The calculating machines of Johann Helfrich Müller

The german engineer and master builder—Johann Helfrich Müller (1746-1830) is a very interesting figure in the world of mechanic calculators, not so for his small calculator, an improved version of the machine of Philipp Hahn, which he created, but for his plans to build a difference engine almost 40 years before Babbage. Who was Johann Helfrich Müller?

He was born in Cleve in Germany, on 16th of January, 1746. After receiving his initial education at little private schools, he was sent to study at the Grammar School in Darmstadt. In the Spring of 1762, he became a cadet in the Artillery Corps at Giessen, where his family had moved, and began to attend lectures in mathematics and physics at the university. His father, Lorenz Friedrich, who was in the building trade, wanted his son to carry on the family tradition, but Johann appears to have found certain other careers equally tempting. While still at the grammar school, he wanted to become a painter; at the university he hoped to be a professor of mathematics and physics. Soon however he discovered that the profession of an engineer was the most interesting of all and he devoted much time to reading books on statistics, hydraulics and mechanics.

In 1769, the Artillery Corps was reduced in strength and Müller was forced to look for work elsewhere. Shortly afterwards, he was employed as an engineer by Prince Georg Wilhelm, the Governor in Giessen, and a man with wide interests in both civil and military engineering and architecture. In April 1772, Prince Wilhelm took Müller with him on a five month trip to Paris. There they visited and studied many of the marvelous buildings and machines in the city and at St. Cloud, Versailles and Marly. On his return from the trip, Müller devoted most of his time to architectural and mechanical design and various financial calculations for his master. In February 1774, Müller became building inspector in Darmstadt and three years later master builder with the responsibility for designing and maintaining buildings. He also was expected to act as an adviser in case of emergencies such as floods.

Müller was a creative mind and started to make inventions from the beginning of 1770s. First he created a theatre, equipped with optical and mechanical effects for the children of the Prince, later on he designed a large and powerful burning mirror.

In the beginning of 1780s Müller was asked by the local superintendent's office to check and recalculate some tables relating to the volumes of trees. To shorten this task he had come up with the idea of a machine for the purpose. Soon he realized however that with a few small changes, he could get the machine to perform subtraction, division and multiplication as well. At this stage, he came across an article about the calculating machine of his compatriot, Philipp Hahn. After about three months of designing, in June, 1782, the project was ready and he gave the drawings to a clockmaker in Darmstadt, with the order to make the machine in metal. The work was taken over by a pair of journeymen in the same trade and on the 20th of June 1784, the machine was ready. On 24th of June, 1784, the 14-digit calculator was demonstrated in the Göttingen Academy of Sciences.

A drawing of the calculating machine of Müller from 1784

Müller's calculating machine is very similar to the machine of Hahn and was based on the stepped rum of Leibniz, but it is larger (285 mm diameter, 95 mm height, weight 15,4 kg). It was in the form of a round box with a handle placed centrally and the number wheels concentrically arranged around the handle. It could calculate with 14 figures and its number and gear wheels could be altered to enable it to operate with non-decimal number systems.
Compared with Hanh's machine, Müller's had 3 important improvements.
1. The axes of stepped drums from the Hahn's machine, which are used for entering of the numbers, are not anymore set by pulling upwards (which requires a great precision), but by means of rotating dials, with inscribed digits from 0 to 9 over its periphery.
2. The axes, along with mounted on them pinion-wheels, can be easily changed with pinion-wheels with different number of teeth, which provides a possibility of calculations in different numeral systems.
3. In the mechanism of the machine is included a bell, which rings in case of overflow during a adding or negative result during a subtraction (if the operator tried to subtract a larger number from a smaller one).

A replica of the calculating machine of Müller from 1784

While still working on his machine, on 22nd of May 1783, Müller wrote a letter to his friend Georg Christoph Lichtenberg, in which he described his invention and set forth his plans. He intended to make a profit from the machine and informed Lichtenberg about his intention to make another copy of it, as soon as the first one was ready. He also wanted to build a couple of simpler calculating machines for addition and subtraction only and hope to sell his machines in England. Later on however the Grand Duke of Hesse-Darmstadt Ludwig I, bought the first machine for 4000 guilders, and incorporated it in his collection of scientific instruments.

Müller intended also to use his machine for calculating of tables. He wrote "How easy it would be, by this means, to correct and extend the tables of logarithms". Later on Müller in fact used the machine to calculate a set of tables—"Tafeln des Kubischen Gehalts des Bauholzes", which was published in Frankfurt in 1788. In a other letter to Lichtenberg from 9th of September 1784, Müller recorded a new thought in a postscript for a printing tabulating machine.
"P.S. If the calculating machine sells well, I would in the future make a machine, which would simultaneously print in printer's ink on paper any arbitrary arithmetical progression in natural numbers or the units, together with the numbers of the terms and the lines in between and which would halt of its own accord, when the side of the paper was full up. After setting the first figures, all one has to do is to turn a handle and after stopping to turn over the paper or to put another sheet in its place. In this way, a sequence of 60 terms can be delivered in a minute."

Certainly Müller never built a machine of this kind. Instead the idea was further perfected and presented together with his universal calculating machine in a book published in Frankfurt and Mainz in 1786. This 50 page booklet was edited by Müller's friend Philipp Engel Klipstein. It contained, besides an exhaustive description of the mode of operation and design of the constructed machine, also a discussion about its advantages over manual computation and a detailed account of the special security (correction) mechanisms incorporated.
Finally, under the heading "Further inventions of superior calculating machines and an arithmetical printing machine" the new machine was presented. In these passages, it is made clear that Müller, perhaps at an even earlier date, had invented a difference engine for the rapid production of error-free tables, by means of "whole series of numbers" and by using "difference-numbers" (Differenzzahlen). A description similar to that in the postscript to Lichtenberg is given of the printing part, which was supposed to print directly onto paper, rather than produce matrices for stereotypes. No mention is made of how the necessary mechanisms to accomplish this were to be designed. Nonetheless it is clear from this document that Johann Müller had conceived the idea of a difference engine by the year 1786. Such a machine will be proposed almost 40 years later by Charles Babbage (see the differential engine of Babbage).