212 Trans. Acad. Sci. of St. Lowis. 
the pendulum is directly proportional to a time interval 
which is ¢ times as great as the time interval ¢ 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 m was greater than 1. 
The result was a curve which made it seem probable that 
tar’. When the values of r were squared and ¢ and r° 
were plotted, the result was a curve which bent in the op- 
posite direction. Therefore » must be less than 2 and 
greater than 1. When it was assumed that » was 3/2 or 
that t?—= ar? and these values ¢? and r* 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 n in the equation ya", the equation may be 
written 
log y=loga+nlog «x. 
When log y and log x are plotted as co-ordinates, a 
stranght line will be obtained whatever the value of » may 
be if the above equation holds. The slope of that line 1s 
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 12 
possession of methods which enable him to discover laws; 
even if he is not the first to discover them. 
Issued January 23, 1919. 
