#
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 7 items.

## Multiplication Table

- Description
- Printed multiplication tables have long been included as parts of general introductions to arithmetic and its applications in business. This table, which shows multiples of integers from 1 times 1 through 25 times 25, contains larger numbers than those found in most elementary texts. It also has no printing on the back. This suggests that it may have been printed as a broadside to be used separately.

- An inscription in pencil on the back reads: Francis Lincoln (/) Fiskale (/) Mass.” Fiskale (known also as Fiskdale) is an area of Sturbridge, Massachusetts. The object was a donation of George H Watson of Sturbridge.

- Location
- Currently not on view

- date made
- ca 1850

- ID Number
- 72.5

- accession number
- 280076

- catalog number
- 72.5

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

## Mathematical Table, Young Rule For Calculating Interest

- Description
- This United States patent model has a rectangular wooden frame with five grooves, each of which holds a bar (made from a different kind of wood) that slides crosswise. Two flat wooden pieces cover much of the bars on the left side, with a gap between them. Each bar has a set of 12 evenly spaced holes that are numbered from 11 down to 1 (the “0” holes are not numbered). Each bar also is indented at the top to hold a slip of paper that slides under the top of the machine. There are 11 further, unnumbered, holes to the right of each slip of paper. Setting up a number on the rods (to represent an amount of money or a length of time) reveals a number on the paper slips that represents an amount of tax or interest.

- A piece of paper glued to the top of the device reads: S.S. Young’s Tax and (/) Interest Rule (/) Red March 18th 1851. The “d” in this mark is a superscript.

- Samuel S.Young of Eaton, Ohio, took out three patents for computing devices. This is the patent model for the second, a rule for calculating interest, patented September 2, 1851 (U.S. Patent 8323). The first was an add to addition or adder, patented July 24, 1849 (U.S. Patent 6602), the third an arithmetical proof rule, patented October 26, 1858 (U.S. Patent 21921). The U.S. Census for 1850 lists an S. S. Young of Eaton, Ohio, who was forty years old that year and living with his wife and two children. His occupation is given as “gardener.” Apparently by 1860 he had moved to the nearby town of Washington and is listed as being a “horticulturalist” by profession. Young assigned his patent to John R. Stephen of Eaton, who is listed in the 1860 Census as a farmer.

- Compare to the model for the first of his inventions, MA*252680.

- References:

- S. S. Young, “Rules for Calculating Interest,” U.S. Patent 8329, September, 1851.

- U. S. Census Records

- Robert Otnes, “Sliding Bar Calculators,”
*ETCetera*, #11, June, 1990, p. 6.

- P. A. Kidwell, “Adders Made and Used in the United States,”
*Rittenhouse*, May, 1994, p. 80.

- Location
- Currently not on view

- date made
- 1851

- patentee
- Young, Samuel S.

- maker
- Young, Samuel S.

- ID Number
- MA*252683

- catalog number
- 252683

- accession number
- 49064

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

## 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 Tables, Automatic Arithmetic: A New System for Multiplication and Division without Mental Labour and without the Use of Logarithms

- Description
- The title of this bound set of tables well describes the goal of the author. Printed in gold color on the front of the reddish brown binding, it reads: AUTOMATIC ARITHMETIC: (/) A NEW SYSTEM (/) FOR (/) MULTIPLICATION AND DIVISION (/) WITHOUT MENTAL LABOUR (/) AND (/) WITHOUT THE USE OF LOGARITHMS. In 1879, when the book was published, London accountants like the author John Sawyer multiplied and divided large numbers by consulting tables of logarithms. They or their clerks also carried out calculations by hand. However, the results obtained from logarithm tables were only approximate, and hand calculations might be erroneous. Calculating machines had sold commercially in England from the 1850s, but they were expensive and required maintenance.

- As an alternative, Sawyer proposed and patented his system of “automatic arithmetic.” This was an unusually designed arrangement of multiplication tables that allowed one to read off the partial products needed to solve multiplication problems.

- Sawyer’s system consisted of eighteen pages of instructions bound with ten sheets of heavier paper. Each of these heavier pieces is cut horizontally to form nine rows. The topmost and shortest strip of paper has the digits from 1 to 9 in one row and then a row of nine 00s. The eight other strips on this page simply have a row of nine 00s. The figures in each succeeding row are shifted one place to the right from those immediately above them.

- On the second page of the tables, the slips are slightly longer and are marked 1 on the right end. The topmost slip has the digits from 1 to 9 in a row, as well as a row of multiples of 1 running from 01 to 09. The slips below it have the multiples of 1, shifted one place to the right in each successive row. Similarly, the third page has still longer slips, marked 2 on the right end. The topmost slip has the digits from 1 to 9 in a row, and then multiples of two. The slips below this have multiples of two, shifted one place to the right from the row above. The remaining slips follow a similar pattern.

- To multiply using the tables, one turns the top pile of slips to the slips for the leftmost digit of the multiplicand, the slips in the second pile to the second digit of the number and so forth. To multiply by a single digit, one adds the partial products found on the slips turned. Further instructions suggest how the slips can be used to multiply by larger number of digits and to divide.

- Sawyer obtained patents for his invention in the United Kingdom in 1877 and in the United States in 1879. It was advertised in the British journal
*The Accountant*, and reviewed there and in*Nature*. This example was from the library of Brooklyn mathematics teacher, collector and historian of mathematics L. Leland Locke. There is no indication that the product proved popular.

- References:

- John Sawyer, “Improvement in the Means of Obtaining Arithmetical Results,” U.S. Patent 208037, September 17, 1878.

*The Accountant*, vol. 4, #189, (July 20, 1878), p. 1; #191 (August 10, 1878), p. 9. The second reference is the review.

*Nature*, July 25, 1878, p. 327.

- Location
- Currently not on view

- date made
- 1878

- ID Number
- 2011.0129.03

- accession number
- 2011.0129

- catalog number
- 2011.0129.03

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

## Patenta Multiplication Table

- Description
- This simple German table was intended as an aid to multiplication and division. Each side of the blue and white paper disc has a blue and white paper arm pivoted from the center. The disc has a diameter of 15 cm.; with the arm, the width is 15.6 cm.

- Going in from the circumference, each arm has printed on it the numbers from 1 to 20 in a column. On each side of the disc there are 19 radial columns. On one side these are numbered from 2 to 20; on the other from 21 to 39. The “2” column contains multiples of 2, the “3” column multiples of 3, etc. To find, say, the product of 35 and 19, one lines up the arm next to the 35 column on the disc. Next to 19 on the arm is 665 (the desired product) on the disc.

- The instrument is marked on both sides: PATENTA. It is also marked on both sides: D.R.G.M. (/) 145,796. The initials D.R.G.M. stand for Deutsches Reichgebrauchmuster, a temporary form of German intellectual property protection (not a full patent). D.R.G.M. numbers were first issued in 1891 and continued to be used through World War II. This number was issued in about 1901. A second form of the instrument, with more numbers, was issued later.

- The object is described briefly in a column in the
*Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht*in 1903. This example was found in the collections of the National Museum of American History’s Division of Transportation around 1980.

- References:

- “Besprechung von Lehrmitteln, Mathematik”
*Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht*34, 1903, p. 67.

- D. von Jezierski, trans. R. Shepherd,
*Slide Rules: A Journey through Three centuries*, Mendham, N.J.: Astragal Press, 2000, p. 102. This reference indicates that D.R.G.M. registered design 148526 was issued in 1901 and 173095 in 1902.

- Location
- Currently not on view

- date made
- ca 1901

- ID Number
- 1988.0579.01

- catalog number
- 1988.0579.01

- accession number
- 1988.0579

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

## Pencil-Multiplier, a Multiplication Table

- Description
- Inventors have arranged multiplication tables on cylinders and on discs to ease use. This set of tables is designed to fit over the end of a pencil.

- Near the top of this red pencil, just below the eraser, is a table of multiples of the numbers from 13 to 24 by the numbers 1 through 12. A metal cap numbered from 13 to 24 fits over the table at the top. A rotating metal cylinder fits into the cap, and is numbered 1 to 12 around the top. There is a small window in the cylinder below each of these numbers; the distance of the hole from the top varies with the size of the number. The “1” hole reveals multiples of 1 in the table, the “2” hole multiples of 2, etc. To find, say, 15 times 9, one sets the 9 column of the cylinder under the 15 of the cap and reads off 135.

- A mark on the rotating cylinder reads: CHICAGO RECORDING SCALE CO. (/) WAUKEGAN. ILL. (/) PAT. PENDING. A mark on the pencil reads: U.S.A. SOUTHERN CROSS - No 2502.

- The Chicago Recording Scale Company was in business in Waukegan, Illinois, from at least 1895 until at least 1910. I have seen no patent assigned to the company that corresponds to this object. The drawings for U.S. patent 613,432 for an improvement in pencil-boxes show something somewhat similar to this device, although the numbers included and the arrangement of windows is different. That patent was taken out by Stanislas Szenhak of “Warshaw, Russia,” and assigned to Julius Witkowski of Yokohama, Japan. Szenhak applied for a patent on August 19, 1898, and received it November 1, 1898. He also obtained a patent in Great Britain, where his invention was called a “toy for teaching arithmetic.”

- This example of the device was given to the Museum by John William Christopher Draper and James Christopher Draper. Several objects in this gift were once the property of the New York meteorologist Daniel Draper, who took an active interest in the improvement of calculating instruments.

- References:

- Stanislas Szenhak, “Pencil-box,” U.S. Patent 613432, November 1, 1898.

- P. A. Kidwell, “American scientists and calculating machines: from novelty to commonplace,”
*Annals of the History of Computing*, 12, 1990, pp. 31–40.

- Location
- Currently not on view

- date made
- ca 1900

- maker
- Chicago Recording Scale Company

- ID Number
- MA*335350

- catalog number
- 335350

- accession number
- 304826

- 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