196 Trans. Acad. Set. of St. Louis. 



In Fig. 3 we find the lengths in centimeters to be 



Unit = 0.5 cm. — 1.0 



x = 1.5 cm. — 3 



y = 2.5 cm. — 5 



xy = 7.5 cm. — 15 



y 2 - 12.5 cm. — 25 



The lengths in terms of the chosen unit are given in the 

 final column. 



The distance x 2 y may be laid off upon the axes, by 

 drawing a line through the points marked 1 and xy. 

 A line parallel to this through the point distant x on 

 the vertical axis of the diagram, Fig. 3, will determine 

 the point distant x 2 y from the origin on the horizontal 

 axis. 



If we increase the width of the two rectangles y z and 

 x 3 of Fig. 2 to y 2 and x 2 , we shall have two squares whose 

 areas are y 2 X y 2 = y* and x 2 X x 2 = x*. 



They are represented in Fig. 4. Figures 1 and 2 will 

 also be identified in this diagram. 



The difference between the areas of these two squares 

 is 



if — x* — (y 2 + x 2 ) (if — x 2 ) 



= (y +x) (y 2 -\-x 2 ) (y — x) 

 = f (y — x) + y 2 x (y — x) 



+ yx 2 (y — x)+x 3 (y — x). (3) 



The first two terms of the second member may be 

 written 



y 2 (y 2 — yx) and y 2 {yx — x 2 ) 



They are represented in Fig. 4 by the large rectangles 

 a and b. 



The two following terms of Eq. (3) may be written 



x 2 (y 2 — yx) and x 2 (yx — x 2 ) 



They are represented by the rectangles marked c and d. 



