GREEK SCIENCE IN ALEXANDRIA 95 



Given magnitudes have a given ratio to each other. 



When two lines given in position cut each other their point of 

 intersection is given. 



When in a circle of given magnitude a line of given magnitude 

 is given, it bounds a segment which contains a given angle. 



A work on Fallacies is designed to safeguard the student against 

 erroneous reasoning. Still other treatises are devoted to Division 

 of Figures, Loci, and Conic Sections ; finally there are works on 

 Phenomena, on Optics, and on Catoptrics dealing with applica- 

 tions of geometry. 



The Phenomena gives a geometrical theory of the universe, the 

 Optics is an unsuccessful attempt to deal with problems of vision 

 on the hypothesis that light proceeds from the eye to the object 

 seen. The fundamental assumptions are, for example : " Rays 

 emitted from the eye are carried in straight lines, distant by an 

 interval from one another," etc. 



The Catoptrics deals in 31 propositions with reflections in plane, 

 concave, and convex mirrors. It is remarked that a ring placed 

 in a vase so as to be invisible from a certain position, may be made 

 visible by filling the vase with water. The authenticity of this 

 work is however questionable. 



These two works constitute the earliest known attempt to 

 apply geometry systematically to the phenomena of light-rays. 

 The law of reflection is correctly applied. Just as geometry is 

 based on a definite list of axioms, so Euclid makes his optics 

 depend on eight fundamental facts of experience. For example, 

 the light rays are straight lines. The figure inclosed by the rays 

 is a cone with its vertex at the eye, while the boundary of the 

 object corresponds to the base, etc. This work, though in very 

 imperfect form, continued in use until Kepler's time. 



ARCHIMEDES. - - The second great name in the Alexandrian 

 school and one of the greatest in the whole history of science is 

 that of Archimedes. He was both geometer and analyst, mathe- 

 matician and engineer. He enriched even the highly developed 

 Euclidean geometry, made important progress in algebra, laid the 

 foundations of mechanics, and even anticipated the infinitesimal 



