MesoCalc was created in March 2013
by Baptiste MÉLÈS
(CNRS, Archives Henri Poincaré, Université de Lorraine) with
the scientific assistance of Christine PROUST
(CNRS, Université Paris-Diderot) in the framework of
the SAW Project
(Mathematical Sciences in the Ancient World), headed
by Karine CHEMLA (CNRS, Université
Paris-Diderot).
MesoCalc includes computations made by Mathieu OSSENDRIJVER
(Humboldt-Universität) and a calendar designed by Bruno GOMBERT (Université Paris 1 Panthéon Sorbonne, ARSCAN),
with their benevolent acceptance.
Contents
- Introduction
- Arithmetics
- Conversion and arithmetical
properties: conversions between decimal and
sexagesimal numbers, regularity test, regular
approximations, list of regular numbers, prime
factors, greatest common divisor;
- Multiplicative operations:
multiplication, quotient, reciprocals, tables...
- Exponential operations:
square, square root, cube, cube root, geometric
progression, tables...
- Additive operations: addition,
subtraction;
- Measures
- Dates
- About MesoCalc
Introduction
Computing systems
The numbers if the calculator below may belong to different
kinds:
Code |
Numerical system |
Examples |
10A |
Decimal place-value notation |
7200 |
60A/F |
Sexagesimal place-value notation |
2.0.0 or 2 |
60A |
Absolute sexagesimal place-value notation |
2.0.0 |
60F |
Floating sexagesimal place-value notation |
2 |
60R |
Regular number in 60F |
2 |
Decimal and sexagesimal place-value notation
(10A
and 60A/F)
Our numerical system (10A)
is a decimal place-value notation. It
means that we use ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8,
9), and that every number containing several digits
(like 1729) is composed of multiples of powers
of 10. As an example, when we write the
number 1729, we mean the number
(1 × 1000) + (7 × 100) + (2 × 10) + 9.
By contrast, Mesopotamian mathematics use
a sexagesimal place-value notation
(60A/F). It means that they
used 59 "digits" (they had no 0), and that the
numbers are decomposed into multiples of powers
of 60, which we can separate with a dot.
As an example, the number 1729 would be noted 28.49, which
means
(28 × 60) + 49.
The number 10000 would be noted 2.46.40, which means
(2 × 3600) + (46 × 60) + 40.
Unlike 10, which has only two divisors (2 and 5), 60 has a lot
of divisors: 2, 3, 4, 5, 6, 10, 12, 15, 20, 30.
What are the absolute and floating notations?
(60A
and 60F)
Whereas our numerical system (10A)
is always absolute, the sexagesimal notation used in
mathematical cuneiform texts can be either
"absolute" or
"floating" (60A/F).
Intuitively, the sexagesimal notation is said to be
floating (60F)
when we neglect all the ".0"s that
we can find at the end of a number,
and absolute
(60A) when we can not neglect
them. For instance, in 60A, the
decimal number 60 is noted "1.0", since it is
(1 × 60) + 0. But
in 60F, it would just be
noted "1", since we have to neglect all
final ".0"s! We can say that
6010A = 1.060A = 160F.
As a consequence, the floating sexagesimal number which is
noted "1" in 60F can
correspond to the decimal numbers 1, 60
(= 1 × 60), 3600
(= 1 × 60 × 60), etc.; but it can
also correspond to the fractions 1/60
(= 1 × 1/60), 1/3600
(= 1 × 1/(60 × 60)), etc.
Similarly, the number which is noted "2"
in 60F can correspond to the
decimal numbers 2, 120, 7200, etc., but also to the fractions
1/30, 1/1800, etc.
Here are some more examples of sexagesimal place-value
notation:
10A |
60A |
60F |
1 |
1 |
1 |
2 |
2 |
2 |
59 |
59 |
59 |
60 |
1.0 |
1 |
61 |
1.1 |
1.1 |
120 |
2.0 |
2 |
3600 |
1.0.0 |
1 |
7200 |
2.0.0 |
2 |
10000 |
2.46.40 |
2.46.40 |
The floating system may seem strange, but it has wonderful
mathematical properties. For instance, the square root of 15
in 60A is not an integer (it is
someting like 3.87..., just like
in 10A); but the square root of 15
in 60F is 30, for 15 can also
represent the number 15.0 = 900, which is the square
of 30.
Similarly, the cube root of 7.30
in 60A is not an integer (just
like the square root of 450
in 10A), but the cube root of 7.30
in
60F is 30, for
7.3060F can represent
the
number 7.30.060A = 2700010A,
which is the cube of 30.
What are reciprocals and regular numbers?
(60R)
The Mesopotamian "division" is performed as
the multiplication of the dividend by what is
called the
"reciprocal" number of the
divisor.
The reciprocal R of
a number N is the number such
that, in floating notation,
R × N = 1 (be it 1, 1.0, 1.0.0 or any
other power of 60 in absolute notation). For instance, the
reciprocal of 1 is 1, since 1 × 1 = 1. The
reciprocal of 2 is 30, since 2 × 30 = 1.0 (which is
noted 1 in floating notation). The reciprocal of 8
is 7.30, since 7.30 × 8 = 1.0.0 (which is also
noted 1 in floating notation).
Some numbers do not have any reciprocal. As a matter of
fact, a number has a reciprocal if and only if all of
its prime factors are prime factors of 60 as well,
which means that it must be reducible to a product of powers
of 2, 3 and 5. Such a number is called a regular
number (60R). The
numbers 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 etc. are regular
numbers, whereas the numbers 7, 11, 13, 14, 17, 19, 21, 22,
23 etc. are not.
Bibliography
If you want to know more about the sexagesimal place-value
notation and mathematical cuneiform texts:
- FRIBERG, Jöran (2007), A Remarkable Collection of
Babylonian Mathematical Texts, New York:
Springer.
- HØYRUP, Jens (2002), Lengths, Widths, Surfaces. A
Portrait of Old Babylonian Algebra and Its Kin,
Berlin & Londres: Springer.
- MELVILLE
Duncan, Bibliography
of Mesopotamian Mathematics.
- NEUGEBAUER, Otto (1957), The Exact Sciences in
Antiquity (2th ed.), Providence: Brown University
Press.
- NEUGEBAUER, Otto and SACHS, Abraham J.
(1945), Mathematical Cuneiform Texts, New Haven:
American Oriental Series & American Schools of
Oriental Research.
- PARKER, R. A. and DUBBERSTEIN, W. H., Babylonian
Chronology, 626 B.C.-A.D. 45, Studies in Ancient Oriental Civilization 24, 1942.
- PROUST, Christine (2007), Tablettes mathématiques de
Nippur, Istanbul: Institut Français d'Etudes
Anatoliennes, De Boccard.
- PROUST,
Christine, Le calcul
sexagésimal en Mésopotamie.
- ROBSON, Eleanor (2008), Mathematics in Ancient Iraq:
A Social History, Princeton: Princeton University
Press.
- THUREAU-DANGIN, François (1932), Esquisse d'une
histoire du système sexagésimal, Paris:
Geuthner.
- Cuneiform Digital
Library Initiative (CDLI).
- Mathematical
Notations: conventions for the translitteration of
Mesopotamian mathematical and astronomical texts.
Arithmetics
Conversion and
arithmetical properties
Conversions
Arithmetical properties
Multiplicative operations
Multiplication and quotient
Reciprocal and regular numbers
Exponential operations
Squares and square roots
Cubes and cube roots
Geometric progression
Additive operations
Measures
Lengths
Units of length:
danna | ← 30
← |
uš | ← 60 ← |
ninda | ← 12
← |
kuš | ← 30
← |
šusi |
10.8 km | |
360 m | |
6 m | |
50 cm | |
17 mm |
Surfaces
Units of surface:
gan |
← 100 ← |
sar |
← 60 ← |
gin |
← 180 ← |
še |
3600 m² | |
36 m² | |
0.6 m² | |
33 cm² |
| |
1 ninda × 1 ninda | |
| |
|
Volumes
Units of volume:
gan |
← 100 ← |
sar |
← 60 ← |
gin |
← 180 ← |
še |
1800 m³ |
|
18 m³ |
|
300 dm³ |
|
1.66 dm³ |
|
|
1 (surface-)sar × 1 kuš |
|
|
|
|
Capacities
Units of capacity:
gur |
← 5 ← |
bariga |
← 6 ← |
ban |
← 10 ← |
sila |
← 60 ← |
gin |
← 180 ← |
še |
300 L |
|
60 L |
|
10 L |
|
1 L |
|
16.6 mL |
|
0.092 mL |
Weights
Units of weight:
gu |
← 60 ← |
mana |
← 60 ← |
gin |
← 180 ← |
še |
30 kg
|
|
500 g
|
|
8.33 g |
|
0.046 g |
System G and system S
System G:
šar-gal |
← 6 ← |
šar'u |
← 10 ← |
šar |
← 6 ← |
bur'u |
← 10 ← |
bur |
← 3 ← |
eše |
← 6 ← |
iku |
64800 |
|
10800 |
|
1080 |
|
180 |
|
18 |
|
6 |
|
1 |
System S:
šar-gal |
← 6 ← |
šar'u |
← 10 ← |
šar |
← 6 ← |
gešu |
← 10 ← |
geš |
← 6 ← |
u |
← 10 ← |
diš |
216000 |
|
36000 |
|
3600 |
|
600 |
|
60 |
|
10 |
|
1 |
Dates
During the Neo Babylonian and Persian periods, which
are concerned by this converter, the year was based on the
revolution of the Earth around the Sun, that is 365 days. It
was beginning in spring, during the Babylonian month of
nisannu (1), and ending during month addaru
(12). Years’ calculation was based on Kings’ reigns which are
well known, thanks to Ptolemy's Canon. This
fundamental document allows us establishing equivalences
between Babylonian and Gregorian dates.
The year in which a king acceded to the throne, is
recorded as “Year 0” in this converter, then subsequent regnal
years are numbered until the next King. A year consisted of
12 lunar months, each of them corresponding to the interval
between two new moons, thus 12 × 29 or 30 days :
- 1: nisannu
- 2: aiāru
- 3: simānu
- 4: dūzu
- 5: abu
- 6: ulūlu
- 7: tašrītu
- 8: arahsamnu
- 9: kislīmu
- 10: ṭebētu
- 11: šabāṭu
- 12: addaru
However, 11 days are lacking to complete the solar year. To
fill this gap, Mesopotamians used empirically to add from time
to time a 13th month. This intercalary month was added either
after the 6th (month 6b), or the 12th (month 12b).
A mathematical scheme enabling a regular cycle of intercalary
months was elaborated during Achaemenid Period, based on the
fact that 19 solar years and 235 lunar months have the same
number of days. Seven intercalary months were added over a
period of 19 years, on year 1, 3, 6, 9, 11, 14, and 17 year.
Only the 1st year had an intercalary month ulūlu (6b), all the
others had an addaru (12b).
Chronological equivalences between Babylonian and Gregorian dates of
this converter have been taken from the book of R. A. Parker and
W. H. Dubberstein, Babylonian Chronology, 626 B.C. - A.D. 45,
Studies in Ancient Oriental Civilization 24, 1942 and encoded by Bruno
Gombert (Université Paris 1 Panthéon Sorbonne, ARSCAN).
About MesoCalc
Can I download MesoCalc?
Yes, you can! You can download MesoCalc and use it offline
on your computer or smartphone.
Can I read MesoCalc's source code?
Yes, you can! MesoCalc is a free software. You can read the
source code of the present web page. You can even modify the
source code and then redistribute your own modifications
under the terms of the GNU General Public License (see
the license below).
You can see the source code of this program. It is written
in HTML, CSS and JavaScript.
Is MesoCalc's source code archived?
Yes, it is. Archiving software is crucial because
programmers are not aware that this part of scientific
culture disappears more easily than clay tablets. The
programs and source codes of the first decades of computer
science are already lost at the beginning of the
XXIth century.
This is the reason why MesoCalc'c source code is archived.
You can find its successive versions (since 2016)
on Github and
Software Heritage,
the universal software archive.
Are there new features in MesoCalc?
Yes, very frequently! Month after month, MesoCalc gets new
features and new bug corrections. In case you downloaded
MesoCalc to use it offline, make sure that you have the
newest version.
Last modifications:
- 10th February 2023: Software Heritage links.

- 16th May 2017: integration of Bruno
Gombert's calendar, designed after R. A. Parker and W. H. Dubberstein, Babylonian Chronology,
626 B.C.-A.D. 45, Studies in Ancient Oriental
Civilization 24, 1942.
- 7th May 2016: metric approximation of
measuring units (metres, litres, etc.). Bug correction in
system G (1 šar-gal = 64800, not 34800). Cleaning of the
HTML code. "Clear" buttons for all operations on measuring
units. Verification of the XHTML 1.0 Strict validity :
- 15th January 2016: two new features. 1)
Regular approximation of a number; 2) a new algorithm to
list regular numbers, made possible by Mathieu
Ossendrijver's huge database of regular numbers.
- 16th September 2015: bug correction in
measurement units. Now, input numbers can be either
decimal or sexagesimal (until recently, only sexagesimal
numbers were accepted).
License
MesoCalc: a Mesopotamian calculator.
Copyright (C) Baptiste
MÉLÈS 2013.
Latest version: 11th April 2024.
This program is free software: you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation, either version 3 of the
License, or any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See
the GNU General Public
License for more details.
Contact
If you see bugs or want new features,
please contact
Baptiste Mélès. All comments and suggestions are
welcome!