Nipher — Graphical Algebra. 201 



The width of these rectangles e, f, g, h and k remain 

 as before, but their length has been made x times 

 greater. 2 



In an arithmetic published by Dr. Daniel Adams, of 

 Keene, N. H., in 1848, and which caused me much trouble 

 in my youthful days, he gives a drawing of a cube, the 

 length of the edges being 3 units. He then presents a 

 drawing of 3 4 , which he represents as three such cubes 

 placed side by side. Then follows a drawing of 3 5 , and 

 3 6 , the latter being represented by 27 such cubes in the 

 form of a cube whose edges are 3 2 in length. He gives 

 a verbal description of the forms of such volumes up to 

 "3 18 , etc.," but he says not a word about space of 18 

 dimensions. 3 



All of the preceding discussion for quantities of the 

 third degree or over, can be most interestingly presented 

 by drawings in three dimensions. If y 3 and x 3 are rep- 

 resented as is usually done by cubical volumes, the geo- 

 metrical significance of the three terms of Eq. 2 differs 

 from that given in Fig. 2. It is unnecessary to explain 

 them here. 



If the lengths of x and y be increased to x 2 and y 2 we 

 shall have two cubes whose volumes are y 6 and x 6 . The 

 difference in the volumes of these cubes will be, 

 y e — x e = y 5 {y — x)+ y*x (y — x)+ y 3 x 2 (y — x) 

 + y 2 x 3 (y — x)+yx 4 {y — x)+x 5 (y — x) (6) 



The second member may be written 



y 3 [y 2 (y—x) +y* (y — %) + % 2 (y — %)] 

 %* [y 2 (y — %)+ y% (y :— «) + x * (y — %) ] 



A reference to Eq. (2) will show that we have between 

 the outer and inner cubes, two sets of rectangular volumes, 



2 The diagram representing Eq. 5 can be laid out upon a lawn. 

 White twine attached to corner stakes may be used to mark boundary 

 lines. 



3 As a matter of historical interest a reproduction of the page in 

 which Dr. Adams gives this discussion, is reproduced at the close of 

 this paper. 



