In the time before electronic calculators, technicians, architects, engineers, etc. used a slide rule whenever they had to compute quickly. Its principle was proposed by the English astronomer Edmund Gunter (1581 - 1626). While he still worked with a scale and picked up number distances with a compass, the English mathematician William Oughtred (1575 - 1660) introduced movable scales. Through many successive improvements arose the computing tool shown above.
You
transfer to a straight line the logarithms in a given scale and
write the arguments against the scale. The modern slide rule has
4 such scales: A, B, C and D, two (A, D) on the main body and two
(B, C) on the slide. The scales on A and B are the same, on C and
D twice as large.
Multiplication and division
In order to add two logarithms, for example, log 1.5 and
log 4, you must advance from log 1.5 by 4 units to log 6. You
move the slide as shown above and read off the result 6 (D) under
4 (C) on the slide. You have log 1.5 + log 4 = Log 6. Since log a
+ log b = log ab, the task 1.5·4 = 6 has been solved
For the
task log 9 - log 4.5, you must move backwards from log 9 by 4.5
units. You read off log 9-log 4.5=log 2. As loga-logb=loga/b, the
task 9:4.5 = 2 has been solved. You must move backwards from 9 by
4.5 units and read log9-log4.5=log2.
All products and quotients can be
computed in the above manner with the accuracy, offered by the
slide rule. In general, since the slide rule, like the table of
logarithms, has only mantissae, you obtain the required sequence of digits; you must
determine the position of the decimal point by mental arithmetic.
For example, in order to multiply 260 by 1.3, you use the slide
rule to find 2.6·1.3 = 3.88. Your mental check tells you that
260·1 = 260, whence 260·1.3 = 338.
In order to multiply 4.5 by 5, the scales C and D are not long enough. Imagine D extended to the left and place 1 at 4.5. For this purpose, you shift the other end of the slide, that is (C) 10, to 4.5 and read off the result 2.25 under 5 on the slide. The figure above shows this process. Your mental check (5·5 = 25) tells you that there are two places before the decimal point, whence 4.5·5 = 22.5.
For the task 3.5 : 7,
you must move from log 3.5 by log 7 units backwards. The scale D
is not sufficient. You place 10 on the slide over 5 of D and read
off 0.5 under 7 on the slide.
In addition, the slide rule has the scales A and B. They have half the scale of C and D, and therefore can accommodate the numbers 1 - 100. The results you obtain by means of A and B are less accurate, while the operations do not require you to move the slide backwards.
Task: Find 2.2 · 1.5 · 2.5 · 0.2.
You do not read off the result 2.2 · 1.5, but fix it with the runner; you then multiply it by 2.5, then by 0.2, each time fixing the result with the runner, enabling you to process the next factor. You can in this manner execute products and divisions involving several terms, and even move the slide in the process to the other end.
You use the scales A and D (or B and C). Since the scale D is twice that of the scale A, every mark on A represents twice the logarithm on the corresponding mark on D, conveniently reached by the runner. Hence you can readily find squares and square roots.
| Find 2² | Place the runner at 2 on (D), read off 4 above on (A) | |
Find  |
Place the runner at 3 on (A), read off below 1.73 on D | |
| Find a3 | Fix a2 with the runner and multiply by a, etc. |
You can set up the quotient a/b on the slide rule. The fourth last figure is set up for 9/4.5. With this you can determine many other numbers which have the same quotient, for example 7/3.5 = 6/3 = 4.4/2.2 = ··· = 2. If 1 on (C) is placed above a certain number on D, then it is the quotient of all numbers on top of each other in D, whence you can determine easily the 4th proportional. For example, if you are given 9/4.5 = 3/x , the fourth last figure gives you x = 1.5 on C above 3.
Task: 3.5 kg cost 7.50 Bath. What is the price of 4.2 kg?
Set up the quotient 7.5/3.5 and obtain on D from 1 on (C) the cost of 1 kg, while below 4.2 on (C) you find on (D) the required number 9, that is, 4.2 kg cost 9 Baht.
Task: Construct a table for y = 1.32·x.
Place (C) 1 above 1.32 on (D). On these two scales, corresponding numbers, x on C and y on D, are in that ratio.For example, if x =2.5, then y = 3.3.
If you reverse the slide and place, for example, 1 on (B) under 9 on (A) , then numbers on top of each other are always the same product, because, for example, (A) log 2 + (B) log 4.5 = log 9, that is, 2·4.5 = 9. If yx = c or y =c/x, then corresponding numbers are in the reverse ratio and you can solve with the slide rule also such tasks. However, since all the time you are only dealing with mantissae, you must allow for powers of 10. Then you can also find the reciprocals of given numbers.