The fifth postulate and the other geometries

Out of nothing I have created a strange new universe.

— János Bolyai, letter to his father (3 November 1823)

Euclid’s first four postulates are short and obvious: draw a line, extend it, draw a circle, all right angles are equal. The fifth is different:

If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side.

It is long, and it speaks of something nobody can check: what happens to two lines extended indefinitely. Euclid seems to have distrusted it too: he avoided using it as long as he could, and invokes it for the first time only in Proposition I.29.

Playfair’s form

There is a version that is easier to remember, popularised by the Scotsman John Playfair in 1795: through a point outside a line there passes exactly one parallel to it. Given the other postulates, the two forms are equivalent: whoever accepts one must accept the other.

The angles of a triangle

The most famous consequence of the fifth postulate is Proposition I.32: the interior angles of a triangle add up to two right angles, $180°$. This site’s old trigonometry course called it the first canon:

The proof: through vertex $C$ of triangle $ABC$, draw the parallel to the base $AB$. The angles this parallel makes with sides $CA$ and $CB$ are equal to the angles $\alpha$ and $\beta$ of the triangle, because they are alternate interior angles (I.29). Together with angle $\gamma$, the three lie side by side on the parallel and form a straight angle, $180°$. The step in which the alternate angles are equal is exactly where the fifth postulate comes in.

Two thousand years of trying to prove it

For two millennia mathematicians thought the fifth postulate should not be a postulate but a theorem: it should be possible to prove it from the other four. Ptolemy tried, as did Proclus, the Persian Omar Khayyam, the Persian Nasir al-Din al-Tusi and the Englishman John Wallis. All failed: every “proof” used, hidden somewhere, an assumption equivalent to the postulate itself.

In 1733 the Jesuit priest Girolamo Saccheri tried another route: assume the postulate is false and reach a contradiction. He deduced dozens of strange theorems, such as triangles whose angles add up to less than $180°$. He found no contradiction at all, but, convinced that one had to exist, he declared the results “repugnant to the nature of the straight line” and published the book under the title Euclid Freed from Every Flaw. He was one step from a discovery and did not see it.

The new geometry

At the beginning of the nineteenth century three people, independently, took the step that was missing: accepting that geometry without the fifth postulate is as consistent as Euclid’s. Gauss reached this around 1820 but did not publish, fearing the reaction. The Russian Nikolai Lobachevsky published in 1829, in a journal of the University of Kazan. The Hungarian János Bolyai, son of a mathematician, published in 1832, in an appendix to his father’s book.

Bolyai’s father, Farkas, had spent his life trying to prove the fifth postulate. When he learned his son was on the same path, he wrote begging him to give up: “For God’s sake, leave the parallels alone. Detest them as you would an indecent passion; they can deprive you of all your time, your health, your peace of mind and the happiness of your life.” János did not give up, and in 1823 he replied with the sentence that opens this post.

In this hyperbolic geometry, through a point outside a line there pass infinitely many parallels, and the angles of a triangle always add up to less than $180°$. The bigger the triangle, the smaller the sum.

The geometry of the sphere

There is also the opposite case. On the surface of a sphere, the “straight lines”, the shortest paths, are the great circles, like the equator and the meridians. Two great circles always cross: there are no parallels. Take a triangle formed by the equator and two meridians leaving the north pole $90°$ apart. All three angles are right angles, and the sum is

On the sphere, the angles of a triangle always add up to more than $180°$, and the excess is proportional to the area of the triangle. For small triangles, like a plot of land or a room, the difference is imperceptible, which is why Euclid’s geometry works so well on Earth. Airline pilots, who fly along arcs of great circles, need the other one.

Sources: Elements, fifth postulate and Proposition I.32, D. E. Joyce’s edition · Roberto Bonola, Non-Euclidean Geometry (Dover, 1955) · Jeremy Gray, Ideas of Space (Oxford, 1989).

All topics on Euclid · ← The primes never run out · The legacy of Euclid →

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