The Stepped Reckoner of Gottfried Leibniz

The great polymath Gottfried Leibniz (biography) was one of the first men (after Raymundus Lullus and Athanasius Kircher), who dreamed for a logical (thinking) device (see the article). Even more—Leibniz tried to combine principles of arithmetic with the principles of logic and imagined the computer as something more of a calculator—as a logical or thinking machine. He discovered also that computing processes can be done much easier with a binary number coding (in his treatises De progressione Dyadica, 1679 and Explication de l'Arithmetique Binaire, 1703). He even describes a calculating machine which works via the binary system: a machine without wheels or cylinders—just using balls, holes, sticks and canals for the transport of the balls!

Leibniz dreamed of inventing the general problem-solver, as well as a universal language—"I thought again about my early plan of a new language or writing-system of reason, which could serve as a communication tool for all different nations... If we had such an universal tool, we could discuss the problems of the metaphysical or the questions of ethics in the same way as the problems and questions of mathematics or geometry. That was my aim: Every misunderstanding should be nothing more than a miscalculation (...), easily corrected by the grammatical laws of that new language. Thus, in the case of a controversial discussion, two philosophers could sit down at a table and just calculating, like two mathematicians, they could say, 'Let us check it up ...'".

Certainly the impressive ideas and projects of Leibniz have to wait some centuries, to be fulfilled (the ideas of Leibniz will be used two and half centuries later by Norbert Wiener, founder of Cybernetics). So, let's ground and examine his famous stepped reckoner.

Leibniz got the idea of a calculating machine most probably in 1670. In 1672 he moved for some years to Paris, where he get access to the unpublished writings of the two greatest philosophers—Pascal and Descartes. Most probably in this year he became acquainted with the calculating machine of Pascal (Pascaline), which he decided to improve in order to be possible to make multiplication and division. At the beginning, he tried to use a mechanism, similar to Pascal's, but soon realized, that for multiplication and division it is necessary to create a completely new mechanism, which will make possible the multiplicand (dividend) to be entered once and then by a repeating action (rotating of a handle) to get the result. Trying to find a proper mechanical resolution of this task Leibniz made several projects, before to invent his famous stepped-drum mechanism. One of his projects (see the nearby sketch) describes something very similar to the pin-wheel mechanism, which will be reinvented in 1709 by Giovanni Poleni. Some of the historians even assume, that the pin-wheel was used in one of the Leibniz's calculating machines, but this cannot be proved.

Starting to create the first prototype, Leibniz soon faced the same obstacles that Pascal had experienced—poor workmanship, unable to create the fine mechanics, required for the machine. First wooden prototype of the stepped reckoner was ready soon and in the beginning of 1673, when Leibniz was sent to London with a diplomatic mission, he succeeded not only to met some english scientists and to present his treatise called The Theory of Concrete Motion, but also to demonstrate the prototype of his calculating machine to the Royal Society. This demonstration was not very successful, because the inventor admitted that the instrument wasn't good enough and promised to improve it after returning to Paris. Nevertheless, the impression of Leibniz must have been very positive, because he was elected as a member of Royal Society.

Back in Paris, Leibniz hired a skilful mechanician—local clockmaker Olivier, who was a fine craftsman, and he made the first metal prototype of the machine. (Olivier used to work for Leibniz as late as 1694. Later on professor Gabriel Wagner and the mechanic Levin from Helmstedt worked on the machine, and after 1715, the mathematician Gottfried Teuber and the mechanic Has in Leipzig did the same). In 1675 the second device was presented to the French Academy of Sciences and was highly appreciated by the most prominent members of the Academy—Antoine Arnauld and Christian Huygens. Leibniz was so pleased by his invention, that he immediately informed some of his correspondents: e.g. Thomas Burnett, 1st Laird of Kemnay—I managed to build such arithmetic machine, which is completely different of the machine of Pascal, as it allows multiplication and division of huge numbers to be done momentarily, without using of consecutive adding or subtraction, and to other correspondent, the philosopher Gabriel Wagner—I managed to finish my arithmetical device. Nobody had seen such a device, because it is extremely original.

One of the old machines of Leibniz

It is unknown how many machines were manufactured by order of Leibniz. It is known however, that the inventor spent on his machine a very large sum at the time—24000 talers according to some historians, so it is supposed the number of the machines to be at least 10. One of the machines was stored in an attic of a building of the University of Göttingen sometime late in the 1670s, where it was completely forgotten. It remained there, unknown, until 1879, when a work crew happened across it in a corner while attempting to fix a leak in the roof. At the present time exist two old machines, which probably are manufactured during Leibniz's lifetime (in the Hanover State Library and in the Deutsches Museum in München), and several replicas (see one of them in the photo below).

A replica of the Stepped Reckoner of Leibniz

The mechanism is 67 cm long, 27 wide and 17 cm high and is housed in a big oak case with dimensions 97/30/25 cm. Let's examine what is the principle of the stepped-drum (see the lower sketch).

The stepped-drum mechanism

The stepped-drum (marked with S in the sketch) is attached to a four-sided axis (M), which is a teeth-strip. This strip can be engaged with a gear-wheel (E), linked with the input disk (D), on which surface are inscribed digits from 0 to 9. When the operator rotates the input wheel and the digits are shown in the openings of the lid, then the stepped drum will be moved parallel with the axis of the 10-teehth wheel (F) of the main counter. When the drum rotated to a full revolution, with the wheel (F) will be engaged different number of teeth, according to the value of the movement, which is defined by the input disk and the wheel (F) will be rotated to the appropriate angle. Together with the wheel (F) will be rotated linked to it digital disk (R), whose digits can be seen in the window (P) of the lid. During the next revolution of the drum to the counter will be transferred again the same number.

Stepped Reckoner without the cover

The input mechanism of the machine is 8-positional, i.e. it has 8 stepped drums, so after the input of the number by means of input wheels, rotating the front handle (which is connected to the main wheel (called by Leibniz Magna Rota), all digital drums will make 1 revolution each, adding the digits to the appropriate counters of the digital positions. The output (result) mechanism is 12-positional. The result (digits inscribed on the digital drums) can be seen in the 12 small windows in the upper unmovable part of the machine.

One of the main flaws of the Stepped Reckoner is that tens carry mechanism is not fully automatic. Let's see why. In the next sketch are shown mechanisms of two adjacent digital positions. The stepped drums are marked with 6, the parts, which formed the tens carry mechanism, are marked with 10, 11, 12, 13 and 14.

The tens carry mechanism (© Aspray, W., Computing Before Computers)

When a carry must be done, the rod (7) will be engaged with the star-wheel (8) and will rotate the axis in a way, that the bigger star-wheel (11) will rotate the pinion (10). On the axis of this pinion is attached a rod (12), which will be rotated and will transfer the motion to the star-wheel (10) of the next digital position, and will increase his value with 1. So the carry was done. The transfer of the carry however will be stopped at this point, i.e. if the receiving wheel was at the 9 position, and during the carry it have gone to 0 and another carry must be done, this will not happen. There is a workaround however, because the pentagonal disks (14) are attached to the axis in such way, that theirs upper sides are horizontal, when the carry has been done, and with the edge upwards, when the carry has not been done (which is the case with the right disk in the sketch). If the upper side of the pentagonal disk is horizontal, it cannot be seen over the surface of the lid, and cannot be noticed by the operator, so manual carry is not needed. I however the edge can be seen over the surface of the lid, this will mean that the operator must rotate manually this disk, performing a manual carry.

The mechanism of the machine can be divided to 2 parts. The upper part is unmovable and was called by Leibniz Pars immobilis. The lower part is movable and is called Pars mobilis (see the sketch below).

An outside sketch

In the Pars mobilis is placed the 8-positional setting mechanism with stepped drums, which can be moved leftwards and rightwards, so to be engaged with different positions of the 12-positional unmovable calculating mechanism. Adding with the machine is simple—first addend is entered directly in the result wheels (windows) (there is a mechanism for zero setting and entering numbers in the result wheels), second addend is entered with the input wheels in the Pars mobilis, and then the forward handle (Magna rota) is rotated once. Subtraction can be made in a similar way, but all readings must be taken from the red subtractive digits of the result wheels, rather than the normal black additive digits. On multiplication, the multiplicand is entered by means of the input wheels in the Pars mobilis, then Magna Rota must be rotated to so many revolutions, which number depends on the appropriate digit of the multiplier. If the multiplier is multidigital, then Pars mobilis must shifted leftwards with the aid of a crank and this action to be repeated, until all digits of the multiplier will be entered. Division is done by setting the dividend in the result windows and the divisor on the setup dials, then a turn of Magna rota is performed and the quotient may be read from the central plate of the large dial.

Upper view sketch of Stepped Reckoner

There is also a counter for number of revolutions, placed in the lower part of the machine, which is necessary on multiplication and division—the large dial to the right of the small setting dials. This large dial consists of two wide rings and a central plate—the central plate and outer ring arc inscribed with digits, while the inner ring is colored black and perforated with ten holes. If for example we want to multiply a number on the setting mechanism to 358, a pin is inserted into hole 8 of the black ring and the crank is turned, this turns the black ring, until the pin strikes against a fixed stop between 0 and 9 positions. The result of the multiplication by 8 may then be seen in the windows. The next step requires that the setting mechanism to be shifted by one place by means of the crank (marked with K in the upper figure), the pin inserted into hole 5, and the crank turned, whereupon the multiplication by 58 is completed and may be read from the windows. Again the setting mechanism must be shifted by one place, the multiplication by 3 is carried out in the same manner, and now the result of the multiplication by 358 appears in the windows.