THE MATHEMATICAL SCIENCES 137 



The discovery of the irrational V2 dealt a first blow 

 to this conception, which was completely shaken by 

 the arguments of Zeno of Elea ; but, before even the 

 theory of proportions had been established by Eudoxus, 

 the Greek geometers had succeeded in generalizing 

 the quantitative study of magnitudes and in creating 

 thus a kind of geometrical algebra. Their method was 

 as follows : 



The representation of a magnitude by the length of 

 a segment can play the same part as the symbolical 

 letters of algebra. This being so, in order to subtract 

 or add two rational or irrational magnitudes, it is 

 sufficient to represent them by segments, and then 

 to place one of these segments on the other or on its 

 extension. 



The quantities which we call imaginary or negative 

 certainly cannot be represented in this way ; still, in 

 many cases, the variations of the figure lend themselves 

 partly to the same generalizations as the use of negative 

 quantities in algebra. 



As to the multiplication of magnitudes, in the direct 

 sense, it is nonsensical, but it is possible to represent 

 it indirectly by means of a rectangle whose sides are 

 formed by the segments representing the two magni- 

 tudes to be multiplied. 



In this manner a second geometrical expression of 

 magnitudes is obtained, that is, as rectangular or 

 square surfaces. To add or subtract them in this new 

 form, it is necessary to give them a common side ; one 

 of the rectangles, whilst keeping the same area, is 

 then transformed in such a way as to enable it to 

 be applied exactly to the other. This operation is 

 performed by means of the following proposition : the 

 lines parallel to the sides of a rectangle, which intersect 

 on one of the diagonals, divide this rectangle into 

 four others, of which two are equal, that is, those which 

 do not cross this diagonal (Fig. 15). 



