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.
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Mathematical Tables, Automatic Arithmetic: A New System for Multiplication and Division without Mental Labour and without the Use of Logarithms
- 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.
- 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.
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- National Museum of American History, Kenneth E. Behring Center