Four Books on Conics (Commandino 1566)

And Apollonius writes of these, not as if they were invented by himself. For he would not have said that these had been treated by him more fully and universally than by others. But what Geminus writes is true. The ancients, he says, defining a cone as the revolution of a right-angled triangle, while one of the sides about the right angle remains fixed, considered that all cones were right cones and that one section was formed in each: in a right-angled cone, they called it a parabola; in an obtuse-angled cone, a hyperbola; and in an acute-angled cone, an ellipse. You may find sections named this way among them everywhere. Therefore, just as those ancients, contemplating the two right angles in each species of triangle, first in the equilateral, then in the isosceles, and afterwards in the scalene, proved the universal theorem of later ages, "The three interior angles of every triangle are equal to two right angles," so also in conic sections, they contemplated the section of a right-angled cone only in a right-angled cone—by cutting the plane at a right angle to one side of the cone—and they demonstrated the section of an obtuse-angled cone as formed in an obtuse-angled cone, and the section of an acute-angled cone in an acute-angled cone, similarly drawing planes at right angles to one of their sides in all cones: which the ancient names of the sections also indicate. But later, Apollonius of Perga looked universally at all sections existing in every cone, whether right or scalene, according to the different inclination of the plane to the cone. For this reason, the men of that time, admiring the wondrous demonstration of the conic theorems, called him the great geometer. These things Geminus left written in the sixth book of his mathematical doctrines.