### The Slide Rule (multiplying with addition)

by Rick Reed

A typical slide rule arranges two equal logarithmic scales (see working simulation below) next to each other so that they can be slid back and forth to align tick marks and take readings.

Figure 1. Curve for Log Function

Figure 2. John Napier

Figure 3. Curve for Log Function showing
how slide rule scales are generated.

Example A. To multiply 2 by 3 using the above slide rule:

1. Slide the top scale so that the 1 lines up over the 2 (of the bottom scale).
2. Move the red indicator line so that it lines up over the 3 on the top scale.
3. Read the answer (6) on the bottom scale under the red line.

So, 2 ¥ 3 = 6.

Try a few more multiplications. Isn't it amazing? It works because of logarithmic scales.

### Logarithms

A logarithmic scale is simply a scale where the values of the log(x) (see figure 1.) are calculated and used to space the x values displayed along a line.

We credit the discovery of logarithms (about 1614) to mathematician, John Napier (see figure 2.), born 1550 in Edinburgh, Scotland1.

A logarithm is the value that a base number (10 for example) would be raised the the power of (2 or squared for example), to result in a specified value (100 for example).

In other words, in base 10, the log(100) = 2, because 10 raised to the second power equals 100. Likewise,

log10(1,000) = 3, because 103 = 1,000

log10(10,000) = 4, because 104 = 10,000

and so on.

It follows from logarithms and exponents that since

1) x a • x b = x (a + b)

and

2) 10log(x) = x for any positive x != 0

we get (in base 10):

10log(c) • 10log(d) = 10(log(c) +log(d)) (see 1)

But

10log(c) = c (see 2)

and

10log(d) = d, (see 2)

so, we can multiply by adding exponents!

c • d = 10(log(c) +log(d))

Example B.

4 ¥ 100 = 10(log(4) + log(100)) or 10(0.602059991 + 2.0) or 10(2.602059991) = ~400

This led the English astronomer Edmund Gunter in 1620 to devise a way to visually construct the operation of adding the logs of two numbers, by adding the distances equal to their log values (steps 1 and 2 in the example A above), and essentially raising 10 to the sum distance (step 3) to get the product of two numbers using his "Gunter line" (logarithmic scale).2

The numbers on a logarithmic scale are spaced according to the log of their value (see figure 3.)

The result of Gunter's ingenious reasoning is fully realized 12 years later by William Oughtred when he described opposing sliding scales in 16323

The final modern form of the slide rule (see simulator at top of page) was established in 1850 by a lieutenant in the French Army named Lt. Amedee Mannheim.4 He set a sliding scale between two fixed scales and established the basic parts, the body, the slide, and the indicator or runner. His slide rule design had several scales including:

Body scales:

K: Cubes and cube-roots
A: Squares and square-roots
D: Logs (same as C scale)
L: Linear (mantissa of C scale)

Slide scales:

B: Sins
S: Angles (for B scale readings)
CI: Inverted Logs (reversed right to left)
T: Angle for reading Tan from C scale
C: Logs (same as D scales)

Except for the addition, elaboration, and modification of the scales included on slide rules, Mannheim's basic design served for about a hundred years (until mechanical calculators came along) as the standard calculating tool for scientist and engineers.

### Subtracting to Divide

You can simply reverse the multiplication operation with a slide rule to divide two numbers.

Example C. To divide 12 by 4 using the above slide rule:

1. Slide the top scale so that the 12 lines up over the 4 (of the bottom scale).
2. Move the red indicator line so that it lines up over the 1 on the bottom scale.
3. Read the answer (3) on the top scale under the red line.

So, 12 / 4 = 3.

Problem 1: How would you use the slide rule above to divide 14.4 by 12?

Problem 2: How would you use the slide rule above to divide 144 by 12?

References:

1Ball, W. W. Rouse. A Short Account of the History of Mathematics. New York, New York: Sterling Publishing Company, Inc., 2001. 195, 235

2,3Ball - 196

4Hebra, Alex. Measure for Measure: The Story of Imperial, Metric, and Other Units. Baltimore, Maryland: Johns Hopkins University Press, 2003. 21