Content of type 'chapter'

Measurement of a Circle (Κύκλου μέτρησις)

The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference, of the circle.

On the Equilibrium of Planes (Ἐπιπέδων ἰσορροπιῶν αʹ βʹ)

Proposition 6. Commensurable magnitudes balance at distances reciprocally proportional to the weights.

Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς)

Lemma of Archimedes. The excess by which the greater of (two) unequal areas exceeds the less can, by being added to itself, be made to exceed any given finite area.

Sphere and Cylinder, On Spirals & Conoids and Spheroids

Proposition 33. The surface of any sphere is equal to four times its greatest circle ().

Book of Lemmas (Liber Assumptorum)

Proposition 4 (Area of the Arbelos). If AB be the diameter of a semicircle and N any point on AB, and if semicircles be described within the first semicircle and having AN, NB as diameters respectively, the figure included between the circumferences of the three semicircles is ‘what Archimedes called ἄρβηλος’ (shoemaker’s knife); and its area is equal to the circle on PN as diameter, where PN is perpendicular to AB and meets the original semicircle in P.

The Method of Mechanical Theorems (Ἔφοδος)

Let ABC be a segment of a parabola bounded by the straight line AC and the parabola ABC. Bisect AC at D, draw DBE parallel to the diameter, and join AB, BC. Then shall the segment ABC be of the triangle ABC.

Semi-Regular Solids, Stomachion & The Sand-Reckoner

If be a geometric progression starting from (specifically powers of ), the product equals the term whose distance from equals the distance of from !