194 Trans. Acad. Sci. of St. Louis. 
If we increase the lengths of the horizontal sides of 
these squares from y and 2, to y® and 2’, the rectangles 
so resulting will have an areayX y’?=y*, andvx v= 
x*. In the diagram, Fig. 2, representing these rectangles 
the values of y and # have been assumed to be 5 and 3. 
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a 
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8 He 
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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’, and in the other 
of three squares each having an area of 3°. We may also 
say that these rectangles represent ee y® and @. 
The binominal y’*— z° being divided by y—w as in 
Kq. 1, we have when y —za is multiplied sate the terms 
of the quotient 
y—a—y (y—a) + ye (y—ax) +2* (y—a2) (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’ and whose width is y—a. The ieceadl term is rep- 
resented by the rectangle whose dimensions are % and 
y’ — yx = y (y—ax). The final term has an area & (y% — 
x*) = 2° (y—a). 
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 gram it Big. 2 and in others that follow may be made by 
1 When one or two factors in a term represent an area, the remain- 
ing factors may be treated as multipliers. Thus «2 (y—z) may * 
2x2 in this case, or x X 2z. 
