Nipher—Graphical Algebra. 207 
and x° = 0.000976 inch. This square would hardly be visi- 
ble to the unaided eye. The width of the two strips marked 
1 and 2 in Fig. 1 would be y® — y*a = 3125 — 156.25 = 
2968.75 inches. Their combined length is y’+ y’a= 
3281.25. The product of these two numbers would be the 
difference between the areas of the two squares, y"° and 
a: 
The width of the two areas which represent the ninth 
and tenth terms would be yx* — a = 0.01953 — 0.000976 
= 0.01856. Their combined length is ya* + 2° = 0.02050. 
The product of these two quantities gives the difference 
between the two squares y’a* and 2”. 
This diagram and the final form given to Eq. (7) shows 
that Eq. (5) is similar to another well known equation 
y—v=y (y—2z) +a (ya). 
Equation (5) may be written: 
y” — a" =y' (ys — y'z) + y* (y’a— 2") 
oa y (yea? — yx) = y? (y?x* — yx") 
+ y° (yxt— 2°) + 2° (y’— y'a) 
ot xe (y*a — ya") oa Lie (yra? — y*x") 
+ a? (y’a* — yx") + 2° (ya* — 2°) (8) 
In order to respond to this change the strips marked 
3, 5, 7, and 9 must be extended to the right side of the 
square y™, their widths remaining unchanged, when 
drawn to scale. They will then have a common length y’. 
The first term in this equation is the same as in Kq. (7). 
The strips marked 2, 4, 6, and 8 must be correspondingly 
reduced in length to 2°, the strip marked 10 remaining 
unchanged. In this arrangement the strips marked 1, 3, 
5, 7, and 9 will represent the first five terms in the equa- 
tion, and those marked 2, 4, 6, 8, and 10 will represent the 
five final terms. As thus changed, the strip marked 9 
would have an area y° (yx*—a*). As drawn in the dia- 
gram the strip marked 5 has an area y°x* (y°a* — y*a"), 
which is equal to the area of strip 9 when thus extended. 
