A Mesopotamian Calculator

MesoCalc is a Mesopotamian calculator. It computes with integers in sexagesimal place-value notation and operates on measures.

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 also includes computations made by Mathieu OSSENDRIJVER (Humboldt-Universität), with his 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
11 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 1.7 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 17 mL 0.1 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 g 0.05 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
34800 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

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 of MesoCalc?

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.

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:

  • 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).


MesoCalc: a Mesopotamian calculator.

Copyright (C) Baptiste MÉLÈS 2013.

Latest version: 16th January 2016.

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.