# Slide rule

The **slide rule** is a mechanical precursor of the pocket calculator. It was invented in 1625 by William Oughtred and was very commonly used until the 1970s when it was made obsolete for most purposes by electronic calculators.

Once widely used for rapid, approximate scientific and engineering calculations, a slide rule is a portable, mechanical, analog computer consisting of three interlocking calibrated strips. The central strip can be moved lengthways relative to the other two. A sliding cursor with one or more vertical alignment lines can record an intermediate result on any of the scales. Some slide rules have scales on both sides of the rule and slide strip, others on the slide strip only.

One slide rule remaining in daily use around the world is the *E-6B*. This is a **circular slide rule** first created in the 1930s for aircraft pilotss to help with dead reckoning. It is still available in all flight shops, and remains widely used. While GPS has greatly reduced the use of dead reckoning for aerial navigation, the E-6B remains widely used as a backup device and the majority of flight schools demand its mastery to some degree.

Table of contents |

2 Materials 3 History 4 See also 5 External links |

## Theory of operation

### Standard linear rules

The rule has logarithmic scales. That is, a number *x* is printed on each rule at a distance *c*·log(*x*) from the 'index', which is marked with the number 1. A logarithm transforms an operation of multiplication or division to one of addition or subtraction to the rules log (a·b) = log a + log b and log (a/b) = log a - log b.

Since addition and subtraction are easily carried out using a number line, the slide rule effectively implements a number line with a sliding scale. By the use of the logarithmic transform multiplication and division can be carried out.

To multiply *x* by *y*, one aligns the index on the sliding scale with the number *x* on the fixed scale. To do this, the sliding scale must be moved over a distance of *c*·log(*x*). The number *y* on the sliding scale has now moved from its old position, *c*·log(*y*), to *c*·log(*y*)+*c*·log(*x*) = *c*·log(*xy*), and is therefore aligned with the number *xy* on the fixed scale.

The illustration below shows the multiplication of 2 with any other number. The index (1) on the upper scale is aligned with the 2 on the lower scale. The numbers on the upper scale (multipliers) correspond with the multiplication on the lower scale. Example: the 3.5 on the upper scale is aligned with the product 7 on the lower scale, the 4 with the 8 etc.

Where operations go 'off the scale' e.g. 2·70 the user has to slide the upper scale to the left and use the index 100 (or 10) instead and remember the 'carry' of ·10.

Division reverses this process. The illustration below shows all divisions by 2.75 as the index (1) on the upper scale is aligned with the 2.75 on the lower scale. Example: the 22 on the lower scale (the mark just to the left of the 22.5 mark) is aligned with the quotient (22.5/2.75=) 8 on the upper scale, the 55 is aligned with 20 etc.

Slide rules calibrated on one side were called "simplex." Slide rules calibrated on both sides were called "duplex."

Typically two significant figures of precision were possible, with three being obtained by expert users who could estimate the fraction between gradations. Some high-end slide rules had magnifying cursors that effectively doubled the accuracy, permitting a 10-inch slide rule to serve as well as a 20-inch.

Slide rules often have other mathematical functionss encoded on other auxiliary scales. The most popular were trigonometric, usually sine and tangent, logarithm of logarithm (base 10) (for taking the log of a value on a multiplier scale), natural logarithm and exponential scales. Some rules included a pythagorean scale, to figure sides of triangles, and a scale to figure circles.

Specialised slide rules were invented for various forms of engineering, business and banking. These often had common calculations directly expressed as special scales, for example loan calculations, optimal purchase quantities, or particular engineering equations.

A number of tricks were used to get more convenience. Trigonometric scales were sometimes dual-labelled, in black and red, with complementary angles, the so-called "Darmstadt" style. Duplex slide rules often duplicated basic scales on the back. Scales were often "split" to get higher accuracy.

### Circular slide rules

Circular slide rules came in two basic types, one with two cursors, and another with a moveable disk and a cursor. The basic advantage of a circular slide rule is that the longest dimension was reduced by a factor of about 3 (i.e. by π). For example, a 10cm circular would have a maximum accuracy equal to a 30 cm ordinary slide rule.

Circular slide rules were mechanically more rugged, smoother-moving and more precise than linear slide rules, because they depended on a single central bearing. The central pivot did not usually fall apart. The pivot also prevented scratching of the face and cursors. Only the most expensive linear slide rules had these features.

The highest accuracy scales were placed on the outer rings. Rather than "split" scales, high-end circular rules used helical (snail-shell-shaped) scales for difficult things like log-of-log scales. One eight-inch premium circular rule had a 50 inch helical log-log scale! Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around."

In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations — the Breitling Navitimer. The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed, rate/time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer–nautical mile and gallon–liter fuel amount conversion functions.

Technically, a real disadvantage of circular slide rules is that less-important scales are closer to the center, and have lower accuracies. Historically, the main disadvantage of circular slide rules was just that they were not standard. Most students learned slide rule use on the linear slide rules, and never found reasons to switch.

## Materials

The best older slide rules were made of bamboo, which is dimensionally stable, strong and naturally self-lubricating. They used scales of celluloid or plastic. Some were made of mahogany. Later slide rules were made of plastic, or aluminum painted with plastic.

All premium slide rules had numbers and scales engraved, and then filled with paint or other resin. Painted or imprinted slide rules are inferior because the markings wear off.

Early cursors were metal frames holding glass. Later cursors were acrylics or polycarbonates sliding on teflon bearings.

Magnifying cursors can both help engineers with bad eyes, and double the accuracy of a slide rule.

Premium slide rules included clever catches so the rule would not fall apart by accident, and bumpers so that tossing the rule on the table would not scratch the scales or cursor.

The recommended cleaning method for engraved markings is light scrubbing with steel-wool. For painted slide rules, and the faint of heart, use diluted commercial window-cleaning fluid and a soft cloth.

## History

Slide rules came into wide use in the 1850s, as engineering became a recognized professional activity. In World War II, bombardiers and navigators who required quick calculations often used specialized slide rules. One office of the U.S. Navy actually designed a generic slide rule "chassis" with an aluminum body and plastic cursor into which celluloid cards (printed on both sides) could be placed for special calculations. The process was invented to calculate range, fuel-use and altitude for aircraft, and then adapted to many other purposes.

Throughout the 1950s and 1960s the slide rule was the symbol of the engineer's profession in the same way that the stethoscope symbolized the medical profession. Some engineering students and engineers actually carried five-inch pocket slide rules in belt holsters, in addition to using a ten- or twenty-inch rule for precision work at home or at the office. All this came to an end in the 1970s, when the advent of miniaturised calculators made slide rules obsolete. The last nail in the coffin was the launch of *scientific* pocket calculators, i.e. having trigonometric and logarithmic functions, of which the HP-35 was the first, in 1972.

Most slide rules are now collectors' items. A very popular model is the Keuffel & Esser *Deci-Lon*, a premium scientific and engineering slide rule available both in a ten-inch "regular" (*Deci-Lon 10*) and a five-inch "pocket" (*Deci-Lon 5*) variant. Another prized model is the eight-inch Scientific Instruments circular rule. At the turn of the millennium, the collectors' market still doesn't seem to be drying up, and as recently as 2002, new slide rules were being located in the back-shelves of university book-stores, even though production ended in 1973.

## See also

- Abacus
- Common logarithm
- Timeline of computing
- Counting rods
- Mathematical tables
- Napier's bones
- Nomogram

## External links

- Sliderule information at the Museum of HP Calculators
*Make your own slide rule*(PDF)- ASA Micro E-6B Flight Computer – A specialized circular slide rule
- ASA E-6B Metal Flight Computer – Metal model of the E-6B
*Make your own*circular*slide rule*- The Slide Rule Universe – A comprehensive slide rule reference and buying/selling site
- Breitling Navitimer info webpage – Wristwatch with circular rule