Nipher—Graphical Algebra. 209 
In planning a diagram involving squares y‘ and 2*, 1/y', 
1/az* upon a sheet of paper having an ordinary size the 
lengths y* and 1/y? may be laid off upon the two axes as 
may be suitable to the size of the paper. If y’ is about 
15 or 16 inches and 1/y? is to be not less than one inch, the 
length which the unit of length must have may be deter- 
mined by purely geometrical means. Draw parallel lines 
through the points marked y? on one axis and 1/y* on the 
other axis, which cross the two axes at points equidistant 
from the origin. Since 
eee 1 
yee y? 
these points will be at unit distance from the origin. If 
the above values of 1/y? and y’ are adopted, the unit of 
length thus assumed will be not far from 3%4 or 4 inches. 
In terms of this unit y? will be about 4 units in length. 
The value of y in terms of this unit may be determined 
in a similar manner, since 1/y=y/y’. By this method 
of similar triangles the lengths Vy, 1/Vy and 1/y may be 
determined, all values being laid off on both axes. 
The length 2 may then be assumed to be about midway 
between 1 and y. Then the lengths xy, 2°, Vary, y/x, V2. 
/y, 1/a, 1/Vxy, 1/2? and 1/ay can be similarly deter- 
mined by graphical methods, and laid off on both axes. 
If cross-lines are now drawn through points so deter- 
mined extending across the square y‘, an interesting col- 
lection of squares and rectangles will be produced. With- 
in the square y* eight hyperbolae may be drawn, passing 
thr ough the corners of these squares and rectangles. The 
Points on the various curves which are so determined 
range from three to thirteen. The constants of these hy- 
Perbolae are yx, y’, ya, a, y, x, i, and 2? /y?. 
The difference between the areas of the two squares 
having sides whose lengths are Vy and Va is 
y¥—x=Vy (Vy— Va) + Va (Vy—V2). 
