Appolonius to Eudemus, greeting.
If you are in good health and matters are in other respects as you wish. it is well: I am pretty well too. During the time I spent with you at Pergamum. I noticed how eager you were to make acquaintance with my work in conics: I have therefore sent to you the first book, which I have revised, and 1 will send the remaining books when I am satisfied with them. I suppose you have not forgotten hearing me say that I took up this study at the request of Naucrates the geometer, at the time when he came to Alexandria and stayed with me. and that. when I had completed the investigation in eight books, I gave them to him at once, a little too hastily, because he was on the point of sailing, and so I was not able to correct them. but put down everything as it occurred to me. intending to make a revision at the end. Accordingly. as opportunity permits, I now publish on each occasion as much of the work as I have been able to correct. As certain other persons whom I have met have happened to get hold of the first and second books before they were corrected, do not be surprised if you come across them in a different form.
Of the eight books the first four form an elementary introduction. The first includes the methods of producing the three sections and the opposite branches of the hyperbola and their fundamental properties, which are investigated more fully and more generally than in the works of others. The second book includes the properties of the diameters and the axes of the sections as well the asymptotes, with other things generally and necessarily used in determining limits of possibility: and what I call diameters and axes you will learn from this book. The third book includes many remarkable theorems useful for the syntheses of solid loci and for determining limits of possibility; most of these theorems. and the most recent are new, and it was their discovery which made me realize that Euclid had not worked out the synthesis of the locus with respect to three and four lines, but only a chance portion of it, and that not successfully; for the synthesis could not be completed without the theorems discovered by me. The fourth book investigates how many times the sections of cones can meet one another and the circumference of a circle; in addition it contains other things. none of which have been discussed by previous writers, namely, in how many points a section of a cone or a circumference of a circle can meet the opposite branches of hyperbolas.
The remaining books are thrown in by way of addition: one of them discusses fully minima and maxima, another deals with equal and similar sections of cones, another with theorems about the determinations of limits, and the last with determinate conic problems. When they are all published it will be possible for anyone who reads them to form his own judgment. Farewell.
If a straight line be drawn from a point to the circumference of a circle, which is not in the same plane with the point, and be produced in either direction, and if while the point remains stationary, the straight line be made to move round the circumference of the circle until it returns to the point whence it set out. I call the surface described by the straight line a conical surface, it is composed of two surfaces lying vertically opposite to each other, of which each extends to infinity when the straight line which describes them is produced to infinity; I call the fixed point the Vertex, and the straight line drawn through this point and the centre of the circle I call the axis.
The figure bounded by the circle and the conical surface between the vertex and the circumference of the circle I term a cone, and by the Vertex of the cone I mean the point which is the vertex of the surface. and by the axis I mean the straight line drawn from the vertex to the centre of the circle, and by the base I mean the circle.
Of cones, I term those right which have their axes at right angles to their bases, and scalene those which have their axes not at right angles to their bases.
In any plane curve I mean by a diameter a straight
line drawn from a curve which bisects all straight lines drawn
in either curve parallel to a given straight line and by the verteces of
the curve I mean the extremity of the straight line on the curve, and
I describe each of the parallels as being drawn ordinatewise to