212 Trans. Acad. Sci. of St. Louis. 



the pendulum is directly proportional to a time interval 

 which is t times as great as the time interval t of one 

 vibration. It does not follow that a time interval may be 

 squared, or that there is such a unit as a square second. 



Many times a class of 50 or 60. pupils who had been 

 with us but a few months, have been given data repre- 

 senting the times of rotation of planets around the sun 

 and their mean distances from the sun, with the request 

 that they find the relation between these quantities. In 

 every case more than nine-tenths of the class would ''dis- 

 cover ' ' Kepler 's third law within twenty-four hours. The 

 plotted values made it certain that n was greater than 1. 

 The result was a curve which made it seem probable that 

 t = ar 2 . When the values of r were squared and t and r 2 

 were plotted, the result was a curve which bent in the op- 

 posite direction. Therefore n must be less than 2 and 

 greater than 1. When it was assumed that n was 3/2 or 

 that t 2 = ar 3 and these values t 2 and r 3 were plotted, the 

 result was a straight line. 



The young discoverers were always greatly interested 

 to learn that Kepler struggled with this problem for sev- 

 enteen years after he was in possession of the data which 

 they had received, before he discovered the relation be- 

 tween these variables. Of course after students have be- 

 come familiar with logarithms, and they wish to find the 

 value of it in the equation y = ax n , the equation may be 

 written 



log y = log a-\-n log x. 



When log y and log x are plotted as co-ordinates, a 

 stranght line will be obtained whatever the value of n may 

 be if the above equation holds. The slope of that line is 

 the value of n. This method of instruction is far better 

 than the one which requires the student to test the equa- 

 tion given in his text-book. The student should be put in 

 possession of methods which enable him to discover laws, 

 even if he is not the first to discover them. 



Issued January 23, 1919. 



