THE CONICS 
129 
well. During the time I spent with you at Pergamum 
I observed your eagerness to become acquainted with my 
work in conics; I am therefore sending you the first book, 
which I have corrected, and I will forward the remaining 
books when I have finished them to my satisfaction. I dare 
say you have not forgotten my telling you that I undertook 
the investigation of this subject at the request of Naucrates 
the geometer, at the time when he came to Alexandria and 
stayed with me, and, when I had worked it out in eight 
books, I gave them to him at once, too hurriedly, because he 
was on the point of sailing; they had therefore not been 
thoroughly revised, indeed I had put down everything just as 
it occurred to me, postponing revision till the end. Accord 
ingly I now publish, as opportunities serve from time to time, 
instalments of the work as they are corrected. In the mean 
time it has happened that some other persons also, among 
those whom I have met, have got the first and second books 
before they were corrected; do not be surprised therefore if 
you come across them in a different shape. 
Now of the eight books the first four form an elementary 
introduction. The first contains the modes of producing the 
three sections and the opposite branches (of the hyperbola), 
and the fundamental properties subsisting in them, worked 
out more fully and generally than in the writings of others. 
The second book contains the properties of the diameters and 
the axes of the sections as well as the asymptotes, with other 
things generally and necessarily used for determining limits 
of possibility {SLopLa-fj-oi)', and what I mean by diameters 
and axes respectively you will learn from this book. The 
third book contains many remarkable theorems useful for 
the syntheses of solid loci and for diorismi; the most and 
prettiest of these theorems are new, and it was their discovery 
which made me aware that Euclid did not work 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 it 
was not possible for the said synthesis to be completed without 
the aid of the additional theorems discovered by me. The 
fourth book shows in how many ways the sections of cones 
can meet one another and the circumference of a circle; it 
contains other things in addition, none of which have been 
discussed by earlier writers, namely the questions in how 
many points a section of a cone or a circumference of a circle 
can meet [a double-branch hyperbola, or two double-branch 
hyperbolas can meet one another]. 
The rest of the books are inore by way of surplusage 
(TrepiovcrLaa-TLKcoTepa): one of them deals somewhat fully with