Nipher — Graphical Algebra. 209 



In planning a diagram involving squares y* and x*, 1/y*, 

 1/x 4 upon a sheet of paper having an ordinary size the 

 lengths y 2 and 1/y 2 may be laid off upon the two axes as 

 may be suitable to the size of the paper. If y 2 is about 

 15 or 16 inches and 1/y 2 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 2 on one axis and 1/y 2 on the 

 other axis, which cross the two axes at points equidistant 

 from the origin. Since 



W i 



y 2 



these points will be at unit distance from the origin. If 

 the above values of 1/y 2 and y 2 are adopted, the unit of 

 length thus assumed will be not far from 3% or 4 inches. 



In terms of this unit y 2 will be about 4 units in length. 

 The value of y in terms of this unit may be determined 

 in a similar manner, since l/y = y/y 2 . 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 x may then be assumed to be about midway 

 between 1 and y. Then the lengths xy, x 2 , Vxy, y/x, Vx, 

 x/y, 1/x, 1/Vxy, 1/x 2 and 1/xy 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 4 , an interesting col- 

 lection of squares and rectangles will be produced. With- 

 in the square y 4 eight hyperbolae may be drawn, passing 

 through 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 y 2 x 2 , y 2 , yx, x 2 , y, x, 1, and x 2 /y 2 . 



The difference between the areas of the two squares 

 having sides whose lengths are Vy and Vx is 



y — x=Vy (Vy—Vx) +V~x (Vy— Vx). 



