The Works of Euclid, Vol. 1

[he] was not studying Geometry in Euclid, was doing the same thing as one who would want to learn Greek and Latin by reading modern works written in those two languages.

The following theorems, which are usually found in every elementary treatise on Geometry, are not found in the Elements of Euclid:

The circumferences of circles are to one another as their diameters.

A circle is equal to a right-angled triangle in which one of the sides of the right angle is equal to the radius, and the other side of the right angle is equal to the circumference.

The convex surface of a right cylinder is equal to a rectangle whose height is equal to the side of the cylinder, and whose base is equal to the circumference of the base of the cylinder, or else to a circle whose radius is the mean proportional between the side of the cylinder and the diameter of its base.

The surface of a right cone, the base excepted, is equal to a right-angled triangle in which one of the sides of the right angle is equal to the side of the cone, and the other side of the right angle is equal to the circumference of the base of the cone, or else to a circle whose radius is the mean proportional between the side of the cone and the radius of the circle which is the base of the cone.

The convex surfaces of right and similar cylinders, and of right and similar cones, are to one another as the diameters of the bases of these cylinders and these cones.

The surface of a sphere is equal to four great circles, or to the convex surface of the circumscribed cylinder.

The surfaces of spheres are to one another as the squares of their diameters.

A sphere is equal to two-thirds of the circumscribed cylinder.

Some persons have thought that these theorems had disappeared from the Elements of Euclid due to the ravages of time; this is a mistake. These theorems, which can only be demonstrated with the help of the first four postulates placed at the beginning of the first book of the Sphere and the Cylinder, could not have been [demonstrated] by Euclid, who had not admitted these postulates of Archimedes.