Four Books on Conics (Commandino 1566)

lines; but only a small part of it, and this not sufficiently well. For it could not be that this composition was correctly performed without those things which were discovered by us. The fourth book hands down in how many ways the conic sections can meet each other and the circumferences of circles; and many other things for a fuller doctrine, none of which has been handed down to memory by those who were before us. To how many points a conic section, and the circumference of a circle, and the opposite sections can meet the opposite sections. The remaining four books, however, pertain to a more abundant knowledge. For the fifth deals for the most part with minimums and maximums. The sixth concerns equal and similar conic sections. The seventh contains theorems that have the power of determining. The eighth, determinate conic problems. But truly, all these having been published, it is permitted for each person who happens upon them in reading to judge according to his own opinion. Farewell.

COMMENTARY OF EUTOCIUS OF ASCALON ON THE FIRST BOOK OF THE CONICS OF APOLLONIUS, FROM HIS OWN EDITION.

An ornamental drop cap 'A' features a landscape with a building and foliage within a square frame.

APOLLONIUS the geometer, dearest companion of Anthemius, was born in Perga, which is a city of Pamphylia, during the time of Ptolemy Euergetes, as Heraclius relates in the life of Archimedes. He also writes that Archimedes was indeed the first to have approached the conic theorems; but that Apollonius, when he had discovered them not yet published by Archimedes, published them as his own. And this is not true, in my opinion. For Archimedes seems to make mention of an older instruction of conics in many places; and Apollonius writes these not as if discovered by himself. For he would not have said that these had been treated more fully and more universally by him than by others. But what Geminus writes is true: "The ancients," he says, "defining the cone by the revolution of a right-angled triangle, with one of the sides around the right angle remaining, thought that all cones were right, and that there was one section in each: in the right-angled cone it was called a parabola a conic section where the plane is parallel to the side of the cone; in the obtuse-angled one, a hyperbola a conic section where the angle exceeds two right angles; and in the acute-angled one, an ellipse a conic section where the angle is less than two right angles; and thus you may find the sections named among them everywhere." Just as, therefore, those ancients, while contemplating two right angles in each species of triangles, first in the equilateral, then in the isosceles, and afterwards in the scalene, those later in age demonstrated a universal theorem of this kind: "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, called the parabola, only in a right-angled cone, namely with the plane cutting at a right angle to one side of the cone; but they demonstrated the section of the obtuse-angled cone made in an obtuse-angled cone, and the section of the acute-angled cone in an acute-angled cone, similarly drawing planes at right angles to one of their sides in all cones. This is also indicated by the ancient names of the sections. But afterwards, Apollonius of Perga examined universally that all sections exist in every cone, whether right or scalene, according to the different inclination of the plane to the cone. For this reason, the people of that time, admiring the wonderful demonstration of the conic theorems, called him a great geometer. Geminus left these things written in the sixth book of his mathematical doctrines. We will make what he says manifest in the figures below. For let the triangle through the axis of the cone be a b c; and from any point e let a line d e f be drawn at right angles to a b; and let a plane passed through d f, at a right angle to a b, cut the cone. Therefore, each angle a e d, a e f is a right angle; and with the cone being right-angled, and the angle b a c being right, as appears in the first figure, the angles b a c, a e f will be equal to two right angles. Wherefore, the line d e f will be equidistant to a c; and there will be formed on the surface of the cone a section called a parabola, so named original: "ἀπὸ τοῦ παραβάλλειν εἶναι" from the fact of being parallel, that is, from the fact that the line d f, which is the common section of the cutting plane and the triangle through the axis, is parallel to the side a c of the triangle.