Nipher—Graphical Algebra. 201 
The width of these rectangles e, f, g, h and k remain 
as before, but their length has been made # times 
greater.” 
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‘, which he represents as three such cubes 
placed side by side. Then follows a drawing of 3°, and 
3°, the latter being represented by 27 such cubes in the 
form of a cube whose edges are 3? in length. He gives 
a verbal description of the forms of such volumes up to 
(318. ete.,”? but he says not a word about space of 18 
dimensions.° 
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* and 2° 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 # and y be increased to #* and y we 
shall have two cubes whose volumes are y’ and 2". The 
difference in the volumes of these cubes will be, 
yo —a=y (y—a) + ya (y— 2) + yx" (y—2) 
+ y?x* (y—a) + ya (y—2) +2 (y— a) (6) 
The second member may be written 
y Ly? (y—a2) + ya (y—#) +2* (y—2)] 
a* [y? (y—a) + yx (y—2) +2 (y—2)] 
A reference to Eq. (2) will show that we have between 
the outer and inner cubes, two sets of rectangular volumes, 
: n a lawn. 
White tine attached to corner stakes may be used to mark boundary 
J 
3As a matter of historical interest a reproduction of the page in 
which Dr. Adams gives this discussion, is reproduced at the: close. of 
8 paper. 
