# Some Original Sources for Modern Tales of Thales - Other Contributions to Geometry

Author(s):
Michael Molinsky (University of Maine at Farmington)

You will often find a collection of five geometric properties associated with Thales in modern history of mathematics textbooks:

• A circle is bisected by its diameter.
• Base angles of an isosceles triangle are equal.
• Vertical angles are equal.
• ASA and AAS triangle congruences hold.
• Angle inscribed in a semicircle is right.

Although most modern authors are careful to state only that Thales knew these facts, some go much further and claim that Thales was the first to prove them.  Some sources even go so far as to give Thales credit for the creation of mathematical proof itself.  An examination of the oldest existing sources does not seem to support these stronger assertions: while the number of ancient works that address Thales’ geometric knowledge are very limited, you will see below that these sources make little or no mention of proof; in fact, they often explicitly credit later mathematicians with the first proofs of these results.

Our earliest source for Thales’ relationship to the first four geometric results is A Commentary on the First Book of Euclid’s Elements by Proclus (c. 412 – 485 CE), which was written approximately a thousand years after the time of Thales.  While no earlier sources have survived, it is clear that Proclus did have access to such sources, and he frequently quotes from an earlier work: History of Geometry by Eudemus of Rhodes (c. 350 – c. 290 BCE), a pupil of Aristotle.

Proclus offers a brief history of geometry in his commentary, stating that geometry began in Egypt, motivated by the necessity of recreating property lines after Nile floods.  He then credits Thales with being the first to transmit Egypt’s knowledge of geometry to Greece, adding that he “made many discoveries himself and taught the principles for many others to his successors, attacking some problems in a general way and others more empirically” [8, p. 52].

In the rest of the Commentary, Proclus makes four more brief mentions of Thales.  The first occurs after Definition 17, where Euclid defines the diameter of a circle.  In addition to stating that the diameter is a line segment passing through the center with endpoints on the circle, Euclid simply includes that a diameter bisects the circle as part of the definition.  Proclus adds, “The famous Thales is said to have been the first to demonstrate that the circle is bisected by the diameter" [8, p. 124]. Proclus goes on to provide such a demonstration, using the superposition of semicircles onto the circle, but he doesn’t explicitly state if this method is the one he is crediting to Thales.

The next mention of Thales is in Proposition 5 (“In isosceles triangles the angles at the base are equal; and if the equal straight lines are produced further, the angles under the base will be equal”), where Proclus gives Thales credit for discovering the first part of the proposition, but does not mention proof [8, p. 195]:

We are indebted to old Thales for the discovery of this and many other theorems. For he, it is said, was the first to notice and assert that in every isosceles [triangle] the angles at the base are equal, though in somewhat archaic fashion he called the equal angles similar.

In his comments on Proposition 15 (“If two straight lines cut one another, they make the vertical angles equal to one another”), Proclus again gives Thales credit for discovery, but this time explicitly says he wasn’t the first to prove this statement [8, p. 233]:

It was first discovered by Thales, Eudemus says, but was thought worthy of a scientific demonstration only with the author of the Elements.

Finally, in Proposition 26, where Euclid shows what modern texts would label as the angle-side-angle (ASA) and angle-angle-side (AAS) triangle congruences, Proclus states that there is only an inference that Thales at least knew this result (there is again no mention of proof) [8, p. 275]:

… Eudemus in his history of geometry attributes the theorem itself to Thales, saying that the method by which he is reported to have determined the distance of ships at sea shows that he must have used it.

Figure 5: How Thales might have measured the distance from shore to ship

The final geometrical idea, that an angle inscribed in a semicircle must be a right angle, which is sometimes even called Thales’ Theorem, is not addressed in Proclus (it does not appear in the Elements until Book 3, Proposition 31).  This idea is connected to Thales instead through Lives of Eminent Philosophers, where Diogenes Laertius credits the information to the historian Pamphila of Epidaurus (1st century CE); unfortunately, her original works have not survived and so we only have Diogenes’ terse statement [7, pp. 26-27]:

Pamphila states that, having learnt geometry from the Egyptians, he was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox. Others tell this tale of Pythagoras, amongst them Apollodorus the arithmetician.

The attribution is quite vague, simply giving Thales credit for successfully inscribing a right triangle in a circle, but not specifically mentioning that the hypotenuse of said triangle would have to be a diameter of the circle.  The reference made to the mysterious “Apollodorus the arithmetician” is also a bit confusing.  In his biography of Pythagoras in the eighth book of Lives, Diogenes again cites “Apollodorus the arithmetician,” but this time saying that Pythagoras sacrificed oxen upon discovering the relationship that the sum of the squares of the sides of a right triangle equals the square on the hypotenuse, so it is seems likely that Diogenes is confusing the two stories.

Michael Molinsky (University of Maine at Farmington), "Some Original Sources for Modern Tales of Thales - Other Contributions to Geometry," Convergence (November 2015)