### Thales, Heron, and Experimental Mathematics

February 12, 2024

Euclidean geometry was taught to my generation during the sophomore year of high school. There was even the argument in those days that the logical reasoning involved in geometrical proofs was worthwhile in its own right, independent of whether the geometry used would be useful in later life. Geometry was very important in my later scientific life; but, for others, probably not at all.

My interest was especially excited by two things learned at that time. One of these was Thales's theorem, named after the Greek philosopher, Thales of Miletus (c. 624 - c. 546 BC). Thale's theorem states that any angle inscribed in a semicircle is a right angle. Thales was one of the first Greek philosophers interested in physics. Unfortunately, his work is known to us only through the works of other philosophers, including Aristotle.

The other thing of interest to me was Heron's formula, named after Heron of Alexandria (c. 10 - 70 AD), a Greek mathematician, engineer, and native of Alexandria in Roman Egypt in the 1st century. Heron's formula gives the area of any triangle as a function of the lengths of its sides. It's quite likely that Heron didn't discover this formula, which is included in his Metrica, but his name is linked to it, nonetheless. I wrote about Hero's many inventions in a previous article (Steam Power, January 28, 2011). The most famous of Heron's inventions, his steam engine, called an Aeolipile, was published in his book, Pneumatica. Heron's steam engine, although primitive, predates James Watt by seventeen centuries.

Left, Heron of Alexandria (c. 10 - 70 AD), and right, Thales of Miletus (c. 624 - c. 546 BC). (Left image, an illustration from the Codex of San Gregorio de Nizance, a ninth century Greek manuscript, from Wikimedia Commons. Right image, a woodcut from the Nuremberg Chronicle (1493), also from Wikimedia Commons.)

Aristotle tells an interesting anecdote about Thales in his Politics.
"He was reproached for his poverty, which was supposed to show that philosophy was of no use. According to the story, he knew by his skill in the stars while it was yet winter that there would be a great harvest of olives in the coming year; so, having a little money, he gave deposits for the use of all the olive-presses in Chios and Miletus, which he hired at a low price because no one bid against him. When the harvest-time came, and many were wanted all at once and of a sudden, he let them out at any rate which he pleased, and made a quantity of money. Thus he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort."[1]

The proof of Thales's theorem is illustrated in the following figure.

Proof of Thales's theorem.

Since the circle radii are equal, two isosceles triangles can be formed.

Thus, ∠BOC = (180-2b), ∠AOC = (180-∠BOC) = 2b, 180 = 2(a+b), and (a+b) = 90.

(Created by the author using Inkscape

A proof such as this is the mathematical gold standard; but, how would you convince yourself that a proof is likely and worthy of your effort? That's where experimental mathematics can be used as a tool. While a Thales's conjecture is easy to validate using a computer program, other conjectures do not appear as obvious, and recourse to experimental mathematics could save needless effort. I've created a simple C language program (source code here) that creates random inscribed triangles in a semicircle and tests the validity of the converse of the Pythagorean theorem; that is, that the square of the longest side, in our case, the diameter, is equal to the sum of the squares of the other two sides. For a circle of radius 1, the mean diameter after a million trials is 2.000000±0.

Heron's formula for the area A of a triangle with sides a, b, and c, is simply,
Where s, known as the semiperimeter, is half the perimeter; viz.,
The formula can be written in terms of the sides, only, as
Checking the validity of Heron's formula for a scalene triangle simply involves decomposing the triangle into two right triangles (see figure) and comparing the sum of their areas to the area given by the formula. I've created a simple C language program (source code here) that does this. For 100,000 trials, the difference of the two values was -0.000001±0.000099, the slight error attributed to floating point precision.

Scalene triangle decomposed into two right triangles.

(Created by the author using Inkscape

### Reference:

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