Nipher — Graphical Algebra. 211 



In the vertical column having y 5 and x 5 at its extremi- 

 ties, are the terms involved in the construction of the dia- 

 gram forming Fig. 1, and representing?/ 10 — x 10 . The 

 difference between the two terms y* and y*x, multiplied 

 by their sum, gives the area of the two strips marked 1 

 and 2 in Fig. 1. It represents the difference between the 

 areas of the two squares'?/ 10 and y s x. Similarly (y*x — 

 y*x 2 ) (y 4 x-\-y z x 2 ) gives the areas of the two strips 

 marked 3 and 4 in Fig. 1. This vertical column is a re- 

 production of the series of values along the co-ordinate 

 axes in Fig. 1. The vertical column to the left contains 

 the terms obtained by dividing y 5 — x 5 by y — x. Start- 

 ing with any term in the triangle, successive terms along 

 the line leading upwards to the right are obtained by mul- 

 tiplying by y. Along a horizontal line the factor is xy. 

 Along a line leading downwards to the right the factor 

 is x. Along a line leading vertically downwards the factor 

 is x/y. Throughout this triangle we find the conditions 

 which exist around the small quadrangle forming its apex 

 where (y°x°) (yx) =y X x. The square of any term is 

 equal to the product of any two terms similarly placed 

 with respect to it, along either of the eight lines of terms 

 diverging from it. 



The methods here outlined have been in use in the 

 physics laboratory of Washington University for forty- 

 four years. The student who knew nothing of analytical 

 geometry was made familiar with the curves represented 

 by the equations y = x n and y = ax n . Curves were drawn 

 for values of n of 3, 2, 1, y 2 , 0, — y 2 , — 1, and — 2. When 

 familiar with the forms of these curves, he would be 

 asked to "discover" by experimental methods such laws 

 as that for the simple pendulum. The form of the curve 

 obtained by plotting the observed values of I and t made 

 it seem probable that the relation was represented by the 

 equation l = at 2 . When the lines representing the values 

 t were elongated until their length was t X t, if the points 

 so determined were in a straight line, the probability be- 

 came a certainty. The conclusion is that the length of 



