Euclid's Elements

Euclid's Elements is the oldest systematic treatise on Euclidean geometry. For more than twenty centuries the Elements was the major textbook model in the study and teaching of mathematics in the West. Also in other fields Euclid's work led the way. The philosopher Spinoza wrote his work Ethics along the lines of the Elements and so did the physicist Newton when he composed his opus magnum Principia.

The Elements is often considered as one of the documents, next to the Bible, that had the most impact on the Western culture. However, according to modern mathematical standards of rigor, the Elements show some shortcomings. These have been repaired as late as the 1890s by the German mathematician Hilbert.

Contents

The work consists of thirteen Books.

Book I starts with definitions of the type: a point is that which has no part and: a line is a breadthless length. (Most of the other books begin with their own collections of definitions.) It then goes on to the formulation of the five postulates of Euclidean geometry:

1. It is possible to draw a straight line from any point to any point.
2. It is possible to extend a finite straight line continuously in a straight line.
3. It is possible to describe a circle with any center and distance (radius).
4. All right angles are equal to one another.
5. 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 on which the angles are less than the two right angles.

After these come axioms or common notions, such as "Things that are equal to the same thing are equal to each other."

The first 26 propositions (theorems or constructions) are based on the first four of the above postulates, and treat mainly congruence of triangles and of other geometric figures. After the parallel postulate is first used, the subject largely shifts to parallelograms.

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Euclid's parallel axiom. In the upper figure the angles α and β are equal. In the lower figure β > α.

One of the most famous propositions is the sixteenth which states that the exterior angle of a triangle is greater than either remote interior angle. See the lower part of the figure on the right, where angle β is larger than angle α by virtue of Euclid's sixteenth proposition.

The sixteenth proposition, together with the fifth postulate, implies the existence of parallel lines, see the upper part of the figure. Indeed, one can prove that if a straight line (the red one) falling on two other straight lines makes two alternate angles that are equal (α = β), then the two straight lines must be parallel. Euclid realized that this result could not be proven on basis of his first four postulates, and therefore he added his fifth most famous postulate. In the present case the fifth postulate reads: if β > α then line 1 and line 2 must cross. To see that this statement is indeed the fifth postulate, we call γ the supplementary angle adjacent to β. That is, β+γ = 180⁰. The fifth postulate states that if α+γ < 180⁰ then line 1 and line 2 cross on the side where that is the case. Now,

$\alpha +\gamma <180^{0}\Longrightarrow \alpha +\gamma -180^{0}<0\Longrightarrow 180^{0}-\gamma >\alpha$

and since β ≡ 180⁰−γ we see that the statement: if β > α then line 1 and line 2 cross, is indeed Euclid's fifth postulate. By proposition 16 and postulate 5 it follows that line 1 and line 2 cross if and only if β > α. Hence the lines do not cross (are parallel) if and only if α = β. (Note that the case β < α is covered by interchanging α and β.)

For many centuries workers have tried to prove the fifth postulate from the other four, but all attempts failed. Indeed, a geometry—a non-Euclidean geometry—without this postulate is possible and logically consistent (in contrast to, for instance, Immanuel Kant's view, who gave a philosopher's proof of the necessity of Euclidean geometry).

Another famous proposition in Book I is number 47: the Pythagorean theorem, which is proved by a technique still used in high school texts. This is followed by its converse, which ends Book I.

Book II is on geometrical algebra. All quantities in Book II are handled geometrically. Numbers are represented by line segments and products of numbers by areas. Most of division is postponed to Book V.

Book III is about circles, chords, inscribed angles and so on.

Book IV deals with figures inscribed in and circumscribed about circles. For example, triangles, squares, regular pentagons and hexagons.

Book V is probably the most important of the books. It is based on Eudoxus' work on proportions. It is about ratios, including incommensurable ratios, but avoids irrational numbers by an ingenious method. The ratio a/b is said to be the same as the ratio c/d if ma >, = or < nb according as mc >, = or < nd, respectively. The approach covers proportions for all kinds of magnitudes. Book V proves twenty-five theorems about magnitudes and ratios of magnitudes.

Book VI, about similar figures, uses the theory of Book V. It gives the solution to the general quadratic equation with positive discriminant.

Book VII deals with theory of numbers, i.e. whole numbers. It covers divisors and multiples, and develops a theory of ratio and proportion of such numbers independently of Book V. This is because the method of proof used in Book V assumes quantities are divisible.

Book VIII continues the treatment of number theory with coverage of geometric sequences and of square and cube numbers.

Book IX continues these topics and also brings in prime numbers, and odd and even numbers. It concludes with the proof that, if a power of 2 is 1 more than a prime number, then the product of that prime number and the previous power of 2 is a perfect number.

Book X classifies many types of irrationals—magnitudes incommensurable with given magnitudes.

Book XI deals with basic solid geometry, a solid being that which has length, breadth and depth. Surfaces are the extremities of solids. Discussed are, among others, pyramid, prism, sphere, cone, cylinder, cube, the regular octahedron, and the regular icosahedron.

Book XII treats the method of exhaustion, the word stemming from the seventeenth century when exhaustion was the immediate forerunner of integration. This method is also due to Eudoxus. Circles, spheres, cones and cylinders are treated as limits of polygons and polyhedra.

Book XIII deals with the five regular solids. It shows how to construct them, inscribed in spheres, classifies the ratios of the lengths of the sides of the regular solids inscribed in the same sphere by the system of Book X, and proves there are only five.

The thirteen books of the Elements contain about 467 propositions (theorems or constructions); the exact number is somewhat uncertain as various ancient editors "improved" the text.

Bibliography

• Artmann, Benno. Euclid - The Creation of Mathematics (2001) excerpt and text search
• Cajori, Florian. A History of Mathematics (1919) complete text online free
• Euclid. The Thirteen Books of Euclid's Elements, Books 1 and 2 ed. by Thomas L. Heath (1956) excerpt and text search
• Euclid, trans. Thomas L. Heath. The Thirteen Books of Euclid's Elements, 3 vols., Cambridge University Press, reprinted Dover Publications.
• Kline, Morris. Mathematical Thought from Ancient to Modern Times, (1972). excerpt and text search
• Mueller, Ian. Philosophy of Mathematics and Deductive Structure in Euclid's Elements (1981).