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Lecture

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York University

Natural Science

NATS 1700

Zbigniew Stachniak

Fall

Description

Lecture 3.
The mechanical calculating machines of the
17th century,
Charles Babbages Analytical Engine
Informal and unedited notes, not for distribution. (c) Z. Stachniak, 2011.
Note: in the case when I was unable to nd the primary source of an image or
determine whether or not an image is copyrighted, I have specied the source as
unknown. I will provide full information about images and/or obtain reproduc-
tion rights when such information is available to me.
Every now and then, an individual is born with an unusual ability to memo-
rize and retain large numbers (40, 59 and more digit long) for a long period of
time. They could recall these numbers even after weeks. Because of this abil-
ity, they could also perform arithmetic operations on large numbers mentally,
without any external aids. They can achieve this by memorizing intermedi-
ate results and using them when needed.
For instance, to multiply 397 by 173 some of these people use the follow-
ing method. First, represent 397 173 as the sum of simple products:
397 173 = 397 (100 + 70 + 3) =
(100 397) + (70 397) + (3 397) =
(100 397) + (70 (300 + 90 + 7)) + (3 (300 + 90 + 7)) =
(100397)+(70300)+(7090)+(707)+(3300)+(390)+(37).
Then perform 100 397, and get the partial sum of 39,700
then add 70 300 = 21,000, and get the partial sum of 60,700,
then add 70 90 = 6,300, and get the partial sum of 67,000,
then add 70 7 = 490, and get the partial sum of 67,490,
then add 3 300 = 900, and get the partial sum of 68,390,
then add 3 90 = 270, and get the partial sum of 68,660,
then add 3 7 = 21, and get the nal number 68,681.
The above calculation requires memorization of a sequence of partial sums:
39,700,60,700,67,000,67,490,68,390,68,660,68,700.
1 At the beginning there was the abacus
Performing arithmetic operations on large numbers mentally, say by the
method of multiplication discussed above, is beyond the capabilities of most
of us. We can typically memorize numbers about 5-digit long and only for a
short period of time. To do every-day arithmetic on large numbers we need
external aids, such as modern calculators, or follow algorithms for perform-
ing such operations using pen and paper. Some calculating aids were devised
almost concurrently with counting.
One of the most ancient and most prevailing counting aids consisted of ver-
tical lines drawn on sand or ground, and pebbles that were placed on these
lines (as shown on Figure 1). Such counters could be made of a slab of
stone with etched grooves, or even of a piece of cloth or a wooden board
with painted lines. These early counters are classies (for rather obvious
reasons) as the dust abacus, the line abacus, and the grooved abacus. Re-
gardless of the material used, historians classify all of these devices as abacus.
The exact origin of these devices is dicult to trace. Historical references
to early use of abacus-like devices in various regions of the world have been
found. Counting-boards were known in Mesopotamia more than 4,000 years
ago (Mesopotamia spanned the area corresponding to the present-day ter-
ritories of Iraq, northeastern Syria, southeastern Turkey and southwestern
Iran). These counters were adopted by Greeks and Romans. Some forms of
abacus were known in ancient China before 1000 B.C.
----ooooo hundreds of thousands
--------- tens of thousands
-------oo thousands
-----oooo hundreds
--------o tens
------ooo units
Figure 1: An example of a decimal counter showing the number 502,413.
The bottom line represented units, the lines above it: tens, hundreds, and
so on. The addition and subtraction were easy to preform on such counters.
Romans placed small marbles along the lines called calculi which is a plural
of calculus or pebble hence the origin of the modern word calculate.
2 Example 1: To add 928 to 502,413, one rst started with the counter set
to 502,413
----ooooo ----ooooo ----ooooo ---ooooo
--------- + 8 --------- + 2 --------- + 9 --------
-------oo units -------oo 10s -------oo 100s -----ooo
-----oooo =====> -----oooo ===> -----oooo ====> -----ooo
--------o -------oo -----oooo ----oooo
------ooo --------o --------o -------o
The last conguration of the counter shows 503,341
Fig. 2. This drawing depicting a counting board appeared in one of Adam Rieses
books on arithmetic (1492-1559).
3 Fig. 3. This drawing depicting the use of a counting board in trade and commerce
appeared in one of Adam Rieses books on arithmetic.
4

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