Pythagoras & Euclid

4.1 Thales of Miletus (c. 624–546 BCE)

Before Pythagoras, before Euclid, there was Thales of Miletus — widely regarded as the first philosopher and the first mathematician in the Western tradition. Born in the prosperous Ionian trading city of Miletus on the coast of modern-day Turkey, Thales is traditionally credited with introducing deductive reasoning into mathematics: the radical idea that geometric truths could be established not merely by measurement or observation, but by logical argument from accepted principles.

Ancient sources attribute several fundamental geometric results to Thales. According to the later commentator Proclus (drawing on the lost history of Eudemus), Thales proved that a circle is bisected by any diameter, that the base angles of an isosceles triangle are equal, that vertical angles formed by intersecting lines are equal, and — most famously — that any angle inscribed in a semicircle is a right angle.

Thales is also credited with one of the earliest applications of mathematical reasoning to practical measurement. According to Plutarch and other ancient writers, Thales determined the height of the Great Pyramid of Giza by measuring the length of its shadow at the precise moment when his own shadow was equal to his own height — that is, when the sun's rays made a 45° angle with the ground.

A more sophisticated version of the story, attributed to other sources, has Thales using the theory of similar triangles. By planting a staff of known height near the pyramid and measuring both shadows simultaneously, one obtains a proportion:

$$\frac{\text{Height of pyramid}}{\text{Shadow of pyramid}} = \frac{\text{Height of staff}}{\text{Shadow of staff}}$$

What makes Thales truly revolutionary is not the individual theorems themselves — some of these facts were known empirically to the Babylonians and Egyptians long before — but the very concept that geometric truths require proof. Thales initiated the transformation of mathematics from a collection of practical techniques into a deductive science. Every mathematician who has written “Proof:” at the start of an argument is following in the footsteps of Thales.

4.2 The Pythagorean Brotherhood

Pythagoras of Samos (c. 570–495 BCE) is one of the most famous — and most mysterious — figures in the history of mathematics. Born on the Aegean island of Samos, he is said to have traveled extensively in his youth, studying under the priests of Egypt, the magi of Babylon, and possibly even the sages of India. Around 530 BCE, fleeing the tyranny of Polycrates, he emigrated to Croton in southern Italy, where he founded a philosophical and religious community that would profoundly influence the development of mathematics, philosophy, and science.

The Pythagorean Brotherhood was part mathematical school, part religious order, and part political society. Members followed strict rules: they practiced communal living, observed dietary restrictions (most famously, the prohibition against eating beans), took vows of secrecy about the community's discoveries, and attributed all mathematical results to “Pythagoras” collectively rather than to individual members.

Figurate Numbers

The Pythagoreans were fascinated by the geometric properties of numbers. They represented numbers as arrangements of pebbles (Greek: psephoi) and studied the patterns that emerged. This led them to discover several important classes of figurate numbers.

Music and Harmonic Ratios

Perhaps the most celebrated Pythagorean discovery was the mathematical basis of musical harmony. According to legend, Pythagoras was passing a blacksmith's shop when he noticed that different hammers produced harmonious sounds when struck together. He investigated and found that the pleasing musical intervals correspond to simple whole-number ratios of string lengths:

• Octave — ratio $2 : 1$ (halving the string length doubles the pitch)
• Perfect Fifth — ratio $3 : 2$
• Perfect Fourth — ratio $4 : 3$
• Whole Tone — ratio $9 : 8$ (the difference between a fifth and a fourth: $\frac{3}{2} \div \frac{4}{3} = \frac{9}{8}$)

This was a stunning confirmation of the “all is number” philosophy. If something as subjective as musical beauty could be reduced to simple numerical ratios, what else might be expressible in the language of mathematics? The Pythagoreans extended this idea to the cosmos itself, proposing the Music of the Spheres — the idea that the planets produce harmonious tones as they move through the heavens, their orbital ratios reflecting the same mathematical harmony found in music.

Perfect and Amicable Numbers

The Pythagoreans classified numbers according to the sum of their proper divisors. A number is perfect if it equals the sum of its proper divisors, abundant if the sum exceeds it, and deficient if the sum falls short. They knew the first four perfect numbers:

$$6 = 1 + 2 + 3$$

$$28 = 1 + 2 + 4 + 7 + 14$$

$$496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248$$

$$8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064$$

Two numbers are amicable if each is the sum of the other's proper divisors. The Pythagoreans knew the pair $(220, 284)$: the proper divisors of 220 sum to 284, and the proper divisors of 284 sum to 220. They considered amicable numbers to be symbols of friendship.

4.3 The Pythagorean Theorem

No theorem in all of mathematics is more famous than the Pythagorean theorem. While the relationship between the sides of a right triangle was known empirically to the Babylonians (who catalogued extensive lists of Pythagorean triples over a thousand years earlier) and to the Chinese and Indians, the Pythagoreans are traditionally credited with providing the first proof — elevating an empirical observation to a demonstrated truth.

Proof 1: Euclid's Proof (Elements I.47)

Euclid's proof, given as Proposition 47 of Book I of the Elements, is a masterpiece of synthetic geometry. It is sometimes called the “Windmill Proof” because of the shape of the figure.

Consider a right triangle $ABC$ with the right angle at $C$. Construct squares on each of the three sides: square $ACGH$ on side $AC = b$, square $BCEF$ on side $BC = a$, and square $ABDE$ on hypotenuse $AB = c$.

Drop a perpendicular from $C$ to side $DE$ of the large square, meeting it at $L$, and extend $CL$ to meet $AB$ at $K$. This line divides the square on the hypotenuse into two rectangles: $ADLK$ and $BKLE$.

Euclid then shows that rectangle $ADLK$ has the same area as the square on side $b$, and rectangle $BKLE$ has the same area as the square on side $a$. The proof uses two key congruences of triangles, established through the Side-Angle-Side (SAS) criterion:

$$\triangle ABG \cong \triangle ADC \quad \text{(both have sides } c, b \text{ with included angle } 90° + \angle BAC\text{)}$$

Since $\triangle ABG$ has half the area of square $ACGH$ (same base and height), and $\triangle ADC$ has half the area of rectangle $ADLK$ (same base and height), we conclude that $\text{Area}(ACGH) = \text{Area}(ADLK)$. Similarly, $\text{Area}(BCEF) = \text{Area}(BKLE)$. Therefore:

$$a^2 + b^2 = \text{Area}(BKLE) + \text{Area}(ADLK) = \text{Area}(ABDE) = c^2 \quad \blacksquare$$

Proof 2: Algebraic Proof by Rearrangement

Consider a large square with side length $(a + b)$. Inside it, arrange four copies of the right triangle (with legs $a, b$ and hypotenuse $c$) so that their hypotenuses form a tilted inner square with side length $c$.

The area of the large square is $(a + b)^2$. It equals the area of the inner square plus the four triangles:

$$(a + b)^2 = c^2 + 4 \cdot \frac{1}{2}ab$$

Expanding the left side:

$$a^2 + 2ab + b^2 = c^2 + 2ab$$

Subtracting $2ab$ from both sides:

$$a^2 + b^2 = c^2 \quad \blacksquare$$

Proof 3: Proof by Similar Triangles

In right triangle $ABC$ with the right angle at $C$, drop the altitude from $C$ to the hypotenuse $AB$, meeting it at point $H$. This creates two smaller triangles, both similar to the original:

$$\triangle ACH \sim \triangle ABC \sim \triangle CBH$$

From $\triangle ACH \sim \triangle ABC$:

$$\frac{AC}{AB} = \frac{AH}{AC} \implies AC^2 = AB \cdot AH \implies b^2 = c \cdot AH$$

From $\triangle CBH \sim \triangle ABC$:

$$\frac{BC}{AB} = \frac{BH}{BC} \implies BC^2 = AB \cdot BH \implies a^2 = c \cdot BH$$

Adding these two results:

$$a^2 + b^2 = c \cdot BH + c \cdot AH = c(AH + BH) = c \cdot c = c^2 \quad \blacksquare$$

Proof 4: President Garfield's Proof (1876)

James A. Garfield, later the 20th President of the United States, published a proof while serving as a congressman. Place two copies of the right triangle together to form a trapezoid with parallel sides $a$ and $b$ and height $(a + b)$.

The area of the trapezoid is $\frac{1}{2}(a + b)(a + b) = \frac{1}{2}(a + b)^2$. It also equals the sum of three triangles: two copies of $\frac{1}{2}ab$ plus a right isosceles triangle with legs $c$:

$$\frac{1}{2}(a + b)^2 = 2 \cdot \frac{1}{2}ab + \frac{1}{2}c^2$$

$$(a + b)^2 = 2ab + c^2$$

$$a^2 + 2ab + b^2 = 2ab + c^2 \implies a^2 + b^2 = c^2 \quad \blacksquare$$

4.4 The Crisis of Incommensurables

The Pythagorean worldview — that “all is number” and that any two magnitudes can be expressed as a ratio of whole numbers — was shattered by one of the most profound discoveries in the history of mathematics: the existence of incommensurable magnitudes. Two lengths are incommensurable if no common unit of measurement, however small, can measure both exactly — that is, their ratio cannot be expressed as a fraction of whole numbers.

The classic example is the diagonal of a unit square. By the Pythagorean theorem, a square with side 1 has a diagonal of length $\sqrt{2}$. The Pythagoreans discovered — to their horror — that $\sqrt{2}$ cannot be expressed as a ratio of integers. Legend has it that this discovery was made by Hippasus of Metapontum, and that the Brotherhood was so disturbed by the result that Hippasus was expelled — or, in more dramatic versions of the story, drowned at sea as divine punishment for revealing this secret.

This proof, one of the earliest and most elegant examples of reductio ad absurdum (proof by contradiction), represents a watershed moment in intellectual history. For the first time, a purely logical argument demonstrated a truth about the nature of numbers that contradicted physical intuition and deeply held philosophical beliefs.

The crisis of incommensurables had far-reaching consequences for Greek mathematics. Since ratios of integers could not describe all geometric magnitudes, the Pythagorean program of reducing geometry to arithmetic was untenable. Greek mathematicians responded by shifting their foundations: geometry became primary, and number theory was subordinated to geometric reasoning. Lengths, areas, and ratios were treated as geometric objects rather than numerical ones. This geometric turn would dominate mathematics for nearly two thousand years, until the development of real analysis in the 19th century finally provided a rigorous arithmetic theory of irrational numbers.

4.5 Euclid's Elements (c. 300 BCE)

Euclid of Alexandria (c. 325–265 BCE) compiled the most influential mathematical textbook ever written: the Elements (Greek: Stoicheia). Working at the great Library of Alexandria under the patronage of Ptolemy I, Euclid organized virtually all known Greek mathematics into a single, magnificent deductive system. The Elementsconsists of 13 books containing 465 propositions, each proved from first principles.

Euclid did not invent most of the mathematics in the Elements — many results were known to Pythagoras, Hippocrates, Eudoxus, and Theaetetus. His genius lay in the organization: selecting the right definitions, postulates, and common notions, and arranging the propositions in a logical sequence where each theorem follows from those before it. The result is a cathedral of pure reason that has served as the model for axiomatic thinking ever since.

The Axiomatic Method

The Elements begins with a series of definitions, followed by five postulates(specific geometric assumptions) and five common notions (general logical principles). Every subsequent proposition is proved using only these starting points and previously proved results.

The fifth postulate stands out immediately: it is far more complex and less self-evident than the other four. For over two thousand years, mathematicians attempted to prove it from the other postulates, convinced it was not truly independent. Proclus, Saccheri, Lambert, Legendre, and many others tried and failed. In the 1820s–1830s, Bolyai and Lobachevsky independently showed that the fifth postulate is independent by constructing consistent geometries in which it is false — the non-Euclidean geometries. This discovery revolutionized mathematics and, through Einstein's general relativity, our understanding of the physical universe.

An equivalent and more intuitive formulation of the fifth postulate is Playfair's axiom (1795): “Through a point not on a given line, there is exactly one line parallel to the given line.” In hyperbolic geometry (Lobachevsky), there are infinitely many such parallels; in elliptic geometry (Riemann), there are none.

4.6 Euclid's Number Theory

Books VII–IX of the Elements develop a sophisticated theory of numbers using purely geometric language. Euclid represents numbers as line segments and divisibility as one segment “measuring” another. Despite this geometric dress, the content is purely number-theoretic and includes some of the most important results in the history of mathematics.

The Euclidean Algorithm for GCD

Proposition VII.2 of the Elements describes an algorithm for finding the greatest common divisor (GCD) of two numbers. In modern terms, given positive integers $a$ and $b$ with$a > b$, we repeatedly replace the larger number by the remainder upon division:

$$\gcd(a, b) = \gcd(b, a \bmod b)$$

This continues until the remainder is zero; the last nonzero remainder is the GCD.

The Euclidean algorithm is remarkable for several reasons: it is the oldest nontrivial algorithm that is still in regular use today (over 2,300 years later), it is extremely efficient (the number of steps is at most about $5 \times$ the number of digits in the smaller input), and it generalizes to polynomials, Gaussian integers, and other algebraic structures.

Infinitude of Primes (Book IX, Proposition 20)

Euclid's proof that there are infinitely many prime numbers is one of the most celebrated proofs in all of mathematics — a masterpiece of logical elegance that has been called “the proof that every mathematician loves.”

Perfect Numbers and the Euclid–Euler Theorem

In Book IX, Proposition 36, Euclid proved that if $2^p - 1$ is prime (what we now call a Mersenne prime), then the number $2^{p-1}(2^p - 1)$ is perfect. For example:

• $p = 2$: $2^2 - 1 = 3$ is prime, so $2^1 \times 3 = 6$ is perfect
• $p = 3$: $2^3 - 1 = 7$ is prime, so $2^2 \times 7 = 28$ is perfect
• $p = 5$: $2^5 - 1 = 31$ is prime, so $2^4 \times 31 = 496$ is perfect
• $p = 7$: $2^7 - 1 = 127$ is prime, so $2^6 \times 127 = 8128$ is perfect

4.7 Eudoxus and Proportion Theory

Eudoxus of Cnidus (c. 408–355 BCE) was arguably the most brilliant mathematician of the generation before Euclid. A student of Plato's Academy, Eudoxus made two contributions that would reshape the foundations of Greek mathematics: the theory of proportions (presented in Book V of the Elements) and the method of exhaustion(used throughout Book XII).

The Theory of Ratios (Book V)

The crisis of incommensurables had left a gaping hole in Greek mathematics: the old Pythagorean theory of proportion worked only for commensurable magnitudes (those whose ratio is rational). What did it mean to say that two ratios of possibly incommensurable magnitudes are “equal”? Eudoxus provided a stunning answer.

This definition is a work of genius. In modern terms, it says that $a/b = c/d$ if and only if for every rational number $m/n$, the quantities $a/b$ and $c/d$ are on the same side of $m/n$ (or both equal to it). This is essentially the same idea as the Dedekind cut construction that Richard Dedekind used in 1872 to give the first rigorous definition of the real numbers — over two thousand years later! Dedekind himself acknowledged his debt to Eudoxus.

The Method of Exhaustion

The method of exhaustion is a technique for computing areas and volumes by approximating them with sequences of polygons (or polyhedra) whose areas (or volumes) are known. It is a rigorous precursor to the limit concept in calculus.

The key idea is captured in a proposition sometimes attributed to Eudoxus and stated explicitly by Euclid (Elements X.1):

Using this principle, Euclid proves (in Book XII) the fundamental results on areas and volumes:

• XII.2: Circles are to one another as the squares on their diameters:$\frac{A_1}{A_2} = \frac{d_1^2}{d_2^2}$
• XII.7: Any pyramid is one-third of the prism with the same base and height
• XII.10: Any cone is one-third of the cylinder with the same base and height:$V_{\text{cone}} = \frac{1}{3}\pi r^2 h$
• XII.18: Spheres are to one another as the cubes of their diameters:$\frac{V_1}{V_2} = \frac{d_1^3}{d_2^3}$

4.8 Legacy

The Elements is, by any measure, the most influential textbook in the history of mathematics — and arguably in the history of any subject. Over 1,000 editions have been published since the invention of printing. For more than two thousand years, studying Euclid was synonymous with studying mathematics itself.

The impact of the Elements extends far beyond mathematics. Its axiomatic-deductive structure became the model for rational inquiry in every field. Spinoza wrote his Ethics in the style of Euclid, with definitions, axioms, and propositions. Newton's Principia follows the same pattern. The Declaration of Independence, with its “self-evident truths” from which political conclusions are derived, echoes the Euclidean method. Abraham Lincoln studied the first six books of the Elements to sharpen his logical thinking; he later said that it taught him the meaning of “demonstrate.”

The Pythagorean-Euclidean tradition established the paradigm that persists to this day: mathematics consists of theorems proved from axioms through logical deduction. Every “Proof:” and every “Q.E.D.” written in mathematics today traces its lineage back to Thales, Pythagoras, Eudoxus, and Euclid.

“There is no royal road to geometry.” — Euclid, to Ptolemy I (attributed by Proclus)

Video Lectures

These video lectures explore Greek mathematical methods, from number theory and the Pythagorean theorem to Euclid's geometric algebra and the golden ratio.