90 A SHORT HISTORY OF SCIENCE 



itself, or a product of prime factors greater than n, either of 

 which suppositions is contrary to the hypothesis that n itself is the 

 greatest prime number. 



Book X deals with the incommensurable on the basis of the 

 theorem : If two unequal magnitudes are given, and if one takes 

 from the greater more than its half, and from the remainder more 

 than its half and so on, one arrives sooner or later at a remainder 

 which is less than the smaller given magnitude. Books XI, XII, 

 and XIII are devoted to solid geometry, leading up to our familiar 

 theorems on the volume of prism, pyramid, cylinder, cone, and 

 sphere, but in every case without computation, emphasizing the 

 habitual distinction between geometry and geodesy or mensura- 

 tion . . . a distinction expressed by Aristotle in the form : "One 

 cannot prove anything by starting from another species, for ex- 

 ample, anything geometrical by means of arithmetic. Where 

 the objects are so different as arithmetic and geometry one cannot 

 apply the arithmetical method to that which belongs to magni- 

 tudes in general, unless the magnitudes are numbers, which can 

 happen only in certain cases." Book XIII passes from the 

 regular polygons to the regular polyhedrons, remarking in con- 

 clusion that only the known five are possible. 



The extent to which Euclid's Elements represent original work 

 rather than compilation of that of earlier writers cannot be deter- 

 mined. It would appear, for example, that much of Books I and 

 II is due to Pythagoras, of III to Hippocrates, of V to Eudoxus, 

 and of IV, VI, XI, and XII, to later Greek writers ; but the work 

 as a whole constitutes an immense advance over previous similar 

 attempts. 



Proclus (410-485 A.D.) is the earliest extant source of informa- 

 tion about Euclid. Theon of Alexandria edited the Elements 

 nearly 700 years after Euclid, and until comparatively recent 

 times modern editions have been based upon his. 



Like other Greek learning, Euclid has come down to later times 

 through Arab channels. There is a doubtful tradition that an 

 English monk, Adelhard of Bath, surreptitiously made a Latin 



