Nipher — Graphical Algebra. 203 



significance as when the 3d power is so represented. 

 Also any numerical value raised to any power, may be 

 represented in space of two dimensions or as a straight 

 line. 



Eq. (6) may also be represented by a diagram analo- 

 gous to that in Fig. 4. There will be three strip areas 

 across the top of the square x 6 , instead of the two areas 

 a and b. Their common length is y s . There will be three 

 rectangles replacing c and d of Fig. 4, the height of each 

 being x 3 . The width of these strips can be found by 

 dividing these factors into the terms of Eq. 6. 



All of the terms of the developed function y n — x n can 

 be identified in a diagram, in which y and x are laid off 

 on the vertical axis, and «/ n_1 and x Uml are laid off along 

 the horizontal axis. The strip a of Fig. 1 will increase in 

 length as n increases, but its width y — x will remain 

 constant. It will be represented by the first term. Below 

 this strip, will be a series of rectangles having a common 

 height x. They will represent the second and succeeding 

 terms of the developed function. 



The value of (y-\-xY may be represented by an area. 

 Assume for example y — 5 and x = 3. Construct a 

 square whose sides are (y + x) 2 = 64. Divide the sides 

 into segments whose lengths are y 2 — 25, 2xy = 30 and 

 x 2 = 9. Draw lines across the square through the points 

 separating these segments. The square {y + xY will 

 then be divided into nine rectangles, representing the 

 six terms of the expanded binominal {y-\-x)*. If x be 

 considered negative, some of these rectangles must be 

 considered negative in value. Their location in the 

 square will be a matter of interest to the young experi- 

 menter. If in addition y = x, the sum of the positive 

 and negative areas will be equal. 



Issued May 9, 1918 



