#
Slide RulesLinear Slide Rules

Between 1614 and 1622, John Napier discovered logarithms, Edmund Gunter devised a scale on which numerals could be multiplied and divided by measuring the distance between two logarithmic numbers with a pair of dividers, and William Oughtred put two such scales alongside each other, moving one on a slide so that the distance between the numbers could be read off directly. The speed with which these developments unfolded suggests the power that logarithms provided for reducing the tedium of calculation. By the 1680s, the English used slide rules in carpentry and in gauging (estimating liquid volumes)—the instruments were quite helpful for determining excise taxes on barrels of liquor.

It was not until the late eighteenth century that slide rules were commonly utilized in the manufacture of machines and machine tools, most notably by James Boulton and James Watt. Several more decades passed before slide rule use became widespread throughout Europe. French artillery officer Amedée Mannheim fostered acceptance of the rectangular or linear form of the instrument in 1851 by standardizing the general types of scales and the order in which the scales were arranged. Mannheim also developed the cursor or indicator, which made it easier to read results from two scales that were not adjacent to each other. By the 1870s, German companies made slide rules one of the first consumer products to be fashioned out of plastic (specifically, sheets of celluloid laminated to a wooden frame), and they built dividing engines that permitted mass production of the scales engraved on slide rules. In the United States in the 1890s, Keuffel & Esser of New York City pioneered first the import and then the domestic manufacture of linear slide rules. Numerous other companies entered the market in the 20th century.

A quick tour of what you can expect to see on a slide rule starts with the C and D scales, which both represent the standard number line. To multiply two numbers, set the 1 on the C scale above the first number to be multiplied on the D scale. Look at the second number to be multiplied on the C scale; the number below it on the D scale is the answer. For example, to multiply 2 by 3, set 1 on C over 2 on D, and then look below 3 on C to see the answer 6 on D. Division is accomplished by reversing the process. To calculate 6 ÷ 3, set 3 on C over 6 on D, then look at 1 on C to see the answer 2 on D. To deal with numbers larger than 10 or smaller than 1, the user must mentally move the decimal point. Slide rule users also had to be able to estimate distances between marks on the scales, since there was no way to include all of the digits needed to solve a problem such as 3.14 X 2.7. (On linear and circular slide rules, the answer is "approximately 8.48." Web sites on the Resources page provide more detailed training in using slide rules.)

The basic process for setting up and solving problems is the same for operations on other scales. If the numbers used in the calculation produce a result off the ends of the scales, a user employs the CI and DI scales, which put the number line in inverse, or reverse, order. If the numbers still extend past the end of the instrument, the user may try the "folded" CF and DF scales, which start numbering at π instead of at 1. The A and B scales are number lines of squares, so they are used with the C or D scales to square (or take the square root of) a number. The K scale provides cubes and cube roots. L scales represent common logarithms, S scales give sines and cosines, and T scales indicate tangents.

"Slide Rules - Linear Slide Rules" showing 2 items.

## Keuffel & Esser 4088-3 Polyphase Duplex Slide Rule

- Description
- This ten-inch duplex linear slide rule is made from boxwood faced with white celluloid and held together with L-shaped metal end pieces. On one side, the base has DF and D scales, with CF, CIF, and C scales on the slide. The bottom of the base is marked: KEUFFEL & ESSER CO. N. Y.; PAT. JUNE 5, '00 DEC. 22, '08. On the other side, there are K, A, D, and L scales on the base and S, T, and CI scales on the slide. Both pieces of the base and the slide are stamped on one end: 13. The indicator is glass in a metal frame with vertical bars on the sides.

- Keuffel & Esser introduced this form of slide rule in 1913 as model 4088-3 and sold it with a leather case for $7.00. This style of indicator was used only from 1913 to 1915, when K&E switched to its "frameless" glass cursor. This example thus dates to 1913–1915. Compare to MA*321778, 1981.0933.03, and MA*318480.

- References: Willie L. E. Keuffel, "Slide-Rule" (U.S. Patent 651,142 issued June 5, 1900) and "Slide-Rule" (U.S. Patent 907,373 issued December 22, 1908);
*Catalogue of Keuffel & Esser Co.*, 34th ed. (New York, 1913), 304a;*Catalogue of Keuffel & Esser Co.*, 35th ed. (New York, 1915), 307b.

- Location
- Currently not on view

- date made
- 1913

- maker
- Keuffel & Esser Co.

- ID Number
- MA*318476

- catalog number
- 318476

- accession number
- 235479

- Data Source
- National Museum of American History, Kenneth E. Behring Center

## Keuffel & Esser 4088-3 Polyphase Duplex Slide Rule

- Description
- This ten-inch duplex linear mahogany slide rule is completely coated with white celluloid and held together with L-shaped metal end pieces. The indicator is frameless glass with metal edges and held together with metal screws. One edge is marked: PAT.APPL'FOR. On one side of the rule, the base has DF and D scales, with CF, CIF, and C scales on the slide. Near the left end, the top of the base is marked in red: PAT. JUNE 5, '00 DEC. 22, '08. The slide is marked: < 4088-3 >. The bottom of the base is marked: KEUFFEL & ESSER CO. N. Y. Both pieces of the base and the slide are stamped on one end: 85.

- On the other side, the base has K, A, D, and L scales, with S, T, and CI scales on the slide. The K scale is divided logarithmically three times in the length of the scale, for use in finding cubes and cube roots. The A scale is divided logarithmically from 1 to 10 twice in the length of the scale and provides the square of numbers on the D scale. The S scale gives the sines of angles from less than 40 minutes to 90 degrees. The T scale gives tangents of angles from somewhat less than 6 degrees to 45 degrees. The CI scale is divided logarithmically from 1 to 10 once the length of the scale; it runs in the opposite (inverse) direction from the D scale below it. The L scale is a scale of equal parts running from 0 to 10 and provides the mantissa for the logarithm of D.

- The Keuffel & Esser Company of New York sold model 4088-3, a polyphase duplex slide rule, from 1913 to 1939. The indicator used in 1913 had vertical metal bars on the sides, while the patent for the frameless indicator was issued in 1915 and mentioned on K&E slide rules almost immediately. Since the markings on this example indicate the patent had not yet been granted, this rule must have been manufactured around 1914. Model 4088-3 sold in 1913 and 1915 for $7.00. Compare to MA*318476 and 1981.0933.03.

- References:
*Catalogue of Keuffel & Esser Co.*, 34th ed. (New York, 1913), 304a;*Catalogue of Keuffel & Esser Co.*, 35th ed. (New York, 1915), 307b; Willie L. E. Keuffel, "Slide-Rule Runner" (U.S. Patent 1,150,771 issued August 17, 1915); Wayne E. Feely, "K & E Slide Rules,"*The Chronicle of the Early American Industries Association*49, no. 5 (June 1996): 50–52; Peter M. Hopp,*Slide Rules: Their History, Models, and Makers*(Mendham, N. J.: Astragal Press, 1999).

- Location
- Currently not on view

- date made
- 1914

- maker
- Keuffel & Esser Co.

- ID Number
- MA*321778

- accession number
- 246883

- catalog number
- 321778

- Data Source
- National Museum of American History, Kenneth E. Behring Center