Euclidean geometry

• 1. Introduction

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  Euclidean geometry has shaped how humans understand space for more than two thousand years, and it begins with a single man and a single book. Euclid, an ancient Greek mathematician, laid out his system in a work called Elements, and the approach he took was strikingly spare: assume a small number of obvious-seeming starting points, then use pure reasoning to derive everything else. What kinds of truths can you reach from almost nothing? How far can logic, unaided by experiment, really go? And what happens when the foundations that seemed unshakeable turn out to be just one option among many?

• 2. The Book That Replaced All Others

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  Elements is thirteen books in one. Books I through IV, and Book VI, cover plane geometry, proving results about triangles, circles, and other flat figures. Book I, proposition 47, contains the Pythagorean theorem, stated in Euclid's own words: the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Books V and VII through X turn to number theory, treating numbers as lengths of line segments. There, Euclid proves that there are infinitely many prime numbers. Books XI through XIII address solid geometry, including the result that a cone has exactly one-third the volume of a cylinder with the same height and base, and the construction of the Platonic solids.

  The improvement Elements offered over earlier treatments was recognized so quickly that earlier works were largely abandoned. They are now nearly all lost. Euclid was not simply reporting what others had found; he was the first to organize those findings into a logical system where each result follows from axioms and previously proved theorems.

  Plane geometry from Elements is still taught in secondary schools today, where it serves as the first example most students encounter of an axiomatic system and of mathematical proof. That continuity, across more than two millennia of education, points to something about the book that goes beyond its content alone.

• 3. Five Postulates

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  Near the beginning of Book I, Euclid states five postulates for plane geometry, as translated by Thomas Heath. The first four are brief and feel self-evident: draw a straight line between any two points; extend a finite straight line continuously; describe a circle with any center and radius; and accept that all right angles are equal. The fifth postulate, the parallel postulate, is longer and more conditional. It states that if a straight line crosses two other straight lines and creates interior angles on one side that together sum to less than two right angles, then those two lines will eventually meet on that side.

  Euclid himself seems to have sensed that the fifth postulate was different in character from the first four. The evidence is in his own ordering: his first 28 propositions in Book I are exactly those that can be proved without the parallel postulate. He held it back as long as possible.

  To the ancient scholars who studied Elements, the parallel postulate seemed to require proof from simpler statements. By the year 1763, at least 28 different proofs of it had been published, and every single one was found to be incorrect. The reason, as later mathematicians would discover, is that proof is impossible. Both the version with the parallel postulate and the version without it are internally consistent systems.

• 4. Compass, Straightedge, and What Cannot Be Built

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  Postulates 1, 2, 3, and 5 are constructive by nature: they do not merely assert that certain geometric figures exist, they provide methods for creating them using only a compass and an unmarked straightedge. This makes Euclidean geometry more concrete than many modern axiomatic systems, which assert the existence of objects without specifying how to build them.

  For centuries, geometers tried to determine the limits of compass-and-straightedge construction. Trisecting an arbitrary angle seemed like a natural problem, since the axioms themselves refer to those tools. Attempts continued until Pierre Wantzel published a proof in 1837 that the trisection is impossible by those means. Wantzel also addressed doubling the cube: the impossibility there comes from the fact that compass-and-straightedge methods involve equations whose order is an integral power of two, while doubling a cube requires solving a third-order equation. Squaring the circle was also proved impossible.

  Euclid often used proof by contradiction alongside construction. Some early Pythagorean proofs that assumed all numbers are rational were nonconstructive and fallacious; Euclid's constructive replacements were both correct and, many considered, more useful in practice.

• 5. A World Built on Measurement

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  Euclidean geometry rests on two fundamental types of measurement: angle and distance. The angle scale is absolute, with the right angle serving as Euclid's basic unit. A 45-degree angle, in Euclid's terms, is half of a right angle. Distance, by contrast, is relative: one line segment is chosen as the unit, and all other distances are expressed in relation to it.

  Some practical results were used long before they were formally proved. The right-angle property of the 3-4-5 triangle, for instance, was put to work in surveying well before anyone had written a proof. Historically, surveyors measured distances using chains, including a device called Gunter's chain, and angles using graduated circles and, later, the theodolite.

  Area and volume measurements in Euclid's system are derived from distances. A rectangle with a width of 3 and a length of 4 has an area representing the product 12. Because this geometric interpretation of multiplication was tied to physical dimensions, it broke down at four or more factors, and Euclid deliberately avoided products of four or more numbers, even when they appeared implicitly in proofs. The distinction between figures that are equal in measure and figures that are congruent, meaning the same size and shape, runs throughout the system: a 2x6 rectangle and a 3x4 rectangle are equal in area but are not congruent to each other.

• 6. Archimedes, Descartes, and Later Builders

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  Archimedes, born around 287 BCE, is remembered alongside Euclid as one of the greatest ancient mathematicians. He proved equations for the volumes and areas of figures in two and three dimensions, and established that a sphere has two-thirds the volume of its circumscribing cylinder, a result Euclid had not reached. His work, unlike Euclid's, is believed to have been entirely original.

  Rene Descartes, who lived from 1596 to 1650, introduced analytic geometry, an alternative approach that turned geometric problems into algebra. In his system, a point on a plane is represented by its Cartesian coordinates, and a line is represented by its equation. The Pythagorean theorem, which follows from Euclid's axioms in the original approach, becomes in Descartes's version a definition embedded in the formula for distance between two points.

  Also in the 17th century, Girard Desargues introduced idealized points, lines, and planes at infinity, motivated by the theory of perspective. His work produced a form of projective geometry that could, in turn, generate proofs in ordinary Euclidean geometry with fewer special cases to handle.

  In the 1840s, William Rowan Hamilton developed the quaternions, and Arthur Cayley used them to study rotations in four-dimensional Euclidean space. Ludwig Schläfli, working around the same period, extended Euclidean geometry to higher dimensions and discovered all the regular polytopes, the higher-dimensional analogues of the Platonic solids. He found six regular convex polytopes in dimension four, and three in all higher dimensions. His work was published in full only after his death, in 1901, and had little influence until H.S.M. Coxeter rediscovered and fully documented it in 1948.

• 7. When Parallel Lines Diverge

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  Around 1830, Janos Bolyai and Nikolai Ivanovich Lobachevsky separately published work on non-Euclidean geometry, in which the parallel postulate does not hold. Because non-Euclidean geometry is provably consistent with Euclidean geometry, the parallel postulate cannot be derived from the other four postulates. The centuries of failed proofs finally had a definitive explanation.

  Albert Einstein's theory of special relativity involves a four-dimensional space-time called Minkowski space, which is non-Euclidean. In general relativity, even the three-dimensional geometry of space is not Euclidean. A triangle constructed from three rays of light will not, in general, have interior angles that sum to 180 degrees, because gravity curves the paths. A relatively weak gravitational field, such as the Earth's or the Sun's, produces a metric that is approximately but not exactly Euclidean.

  Until the 20th century, no technology existed that could detect these deviations. Einstein predicted them. During a solar eclipse in 1919, observations confirmed the slight bending of starlight by the Sun. That confirmation, grounded in non-Euclidean geometry, is now embedded in the software that runs the GPS system.

• 8. Rigor Revisited

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  By the 19th century, mathematicians recognized that Euclid's original ten axioms and common notions were not sufficient to prove all the theorems in Elements. Euclid had assumed implicitly, for instance, that any line contains at least two points, and that when two circles each pass through the other's center they will actually intersect. Neither assumption is guaranteed by his axioms; both require additional axioms of their own.

  Starting with Moritz Pasch in 1882, several improved axiomatic systems were proposed. The best known are those of Hilbert, George Birkhoff, and Alfred Tarski. Hilbert aimed to identify a simple and complete independent set of axioms from which the major geometric theorems could be deduced. Birkhoff proposed four postulates confirmable with a scale and protractor, relying heavily on properties of the real numbers. Tarski, who lived from 1902 to 1983, defined elementary Euclidean geometry as the geometry expressible in first-order logic without depending on set theory. He proved that his formulation is consistent and complete in a specific sense: there is an algorithm that can determine, for every proposition, whether it is true or false. This is equivalent to the decidability of real closed fields, of which elementary Euclidean geometry is one model.

  At the 1900 Paris conference, Alessandro Padoa of the Peano delegation articulated the modern understanding of undefined symbols in a formal system: they are, in his words, completely devoid of meaning until an interpretation is supplied. Bertrand Russell put it more bluntly in his own assessment: mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. That apparent paradox is not a flaw in Euclidean geometry. It is precisely what makes the system work.