The history, theory and use of the engineering slide rule

Until about 1980, the engineering student could be recognized by the slide rule case dangling from his or her belt, in addition to the pocket protector, beanie and white socks. Then came the HP-35 pocket calculator: expensive and hungry for batteries, but very handy, and seemingly requiring little skill to operate, which recommended it to the richer and less able students. The prices for a scientific pocket calculator--the calculators that can do trig functions, exponentials and logarithms--fell rapidly, soon approaching the price of the cheaper slide rules, so that everyone could have one. Now, a perfectly useful scientific calculator can be purchased for less than $20. Slide rules disappeared from the engineering student's kit along with socks.

The slide rule is still a remarkable instrument, small, light and not requiring batteries or sunlight, and not subject to the laws of decay and disintegration. A fifty-year-old slide rule is as good as a new one. If you have one around, it might be interesting to dig it out and examine it. This paper will give you some hints on how to use it. Slide rules multiply and divide, raise to powers and take roots, and take the place of tables of the trigonometric, logarithmic and exponential functions. They do these things practically instantly, and to a constant precision of about three digits. You have to manage the decimal points yourself, however.

The slide rule is based on logarithms. Distances on a number scale are proportional to the logarithms of numbers, instead of to the numbers themselves, as on a linear scale, as found on a ruler, for example. A scale goes from 1 to 10, or 10 to 100, or 0.1 to 1, or any other range of a factor of 10. Natural logarithms (base e = 2.718...) were invented by John Napier in 1614, and common logarithms (base 10) were invented by Henry Briggs in 1617, to facilitate multiplication and division with the then rather new Hindu-Arabic digits 0 to 9. Edmund Gunter (famous for the surveyor's chain) invented a logarithmic scale in 1620, from which distances were taken off with dividers. William Oughtred used two logarithmic scales placed side by side for multiplication and division in 1630. Seth Partridge arranged one scale, the *slide*, so that it was held within the other, the *stock* or *body*, making it much easier to hold a setting. In 1775, John Robertson added the *cursor* or *runner*, which allowed setting to be transferred to any of several parallel scales, as well as holding a position while the slide was moved. P. M. Roget devised the *log-log* scale in 1815, which gave values of e^{x}, permitting the calculation of any power or root of a number. The classic form of the slide rule was due to the 19th-century French engineer Mannheim. There were many different kinds of slide rules, adapted to specialized calculations, but the Mannheim slide rule was a general-purpose instrument for scientific and engineering calculations.

Slide rules were manufactured by the suppliers of drafting equipment, such as Dietzgen, Keuffel and Esser and Post in the United States. The body and slide were of wood, with enameled faces. The upper and lower rods of the body were connected by metal end pieces. Clamping screws allowed the rods to be precisely adjusted so they were in register. The cursor was an etched line in a glass plate held in a slide. It was spring-loaded so that the cursor line was accurately perpendicular to the rods and slide. The typical engineer's slide rule had scales 10" (25 cm) in length, making the slide rule about 12" overall, and 1-3/4" to 2" wide. A *duplex* slide rule had scales on both sides, the usual case for engineering slide rules. Angle divisions were originally in minutes, but decimal divisions gradually became predominant, in *decitrig* slide rules.

In 1950, Post brought out an American version of the bamboo slide rule manufactured by Hemmi of Japan, the *Versalog*. Bamboo is an excellent material for this purpose, and these slide rules were first-rate, as good as any made, but sold for half the price, about $25. I obtained one when I went off to college, and it is still as good as when it was purchased. It was accompanied by a hard-bound, excellent instruction manual. Pickett and Eckel manufactured slide rules made of magnesium alloy, with scales of black on a yellow background. Traditionally, scales had been blue or red on white. These were a bit more expensive than the Post rules, but were very durable and accurate.

Longer slide rules were produced for special purposes, but they were expensive and rather cumbersome to use. The extra significant figure was seldom of much use. If more significant figures were required, 7-place logarithms and an adding machine were the usual choice. Later, mechanical calculators that could multiply and divide were available, but they were very expensive. Shorter slide rules that would fit conveniently in a shirt pocket were, however, quite popular. Pickett brought out a very handy rule with 5" scales.

A final development was the replacement of wood, bamboo or magnesium by high-quality plastic, with the upper and lower rods in permanent adjustment. The British Thornton Model AA010 of 1969, a standard 10" rule, is a good example. Pickett produced very serviceable all-plastic simplex slide rules selling for only a few dollars, such as the Microline 120 series. Such slide rules may still be available.

In the 1950's, the circular slide rule appeared. The scales were circles, the beginnings meeting the ends. As we shall see, this was quite logical. A 10" scale fit on a 4" diameter disc, and longer scales were made more convenient. There was no longer the problem of the end of a scale sticking out one side or the other; all parts of the scales were always in contact. One kind of circular slide rule had rotating discs on each side of the circular body, and a single cursor. It was used exactly like a straight slide rule. An example was the Fullerton No. 1458, from Japan. Another kind had a solid disc and two cursors. The two cursors could be made to rotate together, or separately, as required. The Pickett 101-C was of this type, with a magnesium alloy disc and black on yellow graduations. These were accurate and easy to use, but did not become as popular as the familiar straight slide rules.

Making a slide rule at home is not recommended. Not only are the mechanical arrangements difficult, but the scales must be very accurate. For demonstration purposes, however, logarithmic scales are not too difficult to make, and may help one to understand the principles.

The Pickett ES-600 5" slide rule is illustrated below. It has all the scales usually found on an engineering slide rule, 25 in all. The Microline 120 has 9 scales, the ones labelled K, A, B, S, T, CI, C, D and L. It can do trigonometric calculations and common logarithms, but not natural logarithms or raising to any power. The most basic small slide rule may have only 6 scales: A, B, CI, C, D and K. It can do only multiplication and division, squares and square roots, cubes and cube roots.

The C and D scales are used for ordinary multiplication and division. The CI scale is like the C scale, but is graduated in the opposite direction. To multiply two numbers, set the cursor over one factor on the D scale. Then move the slide until the other factor is under the cursor on the CI scale. The product is then on the D scale at one end or the other of the C scale, whichever one overlaps the D scale. Note how this adds the distances corresponding to the logarithms, assuming that the scales are extended in both directions in decade after decade. A graduation is given for π. The graduation marked R is at 57.3, the number of degrees in a radian, to facilitate interconversion of degrees and radians.

To divide, set the cursor over the dividend on the D scale. Move the slide until the divisor on the C scale is under the cursor, and read the quotient on the D scale at whichever end of the C scale is on the D scale. Since the answer is on the D scale, you can do any number of multiplications or divisions one after the other without writing anything down. This put a premium on expressing solutions in terms of multiplications and divisions, without adding or subtracting, especially in trigonometry. With a little practice, you can easily become proficient in multiplying and dividing with the slide rule. Use the theory to figure out *why* you are doing things; then do them by rule, to save time and avoid errors.

Sines and cosines are found with the S scale, tangents and cotangents with the T scale. If you set an angle on these scales, the trigonometric function is given on the C scale. The range of the sine and cosine, or the tangent and cotangent, are from 0.1 to 1.0. Angles smaller than about 6° are shown on the ST scale (the sine and tangent are about equal in this range). When using this scale, the decimal place is assumed one place to the left, so the range is 0.01 to 0.1. For even smaller angles, you simply assume that the sine or tangent is the angle in radians.

The A and B scales are just like the D and C scales, but compressed by a factor of 2, so that they run from 1 to 100. If you set a number under the cursor on the D scale, its square is under the cursor on the A scale. Similarly, if you set a number under the cursor on the A scale, its square root is under the cursor on the D scale. You can multiply and divide with the A and B scales, but not as accurately as with the longer C and D scales. The K scale gives cubes and cube roots in the same way. The *Versalog* had an R scale twice as long as the D scale, for more accurate square roots.

The L scale is a linear scale graduated from 0 to 1, giving the common logarithms of numbers on the C scale, or antilogarithms if read the other way. The DI scale is an inverted D scale, just as CI is an inverted C scale, and is used the same way for facilitating certain calculations. The DF, CF and CIF are C, D and CI scales simply shifted by log π. You can use them for ordinary arithmetic, but their real convenience is that they give an automatic multiplication by π of numbers set on C or D.

The log-log scales LL_{1} to LL_{3} are calibrated in values of e^{x} (the + scales) or e^{-x} (the - scales) for x from 0.01 to 10 on the D scale. The letter M means 1000. The range is from about 20,000 to 0.00005. If you set a number on the LL scale, its natural logarithm is set on the D scale at the same time. You can multiply or divide this logarithm by another number to find any power or root, and then project the answer back onto the LL scale to read its value. For smaller values of x than 0.01, use the approximation e^{x} = 1 + x + ... . The *Versalog* had an additional log-log scale for x between .001 and .01. The theory of use of the log-log scales is that x = a^{b} can be written ln x = b ln a, and ln ln x = ln b + ln ln a. The addition of logarithms is done on the slide rule as usual.

For electrical engineers, one of the most common calculations done on a slide rule was the conversion of a phasor in rectangular components to a phasor in polar components and vice versa. These conversions were necessary for phasor arithmetic because the rectangular form was adapted to addition and subtraction, while the polar form was convenient for multiplication and division.

Of course, one could simply use the ordinary rules for multiplication and division to do this, but there were better ways to do it quickly on a slide rule. First of all, one never used r = √(x^{2} + y^{2}), which is cumbersome to work out any way you look at it. The angle was first obtained from θ = tan^{-1} (y/x), and then r = y / sin θ or x / cos θ. To go the other way, x = r cos θ and y = r sin θ were used.

The procedure for rectangular to polar conversion was first to set the larger of x or y under the cursor on the D scale. Move the slide so that the other component is under the cursor. Then θ can be read on the T scale above the index of the D scale. Move the cursor back to the component on the D scale, and move the slide until the θ on the S scale is under the cursor. Read r on D at the index of the C scale.

To go the other way, set one end of the C scale over r on the D scale. Then set θ as a sine on the S scale, and read y on the D scale. Now set θ as a cosine on the S scale, and read y on the D scale. One could also set r on the DI scale, and read y and x on the DI scale under the index of the C scale.

These procedures became automatic after a little practice. Go through them for the 3, 4, 5 triangle (angle 36.87°) to see how they work. Bad students would muddle through somehow, but good students profited from learning the quick and reliable methods. Scientific pocket calculators have this conversion built-in, incidentally. The British Thornton slide rule had trigonometric scales on the body, but provided special scales for rectangular-polar conversion.

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Composed by J. B. Calvert

Created 19 January 2001

Last revised 10 January 2004