The History of Calculating Machines  (page 1)

Since the earliest times, scientists, engineers, mathematicians, accountants and tax collectors have needed to carry out mathematical calculations as part of their job. While mental arithmetic, fingers, paper and pencil or papyrus and quills can be used for simple addition, subtraction, multiplication and division, there soon becomes a need to use some form of calculating machine to solve more complex problems involving larger numbers. Like most things when someone invents a machine it is not long before someone else improves on the design, making it difficult to identify all the different types of calculating machines. Below and on page 2 is a list of some of the main inventions created to simplify the process of mathematical calculation.

The earliest recorded calculating machine was the abacus or counting frame.  Abacus is derived from the Ancient Greek “Abax” which means counting board containing grooves for counters. The records indicate that there were many different types of abacus including Mesopotamian, Egyptian, Persian, Greek, Roman, Chinese, Indian, Japanese, Korean, Native American and Russian. The earliest recorded was a dust abacus around 2400 Before Common Era (BCE) used by the Babylonians. This was a stone slab covered with dust or sand, with letters and numbers drawn in the sand. Eventually addition and subtraction calculation were aided with small pebbles.

Salaminische Tafel Salamis Tablet nach Wilhelm Kubitschek Numismatische Zeitschrift Bd 31 Wien 1899 p. 394 ff
Salamis Tablet

Archaeological evidence was found on the Greek island of Salamis that dated from 300 BCE. This was a counting board known as the Salamis Tablet made from marble. However the calculating machine most people would recognise as an abacus is the Saunpan of Chinese origin, which dates from the 2nd century BCE. Saunpan means calculating tray.


How to use a 'Saunpan' abacus by Pelle Lindblå










How to use a Roman abacus by Dasty Jones

The Roman abacus was similar to the Salamis tablet by moving pebbles on a board. By the 1st century BCE the board was covered with a thin layer of wax onto which the columns and figures were inscribed using a stylus. 



Hellenic Republic Ministry of Culture and Tourism






The Antikythera mechanism considered by many to be the world’s first computer was discovered in 1902 by archaeologist’s in a shipwreck off the coast of the island of Antikythera which lies in the straits between the Greek island of Peloponnese and Crete. The clockwork mechanism was probably used in Greece somewhere between 150 and 60 BCE and is thought to have been used to calculate the lunar and solar cycles. Reconstruction of the mechanism it was based on the geocentric model of the Solar System in which the Earth is in the centre with the Sun and the Moon orbiting the Earth.






Gregor Reisch - Margarita Philosophica - Arithmetica
Medieval Counting Table

The medieval counting table or Exchequer table was used to calculate taxes and goods. Unfortunately, there are very few examples of counting tables survive. One exists in the Museum of Notre Dame in Strasburg. In the UK civil service “Her Majesty’s Exchequer”, the central government accounting process takes its name from the Medieval Exchequer table.




image of john napier - scottish mathematician
John Napier (1550 - 1617)


It was the Scottish mathematician John Napier (1550 – 1617), accredited with inventing Napierian logarithms in 1614 who also invented a calculating device known as Napier’s bones. These were also sometimes known as Napier’s rods and consisted of flat rods that could be used to perform multiplication or division of any number between 2 and 9.  He described his work on rods In his book “Rabdologia” published in 1617.

Rabdology Cover Page
Front cover of Rbdologia


How to use Napier's bones











A contemporary of John Napier was Edmund Gunter (1581 – 1626) an English mathematician and clergyman. Gunther created the forerunner to the slide rule using a rule marked with various scales including one or more logarithmic scales and a pair of dividers he was able to carry out mathematical calculations including solving navigational problems.

The Gunter Scale - Image source: "The Museum of HP Calculators"

It was another English mathematician and clergyman, William Oughtred (1575-1660) who replaced the need for a pair of dividers used by Gunter with a second rule. Using the second rule and sliding it past the first, he was able to achieve the same result as Gunter had with a pair of dividers. Oughtred had invented the slide rule that continued to be used by engineers until the 1970s.

modern slide rule

A 6" slide rule in use about circa. 1970 

The basic slide rules are all similar to the image on the left. They consist of three scales. Two of the scales  are fixed together with the third scale on a sliding bar sandwiched between the fixed scales. By aligning the sliding scale the correct answer could be read from the fixed scale. A perspex sliding window with a curser makes reading the results easier. The Museum of HP Calculators page provides a good description of how a modern slide works. The same website also has details of circular and cylindrical slide rules. Many slide rules were designed for specific tasks  including designs specifically for carrying out hydraulic engineering calculations such as hydraulic force, fluid pressure, piston speed, flow, stroke time and volume and fluid speed. I have included a photograph of two of these hydraulic slide rules on my page about Maths Tools.



Continued on page 2



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Web page last updated  24 February 2019

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