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The Elements

The corpus

Thirteen books

From the plane to the five cosmic figures. Books I–VI treat magnitude in the plane; VII–IX treat number; X the irrational; XI–XIII the solid. The last proposition of the last book is that there are exactly five regular polyhedra.

  1. I

    Foundations of the plane

    Definitions, postulates, and common notions, then 48 propositions: congruence, the parallel theory, and the Pythagorean theorem as I.47.

    Triangles, parallels, area

  2. II

    Geometric algebra

    Identities we now write algebraically — (a+b)², completing the square — proved as statements about rectangles.

    Rectangles and squares

  3. III

    The circle

    Central and inscribed angles, the tangent as perpendicular to the radius, intersecting chords, and the power of a point.

    Chords, tangents, angles

  4. IV

    Regular figures

    How to inscribe and circumscribe triangles, squares, pentagons, hexagons, and the 15-gon in a circle.

    Inscription and circumscription

  5. V

    Proportion

    Eudoxus’s theory of proportion, built to handle incommensurable lengths as carefully as whole numbers.

    Magnitudes in ratio

  6. VI

    Similarity

    Similar triangles, the geometric mean, and the general Pythagorean theorem for similar figures on the sides of a right triangle.

    Similar figures

  7. VII

    The numbers

    Units, primes, and the Euclidean algorithm for the greatest common measure of two numbers.

    Elementary number theory

  8. VIII

    Continued proportion

    Numbers in geometric progression, and when a mean proportional exists between two numbers.

    Geometric sequences

  9. IX

    Number and infinity

    Infinitely many primes (IX.20), even and odd, and the construction of even perfect numbers from Mersenne primes.

    Primes and evenness

  10. X

    Incommensurables

    The longest book. A classification of irrational lengths arising from squares and rectangles — the ancestry of √2.

    Irrational magnitudes

  11. XI

    Solids

    Lines and planes in three dimensions, parallelepipeds, and the solid analogue of Book I.

    Planes in space

  12. XII

    Method of exhaustion

    Circles are to one another as the squares on their diameters. Pyramids, cones, and cylinders measured by exhaustion — a precursor of the integral.

    Volumes and area

  13. XIII

    The cosmic figures

    Construction of the five regular polyhedra — tetrahedron, cube, octahedron, dodecahedron, icosahedron — and the proof that there are no others.

    Platonic solids

Start at the beginning — the postulates of Book I.