According to Wikipedia’s web page dedicated to Thales of Miletus, he came to the idea to prove the equality of vertical angles as a theorem when he saw that the Egyptians, after intersecting two lines, would overlay an angle over the opposite to check their equality. But Thales concluded that this measurement was unnecessary and showed this by "proof." Whether he indeed proved it or whether he was the first to do so is not relevant here.

The Wikipedia page gives reference to a book.

Shute, William George; Shirk, William W.; Porter, George F. (1960). Plane and Solid Geometry. American Book Company. pp. 25–27.

Unfortunately, this book has no references.

In Michael Molinsky's Convergence article Some Original Sources for Modern Tales of Thales - Other Contributions to Geometry, there is no reference to this story.

Finally, I did not find such a reference in Proclus’ Commentary on the first book of Euclid's Elements.

Is this story supported by a genuine reference?


1 Answer 1


Yes, it is, but only partially. Moreover, the ancient sources reporting the episode(s) are notoriously not the most reliable, and the story, as reported by Wikipedia and its sources, is probably fabricated merging together different quotes relating to different results.

These are all the ancient sources that report in some way the episode of Thales' stay in Egypt.

Pliny the Elder, Naturalis Historia, XXXVI, 82:

Thales of Miletus discovered how to find the measure of the height [of the pyramids] by measuring their shadow at the moment when it is equal [to that of] the projecting body.

Plutarch, Moralia, Septem sapientium convivium, 2:

[here, one of the wise men, Niloxenus, is addressing Thales directly] and whereas he honors you for divers great accomplishments, he particularly admires you for this invention, that with little labor and no help of any mathematical instrument you took so truly the height of one of the pyramids; for fixing your staff erect at the point of the shadow which the pyramid cast, two triangles being thus made by the tangent rays of the sun, you demonstrated that what proportion one shadow had to the other, such the pyramid bore to the stick.

Diogenes Laertius, Lives and Opinions of Eminent Philosophers, I, 24:

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.

Ibidem, I, 27:

He had no instructor, except that he went to Egypt and spent some time with the priests there. Hieronymus informs us that he measured the height of the pyramids by the shadow they cast, taking the observation at the hour when our shadow is of the same length as ourselves.

Proclus does not says the Thales prove the theorem when he saw that the Egyptians intersecting two lines. Instead, he says that Thales brought geometry to Greece from Egypt (65, 3) and reports other results concerning the fact that the diameter divides the circle into two equal parts (157, 10), that the angles at the base of an isosceles triangle are equal (250, 20), and that two triangles with one side and two adjacent angles are equal (352, 14). About your question, the relevant part is in 299, 1 (see here for the English translation):

Proposition XV, theorem VIII

If two right lines cut one another, they will form the angles at the vertex equal.

We must call successive angles different from such as are vertical. For these last originate from the section of two right lines: but the former from the mere dissection of the one by another. [...] But if the two right lines mutually cut each other, they form vertical angles. [...] This, therefore, is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented (according to Eudemus) by Thales: but was thought worthy of a demonstration producing science by the institutor of the Elements. But it is not exhibited from all the particulars requisite to a perfect proposition. For construction is wanting in the present theorem: but demonstration, which must be necessarily inherent, depends on the thirteenth therem.

In conclusion, even if one wants to rely on Proclus' testimony through Eudemus, the story that Thales thought up the demonstration of the theorem by observing the behaviour of the Egyptians is not based on sources, and (although it may be plausible in some respects) seems rather fabricated by combining Proclus' quotation with the episode cited by Pliny, Plutarch and Diogenes concerning the measurement of the height of the pyramids.

Lastly, observe that ancient sources, when reporting Thales' "scientific" achievements, dwell rather on his contribution to astronomy: so does the Suda, but also Callimachus (in Iambs, Papyrus Oxyrhynchus VII, 33), Aetius (in many passages of De Placita Philosophorum), Dercyllides (cited by Theon of Smyrna) and Apuleius (in Florida).

  • 1
    $\begingroup$ Thanks for the complete list. So the conclusion is that the available historical sources, more or less reliable, do not sustain the story about the vertical angles. Can we classify it as a "didactical concoction"? $\endgroup$
    – bandi
    Oct 10, 2022 at 0:59
  • $\begingroup$ @bandi Yes, I've edited my answer and added a possible explanation $\endgroup$
    – user6530
    Oct 10, 2022 at 11:19

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