The Egyptians and Babylonians knew how to calculate. The Greeks asked a different question: why does it work? Between roughly 600 and 300 BCE, a series of Greek thinkers transformed mathematics from a collection of useful recipes into a subject built on definitions, logic and proof, and that change still shapes every maths lesson you take.
Thales of Miletus: The First Mathematician?
Thales lived around 624–546 BCE in Miletus, a wealthy Greek trading city on the coast of what is now Turkey. Later Greek writers called him the first of the Seven Sages and credited him with being the first person to prove geometric facts rather than simply state them. These attributions come from writers centuries after his death, so we should say they are "traditionally attributed" to him. They include:
- A circle is bisected by its diameter.
- The base angles of an isosceles triangle are equal.
- Vertically opposite angles are equal.
- An angle inscribed in a semicircle is a right angle (still called Thales' theorem).
According to tradition, Thales travelled to Egypt and astonished the priests by measuring the height of a pyramid using only a stick and the shadows it cast.
Worked Example 1: Thales' shadow method
Thales waited until the sun's rays made shadows that were proportional to height. Suppose a vertical stick 2 m tall casts a shadow 3 m long. At the same moment, the shadow of the pyramid, measured from the centre of its base to the tip of the shadow, is 219 m.
Step 1: The sun's rays are parallel, so the stick and its shadow form a triangle similar to the pyramid and its shadow.
Step 2: Corresponding sides of similar triangles are proportional:
$$ \frac{\text{height of pyramid}}{219} = \frac{2}{3} $$
Step 3: Solve:
$$ \text{height} = 219 \times \frac{2}{3} = 146 \text{ m} $$
The Great Pyramid's original height was about 146.6 m, so this method works beautifully. The key idea, similar triangles, is a cornerstone of Matric geometry.
Pythagoras and His Brotherhood
Pythagoras was born on the island of Samos around 570 BCE and later founded a community in Croton, southern Italy. The Pythagoreans were part religious order, part school. Their motto, roughly "All is number", reflected their belief that whole numbers and their ratios explained the universe. They discovered, for example, that pleasing musical intervals correspond to simple length ratios on a string: 2 : 1 for an octave and 3 : 2 for a fifth.
The theorem that bears his name,
$$ a^2 + b^2 = c^2 $$
was known to the Babylonians over a thousand years earlier. What the Pythagorean school is traditionally credited with is a general proof. Because the group kept its work secret and credited everything to its founder, we cannot know who actually proved what.
The crisis of irrational numbers
The Pythagoreans' belief that everything could be expressed as a ratio of whole numbers was shattered when someone, traditionally named Hippasus of Metapontum, showed that the diagonal of a unit square, \( \sqrt{2} \), cannot be written as a fraction. Legend says he was drowned for revealing it, though this story may be invented.
The proof is a model of Greek reasoning. Suppose \( \sqrt{2} = \tfrac{p}{q} \) in lowest terms. Then \( p^2 = 2q^2 \), so \( p^2 \) is even, so \( p \) is even. Write \( p = 2k \). Then \( 4k^2 = 2q^2 \), giving \( q^2 = 2k^2 \), so \( q \) is also even. But then \( \tfrac{p}{q} \) was not in lowest terms, a contradiction. Therefore \( \sqrt{2} \) is irrational.
The Athenian Period
During the 5th and 4th centuries BCE, Athens became the intellectual centre. Plato (around 428–348 BCE) did not make major discoveries himself, but his Academy encouraged mathematics; tradition says the entrance bore the words "Let no one ignorant of geometry enter". Eudoxus of Cnidus (around 408–355 BCE) developed a theory of proportion that handled irrational lengths rigorously, and the method of exhaustion, an early ancestor of limits. Three famous construction problems also emerged in this period: squaring the circle, doubling the cube and trisecting an angle using only straightedge and compass. All three were proved impossible in the 19th century.
Euclid and the Elements
Around 300 BCE, in the new city of Alexandria in Egypt, a teacher named Euclid wrote the Elements, a 13-book work that collected and organised the geometry and number theory known to the Greeks. Almost nothing is known about Euclid's life, but his book became the most influential mathematics textbook in history, used for over 2,000 years.
The genius of the Elements is its structure. It begins with:
- Definitions, such as "A point is that which has no part."
- Five postulates, assumptions about geometry, such as "A straight line can be drawn between any two points."
- Common notions, such as "Things equal to the same thing are equal to each other."
From these, Euclid proves 465 propositions, each built only on what came before. This axiomatic method is how all of modern mathematics is organised. His fifth postulate, about parallel lines, puzzled mathematicians for centuries and eventually led to non-Euclidean geometry in the 1800s.
Book IX contains a famous proof that there are infinitely many prime numbers: multiply any finite list of primes together and add 1, and the result has a prime factor not on the list.
Worked Example 2: Euclid's algorithm
Book VII gives a method for finding the greatest common divisor (GCD) of two numbers, still used in computers today. Find \( \gcd(1071, 462) \).
Step 1: Divide the larger by the smaller and keep the remainder:
$$ 1071 = 2 \times 462 + 147 $$
Step 2: Repeat with 462 and 147:
$$ 462 = 3 \times 147 + 21 $$
Step 3: Repeat with 147 and 21:
$$ 147 = 7 \times 21 + 0 $$
Step 4: The remainder is now 0, so the last non-zero remainder is the GCD:
$$ \gcd(1071, 462) = 21 $$
Check: \( 1071 = 21 \times 51 \) and \( 462 = 21 \times 22 \), and 51 and 22 share no common factor. Correct.
Did You Know?
- After the Bible, the Elements is often said to be the most printed book in the Western world, with over 1,000 editions since the first printing in Venice in 1482.
- The word "mathematics" comes from the Greek mathema, meaning "that which is learned".
- Our oldest complete copies of the Elements come not from Euclid's time but from Byzantine manuscripts copied over 1,000 years later, and Arabic translations helped preserve it.
Key Takeaways
- Greek mathematicians introduced deductive proof, asking why results are true, not just how to use them.
- Thales is traditionally credited with the first geometric proofs and with using similar triangles to measure heights.
- The Pythagoreans linked numbers to nature, and the discovery that \( \sqrt{2} \) is irrational challenged their worldview.
- Euclid's Elements, written around 300 BCE, built geometry from axioms and became the model for all mathematics.
- Euclid's algorithm for the GCD and his proof of infinitely many primes are still taught and used today.
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