Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
by
Stephen Kent Stephenson
BSEE, MEngElect, MEd
[email protected]
12 Villanova Dr.
Westford, MA
USA 01886-1960
978-692-3415
(Individual work.)
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Biographical Sketch of Stephen Kent Stephenson
Mr. Stephenson teaches Calculus at Lowell High, MA, USA. He bought a Soroban in
Tokyo in 1963. After college, he worked as an electrical engineer, computer programmer,
systems analyst, and industrial computer salesman. Spring 2001 he took a History of
Math course. He noted how ancient numbers could easily be registered on a counting
board abacus; the professor asked him to do multiplication and division. Recently he
posted, How to Use a Counting Board Abacus,
http://www.youtube.com/view_play_list?p=545ABCC6BA8D6F44.
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Title
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Abstract in English
Features of The Roman Hand Abacus, including a compromise in engineering, lead to the
conclusion that the Romans used The Salamis Tablet as their counting board abacus. The
Roman Hand Abacus then defines the proper promotion factors between the lines and
spaces on The Salamis Tablet. The Roman Numeral Subtractive Notation (e.g., IV = 4)
indicates that opposite sides of The Salamis Tablet were used for additive and subtractive
parts of each number. From these features, methods are identified that use The Salamis
Tablet to do all four arithmetic operations. Methods that are efficient, robust, and fast.
Abstract in German translated from English by Google
Eigenschaften des römischen Abacus Hand, darunter einen Kompromiss in der Technik,
zu der Schlussfolgerung führen, dass die Römer nutzten die Salamis Tablet als Zählen
Bord Abakus. Die römische Hand Abacus definiert dann eine angemessene Förderung
Faktoren zwischen den Zeilen und Felder auf dem Tablet Salamis. Die römische Ziffer
Subtraktive Notation (z. B. IV = 4) zeigt an, dass gegenüberliegenden Seiten des Salamis
Tablette zur additiven und subtraktiven Teile jede Rufnummer verwendet wurden. Aus
diesen Funktionen sind Methoden festgestellt, dass die Salamis Tablet verwenden, um
alle vier arithmetischen Operationen zu tun. Methoden, die effizient, robust und schnell.
Abstract in English translated from German translated from English by Google
Features of the Roman Hand Abacus, including a compromise on the technology, leading
to the conclusion that the Romans used as the Salamis Tablet Counting board abacus. The
Roman Hand Abacus then defines the proper promotion factors between the lines and
fields on the Tablet Salamis. The Roman numeral notation Subtractive (4 = IV, for
example) indicates that opposite sides of the Salamis tablet was used for additive and
subtractive parts of each number. From these functions, methods are found that the
Salamis tablet used to do all four arithmetic operations. Methods that are efficient, robust
and fast. (Pretty good.)
Keywords
“Roman Hand Abacus”, “Salamis Tablet”, “Counting Board”
Classification Codes
History of mathematics 01A20 Greek, Roman
History of mathematics 01A17 Babylonian
History of mathematics 01A16 Egyptian
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Introduction
Historians know that the Romans and other ancient peoples used counting board abaci.
Without definitive evidence to guide them, historians have published many conjectures
for what the counting board abaci looked like and how they were used for all four
arithmetic operations. Based on those hypothetical configurations and methods, the
conclusion that many historians have reached is that using counting board abaci is an
extremely laborious and cumbersome process.
Such is not the case.
One version of the Roman Hand Abacus contains strong evidence that the Romans used
The Salamis Tablet as their counting board abacus, and reveals the counter promotion
factors between lines and spaces. Roman Numeral subtractive notation, e.g. IV for 4,
indicates that Romans used the two sides of the grids on The Salamis Tablet Abacus to
represent a number in two parts, additive and subtractive.
Methods for using The Salamis Tablet Abacus flow naturally from this structure, and
prove to be extremely efficient, robust, and rapid (compared to other methods of the
time).
Theory
The Japanese Abacus, or Soroban, is still in use, esp. in Japan, and the methods to use it
are well known (e.g., Kojima, Takashi. The Japanese Abacus, It’s Use and Theory. 1963
Printing. Charles E. Tuttle Company, 1954.)
A trained Soroban operator can accomplish arithmetic calculation speeds exceeding those
of a trained electric calculator operator. (Nov 12, 1946, Tokyo Contest, Kojima, p.12).
Figure 1: Author’s Soroban
Every rod on a Soroban represents one decimal digit (Figure 1). The bead above the bar
represents five of the beads below the bar. Each bar can count from zero (no beads next
to the bar) to nine (all beads moved next to the bar).
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Figure 2 (Left): Roman Hand Abacus.
[From the Online-Museum of Prof. Dr. Jörn Lütjens, Hamburg, Germany.
http://www.hh.schule.de/metalltechnik-didaktik/users/luetjens/abakus/ab96.jpg
Permission: http://www.joernluetjens.de/sammlungen/copyright.htm]
Figure 2-B (Right): Source for Construction of Figure 2, Roman Hand Abacus.
[From p.819 of Opera historica et philologica, 1682, by Marcus Welser
(http://openlibrary.org/b/OL23366519M/Opera-historica-et-philologica%2C-sacra-et-profana.). This was
referenced on p.305 of Number Words and Number Symbols, A Cultural History of Numbers, by Karl
Menninger, MIT and Dover Publications. Welser's Abacus was obviously the source for the creation of the
replica in Figure 2.]
On the Roman Hand Abacus (Figure 2), each of the seven decimal digits has four beads
in the lower slot and one bead in the upper slot; functioning exactly like the Soroban.
It would be hard to understand why the Romans would not have developed similarly
efficient methods to use the Hand Abacus as the Japanese did to use the Soroban.
The two rightmost columns both count to twelve, but differently.
The left column counts to five in the lower slot and carries into the upper slot on a six
count, repeats to a count of eleven, then carries into the decimal units column on a twelve
count.
But the rightmost column breaks each six count into two three counts. Why the
difference?
Mapping the Roman Hand Abacus slot symbols onto The Salamis Tablet of Figure 3
results in Figure 4.
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Figure 3 (left): The Salamis Tablet, before 300 BCE, preserved in the Greek National
Museum, Athens.
Image from http://www.historyofscience.com/G2I/timeline/index.php?era=-300, 8/22/2009.
Figure 4 (right): Roman Hand Abacus mapped onto The Salamis Tablet.
The mapping is perfect. It uses the bottom grid’s eleven lines exactly, no more, no less.
However, to do so the Romans had to use a less preferred structure for one of the base-12
digits. Why it’s less preferred we’ll address below. But the fact that they had to make an
engineering compromise is indicative that they used The Salamis Tablet as a design
template for their Hand Abacus.
In the figure, the numbers on the left are the promotion factors that are dictated by the
Roman Numerals mapped to the lines and spaces. For example, a promotion factor of 5
means that 5 pebbles on that line can be replaced by one pebble in the space above. All
the spaces between lines have a promotion factor of 2. (We’ll talk about the unused
dashed line below.)
Placing the Roman Numeral MCMXLVI = 1946 on the mapped Salamis Tablet
demonstrates the use of the left side of the grid as a subtractive side. This is an extremely
important observation as it reduces the number of pebbles needed in total and on each
line or space. It makes the calculations much more efficient, robust, and rapid.
As an example of using the subtractive side, if in 2009 we wanted to calculate the age of
a person born in 1946 we would slide each of the pebbles shown to the opposite side of
the vertical median line, then add 2 pebbles to the right side of the (|) or M line, a pebble
to the right side of the X line, and a pebble to the left side of the I line, thus adding
MMIX to the opposite of MCMXLVI. Merging the pebbles and replacing the C pebble
with two L pebbles and one of the X pebbles with two V pebbles (both operations being
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
demotions), then removing zero-sum pairs on every line and space (one pebble on each
side of the median is a zero-sum pair) we read the answer LXIIV = 63.
While IIV is not considered a proper Roman Numeral expression, there are some rare
examples of documents printed with both IIV and IIX type constructions. Without getting
into a debate over the appropriateness of these constructions as a written form on paper or
stone, they do reduce the pebble count on The Salamis Tablet Abacus, both in total and
on each line and space. Using these constructions, any number can be registered with no
more than two pebbles on any line or space.
Moving the pebbles of an accumulated sum away from the median as far as possible,
there will always be room near the median to place another additive or subtractive
number. Before combining the pebbles the operator can then check his pebble placement
for accuracy without damaging the accumulated sum. This checking, or auditing, feature
adds greatly to the accuracy and robustness of these methods. After checking, the pebbles
are combined and moved to the outside ready for the next number to be added.
The preferred base-12 digit configuration will never have more than two pebbles per line
or space, but the other base-12 digit configuration will have three pebbles on a line for
counts of 3 or 9. That is objectionable, both for pebble count per line or space as well as
the psychological problem of handling two types of entities on the board. The more alike
everything is, the faster and more accurately the operator can perform. The width of the
Salamis Tablet Abacus would also have to be increased by 40% (14 vs. 10 pebble spaces
per line).
In article 26 of the book, The Aqueducts of Rome,
(http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Frontinus/De_Aquis/text*.html)
Sextus Julius Frontinus states, “… the inch ajutage, has a diameter of 1 1/3 digits. Its
capacity is [slightly] more than 1 1/8 quinariae, i.e., 1 1/2 twelfths of a quinaria plus
3/288 plus 2/3 of 1/288 more.”
In Roman Elementary Mathematics: The Operations [The Classical Journal, Vol. 47, No.
2 (Nov. 1951), 63!74 and 106!108], Prof. J. Hilton Turner says that this value results
from ((1 1/3)^2)/((1 1/4)^2), where the squares are calculated before dividing
(http://penelope.uchicago.edu/Thayer/E/Roman/Texts/secondary/journals/CJ/47/2/Roman
_Elementary_Mathematics*.html#note4).
The Romans would not have been able to do this calculation on the Roman Hand Abacus,
nor on a Roman Hand Abacus mapped Salamis Tablet. The number “1 1/2 twelfths plus
3/288 plus 2/3 of 1/288 more” written as a base-12 number is 1;1,7,10. That number has
three base-12 digits and you’d have to calculate a fourth for proper rounding. The Roman
Hand Abacus only has two base-12 digits.
So the Romans would probably have done the calculation on three coupled Salamis
Tablet abaci, each configured with 5 preferred-configuration base-12 digits, as in
Figure 5.
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Figure 5. Three Base-12 Salamis Tablets Arranged for Multiplication or Division.
Figure 5 shows the first step in squaring 1 1/3. The radix shift is shown as a zero-sum
pair for both the Multiplicand and Multiplier to indicate no shift, so the unit line is the top
line of the bottom grid. The radix shift of the Product is the sum of the Multiplicand and
Multiplier radix shifts. In Multiplication a partial product for every Multiplier pebble on a
solid line is added into the Product, and the Multiplier pebble is taken away. If there are
no more Multiplier pebbles on solid lines then a pebble in a space is replaced by the
equivalent number of pebbles on the line below. When there are no more Multiplier
pebbles the product is complete.
In Division appropriate additive or subtractive copies of partial products of the Divisor
are combined with the Dividend until the remaining Dividend is zero, accumulating the
partial product pebbles in the middle grid. When the Dividend is zero, simply slide all the
pebbles in the middle grid to the opposite side and you have the Quotient. The Quotient’s
radix shift will be the radix shift of the Dividend added to the opposite of the radix shift
of the Divisor.
Note that there is no need for multiplication or division tables, printed or memorized.
And the calculations are blindingly fast compared to any other methods of the time.
Frontinus’s calculation, and others, are performed by the author in a set of videos. It’s
much easier and more insightful to watch someone doing the calculations than reading
about how to do them.
Results
Features of the Roman Hand Abacus indicate that the Romans used The Salamis Tablet
for their heavy-duty calculations; they also give us the promotion factors between lines
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
and spaces. The Subtractive Notation of Roman Numerals indicates that one side of the
Salamis Tablet grid is used for the additive part of a number and the other side for the
subtractive part of the number.
The requirement of a four digit base-12 result in the calculation in article 26 of the book,
The Aqueducts of Rome, indicate that the Romans used the Salamis Tablet in multiple
configurations, decimal or duodecimal, or mixed, as needed.
From these factors a consistent and powerful set of methods that the Romans must have
used on the Salamis Tablet Abacus has been identified.
Discussion and Possible Future Research
The Romans borrowed heavily from the Greeks. So it’s not hard to think that the Greeks
used the Salamis Tablet as an Abacus in much the same way as the Romans.
At the time the Greeks sculpted The Salamis Tablet it was the Seleucid Period of
Babylon, 312 BCE - 64 CE, when the Babylonians were doing astronomical calculations
in base-60.
The Salamis Tablets in Figure 5 can easily be converted to base-60 digits that conform to
Babylonian cuneiform numbers by changing the promotion factors in each digit from 2-23 to 2-3-2-5, where the 3 is applied to the dashed line. This seems to indicate a direct
lineage between the Roman’s use of the Salamis Tablet and a possible use by the
Seleucid Babylonians.
If the Seleucid Babylonians did use the Salamis Tablet, they were probably the origin of
the methods that the Greeks and Romans used. If so, the origin of those methods may go
back to the Babylonians’ ancestors, the Akkadians, whom The MacTutor History of
Mathematics says invented the abacus around 2200 BCE (http://www-history.mcs.stand.ac.uk/history/HistTopics/Babylonian_mathematics.html).
One of the author’s videos shows how to use a base-60 Salamis Tablet Abacus with
Heron’s Method to calculate the cuneiform numeral values on Yale tablet YBC 7289, the
famous square root of 2 tablet, from the Old Babylonian Period, 2000-1600 BCE. The
author, a rank beginner abacist, completes the calculations in 25 minutes, all the while
explaining what he is doing. A practiced abacist master, with no requirement to provide
narrative, would probably finish in less than 5 minutes. Blindingly fast compared to table
lookups, writing cuneiform numeral intermediate results and combining them on clay
tablets with reed styluses.
The MacTutor History of Mathematics also states, “The original papyrus on which the
Rhind Papyrus is based … dates from about 1850 BC.” (http://www-history.mcs.stand.ac.uk/history/HistTopics/Egyptian_mathematics.html) This is the source of the
famous Egyptian table of unit fraction decompositions of 2/n, n = 3 to 101, odds. The
time frame is Old Babylonian, 2000-1600 BCE, after the Akkadians invented the abacus.
Roman Hand Abacus is Key to Roman Use of The Salamis Tablet
Is it possible that the answer to the riddle of how the Egyptians decomposed these
fractions involves counting board abaci? Each unit fraction could be a pebble on some
abacus line. Factoring the denominators of the unit fractions composing some of the 2/n
fractions indicates factors of 2, 3, 5, and n. The promotion factors of The Salamis Tablet
in base-60 mode are 2-3-2-5. But there’s no n, so perhaps the Egyptians had some way
of creating a special abacus that included n as a factor in the base along with 2, 3, and 5.