go BROOMALL : 



or decrease according to the magnitude of the quantity itself. 

 Thus, the principal at compound interest increases at a rate 

 proportional to the principal itself, since interest is added to 

 principal ; the rate of cooling of a body becomes less as its 

 temperature differs less and less from the surroundings ; the 

 rate of chemical change becomes slower as there is less sub- 

 stance left unchanged. The applicability of an exponential 

 equation to changes which follow this law is well known to 

 those familiar with the calculus. 



If we have an exponential equation with negative expo- 

 nent containing the square of the variable, the equation, pro- 

 perly modified, represents the probability curve, expressing 

 the law of the facility of error. In other words, the ordinate 

 to the probability curve at any point is a measure of the pro- 

 bability of committing a given error in a series of measure- 

 ments, observations or trials, and the area of the curve 

 between proper limits determines the probability that an error 

 shall not exceed a certain amount. The curve is the basis of 

 the method of least squares by which the most probable value 

 of a given quantity is deduced from a series of observations, 

 a method of immense service in geodetic, astronomical and 

 other measurements. 



In the foregoing the effort has been made to mention a few 

 of the so-called physical curves and their characteristics. 

 The list is very incomplete, and many more examples could 

 be cited. These which have been given, however, it is hoped 

 will serve to suggest to the reader further relationships between 

 mathematics and physics of similar character. 



In conclusion, it is interesting to pause for a moment and 

 consider why there should exist this common boundar)'^ region 

 between mathematics and physics. We have here two branches 

 of science which have developed more or less independently 

 side by side. One is what may be termed a mental science, 

 and might have developed along any line consistent with the 

 fundamental conceptions imagined by its creators. The other 



