484 
DIOPHANTUS OF ALEXANDRIA 
or fractional. It is possible, therefore, that Diophantus was 
empirically aware of the truth of the theorem of Fermat, but 
we cannot be sure of this. 
Conspectus of the Arithmetical with typical solutions. 
There seems to be no means of conveying an idea of the 
extent of the problems solved by Diophantus except by giving 
a conspectus of the whole of the six Books. Fortunately this 
can be done by the help of modern notation without occupying 
too many pages. 
It will be best to classify the propositions according to their 
character rather than to give them in Diophantus’s order. It 
should be premised that x, y, z ... indicating the first, second 
and third ... numbers required do not mean that Diophantus 
indicates any of them by his unknown (y); he gives his un 
known in each case the signification which is most convenient, 
his object being to express all his required numbers at once in 
terms of the one unknown (where possible), thereby avoiding the 
necessity for eliminations. Where I have occasion to specify 
Diophantus’s unknown, I shall as a rule call it except when 
a problem includes a subsidiary problem and it is convenient 
to use different letters for the unknown in the original and 
subsidiary problems respectively, in order to mark clearly the 
distinction between them. When in the equations expressions 
are said to be = u 2 , v 2 , tv 2 , t 2 ... this means simply that they 
are to be made squares. Given numbers will be indicated by 
a, b, c ... m, n ... and will take the place of the numbers used 
by Diophantus, which are always specific numbers. 
Where the solutions, or particular devices employed, arc 
specially ingenious or interesting, the methods of solution will 
be shortly indicated. The character of the book will be best 
appreciated by means of such illustrations. 
[The problems marked with an asterisk are probably 
spurious.] 
(i) Equations of the first degree with one unknown. 
I. 7. x — a = m{x — b). 
I. 8. x + a = m(x + b).