110 A SHORT HISTORY OF SCIENCE 



conic section, these omissions being first filled by Pappus in the 

 third century A.D. It is shown that the normal makes equal angles 

 with the focal radii to the point of contact, and that the latter 

 have a constant sum for the ellipse, a constant difference for the 

 hyperbola. This book, he says in the letter quoted above, " contains 

 many curious theorems, most of them are pretty and new, useful 

 for the synthesis of solid loci. ... In the invention of these, I ob- 

 served that Euclid had not treated synthetically the locus . . . 

 but only a certain small portion of it, and that not happily, nor in- 

 deed was a complete treatise possible at all without my discoveries." 

 These three books, which are indeed based largely on the earlier 

 work of Euclid and others, contain most of the properties of conic 

 sections discussed in modern text-books on analytic geometry. 

 Book IV discusses the intersections of conies, treating tangency 

 correctly as equivalent to two ordinary intersections. In Book V 

 Apollonius even undertakes the difficult problem of determining 

 the longest and shortest lines which can be drawn from a given 

 point to a conic, identifying this with the problem of drawing 

 normals from a given point. He succeeds in discovering the points 

 for which two such normals coincide, i. e. what we call the centre 

 of curvature. Book VI deals with equal and similar conies, reach- 

 ing the problem of passing through a given cone a plane which 

 shall cut out a given ellipse. Book VII deals with conjugate 

 diameters and the complementary chords parallel to them. Book 

 VIII is lost. On the whole, in this remarkable work of some 400 

 propositions he achieved nearly all the results which are included 

 in our modern elementary analytic geometry, even approximating 

 the introduction of a system of coordinates by his use of lines 

 parallel to the principal axes. 



It is noteworthy that Fermat, one of the inventors of modern 

 analytic geometry, was led to it by attempting to restore certain 

 lost proofs of Apollonius on loci. 



Of his other mathematical writings little more than the titles 

 are known. Among these are one on burning mirrors, one on 

 stations and retrogressions of the planets, and one on the use and 

 theory of the screw. In astronomy he is believed to have sug- 



