Category: Timeline Stories

In 1840, Thomas Fowler, an English businessman, self-taught mathematician, and inventor, created a unique ternary calculating machine in his workshop in Great Torrington, a small market town in the north of Devon, England. In 1842 Fowler devised a greatly improved model, which was exhibited in the museum of King’s College London for a time.

During the 1830’s Thomas Fowler became the sole manager and partner of the only bank in Torrington, Devon, Messrs Loveband & Co, and the treasurer of the Torrington Poor Law Union. The tedious nature of the calculation of payments for each of the parishes, which was one of his responsibilities, led him to attempt to automate the calculations by the use of tables. Fowler’s solution was typically brilliant and led, in 1838, to Fowler’s Tables for Facilitating Arithmetical Calculations.

The tables used a method based on Fowler’s realization that any number might be produced by a combination of the powers of 2 or 3. The first section of the booklet is the Binary Table, or a table of indices of the number 2 from 1 to 130048. The second section is the Ternary Table, or a table of the indices of the power of the number 3 from 1 to 3985607.

Soon after Fowler had devised the tables, he used the same ideas to build a mechanical calculating machine, which was exhibited before members of the Royal Society in May 1840. In a subsequent letter to the famous English mathematician and astronomer George Bidell Airy, Fowler writes:
This Machine was constructed entirely with my own hands (principally in wood) with the utmost regard to economy and merely to put my ideas of this mode of calculation into some form of action; It is about 6 feet long, one foot deep and three feet wide. In Brass & Iron it might be constructed so as not to occupy a space much large than a good portable writing desk and with powers such as I have described.

The Astronomer Royal, Professor George Airy was to promote Fowler’s invention to a gathering of the British Society for the Advancement of Science in August 1840. In the minutes of that meeting we read:
“Mr Airy gave an account of Mr Fowler’s new Calculating Machine. The origin of the machine was to facilitate the guardians of a poor-law district in Devonshire in the calculating of the proportions in which the several divisions were to be assessed. The chief peculiarity of the machine was, that instead of our common decimal notation of numbers, in it ternary notation was used; the digits becoming not tenfold but threefold more valuable as they were placed to the left; thus, 1 and 2 expressed one and two as in common, but 10 expressed not ten but 3, 11 four, 12 five; but again 2 can be expressed by three, one taken from it. Now, let T (one with a bar over it), written thus with a small bar above it, mean that it is subtractive; then 12 and 2T are the same in effect, both meaning five; and for a similar reason, by replacing 2 with 1T, we have five written in there several ways; 12, or 2T, or 1TT. The last is the form used it is obvious by an assemblage or unit digits thus positively or negatively written, any number may be expressed. In the machine levers were contrived to bring forward the digits T or 1 as they were required in the process of calculation.”

Fowler writes to Airy:
“I had the honor in May 1840 to submit the machine to the inspection of many Learned Men in London among whom were the Marquis of Northampton, Mr Babbage, W F Baily and A de Morgan Esq with many other Noblemen and Gentlemen, Fellows of the Royal Society etc. and it would have been a great satisfaction to me if I could have had the advantage of your opinion also. They all spoke favorably of my invention but my greatest wish was to have had a thorough investigation of the whole principle of the machine and its details, as far as I could explain them, in a way very different from a popular exhibition: this investigation I hope it will still have by some first rate men of science before it is be laid aside or adopted.
I am fully aware of tendency to overrate one’s own inventions and to attach undue importance to subjects that preoccupy the mind but I venture to say and hope to be fully appreciated by a Gentleman of your scientific achievements, that I am often astonished at the beautiful aspect of a calculation entirely mechanical.
I often reflect that had the Ternary instead of the binary Notation been adopted in the Infancy of Society, machines something like the present would long ere this have been common, as the transition from mental to mechanical calculation would have been so very obvious and simple.
I am very sorry I cannot furnish you with any drawings of the Machine, but I hope I shall be able to exhibit it before the British Association at Devonport in August next, where I venture to hope and believe I may again be favoured with your invaluable assistance to bring it into notice. I have led a very retired life in this town without the advantages of any hints or assistance from any one and I should be lost amidst the crowd of learned and distinguished persons assembled at the meeting without some kind friend to take me by the hand and protect me.”

Charles Babbage, Augustus De Morgan, George Airy, and many other leading mathematicians of the day witnessed his machine in operation. Airy asked that he produce plans of his machine but Fowler, recalling his experience with the Thermosiphon, refused to publish his design.

The only depiction of the calculating machine surviving to the present time is in a stained glass window (see the upper image), in St. Michael’s Church in Torrington, Devon, ordered by Fowler’s son Hugh in 1864. Besides the upper mentioned notices of Fowler, the only other information about his machine is Augustus De Morgan’s description. De Morgan saw the above-mentioned demonstration of Fowler’s machine and described it. Below is the description:

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De Morgan, Augustus. “Description of a calculating machine, invented by Mr Thomas Fowler of Torrington in Devonshire.”  London: The Royal Society, June 1840.
The machine consists of four essentially distinct parts. The first, second and third exhibit the multiplicand, multiplier and product, or quotient, divisor and dividend, according as the question to be worked is one of multiplication or division. The fourth is a carrying apparatus, which though at present detached, and employed to reduce the result to its simplest form after the main operation has been performed, might without much difficulty be attached to the multiplier or divisor, and work with it.
Let us now suppose a question of multiplication, both multiplier and multiplicand being exhibited in the ternary system. The multiplicand consists of nothing but a number of rods, each bearing an index, and each movable backwards and forwards. When the indices are all arranged in line, and in one particular line, this multiplicand is 000…. But if any one of the rods be advanced by a certain space forwards, the digit +1 is indicated as occupying the numeral column which that rod represents, and if it be moved the same space backwards, -1 is the digit indicated. This, which we may call the frame of the multiplicand, is thus a collection of rods, not itself connected with any machinery, but only useful as indicating the manner in which the frame of the multiplier is to act.
The multiplier is a frame movable in the direction perpendicular to the rods of the multiplicand and product, and situated between the planes of the two, in such manner that its extremity can be brought by a sliding motion over each rod of the multiplicand in succession. This multiplier consists of a number of rods in a common system, each furnished with two teeth, one at each extremity, the tooth by which it is acted on being a continuation of the rod, that by which it acts being perpendicular to the axis of the rod. The first set of teeth are dispersed/disposed so as to rest in a frame which has a slight motion round an axis; and each rod can be moved so that its teeth shall touch the frame above, on or below the axis. Those rods, then, which have their teeth on the axis do not receive motion from the frame, while the others receive motion in one direction or another, according as the teeth touch the frame above or below the axis. The perpendicular teeth at the other extremities may thus be made to move in either direction, or to remain stationery: and these last mentioned teeth act upon the rods which make up the frame of the product. This last frame precisely resembles the frame of the multiplicand, with the addition of the connecting part by which the multiplier acts upon it.
The process of multiplication is then as follows; the frame of the multiplier having been set, and also that of the multiplicand, the extremity of the multiplier frame is brought over the first rod of the multiplicand. To this extremity is attached a tooth which acts upon the rod of the multiplicand over which it comes, giving it a motion in one or the other direction, according as the slightly revolving frame of the multiplier is made to move in one or the other direction. The rule is, to move the revolving frame in such a way as to bring the rod of the multiplicand to its zero position; and this one motion multiplies the figure of the multiplicand by the whole of the multiplier, and by the action of the perpendicular teeth, exhibits the result upon the product frame. The lateral motion is then given to the whole of the multiplier apparatus, until the tooth comes upon the next figure rod of the multiplicand, and the revolving frame being then made to bring the new multiplicand rod to zero, the effect upon the product frame is that the new figure of the multiplicand is multiplied by the whole of the multiplier, and the result added to that of the preceding figure. This process is continued until the whole of the figures of the multiplicand are exhausted.
The result is then completely exhibited on the frame of the product, but not in its simplest form, For, whereas +1 or -1 should be the only digits in the final result, this intermediate result may exhibit +2 or -2, +3 or -3 etc on any rod. The carrying frame is a simple apparatus which, like the multiplier, has a lateral motion, and can be brought on any pair of consecutive rods. By one motion of the hand, it advances the left of two rods by a unit, and throws back the right hand rod by 3 units, or vice versa. Some little expertness is here necessary in making the carriages properly, with reference to the simplicity of the result: but there is no possibility of absolute error being introduced, since each process can only consist in altering a lower column by 3 units while the next column is altered in the contrary way by one unit.
The method of performing division is precisely the reverse of the preceding, and will hardly need description.
End of description

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Almost all information, which can be found in the present day for Fowler, has been discovered and classified by the researchers Pamela Vass and David Hogan, from North Devon, England. In 1999 they decided to contact a skillful mechanic—Mark Glusker, in order to make a replica of the machine.

Due to the scarce historical information, Mr. Glusker had a hard time during his work but managed to produce a working device. The model was completed in about eight months and was presented to the Great Torrington Museum in a ceremony in August of 2000. Below you can see some sketches of the replica. All copyrights for the replica and photos are property of Pamela Vass, David Hogan, and Mark Glusker.

While his machine was at King’s College, Thomas Fowler died in 1843 and the machine was taken apart and sent back to his son, Hugh, in pieces. Hugh admitted that he didn’t know how to put it back together again despite having some written instructions dictated by his father on his deathbed.

Biography of Thomas Fowler

Thomas Fowler was born in 1777 in North Devon, England, to Hugh and Elizabeth Fowler of Great Torrington. The Fowler family was poor, Hugh was a cooper, and Thomas received only a rudimentary education at the local school. He was apprenticed to a fell-monger (dealer in hides and skins, especially sheep, who removed the wool or hair from the hides in preparation for tanning) at the age of 13. Despite this unpromising start in life, Thomas Fowler became arguably one of the great thinkers of his age.

His son and biographer, the Reverend Hugh Fowler (28 Feb 1816 – 7 Aug 1877) writes:
“…after a hard day’s work among sheepskins he would spend half the night poring over his mathematics, until he had gone as far as to master Saunderson’s Fluxions, the name of which the method of the differential calculus, as far as it was then known, was designated. There was no one alas! to take him by the hand, and help him to carry on his studies at Cambridge, where alone such talent as he undoubtedly possessed could either have been fully developed or adequately rewarded, so he was left, without help or sympathy, to his solitary studies.”

Fowler was self-taught with the only book Ward’s Mathematician’s Guide—John Ward’s Young Mathematician’s Guide. About 1800 he became a printer and bookseller.

Thomas spent his whole life in Torrington. He married Mary Copp on 21 February 1813. They had at least eleven children, but, as it was common at this time, several died before reaching adulthood; the average life expectancy was only 40 in the 1830s. The genius of Thomas Fowler is evident in some of his children, particularly his daughter Caroline Elizabeth (1814-1853), who was composing books, i.e. taking the print and placing it back to front and upside down in a composing stick ready for printing, by the age of eight.

In 1828 Fowler patented the Thermosiphon (British Patent number 5711). This was to become the modern central heating system. A heating system based on a design by Thomas Fowler was installed at Bicton, then part of the Rolle Estate, and received great acclaim in the Gardener’s Magazine of 1829.
“Mr Fowler has had the good fortune to hit on the idea that water may be heated and made to circulate through a siphon, as well as through horizontal pipes, or by force through pipes in any direction; provided always, that the height of the siphon be not greater than to be counter-balanced by the pressure of the atmosphere; say not greater than 30 feet. Any person might have discovered the same thing by reflection, or in answer to the question asked; but we are not aware that the idea has occurred, either to the original inventor of the hot-water system, Bonnemain; to its introducers into England, Bolton and Watt; to its subsequent introducer; Chabbanes; to Count Romford; and to its reinventors, or English inventors, Atkinson and Bacon; or to any to any of the numerous engineers now occupied in applying this mode of heating.”

The patent laws of the time were flawed. By introducing any small change to the original design, the resulting new version would not be covered by the original patent. This meant that others could steal his invention with impunity, which of course they duly did. Fowler’s son Hugh writes in 1875:
“Unfortunately the invention was soon pirated in all directions. The only remedy was costly legal proceedings, but even if he had had the means to conduct them, success would have been doubtful.”
Fowler became very embittered by this experience and this had an unfortunate side-effect on the history of computer science.
(It is worth noting that the Romans had a convective heating system called the Hypocaust. However, there are fundamental differences. Firstly, heat was conveyed by hot air and secondly, it was not a closed system. The hot air, having passed through channels under the floor, then was allowed to escape into the atmosphere.)

In 1835 Fowler became the Treasurer of the Poor Law Union and partner in the bank Messrs. Loveband & Co. in Torrington. The tedious nature of the calculation of payments for each of the parishes, which was one of his responsibilities, led him to attempt to automate the calculations by the use of tables, and then to devise his astonishing ternary calculator.

Fowler’s talents were recognized by some prominent people in and around Torrington (including Lord Clinton) who did everything they could to encourage him in his work and bring his inventions to public notice. He was also an organist at the parish church.

Thomas Fowler died on 31 March 1843, of dropsy of the chest.

Around 1835 the young Warsaw watchmaker and inventor Izrael Abraham Staffel (1814-1885) commenced building in his workshop at Marszałkowska str. 125 an advanced first calculating machine, spending his meager funds. It took him ten years to finish the device, which was demonstrated to the public as late as 1845.

Staffel was a Polish Jew, who was born and lived in the capital Warsaw (then part of the Russian Empire) all his life. He was a watchmaker and mechanic, who spent most of his life developing various inventions, primarily calculating machines, for which he obtained multiple prizes, but he also invented other devices, such as an anemometer, a powerful fan, a probe for determining the contents of alloys, a counter-fitting machine, etc.

The distinguished Jewish historian Jacob Shatzky (1893-1956) asserts that Staffel was a relative of Abraham Jakub Stern. If this is true, probably Staffel was inspired by Stern and received from him valuable information about the construction of calculating machines. Anyway, it is hard to believe that the young watchmaker Staffel didn’t know Abraham Stern, who was one of the most remarkable figures in the Warsaw Jewish community during the first decades of the 19^th century, and who also used to work as a watchmaker in his youth.

Staffel commenced his occasions with calculating machines around 1835, but since he lived in poverty, his first machine, based on the pin-wheel mechanism of Leibniz and Poleni, was ready in 1842 and was demonstrated at the industrial exhibit in Warsaw as late as 1845 (by recommendation from the Minister Sergey Uvarov (Серге́й Семёнович Ува́ров), and it received a silver medal (now this machine is preserved in the Museum of Technology in Warsaw, see the lower photo).

In the Polish press after the exhibition was written that the machine particularly accelerated the division and multiplication operations. The author of one of the articles even expressed his opinion:
… the thirteen-digit product resulting of multiplying the number by a six-digit multiplier you can get in 50 seconds, while multiplying on paper the usual way even when the multiplier is folded from nines alone (999999), the product is obtained in just 1¼ minute. The same shortening can be achieved also in division.

After the exhibition, the governor of Warsaw gave Staffel 150 rubles on a trip to St. Petersburg to present the machine to the Russian Academy of Sciences. Thus in 1846 Staffel traveled to the Russian capital to demonstrate his machine (accompanied with a handwritten description of 71 Russian and 57 Polish pages plus 3 color drawings) and was very positively assessed by the Russian Academy of Sciences. Two famous mathematicians, Viktor Bunyakovsky (later creator of a calculating device also, Самосчеты Буняковского) and Moritz Jacobi, gave it a very positive opinion and Staffel was awarded the very prestigious Demidov prize. Later on, the machine was presented to the Russian Emperor Nikolai I, who was so astonished by this invention, that he ordered the substantial sum of 1500 silver rubles to be paid to the inventor.

Interestingly, in his presentation to the Russian Academy, Staffel mentioned the names of previous inventors of calculating machines, like Leibnitz, Hahn, Babbage, Müller, and Stern, but didn’t mention where he found the idea of pin-wheel mechanism, which he used in his machine (see the nearby image). It is known, that the first sketch of such a mechanism can be found in a manuscript of Leibniz, the first practical implementation was that of Poleni in 1709, then Braun in 1720s, and Roth and Staffel around 1840. Staffel even mentioned that Leibnitz spent 24000 thalers on his machine, Babbage spent 1700 pound sterling, and Stern spent 10000 thalers.

Later on, Staffel improved his first machine, placing the calculating mechanism on a movable rail, and manufactured a calculating machine for adding, subtracting, multiplication, and division (plus square-root). In 1851, the improved machine was exposed to the Great Exposition in Crystal Palace in London, where it was awarded with a gold medal. The report, describing the calculating machines presented at the Exposition, says the following about Staffel’s machine: “The best machine of this kind exhibited is that of Staffel (Russia, 148), which, on examination, seems to combine accuracy with economy of time, and works easily and directly”.
The machine was evaluated higher than the machine of Thomas de Colmar, which was a real sensation. The newspaper Illustrated London News also published an enthusiastic article, with the description of the machine and its picture: “In the Russian Court, modestly secluded amidst the glitter of malachite doors and vases, jewelry and silver, there is one work, the produce of high intelligence, and intended to assist in certain intellectual labors. This solitary tribute of mind to minds comes not from Petersburg, nor Mexico [meant Moscow], nor from Siberia, nor from the Ural Mountains, but from Poland. We refer to Staffel’s Calculating Machine, №148 in the Catalogue”.

In the Jewish Chronicle newspaper from Nov. 1851 was mentioned that:
Liberality of the Prince Albert. – It must be a source of great delight to our brethren, to be made acquainted with the munificent liberality of the royal consort of our beloved Queen towards a humble mechanic of the house of Israel. The liberality of his Royal Highness has been exercised in the case of J. A. Staffel, of Warsaw, the inventor of the calculating machine, etc. , which was exhibited in the Russian department of the Crystal Palace, who has received from his Royal Highness a cheque for 20 l. as an acknowledgment of his Royal Highness’s appreciation of Mr. Staffel’s ingenious invention.
Since writing the above, we are glad to hear that Baron L. Rothschild, M. P., also presented our scientific brother with a cheque for 10 l. as a due acknowledgment of Jewish talent.

Let’s quote also the description of the machine in the newspaper Illustrated London News, No. 518, 20 Sep. 1851:

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The machine is the size of an ordinary toilet: the mechanism is 18 inches by 9, and about 4 inches high. The external mechanism represents three rows of ciphers. The first and upper row, containing 13 ciphers, is immovable; the second and third, containing 7 ciphers, are movable. To the right is a semicircular ring, containing the words Addition, Subtraction, Multiplication, Division and Extraction. Underneath is a hand, which serves as a regulator for the operation, pointing ad libitum to either of the four rules or the square root, whichever is to be worked. The advantage which this machine has above others are as follows:
1. That the four rules and the square root, with fractions, can be worked by means of a curved handle (which in itself is a piece of mechanism), showing the various sums alternatively, without being obliged to note down any auxiliary figure, as is the case with all other calculating machines.
2. That all compound rules, as the rule of three, of five, etc., can be worked by transposition of the regulator, without shifting any of the figures.
3. That, if by subtraction a larger number is subtracted from a smaller, the sound of a bell is heard, indicating the false proceeding; and when turning the handle a negative number shows itself in the upper row, where, instead of the 13 ciphers, the figure 9 will appear in their place, and which, added to the number given, will prove the inverted position of the number. The bell will also be heard if, by division, the handle is turned once too many. A retractive move of the handle will then retrieve the error.
4. That the entire mechanism is of simple construction, the parts acting without springs, its correctness and accuracy secured, and the efficiency of the mechanism guaranteed.
<<  End of quote  >>

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Let’s see also the description of Staffel’s calculating machine in the official document of the London Exhibition (see below the section Calculating Machines from Reports by the Juries, Great Exhibit, London, 1851):

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Calculating Machines
There have been very many attempts to perform numerical calculations by mechanical means, or at least such parts of them as follow simple and rigid laws. Hitherto instruments have failed to unite correctness in the results, combined with economy of time, and, for the most part, have been limited to the performance of the first two operations of arithmetic.
To make such instruments really useful, they must have the power of executing, by themselves, the operations for the solution of the problem imposed on them, when the simple data for this problem have been introduced, without trial, and without guesswork.
The best machine of this kind exhibited is that of Staffel (Russia, 148), which, on examination, seems to combine accuracy with the economy of time, and works easily and directly. The mechanism is 8 inches in length, 9 inches in breadth, and 4 inches in height, and consists of three rows of vertical cylinders; the first contains 18, the second 7, and the third 7. Upon each of the cylinders in the first row are 10 notches, corresponding with the units 1 to 10. Within each of these cylinders is a small pulley, in connection with a lever, set in motion by a slider which, when the cylinder has turned from either 9 to 0, or 0 to 9, sets in motion the lever, and communicates its action to wheels, which carry over the figures. The pulley connected with the cylinder, the furthest from the handle, is in connection with the hammer of a bell. The purpose of this bell is to give a warning to the operator, on committing an error, and constitutes a most important addition to the machine, particularly in the operation of division.
Upon each of the cylinders in the second row 10 units are placed. These seven cylinders are so fixed upon their axes, that they can bodily be moved right and left, and fixed at any part, so that the in the cyphers in the two cylinders can be made to correspond. This cylinder is with a spike, which lays hold of and works the third row of cylinders.
The internal communication of each of the parts is brought about by means of a wheel, furnished with nine moveable pegs, which are set in motion by means of an eccentric incision in the dial.
The machine is capable of performing addition, subtraction, multiplication, division, and of extracting the square root.
The operation of addition is performed as follows:
By simply placing one line of the numbers upon the second row of cylinders (the index pointing to addition), and turning the handle, till it stops, these numbers are transferred instantly to the first row of cylinders, and so on successively, till all the numbers to be added are transferred, and their sum is shown on the top row.
In performing subtraction, the first part of the operation is the same as in addition, but on placing the second line of figures on the second row of cylinders, the pointer being placed to subtraction, the handle is turned the opposite way, or against the motion of the sun, and the difference of the two numbers is shown on the upper line.
The operation of multiplication is performed by placing the multiplier and the multiplicand on the second and third rows of cylinders, and then, the index pointing to multiplication, the product will be found on the first cylinder.
The operation of division is very similar, except that the handle is turned as in subtraction.
These several operations were performed accurately, and with dispatch.
In the performance of the square root, the following additional mechanism needs explanation. Between every division of the cylinder, in row 2, a small wheel is placed, and near it a projecting piece which acts upon a lever; when the projecting piece is near the word “rad” engraved on the cylinder, on turning the handle, the figures increase by 1. This, by another mechanism, is connected with the other two rows of cylinders. The operation of the square root is performed directly, without any guessing at numbers; but it is, comparatively, rather a long process.
On the whole, it must be considered that Mr. Staffel has made an instrument possessed of considerable powers, and that great praise is due to him. The double motion of the handle as well as the warning bell are important improvements.
Mr. Staffel also exhibits a small mechanical machine for the performance of the addition and subtraction of fractions, whose denominators are 10, 12, and 15. By enlarging the machine, this number would be increased, and the power of the instrument extended. The operations were performed with quickness, and with accurate results. A Prize Medal was voted to Mr. Staffel.
<<  End of quote  >>

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Even many years later, in the Reports of the United States Commissioners to the Paris Universal Exposition, 1867, Staffel’s machine is praised in Chapter XVIII “Metrology and Mechanical Calculation”: Of the numerous calculating machines which have been proposed or constructed since that of Mr. Thomas became an ascertained success, those of Messrs. Maurel & Jayet of France, and of Mr. Staffel of Russia, are the only ones which, so far as is known, have solved the problem in a manner entirely satisfactory.

Besides the above-mentioned calculating machine, Staffel invented also another simpler calculating device (in total he designed four different types of calculating machines). One of them (see the picture below), is now preserved in Braunsweig Landesmuseum, Germany. This model of the calculating machine was demonstrated by Staffel in 1858 and was awarded at an exhibition in Warsaw. Named by Staffel’s liczebnik kieszonkowy (pocket numeral machine), it was a seven-digit calculating device, used for addition and subtraction.

In 1876 Staffel handed over the most famous of his machines, the 13-digit arithmometer to the Physical Cabinet of the Russian Academy of Sciences in St. Petersburg, but it seems the device has been lost.

Biography of Izrael Abraham Staffel

The Polish Jew Izrael Abraham Staffel (Polish: Izrael Abraham Sztafel, Russian: Израиль Авраам Штафель) was born in 1814 to an impoverished Jewish family in Warsaw, Poland, then part of the Russian Empire. He was the son of Lewek Staffel (Лейвик Штафель) and Gryna Izrael Staffel (Грина Израилевна Штафель) and had an elder sister—Estera Lewek Stafell (Эстер Левиковнa Штафель) (died 1875).

His primary education Izrael got at an elementary religious Jewish school, and later entered as an apprentice in a watchmaking factory. There he learned the Polish language, which enabled him to read scientific and technical books and deepen his professional knowledge.

In 1833, when he was only 19, Staffel obtained a concession and opened a watch-maker shop in Warsaw (at Marszałkowska str. 125), and several years later he opened another workshop at Grzybowskiej str. 982, where he worked till the end of his life. Despite being a diligent and talented watchmaker, Staffel obviously was a poor businessman, because his workshop did not prosper.

Staffel was married (at least) twice. We know nothing about his first wife(s), but there is a record from 20 January 1845 for the marriage of Izrael Sztafel (divorced), to Frajda Nuta Amerykaner (b. 1827). No records for children.

Staffel spent most of his life developing various inventions, primarily the calculating machines, for which he obtained multiple prizes, but he also invented other devices, such as an anemometer, Ventilator Helis (a powerful fan, which was installed at the Royal Castle in Warsaw, as well as at the Noble Institute and in the hospital of St. Spirit), a probe for determining the contents of alloys, a counter-fitting machine, an automatic taximeter for cabs, and a two-color printing press, used for printing stamps and banknotes (on this machine in 1860 was printed the first Polish post stamp, the so-called Poland No. 1, in the Stempel Factory in Warsaw). Staffel also made room fans, called helix-shaped blades, intended for removing smoke from kerosene lamps, cigars, etc.

Unfortunately, Staffel did not manage to patent any of his inventions and died extremely poor after a long illness in 1885 in Warsaw. In his obituary is written that he was very modest, and glory and recognition didn’t interest him, so his death remained almost unnoticed.

The first calculating machine put in serial production was the Arithmomètre (arithmometer) of the French entrepreneur Charles-Xavier Thomas de Colmar (1785-1870).

Colmar conceived the idea of the arithmometer during his lengthy stay with the armies of Marchall Soult, where he needed to perform a lot of calculations. This became even more important in his eyes when, in 1819, he was appointed General Manager of the Phoenix insurance company and, later, when he founded the insurance companies Soleil (1829) and Aigle (1843).

Of course, others had tried before him to make calculating machines in quantities: let’s mention only Pascal, Leibniz, Morland, Hahn, Stanhope (especially Hahn tried to manufacture in amount his machines but without success). But these machines, often defective and very expensive, made it impossible to commercialize. Moreover, it was too early to produce in large quantities a calculator in the 17^th or 18^th century. Human society was not needed yet such devices and the technologies, needed for such mass production, have not been invented yet. In the middle of the 19^th century, with the industrial revolution, technological trammels dropped out. More and more enterprises, scientific, military and government institutions became eager to accept a calculator. In the nick of time, then came Thomas de Colmar.

In fact, Thomas commenced the design of his calculating machine in 1818, but it was first made public in 1820 when he was granted a five-year patent (pat. No. 1420, 18 November 1820, see the drawing below). Obviously, the calculating mechanism is based on the stepped drum mechanism of Leibniz. It is clear however that the patent represents only a transient prototype, on which Thomas was still actively working. By 1821, when he was ready to submit an example to the scrutiny of the Société d’encouragement pour l’industrie nationale, the design had already moved on significantly.

In 1821 Thomas de Colmar submitted to the Sociėtė d’Encouragement… in Paris the first copy of the calculating machine he had constructed (which he called an arithmometer), manufactured by the Parisian horloger-mécanicien Jean-Pierre Devrine. From 1822, when the production started, until 1878, were manufactured about 1500 machines, as the last models cost 500 franks, a serious sum for this time. Interestingly, in the book Histoire Des Nombres: Et de La Numeration Mechanique (1855), the author, Jacomy-Régnier claimed that Thomas had spent 300000 francs on developing the arithmometer, setting that figure against Leibniz’s reputed costs of 100000 francs and Babbage’s notorious government subvention of £17000 (reckoned as equivalent to 425000 francs). Admittedly, at least until the end of the 1850s Thomas’s work on the arithmometer is more likely to fall within the category of vanity publishing than profit and mass production.

The series production started really only about 1851 and finished around 1914. As we know, the Thomas workshop completed five hundred machines from 1821 to 1865, three hundred machines from 1865 to 1870, four hundred machines from 1871 to 1875, and three hundred machines from 1876 to 1878. More than 5000 examples of the arithmometer were manufactured during these 90 years, 40% of the production was sold in France and the remainder was intended for export. In fact, up to the time when the calculating machine industry was introduced into Germany by Arthur Burkhardt (1878), Thomas’ workshop was the only company in this line and supplied the whole world with its products.

The 1820 machine had overall measurements 8.2 cm x 29.5 cm x 13.4 cm, and featured a ribbon to pull (instead of a crank as in later models), a second set of result display for subtraction and division, and (most important)—a multiplication gear, set by the first slider from left, which allowed the “multiple add” by one “pull” and actually shows the number of the revolutions of the calculating mechanism. It has a capacity of three digits in the input mechanism and six digits in the result mechanism. It has only clearance of the single result digits.

In the second model of 1848 (see the photo below), the complex and unreliable movement mechanism with ribbon was replaced by a crank at the front side, which can be rotated in two directions so many times, according to the value of the particular digit in the multiplier/divisor. It still has the multiplication gear. The second set of result displays is gone, the switch from addition and multiplication to subtraction and division was done by a lever. Internally, the stepped drum was reduced from 18 to 9 teeth. The capacity was increased to five digits in the input mechanism and ten digits in the result mechanism. Every digital position is shown in one window already because switching between multiplication and division was done by means of the lever (placed to the left of the multiplication lever), which changed the direction of the carry from the calculating to the result mechanism.

In the third version of the machine from 1858, the main improvement was the second counting mechanism without a tens-carry (revolution counting mechanism), which simplifies multiplication and division. The machine was also provided with one zero-setting device that acted on all the windows of the result mechanism and another for all the windows of the revolution-counting mechanism. Previously, all the numeral disks had to be set to zero individually by turning knobs placed below the individual windows. The zero-setting device is in the form of a rotating knob that is turned to the right until all the windows of the respective numeral mechanisms show zero. The zero-setting device of the result mechanism is mounted on the right side of the upper surface of the carriage, whereas the zero-setting mechanism of the revolution-counting register is arranged on the left.

In the fourth version of the machine from 1878, the setting slides were provided with small springs, which, when the slide has been set to a certain digit, causing the slide to slip into a notch opposite that digit so that an accidental movement of the setting knob during operation of the crank is avoided. The tens-carry mechanism was materially improved. Means were provided to prevent overthrow. The capacity of the model is 10 x 11 x 20.

Let’s examine the principle of work of the mechanism:
The machine presents two principal parts (see the drawing below), a fixed setting plate with a series of sliders for inputting numbers (marked with A), and a movable carriage where results appear (M). The number, set with the sliders, is mechanically transferred to the result dials on the carriage (C) by turning the handle (N). This transfer operation, basic to all the arithmometer’s workings, is accomplished using the stepped cylinders of Leibniz.

Each cylinder carries nine teeth whose length increases step-wise (see the drawing below) (marked with A). The cylinder’s teeth engage a pinion (B), whose position is controlled by the setting slider (C). The higher the number set by the slider, the larger the number of teeth on the cylinder engaging the pinion. When the handle is turned the cylinder rotates and as a result, the pinion’s square arbor (F) is turned through an arc proportional to the value set on the slider. It is this rotation that is communicated to the result dial (K) via a bevel wheel (G). One turn of the handle adds the value set on the sliders to the result dials and, since multiplication is simply repeated addition, turning the handle, say, 8 times multiplies the given number by 8. To multiply by 38 it is not necessary to turn the handle 38 times. Rather, after turning it 8 times, the carriage is moved one step to the right and the handle is then turned 3 times.

Using the reversing switch on the setting plate (upper figure, B), the machine can be set to perform subtraction and division. The lower figure shows the result of pushing the switch: M slides forward, disengaging the bevel wheel G from I (on the axis of the result dial K), and bringing H into contact with I. Now when a turn of the handle causes arbor F to turn, I—and thus the result dial rotates in the opposite direction, reducing rather than increasing the displayed value. A turn of the handle thus subtracts the number set on the sliders from a number entered on the result dials.

As multiplication is repeated addition, so division is repeated subtraction, with the quotient appearing in the smaller set of dials on the carriage (upper figure, D). These quotient dials are simply counters: each turn of the handle increments the dial currently in contact with the counting mechanism by one unit. The quotient dials are also useful in multiplication since they provide a visual check on the value of the multiplier. Finally, when a calculation is complete, the carriage dials can be reset to zero: each set of dials has an independent zeroing mechanism operated by twisting one of the two knurled knobs at either end of the carriage (upper figure O and P).

Thomas first ventured into the world of exhibitions when the arithmometer was revived in 1844. A machine was entered in the French national exhibition of industrial products where it was classed amongst precision instruments in a category of diverse measures, counters and calculating machines. If Thomas had hoped for substantial recognition and reward, he was to be disappointed. The arithmometer was granted an honorable mention in the jury report but was clearly considered inferior to the submission of the Austrian emigré doctor Didier Roth, who obtained a bronze medal for his adding and calculating machines and counters. The judgment of the 1844 jury was mirrored in the coverage given to Roth in a separate guide to the exhibition’s highlights, in which Roth’s adding machine was described and illustrated, while the arithmometer was ignored.

The next French national exhibition took place in 1849 and Thomas again tried his luck. On this occasion, he was awarded a silver medal and the jury report devoted three pages to his machine. However, despite this higher honor, he was again eclipsed, for a gold model went to the mechanics Maurel and Jayet for their Arithmaurel, a calculating machine with automatic capabilities, judged to exceed those of the arithmometer.

A further competitive opportunity was soon offered by the 1851 Great Exhibition. But again, Thomas was frustrated. The arithmometer was one of two calculating machines to receive a prize medal, but the jury decided that it was inferior to a Russian entry devised by Izrael Staffel, originally a watchmaker from Warsaw. The arithmometer was illustrated in the official catalog, but it was Staffel’s calculator, already successful at a Polish exhibition and rewarded in St Petersburg, which was featured in the Illustrated London News.

Disappointed again, Colmar decided to prepare well for the next challenge, the 1855 exhibition in Paris. He created a giant machine, especially for the exhibition (see the nearby photo). Some six feet long, equipped with 15 setting sliders and 30 result dials, and encased in fine cabinetwork the result was evidently meant to capture more than technical interest. And again arithmometer with no more than an honorable mention, this time the goals medal was awarded to Scheutz’s difference engine.

As a whole, in his long production history, the arithmometer of Thomas de Colmar received many medals, but very often has been neglected on the account of more sophisticated and advanced machines, which however will never reach the market and production achievements of his rival.

Biography of Charles-Xavier Thomas de Colmar

Charles-Xavier Thomas, also known as de Colmar, was born on 5 May 1785, at number 8 rue Rapp in the town of Colmar, the capital of the Alsace wine region. He was the son of Joseph-Antoine Thomas (1758-1831), a physician, and Françoise-Xavière Entzlen (Anselin) (1759-1817). Joseph-Antoine Thomas studied medicine in Freiburg and married on 12 November 1781, in Rastatt (Baden) to Françoise-Xavier, a native of Carlsruhe, Baden-Württemberg.

The Thomas family, originally from Burgundy, moved to Guebwiller in the Alsace region during the 30-year war, around 1634. Sir Thomas, born on 8 February 1758, in Guebwiller, after graduating from the University of Freiburg in the early 1780s, practiced medicine in Colmar and then at the Hospice in the town of Rouffach, where he died on 11 April 1831 (Françoise-Xavière also died in Rouffach on 1 May 1817). Joseph-Antoine was a member of Rouffach’s town council.

After finalizing his studies and after a quick passage through the administration of the French Regie, Charles-Xavier joined the French army during the 1809-1811 and 1813 campaigns in Portugal and Spain. He was Cashier General for supplies in Portugal and Spain in 1809, then General Manager of the supply store of the army’s headquarters in Seville in 1810. He was then General Manager of the supply store of all the armies located in Spain in 1813. When he arrived in Bayonne just after the defeat of Vitoria he was promoted to Inspector of Supply for the entire French army.

It was during his lengthy stay with the armies of Marchall Soult where he needed to perform a lot of calculations, that he conceived the idea of the calculating device—arithmometer. This became even more important in his eyes when, in 1819, he founded Le Phénix fire insurance company (he was named General Manager for 15 years, but resigned the next year, following a disagreement with his partners and shareholders) and, later, the companies Soleil (Sun), founded in 1829, and Aigle (Eagle), founded in 1843, that became the number one insurance group in France at the beginning of the Second Empire. Thomas became one of the founders of this industry in France (he introduced many innovations in this industry) and his business success in insurance, later on, will allow him to invest in such а non-profitable enterprise, as the production of calculating machines. Thus this remarkable man remained in history best known for designing, patenting, and manufacturing the first commercially successful mechanical calculator in the world.

In early 1811 Charles-Xavier married in Seville to a young woman from one of the oldest and most renowned families of Andalusia— Francisca (Frasquita) Garcia de Ampudia Alvarez (23 September 1794, Marbella, Espagne – 19 Oct 1874, Paris), who became his faithful companion for life and gave him seven sons (two of them died in infancy), and three daughters. Their first child, Joseph Thomas Alvarez (1811-1873), was born in Seville in December 1811. Later they will have Charlotte Marie (1813-1840), Antoine Auguste (b. 1815), Louis François (b. 1816), René (b. 1817), Nicolas Louis (1818-1881), Charles (b. 1821), Frasquita Madeleine Joséphine (1821-1905), Henri (b. 1825), Emmanuel Eugene (1827-1840), Henriette Leontine (1831-1876).

In his long life Thomas de Colmar was decorated with many orders—in 1821, Chevalier of the Legion d’Honor (for his invention of the Arithmometer); 1852, Knight of the Ordre de la Couronne de Chêne; 1852, Commander of the Ordre de Saint Grégoire le Grand; 1853, Croix de Chevalier du Sauveur; 1854, Knight of the Ordre des Saints Maurice et Lazare; 1857, Officier de la Légion d’honneur and others.

Thomas de Colmar died of acute bladder disease on 12 March 1870, at the age of 84, in one of his properties on 156 Boulevard Haussmann in Paris, and was buried in the Père-Lachaise cemetery. The Sun King (as he was dubbed) left a huge fortune of over 24 million francs, not to mention château de Maisons-Laffitte, château de Champfleury in Carrières-sous-Poissy, château and domaine de Mairé in Vienne, etc. By his death, the “Aigle – Soleil” group was the biggest insurance business in France and he owned 81% of it.

The prominent US statistician, romance novelist, and amateur engineer Charles Felton Pidgin (1844-1923) from Boston, Mass., was a holder of quite a few patents (at least ten) for various devices, like indicator, apparatus for compiling statistics (Pidgin’s system for the census, the so-called “chip” system, in the late 1880s was the main rival of the tabulating system of Hollerith), calculating mechanism for typewriter (pat. No. 1044597), motion picture (a very funny idea for adding word balloons to movies by having the actors blow into inflatable tubes that had the words on them), and others, between them two patents for electro-magnetic calculating machines (US pat. No. 284755 and US pat. No. 735291).

The first patent (US pat. No. 284755 from 11 Sep. 1883) for an electro-magnetic adding machine was granted to Charles Pidgin and Francis Leonard. It was probably the first patented electro-magnetic calculating machine in the world. Francis Henry Leonard Jr. (1859-1908) was a Boston salesman and machinist, who also was a holder of several patents. The second patent of Pidgin was granted in 1903 and was for electro-magnetic adding machine of a different type.

To illustrate the mode of operation and manipulation of the machine of Pidgin, let’s refer to the first patent drawing.

In the operation of the machine, to indicate the number “1” on any of the indicators, the circuit-closer h of that indicator is drawn forward to make contact with the first spring, k, and the result appears by the presentation of the figure “1” at the corresponding aperture. If the indicator be the first one, the figures “1” will simply indicate unit 1; if the second, 10; if the third, 100; the fourth, 1000, and so on. When the circuit-closer h is drawn forward to make contact with the first spring, k, the battery-circuit is closed through battery post or screw 0, wire I, plate i, spring l, and screw m of circuit-closer h, contact-spring k, (and supposing we are operating the unit-slide,) wire 3, magnet-coil M, wire 6, common return-wire 11, and terminal o’, thence to the other battery-pole or ground wire. The electromagnet M is thus vitalized and attracts the armature-arm f, causing the escapement to rotate the ratchet-wheel e one space, the dial j, of course, rotating with it and showing the figure “1” at the opening t of the casing. Every time the circuit-closer h touches a spring, k, an electrical impulse is sent through the electromagnet, and the dial j advances, as described.

To further illustrate the manipulation, let’s refer to Fig. 6. We have already described how one number is indicated on the dial j. Let it be supposed that we wish to add 29 to 33, the operation is as follows: To add 29 to 33, the unit-dial indicating “3” and the tens-dial indicating “3”, to add the “9” to the three units, pull the unit-key down to the point numbered 9 and push it back, making the contacts going down and four going back. The unit-dial will then indicate “2” and on the tens-dial “1”; ten will be carried to the “3”, changing it to four tens. To add the two tens, pull the tens-key to 2 and return, making one contact going and one returning. The tens-dial will then indicate “6” and the units-dial “2”, reading from left to right “62”, which is the sum of the two numbers 33 and 29. The machine will of course always show the sum of all numbers registered subsequent to the time it was last set at zero; and any number may be registered from 1 to 990000000, the simple registering of an additional number or numbers immediately changing the reading on the dials to the sum of the number previously registered and the amount added by the last operation.

The machine has an automatic electric carrying device, whereby the figures of one arithmetical denomination are automatically carried to the next, if tens carry should be done.

Biography of Charles Pidgin

Charles Felton Pidgin was born on 11 November 1844, in a house in Felton place at Boston Highlands, Roxbury, Massachusetts, to Mary Elizabeth (Felton) Pidgin and Benjamin Gordon Pidgin (1820-1882). Charles had two younger sisters—Mary Felton (b. 1848) and Nancy, who died young. Benjamin Pidgin was a varnisher, while Mary Elizabeth (born in Roxbury in Jan. 1824 to Joshua Felton (1787-1835) and Hephsibah Skinner (1786-1854), eight generation descendant of Lieutenant Nathaniel Felton, who came to Salem, Mass., from Great Yarmouth, England, in 1633), as it was a general rule at the time, was keeping house.

As a young child, Pidgin was rendered lame by an accident to his hip, and he was also partially blind for a number of years, but he remained an unremitting worker for the rest of his life. After graduating from the English High School in Boston (one of the first public high schools in America, see the image below) in 1863, he worked for ten years as an accountant in the mercantile business.

In 1873 Pidgin was appointed as a chief clerk of the Massachusetts Bureau of Labor Statistics and remained connected with the Bureau for over thirty-four years. In addition to performing his administrative duties as an official of the Bureau, he contributed much statistical material for publication in the Bureau’s reports and labor bulletins, and in 1888 he published, independently, a book entitled Practical Statistics. He was appointed Chief of the Bureau in 1903, and remained there until 1907, leaving to focus on his writing work.

As we already mentioned, Charles Pidgin invented numerous devices and machines for the mechanical tabulation of statistical data, some of which were patented, and two advanced calculating machines. Pidgin developed also a system for the census (in his so-called “chip” system, data from the schedules were transcribed to colored cards, and the cards were counted by hand), which in the late 1880s was the main rival of the tabulating system of Hollerith.

I find it funny that besides the elaborated machines, in 1917, Pidgin patented a very ludicrous idea (patent No. US1240774A)—to display dialogue in silent films, as actors inflate balloons or party favor-like objects with text on them, in order to recreate the act of speaking. He even improved his funny idea, obtaining another patent in 1919 (patent No. 1317725).

His spare time out of office Pidgin devoted not only to inventions but also to literary work. Besides being a frequent contributor to periodicals, he wrote and gained even greater prominence as an author of many books, librettos for cantatas, operas, and musical comedies, than as a statistician and inventor.

Pidgin’s musical comedy adaptation of Peck’s Bad Boy was first produced in 1883 and ran for many years. Pidgin’s first and most popular novel Quincy Adams Sawyer, published in 1900, was aggressively marketed by his publisher, C.M. Clark Publishing, sold over 250000 copies for several years, and later was adapted for stage and films. His next novel, Blennerhassett; or, The decrees of fate (1901), was sold over 60000 copies before even appearing in print. The next novel of Pidgin from 1902, The Climax: or, What Might Have Been: A Romance of the Great Republic, envisioned an alternate history where Aaron Burr did not kill president Alexander Hamilton, and later became president.

Charles Pidgin married three times. He married his first wife Lizzie Abbott Dane on 3 July 1867, but she died the next year, in June 1868. In November 1873, Pidgin married Lucy Sturtevant Gardner (1850-1896), who became a doctor and practiced medicine until her death of heart disease in June 1896. Pidgin’s third wife was Frances Fern Douglas, they married in July 1897.

Charles Felton Pidgin died at his home in Melrose Highlands, Massachusetts, on 3 June 1923.

The first man, who was flashed upon to use electricity for calculation purposes, was probably a German mechanic, showman, electrifier and miracle healer, physicist, chemist, teacher, author, and mail order company owner, named Johann Conrad Gütle (1747-1827). Gütle was a self-taught scientist, who from 1780 worked as a traveling experimenter, organizing demonstrations at fairs and in restaurants in many German towns.
During one of his shows, at the beginning of June 1785, in Heilbronn am Neckar, a town in southwest Germany, Gütle demonstrated several electrical experiments, among them an electric calculating machine. The event was described in the local Journal von und für Deutschland thus:
In our area arrived the former bookbinder from Schwabach, called Gütle, who has knowledge in experimental physics that goes beyond empirical knowledge and knows how to amuse the public with his electrical experiments. However, there were also some gasconade and hand-held games. He showed also his electric calculating machine and cubes, which he pretends to show electrical phenomena of them without the tools being connected to the electric machine, a mere illusion consisting in an internally hidden conductor that cannot be seen when the machine it is on is not disassembled…

Gütle himself described the electric calculator several years later, in his 1792 catalog of devices for sale „Kunstkabinet verschiedener mathematischer und physikalischer Instrumente und anderer Kunstsachen, die theils zur Erleichterung der Lehre in den Wissenschaften, theils zu nützlichen Unterhaltungen und zum Vergnügen gehören“ thus:
The electric accounting machine, a number chosen by a person in secret, is instantly calculated by the electric fire on that machine, in the manner the person requested, either by numbering, adding, multiplication, dividing, or subtracting given numbers. There are different numbers on four boards, Tab. V, Fig. 21, you leave it up to a person to choose one of them, just ask them to put the board on which the number is on the magic table, and now determine what the electric fire should do with the chosen number, whether it should display such numbers, add, multiply, divide or subtract, and determine how much. The product of the displayed invoice type immediately appears in the electric fire, by discharging the bottle. You can see that the electric fire is making a lot of changes again.
The cost of the device was announced as 10 Thaler.

As you can see, Gütle used as an electrical source a “bottle”, i. e. a Leyden jar, an electrical component, invented in the 1740s, which stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. The first true battery (so-called Voltaic Pile) will be made in early 1800 by the celebrated Italian physicist Alessandro Volta (1745-1827).

Obviously, Johann Gütle as a good showman and salesman, decided to create a great impression on his public and readers, combining two interesting but puzzling areas of knowledge at the time—calculating machines and electricity. Certainly, he was not able to create a real electric calculator (the first electro-magnetic adding machine in the world will be created some 100 years later by the US statistician, romance novelist, and amateur engineer Charles Pidgin), but nevertheless, Gütle deserves our attention for his imagination.

Biography of Johann Conrad Gütle

Johann Conrad Gütle was born on 25 March 1747, in Schwabach, a town near Nuremberg in the center of the region of Franconia. He was the son of the local burger Johann Friedrich Gütlein, a bookbinder and fodder maker. As a young, Gütle learned his father’s trade (bookbinding), but over time, acquired self-taught knowledge in physics and mathematics.

Johann Conrad Gütle was a hyperactive man and a jack of all trades, who had a wide variety of professions. From around 1780 he used to work as a traveling experimenter (showman), and he also sold and repaired electrifying machines and accessories, and also healed the sick by electrifying (for the latter, he had problems, and during one of his shows, in Heilbronn, he was scolded as a fraud.) Gütle used to work also as a mechanic, watchmaker, and private teacher of mathematics, physics, and natural science.

Gütle settled in Nuremberg in 1788 and during the next 40 years, he published some 70 different titles (including sales catalogs and books), mainly in the area of electricity, mathematical and physical instruments, mechanics, geometry, etc. He was well known especially as a producer of electrification machines, as a drummer for lightning protection, and in 1788 he installed the first lightning-conductors in a few places in Franconia, and enjoyed an excellent reputation in this area. Gütle was mentioned in international literature as an inventor of “electric lamp” in which a gas lamp was ignited using an electrophor. He ran a lively mail-order business from Nuremberg and also manufactured cleaning, beauty, and hair restoration products as a “chemical industry expert”.

In 1770 Gütle married Sophia Magdalena Kern. Soon was born his first son, Friedrich Christoph Gütle, who became a skillful watch-maker. Sophia Magdalena died in 1782, and in 1788 Gütle married a second time Antonia Steingruber (she died on 26 Feb 1808), granddaughter of the famous Ansbach master builder and architect Johann David Steingruber (1702-1787). From his marriages, Gütle had eight sons, and three daughters, but only six of his children survived to adulthood.

Johann Conrad Gütle died on 18 October 1827, in Nürnberg.

At the end of 1617 in Edinburgh after the death of John Napier, was published (in the Latin language, which was a common practice then) his small book—Rabdologiae seu Numerationis per Virgulas libri duo. Looking to ease his own difficulties in calculating logarithmic tables, and impatient with the tedious and error-prone process of working with large numbers, Napier invented several mechanical methods of simplifying and speeding up multiplication, the most famous being special rods, later known as Napier’s bones. Besides the virgulas or rods, in his book John Napier described multiplicationes promptuario (promptuary of multiplication or lightning calculator), and the scacchiae abaco or chessboard abacus.

Napier published a description of his inventions in Rabdologiae, the title of which he derived from the Greek ραβδoς (rod) and λόγος (word) (incidentally, this section of Napier’s work also contains the first printed reference to the decimal point). The reason for publishing the work is given by Napier in the dedication, where he says that so many of his friends, to whom he had shown the numbering rods, were so pleased with them that they were already becoming widely used, even beginning to be used in foreign countries (one of these friends was Alexander Seton, the Earl of Dunfermline and High Chancellor of Scotland).

As opposed to the title …libri duo (two books), actually the Rabdologiae consists of four parts (books), two basic, and two appendixes. Book I is a description of the calculating tool, Book II offers forty-seven pages of tables, examples, and general problems demonstrating the utility of the rods in solving questions of geometry and mechanics. Book III is an appendix on Napier’s promptuary, a more elaborate calculating device consisting of engraved rods and strips; and Book IV is an appendix of forty-one pages, devoted to so-called arithmeticæ localis (location arithmetic).

Napier apparently based his invention on a popular during this time method for multiplication, described in several books—e.g. in the book of the famous Italian mathematician Luca Paccioli Summa de Arithmetica, Geometrica, Proportioni et Proportionalita, printed in Venice in 1494, and in De arithmetica practica, published in the 1540s by Oronce Fine, a late-Renaissance French mathematician, and cartographer. This method was called gelosia (jealousy) in Italian. Most probably these ancient methods for multiplication were invented by Indian mathematicians, then transferred to China and via Arabian Haliphat—to Europe. The gelosia method is as follows:

A grid of squares, divided into parts by a diagonal, must be cross-ruled, as the number of squares depends on the number of digits in factors, e.g. if we want to multiply 3-digital to 3-digital factor, then the grid must be 3 by 3 squares. To the upper side and right side of the grid must be written the two factors, and intermediate products are written in the squares in such a manner, that the diagonal divides the units from the tens. The units of the partial product (the digit from the right by the digit from the upper) are written on one side and the tens on the other, so that when a multiple consists of two figures they are separated by the diagonal. To get the final product, the numbers along the diagonals are added and the result is written to the left of the grid (senior digital positions) and below the grid (junior digital positions).
Let’s see an example, to multiply 456 by 128 (see the nearby figure).
As the two numbers are 3-positional, we have to draw a 3 x 3 grid, to the upper side we have to write 456 (first factor), and to the right side 128 (second factor). In the squares, divided by a diagonal, we have to write the products of a digit placed on the upper side of a particular column to the digit, placed on the right side, as in the upper left part of the square we have to write tens (if any), while in the lower right part, we have to write units. Then we have to prolong the diagonals and to add digits in every diagonal, starting with the units and if it is necessary, we make a carry to the next diagonal. In such a manner, we get the result—456 x 128 = 058368. The multiplication was done by means of addition.

Essentially, what Napier did (how often simple things are of genius!?), is that he made slips (columns) with all possible nine columns of squares of the gelosia grid, and thus he can put aside manual drawing of a grid and writing in squares. These slips are written on the surface of ten rods, later on, called Napier’s rods (the best sets of Napier’s numbering rods were made of ivory, so that they looked like bones, which explains why they are now known as Napier’s bones).

Let’s make a multiplication by means of Napier’s rods, e.g. 3105 x 6 (see nearby figure).
We arrange side by side four rods (for 3, 1, 0, and 5).
First, we have to take the row for factor 6 (marked with an arrow). We start from the right, taking initially zero from the lower right part of the first cell. It will be the first digit of the result. Then we have to add the three digits from the left part of the first cell with 0 from the right part of the second cell. Continuing to add the digits along the diagonals and we will get the proper result 18630.

During the multiplication of a number, which has identical digits, we have to use identical rods. That’s why Napier suggested rods take the form of a parallelepiped, on the four surfaces of which to be inscribed four digital columns of rods in such a manner, that the four faces of each rod contain multiples of one of the nine digits, and is similar to one of the slips just described, the first rod containing the multiples of 0, 1, 9, 8, the second rod of 0, 2, 9, 7, the third of 0, 3, 9, 6, the fourth of 0, 4, 9, 5, the fifth of 1, 2, 8, 7 (see the nearby figure), the sixth of 1, 3, 8, 6, the seventh of 1, 4, 8, 5, the eighth of 2, 3, 7, 6, the ninth of 2, 4, 7, 5, and the tenth of 3, 4, 6, 5. Each rod, therefore, contains on two of its faces factors of digits that are complementary to those on the other two faces, and the factors of a digit and its complement are reversed in position.

When the second factor is multi-digital, then intermediate products must be written manually, shifting one position leftwards, then intermediate products must be added (see the drawing below). After arranging rods for the multiplicand side by side, we have to multiply the multiplicand (46785399) by the units of the multiplier (96431). This result is 46785399 x 1 = 46785399. Then we have to multiply the multiplicand by the tens of the multiplier 46785399 x 3 = 140356197, shifting the result to one position left and continuing, while all digits of the multiplier will be used. Then we have to add manually the partial factors. It was a matter of time, someone to think about, that if we have an adding machine, the multiplication can be done without any thinking, and this happened only some years later—Wilhelm Schickard, who used Napier’s Rods in his Rechenuhr, made to assist Kepler in his astronomical calculations.

Obviously, the use of Napier’s rods is easy, but tedious when one wants to multiply two numbers each having two or more digits. That’s why Napier went further, and in book III of his Rabdologiae he described a more elaborate calculating device, consisting of engraved rods and strips (so-called promptuary). Promptuary comes from the Latin promptuarium, “a place where things are stored ready for use”.
The device consists of 200 (100 thick and 100 thin) rods, placed in a special box, called promptuary (one set consists of 10 rods for every digit). The tick rod is 5 mm thick, while the thin is 2,5 mm. The side surface of each rod is divided into ten squares and two rectangles (in the upper and lower part of the surface), in which are inscribed digits from 0 to 9. The squares are divided into triangles: these of thick rods—with diagonals from the upper right corner to the left lower, while the squares of the thin rods—from the left upper to the right lower (on the nearby figure you can see the part from two rods to the left is placed the thick rod for the digit 4, to the right is placed thin rod for the 7, cut off (missing) triangles are filled with black color).
Tick rods are filled with digits (or empty for zero), while on the surface of thin rods are cut-off triangle openings (without openings for zero). Squares of the rods, which are inscribed with digits from 1-9 (not 0), are divided into nine smaller squares, and each of these squares is divided in half into 3 triangles by means of a diagonal, parallel to the diagonal in the bigger square. In the triangles are inscribed digits according to the letters a, b, c, d, e, f, g, i, and j, as shown in the lower figure, following the rule:

On the surface of the thick rods are inscribed multiple numbers of digits, marked on the rod, as the digit of the rod is inscribed instead of the letter a, a doubled digit is inscribed instead of the letter b, and so on. If the multiple number has two digits, then tens are inscribed instead of the upper letters, while the units are inscribed instead of the appropriate lower letter (see the lower figure).

On the surface of the thin rod with header 1, the openings are cut off at the place of triangles, marked with the letter a, on the surface of the thin rod with header 2, the openings are cut off at the place of triangles, marked with the letter b and so on (as it is shown in the right square of the upper figure).

During the multiplication, the thick rod is leaned against the thin rod in such a manner, that the diagonals of the two big squares coincide (thin rods are rotated to 90 degrees). Then in the openings of the thin rod can be seen the digits from the product of the numbers, which are inscribed on these rods. In the next figure, you can see the multiplication of 7213 x 524. First must be arranged properly thick rods, according to the multiplicand 7213. Then the thin rods, according to the multiplier 524, are placed over and rotated to 90 degrees to the thick, as the diagonals of squares on the thick and thin rods are concurrent. The result can be shown by adding of visible digits (in the figure the digits are shown in the thick triangles).

In the description of the promptuary Napier specified, that it can be used for the division also. For this action, he suggests to be found first the reciprocal of one of the multipliers, then to be done multiplication with the promptuary.

Napier’s rods rapidly became popular in England. According to one Seth Partridge, a London-based surveyor and mathematical practitioner, these reckoning rods were easy to make in any material whatsoever; they could either be manufactured by oneself or bought in the London instrument shops:
These speaking-Rods may be made either of Silver, Brasse, Ivorie, or Wood, as the maker and user of them best pleaseth, but they are most ordinarily made of good sollid Box, and being thereof made, they are as usefull as those made of any other substance whatsoever, Nay, I hold them more light and nimble then those made of Mettall; …Every practitioner may make them himselfe by cutting the faces of every one of the printed papers of the Rods, and so placed on a square piece of wood as before; or else they are ready made in Wood, by Master John Thompson…

In the next centuries, a lot of inventors tried to improve and facilitate the work with Napier’s rods, starting with the above-mentioned Wilhelm Schickard in the early 1620s.

In the early 1650s, an attempt to make a tool with Napier’s rods made the French physicist, cartographer, and engineer Pierre Petit (1594-1677), a King Counsellor and Intendant des Fortifications. Petit placed paper strips with Napier’s rods and made a mechanism, the so-called Arithmetical Cylinder or Tambour de Petit (Cylinder of Petit) (see the nearby image), allowing the paper strips to be moved along the axes. The device he described in his book Dissertations academiques sur la nature du froid et du chaud. Avec un Discourssur la construction & l’usage d’un Cylindre Arithmetique, inventé par le mesme Autheur (Paris, 1653).

According to Petit, people ceased using Napier’s “beautiful invention” because “the multitude and embarrassment of those sticks, filled with numbers on all sides, proved prolonged and tedious.” Since Petit found this method of calculating still useful, and because it was “easier to improve on inventions than to become an inventor”, he designed long bands or ribbons of paper each containing all the multiples of Napier’s rabdology. Those long bands were then attached end to end and mounted on a wooden cylinder the size of a child’s drum or a hat, and of a length that depended on the number of bands one wished to have in order to make calculations with large numbers.

The reckoning principles were identical to Napier’s bones. Pierre Petit deemed these common enough by then that he wrote only a brief summary of how to proceed toward making a multiplication and a division.

Several years after Petit, in the late 1650s, a device with Napier’s rodes was developed by the famous German scientist Athanasius Kircher, and his pupil and friend Gaspar Schott.

In 1667 Sir Charles Cotterell (1615–1701), an English courtier and translator devised a calculating instrument (called arithmetical compendium) with Napier’s rods, which included a wire-and-bead abacus for adding the partial products.

In 1673 was published the book The Description and Use of Two Arithmetick Instruments of Samuel Morland. In this book are described two calculating devices, one of them, the so-called multiplying machine was based on Napier’s rods.

In the same year (1673) a cylindre arithmetique (Napier’s bones engraved on a cylinder) were used in the adding instrument (called nouvelle machine d’arithmétique) of René Grillet de Roven.

In 1727 in the book Theatrum arithmetico-geometricum of Jacob Leupold, was described a calculating tool (so-called calculating drum), based on Napier’s rods.

In 1728 the German scientist Johann Michael Poetius described in his book “Anleitung zu[r] arithmetischen Wissenschaft, vermittelst einer parallelen Algebra” (Instructions for arithmetic means of science, a parallel algebra) an instrument, composed of concentric moving circles (so-called Mensula Pythagorica), which seems to be a variation of the Napier’s bones and can not render more services than the multiplication table (on the nearby image is shown a sector of Mensula Pythagorica).

In 1789, the German bailiff and mathematician F. X. M. Prahll devised an instrument, which he called Machina Arithmetica Portatilis (portable arithmetical machine), and which was essentially the same as the Mensula Pythagorica of Poetius, except only that the movable circles were much larger and carried the numerals 1 to 100 so that with the aid of that instrument numbers could be added and subtracted up to 100.

An instrument (called Rechenscheibe, see the upper drawing), similar to Mensula Pythagorica of Poetius, was devised in 1790 by the German professor of mathematics in Berlin Johann Philipp Grüson (1768-1857).

One of the recent chapters in the development of Napier’s bones as a calculating instrument took place at the end of 1870s when the French mathematician François Édouard Anatole Lucas (1842–1891) presented to the Académie Française a problem on arithmetic, that caught the attention of Henri Genaille (1839–1903), a French civil engineer, employed by the railway system in Tours. Genaille, who was already quite well known for his invention of several different arithmetic aids (yet in 1878 he gave a lecture, discussing a version of Napier’s Rods, which avoided the need to carry from one position to another), solved Lucas’s problem and, in the process, essentially devised a different form of Napier’s rods. This instrument, presented in 1885, eliminated the need to carry digits from one column to the next when reading off partial products. Genaille demonstrated his instrument (so-called Réglettes multiplicatrices, Reglettes Financieres or Réglettes de Genaille Lucas – Genaille-Lucas rulers) in 1891. Lucas gave these rulers enough publicity that they became quite popular for a number of years and several instruments, based on the rulers have been manufactured.

Lucas gave glowing praise to co-inventor and invention alike:
“An engineer at the State Railways in Tours, Mr. Henri Genaille, obscure yesterday, illustrious tomorrow, had the exceedingly remarkable and ingenious idea of replacing these additions with very simple drawings which allow all these partial products to be instantly read. The maneuvering of these rods is as easy as that which consists in following a path through a labyrinth, by means of indicator arrows on posts placed at the crossroads; that is to say that we learn to use these rods in a minute at most”.

In fact, Genaille and Lucas marketed four boxed sets. The first, the one shown here, was for multiplication; the second, for division; the third, for financial calculations; and the last was a set of the classic Napier rods. The financial set is actually a special derivative of the division set, optimized for the single task of figuring daily interest for a given initial sum and annual interest rate.

The multiplication set of Genaille-Lucas rulers contains 11 strips. The physical size of the original set is: the black cardboard container is 12 x 19 x 1 cm, and the 11 rods each measure 1 x 1 x 17 cm. The first strip (marked with Index in the lower figure) has only one useful side, which corresponds to the multiplier. It has nine rectangles (for the digits from 1 to 9), as the height of each rectangle is proportional to the digit in it. The remaining ten strips have four useful sides, as each side of a particular strip is for a different digit of the multiplicand. In the upper part of the strip is inscribed the digit of the multiplicand, and the lower part of the strip is divided into two vertical columns.

The multiplication can be done, as the strips for all digits of the multiplicand are arranged side by side, then the result can be read from right to left (black triangles in the left column of the strip for each digit of the multiplicand), as first can be read units, then—tens and so on.

Let’s make a simple multiplication with the rulers (3271 by 4) (see the nearby figure). First, we have to arrange side by side the proper rulers for the multiplicand and index ruler, placing also to the left the ruler for 0. Then, starting with the fourth rectangle (this for the digit four) of the rightmost (index) ruler (the multiplier is 4) leftwards, we have to select the digit at the top of the rectangle (4 in this case) and then simply to follow the black arrows leftwards, reading off the digits as we come to them—4, 8, 0, 3, 1, and thus we have the product 13084. In contrast to Napier’s rods, the result was obtained without making any arithmetical operations.

The rulers for division are similar to the multiplication ones, except that the large arrows are replaced by a multitude of smaller ones.

Let’s say some words about Book IV of Rabdologiae, which is an appendix of forty-one pages, devoted to so-called arithmeticæ localis (location arithmetic). This appendix contains one of the first explorations of binary arithmetic as a computation aid. Location arithmetic is a technique to do binary arithmetic using a chessboard-like grid. Using simple moves of counters on the board, Napier showed ways to multiply, divide, and even find the square roots of binary numbers. He was so pleased by his discovery that he said in the preface:
…it might be well described as more of a lark than a labor, for it carries out addition, subtraction, multiplication, division, and the extraction of square roots purely by moving counters from place to place.

Location arithmetic uses a square grid where each square on the grid represents a value. Two sides of the grid are marked with increasing powers of two. Any inner square can be identified by two numbers on these two sides, one being vertically below the inner square and the other to its far right. The value of the square is the product of these two numbers. A very good description of location arithmetic can be found on the site of Mr. Stephan Weiss, www.mechrech.info.

In July 1614 in Edinburgh, Scotland, was published a small book (57 pages of explanatory matter and 90 pages of tables) which will make a key advance in the use of mathematics. The book was Mirifici Logarithmorum Canonis Descriptio (Description of the Marvelous Canon of Logarithms), written by a Scotsman—John Napier. Two years later an English translation of Napier’s original Latin text was published, translated by Edward Wright. What is the history of this remarkable book?

There is indirect evidence that Napier was occupied with logarithms as early as 1594 (in a 1624 letter from Kepler to Petrus Criigerus). Moreover, some historians claim, that there was another man, who invented logarithms before Napier (around 1588), the Swiss mechanical and mathematical genius Joost Bürgi (1552-1632). Bürgi however published his work (Tafeln arithmetischer und geometrischer Zahlenfolgen mit einer gründlichen Erläuterungen, wie sie zu verstehen sind und gebraucht werden können) 6 years after Napier, as late as 1620 in Prague. Described by Bürgi tables distinctly involves the principle of logarithms and may be described as a modified table of antilogarithms. Bürgi’s method is different from that of Napier and was clearly invented independently. Bürgi was also an important contributor to prosthaphaeresis, a technique for computing products quickly using trigonometric identities, which predated logarithms and was introduced circa 1005 by Ahmad ibn Yunus al-Sadafi al-Misri (c. 950-1009), an important Egyptian Muslim astronomer and mathematician, and introduced in Europe by Nuremberg mathematician Johannes Werner (1468-1522) in the late 15th century.

Interestingly, after the Descriptio, published in 1614, appeared another Napier’s book on the logarithms, known as the Constructio (Mirifici Logarithmorum Canonis Constructio), published after his death in 1619 by Robert Napier (1580-1655), his second son by the second marriage. Internal evidence as well as the distinct statement of Robert, make it clear that it was in fact written years before the Descriptio, and it represents in many passages an earlier stratum of thought. In the Preface (Greeting) of Constructio Robert asserts:
You have then (kind Reader) in this little book most amply unfolded the theory of the construction of logarithms, (here called by him artificial numbers, for he had this treatise written out beside him several years before the word Logarithm was invented,) in which their nature, characteristics, and various relations to their natural numbers, are clearly demonstrated.
The Constructio is considered to be the most important of all Napier’s works, presenting as it does in the clearest and simple way the original conception of logarithms.

It seems the concept of a logarithm made its first appearance in ancient Babylonia (just as the concept of the abacus) around 1800 B.C. Baked clay tablets have been found, which contain tables of successive powers of whole numbers. In some of these records, a question is asked: “To what power must a certain number be raised in order to yield a given number?” More than 1500 years later the great Archimedes made an observation that is the basis of modern logarithms—he defined “the order” of a number to be equivalent to the exponent and observed that the addition of orders corresponds to finding their product.

In his book, Napier proposed a method, which allows the more complex arithmetical operations such as multiplication, division, and calculating of a root to be done by means of addition and subtraction. He realized, that all numbers can be expressed in what is now called exponential form, meaning 8 can be written as 2³, 25 as 5^2, and so on. What makes logarithms so useful is the fact that the operations of multiplication and division are reduced to simple addition and subtraction. When very large numbers are expressed as a logarithm, multiplication becomes the addition of exponents.

In the preface of the book Napier explains his thinking behind his great discovery—Seeing there is nothing that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances. And having thought upon many things to this purpose, I found at length some excellent brief rules to be treated of (perhaps) hereafter. But amongst all, none more profitable than this which together with the hard and tedious multiplications, divisions, and extractions of roots, doth also cast away from the work itself even the very numbers themselves that are to be multiplied, divided and resolved into roots, and putteth other numbers in their place which perform as much as they can do, only by addition and subtraction, division by two or division by three.

Napier didn’t mention what kind of sources he used, in order to develop the logarithms. It is known, that he was acquainted with a mathematical problem, solved some 25 years before 1614—multiplying two sines together was solved by the method of prosthaphaeresis, which corresponds to the formula:
sin a x sin b = [cos(a - b) - cos(a + b)]/2
As Napier knew of, and used, the method of prosthaphaeresis, it may well have influenced his thinking, because the first logarithms were not of numbers, but were logarithms of sines.

Another factor in the development of logarithms at the beginning of the 17th century was that the properties of arithmetic and geometric series had been used since at least Greek times and had been studied extensively in the previous century. We now know that any numbers in an arithmetic series are the logarithms of other numbers in a geometric series, in some suitable base. For example, the following series of numbers is geometric, with each number being two times the previous one:
natural numbers 1 2 4 8 16 32 64 128 256 512 1024
And the series below is an arithmetic one whose values are the corresponding base 2 logarithms:
logarithms 0 1 2 3 4 5 6 7 8 9 10
It had long been known that if you take any two numbers in the arithmetic progression, say 3 and 4, their sum, 7, would indicate the position of the term in the geometric series that is the product of the terms in the corresponding positions of the geometric series, e.g., 3 + 4 = 7 and 8 x 16 = 128 (the third times the fourth = the seventh).
This is starting to look very much like our own conception of logarithms as being the powers to which some base number is raised, a concept that was not understood in Napier’s time. Often the use of a good form of notation will suggest some basic mathematical principle. Our use of indices to indicate the power to which a number is being raised seems to have an obvious connection with logarithms, but without this form of notation, the connection is vague at best.
John Napier came at the idea of logarithms not by algebra and indices but by way of geometry. When first thinking about this subject, he used the term artificial number but later created the term logarithm from two Greek words—λoγoς and αριθμός, meaning word, ratio, and number respectively. He decided to use this term because his logarithms were based on the concept of points moving down lines in which the velocity of one point was based on the ratio of the lengths of the line on either side of it.

At the end of 1614 one of the most famous English mathematicians of the day, Henry Briggs (1561-1630), who was a Professor of Geometry at Gresham College, London, obtained a copy of Napier’s Descriptio and, by March of the following year wrote that:
Napier, lord of Markinston, hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.

Briggs immediately began to popularize the concept of logarithms in his lectures and even began to work on a modified version of the tables. Working together with Napier for some time (he was able to visit Napier twice at his baronial estate, staying there for a month in 1616, and again in 1617), later on, Briggs will propose (firstly in his 1617 Logarithmorum Chilias Prima (The First Thousand Logarithms)) that the base of the logarithms should be changed in order to make them easier to use, thus we have 10 based logarithms, and in 1624 published tables (Arithmetica Logarithmica), containing the logs of the numbers from 1 to 20000 and from 90000 to 100000 all calculated to 14 decimal places.

In order to make calculations by means of logarithms, we have to use tables with logarithms. If we have to multiply two numbers, we have to find their logarithms in the table, to add the logarithms, and then to find in the table the number, which logarithm corresponds to the sum of logarithms. But what to do, if this logarithm cannot be find in the table? Let’s for example suppose, that we have only a table with decimal logarithms of the integers up to 1000, but we need a logarithm of a fractional number, e.g. 7,93. As long as
7,93=793/100
then
log10(7,93)=log10(793)–log10(100)
So, the needed logarithm can be found by subtraction of 2 (this is the decimal logarithm of 100) from the logarithm of 793. Of course, the usefulness of the decimal logarithms is due to our decimal numbering system.

It was not only Briggs who was impressed by Napier’s Descriptio. In 1617, Johann Kepler first saw the Descriptio in Prague. He was too busy to pay it much attention, as during this period he was hard at work on his third law of planetary motion, but did acknowledge its existence in a letter to his friend Wilhelm Schickard, the creator of the first mechanical calculator in the world, where he indicated:
A Scottish baron has started up, his name I cannot remember, but he has put forth some wonderful mode by which all necessity of multiplications and divisions are commuted to mere additions and subtractions.

A year later, Kepler wrote to Napier expressing his admiration and letting him know that he must publish the promised Constructio as soon as possible. Unfortunately, Napier had been dead for two years before the letter arrived. The letter may have spurred John’s son, Robert Napier, into putting the finishing touches on his father’s notes and overseeing the publication of the Constructio in 1620. In 1624 Kepler published his book Chilias Logarithmorum ad totidem numeros…, creating the logarithmic tables by a new geometrical procedure, the form thus differing from the logarithms of both Napier and Briggs.

Within twenty years of the time that Briggs’s tables first appeared, the use of logarithms had spread worldwide. From being a limited tool of great scientists like Johann Kepler, they had become commonplace in the schoolrooms of civilized nations. Logarithms were used extensively in all trades and professions that required calculations to be done. It is hard to imagine an invention that has helped the process of computation more dramatically than logarithms.

Very soon after the popularization of logarithms began attempts for removing the nasty end error-prone process of looking into tables and manually adding numbers. First was a colleague of Brigs and professor of astronomy at Gresham College—Edmund Gunter (1581-1626). Briggs’s work naturally came to the notice of Gunter, who had some earlier experience in the development of calculating instruments, having been one of the major figures in the perfection of an instrument known as a sector. This experience soon led him to realize that the process of adding together a pair of logarithms could be partially automated by engraving a scale of logarithms on a piece of wood and then using a pair of compasses to add together two values in much the same way, as he would have done when using a sector. Not only did this method eliminate the mental work of addition, but it also removed the necessity for the error-prone and time-consuming process of looking up the logarithms in a table. Gunter described his instrument in the book Description and use of the Sector, the Crosse-staffe and other Instruments, published in 1623. Such a rule is frequently referred to as a Gunter Line or Gunter Scale, often a combination not only of a logarithmic scale, but also of chords, sines, tangents, and rhumbs. A two-foot-long boxwood ruler inscribed with various scales was a standard navigator’s tool until the end of the 19^th century.

Biography of John Napier

John Napier (John Napier frequently signed his name “Jhone Neper, Fear of Merchiston”, but later on we can find his name also as Napeir, Nepair, Nepeir, Napare, Naper, Naipper, Neperus, etc., the only form of Napier that for sure would not have been used in Napier’s lifetime was the present modern spelling “Napier”!) was born on 1 February 1550, in Merchiston Castle (or Tower as it was usually called in those days), in the vicinity of Edinburgh, Scotland, as the first child of Sir Archibald Napier (1534–1608), 7th Laird of Merchiston and Baron of Edenbellie.

Napiers was one of the most important Scottish families at the time. The second Napier was Controller of the King’s Household and ambassador. The third sat in Parliament, and several received the honor of knighthood. Three of them fell in battle (between them John Napier’s grandfather and great-grandfather). During the 15th century, no fewer than 3 Napiers were provosts of Edinburgh.

Sir Archibald Napier fully maintained the repute of his ancestors for energy and sagacity. He had studied the laws and was proficient in mathematics. In 1549, being only 15 years old, Archibald married Janet Bothwell (1534-1563), a daughter of the member of Parliament Francis Bothwell and sister of Adam Bothwell, Bishop of Orkney. The next year was born his heir John, then two other children—Francis and Janet.

Archibald Napier was a justice-depute and was knighted in 1565. After some troubles during the civil war in the 1560s and 1570s, Archibald was back in favor, and in 1582 he was appointed Master of the Mint in Scotland, with the sole charge of superintending the mines and minerals within the realm, and this office he held till his death in 1608. The family also owned estates at Lennox and at Menteith and a residence at Gartness. Sir Archibald must have been very wealthy, and John inherited from him an estate sufficient to live well on.

Three years before the birth of John, in 1547, Alexander Napier, John’s grandfather, had been slain while fighting in the army of Mary Queen of Scotland against the English at the Battle of Pinkie. Later John Napier would spend most of his life trying not to get involved in the sectarian strife that swept Scotland.

As was the practice for members of the nobility, Napier did not enter school until he was 13, being rather taught at home. When he was 13, in October 1563, he entered St Andrews University (he was registered as Johannes Neaper). His mother arranged for him to live in St Salvator’s College (see the photo below), the oldest of the three endowed collegiate societies of the university, and special arrangements were made for the principal of the college, John Rutherford (1515-1577), the most distinguished teacher of his day in Scotland, to take care of him personally. Rutherford was an able academic and, besides adding to Napier’s anti-Rome views, gave him a good grounding in Latin and the other subjects then taught at the college. Napier wrote many years later that it was in St Andrews that he first became passionately interested in theology, but there he studied also Latin and mathematics.

Unfortunately, on 20 December 1563, only two months after Napier matriculated at St Andrews, his mother Janet died only 29 years old. Later on, in 1571 Sir Archibald married a cousin, Elizabeth Mowbray, by whom he had ten children—(Sir) Alexander (of Lauriston); Archibald (of Wowmet); Walter; William; Susanna; Abellina; Agnes; Helen; Marion; and Elizabeth.

Little is known about John Napier’s early years. One of the few scraps of information that we have is from a letter from the Bishop of Orkney, John’s uncle, to Archibald Napier written when John was eleven years old:
I pray you, schir, to send your son Jhone to the schuyllis; oyer to France or Flandaris; for he can leyr na guid at hame, nor get na proffeitt in this maist perullous worlde …

(Let’s translate to English the old Scots that the Bishop of Orkney actually wrote):
I pray you, sir, to send your son John to school; over to France or Flanders; for he cannot learn well at home nor get profit in this most perilous world—that he may be saved in it;—that he may seek honor and profit as I do not doubt that he will…

For an unknown reason, Napier had not remained long enough at St Andrews to get a degree. His name didn’t appear in the list of determinantes for the year 1566, and of masters of arts for 1568. Most probably in 1567 Napier was forced to study abroad, having left St Andrews University prematurely (probably after his friendship with a Catholic student—according to Napier’s own account, he had there “contracted a loving familiaritie with a certaine gentleman, a Papist”). Even this was thought inadvisable in such sensitive times. Anyhow, Napier left the University without taking a degree and went abroad for several years.

Absolutely no evidence exists in which European country Napier studied (the University of Paris is highly likely and it is also probable that he spent some time in Italy, Germany, and the Low Countries), but when he returned home in 1571, he was a scholar competent in Greek, and applied himself closely to the study of the mathematics.

In 1571 Napier returned to Edinburgh, to find his father imprisoned in Edinburgh Tower by the Queen’s party, while the family home at Merchiston was occupied by the forces of the Regent, then besieging the town. The following year, 1572, when Merchiston Castle was bombarded by the guns of Edinburgh Castle, Napier sought refuge on one of the family estates at Gartness in Stirlingshire.

In 1571, the preliminaries of his marriage were arranged at Merchiston, to Elizabeth Stirling (1554-1579, see the image below), the daughter of a neighboring landowner, Sir James Stirling, 4th Laird of Keir and of Cadder and a friend of his father. The couple married towards the close of 1572. Most of the estates of the Napier family were made over to John Napier and a castle was planned for the estate at Gartness. When the castle was completed in 1574, Napier and his wife took up residence there.

Napier devoted himself to running his estates. This task he took very seriously and, being a great genius as an inventor, he applied his skills to these tasks. He approached agriculture in a scientific way and he experimented with:
… improving and maturing of all sorts of field land with common salts, whereby the same may bring forth in more abundance, both of grass and corn of all sorts, and far cheaper than by the common way of dunging used heretofore in Scotland.

About the end of the year 1579 Elizabeth Stirling died, leaving him one son, Archibald (1576-1645) (who in 1627 was raised to the peerage by the title of Lord Napier), and one daughter, Joan (the twins Agnes and Barbara died in infancy). Next year, 1580, John Napier married again, his second wife being Agnes Chisholm (the second cousin of Napier’s first wife, from the same Chisholm family that had provided his grandfather’s second wife, daughter of Sir James Chisholm of Cromlix, and great-granddaughter of King James IV), who survived him. By her, he had five sons and five daughters. On the death of his father in 1608, Napier and his family moved into Merchiston Castle, where he lived the rest of his life.

Napier’s father had been deeply interested and involved in religious matters, and Napier himself was no different. Because of his inherited wealth, he needed no professional position. He kept himself very busy by being involved with the political and religious controversies of his time. For the most part, religion and politics in Scotland at this time pitted Catholics against Protestants. Napier was anti-Catholic, as evidenced by his 1593 book against Catholicism and the papacy entitled A Plaine Discovery of the Whole Revelation of St. John. This work suggested that the Antichrist of the Book of Revelation was none other than the reigning pope, and urged James VI, the Scottish King, to “purge his house, family, and court of all Papists, Atheists, and Newtrals.” This attack was so popular that it was translated into several languages and saw many editions. Napier always felt that if he attained any fame at all in his life, it would be because of that book.

After the publication of the Plaine Discovery, Napier seems to have occupied himself with the invention of secret instruments of war. There is a document, dated 7 June 1596 and signed by Napier, giving a list of his inventions for the defense of the country against the anticipated invasion by Philip of Spain. The document is entitled “Secrett Inventionis, proffitabill and necessary in theis dayes for defence of this Iland, and withstanding of strangers, enemies of God’s truth and religion,” and the inventions consist of:
• a mirror for burning the enemies’ ships at any distance;
• a piece of artillery destroying everything round an arc of a circle;
• a round metal chariot, so constructed that its occupants could move it rapidly and easily, while firing out through small holes in it.
It has been asserted that the piece of artillery was actually tried upon a plain in Scotland with complete success, a number of sheep and cattle being destroyed 🙂

Besides the above-mentioned book, Napier wrote three other books: Mirifici Logarithmorum Canonis Descriptio, Ejusque uses, in utraque Trigonometria; ut etiam in omni Logistica Mathematica, Amplissimi, Facillimi, expeditissimi explicatio. Authore ac Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, ex ofjicina Andreae Hart Bibliopolae (in 1614, a translation into English by Edward Wright was published in 1616); Rabdologiae, seu Numerationis per virgulas Libei duo: Cum Appendice de expeditissimo Multiplicationis promptuario. Quibus accessit & Arithmeticae Localis Liber unus. Authore & Inventore Ioanne Nepero, Barone Merchistonii, &c., Scoto. Edinburgi, Excudebat Andreas Hart (published posthumously in 1617) and Mirifici logarithmorum canonis constructio (written before the Descriptio, but published posthumously in 1619 by his second son by the second marriage, Robert). In this treatise (which was written before Napier had invented the name logarithm) logarithms are called artificial numbers.

As a person of high energy and curiosity, Napier paid much attention to his landholdings and tried to improve the workings of his estate. Around the Edinburgh area, he became widely known as Marvelous Merchiston for the many ingenious mechanisms he built to improve his crops and cattle. He experimented with fertilizers to enrich his land, invented an apparatus to remove water from flooded coal pits, and bat devices to better survey and measure land.

Napier had a great interest in astronomy, which led to his contribution to mathematics. He was not just a stargazer; he was involved in research that required lengthy and time-consuming calculations of very large numbers. Once the idea came to him that there might be a better and simpler way to perform large number calculations, Napier focused on the issue and spent twenty years perfecting his idea.

It would be surprising if a man of such great an intellect as Napier did not appear rather strange to his contemporaries and, given the superstitious age in which he lived, strange stories began to circulate. Like Johannes Kepler and all his contemporaries Napier believed in astrology, and he certainly also had some faith in the power of magic, for there is extant a deed written in his own handwriting containing a contract between himself and Robert Logan of Restalrig, a turbulent baron of desperate character, by which Napier undertakes to serche and sik out, and be al craft and ingyne that he dow, to tempt, trye, and find out some buried treasure supposed to be hidden in Logan’s fortress at Fastcastle, in consideration of receiving one-third part of the treasure found by his aid.”

Napier first described the decimal point, enabling calculations to be made without the use of complex fractions. He discovered what eventually would be called “Pascal’s Triangle” and placed it in common use long before Pascal was even born.

One of the greatest men of Scotland—John Napier, died on 3 April 1617, apparently of goat, with which he had long been afflicted, and was buried in the graveyard (crypt) of the old church of St Cuthbert’s (formerly known as the West Kirk), then outside the West Port of Edinburgh.

Around 1810, the Galician Jew Abraham Jakub Stern (1768-1842), a remarkable mathematician, inventor, translator, and censor, designed his first computing machine. In 1811 he sent a report to his patron Stanisław Staszic (1755-1826), outlining the device and asking for financial help, and the device was first presented to the public in January 1813. Later on, Stern designed two more calculating devices. His inventions became popular already at the time of their development. In 1816 and 1818, Stern demonstrated his machines to the Russian Tzar Alexander I, who received him cordially and granted him an annual pension of 350 rubles, promising, in case of his death, to pay half of this sum to his widow.

For his inventions, Stern was admitted to the Warsaw Society of the Friends of Science (Warszawskiego Towarzystwa Przyjaciół Nauk, the predecessor of the Polish Academy of Sciences), first as a corresponding member (1817), then as a qualifying member (1821), and finally as a full member (1830). He presented his inventions multiple times at the Society’s meetings.

Stern was a father-in-law and heavily influenced another inventor of calculating machines—Chaim Zelig Slonimski. Stern most probably had a strong influence also over another Polish Jew and inventor—Izrael Abraham Staffel.

In fact, Stern presented to Society three calculating machines. The first machine for four arithmetic operations was designed around 1810 and was presented on 7 January 1813, then a different machine for extracting square roots (presented on 13 January 1817), and finally the combined machine for four operations and square roots (30 April 1818). A lengthy description of Stern’s machines had been given by himself, and you can see it below. Unfortunately, an original of any of his machines did not survive to the present time, only a later replica of one of the machines, shown below. There is also a low-quality reproduction of Stern with one of his calculating machines (see nearby image).

The machine of Stern was described in several publications in the press, the first of them was in 1815 in Dziennik Nauk i Umieietnosci, T.1 luty, pages 125-134 (see the description). Stern himself provided a detailed description of his machines in the treatise, prepared for the presentation to the Warsaw Scientific Society of the combined machine for four operations and square roots on 30 April 1817, which you can see below (translated from the Polish language by Phil Boiarski and Janusz Zalewski):

<<< Treatise on an Arithmetic Machine >>>

For the third time, in this earnest place of gatherings of the Society, comprising a selection of learned and enlightened men, I reveal the fruits of my thought—first in the month of January 1813, I presented an invention of a Machine for four arithmetic operations—secondly, in January of the current year 1817, an invention of the Machine for extracting roots with fractions—and then finally today, the 30 April of the current year 1817, an invention combining both these Machines into a single one.

This memorable day is the anniversary of establishing the glorious Warsaw Society of the Friends of Science and honoring it with the title of a Royal Society. I consider myself to be extremely fortunate that on this celebrated day, I can report concerning my inventions, regarding both their historical development and the thought that inspired them as well as regarding the properties of said Machines.

Remarks that initially led me to this thought are the following:

A man, although he comes into the world without any means to meet his inevitable needs, nevertheless, being above all creatures with his invaluable gift of mind elevated, with his unlimited ingenuity, uncountable for meeting his stressing needs finds means; because every need he feels inspires him to search and devise means corresponding to this need. That Man, feeling his superiority, thinks that all of nature for his benefit and service has been created and to him subdued; as the Psalmist with astonishment about a man says:
For thou hast made him a little lower than the angels, and hast crowned him with glory and honor. Thou modest him to have dominion over the works of thy hands; thou hast put all things under his feet: All sheep and oxen, yea, and the beasts of the field; The fowl of the air, and the fish of the sea, etc.

This feeling of his undetermined power over all of nature, causes him to regard whatever he finds for himself in nature useful for his needs, even though it should be considered more as a luxury. Experience teaches us that many things that initially were luxurious only because of having been used by a small number of people, with time, however, have become so common that they have shifted from the level intrinsic to luxury to the level of essential need.

From all this ensues, that when in the mankind a number of needs increases, then by this very thing, the ingenuity in methods and means to meet these needs has to multiply. Since such means are commonly based on physical acts, that is, works of the body, which often become onerous, or even beyond human power, so in such case the mind, as a primary Leader of the Man, makes every effort to invent intermediary means to replace the work of the body, or at least to ease it. Following this purpose, numerous mechanical tools have been invented to protect human physical power and support it.

From the depths of this convincing truth, another one equally undeniable have I drawn, that while no one spared efforts to bring assistance and relief to the physical condition, it becomes at least equally necessary to launch a search for mechanical means, which would offer help in human mental activities and relieve the intensity of thought; since the intensity of thought, as it is known, not only often impairs the subtlety of organs, deadens the wit, degrades memory but also, even causes weakening of the body.

I considered arithmetic or calculation science such a mental activity, one that was necessary, but through an intensity of thought, one that could be harmful. In it, the first four types of operations, that is, Addition, Subtraction, Multiplication, and Division are the main principles of all calculations, insofar that all other calculi are only the result of combinations of the said four kinds. And even though all four arithmetic operations, in general, require an uninterrupted presence of mind, that if for a moment gets distracted, a calculation cannot be accurate; since Multiplication and Division, for the reason of higher and more continuous intensity of thought, turn out to be the most difficult ones and therefore so often are subjected to errors.

At this point, I think it would not be unusual, regarding calculation errors, to make the following remark:
In regular calculus, we do not have and even cannot have a test convincing us whether any error was made; this is because a test performed by a reverse calculation, for example, Multiplication by Division, or Division by Multiplication, does not yet constitute a sufficient proof, for it means to test a mental activity by another mental activity. Since the repeated mental action, intending to serve as a test, is subjected to an equal error as in the primary calculation, this error in a test could have obstructed an error made in the calculation itself, and made it invisible.

All these remarks became for me the reason to invent an arithmetic Machine based on mechanical and arithmetic principles, with the assistance of which even people knowing only counting and numbers, all four kinds of calculations, and therefore all the other calculi, without the slightest application of thought to it, easily could accomplish. And because I thought it to be just, in such an important subject, not to rely solely on the principles of the theory of the mechanism, on which my invention has been based, insofar the slightest error in these principles could have disproved the entire construction of the invention, therefore for better conviction, I elaborated for testing a model of such an arithmetic Machine that worked. And even though the Machine was not of durable construction, and the required accuracy in the first, rough design, could not have been achieved, however, it exactly performed all the arithmetic calculations, so far that it proved the reality of this important invention.

In the month of December 1812, I submitted this invention for the consideration of the respectable Royal Warsaw Society of the Friends of Science. This Eminent Society, having assessed the invention as corresponding fully to its purpose, designed to deliver to the public a message about it, in its gathering on January 1813.

I have stated then, that I have planned to make another Machine, made of metal, in a way strong and durable. And although such an endeavor in particular at the initial stage, required time and significant funds for covering expenses, which by then a critical war situation of the Polish state, of which I am a compatriot, made it even more difficult for me, however, not saving efforts on my part, this statement of mine I have put into effect, so far that working continuously on this invention, I have finished a Machine for four fundamental arithmetic operations, completely of metal made with the finest work, and performing 13 digit operations.

In conjunction with work on this arithmetic Machine, I also worked on another, by far more difficult, invention of a Machine for extracting roots with fractions. The difference existing between only arithmetic operations and extracting roots already shows the level of difficulty; because in the former, there are always at least two known numbers given and the third unknown is searched for, but in extracting roots, there is only one known number given, and the other one unknown, that multiplied by itself equals the given number. Admittedly I learned that I ventured into such an abyss, from which a recovery is subject to numerous difficulties, both regarding the implementation of a thought as well as the huge costs, which a carefully elaborated plan definitely required. But no difficulty could oppose my keen willingness to finish the invention, which from various points of view seems to be important, both for the intention proper, the relief in the intensity of thought and counteracting unintentional errors, and to create a completely new mechanical means, included in this invention, which could apply great benefits to mechanical tools in other objects.

Thanks to the Almighty, I have passed this difficult and dangerous path, too, and the Machine for extracting roots with fractions I have led to the intended goal. This invention just like the first one, I have submitted for consideration of the glorious Royal Society of the Friends of Science, about which the public has been informed at the past January gathering of this Society.

This way, then, these two inventions, two separate Machines have been formed, one for the four arithmetic operations, and another for extracting roots.

I began thinking further on the ways, in which these two inventions could combine into a single Machine. It seemed to me, initially, to be impossible, indeed. But finally, at this point, mechanical ingenuity showed me the means to put my intent into effect. The importance of this thought so overwhelmed me, that to all unpleasant things stemming from shortages I have been insensitive, doubling my efforts, so I could make this combination sooner.

Thanks to the Highest Being, in this subject I did not fail either. I can say this boldly, since I am referring to the convincing proof, that is, before our eyes: a Machine, which accurately performs all four arithmetic operations, as well as extracting roots.

If I wanted to venture into the details and explanations of all principles of the internal Mechanism of this Machine, the purpose would be missed. Because the Mechanism, comprising several wheels of various kind, rotations of a new type, springs, and levers, by various means connected with each other, requires an extensive description and many figures, which will be the subject of a work planned for a later date, with figures clearly presenting the matter, but in this treatise clarifications would be an excessive boring of the respected public. Therefore I am moving now only to a brief sketch of the Machine and the explanation of the way of using it, in various arithmetic operations and extracting roots, as well as doing a foolproof test.

This Machine has a shape of a parallelepiped, longish and rectangular, in its length by five rows of wheels divided. The first, uppermost row, just like the second one underneath, is composed of 13 wheels based on axles. The wheels of the first uppermost row have discs, on which there are engraved ordinary digits of numbers of which only one number over the aperture is visible. Because the numbers of these wheels correspond to positions of units, tens, hundreds, etc., thus, this row entails trillions. The wheels of the second row, in turn, do not have discs and serve only as the Mechanism offering movement to the uppermost numerical wheels. Both these rows do not change (their place in the Machine. Behind these rows underneath, there are two rows of wheels, which similarly have numerical discs visible through the apertures, and are placed in a separate base in the shape of a carriage. This carriage with its two rows of wheels is so embedded in the Machine, that it can easily move on smaller wheels or rollers. The first row in this carriage has 7 numerical wheels on the axles of which there are as many folding cranks. Besides these cranks, on the diameter of a folding crank, there is another main crank that can be inserted and removed. The second row underneath has 8 wheels. Below this carriage, there is a lowermost row, composed of 7 wheels, equipped with numerical discs visible through apertures. This row has a stable and invariable place in the Machine. In addition to these rows of wheels, at the top of the Machine, there are two more rows of wheels, on which Roman numerals are engraved, visible through apertures. One of these rows has its place above the ordinary numerical apertures of the uppermost row, and the other above the ordinary numerical apertures of the lowermost row.

The way of using the Machine in operations is the following:
When any of the 4 arithmetic operations is to be performed, then one has to move to the left the handle positioned on the right on the carriage at the second row. As a result of this move, on a carriage on the left-hand side, the word Species shows through an aperture, all the numerical apertures of the second row are covered, and thus the Machine is ready for 4 arithmetic operations. If the species of an operation is to be addition or multiplication, then with two handles at the ends of the Machine devised at the right and left-hand sides, one moves up, while at the same time the words: Addition – Multiplication show up on the Machine through apertures, and by this the Machine is ready for these operations. If the species of the calculation were subtraction or division, this is done by moving from top down, the same way, words Addition and Multiplication disappear, and in their place, the words Subtraction – Division are seen and the Machine by this is ready for to the said operations.

In calculations of addition or subtraction, one puts the first number known to participate in the problem, in the uppermost row, and the other one in the first crank row on a carriage. The operation is performed by the main crank in the middle of the carriage base, which gives movement to the entire Machine. If only a single circular rotation is performed, then the brake, located on the left-hand side of the carriage, stops further movement of the Machine, and at the same time the unknown number searched for, appears in the uppermost row through apertures as a result of the operation.

In addition, there is one more convenience in the Machine, that is, because in this type of operation it happens that more than two rows of numbers in the calculation have to be combined, for example, in Registers and Tables, one sets on the Machine the first two given rows, as mentioned above, and by making a single circular rotation of a crank the Machine brakes, one touches the brake with a finger, and the Machine becomes available for rotation. Furthermore, one sets, in the third crank row on a carriage, the third given row, and rotates the crank once again, and so on, acting so until all the given rows are exhausted; at that time, in the uppermost row, there will be a general Sum of all the given numbers. However, to prevent an error from squeezing in, when all these different numbers are being added, which can especially happen when the operator interrupts the work, the Machine shows, through the aperture, the number corresponding to the value of how many given rows have been taken to the operation thus far.

Multiplication is performed in the following way. One of the factors is set on the crank row in a carriage, and the other on the lowermost row, while on the uppermost row, which is designated for the product searched for, zeroes are placed. After that, one moves the carriage from the right to the left side, to the very end of the Machine, by the handle placed on the left-hand side of the carriage. After releasing the handle, the carriage returns by itself, and stops in a position resulting from the nature of the problem. In this position, one begins rotation of the main crank. During the rotation, the carriage moves by itself from one number to the other towards the right-hand side, back through the end of the Machine; over there, the operation lasts until the ring of a bell warns about the operation’s completion, while at the same time, the product searched for appears already on the uppermost row. In this species of operation, the Machine has a particular superiority over calculations in an ordinary manner, that from several given multiplications one can obtain a general product without performing an addition operation, that is, without combining individually calculated products together. This is because in an ordinary calculation, in such a case, one has to first calculate a separate product from every two factors, then collect all individual products together and, by addition, derive the general product. On the Machine, however, one sets the first task and operates as long as the ring of a bell indicates to stop; not paying attention to the value of a product, one sets the second task, third, and so on, and when after the last operation the ring of a bell indicates to stop the rotations, at that time the general product of all the tasks appears on the uppermost row.

In division, one proceeds in the following way. The dividend is set on the uppermost row, and the divisor on the crank row in the carriage, while on the lowermost row, designated for the quotient, zeroes are placed. The carriage moves towards the left-hand side, until the divisor stands straight under the dividend number being greater or at least equal to the divisor. Then a main crank rotation begins and lasts as long as the dividend number becomes smaller than the divisor, at which point one presses with a finger a flap situated on the right-hand side of the carriage, after which the carriage moves by itself towards the right-hand side and stops at the appropriate place, where further operation continues in a similar manner till the end of the job. And when the divisor located on the carriage, standing in its first place, that is, at the end of the Machine on the right-hand side, carries the dividend, then the operation is to stop and the quotient appears on the lowermost row. In case there is a fraction, then the numerator appears on the uppermost row and the denominator on a crank row in the carriage. If on the uppermost row there are only zeroes, this means that the quotient is a whole number, without a fraction.

I am now going to describe the way of extracting roots.

If one wants to extract a square root from a given number, first, one has to move to the right the handle at the second row on the right-hand side of the carriage, so that on the left-hand side of the carriage, the word Species disappears and is replaced by the word Radices in the aperture. Then, numerical apertures of the second row of the carriage open, and the Machine is ready for extracting roots. Next, one has to move the handles at the ends of the Machine from top down, where between the inscriptions Subtraction – Division, one can also see on the Machine the word Extraction. The main crank in the middle of the carriage has to be removed, and the smaller folding cranks replace it in the operation. On the uppermost row, one sets up the known number of a given square, and on the first and second rows of the carriage all zeroes, except at the position of units in the second row, where one places the number 1. At the apertures for ordinary numbers of the uppermost row there are various signs dividing this row into sections, in such an order that for every two numerical wheels there is a sign, that is, at units, hundreds, tens of thousands, millions, and so on. On the said cranks there are identical signs, so that each crank corresponds to two wheels of the uppermost row, for example, the first crank from the right corresponds to units and tens, the second one – hundreds and thousands, and so on. The last sign, at the given number of the square, points to the crank from which the operation has to start, for example, if the given square ends on the wheels of the first sign, then the operation has to be undertaken with the first crank on the right-hand side. If, however, a given square ended on the wheels of the second sign, the operation then begins with the second crank, having the same sign.
Indicated this way the folding crank unfolds, the carriage moves to the left until the unfolding crank stops in front of the last sign of a given square. The rotation is conducted with this unfolding crank and lasts as long as the number on the uppermost row, in front of the rotating crank, becomes smaller than or, at least, equal to the number positioned in front of the same crank on the second row of the carriage. Next, by folding this crank, the crank to the right of it unfolds, and by pressing with a finger a flap on the right-hand side of the carriage, the carriage moves by itself to the right-hand side, until it is stopped by a folded crank, just in front of the previous section. One performs the same operations as above, for each section up to the last one. After completing the operation, if a given number was a full square, then it is replaced by zeroes and the whole root on the crank row in the carriage appears. Otherwise, except for the whole number root, an additional fraction results, namely, the numerator on the uppermost row and the denominator on the second row in the carriage.

To approximate the root in decimal fractions, one has to set on the uppermost row as many sections to zeroes as decimal digits in a fraction we want to have, for example, if the root is to be extracted from the number 7, and a fraction approximated with two decimal digits, one sets two sections, or four wheels, to zeroes, and the given number 7 is set in the third section, that is, on the 5th wheel of the uppermost row. To distinguish between the number actually given and the zeroes attached to it, a moving hand always slides out under this sign where the actual number has been set, which warns the operator how many digits for a decimal fraction he has to cut from the right-hand side on the crank row in the carriage. This way, then, when a given number is under the third sign, one has to unfold the third crank and perform the operations as above. The root will, then, result in the 3 crank wheels in the carriage, as number 264. Cutting off, following the hand’s warning, two digits for a decimal fraction, will mean 2 wholes and 64 hundredths. In addition to that, in the uppermost row there is number 304, as a numerator, and in the second row of the carriage, 529, as a denominator of the ordinary fraction of the tenth units of the first order.

At the beginning of this treatise I explained that in our ordinary calculus, there is no convincing test that in our mental operation, there was not any error, and that a way of testing by reverse calculation does not constitute sufficient proof. The same remark applies to the operation of the Machine. In case the Machine, due to damage, produced a false result, then the test by a reverse operation would not be proven, because the same damage that caused a false result in Multiplication, for example, would have affected a false result in Division, which would be clear even from the composition of the Machine. But to remedy this, I devised for the Machine a completely different kind of test, which is absolute proof.

Two rows of wheels with Roman numerals, mentioned above, located on top of the Machine, are designed for this purpose.

To obtain such a reliable test, one proceeds as follows.

Regarding Multiplication: Since from all the factors, the numbers of the factor on the uppermost row disappear during the work, being replaced by zeroes, so to make it visible after the work, what factor was a part of the problem, one sets in advance, on a Roman numbered row located above the apertures of the lowermost row, the signs corresponding to the digits of the factor which is to disappear. And because after completing the operation, the product results on the uppermost row and on the lowermost row there are zeroes, so one shifts as many zeroes to the number 9 as the number of digits of the remaining factor in the Roman numbered row, except the first digit, being meaningful, at the right-hand side of the factor, where the appearing zero remains. After this, the operation of testing begins. The carriage moves to the left and will stop by itself at the last number 9, but the rotation lasts as long as the number appears that is equal to the Roman numeral right above it, that is, the same which previously disappeared. At that time, one presses the flap on the right-hand side of the carriage, the carriage moves to the right and the rotations proceed further, as before, until a given factor fully appears in its first place, that is, on the lowermost row.

If after this work it turns out that there are as many digits in the factor on the lowermost row, as the number of zeroes on the uppermost row, at the right-hand side, and the numbers following them are equal to the numbers on the crank row of the carriage, then it is an absolute proof that the first product was true, otherwise, it had been false.

In Division, the test is conducted as follows. When the dividend is set on the uppermost row, at the same time on the Roman numerals row above it, the same signs are set, so that the dividend, which disappears during the work, that way be preserved; and when after completion of the division, the quotient appears on the lowermost row, it is then moved to the Roman numerals row directly above it. Then one moves the carriage towards the left-hand side, until the first number of units on the carriage appears in front of the last number of the quotient. The rotation is conducted in this place, until the carriage moves by itself towards the right-hand side, and this continues number by number, until the carriage moves to the first number of units, where one has to rotate as long as a zero appears. After completing this operation, one moves the carriage to the left as long as its first number of units passes all the digits of the preserved quotient. When the carriage is moved thus far, one has to watch if the number on the uppermost row, which has now formed, combined with the number on the crank row of the carriage, will match the dividend preserved on the Roman numerals row, which is a positive proof that the first result was true, otherwise, it had been false.

The test of extracting roots is conducted the same way as in the division, except that right before making the test, one has to adjust the Machine from the state of extracting roots to the state of arithmetic operations, and then set on the lowermost row the number equal to the root resulted in the crank row of the carriage. The remaining steps and the proving test are the same as in the division testing.

Remark: I am ending this treatise with a remark that since the Mechanics is an Opener to [meeting] our needs, insofar that not only our physical power but even that of mental power can replace, thus we should put our strongest effort to propagate ingenuity in such a broad and useful field, not venturing, however, into a search for perpetuum mobile, that is, an eternal motion, since this is an incurable disease of Mechanics, just as a philosopher’s stone and an inextinguishable fire in Chemistry, and a squaring of a circle in Geometry—all thoughts in this area have an attribute of an ineffective stubbornness. Let us better strive to make progress in mechanical matters showing promise, since such conduct paves the way to the well-being and glory of the Nation.

<<< End of Treatise on an Arithmetic Machine >>>

Biography of Abraham Stern

Abraham Jakub Stern (also known as Avraham Yaakow Sztern, Abraham Ya’akov, Avram Yaakov, Авраам Штерн, etc.) was born on 31 December 1768, in Hrubieszów, a town in Galicia (today south-eastern Poland). From the middle 16th century until WWII (the Holocaust) Jews made up a considerable majority (about half in the 19th century) of Hrubieszów’s population and controlled a large part of the trade, industry, and craft. Abraham was the son of Menachem Mendel Stern (Morgenstern) (1740-1787) and Miriam Liba Stern (Halpern). Besides Abraham, Menachem and Miriam had an elder son—Yehuda Leybush Morgenstern (1760-1817).

As the son of poor parents with an uncertain future, Abraham received a traditional religious Jewish education in his hometown and was trained as a watchmaker. He showed, while still very young, a marked fondness for the study of Hebrew books, which inspired him with a love for philosophy and mathematics. Abraham was lucky while working at a clockmaker’s shop in his hometown around 1800, to be noticed by Stanisław Wawrzyniec Staszic (1755-1826, see the nearby image), a leading figure in the Polish Enlightenment: a Catholic priest, statesman, philosopher, geologist, writer, and translator. Staszic, who studied at the Hrubieszów and Lublin secondary school in the early 1770s, bought an estate in Hrubieszów around 1800 and founded there Agricultural Society, the first cooperative organization in Europe.

It was the same Staszic, who could hardly be described as sympathetic towards the Jewish community (“Jewery makes our villages poor and our cities smelly”), but he obviously noticed the extraordinary talent of the humble clock-maker and encouraged him to devote himself to the study of mathematics, Latin, and German, later sending him to Warsaw to continue his studies in mathematics, physics, astronomy, mechanics and foreign languages, and to deepen his Talmudic knowledge. Moreover, Staszic patronized Stern until his death in 1826.

Abraham Stern was a remarkable inventor, and his inventions were known to the public, already at the time of their development. His first invention was a “movable topographic machine”, a kind of range finder for the surveyors. From 1810 to 1820 he designed three calculating machines. Later he invented a topographical wagon for the measurement of level surfaces, an invention of great value to both civil and military engineers. He rendered great services to agriculture through his improvements in the construction of thrashing and harvesting machines, a saw-mill, as well as by his invention of a new form of a sickle. In 1835 he invented a device by which the danger of runaways could be eliminated by means of a detachable tongue and a brake. In 1827 Stern invented a machine for raising cereals, which however did not find use.

In 1817 Stern published the book Sefer Mazor at arufah (A book on wounds and drugs). In 1820 Stern presented together with Józef Karol Skrodzki (professor and rector of the University of Warsaw) a report about the project of a chain bridge over the Vistula River.

In addition to working as an official translator (Stern was a great expert in Hebrew and Aramaic literature, and he wrote poetry in Hebrew and translated books to Hebrew, and he also spoke German, Polish, and Russian), Stern held a series of state administrative posts and functions. Since 1818, he was an assistant professor at the Dozora Elementary School for Moslems and director of the Rabbinical School in Warsaw. From 1822 he served as a referent, and from 1825 was the chairman of the Censorship Committee of Hebrew books and writings. In 1825, Stern was appointed to the Komitet Starozakonnych (Jewish Advisory Council to the Committee for Jewish Affairs), which concluded its work with the recommendation that a reform-oriented rabbinical school is founded. He was a harsh anti-Hasidic critic involved in several government investigations related to Hasidism.

Stern joined the Royal Warsaw Society of the Friends of Science, the predecessor of the Polish Academy of Sciences, first as a corresponding member in 1817, then as a qualifying member in 1821, and finally as a full member in 1830. He had several successful presentations of his calculators to Society, but they were not fully understood by the members. In their understanding, the calculating machine was in fact nothing more than a smart toy and also too expensive to consider popularizing it. Stern himself presented a completely different credo:
Man’s physical weakness proves that nature ordered him to work more with the force of the mind rather than the force of the body. So he should try to expand the boundaries of mechanics as it offers wealth and power to countries where it is pursued. Man should make machines and control them, and they should take his arduous work from him. Nations that developed industry rule the world, while those who neglected it have become weak, backward, poor, and enslaved.

Despite the successful career that Stern made in the capital city of Warsaw, he never lost his identity as a provincial Jew. He used to walk with a long grey beard, dressed in black satin, a silk belt, black stockings, and shoes. His head had always been covered with a black velvet crimson, his brow lined with wrinkles, due to constant thinking. His bushy eyebrows hid a pair of fiery, shrewd, intelligent, and beautiful eyes, and in winter and summer, among the most beautiful weather, an old torn umbrella held under his arm did not leave him. When he was staying at the castle of Prince Adam Czartoriski (yet another patron of Stern), a Jewish cook prepared his meals (kosher dishes). Among his friends were also Count Hans Karl Friedrich von Diebitsch (Генерал-фельдмаршал Дибич-Забалканский), Count Nikolay Novosiltsev (Граф Николай Новосильцев), and Prince Radziwill (Interestingly, Radziwill family was connected to an earlier inventor of calculating machine—Jewna Jakobson, so Stern probably was acquainted with the machine of Jakobson.)

It is unknown how many times was married Stern. To his last wife, the young 20 years old Shaindla Lipshitz (born in 1803, to Moshka Lipszyc and Etli Lipszyc), Stern married in 1823. The new family had three daughters: Sara Gitel (1824-1897), Esther (b. 1826), and Idla, but it seems Stern had (at least) six children from his previous marriage(s): Ber (b. 1799), Izaak (1801-1864), Eliezer, Jacob, Hanoch, and Rachel. There is no doubt that Stern was a genius, but the ability to organize his affairs was completely alien to him, and his numerous family lived in poverty, once in a while supported by patrons and friends.

Abraham Jakub Stern (see his biography in the magazine Tygodnik Illustrowany, 1864, Nr. 248, and in the magazine Przegląd Geodezyjny, 1956, T. 12 Nr. 1 p. 27-30) died in his home at Królewska str. 45 (for his merits he was allowed to live outside the Jewish ghetto), Warsaw, on 3 February 1842, and was buried in Jewish cemetery at Bródno (destroyed during and after World War II).

The German pfarrer Johann Reichold (1753-1798) was a bright-minded parson, who was not so interested in theological problems but preferred to spend his time studying and lecturing sciences, especially philosophy, mathematics, and engineering. In 1792 he devised a very interesting arithmetische maschine (arithmetic machine), rather similar to the device of his compatriot Philipp Matthäus Hahn.

Reichold probably has been acquainted with and inspired by the calculating machine of Hahn, not only because it was a quite popular device, described in several publications, but also because Hahn’s brother-in-law Johann Schuster (who made several machines by Hahn’s design) from the early 1790s ran a workshop in Ansbach, while Reichold worked as a parson in Dottenheim, a village located only 40 km north of Ansbach.

It seems the only reliable information for the calculating machine of Reichold can be found in a book by Johann Paul Bischoff, written in 1804, but published as late as 1990—Versuch einer Geschichte der Rechenmaschine (Attempt at a History of Calculating Machines), publisher: Systhema-Verlag, editor: Stephan Weiss. In the last quarter of the 18^th century, Bishoff undertook several long trips to take a look personally at the calculating machines, which he had heard of, in order to describe them in a book. Admittedly this is the case also with the machine of Reichold, otherwise, Bishoff wouldn’t have gained so detailed information about the mechanism, given in the book. Moreover, Ansbach is only some 40 km south of Dottenheim, where Reichold lived.

Reichold obviously had a rich experience in woodwork, making wooden clocks, and other instruments. That’s why the copy of Reichold’s arithmetic machine, seen by Bishoff, was not a metal one but was made ​​entirely of pear wood (pear wood even now is one of the preferred materials in the manufacture of high-quality woodwind instruments and furniture).

Probably due to its wooden construction, the machine was quite big—it has a cylindrical form, 7.5 cm in height, and about 27 cm in diameter (see the nearby image). Let’s mention only, that almost all other calculating machines from that time were big, for example, the first calculating machine by Anton Braun, although made of metal, was even bigger—40 cm diameter, 21 cm height.

The device has two rows with nine graduated number disks in each row. In the center of outer number disks, used during the entering of the numbers into the machine, are mounted axes, on which are fixed cranks (marked with K in Figure 1) and gear-wheels (marked with A, A’ and A”) (see the Figure 2 below). In the center of inner (smaller) number disks are mounted axes (Q), on which are fixed pointers (marked with S), knobs (m), ratchet-wheels (C, C’, C”), gear-wheels (B) and star-wheels (E).

The motion (rotation) from the input mechanism (bigger disks) to the resulting mechanism (smaller disks) is transferred by engaging the teeth of gear-wheels (A) to the teeth of gear-wheels (B) (Figure 2). The mechanism of each digital position is connected also by means of the star-wheel (E) and teeth L (see Figures 3 and 4) to the mechanisms of the next position (if any), in order to propagate tens carry operations when needed.

The number disks in the first (right) pair are divided into four parts, numbered 0, 1, 2, and 3. The second pair of disks are decimal (base ten), divided into 10 parts. The third pair of disks are divided into six parts (numbered 0, 1, 2, 3, 4, and 5). The pairs from 4^th to 9^th are decimal. The graduation of the disks was made according to the monetary system adopted throughout the southern states of the Holy Roman Empire in the 18^th century (240 Pfennige = 60 Kreuzer = 1 Gulden). The first pair of number disks is used for presenting Pfennige (4 Pfennige made a Kreuzer), the second and the third pairs of disks are used for presenting Kreuzer (60 Kreuzer made a Gulden), and the remaining 6 are used for Gulden and decimal calculations.

The crank (K) in the center of each bigger number disk rotates the axis and the gear wheel (A) below the disk. Under each digit on the bigger number disks, there is a small hole, into which can be inserted a pin, thus limiting (stopping) the rotation of the crank. Under the zero digit of each disk, there is also a pin, which abuts the crank, so that it can be turned only up to this pin and not further in circles. So when we want to enter a number in this position, we have to put a pin in the proper hole and rotate the crank from zero position to the pin and back, from the pin to zero.

The smaller (inner) number disks are divided at the same scale, as the larger disks (i.e. first disk in four parts, second in ten parts, etc.), however, they contain a double row of digits—black digits for addition and multiplication, and red digits for the subtraction and division. Black digits are actually placed as a complement to nine of red digits, e.g. black zero is near red 9, black 1 is near red 8, and so on.

In the middle of smaller disks are mounted pointers, which are used for presenting the result of calculations. The pointers also can be moved with the aid of wooden knobs (marked with m in Figure 1), and thus, can be set to any number.

In the center of the front panel is mounted an immovable ring (marked with R) with nine holes, into which can be inserted a small pin (marked with n). This ring is used to describe the place where one left off during the operation. On this ring also are marked the values of the adjacent number disks.

Let’s examine the modus operandi of the machine, performing the four basic arithmetical operations (addition, subtraction, multiplication, and division).
1. Addition.
Let’s add 365 Gulden, 50 Kreuzer and 3 Pfennige to 219 Gulden, 19 Kreuzer and 1 Pfennig. Firstly we have to enter the first addend (365 G, 50 K, and 3 P) into the smaller number disks, using knobs of the pointers and black digits (black digits are used for addition and multiplication, red digits for subtraction and division). So we set the pointer of the right smaller disk to show 3 (we have 3 Pfennige). The 2nd disk pointer must be set to 0, and the 3rd disk pointer—to 5 (we have 50 Kreuzer). The 4th disk pointer must be set to 5, the 5th—to 6, and the 6th—to 3 (we have 365 Gulden).
Now let’s enter the second addend (219 G, 19 K, and 1 P) using the cranks and the bigger number disks. First, we have to put the pin into the hole below 1 (we have 1 Pfennig) of the first big number disk and rotate the crank until it is stopped by the pin (thus the crank will rotate the axes to 90 degrees), and to return the crank back to zero position. The movement from the bigger disk mechanism will be transferred to the smaller, rotating the pointer from 3 (it was 3 Pfennige) to the next position (0), causing a tens carry to be propagated to the next digital position (small disk), which was set to zero Kreuzer, but now will turn to 1.
Using the same manner, we have to enter Kreuzer (19)—9 into the 2nd big disk (causing the 2nd small disk to rotate from 1 to 0, and a tens carry to be propagated to the third small disk, which was 5, but now will go to zero, and another tens carry to go to 4th small disk, which was 5, but now will show 6), and 1 to the 3rd big disk, causing the 3rd small disk to rotate from 0 to 1. As an intermediate result, before entering the Gulden of the second addend, we have now 366 Gulden (in the 6^th, 5^th, and 4^th small disks), 10 Kreuzer (in the 3rd and 2nd disks), and 0 Pfennige (in the last disk). We have only to enter 219 (Gulden of the 2nd addend) using the cranks, causing 1 tens carry to be propagated from the 4th to 5th small disk, in order to get the final result—585 Gulden, 10 Kreuzer, and 0 Pfennige.
2. Multiplication.
The multiplication was done just like in the other calculating machines from the time, i.e. using consecutive additions. If for example, we have to multiply 815 by 9, we have to put the pin under the digit 5 on the 4th big disk and turn the crank to the pin 9 times. A similar operation must be done for tens and hundreds. If we want to multiply multi-digit numbers, this will require writing down the intermediate multiplication results (multiplying the multiplicand to Ones, Tens, Hundreds, Thousands, etc. of multiplier), and adding them one by one, a rather cumbersome operation.

3. Subtraction.
The subtraction is similar to addition, but the red digits must be used instead of black. The minuend must be set into the smaller disks mechanism, using knobs of the pointers and red digits, and then the subtrahend must be entered into the bigger disks mechanism, using the cranks.

4. Division.
The division operation is even more cumbersome and error-prone operation, than multiplication, because it was done using consecutive subtractions, and some intermediate calculations are needed, in order to get the final result. The dividend is entered using the red numbers in smaller disks. Then the divisor must be placed under the highest digit of the dividend and deducted from this part of the dividend until the same has become smaller than the divisor. Then if the dividend has more digits available, the divisor must be shifted to the right, and the consecutive subtraction to be repeated, and so on, until the entire dividend will become smaller than the divisor.

As you can see, just like the other calculating machines from the time, which are actually simple adding devices, the machine of Reichold is suitable only for addition and subtraction. For multiplication and division, it is much more convenient to use other calculating techniques and methods.

Internal mechanism of the device.
In Figure 2 with letters A, A’ and A” are marked gear-wheels, fixed on the same axes as the cranks K. Wheels A are engaged with gear-wheels with the same diameter and number of teeth B, which are fixed firmly to the axes (Q) of smaller number disks.
Directly above each of the gear-wheels B on the same axis (Q) are attached ratchet-wheels (C), which have locking pins and springs (marked with q), mounted on wheels B. These locking pins are fixing the ratchet-wheels to the gear-wheels B and limit their rotation in one direction regarding wheels B. Ratchet-wheels C are not firmly attached to axes Q, but only to the pointers (S). Thus, when we are entering a number in smaller disks mechanisms, rotating the pointers in counter-clockwise direction, this will only rotate on the ratchet-wheel, without affecting the gear-wheel B and the star-wheel E. However, when a number is entered by rotating the cranks forward and backward, then the locking pins and springs, mounted on wheels B will engage the ratchet-wheels and will rotate them, together with pointers, thus changing the result.

On the same axes Q, but mounted at some distance below wheels B (see Figures 3 and 4), are mounted star-wheels (marked with E). The wheels E have teeth L, which are used for propagating the tens carry to the next axis, as once in each full revolution of the axes Q will engage the next star-wheel and will rotate it. The wheels E have also locking pins and springs (marked with h), which are fixed to the body of the machine and are fixing the wheels and limiting rotation only to one direction.

The star-wheels E, just like the ratchet-wheels C, are not fixed firmly to the axes Q, so during the rotation of the cranks K they can rotate or not, depending on the direction of rotation and the position of the locking pins h.

Let’s imagine, that we are going to enter a number in the rightmost wheel A, moving crank A from position d to f, and then back from f to d (see Figure 2). During the first half of the rotation, from position d to f, the teeth of wheel A will engage the teeth of wheel B and will rotate it. At this moment, however, the star-wheel E will be stopped by the fixing pin h and will not follow the movement. The ratchet-wheel C will not follow the movement of wheel B also because the locking pin p is not limiting the movement of wheel C in the counter-clockwise direction. So only wheel B, but neither wheel C nor wheel E will follow the movement.

During the second half of the rotation of crank K, from position f to d, the teeth of wheel A again will engage the teeth of wheel B and will rotate it, but in the opposite direction. Now, the wheel E will follow the rotation, because in this direction the fixing pin h does not limit its rotation. The same is the case with the ratchet-wheel C, which will be pushed and rotated by the fixing pin p.

In this manner, we will manage to transfer the number (motion) to the calculating mechanism and to turn back the crank into the initial (zero) position.

Obviously, the design of the machine is basically simple but workable, and if the parts had been manufactured precisely and with metal, it could be a much smaller and more reliable device. Nevertheless, Reichold must have been a very good constructor, unfortunately, struck down young and in his prime.

Literature: Johann Paul Bischoff, Versuch einer Geschichte der Rechenmaschine, publisher: Systhema-Verlag, 1990, editor: Stephan Weiss.

Biography of Johann Reichold

Johann Christoph Reichold was born on 17 January 1753, in Equarhofen (a small village, located some 90 km west of Nürnberg), in Bavaria, Germany, where his father, the parson Johann Georg Reichold was in service at that time.

The father—Johann Georg Reichold was born in Berneck, Switzerland, in 1707, studied theology at Universität Halle from 1731 until 1736, then worked at the orphanage in Bayreuth, and in 1739 accepted a position of a parson in Equarhofen and Frauental, where he served 40 years until his death on 31 October 1779. In Equarhofen Georg Reichold married Johanna Katharina Silchmüller, and they had two children — the son Johann Christoph, and a daughter — Maria Auguste Luise Reichold-Hoechstetter (20 Oct 1748 – 25 Jan 1803).

From November 1763 until April 1771, Johann Reichold studied at the Fürstenschule (Prince School) of Neustadt an der Aisch (a small town, located 40 km NW from Nürnberg), under Christian Theodor Oertel, Friedrich Wilhelm Hagen, and Johann Friedrich Amthor. Then (according to the University registers—from 11 May 1771 to 22 September 1773) he studied theology at the University of Erlangen, but for some reason (most probably financial) didn’t manage to earn a degree.

Reichold lived in Erlangen, studying and working as a private teacher and Hofmeister (butler) until July 1784, when he accepted the position of a parson in Dottenheim, a village near Neustadt an der Aisch (on the nearby image you can see the church St. Markus in Dottenheim, where Johann Reichold was in service).

Just like his more famous compatriot Philipp Matthäus Hahn, Reichold was a bright-minded parson, who was not so interested in theological problems but preferred to spend his time studying and lecturing sciences, especially philosophy, mathematics, and engineering.

Besides his calculating machine, Reichold was also engaged in the manufacture of wooden clocks and in the construction of instruments, for example, a pedometer device.

Johann Christoph Reichold died on 16 February 1798, in Dottenheim. It seems only the early death of Reichold didn’t allow him to make a significant contribution to the development of calculating machines.

The German engineer and master builder—Johann Helfrich Müller (1746-1830) is a very interesting figure in the world of mechanic calculators, not only for his small calculator, essentially an improved version of the machine of Philipp Matthäus Hahn, which he created but for his plans to build a difference engine almost 40 years before Charles Babbage. He also wanted to build a machine, that was capable of being configured to use Leibnitz’s arithmetica dyadica.

Johann Müller was a creative mind and started to make inventions from the beginning of the 1770s. Firstly he created a theater, equipped with optical and mechanical effects for the children of his boss—Prince Georg Wilhelm, later on, he designed a large and powerful burning mirror, as well as other devices.

At the beginning of the 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. Most probably, at that time he came across an article (probably that in the magazine Teutschen Merkur) about the calculating machine of his compatriot, Philipp Hahn, which he decided to use as a prototype, and to try to improve somehow. Interestingly, later in the middle 1780s, Müller was accused by Hahn of stealing his design, but flatly denied the accusations, although comparing the machines of Hahn and Müller (they are as like as two peas), it is hard to believe, that Müller didn’t use at least the basic design of Hahn. Moreover, the German author and critic Johann Heinrich Merck, who visited Hahn in 1778, was a friend of Müller.

After about three months of designing, in June 1782, the project was ready and Müller gave the drawings to a local clock-maker 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 20 June 1784, the machine was ready. On 24 June 1784, the 14-digit calculator (see the upper image) was demonstrated in the Göttingen Academy of Sciences, which appointed Müller as a correspondent. In July 1784 the device was described in the journal Göttingische Anzeigen von gelehrten Sachen (p. 1201-1206, see the description), and later in other editions.

Müller’s calculating machine is very similar to the machine of Hahn and was based on the Stepped Drum of Leibniz, but it is larger (285 mm diameter, 95 mm height, weight 15.5 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.

An adding operation can be done as follows:
1. Using the buttons (marked with h), we set the first addend in the windows of dials (f), using the black digits.
2. Set the other addend using the side dials (g).
3. Make one revolution of the crank (K). The sum can be read using the black digits in the windows of dials (f).
4. Third, fourth, … addends can be entered at the side circles (g) and each adding operation is performed using one revolution of the crank.

The subtraction can be performed in a similar way, but the minuend is set according to the red digits in the windows of dials (f), while the subtrahend is set in the side dials (g). After rotating of the handle to one revolution the result can be seen using red digits in the windows of dials (f).

The multiplication can be done by performing successive additions, while the division is done in a way, similar to the multiplication, but are used the red digits of the dials and it is based on successive subtractions (just like using the machine oh Hahn).

Compared with Hanh’s machine, Müller’s had three important improvements:
1. The axes of stepped drums from Hahn’s machine, which are used for entering the numbers, are not anymore set by pulling upwards (which requires great precision), but by means of rotating dials, with inscribed digits from 0 to 9 over their periphery.
2. The axes, along with mounted on them pinion-wheels, can be easily changed with pinion-wheels with a 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 the adding or negative result during the subtraction (if the operator tried to subtract a larger number from a smaller one).

While still working on his machine, on 22 May 1783, Müller wrote a letter to his schoolmate and friend, the German physicist Georg Christoph Lichtenberg (1742-1799), 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 hoped to sell his machines in England and Russia. Later on, however, the Grand Duke of Hesse-Darmstadt Ludwig I, bought the first machine for 4000 Gulden, and incorporated it into 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 (see the lower image). In another letter to Lichtenberg from 9 September 1784, Müller recorded a new thought in a postscript for a printing tabulating machine, as follows:
  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.

Admittedly 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 (capable to operate to the third order of difference) 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. Müller indicates that if someone were willing to finance its construction, he was willing to make the machine. He estimated that a table of x³ (1 ≤ x ≤ 100000) could be produced in just over ten days of effort, even if the operator only worked for eight hours per day.

Such a machine will be proposed almost 40 years later by Charles Babbage. It is known that several chapters of the book have been translated for Charles Babbage by his friend John Hershel. However, the date of this translation is unknown and the question remains whether some of Müller’s ideas have been used for Babbage’s machines.

Biography of Johann Helfrich Müller

Johann Helfrich von Müller was born in Cleve (a town in the northwest of North Rhine-Westphalia in Germany), on 16 January 1746, in the family of Lorenz Friedrich Müller (1715-1796), a colonel of the artillery, who worked as an architect and chief engineer in Giessen, and Maria Magdalena Josepha Hambloch (1726-1800). Johann had a brother, Franz, who later became a lieutenant colonel in the British Army. The German writer and journalist Helfrich Peter Sturz (1736-1779) was their cousin.

In 1753 the family moved to Darmstadt, where Johann received his initial education at small private schools, getting private lessons in languages and drawing. Later he was sent to study at the Darmstadt Grammar School. In the spring of 1762, he became a cadet in the Artillery Corps of the Hessian army at Giessen, where his family had moved, and began to attend lectures in mathematics and physics at the University of Giessen (Interestingly, at the same university some 20 years before lectured as a professor of mathematics one of the creators of the early calculating machine—Christian Ludwig Gersten). However, Johann never got an academic degree, because, at his father’s request, he left the University and started self-study.

Johann’s father, Lorenz, who was in the building trade (architect and engineer), and was a son of the architect and civil engineer Helfrich Müller (1686–1759), wanted his son to carry on the family tradition, but Johann appears to have found certain other careers equally tempting. While still at 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 afterward, he was employed as an engineer by Prince Georg Wilhelm, the Governor in Giessen—a man with wide interests in both civil and military engineering and architecture. In 1769/1770 Müller made an extensive educational journey to Italy (Rome, Naples, and Venice) and Austria (Vienna). In April 1772, Prince Wilhelm took Müller with him on a five-month trip to Paris, where 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 a building inspector in Darmstadt and three years later master builder with the responsibility for designing and maintaining buildings (Ingenieur Hauptman). He also was expected to act as an adviser in case of emergencies such as floods. In 1778 he returned to the military service as a Captain in Artillery Corps, to be raised as a Lieutenant-Colonel in 1797, and a Colonel in 1800.

In the 1770s Müller took part in the design of several public buildings in Darmstadt. The fountain in Market Square in the front of the old Rathaus (Town hall) of Darmstadt, built by design by Johann Müller in 1780, is still preserved (see the upper photo from 1900).

Johann Müller was a person of a very creative mind and started to make inventions at the beginning of the 1770s. First, he created, by an order from the Prince, a theater, equipped with optical and mechanical effects for the children of the Prince. Later on, he designed a large and powerful burning mirror, sun clock, air pump, air gun, barometer, range finder, and even a perpetuum mobile, but the most interesting for us invention was certainly his calculating device in the 1780s.

In the period 1776-1790, Müller was the state architect of Giessen. From 1792 until his retirement in 1820 Müller served as a Hofmeister and Major in Darmstadt. For his efforts and contribution, he was knighted (together with his brother) by the Grand Duke of Hesse on 23 June 1810 (i. e. he became von Müller).

On 19 August 1781, Müller married Catherine Fabrice Johanetta von Westerfeld (1761-1830), a daughter of Esaias Fabrice von Westerfeld (1709–1779) and Elisabeth Katharine Schröder (1724–1765). The family had five children—two daughters and three sons, but unfortunately, all died young, except for one of the daughters—Friederike (1784-1841). Friederike’s daughter is the German novelist and social critic Johanna Udalrike Louise Gerhardine Schücking (1815-1855).

Johann Helfrich von Müller died on 12 June 1830, in Darmstadt.