Slide rule

A slide rule is a hand-operated mechanical calculator made of slidable rulers. It helps people do math problems like multiplication, division, exponents, roots, logarithms, and trigonometry. It is one of the simplest kinds of analog computers.

Slide rules come in many shapes, such as straight lines, circles, or cylinders. Some slide rules are made just for special jobs, like flying airplanes or handling money, and have extra parts to help with those jobs. Even though it looks like a normal ruler, a slide rule is not for measuring length or drawing lines. The best slide rules can be accurate to about three numbers after the decimal point. People use scientific notation to keep track of how big or small the answers are.

Typical ten-inch (25 cm) student slide rule (Pickett N902-T simplex trig)

The slide rule was created in the 1600s by an English mathematician and priest named Reverend William Oughtred and others. They based it on new ideas about logarithms by John Napier. The slide rule made math faster and easier than writing everything out on paper. Before small electronic calculators were invented, the slide rule was the most common tool used in science and engineering. Even after big mainframe digital electronic computers were made, people still used slide rules because they were easy, cheap, and ready to use. But in 1972, a small handheld scientific calculator called the HP-35 was made, and it became cheap a few years later. After that, slide rules were mostly replaced. By the 1980s, when personal desktop computers came along, slide rules were no longer used much.

In the United States, people often called the slide rule a slipstick.

Basic concepts

Simple slide rule made from index cards marked with powers of 2, calculating 8x4 by aligning the bottom ruler to start where the top ruler is 8, and then reading at the number above where the bottom ruler is 4.

A slide rule is a tool that helps with math problems. It has rulers with special marks that show answers to different math jobs. By moving the rulers next to each other, you can find answers to multiplication, division, and other math jobs.

For example, to multiply two numbers, you line up one number on a special ruler with the start of another ruler. Then you can read the answer. Slide rules can also help with square roots, powers, and angles. You can use your brain to decide where the decimal point goes in the answer. You might need to write down some steps on paper while using a slide rule.

Cursor on a slide rule

Components

Most slide rules have three main parts:

Frame or base – two long strips held together to make a frame.
Slide – a middle strip that moves back and forth inside the frame.
Cursor, runner or glass – a piece that slides on the outside and has a line to help read numbers exactly.

Some slide rules have numbers on both sides, while others have numbers on just one side. The cursor helps you line up numbers that are far apart on the rulers. It can also hold an answer you found in the middle of a problem.

Operation

Logarithmic scales

Slide rules use special scales to turn multiplication and division into addition and subtraction. This makes calculations easier.

Multiplication

By moving the top part of the slide rule to line up with a number on the bottom, you can multiply two numbers. For example, to multiply 2 by 3, you line up the 2 on the bottom with the start of the top scale, then find the 3 on the top scale. The number below it on the bottom scale will be 6, the answer.

Division

To divide, you place a number on the top scale over another number on the bottom scale. The answer can then be read where a special mark on the top scale lines up with the bottom scale.

This slide rule is positioned to yield several values: From C scale to D scale (multiply by 2), from D scale to C scale (divide by 2), A and B scales (multiply and divide by 4), A and D scales (squares and square roots).

Other scales

Some slide rules have extra scales for more complex math, like angles and special functions. These help with tasks like finding squares or working with angles.

Roots and powers

Slide rules can also help find square roots and other powers by using special scales. For example, to find the square of a number, you find the number on one scale and read its square on another.

Roots of quadratic equations

Slide rules can help solve certain equations by lining up numbers on different scales.

Future value of money

Quadratic and reciprocal scales

Some slide rules have scales to help figure out how money grows over time with interest.

Trigonometry

Special scales on some slide rules help with trigonometry, which deals with angles and triangles.

Logarithms and exponentials

Some slide rules have scales for logarithms, which help with certain kinds of calculations.

Addition and subtraction

Though not common, addition and subtraction can be done on slide rules using special methods or extra scales.

C, D	single-decade logarithmic scales, single sections of the same length, used together for multiplication and division, and generally one of them is combined with another scale for other calculations
A, B	two-decade logarithmic scales, two sections each of which is half the length of the C and D scales, used for finding square roots and squares of numbers
K	three-decade logarithmic scale, three sections each of which is one third the length of the C and D scales, used for finding cube roots and cubes of numbers
CF, DF	folded versions of the C and D scales that start from pi (π) rather than from unity; these are convenient in two cases. First when the user guesses a product will be close to 10 and is not sure whether it will be slightly less or slightly more than 10, the folded scales avoid the possibility of going off the scale. Second, by making the start π rather than the square root of 10, multiplying or dividing by π (as is common in science and engineering formulas) is simplified.
CI, DI, CIF, DIF	inverted scales running from right to left, used to simplify reciprocal (1⁄x) steps
S	used for finding sines and cosines on the C (or D) scale
T, T1, T2	used for finding tangents and cotangents on the C and CI (or D and DI) scales
R1, R2	square root scales – setting the cursor to any value r {\displaystyle r} on R1 or R2, find π r 2 {\displaystyle {\pi }r^{2}} (area of a circle of radius r {\displaystyle r} ) under the cursor on the DF scale
ST, SRT	used for sines and tangents of small angles and degree–radian conversion
Sh, Sh1, Sh2	used for finding hyperbolic sines on the C (or D) scale
Ch	used for finding hyperbolic cosines on the C (or D) scale
Th	used for finding hyperbolic tangents on the C (or D) scale
L	linear scale used for addition, subtraction, and (along with the C and D scales) for finding base-10 logarithms and powers of 10
LL0N (or LL/N) and LLN	log-log folded e − x {\displaystyle e^{-x}} and e x {\displaystyle e^{x}} scales, for working with logarithms of any base and arbitrary exponents. 4, 6, or 8 scales of this type are commonly seen.
Ln	linear scale used along with the C and D scales for finding natural (base e {\displaystyle e} ) logarithms and e x {\displaystyle e^{x}}
P	Pythagorean scale of 1 − x 2 {\displaystyle {\sqrt {1-x^{2}}}} to (1) solve the Pythagorean theorem and (2) to accurately determine cosine for small angles (with the S scale)

The scales on the front and back of a Keuffel and Esser (K&E) 4181-3 slide rule

Designs

Standard linear rules

The width of a slide rule is measured by the size of its scales. The most common "10-inch" models are actually 25 cm wide because they follow metric measurements, though some have longer scales to make it easier when answers go off the edge. Smaller pocket rules are usually 5 inches (12 cm) wide. Very large models, up to a couple of meters (yards) wide, were made to hang in classrooms for teaching.

These rules usually show numbers to two significant figures, and users can guess the third number. Some better slide rules have magnifying cursors to make the lines easier to read, which can double the accuracy, letting a 10-inch rule work as well as a 20-inch one.

Circular slide rules

Circular slide rules come in two main types: one with two cursors and another with one free-moving part and one cursor. The two-cursor type does multiplication and division by keeping a fixed angle between the cursors as they turn around the circle. The one-cursor type works more like a regular slide rule by lining up the right marks.

The biggest benefit of circular slide rules is that they are smaller—about a third of the width of a straight rule. For example, a 10 cm (3.9 in) circular rule can be as accurate as a 31.4 cm (12.4 in) straight rule. They also don’t need to be turned when answers are close to 1.0 because the numbers “wrap around.” But they can be harder to read, and they have fewer scales. Most students started with straight slide rules.

A normal-sized slide rule sitting inside a 7-foot (2.1 m) teaching model

One circular slide rule still used today is the E6-B. Made in the 1930s for pilots, it helps with flying tasks and other calculations like converting units. Even though GPS and calculators exist now, many flight schools still teach students to use the E6-B.

Proportion wheels are simple circular slide rules used in design to find aspect ratios. By lining up the original and new sizes, they show the ratio as a percent. Though less common now with computers, they are still made and used.

In 1952, Breitling made a special pilot’s watch with a built-in circular slide rule for flight math: the Breitling Navitimer. It could help with speed, time, distance, fuel, and unit conversions.

Cylindrical slide rules

Cylindrical slide rules come in two styles: ones with spiral scales like the Fuller calculator, the Otis King and the Bygrave slide rule, and ones with bars like the Thacher and some Loga models. Either way, they offer longer scales and possibly more precision than straight or circular rules.

Materials

Traditionally, slide rules were made from strong, stable woods like mahogany or boxwood, with glass and metal cursors. Aluminum was also used, and some very precise ones were made of steel.

In 1895, a Japanese company, Hemmi, began making slide rules from celluloid-coated bamboo, which was stable, strong, and slippery. These bamboo rules were introduced in Sweden in September 1933 and likely a bit earlier in Germany.

Scales were sometimes made from celluloid or other plastics, or printed on aluminum. Later cursors were made from acrylics or polycarbonate, sometimes with Teflon for smooth movement.

High-quality slide rules had numbers and lines carved deep and filled with paint or resin. Cheaper ones were just painted or printed, which could wear off. Premium rules also had clever catches to keep them together and bumpers to protect the scales and cursors.

History

The slide rule was invented around 1620–1630, not long after John Napier shared his idea about the logarithm. In 1620, Edmund Gunter from Oxford made a tool with one line for measuring, which could help with multiplying and dividing. Around 1622, William Oughtred from Cambridge put two of these tools together to create what we now recognize as the modern slide rule. There was some disagreement about who should get credit for this invention.

In 1677, Henry Coggeshall made a special folding ruler for measuring wood, which expanded what people could use slide rules for.

In 1722, Warner added more lines to slide rules, and in 1755, Everard added another line. Slide rules with all these lines are called “polyphase” rules.

In 1815, Peter Mark Roget made a slide rule that could help with roots and powers directly, which was very helpful for certain calculations.

In 1821, Nathaniel Bowditch wrote about a “sliding rule” in the American Practical Navigator, which helped solve problems related to travel over long distances.

In 1845, Paul Cameron from Glasgow made a slide rule for sailors to help them figure out the positions of the sun and stars.

Modern form

A more modern slide rule was made in 1859 by a French officer named Amédée Mannheim. His design had some changes that made slide rules easier to use. These changes included new ways to read the lines and make calculations more accurate. Mannheim’s design became the standard for many years.

The growing field of engineering in the late 1800s led to more people using slide rules. A new type called the duplex rule was made in 1891, and in 1881, an American inventor named Edwin Thacher made a round slide rule that could be more precise but was also more expensive.

20th century

In the 1920s, an engineer named Nevil Shute Norway used slide rules for building an airship, and it took many people many months to complete the calculations.

In 1937, a physicist named Lucy Hayner made a slide rule that could be read by touch.

During the 1950s and 1960s, slide rules were very common among engineers. They were even taken on space missions, like by Buzz Aldrin on Apollo 11.

Some students and engineers carried slide rules with them until the 1970s when smaller electronic calculators became popular.

In 2004, two researchers made a new kind of slide rule, but it didn’t get much attention.

Specialized calculators

Slide rules were made for many different jobs, like helping farmers guess the weight of cows or helping photographers decide how long to leave a camera open. There were even slide rules for solving math problems used in factories and for figuring out weather patterns.

The E6-B is a round slide rule that pilots and navigators use.

Decline

Slide rules started to become less important in the 1960s when electronic computers became more common. Even before that, small electronic calculators began to appear. These new tools could do many of the same things as slide rules but faster and more easily. By the mid-1970s, cheap handheld calculators made slide rules almost unnecessary.

Comparison to electronic digital calculators

Slide rules were special tools used before digital calculators became common. They helped with math problems like multiplication and division, but they were slower and not as easy for adding or subtracting numbers. People who used slide rules learned to understand numbers better because they could see how the numbers changed as they moved the rulers.

One big difference is that slide rules did not show where to place the decimal point automatically. Users had to figure this out themselves, which could sometimes lead to mistakes. Slide rules usually gave answers with about three important numbers, unlike digital calculators which can show many more. However, this made users more careful about the size of the numbers they were working with.

Slide rules were also useful for quick estimates, like figuring out percentages or comparing prices, without needing to do extra steps. They did not need electricity or batteries to work, which was helpful for sailors and pilots who kept them as backup tools. Even today, some pilots and special watches still use slide rules for fast, easy calculations.

Main article: reverse Polish notation

Main articles: order of magnitude, arithmetic precision, significant digits, false precision, speed-time-distance, grid electricity, E6-B

Contemporary use

Even today, some people still like to use slide rules instead of electronic calculators. Others keep their old slide rules because they remember them from the past or because they collect them as a hobby.

There are still a few places where you can buy brand new slide rules. The Concise Company in Tokyo has been making circular slide rules since 1954 and still sells them. In 2009, the online store ThinkGeek sold straight slide rules that were made by hand, but they are no longer available. Faber-Castell also had slide rules for sale online until 2018.

Collections

The MIT Museum in Cambridge, Massachusetts has many slide rules, nomograms, and mechanical calculators. The Keuffel and Esser Company collection was donated to MIT around 2005, adding many more items to the museum’s display.

The International Slide Rule Museum is said to have the largest collection of slide rules and logarithmic calculators in the world. Its website has a "Slide Rule Library" section with lots of information about slide rules.