Page 4 - Kotov

Before Computers: Mechanical Arithmetic
Time Frame
In 18th and 19th centuries, Russia was emerging from the self-isolation of
the past. Peter the Great “cut the window into Europe”, started domestic
manufacturing, and founded the Russian Academy of Sciences with strong
mathematical bias (Euler, Bernoulli, Lobachevsky, Chebyshev,...). After ab-
olition of serfdom in 1861, Russia started slowly transform herself from a
mostly agricultural feudal country into an industrial capitalistic society. De-
veloping manufacturing, commerce, and banking required replacement of he
traditional Russian accounting device called “schoty” (a sort of abacus) by
some more elaborate calculators.

Jakobson’s adding machine

The adding machine was invented by Jevno Jakobson, a mechanic
and clock master from town Nesvizh, sometime around 1770. The
machine operated with numbers up to 109 and was mainly intended for
adding and subtraction. It included a combination of pinion wheels for
addition,subtraction and tens carry. It was also possible to do multipli-
cation in the following way:
• add the first multiplicand to itself the number of times equal to the
digit in the rightmost position of the second multiplicand and write
down the result;
• add the first multiplicand to itself the number of times equal to the
digit in the 10’s position of the second multiplicand, multiply it to 10 and write down the result;
• add the first multiplicand to itself the number of times equal to the digit in the 100’s position of the second mul-
tiplicand, multiply it to 100 and write down the result;
• and so on;
• add all intermediate results.

Slonimski’s multiplication machine
Hayyim Z. Slonimski from Bialystok (then Russia) was a deeply knowledgeable
Talmudist and a self-taught mathematician.He designed a machine was based on the
theorem discovered by him. The machine allowed to receive products of any number
(whose digit capacity did not exceed a digit capacity of the device) on 2, 3, 4..., 9. It
was something like the mechanical table of multiplying of any number by 2, 3, 4..., 9.
Since the amount of related numbers was not that large, they were put on the cylinders,
which – when moved appropriately – were showing the multiplication results in small
windows.
In 1845, Slonimski presented the machine to the Russian Academy of Sciences in St.
Petersburg, and obtained their recommendation for the Demidov Prize, which was
Hayyim Z . Slonimski awarded to him (2,500 rubles). He was also granted patent for this machine in Russia
(1810–1905) for the period of ten years. The Slonimsky theorem is derived from the Farey sequenc-
es. Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends
with the value 1, denoted by the fraction 1⁄1 (although some authors omit these terms).the Farey sequence of order
n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denomina-
tors less than or equal to n, arranged in order of increasing size.
F2 = {0⁄1, 1⁄2, 1⁄1}
F3 = {0⁄1, 1⁄3, 1⁄2, 2⁄3, 1⁄1}
F4 = {0⁄1, 1⁄4, 1⁄3, 1⁄2, 2⁄3, 3⁄4, 1⁄1}
F5 = {0⁄1, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 1⁄1}
F6 = {0⁄1, 1⁄6, 1⁄5, 1⁄4, 1⁄3, 2⁄5, 1⁄2, 3⁄5, 2⁄3, 3⁄4, 4⁄5, 5⁄6, 1⁄1}
F7 = {0⁄1, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 1⁄1}
F8 = {0⁄1, 1⁄8, 1⁄7, 1⁄6, 1⁄5, 1⁄4, 2⁄7, 1⁄3, 3⁄8, 2⁄5, 3⁄7, 1⁄2, 4⁄7, 3⁄5, 5⁄8, 2⁄3, 5⁄7, 3⁄4, 4⁄5, 5⁄6, 6⁄7, 7⁄8, 1⁄1}

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