Cylindrical Slide Rules

Writing a logarithmic scale in a spiral that is then imprinted around the outside of a cylinder allows instrument makers to lengthen the scale. Since a logarithmic scale only needs to run to 10 twice to include all possible results from adding or subtracting two logarithmic numbers, this means that the distance between the points on the scale may be increased. When any two numbers on the scale are further apart, the user may read a fractional position between these numbers (expressed as a decimal) to a finer level of granularity. For example, on a 10" linear slide rule, results generally may be calculated to only three significant figures (0.123, 1.23, 12.3, 123, 1230, and so on). A cylindrical slide rule may provide results of up to seven significant digits (0.1234567, 1.234567, 12.34567, 123.4567, 1234.567, and so on).

On the other hand, cylindrical slide rules were typically about twice as expensive to produce as linear slide rules, and the provenances for the objects on this page indeed suggest that, in the main, only corporate and government offices could afford them. A number of the objects in this category were received with instruction manuals and advertising pamphlets, which may be viewed in the Index by Makers & Retailers. The collection includes multiple examples for several of the objects, as the 23 items below represent only eight different slide rules, half developed in Europe and half invented by Americans. The cylindrical slide rule by Edwin Thacher, a Pennsylvania railroad bridge designer, also illustrates the 19th-century shift in production from Europe to the United States, as originally William Ford Stanley's London firm made the entire instrument. Then, Keuffel and Esser of New York City began constructing the wooden drum and brass and wood stand while continuing to import the scales printed on paper and pasted around the drum. Finally, K&E developed its own dividing engine for printing the scales and henceforth manufactured the entire instrument in the United States.

This instrument consists of a wooden cylinder covered with paper scales, wooden handles at the ends of the cylinder, and a metal sleeve lined with felt. The sleeve, which is painted maroon, holds the cylinder.
Description
This instrument consists of a wooden cylinder covered with paper scales, wooden handles at the ends of the cylinder, and a metal sleeve lined with felt. The sleeve, which is painted maroon, holds the cylinder. Running the length of the sleeve are a slot 1.5 cm wide and a paper scale.
The instrument is marked on the paper covering the cylinder: “WEBB’S STADIA SLIDE RULE”, (/) DESIGNED BY WALTER LORING WEBB, C. E. (/) MANUFACTURED BY KEUFFEL & ESSER CO., N.Y. It also is marked there: DIRECTIONS. SLIDE THE CYLINDER UNTIL ONE END OF THE CYLINDER IS SET AT THE DISTANCE MARK ON THE SCALE AND SO THAT THE GIVEN ANGLE OF ELEVATION ALSO COMES TO SOME PART OF THE SCALE. THE REQUIRED QUANTITY IS 1/10 (1/100 or 1/1000, AS SHOWN BY THE MARK ON CYLINDER) OF THE SCALE READING AT THAT ANGLE MARK.
The stadia slide rule was used in topographical surveying to determine the elevation and geographical position of points and objects. Initially, a chain and compass or transit had been used to determine geographical position, with a level employed to obtain relative elevations. Greater efficiency in these measurements was then found by using a plane-table.
In about 1864, the U.S. Lake Survey adopted a third system, first used in Italy about 1820. A stadia rod was placed at the point of interest and sighted through the telescope of a transit. The distance to this point was found by observing the portion of the graduated rod shown between certain cross-hairs of the telescope. To find the elevation of the point, one examined the vertical angle on the vertical circle of the transit when the telescope was aimed at a point on the stadia rod that was the same height off the ground as the telescope. A stadia slide rule was then used for data reduction.
Keuffel & Esser of New York introduced a 20-inch linear stadia slide rule in 1895. It sold under various model numbers (1749, 4101, N-4101) until 1952. In 1897, the firm introduced a 50-inch linear stadia slide rule designed by Branch H. Colby of St. Louis. Colby's stadia slide rule (model number 1749-3, later 4125) sold until 1903. Textbook authors such as John Butler Johnson endorsed the rule, but it was awkward to carry in the field.
Walter Loring Webb (1863–1941), a civil engineer who graduated from Cornell University and taught there and at the University of Pennsylvania, proposed a rule that had parallel scales arranged on a cylinder, reducing the length of the instrument to about 16 inches. K&E sold Webb's stadia slide rule for $5.00 from 1903 to 1923.
One end of the sleeve is painted: 1803. This may be an inventory number from the University of Missouri's Department of Civil Engineering, which donated the instrument in 1972. The university began teaching civil engineering in 1859, and its School of Engineering was renamed the College of Engineering in 1877.
See also 1983.0472.01. For circular stadia slide rules, see MA.336425, 1987.0221.01, and 2002.0282.01.
References: John Butler Johnson, The Theory and Practice of Surveying, 16th ed. (New York: John Wiley & Sons, 1908), 237ff; Walter Loring Webb, Railroad Construction: Theory and Practice, 7th ed. (New York: John Wiley & Sons, 1922), 22–23; Wayne E. Feely, "K & E Slide Rules," The Chronicle of the Early American Industries Association 49, no. 5 (1996): 50–52; Catalogue and Price List of Keuffel & Esser Co., 31st ed. (New York, 1903), 308; Mark C. Meade, "A History of the College of Engineering at the University of Missouri – Columbia," Archives of the University of Missouri, http://muarchives.missouri.edu/c-rg9-eng.html.
Location
Currently not on view
date made
1903-1923
maker
Keuffel & Esser Co.
ID Number
MA.333636
accession number
300659
catalog number
333636
This instrument consists of a wooden cylinder covered with paper scales, wooden handles at the ends of the cylinder, and a metal sleeve lined with felt. The sleeve, which is painted maroon, holds the cylinder. A slot 1.5cm wide and a paper scale run the length of the sleeve.
Description
This instrument consists of a wooden cylinder covered with paper scales, wooden handles at the ends of the cylinder, and a metal sleeve lined with felt. The sleeve, which is painted maroon, holds the cylinder. A slot 1.5cm wide and a paper scale run the length of the sleeve. For detailed information, see MA.333636. The only difference between the instruments is that the sleeve on this example has been decoratively painted in gold. Keuffel & Esser of New York sold this model for $5.00 from 1903 to 1923.
The School of Forestry of the University of Montana in Missoula donated this example to the Smithsonian in 1983. The School of Forestry was established at the State University of Montana (later Montana State University, now the University of Montana) in 1913. By 1921, courses taught in the school of forestry included—subject to student demand—a course on the slide rule. The course covered the theory and use of the Mannheim, Stadia, Polyphase, and Log rules.
References: State University of Montana, Annual Catalogue (Missoula, 1921), 125; "The Jack Burton Collection," Journal of the Oughtred Society 2, no. 1 (1993): 15–18; Bob Otnes, "The 31st (1903) and 32nd (1906) Editions of the K&E Catalogue" Journal of the Oughtred Society 11, no. 2 (2002): 24–32; Bob Otnes, "Keuffel & Esser Slide Rules 1909" Journal of the Oughtred Society 12, no. 1 (2003): 25–32.
Location
Currently not on view
date made
1903-1923
maker
Keuffel & Esser Co.
ID Number
1983.0472.01
catalog number
1983.0472.01
accession number
1983.0472
The brass core of this cylindrical slide rule is covered with paper marked with forty A scales. The core fits in an open rotating frame that holds twenty metal slats; each slat is lined with cloth, covered with paper, and marked with two B and two C scales.
Description
The brass core of this cylindrical slide rule is covered with paper marked with forty A scales. The core fits in an open rotating frame that holds twenty metal slats; each slat is lined with cloth, covered with paper, and marked with two B and two C scales. Wooden knobs on each end of the core rotate the instrument. The frame is attached to a mahogany base.
The first A scale runs from 100 to 112; the fortieth runs from 946 to 100 to 105. The first B scale runs from 100 to 112, the last from 946 to 100 to 105. The first C scale runs from 100 to 334, the last from 308 to 305. The paper covering the core is also printed in italics on the right side: Made by Keuffel & Esser Co., New York; Patented by Edwin Thatcher [sic], C.E. Nov. 1st., 1881.
A paper of instructions and rules for operating THACHER'S CALCULATING INSTRUMENT is glued to the top front of the base. The top back of the base is stamped: KEUFFEL & ESSER CO. (encircling N.Y.); 4012 (/) 4917; TRADEMARK (below the K&E logo of a lion).
The instrument is stored in a mahogany rectangular case. A square off-white label inside the lid is printed: KEUFFEL & ESSER CO. (/) NEW YORK HOBOKEN, N.J. (/) CHICAGO ST. LOUIS SAN FRANCISCO MONTREAL (/) DRAWING MATERIALS, SURVEYING INSTRUMENTS (/) MEASURING TAPES (/) No. 4012 (/) SERIAL 4917.
Keuffel & Esser sold versions of the Thacher cylindrical slide rule from about 1883 until about 1950. There were two models, one with a magnifying glass (K&E model 1741 before 1900, K&E model 4013 after) and one without (K&E model 1740 before 1900, K&E model 4012 after). This is a model 4012; the serial number suggests it was manufactured around 1920. Model 4012 sold for $35.00 in 1916, $60.00 in 1922, and $70.00 in 1927.
The National Bureau of Standards (now the National Institute of Standards and Technology) declared this object excess property and transferred it to the Smithsonian in 1965. In 1968, the Department of Commerce borrowed the slide rule to exhibit in the U.S. Pavilion at HemisFair, an international exposition held in San Antonio, Tex. According to the accession file, a staffer replaced four missing screws on the base before returning the rule.
See also MA.312866; MA.323504; and MA.322730.
References: Wayne E. Feely, "Thacher Cylindrical Slide Rules," The Chronicle of the Early American Industries Association 50 (1997): 125–127; Keuffel & Esser Co., Slide Rules and Calculating Instruments (New York, 1916), 22; Keuffel & Esser Co., Slide Rules and Calculating Instruments (New York, 1922), 21; Keuffel & Esser Co., Slide Rules and Calculating Instruments (New York, 1927), 20.
Location
Currently not on view
date made
ca 1920
maker
Keuffel & Esser Co.
ID Number
MA.326628
accession number
261654
maker number
4917
catalog number
326628
This large cylindrical slide rule consists of an aluminum frame supporting a horizontal rotating aluminum drum. Paper covering the drum is marked with 60 logarithmic C and D scales that are 20.5" long and run from 1,000 to 10,000.
Description
This large cylindrical slide rule consists of an aluminum frame supporting a horizontal rotating aluminum drum. Paper covering the drum is marked with 60 logarithmic C and D scales that are 20.5" long and run from 1,000 to 10,000. Each scale repeats part of the previous scale, so the instrument is approximately equivalent to a linear slide rule 50 feet (or 15m) long. A slotted cylindrical sleeve that is 13" long fits over the drum. It slides back and forth. The right end of the sleeve is secured in an aluminum ring. The ring and sleeve can rotate independently of each other.
The sleeve's 60 slats are marked with logarithmic scales that run from 100 to 1,000. Four small clear celluloid triangles attached to the slats serve as indicators. Black bands on either side of the drum and on the left side of the sleeve are marked with numbers from 1,000 to 9,623. The bands on the sleeve were originally covered with clear celluloid.
The side pieces of the frame are both marked: LOGA. The ends of the drum are both marked: LOGA-CALCULATOR • ZURICH. The band on the right side of the sleeve is marked: Loga - Calculator 15m Patente Daemen Schmid, Uster - Zürich.
The Swiss firm of Heinrich Daemen-Schmid manufactured approximately 30,000 cylindrical Loga-Calculators between 1900 and 1935. Daemen-Schmid patented the device in the United States in 1912. The donor, Jacques Kayalaff (1898–1983), was an international banker who used this instrument for calculations relating to currency exchange. He purchased it around 1925 for $500.
References: Accession File; Dieter von Jezierski, Slide Rules: A Journey Through Three Centuries, trans. Rodger Shepherd (Mendham, N.J.: Astragal Press, 2000), 42, 44; Heinrich Daemen-Schmid, "Computing Device" (U.S. Patent 1,036,575 issued August 27, 1912); Heinz Joss, "350 Jahre Rechenschieber, und was die Region Zürich dazu beigetragen hat (350 Years of Slide Rules, and What the Zurich Region Has Contributed)," Vierteljahreszeitschrift der Naturforschenden Gesellschaft in Zürich 146, no. 2–3 (2001): 75–82, http://www.rechenschieber.org/zurich.html.
Location
Currently not on view
date made
ca 1925
maker
Daemen-Schmid
ID Number
MA.335255
catalog number
335255
accession number
316433

Our collection database is a work in progress. We may update this record based on further research and review. Learn more about our approach to sharing our collection online.

If you would like to know how you can use content on this page, see the Smithsonian's Terms of Use. If you need to request an image for publication or other use, please visit Rights and Reproductions.