Monday, 2 July 2018

The Mathematics of Thales

Most of us who have gone to school, know of Pythagoras, Archimedes, and Euclid as the famous mathematicians of ancient Greece. Some of us have heard of other great mathematicians like Eratosthenes, Hipparchus, Apollonius, Aristarchus, etc. But, Thales?

Most schools of the world today, I suspect, teach science and mathematics from a predominantly European syllabus. This is partly an effect of European colonization of most of Asia and Africa, and dimunition of native populations in the Americas and Australia,  in the eighteenth, nineteenth and twentieth centuries of the Christian calendar.

I am currently reading a book titled “Archimedes” by Thomas Little Heath, originally published in 1920, on Kindle. It is a free download, and part of a Men of Science series.

“Greek authors from Heredotus downwards (meaning, after him) agree in saying that geometry was invented by the Egyptians and that it came into Greece from Egypt,” writes Heath.

He quotes an account : “Geometry is said to have been invented among Egyptians, its orgin being due to the measurement of plots of land. This was necessary because of the rising of the Nile, which obliterated (erased) boundaries appertaining to separate owners…. Thales first went to Egypt and thence introduced this study into Greece.”

What we know today as the Pythagoras theorem, about the hypotenuse of right angled triangles, is listed as Proposition 47 in Volume I of Euclid’s “Elements”, the standard European and Arab book of mathematics from the first to eighteenth centuries. The word Elements is the Greek word for Numbers, which in the eighteenth century was adopted into French, English etc for the most basic objects in Chemistry. The word Geometry is formed from two words Geo (Earth) and Metry (measurement). The Sanskrit word for measurement is Maatra. Greek and Sanskrit are part of the Indo European language family, so these words originate from the same root. The Sanskrit word for geometry is Shulba Sutra. Shulba is the Sanskrit word for rope or string; the earliest surviving books are not about  land measurement or business but measurement of altars for yajnas. Parallelly, jyotishaas (astronomers) developed a different stream of mathematics to determine time based on the movement of celestial objects.

“Thales, who had travelled in Egypt and there learnt what the priests could teach him on the subject, introduced GeoMetry into Greece. Almost the whole of Greek science and philosophy begins with  Thales. His dae was about 624-547 B.C. First of the Ionian philosophers, and declared one of the Seven Wise Men in 582-581, he shone in all fields, as astronomer, mathematician, engineer, statesman and man of business,” says Heath.

What fascinated me is what follows, which is Heath’s listing of the contributions of Thales to mathematics and astronomy.

In Astronomy, Thales:
  • Predicted the solar eclipse of 28 May, 585 BC
  •  Discovered the inequality of the four astronomical seasons
  • Counselled the use of the Little Bear instead of the Great Bear as a means of finding the pole (i.e. North Pole)

In Geometry, the following theorems are attributed to Thales:
  1. That a circle is bisected by any diameter
  2. That the angles at the base of an isoceles triangle are equal
  3. That if two straight lines cut one another, the vertically opposite angles are equal
  4. That if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (what we now called Side-Angle-Side congruency)
  5. Was first to inscribed a right angled triangle in a circle, which means he was first to discover that the angle in a semi circle is a right angle.

Thales also solved two problems in practical geometry
  •         He showed how to measure the distance from the land, of a ship at sea (using Proposition 4 above)
  •         He measured heights of pyramids by means of the shadow thrown on the ground

Heath adds, “Their character (of the theorems) shows how the Greeks had to begin at the very beginning of the theory.”

Thales was a practical man, a businessman, and indulged in theoretical excursions perhaps as a pasttime. Pythagoras came a generation after him, and founded a school of mathematics. Euclid’s famous Elements is dated to the first century, AD, six hundred years after Thales. Pythagoras was interested in Mathematics as a leisure activity and an intellectual pursuit. So were most of this followers; his students and such people who pursued Geometry as an intellectual pursuit were collectively called the Pythagoreans. They coined the Greek word Mathemata (μάθήμάτα) from which comes the English tatbhava word Mathematics. The Greek word Mathemata literally means “Subjects of Instruction”.

Both Thales and Pythagoras traveled extensively around and past the shores of the Mediterranean sea, not just Egypt. Other historians of mathematics conjecture that Pythagoras traveled to Persia and perhaps even India. Ancient Greeks themselves acknowledge the Egyptian origins of their mathematics. Through formal and inductive methods, and intellectual pursuits, the Greeks elevated mathematics to a much higher levels, especially in Geometry. Strangely very few people of the Roman Empire that succeeded Hellenic Greece did not continue in this path, though the Persians and central Asians who acquired Greek books via Alexander of Macedonia’s conquest, did. Europe almost entirely abandoned mathematics and science for a millennium, until the emergence of Italian city states and Fibonacci’s ventures to Baghdad.

Pythagoras is introduced to us as the fountainhead of Greek mathematics, just as Aryabhata or some Vedic rishis are mentioned as the fountainhead of Indian mathematics. No mention is made at all of the Egyptian origins of Greek mathematics. Similarly while there are some references to Greeks and Romans in Indian mathematical texts, especially of Varahamihira, in those times, there was no attempt at studying the history of mathematics or acknowledging foreign elements borrowed.

I find Thales a fascinating character, more so perhaps than even Varahamihira. He reminds me of Benjamin Franklin, Antoine Lavoisier and Sir William Jones. And Mahendra Varma Pallava.

I strongly recommend Heath’s biography of Archimedes, from which I have excerpted. The list of his books and their titles also tell you what the level of mathematics was in Hellenic Greece. I also hope to read some biography of Apollonius, whom Simon-Pierre Laplace equates with Archimedes as the great mathematicians of ancient times.

On a linguistic note, you can read each letter in the word μάθήμάτα based on Greek letters use in modern mathematics texts – mu, alpha, theta, eta, mu, alpha, tau, alpha. Notice that Greek has separate letters θ and τ for tha and ta like Sanskrit and Tamil, whereas English doesn’t. So English has to use two letters th for the equivalent of theta or த. That’s another story for another occasion, another blog.

There seem to be some pictures or sculptures of Thales, though I don’t know if they are authentic.
Thales - source : Wikipedia
This one is from Wikipedia. Several Greek philosophers kings and others were depicted in paintings on vases, stone sculptures, bronzes, etc. Unfortunately such depictions in India were rare before the Gupta period; any pictures you see of mathematicians like Aryabhata, Bhaskara, or Varahamihira are entirely the product of recent artists’ imaginations.

Related Blogs and Videos

  1. Archimedes by Thomas Little Heath (free Kindle edition)
  2. Detailed article on Thales
  3. Video of my talk on Astronomy and Mathematics of Ancient Cultures
  4. My essay on Aryabhata (The Week magazine)
  5. Notes from Manjul Bhargava lecture on history of Indian mathematics
  6. Antoine Lavoisier


  1. Interesting article, as usual.

    "4. That if two triangles have two angles and one side respectively equal, the triangles are equal in all respects (what we now called Side-Angle-Side congruency)"

    Thales also solved two problems in practical geometry
    - He showed how to measure the distance from the land, of a ship at sea (using Proposition 4 above)
    - He measured heights of pyramids by means of the shadow thrown on the ground"

    These, in particular, are interesting. I suppose, it was done using pure triangular geometry and no trigonometry yet.

  2. I always thought it's trigonometry that we use for measuring large distances and heights, such as distance of a remote object such as a ship at sea or height of tall structures such as pyramids mentioned in this article. At least, that was the perception from my school study.
    Good to see this article shows a different view - the use of Geometry. It would be nice to teach children concepts such as congruency using these stories of Thales.