This article will provide a brief outline of some stories about Thales of Miletus that often appear in modern publications, tracing each item back to the oldest existing historical sources. Interactive applets illustrate methods of measurement attributed to Thales.

There are many recent history of mathematics books written for general audiences that provide information about the ancient Greek mathematician, astronomer, and philosopher Thales of Miletus. While these general resources tend to repeat the same anecdotes and the same brief list of mathematical accomplishments, they do not always include citations of the ancient sources from which these stories are being drawn. In some cases these books even include quotations which are ascribed to Thales, which is odd, since these modern sources will also often mention that no works written by Thales exist – and, in fact, it is openly debated whether he wrote any at all.

The truth is, as for many figures of the distant past, the information we have about Thales is being drawn from a wide range of ancient sources, some of which contain only terse references and almost all of which were written centuries or even millennia after the time in which Thales lived. Although these sources differ significantly in terms of content and sometimes contradict one another, there seems to be general agreement on at least the following:

- Thales lived sometime in the 6
^{th}century BCE. - He lived in the Greek city-state Miletus (located on the Mediterranean coast of what is present-day Turkey).
- He had a reputation in the Greek world for accomplishments in philosophy, politics, engineering, astronomy and geometry.

In this article, we will examine some of the achievements commonly attributed to Thales, and in each case explore the oldest existing historical sources that form their basis.

While there exist today many different artists' depictions of Thales, it should be noted that no surviving source from antiquity provides any physical description of Thales. If any artwork of Thales was made during his lifetime, it hasn’t survived (and, even if such artwork did exist, it wouldn’t provide an accurate likeness of Thales anyway, since the beginning of the development of faithful, accurate likenesses in Greek art and sculpture took place years later).

For the period that includes the sources discussed in this article (roughly before 600 CE), there are only two surviving works of art that are clearly labeled as being depictions of Thales. One is a wall painting (see Figure 1 below). The painting was made during the first half of the second century CE, and is located in the Baths of the Seven Sages in the Roman river port Ostia.

**Figure 1:** “Ostia, Baths of the Seven Sages (II),” photographed by George Houston, is licensed under CC BY 2.0

The other artwork is part of a mosaic that decorated the floor of the dining room of a Roman villa constructed in the second century CE near Baalbek, in present-day Lebanon. The mosaic shows the muse Calliope, Socrates and the Seven Wise Men of ancient Greece, including Thales (see Figure 2 below).

**Figure 2:** "Thales. Roman mosaic from Suweydie near Baalbek," photographed by Marco Prins, is licensed under CC BY-NC-SA 4.0

No existing Roman or Greek sculpture can be identified with any certainty as being an intended depiction of Thales. The only piece of sculpture inscribed with Thales’ name now resides in the Vatican museum: a partial *herm* (a squared pillar, often with inscriptions, topped by a sculptured head), which was found near the Porta Maggiore in Rome; unfortunately, the top portion including the head was never recovered.

One of the most commonly repeated stories about Thales is that, in order to prove the value of learning and philosophy, he made himself rich by controlling access to olive presses needed after the harvest. The story can be traced all the way back to the Greek philosopher Aristotle (384 – 322 BCE), who details it in Book I, Section 11 of his work *Politics* [2, pp. 1997-1998]:

There is the anecdote of Thales the Milesian and his financial scheme, which involves a principle of universal application, but is attributed to him on account of his reputation for wisdom. 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. He is supposed to have given a striking proof of his wisdom, but as I was saying, his scheme for getting wealth is of universal application, and is nothing but the creation of a monopoly. It is an art often practiced by cities when they are in want of money; they make a monopoly of provisions.

While some modern sources aimed at general audiences do mention Aristotle as the original source of the story, they rarely seem to repeat his obvious skepticism. He points out that the concept of a monopoly was well known, and suggests that the entire story was being attributed to Thales simply based on his reputation as a wise man.

As Aristotle indicates, Thales had a wide-ranging reputation for wisdom in the ancient Greek world and beyond, which led to his name being used in much the same way that the name “Einstein” is used in popular media today: as a generic term used in conversation to indicate a smart person. For example, in the play *The Birds*, the Greek playwright Aristophanes (c. 446 – c. 386 BCE) had one character exclaim, “Why, the man’s a Thales!" [9, p. 229]. Similarly, the Roman playwright Titus Maccius Plautus (c. 254 – 184 BCE), in his work *The Rope,* has one character sarcastically tell another, “Hello, there, you Thales" [12, p.71].

While the story of the olive presses emphasized the supposed real-world applications of the wisdom of Thales, the common tale of Thales and the well presents the opposite story, one of practical problems associated with too much learning. The oldest existing source for this story is the dialogue *Theatetetus* written by Plato (c. 429 – c. 347 BCE) [4, pp. 301-302]:

Socrates:Well, here’s an instance: they say Thales was studying the stars, Theodorus, and gazing aloft, when he fell into a well; and a witty and amusing Thracian servant-girl made fun of him because, she said, he was wild to know about what was up in the sky but failed to see what was in front of him and under his feet.

While of course it is perfectly possible that Thales really did fall down a well because he was watching the stars rather than the ground, it seems a bit more probable that Plato simply invented the story in order to make a point. As we’ve already seen, Thales had a wide reputation in the ancient world, and so it would make sense for Plato to select his very recognizable name to use in a story of a philosopher too absorbed in learning for his own good.

Many of the surviving ancient sources mention that Thales traveled to Egypt at some point in his life. While it is certainly possible for him to have done so, historians have suggested that it is possible that Thales is linked to Egypt simply to strengthen the tradition of tracing Greek mathematics back to Egypt. In any case, one of the earliest tales regarding Thales and Egypt is that while there, he used shadows to measure the heights of the pyramids.

The earliest reported source for the tale of the pyramids is the philosopher Hieronymus of Rhodes (3rd century BCE)**.** Unfortunately, none of the works of Hieronymus survive, and we know what he wrote only through brief quotations and summaries that appear in later sources. The connection to Hieronymus is made in *Lives of Eminent Philosophers* by Diogenes Laertius. (This work is something of an enigma: we know nothing of the author other than his name, and it isn’t even known with much precision when the work was written, although many date it to approximately 3rd century CE.)

**Figure 3:** The first page of the biography of Thales in an edition of *Lives of Eminent Philosophers* published in France in 1761 (Wikimedia Commons, public domain)

Diogenes Laertius includes a substantial biography of Thales, and he credits the work *Scattered Notes* by Hieronymus for many of the details, including the following [7, p. 29]:

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.

There are some other surviving sources that predate *Lives of Eminent Philosophers,* but come after Hieronymus, that include the same story. One example is the Roman author Gaius Plinius Secundus (c. 23 – 79 CE), also known as Pliny the Elder. In Book 37, Chapter 17 of his work *Natural History*, Pliny states essentially the same version of the story (although he makes no mention of Hieronymus, and explicitly makes the much bolder claim that Thales was the first to ever measure heights in such a fashion) [3, p. 338]:

The method of ascertaining the height of the Pyramids and all similar edifices was discovered by Thales of Miletus; he measuring the shadow at the hour of the day at which it is equal in length to the body projecting it.

This tale is also told in the *Moralia*, written by the Greek historian and biographer Plutarch (c. 45 – 120 CE). The second book of the *Moralia* includes the *Dinner of the Seven Sages*, a work of historical fiction depicting a dinner for seven wise men of antiquity, including Thales. Before the dinner begins, the character Neiloxenus, one of the additional characters over and above the seven wise men, says the following to Thales [1, p. 353]:

In your case, for instance, the king finds much to admire in you, and in particular he was immensely pleased with your method of measuring the pyramid, because, without making any ado or asking for any instrument, you simply set your walking-stick upright at the edge of the shadow which the pyramid cast, and, two triangles being formed by the intercepting of the sun's rays, you demonstrated that the height of the pyramid bore the same relation to the length of the stick as the one shadow to the other.

Unlike the other two versions above, Plutarch does not specifically limit Thales’ method to shadows that exactly match the given heights to be measured, but allows that the heights could be computed with shadows of any length. You may also have noted that while all three ancient sources give a basic description, the lack of detail somewhat glosses over practical issues, such as how to measure the entire length of the pyramid’s shadow from tip to the center of the pyramid (see [10] for a recent article that considers these missing details).

**Figure 4.** A method for measuring the height of a pyramid attributed to Thales

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.

Many modern works provide quotations that are attributed to Thales. For example, in his book *In Mathematical Circles*, Howard Eves gives many such quotations, including the following: “I will be sufficiently rewarded if, when telling it to others, you will not claim the discovery as your own, but will say it was mine. [5, p. 24]” As mentioned earlier, we have no existing works written by Thales, and unfortunately Eves provides no citation or source, so the question remains: where exactly does this “quotation” come from?

It can actually be traced back to the *Florida*, a collection of speeches by the author Apuleius (2nd century CE). In the eighteenth speech of the collection, Apuleius addresses the residents of Carthage and offers anecdotes of the achievements of both Thales and the Greek philosopher Protagoras (c. 490 – c. 420 BCE). After listing a wide range of other achievements attributed to Thales, Apuleius concludes [12, p. 159]:

He also, when far advanced in old age, devised a divine theory about the sun, which I have not only learned but confirmed by experience, namely, by what multiple of its size the sun measures its own orbit. Thales is reported to have explained this when he had just discovered it to Mandraytus of Priene, who was delighted by the new and unexpected knowledge, and asked him to name whatever amount of money he wanted to receive as a reward for so wonderful a proof. ‘It will be enough of a reward,’ said Thales the Sage, ‘if, when you begin to make known to others what you have learned from me, you do not attribute it to yourself, but declare that I, and no one else, is responsible for the discovery.’ A handsome reward indeed, worthy of such a man, and everlasting!

In this context, the discovery for which Thales expects acknowledgement appears to be measuring the angular diameter of the sun as it appears from the Earth. It is interesting to note that no other existing source before Apuleius credits Thales with this discovery; in fact, the only other ancient author who credits Thales with this computation is Diogenes Laertius, who states in *Lives of Eminent Philosophers* that “… according to some [Thales was] the first to declare the size of the sun to be one seven hundred and twentieth part of the solar circle …" [7, p. 25], which is indeed a relatively accurate approximation of the Sun’s apparent angular diameter.

**Figure 6: **Angular diameter of the Sun (or other distant object)

Unfortunately, neither Apuleius nor Diogenes give any indication of any older authorities from which they might be drawing this information. Also, there are other ancient authors who credit the discovery differently. For example, in *The Sand Reckoner*, Archimedes (c. 287 – 212 BCE) attributes the calculation to Aristarchus of Samos (c. 310 – c. 230 BCE), saying “… Aristarchus discovered that the sun appeared to be about one seven hundred and twentieth part of the circle of the zodiac …" [6, p. 223]. On the other hand, the Greek astronomer Cleomedes (who lived sometime in the 2nd - 4th centuries CE) makes no mention of Thales or Aristarchus and instead credits the Egyptians, claiming that they used a water clock to approximate the Sun’s angular diameter to one seven hundred and fiftieth part of a complete orbit (see [11] for an evaluation of this claim).

Given that it is open to question whether or not Thales really did measure the sun’s apparently angular diameter, it seems fair to also express doubt about the quotation Apuleius attributes to Thales. Of course it is possible that Apuleius had access to sources no longer available to us today, including ones that might contain the exact words of Thales; however, it seems more reasonable to posit that Apuleius wanted to make a speech about giving credit where credit was due, and the wide fame of Thales made him a natural choice for the protagonist of the story.

While this article has discussed a few of the most common stories involving Thales that appear in modern sources, it has by no means covered them all. Readers are encouraged to perform their own explorations, tracing modern statements about Thales (or any other mathematicians of interest, for that matter) back to their original sources. Listed below as references are some useful resources on Thales, many of which include both the original Greek or Latin texts as well as English translations. The growth of digital libraries means it has never been easier to gain access to historically important works. Finally, always remember: Reading is Fundamental!

- Babbitt, Frank Cole (trans.).
*Moralia*. Volume 2. Cambridge, MA: Harvard University Press, 1928. Perseus Digital Library. Web. 14 Jan. 2015. - Barnes, Jonathan (ed.).
*The Complete Works of Aristotle: The Revised Oxford Translation.*Volume 2. Princeton, N.J: Princeton University Press, 1995. Print. - Bonstock, John (trans.) and Henry T. Riley (trans.).
*The Natural History of Pliny*. Volume 6. London: H. G. Bohn, 1855-57. HathiTrust Digital Library. Web. 22 Aug. 2015. - Burnyeat, Myles (ed.) and M. J. Levett (trans.).
*The Theaetetus of Plato*. Indianapolis: Hackett, 1990. Print. - Eves, Howard.
*In Mathematical Circles*. Boston: Prindle, Weber and Schmidt, 1969. Print. - Heath, T. L. (ed.).
*The Works of Archimedes*, London: C. J. Clay and Sons, 1897. Online Library of Liberty. Web. 23 Aug. 2015. - Hicks, Robert D. (trans.).
*Lives of Eminent Philosophers*. Volume 1. London: W. Heinemann, 1980. Print. - Morrow, Glenn R. (trans.)
*. A Commentary on the First Book of Euclid's Elements*. Princeton, NJ: Princeton University Press, 1970. Print. - Rogers, Benjamin B. (trans.).
*Aristophanes*. Volume 2. Cambridge, MA: Harvard University Press, 1979. Print. - Redlin, Lothar, Ngo Viet and Saleem Watson. “Thales’ Shadow.”
*Mathematics Magazine*73:5 (Dec. 2000): 347-353. JSTOR. Web. 22 Dec. 2014. - Wasserstein, A. “Thales’ Determination of the Diameters of the Sun and the Moon.”
*The Journal of Hellenic Studies*75 (1955): 114-116. JSTOR. Web. 23 Aug. 2015. - Wöhrle, Georg (ed.), Richard McKirahan (trans.), Ahmed Alwishah, and Gotthard Strohmaier.
*The Milesians: Thales*. 2014. Print.*Traditio Praesocratica*; Volume 1.