Nipher — Graphical Algebra. 



195 



purely geometrical methods, without assuming any 

 known values for x and y. If a straight line is drawn 

 upon a sheet of paper, the square of which that line is a 

 side, and the cube of which that square is to be a face, can 

 be drawn, without any measurement of the length of the 

 original line in terms of the centimeter or the inch. 



Similarly if vertical and horizontal lines are drawn in- 

 tersecting at a point which is to be common to the corners 

 of two squares, having areas y u and x n we may mark 

 upon these two axes points distant 1, x and y from the 

 origin. These distances may have any length. The unit 

 of length chosen may be any arbitrary and unknown 

 length. 



Fig. 3. 



In Fig. 3 the unit when finally measured is found to 

 be 0.5 cm. and x and y to be 1.5 and 2.5 cm. 



Draw a line through the point marked 1 on the ver- 

 tical axis, to the point marked y on the horizontal axis. 

 Lines drawn parallel to this line through points marked 

 x and y on the vertical axis, will pass through points 

 distant xy and y 2 from the origin on the horizontal axis. 

 These distances may then be laid off upon the vertical 

 axis and the operation may be continued. All of the 

 lengths in the preceding diagram can be thus laid off. 



