Experimental Mathematics. 401 



system/' and is a sector having an angle o£ 2*302 radians, 

 divided into tenths. If we apply this modulus to the natural 

 logarithms in fig. 2 it will transform them into common 

 logs, and we are able to read off their values, just as we did 

 with the protractor graduated in radians. In adopting a 

 unit angle 2*3 times as big as before, we make all angles 

 measured by it appear 1/2*3 of their previous value. This 

 affords a visible explanation of the method given in text- 

 books for changing from the base e to the base 10. 



The beginner will doubtless ask why we take the trouble 

 to convert natural into common logs, and the model of the 

 watch calculator, which you have seen, provides a satisfactory 

 answer. After making half a dozen calculations (2 x 3, 

 4 x 5, 6 X 7, ... ) it will easily be grasped that 



log 2= * 301 of a revolution. 



log 20 '= 1 revolution + *301 „ „ 

 log 200 = 2 revolutions + *301 „ 



Had the dials been derived from fig. 2 — the natural system 

 — there would be no recurrence, and we should be unable, in 

 this simple manner, to adapt a small range of logarithms to 

 cover an unlimited range of values. The state of affairs 

 might be compared to a clock in which the small hand moved 

 over 2*302 divisions during an hour. 



Instead of obtaining common logs bv transformation we 

 may entirely discard the natural system, strike out a new 

 course, and with a fresh setting of the roller describe the 

 spiral, shown by the dotted line, in fig. 2. This curve meets 

 unit angle — one radian — in the point X, distant 10 inches 

 from 0, and yields logs to the base 10, by direct measure- 

 ment with the standard protractor graduated in radians. 

 Along this curve, e occurs at an angle of 1/2*302 radians, thus 

 proving experimentally that log e 10 is the reciprocal of 

 logiotf. 



The value 10 may be reached by growing from unity, 

 either along the e spiral at the standard rate for 2*3 radians, 

 or along this new curve at 2*3 times that rate through one 

 radian. We may regard this alternatively as the proof, or the 

 consequence, of the new curve being drawn with an angle 

 whose tangent is 1/2*302. In any case it is clear that 

 tana=M, and that all equiangular spirals transform from 

 one into the other, by a suitable change in the unit angle. 



Owing to the rather prevalent idea that common logs can 

 be calculated only by derivation from the natural system, I 

 may perhaps be excused for alluding to the independent, 

 though obvious, method of extracting square roots. 



