The New Thoughts of Galileo

upon A C at right angles, and that A B C is a conoid described by the curved line A B or B C, moved circularly around the axis B D: so that the base A C is a circle, one will find the center of this body A B C D, when the curve A B C is that whose segments of the diameter are as the cubes of the ordinates, if one makes B M to M D as five to three. If it is the following one, one must make it as six to four: if the other following one, as seven to five: if the other, as eight to six, and so on to infinity.

Furthermore, if one wishes to know the areas of these figures, in the first the surface comprised by this curve, and the straight line A C, is to the inscribed triangle A B C, as six to four. And as eight to five in the second; as ten to six in the third, and as twelve to seven in the fourth, and so on to infinity.

And if A B C is the first conoid, that is to say the one which is described by the first of these lines, it is to the inscribed cone as nine to five: if it is the second, it is to this cone as twelve to six: if it is the third, as fifteen to seven: if the