NEW YORK – The ancient Greek philosopher Thales was featured in Big Think (BT) on July 12 for his “discovery” of geometrical propositions. Professor Robert Hahn writes in the article titled Thales: Ancient Greeks Built the Cosmos with Right Triangles that “Thales is credited by the late commentator Proclus, on the authority of Aristotle's student Eudemus, with ‘discovering’ geometrical propositions, some of them more generally and others more practically.”

The article includes diagrams demonstrating “practical examples of right-angled triangles,” BT reported, adding that “originating with Diogenes Laertius of the 3rd century BCE on the authority of the mathematician Pamphila, it [a report] says that Thales made a splendid ritual sacrifice upon inscribing a right triangle in a circle.”

“The first thing Thales had to know is that the angles of every triangle sum to two right angles,” BT reported, noting that “the angles inside every triangle sum to 180 degrees” and “two right angles, each of which is 90 degrees, also sum to 180 degrees.”

Prof. Hahn writes: “We have an ancient report that credits Thales' generation of geometers with having grasped this fact in all species of triangles — equilateral, isosceles, and scalene. How might Thales and his geometers have done it?”

“By dropping a perpendicular from a vertex to the opposite side in each species of triangle, and then completing the two rectangles formed, one can see immediately that each rectangle (containing four right angles) is halved by the diagonal created by each side of the triangle,” Hahn writes, adding that “therefore, each half-triangle contains two right angles. And if the two right angles at the base are removed, leaving the three angles of one large triangle, the angles sum to two right angles.”

“Now, consider how Thales may have proved that every triangle inscribed inside a circle on its diameter must be a right triangle,” Hahn continued, noting that “perhaps these lines of proof persuaded Thales and his companions that every triangle inscribed in a circle on its diameter is right. But why the great ritual sacrifice?”

“The ancient traditions do not give us more insight, and we are left only to speculate,” Hahn writes, adding that “Aristotle claims that Thales posited an underlying unity, water, that alters without changing. Although things look different, water is the substrate of all appearances. Water is merely altered without changing substantially. Had Thales been looking into geometry to try to discover the underlying structure of water, perhaps he followed a similar line of thought as Plato did when he identified the four elements (fire, air, water, and earth) with geometric shapes.”

“Thales may have identified the right triangle as the fundamental structure of water,” Hahn writes, noting that “moreover, he now had a way to produce an unlimited number of them for further investigation simply by making a circle, drawing its diameter, and inscribing a triangle inside it.”

“But there is perhaps another reason for his splendid sacrifice, seen in this metaphysical light,” Hahn continues, adding that “I can imagine one of his compatriots objecting, upon hearing Thales' idea that water was the underlying nature or unity of all things and that the right triangle was its structure. The objection may have gone like this: right triangles may form the basis of every rectilinear figure, but they certainly don't form the basis of the circle. The circle is not constructed out of right triangles, is it? Thus, the right triangle is not the fundamental figure of all appearances.”

“Thales' reply must have been as astonishing to his compatriots as it is to many of us today,” Hahn writes, noting that “indeed, the circle too is built out of right triangles!”

“If we plot on the circle's diameter all the possible triangles inscribable inside a circle — starting from one end of the diameter, touching the circle, and then finishing at the other end of the diameter — we produce what modern mathematicians call a ‘geometrical loci,’” Hahn writes, adding that “the circle itself is constructed out of right triangles!”

Prof. Robert Hahn is the director of the Ancient Legacies program in the Department of Philosophy at Southern Illinois University Carbondale. His interests include the history of ancient and modern astronomy and physics, ancient technologies, the contributions of ancient Egypt and monumental architecture to early Greek philosophy and cosmology, and ancient mathematics and geometry of Egypt and Greece. His book, The Metaphysics of the Pythagorean Theorem: Thales, Pythagoras, Engineering, Diagrams, and the Construction of the Cosmos out of Right Triangles (SUNY series in Ancient Greek Philosophy) is available online.