What are the titles of Euclid's Element's books?

Answer 1

The 13 books of Euclid's Elements are generally referred to by their book number.

The main subject matter of each book:

Book 1 contains the basic properties of geometry: the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area).

Book 2 is commonly called the "book of geometrical algebra," because the material it contains may easily be interpreted in terms of algebra.

Book 3 deals with circles and their properties: inscribed angles, tangents, and the power of a point.

Book 4 is concerned with inscribing and circumscribing triangles and regular polygons.

Book 5 is a treatise on proportions of magnitudes.

Book 6 applies proportions to geometry: Thales' theorem, similar figures.

Book 7 deals strictly with elementary number theory: divisibility, prime numbers, greatest common divisor, least common multiple.

Book 8 deals with proportions in number theory and geometric sequences.

Book 9 applies the results of the preceding two books: the infinitude of prime numbers, the sum of a geometric series, perfect numbers.

Book 10 attempts to classify incommensurable (in modern language, irrational) magnitudes by using the method of exhaustion, a precursor to integration.

Book 11 generalizes the results of Books 1-6 to space: perpendicularity, parallelism, volumes of parallelepipeds.

Book 12 calculates areas and volumes by using the method of exhaustion: cones, pyramids, cylinders, and the sphere.

Book 13 generalizes Book 4 to space: golden section, the five regular (or Platonic) solids inscribed in a sphere.