The Elements of Euclid, written at Alexandria about 300 BCE, sets out plane geometry, the theory of proportion, the arithmetic of whole numbers, the classification of irrational lines and solid geometry in thirteen books of propositions, each proved from what precedes it, with definitions at the head of most books. Book I opens with the definitions, postulates and common notions and ends with the theorem on the square on the side opposite a right angle; Books II to IV treat rectangles on divided lines, the circle, and figures inscribed in and about circles. Book V gives the theory of ratio between magnitudes and Book VI applies it to similar figures. Books VII to IX are arithmetic: repeated subtraction to find the greatest common measure, numbers in continued proportion, the proof that the primes are more than any assigned multitude, and the rule for perfect numbers. Book X classifies irrational straight lines, and Books XI to XIII treat solids, the measure of pyramids, cones and spheres, and the construction of the five regular solids, ending with the proof that there is no sixth.
The Greek survives whole. Most manuscripts carry the edition made by Theon of Alexandria in the fourth century CE; one Vatican manuscript keeps an earlier form of the text.
Euclid of Alexandria Elements, Books I–XIII: summary and contents (PDF)
Euclid of Alexandria Elements, Books I–XIII: summary and contents (Markdown)
Contents
-
Αʹ Book I
Definitions of point, line, angle and circle, with the postulates and the common notions, open the Elements. The propositions build the geometry of triangles and parallels, from constructing an equilateral triangle through the congruence theorems and the angles of a triangle to parallelograms and the application of areas, and end with the theorem that the square on the side opposite a right angle equals the squares on the other two sides, and its converse.
-
Βʹ Book II
Only two terms are defined here, the rectangle contained by two straight lines and the gnomon. The propositions cut a line at random, into equal and unequal parts, or extend it, and compare the rectangles and squares on the pieces. They extend the right-angle theorem to obtuse and acute triangles, and the book closes by constructing a square equal to any given rectilinear figure.
-
Γʹ Book III
Equal circles, tangents, segments and sectors are defined before the propositions on the circle: finding its centre, chords and their distance from the centre, the tangent at right angles to the radius, angles at the centre and in segments, the angle in a semicircle, and the rectangles on intersecting chords. The book closes with a test for a tangent: a line from an outside point whose square equals the rectangle on a secant and its outer part.
-
Δʹ Book IV
Inscribing and circumscribing figures is the whole subject. After defining when one figure is inscribed in or about another, the book fits triangles, squares, regular pentagons and hexagons in and about circles, and circles in and about them. The pentagon rests on an isosceles triangle whose base angles are each double the vertex angle, and the last proposition combines triangle and pentagon to inscribe a regular figure of fifteen sides.
-
Εʹ Book V
The book defines ratio between magnitudes of the same kind and says when four magnitudes are in the same ratio, by comparing equal multiples of the first and third with equal multiples of the second and fourth. The propositions derive the rules for handling proportions, alternation, inversion, composition, separation and the ratio through equals, and end by proving that of four proportionals the greatest and least together exceed the other two.
-
Ϛʹ Book VI
Similar rectilinear figures, division in extreme and mean ratio, and height are defined, and the theory of proportion is applied to plane figures. Triangles and parallelograms under the same height are as their bases; the book proves the tests for similar triangles, finds third and mean proportionals, describes on a given line a figure similar to a given one, and ends with angles in equal circles standing in the ratio of the arcs they stand on.
-
Ζʹ Book VII
Arithmetic begins with definitions of unit and number, even and odd, prime and composite, and numbers in proportion. Subtracting the less from the greater in turn shows whether two numbers are prime to one another and, if not, finds their greatest common measure; the book goes on to proportion among numbers, least numbers in a given ratio and the least number measured by given numbers, closing with the least number that has given parts.
-
Ηʹ Book VIII
Numbers in continued proportion whose extremes are prime to one another are the least of those in their ratio, the book's first result on continued proportion among whole numbers. It then determines when mean proportionals fall between two numbers: one mean between two squares, two between two cubes. The closing propositions show that similar plane numbers are to one another as square to square, and similar solid numbers as cube to cube.
-
Θʹ Book IX
Opening with the product of two similar plane numbers, which is a square, the last arithmetical book of the Elements continues with squares, cubes and continued proportion. It includes the proof that the primes are more than any assigned multitude, and propositions on even and odd numbers, and ends by showing that if the sum of the doubling series from one is prime, that sum multiplied by the last term is a perfect number.
-
Ιʹ Book X
Irrational straight lines are classified here. After defining magnitudes commensurable and incommensurable, in length and in square, and calling rational a line commensurable with one assigned, the book gives the test for incommensurability by repeated subtraction and builds up the medial line, then the binomial and the apotome, each of these two in six orders. It closes by showing that infinitely many irrationals arise from the medial, none the same as any before it.
-
ΙΑʹ Book XI
Solid geometry begins with definitions of the solid, of lines and planes at right angles and inclined, of parallel planes, and of the pyramid, prism, sphere, cone, cylinder, cube, octahedron, icosahedron and dodecahedron. The propositions treat lines perpendicular to planes, parallel planes and solid angles, then parallelepipeds and their equality and ratios, and end with two prisms of equal height, one on a triangle and one on a parallelogram double it, shown to be equal.
-
ΙΒʹ Book XII
Circles, pyramids, cones and spheres are compared by ratio. From similar polygons in circles, which are as the squares on the diameters, the book proves the same of circles, then shows that a pyramid is a third of the prism and a cone a third of the cylinder on the same base and height, and closes with spheres, which are to one another in the triple ratio of their diameters.
-
ΙΓʹ Book XIII
Propositions on a line cut in extreme and mean ratio lead through the regular pentagon to the construction of the five regular solids, pyramid, octahedron, cube, icosahedron and dodecahedron, each enclosed in a given sphere and its edge compared with the sphere's diameter. The last book of the Elements ends by proving that no other solid can be bounded by equilateral and equiangular figures, with a lemma on the angle of the regular pentagon.