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THE WORLD'S ADVANCE 



368, it is between TOO and IOOO, or be- 

 tween io 2 and io 3 , it actually being io 2<B9 . 

 This number, which expresses what 

 power of io the number under con- 

 sideration is, is called the logarithm of 

 that number with respect to base io. 

 Thus 1.65 is the logarithm of 45 with re- 

 spect to base io, and 2.59 is the logarithm 

 of 368 with respect to base io. Hence 

 it is at once apparent that we can take 

 any number as our base and express 

 every other number as a power of that 

 base ; that is to say, every number has a 

 logarithm with respect to every other 

 number. Many of our every day com- 

 putations are made by first expressing 

 the numbers as powers of io, then work- 

 ing out the computations and finally con- 

 verting the numbers back again. Tables 

 known as logarithm tables are found in 

 many mathematical books, and express 

 all numbers in powers of io, so that by 

 means of these tables much laborious 

 work is saved. 



For such calculations as the logarith- 

 mic decrement there is another base other 

 than io which is used. This base is desig- 

 nated by the letter e and is numerically 

 equal to 2.718. It is known as the hyper- 

 bolic or natural base as contrasted with 

 the io base which is called the common 

 or Briggs' base. It is beyond the scope 

 of this article to take up the derivation 

 of the number e, but it seems well to add 

 just a word about it lest the subject of 

 logarithms be left in too hasy a state for 

 those not already familiar with it. Many 

 mathematical curves, as for example the 

 one under consideration, appear to have 

 peculiar properties when considered from 

 our decimal system of counting, but 

 which are perfectly regular when con- 

 sidered with respect to the irrational 

 number designated by e. Thus in order 

 to simplify many computations they 

 must be based on e rather than on our 

 decimal system with io as the base, e 

 gets its value from the expansion of the 

 convergent series (i+h)^ where h ap- 

 proaches O. Expanding by the binomial 

 theorem we get 



1X2 1X2X3 1X2X3X4 



.. = 2.718... 



If we consider e as our base, the loga- 

 rithm of 45 would be 3.81 instead of 1.65 



when we considered io as our base. Its 

 meaning is similar in that 45 is 2.718 to. 

 the 3.81 power just as we saw it to be 

 io to the 1.65 power. 



But let us now return to our curve- 

 Instead of stopping with the simple ratio 

 of AB/DE which we found to be equal 

 to 1.09 we are to take the natural log- 

 arithm of this ratio. From a table of 

 natural logarithms we find that 0.09 is 

 the natural logarithm corresponding to 

 the number 1.09. Our logarithmic de- 

 crement per half period is therefore 0.09. 

 To get it for a whole period it is merely 

 necessary to multiply by 2 and get 0.18 

 as the decrement, which latter figure is 

 the logarithm of the ratio obtained by 

 taking AB/GJ. Here is one place where 

 our logarithmic expression is an aid to 

 us. The logarithm of any ratio of cor- 

 responding perpendiculars taken any 

 number of loops apart is equal to the 

 logarithm per half period multiplied by 

 the number of loops below the one in 

 which the initial perpendicular is meas- 

 ured. No such relationship exists in 

 the simple ratio. 



When we come to radio telegraphic 

 work we have a condition similar to that 

 illustrated by the string. A condenser is 

 charged until the voltage is sufficient to 

 break down the spark gap, then we get a 

 rush of energy through the closed circuit 

 which sets up a current in the open an- 

 tenna circuit. This current in the an- 

 tenna circuit may be likened to the swing 

 which we gave the bob. It is greatest at 

 the instant we start it, swings back and 

 forth, gradually dying out to o. The time 

 of swing is constant but the amplitude is 

 decreasing. While we cannot measure 

 it with a yard stick, the way we did for 

 the pendulum, it is possible to photo- 

 graph the discharge, and the result will 

 show a curve exactly the same as de- 

 veloped by the string. The perpendicu- 

 lars such as AB representing the currents 

 and the distances along OX the time re- 

 quired per oscillation. Instead of requir- 

 ing two seconds for a complete oscil- 

 lation, or cycle, the time is now but a 

 small fraction of a second, the actual 

 time depending on the wave length. For 

 200 meters this time per complete oscil- 

 lation is 1/1,500,000 sec. Compare this 

 frequency of 1,500,000 cycles with 0.5 



