Experimental Mathematics. 



397 



the distance from the pole is respectively 2, 3, 4, &c. inches. 

 Here also are several sectors of cardboard which have been 

 cut out to fit the various angles 102, 103, 104, &c, and are 

 therefore the Naperian logarithms of the natural numbers. 



By means of these sectors, it may be shown experimentally 

 that if we add to the angle 103 the angle 102, we arrive at 

 the angle 106 — (3 x2 = 6), and, conversely, if we take away 

 from the angle 108 the angle 102,, we shall be left with the 

 direction 04 — (8 -r- 2 = 4). Without further examples, it may 

 be demonstrated in the most general way, that by adding the 

 angle under any one value of the radius vector to that under 

 any other value, we obtain a direction giving their product : 

 or by subtraction their quotient. Evolution and involution 

 follow naturally. For instance, to find the value of 5 1 ' 6 : 

 we divide the angle 105 into ten equal parts, and then by 

 taking 16 such parts we get a direction which at once gives 

 us the desired result. By ordinary arithmetic even this 

 simple calculation is quite impracticable, and it is given to 

 show how the real utility and importance of logarithms may 

 be impressed on the student. 



The properties mentioned above are common to all curves 

 drawn with this instrument, and not merely to the special 

 case in which the roller is set at 45°. 



Instead of working direct from the curve, by means of 

 cardboard sectors, or compasses, the values of the various 

 angles may be expressed in terms of some standard angle, 

 such as the radian, and the results printed, in tabular form, 

 for future reference. 



Here is a protractor, graduated in radians, which we will 

 apply to fig. 2, and by means of which we can read off the 

 values given below. 



Length of radius vector 

 in units of 1 inch, 



1 

 2 

 3 



4 



5 

 6 



7 



8 



9 



10 



&c. 



Corresponding angle in 

 radians. 







•693 

 1-098 

 1-386 

 1-609 

 1-791 

 1-946 

 2-079 

 2-197 

 2-302 



&c. 



This constitutes a table of natural logarithms, which may 

 Phil. Mag. S. 6. Vol. 14. No. 81. Sept. 1907. 2 E 



