140 THE RATE OF GROWTH [ch. 



capable of explanation, up to a certain point, without mathe- 

 matics. 



(1) If in our coordinate diagram we have merely to pass from 

 one isolated jpoint to another, a straight line joining the two points 

 is the shortest — and the hkeliest way. 



(2) To rise and fall alternately, going to and fro from maximum 

 to minimum, a zig-zag rectilinear path would still be, geometrically, 

 the shortest way; but it would be sharply discontinuous at every 

 turn, it would run counter to the "principle of continuity," it is not 

 likely to be nature's way. A wavy course, with no more change of 

 curvature than is absolutely necessary, is the path which nature 

 follows. We call it a simple harmonic motion, and the simplest of 



The Sine -curve 



The S- shaped curve 



The bell-shaped curve 

 Fig. 27. Simple curves, representing a change from one magnitude to another. 



all such wavy curves we call a sine-curve. If there be but one 

 maximum and one minimum, which our variant alternates between, 

 the vector pathway may be translated into jpolar coordinates; the 

 vector does- what the hands of the clock do, and a circle takes the 

 place of the sine-curve. 



(3) To pass from a zero-line to a maximum once for all is a very 

 different thing ; for now minimum and maximum are both of them 

 continuous states, and the principle of continuity will cause our 

 vector-variant to leave the one gradually, and arrive gradually at 

 the other. The problem is how to go uphill from one level road to 

 another, with the least possible interruption or discontinuity. The 

 path follows an S-shaped course; it has an inflection midway; and 

 the first phase and the last are represented by horizontal asymptotes. 

 This is an important curve, and a common one. It so far resembles 



