A guide to all 13 books of Euclid's Elements
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Foundations, Parallel Lines & the Pythagorean Theorem
From definitions and postulates to constructions, triangles, parallel lines, equal-area transformations, and the Pythagorean theorem and its converse.
BOOK 2Geometric Algebra & Areas
Develops geometric algebra through rectangles, area decomposition, and classical identities expressed as geometry.
BOOK 3Circles, Tangents & Inscribed Angles
Covers centers, chords, tangents, inscribed angles, intersecting chords, and classical circle power relations.
BOOK 4Inscribed, Circumscribed & Regular Polygons
Explores inscribed and circumscribed figures and constructions of regular polygons including pentagons and hexagons.
BOOK 5Eudoxian Theory of Proportion
Builds a rigorous theory of proportion for commensurable and incommensurable magnitudes and develops operations on ratios.
BOOK 6Similarity, Ratios & Applications of Areas
Connects line ratios, triangle similarity, mean proportionals, area ratios, and applications of areas.
BOOK 7The Euclidean Algorithm & Number Theory
Introduces numbers, primes, the Euclidean algorithm, coprimality, LCM, and Euclid's lemma.
BOOK 8Geometric Sequences & Mean Proportionals
Studies continued proportions, geometric sequences, mean proportionals, and ratios of plane and solid numbers.
BOOK 9Infinitely Many Primes & Perfect Numbers
Covers prime-number results, infinitely many primes, geometric-series relations, and the classical construction of even perfect numbers.
BOOK 10Irrational Magnitudes & Incommensurability
Systematically classifies incommensurable magnitudes, binomials, apotomes, and related irrational constructions.
BOOK 11Lines, Planes & Solid Geometry
Develops spatial relations among lines and planes, solid angles, parallelepipeds, and proportional properties of solids.
BOOK 12Exhaustion, Circles, Cones & Spheres
Uses the method of exhaustion to study areas and volumes of circles, pyramids, cones, cylinders, and spheres.
BOOK 13Golden Ratio & the Five Platonic Solids
Concludes the Elements with golden-ratio geometry, pentagons, and constructions and uniqueness of the five Platonic solids.