Euclidean Geometry

Four of the postulates are short and obvious. You can draw a line between two points, extend it, draw a circle, and all right angles are equal. The fifth is long and awkward:

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 produced indefinitely, meet on that side.

In symbols: if those two interior angles satisfy $\alpha + \beta < \pi$, the lines eventually cross. Euclid himself seems to have been uneasy about it. He proves the first twenty-eight propositions without using it and calls on it only at Proposition 29.

Readers noticed. The postulate reads like a theorem that has not been proved yet, and for the next two millennia many people tried to prove it. In 1795 John Playfair gave the version most people learn today: through a point not on a given line, exactly one parallel line can be drawn.

The attempts form a long, international chain. Ptolemy and Proclus tried in antiquity. Ibn al-Haytham, Omar Khayyam and Nasir al-Din al-Tusi tried in the medieval Islamic world, and John Wallis in seventeenth-century England. Each proof assumed something that turned out to be the postulate in disguise: that rectangles exist, that similar triangles of different sizes exist, that parallel lines stay the same distance apart.

Girolamo Saccheri took the boldest route. In Euclid Freed of Every Flaw (1733), he assumed the postulate was false and tried to derive a contradiction. What he actually derived was a long run of valid theorems about a geometry in which triangle angles add up to less than $\pi$. He had no contradiction, so he declared one and stopped. Johann Lambert (1766) and Adrien-Marie Legendre went down the same road. Without accepting it, they had been mapping the ground that the non-Euclidean geometers would later claim.

Two other threads ran through the same centuries. Greek geometers had posed three construction problems they could not solve using only a straightedge and compass: trisect any angle, double the volume of a cube, and draw a square with the same area as a given circle. For two thousand years no one could do them, and no one could prove they were impossible.

The tool that settled them came from outside geometry. In 1637 René Descartes published La Géométrie, naming every point by coordinates and every curve by an equation. Pierre de Fermat had found the same method independently. Geometry problems became algebra problems.

That translation let Pierre Wantzel prove in 1837 that a straightedge and compass can only produce lengths built from repeated square roots. Doubling the cube needs $\sqrt[3]{2}$, so it is impossible, and so is trisecting a general angle. Ferdinand von Lindemann completed the list in 1882 by proving that $\pi$ is transcendental. It is not the root of any polynomial with whole-number coefficients, so no construction can reach it. Like the parallel postulate, the problems ended not with a construction but with a proof that none exists.

Book I, Proposition 47 of the Elements proves that in a right triangle the square on the hypotenuse equals the squares on the other two sides. Euclid's proof uses no algebra and no numbers, only areas, and it shows the axiomatic method at work.

Draw a right triangle $ABC$ with the right angle at $A$, and build a square outward on each side. From $A$, drop a line perpendicular to the hypotenuse $BC$ and extend it across the big square on $BC$, cutting that square into two rectangles. The claim is that each rectangle equals one of the smaller squares.

Take the square on $AB$. Join $C$ to the far corner of that square, and join $A$ to the far corner of the big square beyond $B$. The two triangles this makes are congruent: they have two sides equal (sides of the same squares) and the angle between those sides equal, a right angle plus the angle at $B$ in each case. Now use a fact proved earlier (I.41): a triangle has half the area of a parallelogram on the same base between the same parallels. One triangle is half the square on $AB$. The other is half the rectangle next to $B$. So the square on $AB$ equals that rectangle.

The same argument on the other side shows the square on $AC$ equals the other rectangle. Together the two rectangles make the whole square on $BC$, so

$$AB^2 + AC^2 = BC^2 .$$

Every step cites an earlier proposition, and those rest on the postulates. In particular I.41 depends, through the theory of parallels, on the fifth postulate. On a sphere or a saddle, where that postulate fails, the theorem is false. The most famous theorem in geometry is secretly a theorem about flat space.

When non-Euclidean geometry was finally accepted in the nineteenth century, the old question turned around. If the fifth postulate is optional, what exactly are the others doing? Looking closely, mathematicians found that Euclid's proofs also relied on facts he never stated. One example is that a line entering a triangle through one side must leave through another.

David Hilbert's Grundlagen der Geometrie (1899) filled those gaps. His axioms come in groups (incidence, order, congruence, parallels, continuity), and he proved results about the system itself: which axioms are independent of the others, and that the whole system is consistent if the real numbers are. After Hilbert, "Euclidean geometry" means one precise system, chosen from among several.

The field did not end there. The questions it still asks are about arrangements in flat space: how tightly shapes can be packed, and which configurations every curve must contain. Some of these questions have turned out to be among the hardest in mathematics.