A Mesopotamian Calculator

MesoCalc is a Mesopotamian calculator. It computes in sexagesimal place-value notation, operates on measures and converts dates.

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.



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


Conversion and arithmetical properties


Conversion from decimal to sexagesimal (10A → 60A/F):

Conversion from sexagesimal to decimal (60A → 10A):

Arithmetical properties

Regular number? (60A/F → yes/no)

Regular approximation
of (60A/F) up to  sexagesimal digits:

Prime factors (60A → 60A/F):

Greatest common divisor (60A and 60A → 60A)

Multiplicative operations

Multiplication and quotient

Multiplication (60A/F):


Multiplication table (60A/F):

Quotient (60F÷60R → 60F):

Reciprocal and regular numbers

Reciprocal number (60F):


Table of regular numbers (60A/F and 60A/F)
between and
up to  sexagesimal digits:

Table of regular and reciprocal numbers
from to (60A/F)
up to  sexagesimal digits:

Exponential operations

Squares and square roots

Square (60A/F):

Square root (60A/F):



Cubes and cube roots

Cube (60A/F):

Cube root (60A/F):


Geometric progression

Geometric progression (a × bn):
a = 60A/F (first term)
b = 60A/F (common ratio)
n = 10A (number of terms)

Additive operations

Addition (60A):


Subtraction (60A):



Units of length:

danna ← 30 ← ← 60 ← ninda ← 12 ← kuš ← 30 ← šusi
10.8 km 360 m 6 m 50 cm 17 mm

Add lengths:

danna ninda kuš šusi

Subtract lengths:

danna ninda kuš šusi

Multiply a length:

danna ninda kuš šusi
× times


Units of surface:

gan ← 100 ← sar ← 60 ← gin ← 180 ← še
3600 m² 36 m² 0.6 m² 33 cm²
1 ninda × 1 ninda

Multiply lengths to get a surface:

danna ninda kuš šusi

Add surfaces:

gan sar gin še

Subtract surfaces:

gan sar gin še

Multiply a surface:

gan sar gin še
× times


Units of volume:

gan ← 100 ← sar ← 60 ← gin ← 180 ← še
1800 m³ 18 m³ 300 dm³ 1.66 dm³
1 (surface-)sar × 1 kuš

Multiply lengths to get a volume:

danna ninda kuš šusi

Multiply length and surface to get a volume:

danna ninda kuš šusi
× gan sar gin še

Add volumes:

gan sar gin še

Subtract volumes:

gan sar gin še

Multiply a volume:

gan sar gin še
× times


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

Add capacities:

gur bariga ban sila gin še

Subtract capacities:

gur bariga ban sila gin še

Multiply a capacity:

gur bariga ban sila gin še
× times


Units of weight:

gu ← 60 ← mana ← 60 ← gin ← 180 ← še
30 kg 500 g 8.33 g 0.046 g

Add weights:

gu mana gin še

Subtract weights:

gu mana gin še

Multiply a weight:

gu mana gin še
× times

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


Date conversion:
King Year Month Day

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! To download MesoCalc and use it offline on your computer or smartphone,

Can I read the 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).

If you want to see the HTML, CSS and JavaScript source code of this program, you will have to find on option like "View > Page Source" or "Tools > Source Code" in your web browser while displaying the present web page.

Is MesoCalc's source code archived?

Are there new features in MesoCalc?


MesoCalc: a Mesopotamian calculator.

Copyright (C) Baptiste MÉLÈS 2013.

Latest version: 19th May 2017.

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.