Believe nothing you cannot check.

The most consequential sentence in the history of measurement. Note that the Greeks proved a fact that several older civilisations already used — the novelty was the obligation to give a reason.

Demonstration

1. Let triangle ABC have its right angle at C, with a = BC, b = CA, c = AB. Drop the perpendicular from C to AB, meeting it at D; let AD = q and DB = p, so that p + q = c.
2. Triangles ACD and ABC share the angle at A, and each contains a right angle; being equiangular they are similar (Elements VI.4).
3. From that similarity, AC : AB = AD : AC, that is b : c = q : b, and therefore b² = cq.
4. Triangles BCD and BAC likewise share the angle at B and each contains a right angle, so a : c = p : a, and therefore a² = cp.
5. Adding the two results: a² + b² = cp + cq = c(p + q) = c · c = c².

The square on the hypotenuse equals the sum of the squares on the legs.

What it costs

Babylonian tablet YBC 7289 gives √2 correct to six decimal places, roughly 1,800 BC. So the fact is not Greek. What Euclid contributed is the reason — and once a reason exists, authority becomes optional. Every later act of intellectual independence is downstream of this.

Where it stops being true

It is true of Euclidean space, not of space. For a right triangle on a sphere the relation becomes cos c = cos a · cos b, and long triangles on the Earth disobey the flat version measurably. Surveyors correct for it daily.

The proof that destroyed the metaphysics of the school that discovered it. Ten lines long, never once refuted, and still the cleanest demonstration in mathematics.

Demonstration

1. Suppose the diagonal d and the side s did have a common measure. Then d : s = p : q for whole numbers p and q, and we may take the fraction in lowest terms, so that p and q are not both even.
2. By Proposition I, d² = 2s². Hence p² = 2q².
3. So p² is even. But the square of an odd number is odd; therefore p is even, say p = 2m.
4. Substituting: (2m)² = 2q², so 4m² = 2q², so q² = 2m². By the same step, q is even.
5. Then p and q are both even — contradicting the choice of lowest terms. The supposition destroys itself.

No common measure exists. The ratio of diagonal to side is not a ratio of whole numbers.

What it costs

The Pythagorean creed was that all things are number, meaning whole number and their ratios. Their own emblem, the square, contained a magnitude the creed forbade — and the proof was theirs. The legend that Hippasus was drowned at sea for divulging it is late and unreliable (Iamblichus, five centuries downstream), but the pressure it records is real. Every worldview eventually meets a proof it did not order.

Where it stops being true

Nothing here says the diagonal is unreal or unmeasurable; it is exactly as long as it is. What fails is a hidden premise about what magnitudes must be. Hear the consequence: √2 is precisely the half-octave. The equal-tempered tritone, later nicknamed diabolus in musica (the name is first recorded centuries after the Middle Ages), is the interval this proof says can never be written as a ratio of whole numbers.

Read Euclid's actual phrasing. He does not say "there are infinitely many"; he refuses to speak of a completed infinity and claims only that no finite list can be complete. Precision about what you are not asserting is part of the proof.

Demonstration

1. Take any finite list of primes p₁, p₂, …, pₙ.
2. Form N = (p₁ · p₂ · … · pₙ) + 1.
3. N is greater than 1, so it has some prime divisor q (Elements VII.31).
4. If q were one of the listed primes, it would divide the product and also divide N, hence divide their difference, which is 1. No prime divides 1.
5. Therefore q is a prime absent from the list. Since the list was arbitrary, no finite list of primes is complete.

The primes exceed any assigned multitude.

What it costs

This is the earliest surviving proof about an unbounded collection, and it works by construction rather than by contemplation of infinity. Two millennia later Gauss urged the same caution: the infinite is "a manner of speaking" about limits. Rigour is often a matter of saying less.

Where it stops being true

The construction does not find the next prime, and N is frequently composite: 2·3·5·7·11·13 + 1 = 30031 = 59 × 509. Existence is not a method. That gap — between proving that something is and being able to produce it — separates classical from constructive mathematics, and it matters enormously in computing.

The most practically consequential theorem ever proved about music, and the reason every fixed-pitch instrument you have ever touched is a negotiated compromise.

Hear and see it: the spiral of fifths on the Science page.

Demonstration

1. A perfect fifth multiplies frequency by 3/2; an octave multiplies it by 2.
2. Twelve fifths therefore multiply by (3/2)¹² = 3¹² / 2¹² = 531441 / 4096, while seven octaves multiply by 2⁷ = 128.
3. Suppose these were equal. Then 3¹² / 2¹² = 2⁷, that is 3¹² = 2¹⁹.
4. But 3¹² is odd and 2¹⁹ is even; by unique factorisation no power of three can equal a power of two. The supposition is impossible.
5. The excess is 531441 / 524288 = 1.0136433…, which in cents is 1200 · log₂(1.0136433) = 23.46 cents: the Pythagorean comma.

The spiral of fifths overshoots its own beginning by 23.46 cents, for ever.

What it costs

Every keyboard, every fret, every fixed tuning in the world is a political settlement over these 23.46 cents: who receives the pure interval and who pays for it. Equal temperament distributes the loss evenly — each fifth narrowed by 1.955 cents — so that no key is home and none is exile. The instrument in your hands is a compromise, and every chord you play is a small managed error.

Where it stops being true

None of this is an argument against equal temperament; it is an argument against pretending the compromise is not there. Just intonation repairs the fifths and ruins modulation; meantone repairs the thirds and produces a wolf. There is no tuning without a loss — only a choice of which loss, made by someone, usually long before you were born.

Drag the point. The angle does not move by one hundredth of a degree, wherever you take it. This invariance under change is the first taste of what mathematicians mean by beauty.

Demonstration

1. Let AB be a diameter of a circle with centre O, and let C be any other point on the circumference.
2. OA, OB and OC are radii of the same circle and therefore equal.
3. Triangle OAC is isosceles, so the angles at A and at C are equal; call each α. Triangle OBC is isosceles, so the angles at B and at C are equal; call each β.
4. The three angles of triangle ABC sum to two right angles (Proposition VI): α + β + (α + β) = 180°.
5. Hence 2(α + β) = 180°, so the angle ACB = α + β = 90°, for every position of C.

The angle subtended by a diameter is right, always.

What it costs

Thales is called the first philosopher rather than the first surveyor because of this move: from "this triangle has a right angle" to "every triangle so constructed must". The leap from instances to necessity is the birth of theory, and it happened, as far as we can tell, once.

Where it stops being true

It presumes Proposition VI, and therefore Euclid's fifth postulate. On a sphere the angle exceeds a right angle. And it is a fact about circles, not about drawings: every physical circle ever made is wrong, which is precisely why the theorem is about something else.

Keep this proposition as the memento of the whole site. It is rigorously proved, universally taught, and false in the universe you are sitting in.

Demonstration

1. Let ABC be a triangle. Through C draw CE parallel to AB — possible by I.31, which rests on Postulate V.
2. AC falls across the two parallels AB and CE, so the alternate angles BAC and ACE are equal (I.29).
3. Produce BC to D. BD falls across the same parallels, so the corresponding angles ABC and ECD are equal (I.29).
4. The three angles at C — namely ACB, ACE and ECD — together make a straight line, that is, two right angles.
5. Substituting the equalities of steps 2 and 3, the angles BAC, ABC and ACB together equal two right angles.

The interior angles of a triangle sum to 180°.

What it costs

Everything you were taught to trust most is conditional on a request made at the door, which you probably did not read. Rigour does not protect you from choosing your premises badly; it only guarantees that the consequences of your choice are visible.

Where it stops being true

False on a curved surface. Take the north pole and two points on the equator a quarter-turn apart: three right angles, 270°. Gauss, Bolyai and Lobachevsky showed around 1830 that denying Postulate V yields perfectly consistent geometries, and in 1915 Einstein found that spacetime uses one of them. Two thousand years of failed proofs of the parallel postulate failed because it is not a theorem — it is a choice.

The whole of Book XIII exists to reach this result, which is why some ancient readers thought the Elements was written for it. An exhaustive proof: not "we have found five" but "there cannot be a sixth".

Demonstration

1. At each vertex of a convex solid at least three faces must meet, and the face-angles there must sum to strictly less than four right angles (360°); otherwise the surface cannot close.
2. Equilateral triangles have 60° angles: three, four or five may meet (180°, 240°, 300°), but six give exactly 360° and lie flat. → tetrahedron, octahedron, icosahedron.
3. Squares have 90°: three meet at 270°, four give exactly 360° and lie flat. → cube.
4. Regular pentagons have 108°: three meet at 324°, four exceed 360°. → dodecahedron.
5. Regular hexagons have 120°, so even three reach exactly 360°; anything with more sides exceeds it. No further solid is possible.
6. Five cases, therefore five solids — and each obeys Euler's relation V − E + F = 2: 4−6+4, 8−12+6, 6−12+8, 20−30+12, 12−30+20.

Exactly five, in three dimensions, for ever.

What it costs

Plato assigned four of them to the elements and the dodecahedron to the cosmos (Timaeus 53c–55c). The physics is dead. Kepler built a planetary model from nesting them (Mysterium Cosmographicum, 1596) — beautiful, ingenious and wrong — and then abandoned it when Tycho's observations refused it. That abandonment, not the model, is why Kepler is a scientist and the Timaeus is literature.

Where it stops being true

"Regular" is carrying the whole argument: convex, with identical regular faces and identical vertices. Relax convexity and four more appear (the Kepler–Poinsot solids). In four dimensions there are six; in five dimensions and above there are only ever three. The number five is a fact about three-dimensionality, not about beauty.

Three lines of school arithmetic, and the conclusion is that almost everything that exists is permanently unnameable.

Demonstration

1. Suppose every real number between 0 and 1 could be listed: r₁, r₂, r₃, … with nothing left out.
2. Write each as an infinite decimal, one per row, forming an infinite square array of digits.
3. Construct a new number d by taking its nth decimal digit to differ from the nth digit of rₙ — say 5 when that digit is not 5, and 4 when it is.
4. Then d differs from r₁ in the first place, from r₂ in the second, and from rₙ in the nth: d appears nowhere on the list.
5. But d is a real number between 0 and 1, so the list was incomplete. Since the list was arbitrary, no list can succeed.

The reals are uncountable: strictly more numerous than the counting numbers.

What it costs

Names, sentences, formulas and computer programs are all countable. Therefore almost every real number has no name, no formula and no program — it cannot be specified by any finite means, ever, by anyone. Any theory of knowledge that assumes the knowable and the existent are the same size has already been refuted, in three lines, in 1891.

Where it stops being true

Care is needed because 0.4999… = 0.5000…; the standard construction avoids 0s and 9s for exactly this reason. "More" means the absence of a one-to-one correspondence — a technical relation, not a bigger heap. And the argument stands inside set theory, which is itself axiomatic. Kronecker rejected the whole apparatus. Postulate V, again, all the way down.

The proof sketch below is honest but compressed; the full argument is a term's work. What matters here is the shape, and the fact that it ended a programme rather than a debate.

Demonstration

1. Assign a unique number to every symbol, formula and proof of the system (Gödel numbering), so that statements about formulas become statements about numbers.
2. Show that "the sequence with number m is a proof of the formula with number n" is itself an arithmetical relation the system can express.
3. Construct a formula G which says, of its own Gödel number g: "the formula numbered g has no proof in this system."
4. If the system proved G, then G would have a proof while asserting it has none: the system proves a falsehood and is inconsistent.
5. If the system proved not-G, it would be asserting that G has a proof, while (by step 4) no actual proof exists. Gödel excluded this by assuming ω-consistency, a slightly stronger condition than consistency. In 1936 Rosser altered the sentence so that plain consistency suffices.
6. So if the system is consistent, neither G nor not-G is provable. And G is true in the ordinary natural numbers, since it really has no proof. A second theorem follows: such a system cannot prove its own consistency.

Truth outruns provability, permanently, in any system strong enough to be interesting.

What it costs

Hilbert's programme — a finite mechanical basis settling every mathematical question — is not merely hard but impossible. Which leaves us exactly where Euclid stood in 300 BC: the honest currency is not certainty but demonstration from premises stated in the open. Twenty-three centuries of effort have not improved on the arrangement.

Where it stops being true

It applies to consistent, effectively axiomatised systems expressing enough arithmetic. It says nothing about vague human intuition, licenses no mysticism about consciousness, and does not mean "anything goes" — the theorems are themselves proved. Gödel was a mathematical Platonist, not a relativist, and would have detested most of what is said in his name.

Every crossing of the Pythagorean pentagram divides a line in this ratio, which is why the emblem and the crisis of Proposition II are the same object.

Demonstration

1. Cut a line so that whole : greater = greater : less. Setting the lesser part to 1 and the greater to φ gives (φ + 1) : φ = φ : 1.
2. Cross-multiplying: φ² = φ + 1.
3. Solving and keeping the positive root: φ = (1 + √5)/2 = 1.6180339887…
4. If φ were rational, then √5 = 2φ − 1 would be rational too; but 5 is not a perfect square and the parity argument of Proposition II applies verbatim. So φ is irrational.
5. Its continued fraction is [1; 1, 1, 1, …] — every term as small as possible — which makes it the slowest of all numbers to be approximated by fractions: in that precise sense (Hurwitz, 1891), the "most irrational" number.

φ² = φ + 1, and no fraction equals φ.

What it costs

Euclid never mentions beauty, and neither should we without evidence. The claims that φ governs the Parthenon, the Mona Lisa or the human face are largely retrofitted — see Markowsky, "Misconceptions about the Golden Ratio" (1992). What is genuinely true is stranger: sunflowers and pine cones pack at the golden angle because the most-irrational rotation never repeats, so no two seeds land in line.

Where it stops being true

Aesthetic claims: unsupported, and repeated mostly by people selling something. Growth-and-packing claims: well supported. Keep the two rigidly apart — this is the cheapest available lesson in how a true fact becomes a false industry.

This is the proposition that explains the monochord — and note carefully that its status is Established, not Demonstrated. It is measured, not derived from axioms, and it is therefore a lesser and different kind of certainty.

Try it: the monochord.

Demonstration

1. Halving the sounding length of a uniformly stretched string doubles its frequency. This is audible to any ear and measurable with any instrument, in any century.
2. In general f = (1 / 2L)·√(T/μ), where L is length, T tension and μ mass per unit length: frequency is inversely proportional to length, proportional to the square root of tension, inversely proportional to the square root of density.
3. Hence the whole-number ratios of the monochord are consequences of a physical law: 2:1 is 1/(1/2), 3:2 is 1/(2/3), 4:3 is 1/(3/4).
4. Consonance follows from coincident partials (Helmholtz, 1863): when the ratio is simple, upper harmonics line up instead of beating against one another.

Established by measurement and explained by mechanics — and labelled as such.

What it costs

The Pythagorean mystery dissolves into a law and loses nothing. The ratios are not sacred; they are what a vibrating line under tension must do. But notice what survived the dissolution: after twenty-five centuries, the numbers were right. That is the best argument available for taking careful observation seriously even when its metaphysics is nonsense.

Where it stops being true

Idealisations everywhere: a perfectly flexible, uniform, infinitely thin string at small amplitude. Real strings are stiff, which is why a guitar needs saddle compensation and why piano tuners deliberately stretch octaves. Stiffness also makes partials inharmonic — so even "consonance = simple ratio" is an approximation with a residue.

Kepler spent twenty years looking for the harmony of the spheres in Tycho Brahe's observations. He found this instead, published it, and kept the title of the book.

Demonstration

1. From Tycho's observational record, for every planet the quantity T²/a³ is constant, with T the orbital period and a the mean distance.
2. Test it. Earth: T = 1 year, a = 1 AU, so T²/a³ = 1. Jupiter: a = 5.204, a³ = 140.9; T = 11.862, T² = 140.7. Agreement to better than two parts in a thousand.
3. In 1687 Newton derived the same relation from an inverse-square attraction (Principia, Book I, Proposition 15), turning an empirical regularity into a consequence of a deeper law.

T² ∝ a³ — established by measurement, later explained by theory.

What it costs

Kepler wanted musical ratios in the heavens and published the arithmetic that replaced them. He had earlier built and then destroyed his own beloved Platonic-solid model on the same grounds. Preferring the result to the hypothesis is the highest act available to a mind, and it is available to you this afternoon, at a smaller scale, for free.

Where it stops being true

Strictly T² = 4π²a³ / G(M + m); the simple proportion holds only when the orbiting mass is negligible, and fails for binary stars. Kepler's laws are corollaries of Newton's theory, and Newton's theory is a limiting case of general relativity. Every floor turns out to be a ceiling.

Archimedes considered this his finest result and asked for the figure on his gravestone. Cicero, serving as quaestor in Sicily around 75 BC, found the monument overgrown with thorns and recognised it by the sphere and cylinder carved on top.

Demonstration

1. A sphere of radius r is circumscribed by a cylinder of radius r and height 2r, whose volume is πr² · 2r = 2πr³.
2. By the method of exhaustion — inscribing and circumscribing solids of revolution whose volumes converge on the sphere from both sides — the sphere's volume is shown to be exactly (4/3)πr³.
3. The ratio is therefore (4/3)πr³ : 2πr³ = 2 : 3. The surface areas stand in the same ratio, 4πr² : 6πr².

Sphere to cylinder, 2 : 3, exactly.

What it costs

This is calculus in everything but notation, eighteen centuries early, and it was obtained by a man who then asked for a ratio rather than a battle on his tomb. Choose, in advance, what you would want carved.

Where it stops being true

The method of exhaustion is airtight but cannot discover; it can only confirm. Archimedes' own Method — rediscovered on a palimpsest in 1906 after being scraped clean and written over by a prayer book — shows he first found these results by an informal mechanical balancing argument involving indivisibles, and only then proved them rigorously. Discovery and demonstration are different acts, and confusing them has ruined many students.