viii PREFACE 
consider the way in which he may have discovered the conic war, 
sections and their fundamental properties. It seems to me enfor 
much better to give the complete story of the origin and to tin 
development of the geometry of the conic sections in one them 
place, and this has been done in the chapter on conic sections replie 
associated with the name of Apollonius of Perga. Similarly wishe 
a chapter has been devoted to algebra (in connexion with Creel 
Diophantus) and another to trigonometry (under Hipparchus, Muse 
Menelaus and Ptolemy). and : 
At the same time the outstanding personalities of Euclid helpf 
and Archimedes demand chapters to themselves. Euclid, the Tr 
author of the incomparable Elements, wrote on almost all 
the other branches of mathematics known in his day. Archi 
medes’s work, all original and set forth in treatises which are 
models of scientific exposition, perfect in form and style, was 
even wider in its range of subjects. The imperishable and 
unique monuments of the genius of these two men must be 
detached from their surroundings and seen as a whole if we 
would appreciate to the full the pre-eminent place which they 
occupy, and will hold for all time, in the history of science. 
The arrangement which I have adopted necessitates (as does 
any other order of exposition) a certain amount of repetition 
and cross-references; but only in this way can the necessary 
unity be given to the whole narrative. 
One other point should be mentioned. It is a defect in the 
existing histories that, while they state generally the contents 
of, and the main propositions proved in, the great treatises of 
Archimedes and Apollonius, they make little attempt to 
describe the procedure by which the results are obtained. 
I have therefore taken pains, in the most significant cases, 
to show the course of the argument in sufficient detail to 
enable a competent mathematician to grasp the method used 
and to apply it, if he will, to other similar investigations. 
The work was begun in 1913, but the bulk of it was 
written, as a distraction, during the first three years of the