OF AETS AND SCIENCES. 223 



With the exception perhaps of the circle, the straight line is the only- 

 curve of whose correctness every one is a judge ; if, then, by any device 

 we can so transform a curve that it shall become a straight line, a 

 moment's inspection will show whether the agreement witli observation 

 is real. 



Let us first take a special case, and then proceed to the more general 

 discussion. A great many 2:)hysical laws may be expressed by the 

 equation y = m x", or that one quantity, y, is always proportional to 

 some power of the other, x. For example, the variation of gravity, 

 of the intensity of light, heat, and electric attraction, with the distance, 

 may be stated y = m x~'^, or n = —2. For elastic forces, w = 1, or 

 is proportional to the distance; for Mariotte's law, ?i = — 1 ; for the 

 deflection of a beam in terms of its length, n = 3, and so on. Some- 

 times « is a fractional number, or has a much larger value ; thus Wer- 

 theim suggests that the laws of elasticity may be explained by assum- 

 ing that the force of attraction of the particles varies at the 14th power 

 of their distance apart. In the same way, Rankine adopts the expo- 

 nents n = — IJ and n=: — y*- for the variations of the pressure and 

 volume of steam in the cylinder of an engine. Suppose now that we 

 have a number of points constructed, and wish to see if they can be rep- 

 resented by any curve of the form y = mx\ By drawing curves tak- 

 ing various values of n, as 1, 2, 3, i, &c., we may find one which will 

 agree, but it will be difficult to be sure whether some other value will 

 not give a more exact concordance. If, however, we take logarithms 

 of both sides, and write log y = log m -\- n log n, and calling log y = 

 T, log n = X, and log m = M, construct a curve with Zand X as co- 

 ordinates, we obtain T^= 31 -\- n X. If now the result is a straight 

 line, or differs from a straight line only by the accidental errors, — ■ that 

 is, if there is no curvature to one side more than on the other, — we 

 know that y varies as some power of x, and the value of n is readily 

 determined from the tangent of the angle the line makes with the axis 

 of X. In the same way m is obtained by finding the number whose 

 logarithm is M, the ordinate of the point where the line meets the axis 

 of T. On the other hand, if the line is not straight, but curved, we 

 may be sure that there is no value of n which will satisfy the observa- 

 tions, or that y is not proportional to any power of x. Let us next see 

 how far this method may be generalized. In the first place, instead 

 of X and y we may use any functions / and /' which include only x, y, 

 and known constants ; that is, which do not include m or n. For ex- 

 ample, the equation y^ = mx^ -\-n x^ may be written ^ = w a;^ -|- «, 



