176 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1914. 
see in them a kind of secret conferring on the initiated a mysterious 
power over numbers. 
The first are at fault in not sufficiently appreciating the great 
ingenuity of this efficient means of simplifying calculations, the sec- 
ond in attributing to it a character somewhat cabalistic. 
For the uses to which they lend themselves there is nothing so 
simple as logarithms. These uses are founded entirely on a prop- 
erty which we will explain: For every number, the logarithmic tables 
have another corresponding number which is called the logarithm 
of the first, and when a number A is equal to the product of two 
other numbers, B and C, the logarithm of A is equal to the sum of 
the logarithms of B and C. 
It is this faculty which logarithms confer, of replacing all multi- 
plication by a simple addition, that is the source of all the simplifi- 
cations attending their use. To obtain the product of B multiplied 
by C, one looks in the table for the logarithms of B and C (which 
may be represented by } and c) and performs the addition 6 plus c. 
If a is this sum, the table shows the number A of which a is the 
logarithm; this number A is equal to the product desired, B multi- 
plied by C. 
Inversely, if it is desired to divide A by B, the difference, a minus 
b, of their logarithms is obtained, and if it is found to be ¢, it is only 
necessary to read in the table the number C of which the logarithm 
is c; this number C is the quotient desired. 
As a general rule, in order to obtain the product of any number of 
factors, it is sufficient to take the sum of the logarithms of those 
factors; this sum is the logarithm of the product which is then read 
in the table, opposite its logarithm. 
In particular, the nth power of a number, which is the product of 
n factors equal to this number, has for a logarithm 7 times the loga- 
rithm of the given number. Inversely, if the nth root of a number 
is desired, it is only necessary to divide by n the logarithm of that 
number; the quotient obtained is the logarithm of the desired root. 
Is there need of insisting on the simplicity of this method of working 
compared with that which made the poor scholars grow pale who 
were forced to apply the arithmetic rule to the extraction of square 
and cube roots? It is not uncommon even to find certain students 
who, deceived by a false appearance, imagine that if they work 
deeper in the realm of mathematics, they would find need for calcu- 
_ lations even more dry and repelling, when, on the contrary, the fur- 
ther one advances, the more are methods discovered which are simple, 
neat, and apt, designed not only to satisfy but even to delight the 
mind. 
And on this occasion one may be led to ask himself whether, 
instead of holding young scholars down to the application of processes 
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