Gottfried Leibniz

Genius is one percent inspiration and ninety-nine percent perspiration.
Thomas Alva Edison

Gottfried Wilhelm (von) Leibniz (1646-1716), portrait from Johann Friedrich Wentzel (circa 1700)

The great polymath Gottfried Leibniz and Charles Babbage are (to my mind) the two greatest persons in the history of computers and computing because only they managed to anticipate events in this area for centuries. In the second half of the 17th century, Leibniz not only created the first mechanical calculator, suitable for addition, subtraction, and multiplication but also dreamed about the logical machine and binary calculator.

Leibniz’ Instrumentum Arithmeticum

Leibniz got the idea of a calculating machine at the end of the 1660s, seeing a pedometer device. The first mention of his Instrumentum Arithmeticum is from 1670, as the breakthrough happened in 1672 when he moved for several years to Paris, where he got access to the unpublished writings of the two greatest philosophers—Pascal and Descartes.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 the same year, he became acquainted (reading Pascal’s Pensees) with Pascaline, which he decided to improve in order to be possible to make not only addition and subtraction but also multiplication and division.

At the very beginning, Leibniz tried to devise a mechanism, similar to Pascaline, but soon realized, that for multiplication and division, it is necessary to create a completely new mechanism, which will make it possible for the multiplicand (dividend) to be entered once and then by a repeating action (e.g. rotating of a handle) to get the result. Trying to find a proper mechanical resolution to this task Leibniz made several projects before inventing his famous stepped-drum mechanism (called also Leibniz gear).

A sketch is from a Leibniz's manuscript from 1672
A sketch is from Leibniz’s manuscript from 1685

The upper sketch is from Leibniz’s manuscript from 1685 (the full text is given below) and shows probably an early design 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 one can see the result of multiplication. The movement from the input wheels to the calculating wheels is transferred by means of chains. The calculating mechanism seems based on pin-wheel, not on the stepped drum.

Leibniz pin wheel, circa 1670
Leibniz pinwheel, circa 1670

Something like a pin-wheel mechanism is described also in a sketch (see the upper drawing) from another Leibniz’s manuscript (written around 1670), which throws light on his initial idea for the calculating mechanism. The un-dated sketch is inscribed “Dens mobile d’une roue de Multiplication” (the moving teeth of a multiplier wheel). Interestingly, Leibniz’s pin-wheel mechanism was reinvented in 1709 by Giovanni Poleni, and improved later by Braun, Roth, and Staffel.

Obviously, the prototype and first designs of the calculator were based on one of the above-mentioned pin-wheel 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 for 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 at the end of 1672 and beginning of 1673, it was demonstrated to some of his colleagues at the French Academy of Sciences, as well as to the Minister of Finances Jean-Baptiste Colbert.

In January 1673 Leibniz was sent to London on a diplomatic mission, where he succeeded not only in meeting some English scientists and presenting his treatise called The Theory of Concrete Motion but also in demonstrating the prototype of his calculating machine to the Royal Society on 1 February 1673. Leibniz was recommended by Huygens, who called his machine a promising project in a letter to Henry Oldenburg, the secretary of the Royal Society. Oldenburg knew Leibniz as a friend of Boineburg and fellow countryman and was committed to helping Leibniz, who expected to make a splash in London with his calculating machine.

During the demonstration, Leibniz stated, that his arithmetic tool was invented for the purpose of mechanically performing all arithmetic operations reliably and quickly, especially multiplication. Leibniz explained it very well, but the demonstration was obviously 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 the Royal Society in April 1673. It is known also, that during his trip to London, Leibniz met Samuel Morland and saw his arithmetic engine.

Particularly unimpressed by the demonstration was the famous scientist and ingenious inventor Robert Hooke, who was the star of the Royal Society at the time when Leibniz came to show his machine. Hooke was infamous for engaging in brutal disputes (not always within the boundaries of fair debate) with his rivals, like Huygens and Newton. After looking carefully at all sides of the machine, and examining it in detail during the demonstration on 1 February 1673, Hooke expressed a desire to take it apart completely to examine its insides. Moreover, several days after the demonstration, Hooke attacked him in public, making derogatory comments about the machine and promising to construct his own superior and better-working calculating machine, which he would present to the Royal Society. Hooke stated that it seems to me so complicated with wheels, pinions, cantrights, springs, screws, stops, and truckles, that I could not perceive it ever to be of any great use… It could be only fit for great persons to purchase, and for great force to remove and manage, and for great wits to understand and comprehend. In contrast, Hooke announced that I have an instrument now making, which will perform the same effects (and) will not have a tenth part of the number of parts, and not take up a twentieth part of the room. Leibniz was not in London at that time to defend himself and had to hear about the attack from Oldenburg, who assured him that Hooke was quarrelsome and cantankerous, and urged him that the best course of action will be to finish his machine as quickly as possible.

In a letter of 26 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 skillful mechanic—the local clock-maker 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 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 Christian 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 from 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 another correspondent, the philosopher Gabriel Wagner—I managed to finish my arithmetical device. Nobody had seen such a device until now, because it is extremely original.

One of the old machines of Leibniz
One of the old machines of Leibniz

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 1694.

It is unknown how many machines were manufactured by the 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 the 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 in Glashütte, and it has been kept at the Niedersächsische Landesbibliothek for some time.

A replica of the Stepped Reckoner of Leibniz form 1923 (original is in the Hannover Landesbibliothek)
A replica of the Stepped Reckoner from 1923 (the original is in the Hannover Landesbibliothek)

At present time exist two old machines, which probably are manufactured during Leibniz’s lifetime (around 1700) (in the Hannover Landesbibliothek and the Deutsches Museum in München), and several replicas (see one of them in the upper photo).

The stepped-drum mechanism
The stepped-drum mechanism

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 nearby sketch).

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-teeth wheel (F) of the main counter. When the drum is rotated to a full revolution, with the wheel (F) will be engaged a 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 the 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 (image from Leporello album on investigation in Glashütte, 1894)
Stepped Reckoner without the cover (image from Leporello album on investigation in Glashütte, 1894)

The input mechanism of the machine is 8-positional, i.e. it has 8 stepped drums, so after the input of the number using 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 the tens carry mechanism is not fully automatic (at least this of the survived until now machine). Let’s see why. In the next sketch are shown mechanisms of two adjacent digital positions. The stepped drums are marked with 6, and the parts, that formed the tens carry mechanism, are marked with 10, 11, 12, 13, and 14.

The tens carry mechanism (© Aspray, W., Computing Before Computers)
The tens carry mechanism (© Aspray, 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 its value by 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 has gone to 0 and another carry must be done, that will not happen. There is a workaround however because the pentagonal disks (14) are attached to the axis in such a way, that their 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. If however the edge can be seen over the surface of the lid, this will mean that the operator must rotate this disk, performing a manual carry.

It seems the problem with the tens carry mechanism was resolved by the German scientist Christian Gottlieb Kratzenstein, who in April 1765, presented to the Russian Academy of Sciences in Saint Petersburg with a perfected version of the calculating machine of Leibniz. Kratzenstein claimed that his machine solved the Leibniz machine’s problem with calculations over four digits and perfected the error that the machine is “prone to error when it is necessary to move a number from 9999 to 10000”, but the machine was not developed further.

The mechanism of the machine can be divided into two 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)
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 as to be engaged with different positions of the 12-positional unmovable calculating mechanism. Adding with the machine is simple—the first addend is entered directly in the result wheels (windows) (there is a mechanism for zero setting and entering numbers in the result wheels), the 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 similarly, 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 employing 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 multi-digital, then Pars mobilis must be shifted leftwards with the aid of a crank and this action is to be repeated, until all digits of the multiplier are entered. The 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
Upper view sketch of Stepped Reckoner

There is also a counter for the number of revolutions, placed in the lower part of the machine, which is necessary for 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 be shifted by one place employing 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.

In 1685 Leibniz wrote a manuscript, describing his machine—Machina arithmetica in qua non aditio tantum et subtractio sed et multiplicatio nullo, divisio vero paene nullo animi labore peragantur. As the 1685 design was based on wheels with a variable number of teeth, not on a stepped drum, obviously survived to our time devices are later work. In English, the manuscript sounds like this:
When, several years ago, I saw for the first time an Instrument that, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only counting but also addition and subtraction, multiplication and division could be accomplished by a suitably arranged machine easily, promptly, and with sure results.
The calculating box of Pascal was not known to me at that time. I believe it has not gained sufficient publicity. When I noticed, however, the mere name of a calculating machine in the preface of his “posthumous thoughts” (his arithmetical triangle I saw first in Paris) I immediately inquired about it in a letter to a Parisian friend. When I learned from him that such a machine exists I requested the most distinguished Carcavius by letter to give me an explanation of the work which it is capable of performing.
He replied that addition and subtraction are accomplished by it directly, the other (operations) in a round-about way by repeating additions and subtractions and performing still another calculation. I wrote back that I ventured to promise something more, namely, that multiplication could be performed by the machine as well as addition, and with the greatest speed and accuracy.
He replied that this would be desirable and encouraged me to present my plans before the illustrious King’s Academy of that place.
In the first place, it should be understood that there are two parts of the machine, one designed for addition (subtraction) and the other for multiplication (division), and that they should fit together.
The adding (subtracting) machine coincides completely with the calculating box of Pascal. Something, however, must be added for the sake of multiplication so that several and even all the wheels of addition could rotate without disturbing each other, and nevertheless, any one of them should precede the other in such a manner that after a single complete turn unity would be transferred into the next following. If this is not performed by the calculating box of Pascal it may be added to it without difficulty.
The multiplying machine will consist of two rows of wheels, equal ones and unequal ones. Hence the whole machine will have three kinds of wheels: the wheels of addition, the wheels of the multiplicand, and the wheels of the multiplier. The wheels of addition or the decadic wheels are now visible in Pascal’s adding box and are designated in the accompanying figure by the numbers 1, 10, 1(X), etc. Each of these wheels has ten fixed teeth.
The wheels that represent the multiplicand are all of the same size, equal to that of the wheels of addition, and are also provided with ten teeth which, however, are movable so that at one time there should protrude 5, at another 6 teeth, etc., according to whether the multiplicand is to be represented five times or six times, etc. For example, the multiplicand 365 consists of three digits 3, 6, and 5. Hence the same number of wheels is to be used.
On these wheels, the multiplicand will be set, if from the right wheel there protrude 5 teeth, from the middle wheel 6, and from the left wheel 3 teeth.
So that this could be performed quickly and easily a peculiar arrangement would be needed, the exposition of which would lead too far into details. The wheels of the multiplicand should now be adjoined to the wheels of addition in such a manner that the last corresponds to the last, the last but one to the last but one, and that before the last but one to that before the last but one, or 5 should correspond to 1, 6 to 10, and 3 to 100. In the addition box itself, there should be small openings with the number set as 0, 0, 0, etc., or zero. If after making such an arrangement we suppose that 365 be multiplied by one, the wheels 3, 6, and 5 must make one complete turn (but while one is being rotated all are being rotated because they are equal and are connected by cords as it will be made apparent subsequently) and their teeth now protruding will turn the same number of fixed teeth of the wheels 100, 10, 1 and thus the number 365 will be transferred to the addition box.
Assuming, however, that the number 365 is to be multiplied by an arbitrary multiplier (124) there arises the need for a third kind of wheels or the wheels of the multiplier. Let there be nine such wheels and while the wheels of the multiplicand are variable so that the same wheel can at one time represent 1 and at another time 9 according to whether there protrude less or more teeth, the wheels of the multiplier shall, on the contrary, be designated by fixed numbers, one for 9, one for 1, etc.
This is accomplished in the following manner: Each of the wheels of the multiplier is connected using a cord or a chain to a little pulley which is affixed to the corresponding wheel of the multiplicand: Thus the wheel of the multiplier will represent many units equal to the number of times the diameter of the multiplier-wheel contains the diameter of the corresponding pulley.
The pulley will turn namely this number of times while the wheel turns but once. Hence if the diameter of the wheel contains the diameter of the pulley four times the wheel will represent 4.
Thus at a single turn of the multiplier-wheel to which there corresponds a pulley having a quarter of its diameter the pulley will turn four times and with it also the multiplicand-wheel to which it (the pulley) is affixed. When, however, the multiplicand-wheel is turned four times its teeth will meet the corresponding wheel of addition four times, and hence the number of its units will be repeated as many times in the box of addition.
An example will clarify the matter best: Let 365 be multiplied by 124. In the first place, the entire number 365 must be multiplied by four. Turn the multiplier-wheel 4 by hand once; at the same time the corresponding pulley will turn four times (being as many times smaller) and with it, the wheel of the multiplicand 5, to which it is attached, will also turn four times. Since wheel 5 has five teeth protruding at every turn 5 teeth of the corresponding wheel of addition will turn once and hence in the addition box there will be produced four times 5 or 20 units.
The multiplicand-wheel 6 is connected with the multiplicand wheel 5 by another cord or chain and the multiplicand-wheel 3 is connected with wheel 6. As they are equal, whenever wheel 5 turns four times, at the same time wheel 6 by turning four times will give 24 tens (it namely catches the decadic addition wheel 10), and wheel 3 catching the addition-wheel 100 will give 12 hundred so that the sum of 1460 will be produced.
In this way 365 is multiplied by 4, which is the first operation. So that we may also multiply by 2 (or rather by 20) it is necessary to move the entire adding machine by one step so to say, SO that the multiplicand-wheel 5 and the multiplier-wheel 4 are under addition-wheel 10, while they were previously under 1, and in the same manner 6 and 2 under 100 and also 3 and 1 under 1000.
After this is done let the multiplier-wheel 2 be turned once: at the same time 5 and 6 and 3 will turn twice and 5 catching twice (the addition-wheel) 10 will give 10 tens, 6 catching 100 will give twelve hundred and 3 catching 1000 will give six thousand, together 7300. This number is being added at the very same turn to the previous result of 1460. To perform the third operation, the multiplication by 1 (or rather by 100), let the multiplication machine be moved again (of course the multiplicand-wheels together with the multiplier-wheels while the addition-wheels remain in their position) so that the wheels 5 and 4 are placed under 100 and in the same way 6 and 2 under 1000 and 3 and 1 under 10,000, If wheel 1 be turned once at the same time the wheels 3, 6, and 5 will turn once and thus add in the addition box that many units, namely, 36,500. As a product we obtain, therefore:


It should be noted here that for the sake of greater convenience, the pulleys should be affixed to the multiplicand-wheels in such a manner that the wheels must move when the pulleys move but that the pulleys do not need to move while the wheels are turned.
Otherwise, when one multiplier-wheel (e.g., 1) is turned and thus all the multiplicand-wheels moved, all the other multiplier wheels (e.g., 2 and 4) would necessarily move, which would increase the difficulty and perturb the motion.
It should be also noted that it does not make any difference in what order the multiplier-wheels 1, 2, 4, etc. be arranged but they could very well be placed in numerical order 1, 2, 3, 4, 5. For even then one is at liberty to decide which one to turn first and which afterward.
So that the multiplier-wheel, e.g., the one representing 9 or whose diameter is nine times as great as the diameter of the corresponding pulley, should not be too large we can make the pulley so much smaller preserving the same proportion between the pulley and the wheel.
So that no irregularity should follow the tension of the cords and the motion of pulleys tiny iron chains could be used in place of the cords and on the circumference of the wheels and pulleys where the chains would rest there should be put little brass teeth corresponding always to the individual links of the chain; or in place of cords, there could be teeth affixed to both the pulleys and the wheels so that the teeth of the multiplier-wheel would immediately catch the teeth of the pulley.
If we wanted to produce a more admirable machine it could be so arranged that it would not be necessary for the human hand to turn the wheels or to move the multiplication machine from operation to operation: Things could be arranged in the beginning so that everything should be done by the machine itself. This, however, would render the machine more costly and complicated and perhaps in no way better for practical use.
It remains for me to describe the method of dividing on the machine, which (task) I think no one has accomplished by a machine alone and without any mental labor whatever, especially where great numbers are concerned.
But whatever labor remains to be done in (the case of) our machine it could not be compared with that intricate labyrinth of the common division which is in the case of large numbers the most tedious (procedure) and (the one) most abundant in errors that can be conceived. Behold our method of division! Let the number 45,260 be divided by 124. Begin as usual and ask for the first simple quotient or how many times 452 contains 124.
It is very easy for anyone with a mediocre ability to estimate the correct quotient at first sight. Hence let 452 contain 124 thrice. Multiply the entire divisor by this simple quotient which can be easily accomplished by one simple turn of the wheel.
The product will be 372. Subtract this from 452. Combine the remainder 80 with the rest of the dividend 60. This gives 8060.
(But that will be effected by itself in the machine during the multiplication if we arrange in it the dividend in such a manner that whatever shall be produced by multiplication will be automatically deducted. The subtraction also takes place in the machine if we arrange in it the dividend in the beginning; the performed multiplications are then deducted from it and a new dividend is given by the machine itself without any mental labor whatever.)
Again divide this (8060) by 124 and ask how many times 806 contains 124. It will be clear to every beginner at first sight that it is contained six times. Multiply 124 by 6. (One turn of the multiplier wheel) gives 744. Subtract this result from 806, there remains 62. Combine this with the rest of the dividend, giving 620. Divide this third result again by 124. It is clear immediately that it is contained 5 times. Multiply 124 by 5; (this) gives 620. Deduct this from 620 and nothing remains; hence the quotient is 365.
The advantage of this division over the common division consists mostly in the fact (apart from infallibility) that in our method there are but few multiplications, namely as many as there are digits in the entire quotient or as many as there are simple quotients.
In common multiplication, a far greater number is needed, namely, as many as (are given by) the product of the number of digits of the quotient by the number of the digits of the divisor. Thus in the preceding example, our method required three multiplications, because the entire divisor, 124, had to be multiplied by the single digits of the quotient 365,—that is, three.
In the common method, however, single digits of the divisor are multiplied by single digits of the quotient and hence there are nine multiplications in the given example.
It also does not make any difference whether the few multiplications are large, but in the common method there are more and smaller ones; similarly one could say that also in the common method few multiplications but large ones could be done if the entire divisor be multiplied by an arbitrary number of the quotient.
But the answer is obvious, our single large multiplication being so easy, even easier than any of the other kind no matter how small. It is effected instantly by a simple turn of a single wheel and at that without any fear of error. On the other hand in the common method the larger the multiplication the more difficult it is and the more subject to errors. For that reason, it seemed to the teachers of arithmetic that in division there should be used many and small multiplications rather than one large one. It should be added that the largest part of the work already so trifling consists in the setting of the number to be multiplied, or to change according to the circumstances the number of the variable teeth on the multiplicand-wheels. In dividing, however, the multiplicand (namely the divisor) remains always the same, and only the multiplier (namely the simple quotient) changes without the necessity of moving the machine. Finally, it is to be added that our method does not require any work of subtraction; for while multiplying in the machine the subtraction is done automatically. From the above, it is apparent that the advantage of the machine becomes more conspicuous the larger the divisor.
It is sufficiently clear how many applications will be found for this machine, as the elimination of all errors and of almost all work from the calculations with numbers is of great utility to the government and science. It is well known with what enthusiasm the calculating rods of Napier, were accepted, the use of which, however, in the division is neither much quicker nor surer than the common calculation. For in his (Napier’s) multiplication, there is a need for continual additions, but the division is in no way faster than by the ordinary (method). Hence the calculating rods soon fell into disuse. But in our (machine) there is no work when multiplying and very little when dividing.
Pascal’s machine is an example of the most fortunate genius but while it facilitates only additions and subtractions, the difficulty of which is not very great in themselves, it commits the multiplication and division to a previous calculation so that it commended itself rather by refinement to the curious than as of practical use to people engaged in business affairs.
And now that we may give final praise to the machine we may say that it will be desirable to all who are engaged in computations which, it is well known, are the managers of financial affairs, the administrators of others’ estates, merchants, surveyors, geographers, navigators, astronomers, and (those connected with) any of the crafts that use mathematics.
But limiting ourselves to scientific uses, the old geometric and astronomic tables could be corrected and new ones constructed by the help of which we could measure all kinds of curves and figures, whether composed or decomposed and unnamed, with no less certainty than we are now able to treat the angles according to the work of Regiomontanus and the circle according to that of Ludolphus of Cologne, in the same manner as straight lines. If this could take place at least for the curves and figures that are most important and used most often, then after the establishment of tables not only for lines and polygons but also for ellipses, parabolas, hyperbolas, and other figures of major importance, whether described by motion or by points, it could be assumed that geometry would then be perfect for practical use.
Furthermore, although optical demonstration or astronomical observation or the composition of motions will bring us new figures, it will be easy for anyone to construct tables for himself so that he may conduct his investigations with little toil and with great accuracy; for it is known from the failures (of those) who attempted the quadrature of the circle that arithmetic is the surest custodian of geometrical exactness. Hence it will pay to undertake the work of extending as far as possible the major Pythagorean tables; the table of squares, cubes, and other powers; and the tables of combinations, variations, and progressions of all kinds, so as to facilitate the labor.
Also, astronomers surely will not have to continue to exercise the patience which is required for computation. It is this that deters them from computing or correcting tables, from the construction of Ephemerides, from working on hypotheses, and from discussions of observations with each other. For it is unworthy of excellent men to lose hours like slaves in the labor of calculation, which could be safely relegated to anyone else if the machine were used.
What I have said about the construction and future use (of the machine), should be sufficient, and I believe will become absolutely clear to the observers (when completed).

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

Brevis descriptio Machinae Arithmeticae, cum Figura
Brevis descriptio Machinae Arithmeticae, cum Figura

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 complicated calculations!”

In 1764, forty-eight years after Leibniz’s death, a Reckoner was turned over to a clockmaker in Göttingen for repair. The job wasn’t done, and the machine wound up in the attic of the University of Göttingen, where a leaky roof led to its rediscovery in 1879. Fourteen years later, the University gave the machine to the Arthur Burkhardt Company, one of the country’s leading calculator manufacturers, for repair and analysis. Burkhardt reported that, while the gadget worked in general, it failed to carry tens when the multiplier was a two- or three-digit number. As it was mentioned earlier, the carrying mechanism had been improperly designed. It’s unknown whether Leibniz has designed a machine without the above-mentioned flaw.

Leibniz’ Logical Machines

Dissertatio De Arte Combinatoria of Leibniz
Dissertatio De Arte Combinatoria of Leibniz

Gottfried Leibniz was one of the first men (after Ramon Llull), who dreamed of a logical (thinking) device.

In 1666 Leibniz published his first book (see the lower image), also his habilitation thesis in philosophy, Dissertatio De Arte Combinatoria (On the Art of Combinations, see the treatise), partly inspired by the Ars Magna of Ramon Llull (Leibniz was still a teenager when he encountered the works of Ramon Lull).

Though the design Leibniz places at the front of his book (see the lower figure) appears to be a very simple and even trivial diagram, compared to the copperplates of Kircher’s books, the entire text that follows speaks a new and different language. Leibniz was only 20, but he analyzed as an advanced mathematician the potential power and limits of the art of combination.

In contrast with Llull and Kircher, Leibniz was not at all interested in any esoteric applications of this method, but rather in a way of reproducing the totality of the universe within one science. After reading his very famous treatise on the monads (Monadology from 1714) (monads are something like atoms, situated in the metaphysical realm) as a model for the art of combination, his new, radical perspective is at once comprehensible.

In 1674 Leibniz described a machine for solving algebraic equations. A year later, he wrote comparing logical reasoning to a mechanism, thus pointing to the goal of reducing reasoning to a kind of calculation and of ultimately building a machine, capable of performing such calculations.

Leibniz Circle
Leibniz Circle from Dissertatio De Arte Combinatoria

There is a letter written by Leibniz to Johann Friedrich, Duke of Hanover, in April 1679, which offers the whole ambitious program of the philosopher. In that letter, we find initially a confession about the source of the method of combination. But then Leibniz starts to criticize Llull and Kircher because, in his view, they did not go far enough in using this art of combination. Regarding his own idea of its use, he says:
“My invention contains the application of all reason, a judgment in each controversy, an analysis of all notions, a valuation of probability, a compass for navigating over the ocean of our experiences, an inventory of all things, a table of all thoughts, a microscope with which to prove the phenomena of the present and a telescope with which to preview those of the future, a general possibility to calculate everything. My invention is an innocent magic, a non-chimerical Cabbala, writing, which everyone can read and which everyone can very easily learn…”

It is quite a pathetic proclamation, but that was the style of the great philosopher 🙂 Leibniz apparently believed that he had invented a general problem-solver, like those in the computer sciences have always dreamed of. But of course, his whole super-ambitious program was not to be realized. Only some aspects of that proclamation were really transposed into useful applications. At first, Leibniz made a few essential steps toward the calculation of probability, which is obviously a very important problem for modern AI (artificial intelligence) applications. He then attempted to transcribe the whole art of combination into a system of formulas because he wanted to calculate every single part of the process, each step, and each result of an interval. Thus he used consequently his mathematical skills to produce a new kind of combination by transposing meanings into figures and values.

Even more—Leibniz was also one of the first men, who realized the importance of the binary system (certainly, he is not the inventor of the binary system. Pingala, an otherwise-obscure Indian author, wrote about 300 B.C. Chandahsastra, or Science of Meters. In this treatise, the author used a binary logic system (short-long, in this case, rather than 0-1) to explore meter in poetry.)

Leibniz discovered that computing processes can be done much easier with binary number coding (in his treatises De progressione Dyadica, dated 15 March 1679 (see the treatise) and Explication de l’Arithmetique Binaire from 1703). In these clear and lucid treatises (in the lower figure is shown the second page of the original manuscript “De Progressione Dyadica”), Leibniz analyzed the possibilities of the binary system and, demonstrated its four fundamental operations of calculation—addition, subtraction, multiplication, and division—he expressed the conviction that one day in future the machines would use this system.

The second page of De Progressione Dyadica
The second page of De Progressione Dyadica

Though hard to believe, in his 3-pages treatise De progressione Dyadica, Leibniz even outlines a calculating machine that works via the binary system: a machine without wheels or cylinders—just using balls, holes, sticks, and channels for the transport of the balls—This (binary) calculus could be implemented by a machine. The following method would certainly be very easy to implement. (A machine with) holes, which 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 channels, through the others nothing falls. It (the gate array) is to be shifted from column to column as required for the multiplication. The channels should represent the columns, and no ball should be able to get from one channel to another except when the machine is put into motion. Then all the marbles run into the next channel, and whenever one falls into an open hole it is removed. Because it can be arranged that two always come out together, and otherwise they should not come out.

In a note, written later in his life, when he was reflecting on his works, Leibniz remembered the old program of the universal art of combination:
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 a 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…”.

Another remarkable idea of Leibniz, announced in his February 1678, essay “Lingua Generalis”, was connected closely with his binary calculus ideas. Leibniz spoke for his lingua generalis or lingua universalis as a universal language, aiming it as a lexicon of characters upon which the user might perform calculations that would yield true propositions automatically, and as a side-effect developing binary calculus.