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 5 — y 4 x = 3125 — 156.25 = 

 2968.75 inches. Their combined length is y 5 + y 4 x — 

 3281.25. The product of these two numbers would be the 

 difference between the areas of the two squares, y 10 and 

 y s x 2 . 



The width of the two areas which represent the ninth 

 and tenth terms would be yx 4 — x 5 = 0.01953 — 0.000976 

 = 0.01856. Their combined length is yx 4 + x 5 = 0.02050. 

 The product of these two quantities gives the difference 

 between the two squares y 2 x 8 and x 10 . 



This diagram and the final form given to Eq. (7) shows 

 that Eq. (5) is similar to another well known equation 



y* — a* = y(y — X )+ X (y—x). 



Equation (5) may be written: 



y 10 — # 10 = y 5 (y 5 — y*x) + y 5 (y*x — y 3 x 2 ) 



+ y 5 (y 3 x 2 — y 2 x 3 ) + y 5 (y 2 x 3 — yx 4 ) 

 + y" (yx 4 — x z ) + x 5 (y 5 — y 4 x) 

 + x 5 (y 4 x — y 3 x 2 ) + x 5 (y 3 x 2 — y 2 x 3 ) 

 + x 5 (y 2 x 3 — yx 4 ) + x 5 (yx 4 — x 5 ) (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 10 , 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 Eq. (7). 

 The strips marked 2, 4, 6, and 8 must be correspondingly 

 reduced in length to x 5 , 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 5 (yx 4 — x 5 ). As drawn in the dia- 

 gram the strip marked 5 has an area y 3 x 2 (y 3 x 2 — y 2 x 3 ), 

 which is equal to the area of strip 9 when thus extended. 



