Nipher—Graphical Algebra. 211 
In the vertical column having y’ and 2° at its extremi- 
ties, are the terms involved in the construction of the dia- 
gram forming Fig. 1, and representing y’°—a'*. The 
difference between the two terms y° and y‘z, 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 y” and y’x. Similarly (y*a— 
yx") (y*x + y*a*) 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* —a* by y—a. 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 ay. 
Along a line leading downwards to the right the factor 
isa. Along a line leading vertically downwards the factor 
is v/y. Throughout this triangle we find the conditions 
Which exist around the small quadrangle forming its apex 
where (y°x°) (yx) =y Xa. 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 
physies laboratory of Washington University for forty- 
four years. The student who knew nothing of analytical 
Seometry was made familiar with the curves represented 
by the equations y = 2" and y==acx", Curves were drawn 
for values of n of 3, 2,1, 14, 0, — 1%, —1, and —2. When 
familiar with the forms of these curves, he would be 
asked to ‘‘discover’’ by experimental methods such laws 
4s that for the simple pendulum. The form of the curve 
obtained by plotting the observed values of / and ¢ made 
it seem probable that the relation was represented by the 
equation 1 = aq ¢?, When the lines representing the values 
t were elongated until their length was t X t, if the paints 
So determined were in a straight line, the probability be- 
came a certainty. The conclusion is that the length of 
