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20 THE NATURE OF MATHEMATICS 

denoted by a conventional sign, as if it were known, and the 
deduction at last, of some relation which determines what the 
entity must be. 

And this brings us to the second branch spoken of: algebra 
with the later Greeks. Diophantus of Alexandria, who pro- 
bably lived in the early half of the fourth century after Christ, 
and probably was the original inventor of an algebra, used 
letters for unknown quantities in arithmetic and treated 
arithmetical problems analytically. Juxtaposition of symbols 
represented what we now write as " + ," ar >d " " and " = " 
were also represented by symbols. All these symbols are 
mere abbreviations for words, and perhaps the most important 
advantage of symbolism the power it gives of carrying out 
a complicated chain of reasoning almost mechanically was not 
made much of by Diophantus. Here again we come across 
the economical value of symbolism : it prevents the wearisome 
expenditure of mental and bodily energy on those processes 
which can be carried out mechanically. We must remember 
that this economy both emphasises the unsubjugated that is 
to say, unsystematised problems of science, and has a charm 
an aesthetic charm, it would seem of its own. 

Lastly, we must mention "incommensurables," "loci," and 
the beginnings of " trigonometry." 

Pythagoras was, according to Eudemus and Proclus, the 
discoverer of "incommensurable quantities." Thus, he is said 
to have found that the diagonal and the side of a square are 
" incommensurable." Suppose, for example, that the side of 
the square is one unit in length ; the diagonal is longer than 
this, but is not two units in length. The excess of the length 
of the diagonal over one unit is not an integral submultiple of 
the unit. And we can proceed in this way without end. Ex- 
pressing the matter arithmetically, the remainder that is left 
over after each division of a remainder into the preceding 
divisor is not an integral submultiple of the remainder used as 
divisor. That is to say, the rule given in text-books on arith- 
metic and algebra for finding the greatest common measure 
does not come to an end. This rale, when applied to integer 
numbers, always comes to an end ; but, when applied to certain 
lengths, it does not. Pythagoras proved, then, that if we