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The Actuary The magazine of the Institute & Faculty of Actuaries

The arithmometer

Robin Michaelson has generously given the Worshipful Company of Actuaries his arithmometer. This fine early calculating machine is currently on display in the Council Chamber at the Institute.
Aids to calculation go back at least to the 17th century. John Napier, for example, invented not only logarithms but also some simple calculating devices, the best known of which is Napier’s Bones. In 1642, Pascal completed the first known metal calculating machine, which did addition and subtraction. Samuel Moreland, later in the century, invented a simple multiplying machine, though it still necessitated the use of mental addition for part of the process. In 1694, Leibniz completed a machine which carried out the full operation of multiplication mechanically. There were also one or two attempts at a calculating machine in the 18th century, for example, by Muller in 1783. However, with the exception of Napier’s Bones, these devices did not catch on in a widespread way probably because the technologies of the day did not allow the manufacture of sufficiently reliable machines.

Charles Xavier Thomas
It was not until 1820 that the first practical calculating machine appeared. This was the arithmometer, invented by Charles Xavier Thomas of Colmar in Alsace, who lived from 1785 to 1870. It is a later version of this machine which will be displayed at Staple Inn. The arithmometer could cope with addition, subtraction, multiplication and division. Instead of a keyboard, the machine had a series of slides for entering numbers. For addition, the turn of a handle added these numbers into the row of ‘total’ dials shown on a moving carriage on top of the machine. A reversing switch enabled the machine to perform subtraction while turning the handle in the same direction. Multiplication was performed by repeated addition and division by repeated subtraction. The mechanical means, by which digits which had been ‘set’ into the machine could be added to the number in the ‘total’ dials, was based on the same ‘stepped reckoner’ Leibniz had used in 1694.
Thomas’s machine was not an instant success. It was remodelled numerous times, presumably to increase its reliability and reduce its heavy cost. Early versions did not have a handle instead there was a silk ribbon which had to be pulled outwards to start the machine!
Thomas promoted his machine via various exhibitions. Starting in 1844 with the French national exhibition of industrial products, the arithmometer was granted an honourable mention, but was considered inferior to a rival machine. Five years later, Thomas tried again. This time he was awarded a silver medal and the jury report devoted three pages to his machine. The gold medal, however, went to a machine by Maurel and Jayet. In the 1851 Great Exhibition in London, once again the arithmometer received a prize medal, but was judged inferior to a Russian entry.
Thomas then embarked on an ambitious marketing campaign for his machine, in which he gave examples to numerous royal families in Europe. As a result, he received many decorations and endorsements that he was able to use in his promotional material. Then came the Paris exhibition of 1855, for which Thomas created a special giant machine which looked like a piano some six feet long, complete with 15 setting sliders and 30 result dials. Despite this immense effort, the machine once again failed to achieve first place, this honour going to the Scheutz difference engine, which calculated and printed mathematical tables.

Extending the scope
The civil engineer, GA Hirn was one of the arithmometer’s enthusiastic advocates. He reported that observers of the machine often doubted its capabilites and he wrote a paper giving advanced techniques to extend the machine’s role beyond the four basic mathematical operations.
Thomas was the general director of the Soleil insurance company in Paris and in the 1850s the arithmometer could be obtained only from the company’s offices. By the 1860s, however, agents were being appointed outside France and from a technical viewpoint the machine had reached the end of its major development phase. Early satisfied purchasers included the Preussische statistische Buro, which bought two machines in 1864 and 1865. Further pairs of machines were added in 1878 and 1879. A promotional leaflet of about 1872, when international sales were still less than a hundred a year, prints a list of the machine’s users in England. This included 33 corporate bodies such as government departments, colleges, observatories, and insurance companies. There were also 40 individuals and firms representing engineers, bankers, actuaries, doctors and chemists. The list also included the engineer, Henry Brunel, son of Isambard Kingdom Brunel, who wrote in 1866 that his ‘new toy’, price £12, did all the common operations of arithmetic in the twinkling of an eye.
The arithmometer was particularly useful in insurance. Actuaries may have been the first professional group to adopt it. From the 1860s there was a steady stream of articles showing how the machine could be used in the construction of actuarial tables. A letter in JIA 12 of 1865 indicates that Major-General Hannyngton was experimenting with the use of the arithmometer in actuarial work, when he wrote; ‘I am convinced that the days of hard work in the actuary’s craft are coming to an end’. He followed this up by a paper to the Institute in 1873 (JIA 16), fully describing the use of the machine for calculating actuarial functions. During the actuarial valuation of the Prudential Insurance Company in 1877, they had 24 arithmometers at work in valuing the 2.5m policies then in force. Unfortunately, the machine was not always reliable in such heavy use. Springs would sometimes snap and mechanical defects would develop from time to time.
In 1892, the Council of the Institute decided to buy an arithmometer (costing £36) as a gift to Peter Gray, for his ‘very valuable services in connection with the Mortality Experience Investigation’.

Calculating the age
The age of Robin Michaelson’s arithmometer is uncertain, but it is accompanied by an instruction booklet written in French and published in Paris in 1884, so the machine may date from around then. The booklet bears the stamp of GF Redfern of 4 South Street, Finsbury, London, who is described as the sole agent for the arithmometer in England, America, and the Colonies. The machine is in first-class original condition and in full working order, but its history is obscure. Robin Michaelson purchased it in the UK and it came with a photocopy of a page from a minute book for an unspecified British financial institution dealing with property investments. The relevant minute reads:
‘Calculating Machines.
The Secretary read a correspondence which had passed between Messrs Elliott Bros and himself about the £300 award in respect of the Calculating Machines and it was
‘That a sum of £200 be awarded to Messrs Elliott Bros and the sum of £100 to Mr Brooke upon his completing the Machines he had in hand.’
Whatever this particular machine’s history, it is a forceful reminder of a time when the actuarial profession and the expanding insurance industry were starting to get to grips with much greater volumes of calculations than had been necessary in the past. The arithmometer was eventually superseded by more modern hand-calculating machines of the Brunswiga type, which came into existence from about 1910 onwards. These, in turn, were replaced by electro-calculating machines, and then eventually by electronic desk calculators. The arithmometer was therefore a true pioneer and it is pleasing that there is now an opportunity for modern actuaries to inspect this fine relic of the Victorian age.

Much of the information in this article was extracted from a longer article by Stephen Johnston, ‘Making the Arithmometer Count’ printed in the Bulletin of the Scientific Instrument Society, No 52 (1997), 1221. Other works consulted were:
Lewin, CG; Evans, JV; Goodare, KJ; and Packer, LR. ‘Calculating Devices and Actuarial Work’, JIA, CXVI (1989), 215287.
Hannyngton, JC. Correspondence ‘On the Adaptation of Assurance Formulae to the Arithmometer of M Thomas’. JIA 12 (186466), 184.
Hannyngton, JC. ‘On the Use of M Thomas de Colmar’s Arithmometer in Actuarial and other Computations’, JIA 16 (187072), 244253.
Hancock, WJ. ‘Observations on the use of the Arithmometer’, JIA 16 (187072), 265269.
Gray, Peter. ‘On the Arithmometer of M Thomas (de Colmar) and its application to the construction of Life Contingency Tables’, JIA 17, 249266;
JIA 18 (187375), 2032 and 12332.
Carment, D. ‘On the Application of the Arithmometer to the Construction of Tables of the Values of Endowment Assurance Policies’, JIA 22 (187981), 368380.


Have you taken into consideration that J.C.Hannyngton a fellow of the Institute of Actuaries had also invented an ingenious slide-rule himself (in the Science Museum) known for it’s accuracy. Nominated by fellow actuaries to The Royal Astronimical Soc. Being a great Mathamatician himself he was in a position to be able to correct Babbage’s work. You state that Hannyngton was experimenting with the use of the Arithmometor in actuarial work when he wrote ’I am convinced that the days of hard work in the Actuaries craft are coming to an end in 1873 fully discribing the use of the machine for calculating Acturial functions. (Something he would have recognised and appreciated from his own busy past in India in which he was responsible for the whole army’s payrole, widows and orphans fund for the whole of Bengal and many other things which he was involved in). A true gentleman of the time he did not promote himself and for this reason has largely gone unrecognised until the last couple of decades. My detailed notes of his work are not available at present but he should be remembered as he was the mathematician of his day.

P. Thomas
8 January 2010