The triangle that a globe gets wrong

Draw a right angle where the equator crosses a meridian. Go fifty degrees east along the equator, and sixty degrees north up the meridian. The flat theorem says the two endpoints are $\sqrt{50^2+60^2} = 78.1$ degrees of arc apart.

They are $71.3$ degrees apart.

Nothing has been measured badly and no approximation has been made. The two legs really are fifty and sixty degrees along the shortest paths available, the angle between them really is a right angle, and the third side really is the shortest path between its endpoints. Every ingredient of the theorem is present, and the conclusion is false.

What replaces it #

The relation that holds instead is short enough to be startling:

$$\cos c = \cos a \cos b,$$

with all three sides measured as angles at the centre of the sphere. For $a = 50°$ and $b = 60°$ that gives $\cos c = 0.643 \times 0.5 = 0.321$, so $c = 71.25°$, against the $78.10°$ the sum of squares asks for.

There are no squares in it. There is no sum. The two legs are combined by multiplication of their cosines, which is not an operation the flat statement contains anywhere, and the right angle enters only by being the case where the general spherical law of cosines loses its last term.

The formula is also symmetric in $a$ and $b$, as it must be, and it has the right degenerate behaviour: if $b = 0$ the relation collapses to $c = a$, which is a triangle flattened onto a single arc.

Why the sphere shortens things #

The mechanism is worth having in a sentence, because it explains the direction of the error as well as its size.

Both legs are arcs of great circles through the corner. Their far ends are two points on the sphere, and the hypotenuse is the great circle joining those. That third great circle does not stay near the two legs; it cuts across, and cutting across a curved surface takes a shortcut that a flat surface does not offer.

Put the other way: the sphere has less room than the plane. A circle of radius $r$ drawn on a sphere has circumference $2\pi \sin r$ rather than $2\pi r$, which is less, so the points at distance $r$ from the corner are packed closer together than the plane would pack them, so the two leg-ends are nearer each other than flat geometry predicts.

That last figure makes the failure structural rather than numerical. On a sphere no two points are more than $180°$ apart, so any formula that can return $200°$ has left the surface behind. The flat theorem is not a good approximation that degrades; it is a statement about a different space, and at large sizes it reports quantities that do not exist.

The small triangle, which lies #

So why did nobody notice for two thousand years?

Because small triangles are almost flat, and the almost improves extremely fast. Expanding both sides of $\cos c = \cos a \cos b$ in powers of the side lengths gives

$$c^{2} = a^{2} + b^{2} - \tfrac{1}{3}a^{2}b^{2} + \cdots,$$

so the flat theorem is not merely the leading term — it is correct to third order, and the first correction is fourth order in the size of the triangle. Halving a triangle divides its error by sixteen.

For a builder’s triangle six metres across on the Earth’s surface, $a$ and $b$ are about $10^{-6}$ radians, and the hypotenuse comes up short by around $2\times10^{-13}$ metres — a thousandth of the width of an atom. The knotted rope is not approximately right; it is right to a precision that has no physical meaning.

The lying starts at the scale of countries. A triangle whose legs are six hundred and eight hundred kilometres has a hypotenuse nearly a kilometre shorter than the flat theorem gives, which is why geodesy — the business of surveying at continental scale — was the first discipline forced to take spherical trigonometry seriously, and why the metre was originally defined by a survey arc rather than by a stick.

The other direction #

There is a second geometry where the theorem also fails, and it fails the opposite way.

In hyperbolic geometry the relation is

$$\cosh c = \cosh a \cosh b,$$

the same statement with the circular cosine replaced by the hyperbolic one, and the same expansion now gives $c^{2} = a^{2}+b^{2}+\tfrac{1}{3}a^{2}b^{2}$. The hypotenuse is longer than the flat answer. There is more room in the hyperbolic plane than in the flat one, not less: a circle of radius $r$ has circumference $2\pi\sinh r$, which grows exponentially, so the ends of the two legs are pushed apart rather than drawn together.

Three geometries, three formulas, one pattern:

$$\cos c = \cos a\cos b \qquad c^{2}=a^{2}+b^{2} \qquad \cosh c = \cosh a\cosh b,$$

and the middle one is the boundary case between them — the case where the correction term vanishes, in the same sense that a right angle is the case where the law of cosines loses its last term.

Curvature is the quantity being measured #

The three formulas are not three unrelated facts. They are one formula at three values of a single number, and that number is the curvature $K$ of the surface: positive on the sphere, zero on the plane, negative in the hyperbolic plane. The correction to $c^2$ at leading order is $-\tfrac{1}{3}K a^{2}b^{2}$, and everything above is that expression at $K = 1$, $0$ and $-1$.

This is what makes the failure of the theorem informative rather than merely inconvenient. A creature confined to a surface, with no access to a third dimension and no view from outside, can find out which surface it is on by drawing a right triangle and comparing the sum of the squares with the square of the hypotenuse. The discrepancy, divided by $\tfrac13 a^2b^2$, is the curvature — measured entirely from inside.

That is Gauss’s Theorema Egregium in its most usable clothing: curvature is intrinsic. It is not a fact about how a surface sits in space but about distances measured within it, and it can be detected by a survey. Gauss, who was running exactly such a survey in Hanover in the 1820s, is reported to have measured the triangle formed by three mountain peaks partly to see whether physical space itself was flat. The measurement was not accurate enough to settle anything, and the ambition was the right one; the same question in the same form is what a modern cosmological survey asks with rather better instruments.

No flat map of it #

One more consequence belongs here, because it is the same theorem read backwards.

If some flat map of a region of the sphere preserved every distance, then triangles drawn on the sphere would have flat triangles’ side lengths, and the sum of squares would hold. It does not, so no such map exists — not for any region, however small, and not for any cleverness of construction. Every map of any part of a sphere distorts distance somewhere, and this is why, rather than because mapmakers have not tried hard enough.

That impossibility has the same source as everything above: the curvature is intrinsic, so it cannot be flattened away, so it obstructs any distance-preserving correspondence with a plane. A sphere is a plane plus one point as far as which points are near which is concerned; it is nothing like a plane as far as how far apart they are is concerned.

What it costs to give the theorem up #

The convenience lost is larger than one formula.

On a sphere there are no similar triangles. Two triangles with the same three angles are congruent — the angles determine the sides, and there is no scaling. Everything the flat subject does with proportion, from the tangent ratio to the whole of trigonometry as a table of shape-dependent ratios, stops working, because a spherical triangle’s angles depend on its size.

Nor is there a rectangle. A quadrilateral with four right angles does not exist on a sphere, so the tiling arguments that prove the flat theorem by rearrangement have nothing to rearrange. The Greek proofs are not merely harder to adapt; their objects are gone.

And the angle sum is no longer a constant. A spherical triangle’s angles add to more than $180°$, by an amount equal to its area, so the triangle with three right angles — an octant of the sphere, an eighth of the whole surface — is perfectly ordinary there and impossible here.

How the spherical version was actually found #

The flat theorem is Greek and famous. The spherical one is Greek and obscure, and its history says something about what mathematics gets remembered.

Menelaus of Alexandria wrote the Sphaerica around AD 100, which contains the first systematic treatment of triangles on a sphere and the theorem that bears his name. His motive was astronomy: the positions of stars are angles, the sky is a sphere, and every question about where a star rises is a question about a spherical triangle. The subject was developed much further by Islamic astronomers — Abū al-Wafā’ in the tenth century had the law of sines for spheres and something close to the relation above, and al-Bīrūnī used spherical trigonometry to compute the direction of Mecca from an arbitrary city, which is a genuinely hard instance of the problem.

The compact form $\cos c = \cos a \cos b$ is one of a set of ten relations for right spherical triangles, packaged in 1614 by John Napier — the same Napier as the logarithms — into a mnemonic circle of five parts, from which any of the ten can be read off by a rule about adjacent and opposite pieces. Napier’s rules were memorised by navigators for three centuries and are now nearly forgotten, because the questions they answered are answered by machines.

What is worth noticing is the order of events. Spherical trigonometry was worked out for use, by people who needed the sky and the sea, and the flat theory was the one that was studied for its own sake. The elegant limiting case became the famous one, and the general case that contains it stayed in the manuals.

What the picture cannot show #

The spherical figures are drawings of a sphere on a flat page, which means every arc in them is distorted by the projection used to draw it. The hypotenuse looks shorter than the legs in the first figure partly because it is shorter and partly because of where it sits relative to the viewer, and no reader can separate the two effects by looking. The numbers in the captions are computed on the sphere; the picture is a photograph of it.

The hyperbolic figure is worse in a specific and famous way. The Poincaré disc is a model: the whole infinite hyperbolic plane is squeezed inside a finite circle, distances near the rim are enormously compressed, and the two sides of a triangle that look short may be far longer than the one that looks long. Every drawing of hyperbolic geometry has this property, because the surface cannot be embedded in ordinary three-dimensional space without distortion — a theorem of Hilbert, and the reason there is no hyperbolic globe to sit next to the spherical one.

So neither picture shows lengths. Both show incidence — what meets what, in what order — and both hand the lengths over to the arithmetic in the caption. That is a real limitation on the site’s usual claim, and it is where a figure stops being a proof and starts being a diagram.

The ladder from here #

Below: the rearrangement proof, Euclid’s shear, the whole-number solutions, the converse and the third dimension. This rung is where the ladder leaves the theorem behind, and what stands above it belongs to a different subject: the metric tensor, geodesics as the paths a metric makes straight, and Gauss–Bonnet, which ties the total curvature of a surface to a number that counts its handles rather than its shape.

Sideways, the failure of the parallel postulate is the same discovery arriving from the direction of logic: a consistent geometry in which Euclid’s fifth axiom is false, which is what makes it an axiom rather than a theorem. The two routes reached the same place in the 1820s, one by surveying and one by trying to prove an axiom and failing usefully.

A theorem is a statement about a space #

The lasting point is what the failure identifies.

For two thousand years the sum of the squares was taken to be a fact about triangles. It is not. It is a fact about flat triangles, and the flatness was invisible because there was nothing else on offer — the only surface anyone was measuring was one that is flat to twenty decimal places at the scale of a courtyard.

What the spherical and hyperbolic versions show is that the theorem was always carrying a hidden parameter, and that the parameter can be measured from inside by using the theorem and seeing how badly it does. A statement believed to be universal turned out to be the boundary case of a family, and its error term turned out to be a measurement of the thing nobody knew was there.

That is a pattern worth carrying, because the same shape recurs whenever an assumption is finally noticed: a rule that seems forced turns out to be a choice, and the alternatives are consistent, and the original is the special case in the middle. Euclid’s fifth postulate, the excluded middle, the commutativity of multiplication and the flatness of space were all discovered to be assumptions in the same way — by someone building the world in which they are false and finding that it does not collapse.