The Elements of geometry. Thirteen books building from basic definitions and axioms to the theory of solid geometry and number theory. The most successful textbook ever written — it was used for two thousand years.
Start ReadingThe foundational definitions, postulates, and common notions that underpin all of Euclidean geometry, including the construction of equilateral triangles and the properties of angles.
Geometric algebra: the transformation of areas and the construction of figures equivalent to given figures — the Greek approach to what we now solve with equations.
The geometry of circles: inscribed and circumscribed figures, tangent lines, and the properties that define circular forms.
Inscribing and circumscribing regular polygons within circles, including the construction of the regular pentagon.
The theory of proportion applied to magnitudes of all kinds — Eudoxus's brilliant solution to the crisis of incommensurables.
Proportion theory applied to plane geometry: similar figures and the fundamental relationships between geometric forms.
Number theory begins: prime numbers, the Euclidean algorithm for finding greatest common divisors, and the fundamental theorem of arithmetic.
Continued number theory: proportions among numbers and the properties of even and odd numbers.
The theory of numbers concludes with results on prime numbers, including the proof that there are infinitely many primes.
The theory of irrational magnitudes — the classification of incommensurable line segments into thirteen distinct types.
Solid geometry begins: the properties of planes, lines in three-dimensional space, and the angles between them.
The volumes and surface areas of pyramids, prisms, cones, and cylinders — the mathematics of three-dimensional measurement.
The construction of the five Platonic solids — tetrahedron, cube, octahedron, dodecahedron, and icosahedron — and the proof that no others exist, crowning Euclid's mathematical edifice.