#
Scale RulesCalculating Rules

In addition to length measurements, scale rules could be marked with aids for calculation. Perhaps the most notable rule of this type was Gunter's scale, which was similar to a sector or a slide rule. Gunter's scales are named after Edmund Gunter, a 17th-century English mathematical practitioner who figured out how to put a table of logarithms on a rule so that logarithmic calculations could be made with the aid of dividers. These instruments were especially handy for mathematicians and navigators.

The mathematics collections contain several other objects, ranging from the 18th to the 20th centuries, that were used to simplify computations for tasks including designing sundials, keeping track of calendar dates, and plotting data for aeronautical engineering. A few of these rules were designed specifically for positioning artillery.

"Scale Rules - Calculating Rules" showing 8 items.

## Gunter's Scale Signed Merrifield & Co.

- Description
- In 1614 Scottish mathematician John Napier announced his discovery of logarithms. Within eight years, Edmund Gunter, an English clergyman who was interested in mathematics, had devised a scale on which logarithms could be multiplied and divided, by measuring the distance between two logarithmic numbers with a pair of dividers. Shortly thereafter, instrument makers were manufacturing wooden rules with standard (or "natural") scales typically used in navigation on one side and Gunter's logarithmic (or "artificial") scales on the other side.

- This instrument, a precursor of the slide rule, became known as Gunter's scale. Since it was made of one piece of wood, the expansion and shrinking that happened at sea did not impede its operation. Thus, Gunter's scale remained popular with ship's navigators until the end of the 19th century, when new materials were available for the manufacture of slide rules. Surveyors, mechanics, craftsmen, and retailers also used Gunter's scales to make logarithmic and trigonometric calculations.

- This 2' boxwood rule is identical to MA*319077 and MA*333945. The top of one side has a scale of inches, divided to tenths of an inch and numbered by ones from 23 to 1. On the left are 10" and 9" (divided to 1/2") plotting scales with diagonal scales at each end. In the middle are scales for rhumbs, chords, sines, tangents, and semitangents. On the right are scales for leagues, rhumbs, miles of longitude, and chords. Brass pins at the zero and 60° marks reduce wear from the points of dividers, which were used to transfer measurements between the scale and the user's drawing.

- The other side has logarithmic scales: sines of rhumbs, tangents of rhumbs, line of numbers, sines of degrees, versines of degrees, and tangent of degrees. At the bottom edge are a meridional line and a scale of equal parts that divides 23" into 17 sections. The sections are numbered by tens from 60 to 10 and from 100 to 0.

- On the side with the scale of inches, the rule is marked in the lower right corner: MERRIFIELD & C
^{O}(/) NEW-YORK. Merrifield & Co. sold Gunter's scales in Boston and New York in the early 19th century.

- References: Adler Planetarium,
*Webster Signature Database*, http://historydb.adlerplanetarium.org/signatures/; Otto van Poelje, "Gunter Rules in Navigation,"*Journal of the Oughtred Society*13, no. 1 (2004): 11–22; George Curtis,*A Treatise on Gunter's Scale, and the Sliding Rule*(Whitehall, N.Y., 1824); Florian Cajori, "On the History of Gunter's Scale and the Slide Rule During the Seventeenth Century,"*University of California Publications in Mathematics*1, no. 9 (February 17, 1920): 187–209.

- Location
- Currently not on view

- date made
- ca 1825

- retailer
- Merrifield & Co.

- maker
- Merrifield & Co.

- ID Number
- MA*318174

- catalog number
- 318174

- accession number
- 232132

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

## Belcher Brothers Gunter's Scales, Group of 2

- Description
- These two 2' boxwood rules are identical to each other and to MA*333945 and MA*318174. The top of one side has a scale of inches, divided to tenths of an inch and numbered by ones from 23 to 1. On the left are 10" and 9" (divided to 1/2") plotting scales with diagonal scales at each end. In the middle are scales for rhumbs, chords, sines, tangents, and semitangents. On the right are scales for leagues, rhumbs, miles of longitude, and chords. Brass pins at the zero and 60° marks reduce wear from the points of dividers, which were used to transfer measurements between the scale and the user's drawing.

- The other side has logarithmic scales: sines of rhumbs, tangents of rhumbs, line of numbers, sines of degrees, versines of degrees, and tangent of degrees. At the bottom edge are a meridional line and a scale of equal parts that divides 23" into 17 sections. The sections are numbered by tens from 60 to 10 and from 100 to 0.

- The first rule is marked on the first side in the lower right corner: BELCHER BROTHERS MAKERS (/) New York. ("New York" is in script.) On the other side, this rule is marked at the top center: J. J. WARREN. The second rule is marked on the first side in the lower right corner: BELCHER BRO
^{S}. & C^{o}. NEW – YORK.

- Thomas Belcher began making rules in New York in 1821 and was joined by his brother, William, in 1825. Around 1831, another brother, Charles, joined the firm, and the company's name was changed from T. & W. Belcher to Belcher Brothers. The first rule dates to this period. Around 1843, manufacturing of rules moved to New Jersey and the firm went by the name Belcher & Bros. Around 1853 some of the men's sons joined the firm, and the name was changed to Belcher Bros. & Co. The second rule dates to this period. The company had become the largest American manufacturer of rules, but it was surpassed by Stanley in the second half of the 19th century. Belcher stopped manufacturing rules around 1877. The Smithsonian purchased these rules in 1961, and no information is known about J. J. Warren, the name on the first rule.

- References: Otto van Poelje, "Gunter Rules in Navigation,"
*Journal of the Oughtred Society*13, no. 1 (2004): 11–22;*Belcher Bros. & Co.'s Price List of Boxwood & Ivory Rules*(New York, 1860; reprint, Fitzwilliam, N.H.: Ken Roberts Publishing Co., 1982); Philip A. Cannon II, "The Makers and Markers of Gauges, Rules, Squares, and Tapes," http://www.pactu.com/makers.htm.

- Location
- Currently not on view

- date made
- ca 1853–1877

- ca 1831–1843

- maker
- Belcher Brothers

- ID Number
- MA*319077

- accession number
- 236088

- catalog number
- 319077

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

## Gunter's Scales, Group of 2

- Description
- These two 2' boxwood rules are identical to each other and to MA*319077 and MA*318174. The top of one side has a scale of inches, divided to tenths of an inch and numbered by ones from 23 to 1. On the left are 10" and 9" (divided to 1/2") plotting scales with diagonal scales at each end. In the middle are scales for rhumbs, chords, sines, tangents, and semi-tangents. On the right are scales for leagues, rhumbs, miles of longitude, and chords. Brass pins at the zero and 60° marks reduce wear from the points of dividers, which were used to transfer measurements between the scale and the user's drawing.

- The other side has logarithmic scales: sines of rhumbs, tangents of rhumbs, line of numbers, sines of degrees, versines of degrees, and tangent of degrees. At the bottom edge are a meridional line and a scale of equal parts that divides 23" into 17 sections. The sections are numbered by tens from 60 to 10 and from 100 to 0.

- The first rule is marked on the first side in the lower right corner: BELCHER BRO
^{S}. & C^{o}. NEW – YORK. Thomas Belcher began making rules in New York in 1821 and was joined by his brother, William, in 1825. Around 1831 another brother, Charles, joined the firm, and the company's name was changed from T. & W. Belcher to Belcher Brothers. Around 1843 manufacturing of rules moved to New Jersey and the firm went by the name Belcher & Bros. Around 1853 some of the men's sons joined the firm, and the name was changed to Belcher Bros. & Co. The first rule dates to this period. The company had become the largest American manufacturer of rules, but it was surpassed by Stanley in the second half of the 19th century. Belcher stopped manufacturing rules around 1877.

- The second rule is unmarked. Key points around the scales are marked with suns, unlike the asterisks on the first rule; the abbreviations for the scale labels are different; and the bottom edge is beveled at a sharper angle than the bottom edge of the first rule. These differences indicate that the second rule was manufactured by a different firm. This rule also has pencil marks on the top and bottom edges.

- References: Otto van Poelje, "Gunter Rules in Navigation,"
*Journal of the Oughtred Society*13, no. 1 (2004): 11–22;*Belcher Bros. & Co.'s Price List of Boxwood & Ivory Rules*(New York, 1860; reprint, Fitzwilliam, N.H.: Ken Roberts Publishing Co., 1982); Philip A. Cannon II, "The Makers and Markers of Gauges, Rules, Squares, and Tapes," http://www.pactu.com/makers.htm.

- Location
- Currently not on view

- date made
- ca 1853–1877

- maker
- Belcher Brothers

- ID Number
- MA*333945

- accession number
- 296611

- catalog number
- 333945

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

## Gunter's Scale

- Description
- The top edge of one side of this 12-inch wooden rule has a scale of inches, divided to tenths of an inch and numbered by ones from 24 to 13, with a "1" visible at the right for the 12" mark. Underneath this scale are 10" and 9" (divided to 1/2") plotting scales with diagonal scales at each end. On the other side are unlabeled, partial scales for logarithms, logarithmic sines, logarithmic versines, logarithmic tangents, meridional lines, and equal parts. On the scale of equal parts, ten units are equivalent to 3 cm.

- This object appears to be a Gunter's scale that was cut in half. Compare to MA*319077, MA*318174, and MA*333945. The Smithsonian's Division of Ethnology received this rule in 1951 and believed it to be of American manufacture.

- Location
- Currently not on view

- ID Number
- MA*388991.01

- catalog number
- 388991

- accession number
- 182022

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

## Square Wood Rule or Calculating Stick

- Description
- This large wood rule has a square cross-section and rounded edges. Metal plates are screwed into both ends. The rule is covered with numbers placed at one-inch intervals. One corner is numbered by ones from 1 to 60 and labeled "Height." "Wood Rule" is written above the scale. Adjacent to the 11 on this scale are listed 15 "Widths" in intervals of one inch, ranging from 3'2" to 4'4". Next to each Width, and adjacent to the numbers 12–60 on the Height scale, are lists of numbers. Each number is eight times the product of the height and width, with both of these taken in feet (i.e., the 12 is assumed to represent one foot). The numbers in the lists are rounded off to the nearest whole number.

- The National Museum of History and Technology, now the National Museum of American History, acquired this calculating stick, probably between 1962 and 1965, for its Growth of the United States exhibition, which opened in 1967 and closed in 1982. What the original owner was measuring with these calculations is not known.

- Reference: William S. Walker, "A Living Exhibition: The Smithsonian, Folklife, and the Making of the Modern Museum" (Ph.D. diss., Brandeis University, 2007).

- Location
- Currently not on view

- date made
- 19th century

- ID Number
- 1987.0107.05

- accession number
- 1987.0107

- catalog number
- 1987.0107.05

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

## Complete Mathematical Chart Designed by C. W. Goodchild

- Description
- This object consists of paper laminated to both sides of two wooden boards that are held together by two brass hinges and fastened with a brass hook. The front gives instructions for using the 100-line logarithmic table that appears on the inside two pages. To multiply, the user looked up the first multiplicand, noted the number of the line on which the multiplicand appeared, and measured the distance from the left of the line to the multiplicand. Then, the user repeated the process with the second multiplicand. The product appeared on the sum of the line numbers at the sum of the distances. (For instance, the number 4 is on line 60 and the number 2 is on line 30, so the number 8 is on line 90.)

- The table could also be used for division, calculations of interest, finding logarithms, and finding the numbers when the logarithm is known. A diagonal scale at the bottom of page three allowed for interpolation of values. A card or ruler was necessary for recording the distances. The back of the object has a 31-line chart of trigonometrical ratios for finding logarithmic sines and cosines. The bottom of the back is marked: Copyrighted September, 1893, by C. W. GOODCHILD.

- Cecil Wray Goodchild (1847–1900) was born in England but lived in central California by 1880. By 1893, he was a civil engineer and attorney in San Luis Obispo. He designed this chart to meet the needs of those surveyors, engineers, and accountants who required greater accuracy in their work than that provided by an ordinary slide rule, but who did not wish to purchase an expensive instrument such as the Thacher cylindrical slide rule.

- In 1903 and 1906, Keuffel & Esser advertised his invention as the Goodchild Mathematical Chart, model 4019. It sold on paper for 75¢ and on a flat board for $2.75. For an additional $5.00, K&E offered a sliding triangular rule for recording and adding the line numbers and distances.

- References: Library of Congress,
*Catalogue of Title-Entries of Books and Other Articles Entered . . . Under the Copyright Law*, no. 116 (18–23 September 1893): 19; "A Slide Rule Fifty Feet Long,"*The Cornell Daily Sun*14, no. 83 (31 January 1894);*Catalogue and Price List of Keuffel & Esser Co.*, 31st ed. (New York, 1903), 298;*Catalogue and Price List of Keuffel & Esser Co.*, 32nd ed. (New York, 1906), 317.

- Location
- Currently not on view

- date made
- 1893–1903

- inventor
- Goodchild, C. W.

- ID Number
- MA*318472

- catalog number
- 318472

- accession number
- 235479

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

## Gerber Variable Scale, Model TP007100B

- Description
- This rectangular aluminum instrument has a scale of reciprocal inches, unevenly divided and labeled R
_{I}; a scale of equal parts, labeled I and numbered by ones from 1 to 10 and by tens from 15 to 95; and a logarithmic scale labeled L_{10}. An aluminum slide on top of the scales has a plastic indicator with a hairline. The slide is attached to a spring that expands and contracts within a clear plastic housing above the scales. It has 103 coils, 100 of which are calibrated. Every tenth coil is colored red, every fifth coil is blue-green, and the rest are white. A second spring is numbered by twos from 0 to 10.

- Below the scales is marked: THE GERBER SCIENTIFIC INSTRUMENT COPMANY HARTFORD, CONN. GERBER VARIABLE SCALE ® MODEL TP007100B U.S. PAT. NO. 2843935 U.K. PAT. NO. 845215 MADE IN U.S.A. A brown leather case is lined with black velvet marked: The Gerber Scientific Instrument Co. (/) Hartford, Connecticut. A small manila envelope inside the case holds an Allen wrench. The case fastens with Velcro and slides into a white cardboard box.

- The instrument allows users to read off points along a plotted equation without having to draw the curve. It can also be used to convert between proportional scales, for instance when enlarging or reducing an engineering drawing. Heinz Joseph Gerber (1924–1996) invented the device while he was studying aeronautical engineering at Rensselaer Polytechnic Institute in 1945, a few years after escaping Nazi-controlled Austria with his mother. He and a partner established the Gerber Scientific Instrument Company in Hartford, Conn., to manufacture the Gerber Variable Scale. Gerber ultimately held about 650 U.S. and foreign patents for calculating instruments, digital drafting machines, and robotic and electronic manufacturing systems for products from electronics to textiles. The firm was renamed Gerber Scientific, Inc., in 1978.

- Compare to 1994.3104.01. For an instruction manual, see 1994.0113.04.

- References: Arthur Bartlett, "A Quick Spring to Success,"
*Nation's Business*(October 1949): 43–45, 62–64; Heinz Joseph Gerber, "Instrument for Measuring, Interpolating, Plotting and the Like" (U.S. Patent 2,843,935 issued July 22, 1958); "Our Founder," Gerber Scientific, http://www.gspinc.com/default.asp?contentID=192.

- Location
- Currently not on view

- date made
- ca 1980

- maker
- Gerber Scientific Instrument Company

- ID Number
- 1994.0113.01

- accession number
- 1994.0113

- catalog number
- 1994.0113.01

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

## Instruction Manual for Gerber Variable Scale

- Description
- The citation information for this 40-page stapled booklet is: H. Joseph Gerber,
*The Gerber Variable Scale: An Application and Instruction Manual*(Hartford, Conn.: The Gerber Scientific Instrument Company, 1981). Gerber wrote these instructions in 1953 to accompany his invention; see 1994.0113.01. He inscribed this copy to Steve Lubar, who was chair of the division of history of technology when Gerber donated examples of several mathematical instruments he had invented.

- The booklet explains the construction and use of the Gerber Variable Scale. It then describes typical problems that could be solved with the instrument, including finding points in a family of curves, translating curves, reading graphs more precisely, converting between proportional scales, enlarging and reducing engineering drawings, normalizing curves, interpolating points, counting cycles of frequencies, dividing one curve by another, determining the center of gravity, reading oscillograms, and mapping aerial photographs.

- Location
- Currently not on view

- date made
- 1981

- maker
- Gerber Scientific Instrument Co.

- ID Number
- 1994.0113.04

- accession number
- 1994.0113

- catalog number
- 1994.0113.04

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