138 THE RATE OF GROWTH [ch. 



whether at free hand or by help of m^athematical rules; it is one 

 way of getting rid of non-essentials — and to do so has been called 

 the very key-note of mathematics*. A simple rule, first used by 

 Gauss, is to replace each point by a mean between it and its two 

 or more neighbours, and so to take a "floating" or "running 

 average." In so doing we trade once more on the "principle of 

 continuity"; and recognise that in a series of observations each 

 one is related to another, and is part of the contributory evidence 

 on which our knowledge of all the rest depends. But all the while 

 we feel that Gaussian smoothing gives us a practical or descriptive 

 result, rather than a mathematical one. 



Some curves are more elegant than others. We may have to rest 

 content with points in which no. order is apparent, as when we plot 

 the daily rainfall for a month or two; for this phenomenon is one 

 whose regularity only becomes apparent over long periods, when 

 average values lead at last to "statistical uniformity." But the 

 most irregular of curves may be instructive if it coincide with another 

 not less irregular : as when the curve of a nation's birth-rate, in its 

 ups and downs, follows or seems to follow the price of wheat or the 

 spots upon the sun. 



It seldom happens, outside of the exact sciences, that we com- 

 prehend the mathematical aspect of a phenomenon enough to define 

 (by formulae and constants) the curve which illustrates it. But, 

 failing such thorough comprehension, we can at least speak of the 

 trend of our curves and put into words the character and the course 

 of the phenomena they indicate. We see how this curve or that 

 indicates a uniform velocity, a tendency towards acceleration or 

 retardation, a periodic or non-periodic fluctuation, a start from or an 

 approach to a limit. When the curve becomes, or approximates to, 

 a mathematical one, the types are few to which it is Kkely to 

 belong f. A straight line, a parabola, or hyperbola, an exponential 

 or a logarithmic curve (like x'=ay^), a sine-curve or sinusoid, damped 

 or no, suffice for a wide range of phenomena; we merely modify our 

 scale, and change the names of our coordinates. 



* Cf. W. H. Young, The mathematic method and its limitations, Atti del Congresso 

 dei Matematici, Bologna, 192/8, i, p. 203. 



t Hence the engineer usually begins, for his first tentative construction, by 

 drawing one of the familiar curves, catenary, parabola, arc of a circle, or curve of, 

 sines. 



