### The *Stepped Reckoner* of Gottfried Leibniz

The great polymath Gottfried Leibniz (see biography of Leibniz) was one of the first men (after Raymundus Lullus and Athanasius Kircher), who dreamed for a logical (thinking) device (see The Dreamer Leibniz). 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*, March, 1679, and *Explication de l'Arithmetique Binaire*, 1703).

In the *De progressione Dyadica* Leibniz 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—*This [binary] calculus could be implemented by a machine (without wheels)... provided with holes in such a way that they can be opened and closed. They are to be open at those places that correspond to a 1 and remain closed at those that correspond to a 0. Through the opened gates small cubes or marbles are to fall into tracks, through the others nothing. It [the gate array] is to be shifted from column to column as required...*!

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 had to wait some centuries, to be fulfilled (the ideas of Leibniz will be used two and half centuries later by Norbert Wiener, the 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 or 1671, seeing a pedometer device. The breakthrough happened however in 1672, when he moved for several years to Paris, where he got access to the unpublished writings of the two greatest philosophers—Pascal and Descartes. Most probably in this year he became acquainted (reading Pascal's *Pensees*) with the calculating machine of Pascal (Pascaline), which he decided to improve in order to be possible to make not only addition and subtraction, but also multiplication and division.

A sketch of Leibniz from 1672

At the beginning, Leibniz 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.

The upper sketch is from a manuscript from 1672 and shows probably one of the first Leibniz designs for the calculating machine. There is an input mechanism, the lower circles, inscribed *Rota multiplicantes*, where must be entered the multiplier; there is a calculating mechanism, inscribed *Rota multiplicanda*, where must be entered the multiplicand; and there is a result mechanism, the top circles, inscribed *Rota additionis*, where can be seen the result of multiplication. The movement from the input wheels to the calculating wheels is transferred by means of chains. From this sketch is not clear however what is the structure of the calculating mechanism, most probably it is neither based on the *stepped drum*, nor based on the *pin-wheel*, because the chain is not needed in these cases.

There is a sketch (see the nearby sketch) from another Leibniz's manuscript (most probably a later one), which throw light on his initial idea for the calculating mechanism. This sketch describes something very similar to the pin-wheel mechanism, which will be reinvented in 1709 by Giovanni Poleni, and improved later by Braun, Baldwin and Odhner. The undated sketch is inscribed "Dens mobile d'une roue de Multiplication" (the moving teeth of a multiplier wheel).

Most probably the prototype and first designs of the calculator were based on one of the above-mentioned calculating mechanisms, before the development of the *stepped drum* mechanism, which was successfully implemented into the survived to our time devices (the machine was under continuous development more than 40 years and several copies were manufactured).

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. He complained: "If only a craftsman could execute the instrument as I had thought the model."

The first wooden 2-digital prototype of the *Stepped Reckoner* (this is a later name, actually Leibniz called his machine *Instrumentum Arithmeticum*), was ready soon and in the end of 1672 and beginning of 1673 it was demonstrated to some of his colleagues at French Academy of Sciences, as well as to the Minister of Finances Jean-Baptiste Colbert.

In January 1673 Leibniz was sent to London with a diplomatic mission, where 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 on 1st of February, 1673. The demonstration was probably 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 had been very positive, because he was elected as a member of Royal Society in April, 1673. It is known also, that dirung his trip to London, Leibniz met Samuel Morland and saw his arithmetic engine.

In a letter of 26th of March 1673 to one of his correspondents—Johann Friedrich, mentioning the presentation in London, Leibniz described the purpose of the *arithmetic machine* as making calculations *easy, fast, and reliable*. Leibniz also added that theoretically the numbers calculated might be as large as desired, if the size of the machine was adjusted: *a number consisting of a series of figures, as long as it may be (in proportion to the size of the machine)*.

Back in Paris, Leibniz hired a skilful mechanician—the local clockmaker Olivier, who was a fine craftsman, and he made the first metal (brass) prototype of the machine. It seems the first working properly device was ready as late as in 1685 and didn't manage to survive to the present day, as well as the second device, made 1686-1694. (Olivier used to work for Leibniz up to 1694. Later on professor Rudolf 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 machine 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*.

In 1676 Leibniz demonstrated the new machine again to the Royal Society in London. Let's clarify however, that this was a small device with several digital positions only. The full-scale workable machine will be ready as late as in 1694.

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 great scientist was interested in this invention all his life and that he spent on his machine a very large sum at the time—some 24000 talers according to some historians, so it is supposed the number of the machines to be at least 10. One of the machines (probably third manufactured device), produced 1690-1720, was stored in an attic of a building of the University of Göttingen sometime late in the 1770s, 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. In 1894-1896 Arthur Burkhardt restored it, and it has been kept at the Niedersächsische Landesbibliothek for some time. 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 of the machine 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 (based on the drawing from *Theatrum arithmetico-geometricum* of Leupold)

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.

Leibniz did manage to create a machine, much better than the machine of Pascal. The *Stepped Reckoner* was not only suitable for multiplication and division, but also much easier to operate. In 1675 during the demonstration of the machine to the French Academy of Sciences, one of the scientists noticed that "...using the machine of Leibniz even a boy can perform the most complicate calculations!"

The first description of Leibniz's stepped-drum calculator appeared in 1710, made by Leibniz himself, in *Miscellanea Berolinensia*, the journal of the Berlin Academy of Sciences. It was a 3-pages short description (see the images bellow), entitled "Brevis descriptio Machinae Arithmeticae, cum Figura", and the internal mechanism of the machine is not described.

Brevis descriptio Machinae Arithmeticae, cum Figura