40 GRAPHICAL AND MECHANICAL AIDS TO CALCULATION 
The mathematical principle is equally simple, and is 
based on the theory of logarithms, in accordance with which 
multiplication of numbers is simplified into the addition 
of their logarithms, division into subtraction, the raising 
of a number to the mth power by multiplying the logarithm 
by n, etc. Ona slide rule the two principles are combined, 
the scales are divided in accordance with the logarithms 
of numbers, and we find at once that the mode of division 
is exactly the same as on Lalanne’s table of multiplication. 
If we have now two such logarithmic scales in contact 
with each other, and place for instance the index 1 >f the 
lower scale under the 2 on the upper scale, we find that 
in all pairs of numbers in coincidence the number on the 
upper scale is the product of the number of the lower scale 
multiplied by two. Again we find that all pairs of numbers 
are in direct proportion with each other, in our case 2+1= 
4+2—6+3=8+4=10+5. 
If we invert the slide or lower scale, we willj find in 
accordance with the mechanical principle that the products 
of all numbers in coincidence are constant, in our case 
5X 1=2x2.5=2.24x2.24=7.07 X 7.07, which latter num- 
bers are on the square roots of 5 and of 50. 
On the ordinary Mannheim sliderule of 25 centimeter 
or about 10 inches length we find two scales of divisions 
from 1 to 10 each on the upper part of the rule, and a scale 
from 1 to 10, but of double length on the lower part of 
the rule, and similar scales on the upper and lower part 
ofthe movable slide. With this rule only approximate 
values (A, Plate III), can be obtained, which for most 
calculations is quite sufficient. The reading of the sub- 
division requires some practice. Ona sliderule all operations 
of calculations are made irrespective of the decimal point, 
and for instance the values of .0265 or 2.65 or 2650 are taken 
on the same place on the scale. Rules exist to ascertain 
the position of the decimal point, but they are rarely re- 
quired, as in practical calculations the position of the decimal 
point is generally self-evident. 
As the accuracy of the results of operations made by 
the aid of a slide rule depends entirely upon the number of 
subdivision on a given length of scale, sliderules of great 
length up to three feet have been constructed, which how- 
