Thales of Miletus

This is about Thales of Miletus and what it means to study him. I am moved to ask what history is in the first place. It is a study of the freedom in which we face our conditions. Thales had his way of understanding the world, and we may benefit from trying to learn it.

“The Thaleses of the future are meeting in Didim, September 24,  2016”

“The Thaleses of the future are meeting in Didim, September 24, 2016”

I studied Thales in preparation for the Thales Buluşması (Thales Meeting), September 24, 2016, in the ancient Roman theater in the ruins of Miletus. The event had been organized by the Tourism Research Society (Turizm Araştırmaları Derneği, TURAD) and the office of the mayor of Didim. Afterwards I organized the notes that I had prepared and wrote what was nominally an introduction; the following is based on that.

On June 15, 2016, I had just learned of the possibility of speaking about Thales of Miletus, in Miletus. I was still recovering from inguinal hernia repair at the time. Having looked up Thales in Heath’s History of Greek Mathematics and Proclus’s Commentary on the First Book of Euclid’s Elements, and having remembered that Kant mentioned Thales in the Preface of the B Edition of the Critique of Pure Reason, I noted in my journal the attribution to Thales of the theorems that

  1. a diameter bisects a circle, and
  2. the base angles of an isosceles triangle are equal.

These theorems may seem obvious, from symmetry; but what if the symmetry is broken, by distorting the circle into an ellipse, or sliding to one side the apex of the triangle?

Thales’s theorems concern equality, which students today confuse with sameness. But Thomas Jefferson wrote that all of us were created equal; this does not mean we are all the same. Robert Recorde invented the equals sign out of two equal, parallel (but distinct) straight lines. “If all men are created equal,” I wrote,

does this mean women too? And Africans? What about—animals? Citing Aristotle, Collingwood says Thales conceived of the world as an animal.

Equations are thought to scare people. Perhaps an equation like A = πr² should scare people: it is a modern summary of the most difficult theorem in Euclid.

From the hotel in Didim where we were acccommodated after the Thales Meeting

From the hotel in Didim where we were acccommodated after the Thales Meeting

Such were my early thoughts of what might be talked about on the theme of Thales. By August 15, my own talk had become definite. I would speak for 20 minutes in Miletus. I noted some additional possibilities. I had a memory from childhood of playing with a rubber band and wondering if, when unstretched, it always surrounded the same area, no matter what shape. I proved that it did not, since the shape surrounded by the rubber band could be collapsed to nothing. In my talk, I might display several shapes, all having the same perimeter, and ask whether they all had the same area. If the circle has the largest area, why is this? How can it be proved?

We were given lunch before the meeting at a cafe opposite the temple of Didyma

We were given lunch before the meeting at a cafe opposite the temple of Didyma

That the circle has maximal area among shapes with the same perimeter: I am as confident of this as of anything else in mathematics, and yet I do not know how to prove it. If I needed a proof, I would look up the calculus of variations; but then the question would remain: what is the use of a proof of a theorem that is obviously true?

Herodotus names the temple for the family of priests, the Branchidae

Herodotus names the temple for the family of priests, the Branchidae

To measure the area of a rectangle, you multiply the length by the width. If you need to measure the area of an arbitrary quadrilateral, you might think it reasonable to multiply the averages of the opposite sides. This is how the ancient Egyptians measured their fields, according to Fowler in The Mathematics of Plato’s Academy; but the rule is not exact. Herodotus says the Greeks learned geometry from the Egyptians; and yet what we see in Book I of Euclid is how to measure a field with straight sides exactly.

Our guide had fetched us from the Nesin Mathematics Village in the morning; now she showed everybody around the temple

Our guide had fetched us from the Nesin Mathematics Village in the morning; now she showed everybody around the temple

Some of these thoughts made it into the talk that I actually gave in Miletus on September 24. The rest continued to be relevant, but there was only so much that could be said in 20 minutes. Meanwhile, I collected a lot of notes. Ultimately I collected these in a single LaTeX document. An electronic file seemed the best way to edit, rewrite, and rearrange; and the LaTeX program allowed easy cross-referencing.

Thales is said to have recognized the theorem that the diameter bisects the circle.  Circles divided by several diameters are said to have been common on monuments in Egypt, where Thales could have seen them

Thales is said to have recognized the theorem that the diameter bisects the circle. Circles divided by several diameters are said to have been common on monuments in Egypt, where Thales could have seen them

If there are ancient sources on Thales that I have not now seen, I am not likely to be in a position to find them. I can go on looking up what modern philosophers and historians say about Thales. I shall go on thinking my own thoughts.

We prepare to enter the inner sanctum of Didyma

We prepare to enter the inner sanctum of Didyma

At this point, I do seem to have something new to suggest about Thales, and I said it in my talk. By saying all was water, Thales recognized a kind of unity in the world. Collingwood noted this in The Idea of Nature; the Frankforts noted it in Before Philosophy: The Intellectual Adventure of Ancient Man. But I have not seen this unity connected to the unity of mathematics. A theorem is the recognition that a single principle can explain many observations. This recognition may well come before the possibility of proving the theorem. Merely asserting the theorem can be a great advance, comparable to the advance of explaining the world without appeal to gods. It looks as if Thales achieved both of these advances.

Apparently this inner sanctum was never intended to be roofed over

Apparently this inner sanctum was never intended to be roofed over

One can read the fragments of Heraclitus of Ephesus as if they were a poem. They may be translated and presented as poetry, as by Guy Davenport in Herakleitos and Diogenes. The fragments may be used in a new poem, as by T. S. Eliot in “Burnt Norton,” the first of the Four Quartets. But the term “fragment” is misleading if it suggests (as it once did to me) random utterances. The fragments are quotations made by writers who knew Heraclitus’s whole book. The serious scholar will read those writers, to get a sense of how they were reading Heraclitus.

The theater of Miletus

The theater of Miletus

We have no clear fragments of Thales himself, but we have ancient mentions of him. The mentions collected by Diels and Kranz in Die Fragmente der Vorsokratiker and by Kirk, Raven, and Schofield in The Presocratic Philosophers are often comparable in length to the fragments of Heraclitus. Again the serious scholar will read these mentions in context. I provided some of this context in my notes, at least by quoting Aristotle at greater length. But still one should read more. One should study the other Milesians, Anaximander and Anaximenes. One should study the Egypt that Thales is said to have been visited.

The view from the speakers' table

The view from the speakers’ table

There is no end to what one should study. This brings on a lament by Collingwood in Speculum Mentis. The chapters of this book are

  1. Prologue,
  2. Speculum Mentis,
  3. Art,
  4. Religion,
  5. Science,
  6. History,
  7. Philosophy, and
  8. Speculum Speculi.

In §5, “The Breakdown of History,” in the chapter “History,” Collingwood observes that, since we cannot know everything, in history at least, it would seem that we cannot really know anything:

History is the knowledge of the infinite world of facts. It is therefore itself an infinite world of thought: history is essentially universal history, a whole in which the knowledge of every fact is included.

This whole, universal history, is never achieved. All history is fragmentary. The historian—he cannot help it—is a specialist, and no one takes all history for his province unless he is content to show everywhere an equal ignorance, and equal falsification of fact. But this is a fatal objection to the claims of historical thought as we have, without favour or exaggeration, stated them. History is the knowledge of an infinite whole whose parts, repeating the plan of the whole in their structure, are only known by reference to their context. But since this context is always incomplete, we can never know a single part as it actually is.

If we are to escape this inference it can only be by withdrawing everything we have said about the structure of historical fact and substituting a new theory.

The bold emphasis (here and below) is mine. The theory that is supposed to solve the problem of history will be referred to by Collingwood as “historical atomism.” I think he must be alluding to his contemporary Wittgenstein, whose Tractatus Logico-Philosophicus had come out a few years earlier.

Ayşe did move away from the lectern to be closer to the audience, as did I

Ayşe did move away from the lectern to be closer to the audience, as did I

In its own poetic or gnomic style, the Tractatus begins:

1  The world is everything that is the case.

1.1  The world is the totality of facts, not of things.

1.2  The world divides into facts.

1.21  Any one can either be the case or not be the case, and everything else remain the same.

2  What is the case, the fact, is the existence of atomic facts.

This new theory does not work. It is only a declining to do history, since nothing studied by history happens in isolation. Collingwood continues:

It is easy to see what this new theory will be. The individuality of historical facts, we must now say, is not systematic but atomic…

…Now these atoms, as the word suggests, are nothing new in the history of thought. On the contrary, they represent a very old idea, and one from which the modern conception of critical history has but lately and with difficulty emerged. They are precisely the instances of an abstract law: that is to say, they belong, together with the allied notions of agglomeration or addition, infinite series, and externality, to the sphere of science. Historical atomism saves history by surrendering the whole thing and plunging back into the scientific consciousness. After recognizing this, we are the less surprised to see that its advocates are almost exclusively mathematicians. For it is simply a proposal to purge history of everything historical and reduce it to mathematics.

If Collingwood means to describe Wittgenstein and Russell as mathematicians, I don’t know that the description fits. The mathematician as such is not interested in solving problems outside of mathematics.

After the presentations, these young women tried to spell out the name of Thales with their shadows

After the presentations, these young women tried to spell out the name of Thales with their shadows

Collingwood goes on to assert his own interest as an historian, on his pages 235–6:

To translate our difficulty into empirical terms, we have already seen that periods of history thus individualized are necessarily beset by ‘loose ends’ and fallacies arising from ignorance or error of their context. Now there is and can be no limit to the extent to which a ‘special history’ may be falsified by these elements. The writer insists upon this difficulty not as a hostile and unsympathetic critic of historians, but as an historian himself, one who takes a special delight in historical research and enquiry; not only in the reading of history-books but in the attempt to solve problems which the writers of history-books do not attack. But as a specialist in one particular period he is acutely conscious that his ignorance of the antecedents of that period introduces a coefficient of error into his work of whose magnitude he can never be aware…Ancient history is easy not because its facts are certain but because we are at the mercy of Herodotus or some other writer, whose story we cannot check; contemporary history is unwritable because we know so much about it…

“Contemporary history is unwritable because we know so much about it.” I first read this in a book whose first paragraph is a quotation of most of page 82 of Speculum Mentis. The book is A Concise History of Modern Painting, and here, in his second paragraph, Herbert Read quotes from page 236 as I am doing, before saying of himself,

The present writer can claim to have given close and prolonged reflection to the facts that constitute the history of the modern movement in the arts of painting and sculpture, but he does not claim that he can observe any law in this history.

As I read Collingwood—and I agree—, it is science that is a search for laws; history is something else. What Collingwood concludes on his page 238 is that

history is the crown and the reductio ad absurdum of all knowledge considered as knowledge of an objective reality independent of the knowing mind.

That is a grand claim, but one that I can agree with, while recognizing that it can be taken the wrong way. Thales had his thoughts, whether we know what they were or not. In this sense, those thoughts are, or were, an “objective reality.” However, as thoughts, they can make no difference to us, unless we do manage to think them for ourselves; and doing this is entirely up to us. I think this is Collingwood’s point. It is a point that he continued to sharpen over the course of his career, as in The Principles of History, drafted fifteen years later:

I do not doubt, again, that the purely physical effects produced in man’s organism by its physical environment are accompanied by corresponding effects in his emotions and appetites; although this is a subject on which information is very difficult to procure, because what has been written about it has mostly been written by men who did not understand the difference between feelings and thoughts, or were doing their best, consciously or unconsciously, to obscure that difference. I will therefore take a hypothetical example. Suppose that there are two varieties of the human species in one of which sexual maturity, with all its emotional accompaniments, is reached earlier than in the other; and that this can be explained physiologically as an effect of climate and the like. Even so, this early sexual development is in itself no more a matter of historical interest than skin-pigmentation. It is not sexual appetite in itself, but man’s thought about it, as expressed for example in his marriage-customs, that interests the historian.

It is on these lines that we must criticize the all too familiar commonplaces about the ‘influence’ of natural environment on civilizations. It is not nature as such and in itself (where nature means the natural environment) that turns man’s energies here in one direction, there in another: it is what man makes of nature by his enterprise, his dexterity, his ingenuity, or his lack of these things…

The distinction between thinking and feeling corresponds to the distinction between The Principles of History and Collingwood’s earlier Principles of Art. Collingwood was interested in both.

After a reception in the courtyard of the museum at Miletus, on the way back to Didim, we paused to enjoy the sunset

After a reception in the courtyard of the museum at Miletus, on the way back to Didim, we paused to enjoy the sunset

In Speculum Mentis, Collingwood’s next section is called “The Transition from History to Philosophy.” I note an objection by Leo Strauss in his paper “On Collingwood’s philosophy of history”:

He justly rejected Spengler’s view that “there is no possible relation whatever between one culture and another.” But he failed to consider the fact that there are cultures which have no actual relations with one another, and the implications of this fact: he dogmatically denied the possibility of “separate, discrete” cultures because it would destroy the dogmatically assumed “continuity of history” as universal history.

I shall only suggest that it is not very good criticism to complain that a writer “failed to consider” something. Perhaps he did consider it, but thought it not worth getting into. It might be something that the reader could work out for himself.

A banquet was held at the Didim marina

A banquet was held at the Didim marina

Strauss was reviewing The Idea of History, a posthumous publication that incorporated only parts of The Principles of History. One of those parts addresses a possible objection to the last quotation from that book:

With the disappearance of historical naturalism, the conclusion is reached that the activity by which man builds himself his own constantly-changing historical world is a free activity…

This does not mean that a man is free to do what he wants. All men, at some moments in their lives, are free to do what they want: to eat, being hungry, for example, or to sleep, being tired. But this has nothing to do with the freedom to which I have referred…

Nor does it mean that a man is free to do what he chooses: that in the realm of history proper, as distinct from that of animal appetite, people are free to plan their own actions as they think fit and execute their plans, each doing what he set out to do and each assuming full responsibility for the consequences, captain of his soul and all that. Nothing could be more false…

The rational activity which historians have to study is never free from compulsion: the compulsion to face the facts of its own situation…With regard to this situation he is not free at all…

The freedom that there is in history consists in the fact that this compulsion is imposed upon the activity of human reason not by anything else, but by itself…

This is from pages 98–100 of Principles, and also pages 315–7 of Idea, though the beginning is different there. Collingwood will work more on the distinction between wanting and choosing in New Leviathan, particularly in chapters XI and XIII, “Desire” and “Choice,” from which the following fragments are taken:

11. 1.  …In appetite or mere wanting a man does not know what he wants, or even that he wants anything; in desire or wishing, he not only knows that he wants something, but he knows what it is that he wants.

13. 14.  Choice is not preference, though the two words are sometimes used as synonyms. Preference is desire as involving alternatives. A man who ‘prefers’ a to b does not choose at all; he suffers desire for a and aversion towards b, and goes where desire leads him.

13. 2.  The problem of free will is not whether men are free (for every one is free who has reached the level of development that enables him to choose) but, how does a man become free? For he must be free before he can make a choice; consequently no man can become free by choosing.

13. 21.  The act of becoming free cannot be done to a man by anything other than himself. Let us call it, then, an act of self-liberation. This act cannot be voluntary.

13. 22.  ‘Liberation from what?’ From the dominance of desire. ‘Liberation to do what?’ To make decisions.

Returning to Thales, I note the difficulty of knowing with any confidence what he thought. There are doubts that Collingwood’s thought was understood by his pupil, friend, and literary executor, Knox, the original editor of The Idea of History, who did not think most of The Principles of History worth publishing.

The yacht club at Didima

The yacht club at Didima

Why try to know somebody else’s thought? Thales is a legend, even today, particularly in the region of Miletus. This is why I found myself speaking about him. His legendary status may be used to provoke curiosity and thought.

Photo of the Ionia: A Quest, by Freya Stark

Miletus and Ephesus were Ionian cities. They may still be a locus for thoughts worth thinking. Visiting the ruins of the Ionian cities on the west coast of Turkey in 1952, Freya Stark wrote in Ionia: A Quest (pp. 2–4):

Curiosity ought to increase as one gets older…Whatever it was, the Ionian curiosity gave a twist for ever to the rudder of time. It was the attribute of happiness and virtue. To look for the causes of it is a hopeless quest in Greece itself; the miracle appears there, perfect, finished and inexplicable. But in Asia Minor there may be a chance, where Thales of Miletus, ‘having learnt geometry from the Egyptians, was the first to inscribe a right-angled triangle in a circle, whereupon he sacrificed an ox.’

…I am looking not for history but for happiness, a secret to be pursued with the accuracy of a different mood; and surely to be found; for—out of most hard and barbarous times, out of strangely modern vicissitudes, sacking of cities, emigration, slavery, exile—it still hangs unmistakable, elusive, like a sea-spray in the sun, over the coastline of Ionia.

Stark does go on to note the paradox of finding happiness in a place where the Persians might capture a city, kill most of the men, and enslave the women and children.

Collingwood finds happiness in medieval Europe, because of its institutions: the guild, the church, the monastic order, the feudal hierarchy. Writing in the Prologue of Speculum Mentis, he does offer a disclaimer, as Stark does:

we do not idealize medieval life or hold it free from defect. We do not forget either the corruptions to which these institutions too often succumbed, their tendency to level downwards, or the hideous fate of those adventurous souls who found their limits too narrow. But the very tendency to level downwards, the very narrowness of medieval institutionalism, secured one great benefit, namely the happiness of those humble ordinary men and women who ask not for adventure or excitement, but for a place in the world where they shall feel themselves usefully and congenially employed.

Ionia is a place for adventure. In writing of Clazomenae, Stark notes how Anaxagoras left it and settled in Athens, possibly even having travelled in the train of Xerxes. However, in the end,

Athens, different and old-fashioned, was outraged by Anaxagoras who announced the sun to be a red-hot stone and the moon made of earth with hills and valleys, and who spoke in a friendly way, perhaps, of the easy-going Persian rule, under which his youth had grown; so that he was brought up for trial, and rescued by Pericles, who seems to have smuggled him out of prison and away. He came back to the liberal atmosphere of Ionia, not to Clazomenae but to Lampsacus, a colony from Miletus, where he taught and died, and asked that school-children be given an annual holiday to remember him by when he was dead. This was still done many years later, and the citizens also put up in their market-place, in his memory, an altar to Mind and Truth.

In studying Thales, I seek happiness through history. Freya Stark quotes Diogenes’s fanciful story about how Thales performed a sacrifice after proving a theorem. In the Nesin Mathematics Village near Ephesus in the summer of 2016, I collaborated successfully to prove a couple of theorems, and I studied Thales. It might be pleasant to establish a ritual upon proving a theorem; but there is little reason to think that Thales actually sacrificed an ox on such an occasion. I want to know Thales, as far as possible, as he is, and not just as legend makes him to be.

Speakers at the Thales Buluşması, from the HaberTürk story at the TURAD website

Speakers at the Thales Buluşması, from the HaberTürk story at the TURAD website


  1. R. G. Collingwood. Speculum Mentis or The Map of Knowledge. Clarendon Press, Oxford, 1924. Reprinted photographically in Great Britain at the University Press, Oxford, 1946.

  2. R. G. Collingwood. The Principles of Art. Oxford University Press, London, Oxford, and New York, paperback edition, 1958. First published 1938.

  3. R. G. Collingwood. The Idea of Nature. Oxford University Press, London, Oxford, and New York, paperback edition, 1960. First published 1945.

  4. R. G. Collingwood. The Idea of History. Oxford University Press, Oxford and New York, revised edition, 1994. With Lectures 1926–1928. Edited with an introduction by Jan van der Dussen.

  5. R. G. Collingwood. The New Leviathan, or Man, Society, Civilization, and Barbarism. Clarendon Press, revised edition, 2000. With an Introduction and additional material edited by David Boucher. First edition 1942.

  6. R. G. Collingwood. The Principles of History and other writings in philosophy of history. Oxford, 2001. Edited and with an introduction by W. H. Dray and W. J. van der Dussen.

  7. Guy Davenport. Herakleitos and Diogenes. Grey Fox Press, San Francisco, 1979. Translated from the Greek. Fourth printing, 1990.

  8. Hermann Diels, editor. Die Fragmente der Vorsokratiker. Weidmannsche, 1960. Revised by Walther Kranz.

  9. T. S. Eliot. The Complete Poems and Plays 1909-1950. Harcourt Brace Jovanovich, San Diego, 1971.

  10. David Fowler. The Mathematics of Plato’s Academy: A new reconstruction. Clarendon Press, Oxford, second edition, 1999.

  11. Henri Frankfort, Mrs H. A. Frankfort, John A. Wilson, and Thorkild Jacobsen. Before Philosophy: The Intellectual Adventure of Ancient Man. An Oriental Institute Essay. Penguin, Baltimore, 1949. First published 1946 as The Intellectual Adventure of Ancient Man. Reprinted 1971.

  12. Thomas Heath. A History of Greek Mathematics. Vol. I. From Thales to Euclid. Dover Publications Inc., New York, 1981. Corrected reprint of the 1921 original.

  13. Thomas Heath. A History of Greek Mathematics. Vol. II. From Aristarchus to Diophantus. Dover Publications Inc., New York, 1981. Corrected reprint of the 1921 original.

  14. Heracleitus. On the universe. In Hippocrates Volume IV., volume 150 of the Loeb Classical Library. Harvard University Press, Cambridge, Massachusetts and London, England, 1931. With an English translation by W. H. S. Jones.

  15. Herodotus. The Persian Wars, Books I–II, volume 117 of the Loeb Classical Library. Harvard University Press, Cambridge, Massachusetts and London, England, 2004. Translation by A. D. Godley; first published 1920; revised, 1926.

  16. Immanuel Kant. Critique of Pure Reason. The Cambridge Edition of the Works of Kant. Cambridge University Press, Cambridge, paperback edition, 1999. Translated and edited by Paul Guyer and Allen W. Wood. First published 1998.

  17. G. S. Kirk, J. E. Raven, and M. Schofield. The Presocratic Philosophers. Cambridge University Press, second edition, 1983. Reprinted 1985. First edition 1957.

  18. Proclus. A Commentary on the First Book of Euclid’s Elements. Princeton Paperbacks. Princeton University Press, Princeton, NJ, 1992. Translated from the Greek and with an introduction and notes by Glenn R. Morrow. Reprint of the 1970 edition. With a foreword by Ian Mueller.

  19. Herbert Read. A Concise History of Modern Painting. Thames and Hudson, London, new and augmented edition, 1974. Reprinted 1986.

  20. Robert Recorde. The Whetstone of Witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers. Jhon Kyngston, London, 1557. Facsimile from, March 13, 2016.

  21. Freya Stark. Ionia: A Quest. Tauris Parke Paperbacks, London, 2010. First published in 1954 by John Murray.

  22. Leo Strauss. On Collingwood’s philosophy of history. The Review of Metaphysics. V(4), June 1952.

  23. Ludwig Wittgenstein. Tractatus Logico-Philosophicus. Routledge, London, 1922. Reprinted 1998.



  1. Posted December 24, 2016 at 6:37 pm | Permalink | Reply

    why is A = pI*r^2 the most difficult?

    • Posted December 24, 2016 at 7:50 pm | Permalink | Reply

      Thanks for reading, Bill. The proposition is XII.2: circles are to one another as the squares on their diameters. The proof needs some kind of infinitesimal calculus (which Euclid provides).

  2. australiannumerals
    Posted December 28, 2016 at 10:39 am | Permalink | Reply

    Thank you for this essay David.

    I felt like a tourist through time while reading it. Given Thales would have seen surveyors using rope whilst learning geometry from the Egyptians, is it possible Thales could have squared the circle as shown in the constructions below? The first proof by Murray Bourne is via Pythagoras and assumes knowledge post-Thales Yet my proof is via Thales’ similar triangles, might have been within his grasp. My proof is at My cartoon animation at

    As for Euclid’s proposition XII.2 ‘circles are to one another as the squares on their diameters’, that means we have another unexplored construction. Pythagoras said a□ + b□ = c□ yet we also have a◯ + b◯ = c◯ as shown at

    I look forward to more writing on ancient Greek mathematics in 2017! In the meantime your readers might enjoy this fun article which incorporates another theorem of Thales:

    Best wishes
    Jonathan Crabtree

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