#
Science & Mathematics

The Museum's collections hold thousands of objects related to chemistry, biology, physics, astronomy, and other sciences. Instruments range from early American telescopes to lasers. Rare glassware and other artifacts from the laboratory of Joseph Priestley, the discoverer of oxygen, are among the scientific treasures here. A Gilbert chemistry set of about 1937 and other objects testify to the pleasures of amateur science. Artifacts also help illuminate the social and political history of biology and the roles of women and minorities in science.

The mathematics collection holds artifacts from slide rules and flash cards to code-breaking equipment. More than 1,000 models demonstrate some of the problems and principles of mathematics, and 80 abstract paintings by illustrator and cartoonist Crockett Johnson show his visual interpretations of mathematical theorems.

"Science & Mathematics - Overview" showing 19 items.

Page 1 of 2

## Book, Logarithmic and Trigonometric Tables to Five Places

- Description
- This paperback book is part of the College Outline Series published by Barnes & Noble. It contains five-place tables of the common logarithms of numbers and of logarithms of sines, cosines, tangents and cotangents for every second of arc. There is also a shorter table of natural trigonometric functions from minute to minute. Further short tables assist in conversion from common logarithms to natural logarithms, give values and logarithms of haversines, and assist in converting between degrees and radians.

- The mathematician Kaj L. Nielsen (1914-1992) was born in Denmark in 1914, came to the United States in 1926, and studied at the University of Michigan and Syracuse University before obtaining a PhD. at the University of Illinois. In addition to teaching at Syracuse, Illinois, Brown, Louisiana State University, and Butler University; he worked in the Mathematics Division of the Naval Ordnance Plant in Indianapolis and also at the Battelle Memorial Institute there. He published a wide array of books relating to practical mathematics, especially numerical analysis. The first edition of this book appeared in 1943. This is a reprint from 1965 of the second edition of 1961.

- Mechanical engineer Edward L. Heller (1912–2007) donated this book of tables to the Smithsonian. From 1956 to 1959, Heller worked as a nuclear project engineer for H. K. Ferguson Co. He was a technical manager for General Dynamics Corporation from 1959 to 1967.

- References:

*American Men and Women of Science*, 12th ed., New York: J. Cattell Press, 1972, iii: p. 2620 (on Heller).

- “Kaj L. Nielsen,”
*Math Times*, Department of Mathematics, University of Illinois, Fall 1992, p. 7.

- Location
- Currently not on view

- date made
- 1943

- 1965

- maker
- Nielsen, Kaj L.

- ID Number
- 1984.3078.01

- nonaccession number
- 1984.3078

- catalog number
- 1984.3078.01

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

## Numbers In Colour, A New Method of Teaching Arithmetic In Primary Schools

- Description
- During the 1950s, the Belgian teacher Emile-Georges Cuisenaire designed a set of rods to teach about numbers and basic arithmetic. Caleb Gattegno popularized his methods in Great Britain and the United States. This small paperbound book by Cuisenaire and Gattegno first appeared in 1954, was in its third edition by 1958, and was reprinted frequently in the next few years. This is a 1961 printing.

- For a set of Cuisenaire rods, see 1987.0542.01. For other related documentation see 1987.0542.03 through 1987.0542.07.

- Location
- Currently not on view

- date made
- 1961

- maker
- Cuisenaire, G.

- ID Number
- 1987.0542.02

- accession number
- 1987.0542

- catalog number
- 1987.0542.02

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

## Book - Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables

- Description
- From the sixteenth century, computing people relied on printed mathematical tables in performing routine mathematical calculations. This volume, compiled by the Applied Mathematics Division of the National Bureau of Standards in Washington, D.C., was conceived in 1954 as “a Handbook of Tables for the Occasional Computer” (at the time a computer was usually a person). It includes formulas and graphs used in computation as well as a wide range of tables.

- The first editor, Milton Abramowitz (1913-1958), began working with tables as a member of the Mathematical Tables Project in New York City. This was a program of the U.S. government’s Works Project Administration, begun in 1937 to provide work for the unemployed. Abramowitz had studied physics at Brooklyn College, but initially had no particular background in table making or numerical analysis. He would go on to earn a PhD. in mathematics from New York University. The second editor, Irene A. Stegun (1919-2008), joined the program in 1943, after she had received a master’s degree from Columbia University. By that time, the WPA had been terminated, and staff from project were doing computations needed by U.S. Navy and the Office of Scientific Research and Development for World War II efforts.

- After the war ended, the Mathematical Tables Project moved to Washington, D.C., where it was incorporated into the Computation Laboratory of the newly established Applied Mathematics Division of the N.B.S. Project staff hoped to produce a compact set of tables that would provide a digest of work they had carried out over the past ten years. With advice from an outside panel and sponsorship from the National Science Foundation, Abramowitz, Stegun, and the collaborators began this volume, which was finally published in June of 1964, some years after Abramowitz had died.

- The
*Handbook*proved popular. A second printing appeared in November of 1964. This is an example of that printing. It was owned by Professor Charles T. G. Looney, who taught engineering at the University of Maryland. Looney stamped the edges of the book with his name and signed it just inside the cover, but did not add further annotations.

- References:

- Scans of various editions of the
*Handbook*are available online.

- Ronald F. Boisvert and Daniel W. Lozier, “Handbook of Mathematical Functions,” in David R. Lide, ed.,
*A Century of Excellence in Measurements, Standards, and Technology: A Chronicle of Selected NBS/NIST Publications, 1901-2000*, NIST Special Publication 958, 2001, Washington, D.C.: Government Printing Office, pp. 135-139.

- David Alan Grier, “Irene Stegun, the
*Handbook of Mathematical Functions*, and the Lingering Influence of the New Deal,”*American Mathematical Monthly*, 113 #7, August-September 2006, pp. 585-597.

- Location
- Currently not on view

- date made
- 1964

- maker
- Abramowitz, Milton

- Stegun, Irene A.

- ID Number
- 1988.3092.01

- nonaccession number
- 1988.3092

- catalog number
- 1988.3092.01

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

## Book, Georg's Freiherrn von Vega Logarithmisch-Trigonometrissches Handbuch

- Description
- By the late eighteenth century, those carrying out extensive calculations routinely multiplied and divided numbers by adding and subtracting logarithms of those numbers. Mathematicians prepared tables of logarithms of ordinary numbers as well as trigonometric functions such as sines, cosines, and tangents. The Slovenian-born mathematics teacher and artillery officer Georg Vega (1754-1802) published his first book of tables in 1793. Ten years later, he published in both German and Latin the first edition of this book, the
*Logarithmisch trigonometrisches Handbuch*. His publisher, the firm of Weidemann, would publish German editions of it into the 1960s. This example is the seventy-fifth edition, published in Berlin in 1894. The title page mentions not only Vega but two later authors, Karl Bremiker (1804-1877), who edited the volume from at least 1856 until his death, and Friedrich Tietjen (1834-1895). Bremiker was an official at the Prussian Board of Trade and later director of the Prussian Geodetical Institute. Tietjen was a longtime associate of the Berlin Observatory, rising to the rank of University Professor of Astronomy and director of the Computing Bureau there.

- The volume contains an introduction describing the procedures used in computing the tables, a table of common logarithms of the integers from 1 to 100,000 (to seven places), and a table of the logarithms of sines and tangents (and cosines and cotangents) from second of arc to second of arc from 0 to 45 degrees. A brief appendix gives tables relating to the refraction of starlight.

- This example of the publication is from the library of the donor, computer pioneer Carl Hammer.

- References:

- MacTutor History of Mathematics website.

- Eberhard Knobloch, “Vega and the Royal Prussian Academy of Sciences in Berlin,”
*Archives Internationales d’Histoire des Sciences*, 2008, vol. 58, pp. 171-184.

- Accession file.

- Location
- Currently not on view

- date made
- 1894

- maker
- Vega, Georg

- Bremiker, Karl

- Tietjen, Friedrich

- ID Number
- 1988.3105.02

- nonaccession number
- 1988.3105

- catalog number
- 1988.3105.02

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

## Book, Seven-Place Values of Trigonometric Functions for Every Thousandth of a Degree

- Description
- In the early twentieth century, calculating machines began to replace logarithms as a common tool of those doing computations. Rather than adding and subtracting logarithms of functions to multiply and divide, people entered values of the functions themselves into machines. This encouraged the production and sale of new books of mathematical tables. Some of these were compiled by Johann Theodor Peters (1869-1941) of the astronomical computing institute in Berlin. In 1918, Peters published
*Siebenstellige Werte der trigonometrischen Funktionen von Tausendstel zu Tausendstel des Grades*. It would be reprinted in German in 1930 and 1938. This is the 1942 English edition of the book.

- The volume contains values of trigonometric functions for every thousandth of a degree, to seven places. The first table give sines and cosines, the second tangents and cotangents. Information in the margins assists in interpolating between values in the tables, and the introduction describes how to do this.

- According to the title page, the volume was “published and distributed in the public interest by authority of the alien property custodian under license #A1.” The Alien Property Custodian was an official of the United States Government responsible for assets of enemy powers seized during World War I and again during World War II. This included the copyright to the trigonometric tables of Peters.

- This example of the publication is from the library of the donor, computer pioneer Carl Hammer.

- Reference:

- R.C. Archibald,
*Mathematical Table Makers*, New York: Scripta Mathematica, 1948, pp. 68-71.

- Location
- Currently not on view

- date made
- 1942

- ID Number
- 1988.3105.21

- nonaccession number
- 1988.3105

- catalog number
- 1988.3105.21

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

## Book, The Ancient Quipu or Peruvian Knot Record

- Description
- From ancient times, bureaucrats have kept numerical records of people and property. Those working for the Inca emperor in 16th-century Peru recorded data on arrangements of knotted strings known as quipus. The devices may also have been used as aids to memory in recounting histories of Inca exploits—the Incas had no written language.

- Particularly from the 1880s, collections in South America, Europe, and the United States began to include quipus, usually recovered from the graves of makers or users. The devices were of great interest to historian of mathematics Leslie Leland Locke, a charter member of the Mathematical Association of America. Locke (1875–1943), a native of Grove City, Pennsylvania, received his A.B. from Grove City College in 1896 and his M.A. in 1900. He studied further at Pennsylvania State University, Cornell University, and Columbia. He came to Brooklyn to teach at Adelphi College and then worked from 1908 to 1933 at Maxwell Training School for Teachers, later moving to Brooklyn Technical High School. He also taught in the evening session at Brooklyn College.

- Around 1909, while studying under historian of mathematics David Eugene Smith at Teacher’s College of Columbia, Locke became interested in the quipu. He examined several examples at the American Museum of Natural History in detail. In a 1912 paper, Locke argued that the knots on the strings of a quipu represented the decimal digits of numbers, arranged vertically by place value. He extended this research in this volume, published by the American Museum of Natural History in 1923. It includes excerpts of numerous early Spanish and other European texts relating to the quipu, and lists forty-five surviving examples. Locke deemed five of these modern or spurious. He obtained and published illustrations of over thirty of the objects, laying the foundation for further studies.

- Locke also took great interest in more modern innovations in computing, particularly the calculating machine. He inscribed and presented this copy of his book to Dorr E. Felt, the head of Felt & Tarrant Manufacturing Company. Felt had invented, and Felt & Tarrant manufactured, the Comptometer, a leading adding machine of its day. The book, along with the rest of Felt’s library relating to the history of mathematical instruments, was given to the Smithsonian Institution by Victor Comptometer Corporation, the successor to Felt & Tarrant.

- In addition to joining the MAA when it was established, Locke was a charter member of the History of Science Society and active in the National Council of Teachers of Mathematics.

- References:

- Marcia Ascher and Robert Ascher,
*Mathematics of the Incas: Code of the Quipu*, Mineola, New York: Dover, 1997. This is a corrected republication of the book*Code of the Quipu: A Study in Media, Mathematics, and Culture*, published in 1981 by the University of Michigan Press.

- Stefanie Gaenger.
*Relics of the Past: The Collecting and Study of pre-Columbian Antiquities in Peru and Chile,1837–1911*, Oxford: Oxford University Press, 2014, esp. 101–159.

- L. L. Locke, “The Ancient Quipu, A Peruvian Knot Record,”
*American Anthropologist*, n.s. vol. 14, #2 (1912), pp. 325–332.

- “Teacher 45 Years: Educator in Mathematics and an Expert on Calculating Machines Dies. . .,”
*New York Times*, August 30, 1943, p. 15 (this is an obituary of Locke).

- For a recent database of quipus, see the Khipu Database Project at http://khipukamayuq.fas.harvard.edu.

- date made
- 1923

- maker
- Locke, L. Leland

- ID Number
- 1991.3107.08.15

- nonaccession number
- 1991.3107

- catalog number
- 1991.3107.08.15

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

## Handwritten Cipher Book of Jesse Harmon Alexander

- Description
- In the 18th and early 19th centuries, when paper was still expensive and textbooks not generally available, students who learned arithmetic sometimes wrote out their own texts by hand. This volume, covered with yellow, black, and red wallpaper, is such a “cipher book.”

- Jesse Harmon Alexander, who was born in 1810 and lived in Rockland, Delaware, prepared this manuscript exercise book in 1825. Each page contains one or two columns of problems and commentary. The first topic considered was the Single Rule of Three. Alexander went in the wrong direction on the very first problem, but he did reach the solution on subsequent problems. His style was typical of the 18th and early 19th centuries. For example, in division problems, he wrote the divisor, dividend, and quotient in a row from left to right. The remaining topics were also typical of the time: tare and tret (adjustments to the price of goods for the weight of the container and for imputities), the Double Rule of Three, direct and inverse proportion, interest, insurance commission and brokerage, compound interest, discount, equation of accounts, barter, loss and gain, foreign exchange, measurement of surfaces, vulgar fractions, reduction of decimals, alligation, single and double position, involution and evolution, extraction of square roots, cube roots, arithmetical and geometrical progression, annuities, and promiscuous questions.

- Alexander's teacher was likely guided by one of the many old-fashioned arithmetic textbooks still in wide use in the 1820s. He (in 1825 most teachers teaching boys in the mid-Atlantic states were men) would have expected Alexander to memorize rules and examples and thus learn how to carry out the mathematical operations Alexander would use in business. The "vulgar fractions" section is one example of this teaching method, as Alexander copied down case after case rather than any general principles governing all fractions. The teacher probably also hoped Alexander would copy the material neatly. This not only offered practice his penmanship but prepared a reference book the boy could use as an adult.

- Alexander clearly returned to this exercise book later in his life, for dates from the 1830s dates are scattered throughout the book. New computations are written on the fourth page of the "interest" section. It is not clear, though, that Alexander's cramped handwriting and failure to clearly mark many of the solutions were any easier to read in the 19th century than now, when bleeding ink and worn page corners have also damaged the textual content.

- Several pages that are not from the original exercise book have been placed inside the front cover. These contain newspaper clippings (mainly poems and obituaries) pasted over accounts from 1831 and 1832.

- Reference:

- For an appreciation of the importance of cipher books in arithmetic education, see: Nerida Ellerton and M.A. (Ken) Clements,
*Rewriting the History of School Mathematics in North America 1607-1861: The Central Role of Cypher Books*, Dordrecht: Springer, 2012.

- Location
- Currently not on view

- date made
- 1825

- maker
- Alexander, Jesse Harmon

- ID Number
- MA*322685.01

- accession number
- 322685

- catalog number
- 322685.01

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

## Post Instruction Manual for Versalog Slide Rule

- Description
- This is a later printing of 1978.0800.02. Its citation information is: E. I. Fiesenheiser,
*Versalog Slide Rule Instruction Manual*, with R. A. Budenholzer and B. A. Fisher (Chicago: Frederick Post Company, 1963). The text appears not to have been revised since these three Illinois Institute of Technology engineering professors helped invent the Versalog slide rule and wrote instructions for using it in 1951. Marks inside the front cover indicate this copy was offered for sale in January 1969 for $1.00.

- date made
- 1963

- maker
- Frederick Post Co.

- ID Number
- 1980.0097.03

- accession number
- 1980.0097

- catalog number
- 1980.0097.03

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

## Pickett Manual for Dual Base Log Log Slide Rules

- Description
- Pickett, Inc., was a slide rule manufacturer that started in Chicago in 1943, shifted most of its operations to Alhambra, Calif., in 1946, and moved to Santa Barbara, Calif., in 1964. Maurice L. Hartung, a mathematics professor at the University of Chicago, wrote several instruction manuals for the company, including
*How to Use Dual Base Log Log Slide Rules*. This 93-page booklet was intended for use with Pickett models 2, 3, and 4. It contains sections on the general operation of a slide rule, the use of scales for trigonometry and roots, elementary vector methods, the use of logarithmic scales, practice problems, hyperbolic functions, and circular functions. Hartung also showed how the double T scales could solve side-angle-side triangle problems in one step. Model 600 was advertised at the back of the manual, and instructions for caring for Pickett slide rules were provided inside the back cover.

- Although Hartung wrote the manual in 1947, this printing was made after the company moved to Santa Barbara in 1964. See the associated items, 1980.0097.01 and 1980.0097.06.

- Location
- Currently not on view

- date copyrighted
- 1947

- date printed
- ca 1965

- author
- Hartung, Maurice L.

- printer
- Pickett Industries

- ID Number
- 1980.0097.05

- accession number
- 1980.0097

- catalog number
- 1980.0097.05

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

## Instructions for Otis King's Pocket Calculator

- Description
- This trifold pamphlet reprints the text found in 1987.0788.06, except for the section on [Pounds] Sterling Calculations, which is omitted. The columns of text are also laid out slightly differently, so that the page length is shortened from eight pages to six. These instructions accompanied 1989.3049.02. See also 1989.3049.04.

- Location
- Currently not on view

- date made
- ca 1965-1968

- maker
- Carbic Limited

- ID Number
- 1989.3049.03

- nonaccession number
- 1989.3054

- catalog number
- 1989.3054.01

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