REVIEWS 34i 



to recognise this — and the logician is naturally concerned with mathematics as a 

 logical structure. It is no disparagement to him as such— though Prof. Shaw 

 seems to think so — that the psychological incidents of discovery are excluded by 

 the logician as irrelevant, for the same reason at bottom as information about the 

 diet of mathematicians is excluded from a book on mathematical theory. 



Philip E. B. Jourdain. 



Empirical Formulas. By Theodore R. Running, Associate Professor of 

 Mathematics, University of Michigan. [Pp. 144.] (New York: John 

 Wiley & Sons ; London : Chapman & Hall, 1917. Price 7s. net.) 

 This is No. 19 of the Mathematical Monographs edited by Merriman and 

 Woodward. " In the results of most experiments of a quantitative nature, two 

 variables occur . . . [such that] on plotting the sets of corresponding values . . . 

 the points so located lie approximately on a smooth curve. In obtaining a 

 mathematical expression which shall represent the relation between the variables 

 so plotted, there may be two distinct objects in view, one being to determine 

 the physical law underlying the observed quantities, the other to obtain a simple 

 formula, which may or may not have a physical basis, and by which an approxi- 

 mate value of one variable may be calculated from a given value of the other 

 variable. In the first case correctness of form is a necessary consideration. In 

 the second correctness of form is generally considered subordinate to simplicity 

 and convenience. It is with the latter of these that this volume is mostly 

 concerned" (p. 9). The first five chapters are on the simple tests which may be 

 applied to a set of data, " and which will enable us to make a fairly good choice of 

 equation" (p. 11), and the calculations of the values of constants in these empirical 

 formulas, and the methods employed, are almost wholly graphical. Chapter VI is 

 devoted to the evaluation of the constants in empirical formulae by the method of 

 least squares ; in Chapter VII formulas for interpolation are developed and their 

 applications briefly treated ; and Chapter VIII is devoted to approximate formulas 

 for areas, volumes, centroids, moments of inertia, and a number of examples are 

 given to illustrate their application. 



The purpose of the sixth chapter "is not to develop the method of least 

 squares, but only to show how to apply the method to observation equations so as 

 to obtain the best values of the constants" (p. 90). We have the theorem (p. 19) : 

 If two variables, x and y, are so related that, when values of x are taken in an 

 arithmetical series, the «th differences of the corresponding values of y are 

 constant, the law connecting the variables is expressed by the equation 

 y = a + ox + ex* + . . . + qx n . A case {n = 2) of such a relation is worked 

 out (pp. 90-4), so that ten observation equations are found for various x's and 

 the corresponding y's, from which we have to obtain three equations which will 

 yield the most probable values of the three unknowns a, b, and c. By the method 

 of least squares such a way of combination is known (see p. 97), and there remains 

 only the problem of calculation. 



Frankly, it is difficult to see what scientific purpose is served by an investi- 

 gation of an (x,y) graph when "simplicity and convenience" are more important 

 than " correctness of form " (p. 9) : surely the object in all such investigations is to 

 find, at least approximately, the form of the functional relation connecting x and y. 

 However, this book will undoubtedly be found useful to those confronted with the 

 problem of deciding upon a suitable relation between x and y and determining 



the constants. 



Philip E. B. Jourdain. 



