#
Mathematical Charts and TablesTables for General Reckoning

**Tables for General Reckoning**

Multiplication tables were among those most frequently produced for general use. Some people apparently just cut tables out of books for their particular purposes. In 1857, James D. Smith of Brantingham, New York, patented a “machine for multiplying numbers” that consisted of a wooden disc engraved with tables and a rotating stylus. At the turn of the century, a patented paper instrument of this type actually sold. At about the same time, Stanislas Szenhak of Warsaw (now in Poland) took out a U.S. patent for an especially designed multiplication table that could be fit around the eraser end of a pencil, with a metal cover that made it easier to find results. Such a pencil multiplier actually was manufactured in Illinois.

Some of these tables could by quite elaborate. In the 1870s, British accountant John Sawyer devised a set of bound tables with slips that could be turned to set up problems of interest. His procedures replaced multiplication and division by addition and subtraction. Most of those doing extensive multiplications and divisions at the time preferred to add and subtract logarithms of numbers, but Sawyer’s process allowed one to get results exactly. In the 1930s the American mathematician John Perry Ballantine proposed replacing slide rules with a set of tables in which the results of multiplication, division, and taking square roots could be read off directly, rather than requiring an awareness of significant figures.

"Mathematical Charts and Tables - Tables for General Reckoning" showing 2 items.

## Mathematical Table, J. D. Smith Machine For Multiplying Numbers

- Description
- This is the United States patent model for a multiplication table. It consists of a wooden disc pivoted to a wooden handle on which it revolves. The front of the part of the handle above the disc is a metal rod with the numbers 1/4, 1/2, 3/4, 1 through 10, and 20 engraved on it. The top of the disc has numbers engraved over its surface such that one can line up the handle with a number on the edge of the disc and find multiples of that number on the disc next to the engraved numbers on the handle.

- A mark painted on the back of the handle and written on the back of the disc reads: J.D. SMITH.

- This invention was patented in 1857 by James D. Smith (1834-1908), a native of Chatham, New York, who had moved to Brantingham in that state in 1841. He worked there in various businesses. In addition to this patent, Smith took out patents for an improvement in tool sharpeners (#87,212, February 12, 1869) and an improvement in station-indicators (#161170, March 23, 1875). No evidence has been found indicating that any of these inventions led to products.

- In 1881, Smith moved to Albany to study law. He spent the rest of his career as an attorney.

- References:

- James D. Smith, “Machine for Multiplying Numbers,” U. S. Patent 18711, November 24, 1857.

- “James D. Smith,”
*The Journal and Republican*, Lowville, New York, June 4, 1908, p. 1.

- Location
- Currently not on view

- date made
- 1857

- patentee
- Smith, James D.

- maker
- Smith, James D.

- ID Number
- MA*252687

- catalog number
- 252687

- accession number
- 49064

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

## Mathematical Table, The Macmillan Table Slide Rule

- Description
- John Perry Ballantine (1896–1970), a mathematician on the faculty of the University of Washington, published this set of tables in 1931 as an inexpensive alternative to the slide rule. The paper instrument includes two 8-1/2” x 11” (22.3 cm. x 28 cm) cards which have printed tables on both sides. These are for multiplication, finding powers of numbers, sines, and tangents. Four narrower tables are placed next to these. Two of these are for multiplication, one for division and one for square root. Each of the wider tables has 20 columns of numbers in 100 rows. The narrower ones have ten columns of numbers in ten rows. Tables are based on antilogarithms to base 10. A leaflet of instructions and a paper dust cover are included.

- This example was the property of Oscar W. Richards of the Osborn Zoological Laboratory of Yale University. It is marked with his stamp. A mark on the corner reads: THE MACMILLAN (/) TABLE SLIDE RULE. Another mark there reads: New York (/) THE MACMILLAN COMPANY (/) 1931.

- Ballantine was born in Rahuri, India, the son of a medical missionary and a teacher. He graduated from Harvard in 1918 and then taught briefly at the University of Maine, Pennsylvania State College, and the University of Michigan. He attended graduate school at the University of Chicago, where he met and married fellow graduate student and mathematician Constance Rummons. They both received doctorates from Chicago in 1923. J. P. Ballantine then spent three years teaching at Columbia University before joining the faculty of the University of Washington in 1926. He stayed there, except for a stint in American military schools, until his retirement in 1966.

- Ballantine’s slide rule was reviewed in the
*Journal of the American Statistical Association*, the*American Mathematical Monthly*, and the British educational journal*Mathematical Gazette*. It cost only fifty cents, but, as reviewers pointed out, was less portable and less durable than a conventional slide rule. No second edition was required.

- Ballantine did not limit his interest in technical improvement to classroom devices. In 1932, he applied for a patent relating to electric power meters, receiving it in 1935. In 1938, he published the textbook
*Essentials of Engineering Mathematics*. Neither of these projects was particularly influential.

- References:

- Advertisement,
*The American Mathematical Monthly*, 38 (May 1931), unnumbered page.

- E. J. Atkinson, “The Macmillan Table Slide Rule,” reviewed in
*The Mathematical Gazette*, 16 (May 1932), pp. 140–141.

- Dorothy C. Bacon, “The Macmillan Table Slide Rule,” reviewed in
*Journal of the American Statistical Association*, 26 (Sept 1931), p 373–374.

- J. P. Ballantine, “Multiple-rate Power Metering,” U.S. Patent #2000736, May 7, 1935.

- R. E. Gilman, “The Macmillan Table Slide rule,” reviewed in
*The American Mathematical Monthly*, 39 (May 1932), pp. 295–296.

- J. Green and J. LaDuke,
*Pioneering Women in American Mathematics: the Pre-1940 PhD’s*, Providence: American Mathematical Society, 2009, pp. 131–132.

- Location
- Currently not on view

- date made
- 1931

- maker
- MacMillan

- ID Number
- 1979.3074.08

- nonaccession number
- 1979.3074

- catalog number
- 1979.3074.08

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