400 
PAP,PUS OF ALEXANDRIA 
book, practically constitutes our only source of information. 
The Book begins (p. 634) with a definition of analysis and 
synthesis which, as being the most elaborate Greek utterance 
on the subject, deserves to be quoted in full. 
‘The so-called AvaXvoyevos is, to put it shortly, a special 
body of doctrine provided for the use of those who, after 
finishing the ordinary Elements, are desirous of acquiring the 
power of solving problems which may be set them involving 
(the construction of) lines, and it is useful for this alone. It is 
the work of three men, Euclid the author of the Elements, 
Apollonius of Perga and Aristaeus the elder, and proceeds by 
way of analysis and synthesis.’ 
Definition of Analysis and Synthesis. 
‘ Analysis, then, takes that which is sought as if it were 
admitted and passes from it through its successive conse 
quences to something which is admitted as the result of 
synthesis: for in analysis we assume that which is sought 
as if it were already done (yeyoro?), and we inquire what it is 
from which this results, and again what is the antecedent 
cause of the latter, and so on, until by so retracing our steps 
we come upon something already known or belonging to the 
class of first principles, and such a method we call analysis 
as being solution backwards {avanaXiv Xvcnv). 
‘ But in synthesis, reversing the process, we take as already 
done that which was last arrived at in the analysis and, by 
arranging in their natural order as consequences what before 
were antecedents, and successively connecting them one with 
another, we arrive finally at the construction of what was 
sought; and this we call synthesis. 
‘ Now analysis is of two kinds, the one directed to searching 
for the truth and called theoretical, the other directed to 
finding what we are told to find and called 'problematical. 
(1) In the theoretical kind we assume what is sought as if 
it were existent and true, after which we pass through its 
successive consequences, as if they too were true and established 
by virtue of our hypothesis, to something admitted; then 
(a), if that something admitted is true, that which is sought 
will also be true and the proof will correspond in the reverse 
order to the analysis, but (h), if we come upon something 
admittedly false, that which is sought will also be false. 
(2) In the problematical kind we assume that which is pro 
pounded as if it were known, after which we pass through its