GREEK SCIENCE IN ALEXANDRIA 109 



ment. Archimedes, indeed, and Euclid obtained ellipses by passing 

 other planes through right cones, but Apollonius first showed that 

 any cone and any section could be taken, and introduced the 

 names ellipse, parabola, and hyperbola. In the prefatory letter to 

 Book I, Apollonius says to the friend to whom it is addressed : 



'Apollonius to Eudemus, greeting. When I was in Pergamum 

 with you, I noticed that you were eager to become acquainted with 

 my Conies ; so I send you now the first book with corrections and will 

 forward the rest when I have leisure. I suppose you have not for- 

 gotten that I told you that I undertook these investigations at the 

 request of Naucrates the geometer, when he came to Alexandria and 

 stayed with me; and that, having arranged them in eight books, I 

 let him have them at once, not correcting them very carefully (for 

 he was on the point of sailing) but setting down everything that 

 occurred to me, with the intention of returning to them later. Where- 

 fore I now take the opportunity of publishing the needful emendations. 

 But since it has happened that other people have obtained the first 

 and second books of my collections before correction, do not wonder 

 if you meet with copies which are different from this.' Gow. 



Of the eight books, the first four are devoted to an elementary 

 introduction. In Book I he defines the cone as generated by a 

 straight line passing through a point on the circumference of a 

 circle and a fixed point not in the same plane ; he fixes the manner 

 in which sections are to be taken and defines diameters and ver- 

 tices of the curves, also the latus rectum and centre, conjugate 

 diameters and axes. The other branch of the hyperbola is taken 

 due account of for the first time. In Book II asymptotes are 

 defined by the statement : " One draws a tangent at a point of the 

 hyperbola, measures on it the length of the diameter parallel to it, 

 and connects the point thus determined with the centre of the 

 hyperbola." Book III contains numerous theorems on tangents 

 and secants and introduces foci with the definition : " A focus is 

 a point which divides the major axis into two parts whose rectangle 

 is one-fourth that of the latus rectum and the major axis," or the 

 square on the minor axis. The focus of the parabola however is 

 not recognized, nor has he any knowledge of the directrix of a 



