclid as the source of the Original Sin of mathematicians, our careful and proud presentation of ideas in the most logical order, which leaves our students spending most of their time asking, "Why are we doing this?"

Antiphon, Eudoxus, and Archimedes.

In his Classical/Hellenistic Exercise 5 Allen raises the issue of whether Archimedes deserves to be called the inventor of calculus. That accolade is presumably based on his calculation (really, definition) of the area of circles by means of approximating polygons -- effectively a sort of Riemann sum. Although Archimedes was the first to make this topic quantitative -- thereby dragging Greek geometrical thinking a big step in the modern direction -- Allen points out elsewhere that Eudoxus had earlier made extensive use of this "method of exhaustion" and would even have understood "volumes of solids of revolution by slicing".

It seems worthwhile to note (from Heath, Vol. I, pp. 221-223) an even earlier reference to the method. Antiphon, a contemporary of Socrates, discussed the squaring of the circle this way: Inscribe a square in the circle; on each side of the square, as base, put an isosceles triangle with its vertex on the nearby arc of the circle; on each side of this octagon put an isosceles triangle; etc. According to the commentary of Simplicius, "Antiphon thought that in this way the area [of the circle] would be used up.... And, as we can make a square equal to any polygon ... we shall be in a position to make a square equal to a circle." Aristotle mentioned Antiphon's argument only in order to denounce it as a well-known fallacy, and of course it is a fallacy within the Euclidean ruler-and-compass framework. But Antiphon was clearly ahead of his time! Heath says, "But the objection [that the whole area will never be used up] is really no more than verbal; Euclid [presumably following Eudoxus] uses exactly the same construction in XII.2, only he expresses the conclusion in a different way, saying that, if the process be continued far enough, the small segments left over will be together less than any assigned area. Antiphon in effect said the same thing, which again we express by saying that the circle is the limit of such an inscribed polygon when the number of its sides is indefinitely increased." How close Antiphon, Eudoxus, and Euclid came, not only to Archimedes, but to Leibniz, Cauchy, Riemann, and Weierstrass!


Our books barely mention Proclus (5th c.), but in TAMU's course Math 467, "Modern Geometry", he plays a significant role. The textbook used is M. J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History. Like Theon and Hypatia shortly before him, Proclus made his name by writing a commentary on Euclid, not by creating any particularly original mathematics. He did, however, introduce two major simplifications into Euclid's deductive structure:
  1. He showed that Euclid's 4th postulate, "All right angles are congruent to each other", can be proved from the previous three postulates. ("Right angle" is defined as "angle congruent to its supplement".)
  2. He replaced Euclid's cumbersome statement of his 5th postulate (see Allen, "Euclid", p. 8; change the last "to" to "two") by the conceptually clearer statement familiar to us today: "For any line L and any point P not on L, there is a unique line M through P that is parallel to L" [i.e., does not intersect L (even if the lines are indefinitely extended)]. This postulate was reintroduced by John Playfair in 1795 and is called the Proclus-Playfair version of the parallel postulate. Spoiler alert: Stillwell Chap. 18.

Mean proportion

Several of our homework problems concern numbers or line segments in "mean proportion". The simplest type of such problem is to find x such that a/x = x/b. Later the problem is extended to two intermediate numbers:

a/x = x/y = y/b.

Why were the Greeks so interested in these problems? Here is my guess: In the first case, let b=2a. Then you quickly see that x= sqrt(2) a. So, if we can solve the mean proportion problem, we can construct a line segment that is sqrt(2) times another segment. Square roots are called "constructible numbers" because Euclidean constructions yield them via the Pythagorean theorem. Now consider the problem of two numbers in mean proportion and consider the case where b = sqrt(2) a (not a big deal, in view of the previous result). You should find that y3 = 2a3. Voila! We have duplicated the cube! Of course, the end of the story is that irrational cube roots are not constructible.