194 



Trans. Acad. Sci. of St. Louis. 



If we increase the lengths of the horizontal sides of 

 these squares from y and x, to y 2 and x 2 , the rectangles 

 so resulting will have an area yXy 2 = y 3 , and x X x 2 = 

 x 3 . In the diagram, Fig. 2, representing these rectangles 

 the values of y and x have been assumed to be 5 and 3. 



Fig. 2. 



We may also describe these rectangles as being in the 

 one case composed of five squares placed side by side, 

 these squares each having an area of 5 2 , and in the other 

 of three squares each having an area of 3 2 . We may also 

 say that these rectangles represent values y 3 and x 3 . 

 The binominal y z — x 3 being divided by y — x as in 

 Eq. 1, we have when y — x is multiplied into the terms 

 of the quotient 



y * _ x 3 = y 2 {y — x )+ yx (y — x)+ x 2 (y — x) (2) 



The first term in the second member is represented by 

 the area of the rectangle at the top of Fig. 2, whose length 

 is y 2 and whose width is y — x. The second term is rep- 

 resented by the rectangle whose dimensions are x and 

 y 2 — yx = y (y — x) . The final term has an area x (yx — 

 x 2 ) =af (y — x). 1 



Following a suggestion made by Professor Roever of 

 the department of mathematics of Washington Univer- 

 sity, it may be pointed out that the construction of the dia- 

 gram in Fig. 2 and in others that follow may be made by 



iWhen one or two factors in a term represent an area, the remain- 

 ing factors may be treated as multipliers. Thus x* (y — x) may be 

 2x 2 in this case, or a; X 2x. 





