202 Trans. Acad. Sci. of St. Louis. 



one set lying outside of the other. They are similar to 

 those represented by the terms of Eq. (2). Each of the 

 three dimensions of each block or volume in the outer 

 layer has been made y times as long as in the volumes 

 represented by the terms of Eq. (2). The similar dimen- 

 sions of the inner layer have been multiplied by x. 



A model of these cubes may be made as follows, assum- 

 ing that y = 5 and x — 3. 



The inner cube x* which is made of wood has dimen- 

 sions 9X9X9 cm. Its faces are each divided by lines 

 into nine squares. Three blocks of wood are then to be 

 applied to three adjoining sides. By the last three terms 

 of Eq. (6) their dimensions are 



xy Xxy X x (y — x)= 15 X 15 X 6, 



xy X x 2 X x (y — x)= 15 X 9X6, 



x 2 X x 2 Xx(y — x)= 9X 9X6. 



This forms a cube whose volume is 



[x 2 + x(y-x)Y=(xyy=(15y. 



Outside of this a similar set of blocks is placed, repre- 

 sented by the first three terms of Eq. (6). The dimen- 

 sions of these blocks are 



y 2 Xy 2 Xy(y — x)= 25X25X10, 



y 2 Xyx Xy(y — x)=25Xl5XlO, 



yx Xyx X y (y — x)= 15 X 15 X 10. 



The result is a cube whose volume is 



[x 2 + x(y — x)+y(y — x)Y = y\ 



The faces of the resulting cube should each be ruled 

 into twenty-five equal squares, whose sides are y = 5. 



Dr. Daniel Adams might have been more emphatic in 

 his statements without going astray. "The 6th, 9th, 

 12th, 18th, etc., powers may be represented by cubes" 

 and when so represented they have the same geometrical 



