

Slide Rule 

The slide rule
is extensively used in engineering, science and commerce for
rapidly performing calculations involving multiplication and
division which have to be accurate to not more than three
or four decimal places. It can also be used for such operations
such as involution (raising to a power) and evolution (extraction
of a root) and for calculations with trigonometric functions
(sine, cosine, tangent, cotangent).


In
addition to those for general use there are many different
types of specialpurpose slide rules. What they all have in
common is logarithmic scales. The slide rule, also known as
a slipstick, is a mechanical analog computer. The slide rule
is used primarily for multiplication and division, and also
for scientific functions such as roots, logarithms and trigonometry,
but does not generally perform addition or subtraction.


To
understand how a slide rule works it is essential to know
about logarithms. In general, the logarithm of a value a to
base b is the exponent n denoting the power to which b must
be raised in order to obtain a; i.e., if a=bn , then log ba
=n, where a itself is called the antilogarithm. Hence logarithms
are exponents. In particular, socalled common, or Briggsian,
logarithms are exponents of the base 10, this being the most
convenient base for purposes of general computation. (In theory
any other base could be adopted for a system of logarithms.
Of special important besides common logarithms, are the natural,
or naperian, logarithms, which have the value e=2.718 …
for their base and are important in higher mathematical analysis).


Since
1=100, 10 = 101, 100=102, 1000=103 etc., the common logarithms
of 1,10,100,1000 etc. are equal to 0,1,2,3, etc. This is set
forth in Table 1. Any number between 1 and 10 has a logarithm
which lies between 0 and 1; numbers between 10 and 100 have
logarithms between 1 and 2 etc. Whereas the logarithms of
whole powers of 10 are whole numbers, the logarithms of intermediate
values are decimal fractions, as exemplified by the logarithms
of numbers between 0 and 10 listed in table 2.


When
two or more values which are powers of the same base are multiplied,
their product is obtained by adding the exponents. For example:
102x105 = 100x1000 =100,000 = 105. Similarly, division is
done by subtraction of exponents.




For
e.g. 105 /103= 1,00,000 /1000= 100 =102 In general 10m x 10n
= 10m+n and 10m/10n =10mn. Thus multiplication and division
can be reduced to the simpler operations of addition and subtraction
if, instead of the actual numbers, the logarithms of those
numbers are employed. For this purpose the logarithms of the
numbers to be multiplied or divided are looked up in suitable
tables and are added or subtracted. The result thus obtained
is the logarithm of the required answer, which can be looked
up (as the antilogarithm) in the same logarithm tables. Example:
to calculate 2 x 3 using logarithms: log 2x3 = log 2 + log
3 = 0.301 +0.477 = 0.778; the antilogarithm of 0.778 is 6.


An
ordinary slide rule consists of the actual rule, the slide,
and the transparent cursor with a hairline. Various logarithmic
scales are engraved on the rule and the slide (Fig.2). When
the rule is closed the pairs of scales A and B, and C and
D, respectively coincide. The divisions on A and B extend
from 1 to 100; those on C and D extend from 1 to 10.


To
determine the square of a value, the hairline is moved to
that value on Scale C and D, and the square is indicated by
the hairline on scale B or A (Fig.3). Square roots are obtained
by the reverse procedure (Fig.4). The scale K (above A in
Fig.2) gives the cubes of the values on scale D (Fig.3). Conversely,
scale D gives the cube roots of the values on K (Fig.4). 

For
multiplication, the scales C and D are preferably used (Fig.5).
It may occur that the slide has to be moved too far to the
right to give a reading of the answer. In that case the slide
should be moved back to the left until the figure 10 instead
of the figure 1 on scale C is opposite the first factor of
the multiplication. The result is then obtained in the usual
way on scale D, opposite the second factor on scale C. Alternatively,
the scales A and B may be used, though with reduced accuracy
of the readings.


Division
is performed as indicated in Figs 7 and 8. Operations involving
both multiplication and division may be carried out in accordance
with Fig.9. On a standard slide rule there is, between the
scales C and D, another scale, usually marked CI. It gives
the reciprocals of the values on scale C (Fig.10). For the
values on scale B the scale CI gives the reciprocals of the
square roots; for example, a value a on scale B corresponds
to 1/va on scale CI. Reciprocals can be used advantageously
for multiplying (dividing by the reciprocal value) and also
for complex multiplications and divisions involving several
factors (Fig.11).


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