The Works of Euclid, Vol. 1

[he] was not studying geometry, [is] acting just as if someone wished to learn the Greek or Latin language from modern works written in Greek and Latin.

The following theorems, which are accustomed to be present in any treatise on Geometry, are missing from the Elements of Euclid:

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

Any circle is equal to a right-angled triangle, one of the sides of which containing the right angle is equal to the semi-diameter, and the other is equal to the circumference.

The convex surface of any 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 to a circle whose semi-diameter is a mean proportional between the side of the cylinder and the diameter of the base of the cylinder.

The convex surface of any right cone, excluding the base, is equal to a right-angled triangle, one of the sides of which containing the right angle is equal to the side of the cone, and the other is equal to the circumference of the base of the cone, or to a circle whose semi-diameter is a mean proportional between the side of the cone and the semi-diameter of the circle which is the base of the cone.

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

The surface of any sphere is equal to four of the greatest circles of that same sphere, or to the convex surface of the circumscribed cylinder.

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

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

Some have believed that these theorems have vanished from Euclid’s Elements through the inclemency of time; but [they are] mistaken. For these theorems, which cannot be demonstrated except by the aid of the four first postulates placed at the beginning of the first book On the Sphere and Cylinder, could not have been demonstrated by Euclid, who had not admitted these postulates of Archimedes.