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25 July 2013
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As with many others I have discovered (belatedly) that the act of forming a collection carries with it an insidious if not addictive desire. Not the least that is because there is always one more thing you could add which would make it all so much better, and maybe a wonderful opportunity to do so lurks just around the corner... This aspect needs some reflection - so I will write about it later. But putting that awkward fact aside, at least explicitly the aim of this collection has been to bring together some nice examples which focus primarily on manual mechanical means of calculation. A couple of electric motorised calculators have been acquired also as examples of the evolutionary direction of innovation during the transition period between manual machines, and the advent of electronic machines based on integrated circuitry.
Rather than seek to be comprehensive,[^For a comprehensive listing see http://www.rechenmaschinen-illustrated.com/^] which in any case turns out to be impossible because of the wide range of devices that were built over the last few centuries, the items in this collection have been acquired with the idea of illustrating the dynamics and trajectory of development. I have tried to acquire interesting machines which were both significant in their time, work as they are supposed to, and act as "bookends" to form examples of directions in which development was evolving. So, above all else, these objects are here because they please me![^Most, but not every acquisition was a success. However, even [[Fakes and False Promises|fakes]] can be interesting.^]
What follows on this page is a first layer story - that of a developing solution to the problem of calculation. I should point out at once that this story is both a common theme amongst collectors' websites, and from another point of view rather superficial. After all the "problem" of how to calculate does not arise from nowhere. Facilitating calculation is a problem to the extent that calculation is proving necessary in people's lives, or institutions' operations. This has been a changing story too over the centuries in which these have developed, moving as it has from human pre-history through to the dynamic period of invention, construction and intensifying trade which is increasingly visible from the 1600s. It is also but a shallow account to consider calculators and calculation outside the challenges arising through the evolution of mathematical thinking. Some examination of that, over the last 10,000 years, is sketched in the section dealing with the context of the history of calculating technology (including the evolving mathematical context). Still, it can be helpful to start with the objects and some simple-minded thoughts about 'why' they have developed.
!! Abacus (Pre-Historic - C21)
The first item may seem prosaic, but the abacus has the most venerable history of all, and is believed to have been introduced into Sumeria during the Akkadian occupation of ~2100-2000 BCE. The demand to compute all four operations (or as is often referred to in discussions of calculators - "four functions" or "four species"), being addition, subtraction, multiplication and division, has been long-standing, especially as out of these all computable functions can in principle be constructed. For reliable computation it is of course attractive to represent the numbers digitally (whether in cogs or beads or some other mechanism) and to be able to manipulate each pair of numbers precisely. The earliest (and highly effective) mechanism for achieving this is the abacus - which was used in ancient Rome, and analogically lies at the conceptual heart of all of most of the calculating machines (including the computer).
In this collection there is a [[site.abacus|contemporary Chinese abacus]]. There were of course different styles of such machines, each with their own particular capabilities and one day there should be added to this collection an early abacus (or Japanese Soroban). Beyond these the collection includes bookends to a series of branching devices designed to make these functions more simply utilised (if not necessarily any faster).
!! Sectors (C16-20)
From the beginning a more general central problem, has been how to add, subtract, multiply and divide quickly. Considering the technical aspects of this (rather than why the problem mattered) we can note that addition and its inverse, subtraction, seem relatively simple compared with the task of multiplication and division. True, multiplication and division can be performed through repeated addition and subtraction, but this is often very laborious. An early methodology which could be used for a more direct form of multiplication and division, invented in the sixteenth century was embodied in the sector. This instrument utilised the principle that the ratio of the base to side of two similar triangles is the same, to make multiplication or division easy. The device consisted of a hinged pair of rules with a variety of scales and was in use, particularly in gunnery, architecture and navigation, up to the early twentieth century. In this collection there are two sectors.
The first from c. 1700 was made by [[site.ButterfieldSector1700|Michael Butterfied]] (1635-1715 [or 1724 depending on source]) an English instrument maker who settled in Paris c. 1675. It is a gunnery sector of French design with additional scales for canon priming and orientation. Other very similar Butterfield sectors are held by the Metropolitan Museum of Art, the British Museum, and the National Museum of Scotland.
The second sector in this collection is an [[site.DoublettSector1830|English architect's sector]] made in c. 1830 from oxbone by the well known instrument makers T. and H. Doublett, London.
!! Early Logarithms (C17)
The discovery of logarithms which transform the problem of multiplication to that of addition (and division to subtraction) was a particularly helpful development for a time. The principle had been noted by the ancient Greeks but use of the principle required an easy means of evaluating the logarithms and their inverses. This was achieved with the first publication of logarithmic tables by Napier in 1614 (only nine years after the unsuccessful gunpowder plot against King James I in 1605).
The earliest bookend here is a set of logarithm and other tables published 12 years after Napier by [[site.Henrion1626|Henrion]] (a French mathematician) in 1626. This was the first French set of logarithm tables (but using the more practical decimal form than Napier's which had been published in 1624 by Briggs). It also contains [[site.Henrion1662|the design of the Gunter calculating rule]], published in England in 1624 (of which, more later).
The Henrion logarithm tables were the first to be introduced in French into Europe. The process of presenting tables of logarithms and associated functions was beset by the problem of errors that would easily creep in. In addition, different levels of accuracy were required in different situations.
Other publications of logarithms, with improvements, and always some errors, followed over subsequent centuries.
In addition to the very early logarithm tables by [[site.Henrion1662|Henrion (1626)]] this collection also contains a set of [[site.Gardiner1783|tables by Gardiner]] (with improvements by Callett) one and a half centuries later in England in 1783.
!! Mechanisation of multiplication, Napier's Rods and Schickard's calculator (C17)
Napier had also tackled the question of how to multiply and divide by creating in 1617 his "rabdologiae" a clever way of encoding multiplication tables onto 9 rods (or as they became known "[[Site.EncyclopediaBritannica1797|Napier's bones]]"). The moment (of 1617) is also therefore near what is regarded by some as a turning point.
Only 6 years after the publication of radbologiae, in 1623 Wilhelm Schickard constructed his "Calculating Clock" which is regarded as the first arithmetical machine since the abacus. It embodied Napier's rabdology but with the rods incorporated in a way that the operations including the necessary additions and subtractions could be carried out with the help of the machine. Schickard's machine was notable for not only its sophisticated use of Napier's rods through a set of sliders set in a rack, but also a register for storing intermediate numbers and a machine using wheels and cogs to enable carry forward to add the partial products obtained from the rods.
The Schickard approach would later be further developed in the [[site.BambergerOmega1904|Bamberger Omega]] (see later) which enjoyed a brief commercial appearance in an art nouveau design, almost three centuries later (1903-5). Unfortunately we only know of the Schickard machine through his correspondence with his friend Kepler (for whom he says he constructed a machine only for it to be destroyed in a fire.) All that is left of the machine is a rough drawing and description of what it does.
Another approach by Gaspar Schott (1668) turned the rods into cylinders and fixed them in a box ("cistula") so that the right one could be got simply with a turn of the knob. To this he had added a table of addition and subtraction to the inside cover of the cistula.[^"Habits of Knowing" pp 182-3.^] This approach is not unlike the Napier rod part of the Schickard.
In 1957 mathematician Bruno Baron von Freytag Löringhoff constructed a working replica from the Schickard drawings and description. A small number of other replicas now exist, and a reconstruction of [[site.Schickard1623|Schickard's Calculating Clock]] based on his sketch and description in a letter to Keppler is in this collection. (This reproduction Schickard was manufactured in a small set in Germany through the collaboration of three public, and one private, museums in Europe).
!! Slide rules (C17-C21)
The mechanical means of adding together logarithms (as first expounded by Gunter) in 1624 represented the logarithms as lengths on an appropriately calibrated scale (the so called "Gunter Line" or "Gunter Rule"). The [[site.Henrion1662|graphical calculation of the Gunter scale]] accompanies the [[site.Henrion1662|Henrion tables]] of 1626 in this collection.
The term "sliding rule" was coined by Everard who developed an instrument based on Gunter's representation of logarithms, but incorporated a slide to allow ease of calculation by adding lengths directly (rather than with dividers). The instrument was designed to support the work of excise officers as they sought to work out the tax payable on different containers of spirits. Logarithmic as well as trigonometric scales were used to calculate quantities in various shaped containers. The earliest such bookend here (beyond Henrion's graphical design for a slide rule based on the work of Gunter, already mentioned) is an [[site.Everard1759|early "sliding rule" gauger based on Everard's design]], but made by Edward Roberts (snr) of Jewery Lane, London, between 1759-69.
Slide rules of course developed in many ways - accuracy, the incorporation of a sliding cursor, the range of computable functions, use of different materials, minituarisation and other design innovations, represented here by a little series through a [[site.KeuffelandEsser1908|Keuffel and Esser (1908)]] to the widely regarded finest slide rule - the [[site.FaberCastell1967|Faber and Castell 2/83N Novo Duplex]].
There was much experimentation with how to get the most computational value out of different shapes and arrangements. In this collection there is a series of circular and cylindrical slide rules - the [[site.Thacher1911|Thacher Calculation Instrument]] from 1911 (the most accurate slide rule ever made, equivalent to a 60 foot slide rule), and then on the path to minituarisation the Fuller (40 foot equivalent), [[site.OtisKing1960|Otis King]], and the watch style pocket circular slide rules - the [[site.KL11960s|Russian KL-1]] and the [[site.Fowler1948|Fowler Jubilee Long Scale]] (1948).
!! Conversion devices
The bane of earlier centuries, and even still, is the fact that quantities were often measured in different incompatible units. Special purpose devices, basically just forms of mechanical look-up tables were devised to deal with this sort of issue, since performing the calculation each time was time consuming, and many did not have the skills in any case to do it. The need to be able to measure and compare occurred extensively in shipping and commerce rather early. In this collection we have a [[site.CHAIX1790|"Convertisseur"]] from (1780-1810) designed by Gabrielle Chaix in Paris, to assist merchants and artisans to convert between the old measure of distance (the "Aune") and the new measure introduced by the Revolutionary Government of the metre in 1791.
!! The first cogged calculation - Pascale and Morland (C17)
Following Schickard's embodiment of Napier's radibology into a machine in 1623 mathematical prodigy Blaise Pascale in 1642 invented a much more robust calculating device focussed on addition. The young Pascale embarked on this to assist his father in adding taxes. There exists a letter (of which I have only the text) from Pascale which explains how this device is used. Nine versions of the machine remain in existence in museums and other collections. Several replica have been constructed of which one such replica "[[Site.Pascaline1652|Pascaline]]" is in this collection.
Around 1666 Sir Samuel Morland invented two cogged machines - one for doing additions and the other for multiplication. In each case the machines were designed to make basic arithmetical operations more accessible for those not versed in them. He published descriptions of the machines in what is regarded as (arguably with Pascale's letters) the first calculating machine (or even "computer") manual. A copy of the [[site.Morland1672|first edition of Morland's book published in 1672]], the book itself being exceptionally rare, is in this collection.
!! Adders (C17-C20)
Other simple mechanisms for adding and subtracting were developed in a series of devices whose basic mechanism using metal strips or cogs which can be used by means of a stylis can be moved linearly or circularly the distance displaced standing for added numbers. This type of device, first developed in the 17th Century by inventors such as Claude Perrot (1613-88) was popularised by J. Louis Troncet, a French inventor who created his Arithmographe,[^for more, including illustrations from his patents see http://www.ami19.org/Troncet/Troncet.html.^] later to be known as the [[site.Addiator1920|Addiator]] (of which there is one in this collection).
Other devices used variations of this principle start with the [[site.Webb1869|Webb Patent Adder]] (1869) and [[site.Stevenson1890|Stevens Adder]] (1890s), and then proceeding through the [[site.GoldenGem1915|Golden Gem]], [[site.Addiator1920|Addiator]], and [[site.Lightning1946|Lightning]] (1940s) - just illustrations of course of the wide range of different types and shapes of such devices that were sold. All of these work by adding movements (of cogs, chains, or straps) imprinted on which are the equivalent numbers. One using rods - a [[site.comptator1910|Comptator]] from 1910 is in this collection.
In a sense the above devices are linear digital sliding rules. One of the key innovations achieved by most of these devices (but not by the 'Troncets') was to develop a mechanical means of 'carrying' (interestingly already solved by Shickard) when a sum of two numbers in the same 'column' was greater than 9. A particularly simple approach to the issue of carrying was addressed in the [[Site.Adall1910|Addall]] from 1910, a circular proportional adder which simply used successive rotations of a wheel to wind on successive numbers whilst a ball bearing, moving in a spiral groove, tracked the successive rotations.
Such adders, whilst simple in concept and cheap to manufacture, are slow and virtually useless for efficient multiplication and division. Early experiments with keyboards can be seen in the [[site.Adix1903|Adix]] adding machine of 1903, but as the one in this collection shows, it can only add and is very hard even to zero.
!! The first commercial calculators (C19-C20)
More elaborate machines were developed in the nineteenth century each aiming to allow all four operations to be successfully carried out. Early examples in this collection include a Felt and Tarrant [[site.Comptometer"Woodie"1896|"woodie" Comptometer]] (1896) - one of the first 40 known to still exist; a highly sought after [[site.ThomasDeColmar1884|Thomas de Colmar arithmometer]] (1884) - also in the first 180 of these known to still exist, and a very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster pinwheel]] (1896) calculator.
Each represents a different mechanical approach to digitally carrying out the four functions, with an easy input, and a digital readout. Each then was improved through a series of innovations, first by the original inventors, and then as more players across the world competed to produce and market better variants.%0a %0aThomas de Colmar was the first person to produce a commercially useable calculator based on Leibnitz's stepped drum principle, and he named his machine the "arithmometer". The Thomas design was copied and improved by a number of other engineers and marketed from different countries.
As well as an [[site.ThomasDeColmar1884|original Thomas de Colmar]], this collection has a [[site.TIM1909|TIM ("Time is Money" )]] from 1909 which was by now designed as a more robust arithmometer, mounted now in an all steel frame. Burkhardt in Austria produced the first Austrian version of the Thomas machine, and later improvements were marketed in association with Bunzel.
In this collection there is a rare and probably unique [[site.BunzelPrototype1913|Bunzel-Delton arithmometer]] (1913) - the only known one existing with a forward crank, and from this fact, and the expert way in which the modification has been made, almost certainly a prototype created in the Bunzel workshops to match a patent, also reproduced int his site.
The pinwheel calculator was developed as an improvement on the arithmometer, utilising a clever 'counting gear' in which the number of teeth could be adjusted by sliders (and later a push-down keyboard). In this collection there is the already mentioned very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster]] pinwheel calculator from 1896 (an Ohdner Brunsviga which had been re-badged by Schuster who on-sold them). There is also a later [[Site.OriginalOhdner1938|Ohdner]] from 1938 and a [[site.Facit1945|Facit calculator]] from around 1945, and then from near the end of production of such machines, from the 1950s a very nice [[site.waltherDemonstrator1957|demonstration Walther 160]] calculator, showing its mechanism, complemented by a complete and [[site.Walther1957|fully operational Walther]] of the same model.
A further approach to the problem of the four functions is found in the American invention by Felt, of a device that he called the comptometer and patented in 1887. It performs the task of addition by a system of keys and levers. Notably no crank handle needs to be turned in order to perform the addition. Addition of a number automatically occurs whenever a corresponding key is pressed. This is fast. But the difficulty is that subtraction must be carried out by a process of addition of complementary numbers. Carrying of "tens" must be done by addition also. And the process of multiplication is done by multiple presses of the number. As already mentioned, in this collection we have one of the first models of the comptomoter - [[site.Comptometer"Woodie"1896|one from 1896 cased in wood]] and one of the 40 oldest known to still be in existence, as well as a later [[site.SumlockDemonstrator1955|demonstration machine from Bell Punch]], and then its embodiment in a fully working [[site.SumlockComptometer1950s|Bell Punch Sumlock]] (from the 1950s).
!! Mechanical minituarisation at its best - the Curta Calculator (1947-1970)
Finally there is the last and most beautifully mintuarised of the four function manual mechanical calculators buit by Curt Hertzstark, the son of Samuel Hertzstark (mentioned earlier) which he developed when a prisoner in Buchenwald concentration camp. The first model (the Model 1) began production in 1947. A (rare) example of the [[site.Curta-1-1948|Model I Curta ( with pin sliders which were soon improved upon)]] in mint condition coming from the first 5500 made in ~July 1948, is in this collection along with [[site.Curta-1-1967|another Model I (complete with its packing box and instructions)]] from 1967, and a [[site.Curta-2-1963|Model II Curta]] from 1962 (with an 'extra' leather carrying case). These machines constitute the pinacle reached in the development of the personal hand-operated mechanical digital calculator - able to carry out all four functions by means of a crank operated machine of exquisite minituarised workmanship and design. Production ceased in November 1970 although sales continued through 1973.
!! Building on Napier's rods: the search to automate multiplication and division
An entirely different evolutionary path which has already been mentioned was attempted to solving the problem of mechanising the four arithmetic operations (+, -, x, /), with emphasis on finding ways to directly perform the more difficult two operations of multiplication and division. This developed from the Napier's rods - or "bones" (developed by John Napier (1550-1617) to which we have already referred.
As already mentioned calculational approaches were designed around them by Wilhelm Schickard in 1623, Charles Cotterel in 1667, Gaspard Schott in 1668, and Samuel Morland between 1625 and 1695. A late and unique expression of these in this collection is [[site.BambergerOmega1904|Justin Bamberger's Omega Calculating Machine (1903-6)]]. It combines a stylus operated adding machine (complete with provision for automatic carry) with a sophisticated Napier bones based machine for discovering the partial products of two multiplied numbers (a bit like that in [[[site.Schickard1623|Schickard's much earlier device]]), which can then be added in the adding machine to find corresponding products and quotients. I am working on an English language set of instructions for its use have been devised and are available. For that, watch this space.
One helpful development in design was the development of a //proportional rack// which moved cogs through a series of increments depending on the length of a toothed rack with which they engaged. Different lengths thus represented different numbers to be added. In 1906 Christel Haman founded the Mercedes-Euklid company who adopted this principle to create a machine that after a process of development eventually could automate the process of division. There is both a [[site.Mercedes29Demonstrator1923|demonstration Mercedes-Euklid (model 29 from 1934)]] and then a [[site.Mercedes291923|fully working Mercedes-Euklid 29]] in this collection.
An arithmometer, the MADAS (standing for "Multiplication, Addition, Division - Automatically, Substraction") was developed by Erwin Janz and manufactured by Egli in Zurich, starting in 1913. As the acronym suggests, this machine achieved an entirely automatic division process - the first ever for an arithmometer style calculator. In this collection there is a [[site.Madas-IX-Maxima|MADAS IX Maxima calculator]] (produced in 1917) which could display 16 numbers in its results register.
A much heavier and complex mechanical approach was also explored. First it was embodied in Léon Bollée's calculating machine which won a gold medal at the Paris Exposition of 1889. One surviving example of this bulky but beautiful machine can be seen at the Musée des Arts et Métiers in Paris. This collection has only an article on this [[site.Bollee1890|"New Calculating Machine of very General Applicability"]] from the //Manufacturer and Builder// of 1890. Similar principles were however utilised by Otto Steiger who patented a rather more practical "Millionaire calculating machine" which had a simple enough mechanism to enable production on a commercial scale. The Millionaire calculating machine combines the idea of the physical embodiment of multiplication tables with that of the proportional rack. It is able to interrogate a multiplication table represented by metal rods, and in a single crank of the handle multiplies the multiplicand by a number set between 1 and 9 then advancing its internal carriage one place ready for the next multiplier to be set and applied. Manufactured by H.W. Egli, some 4,655 of these were sold. The [[site.Millionaire1912|Millionaire calculating machine]] in this collection is from 1912, and is rather rare since it is capable of greater accuracy - 10 column - than the more common 8 column ones.
!! One step backward, many steps forward: applying the electric motor (C20)
The electric motor marked the beginning of the end for all forms of mechanism more ingenious than those depending on the simple minded operation of addition and its inverse, subtraction. The greatest gains in efficiency could be obtained by simply increasing the speed with which these operations were repeated and controlled. Speed gains followed from simpler rather than more complex basic mechanisms. The control mechanisms that utilised these simple repeated basic operations, however did become more complex in the interests of using them to produce more complex and accurate outputs.
In this collection there is an [[site.Hertstark1929|arithmometer branded by Samuel Hertzstark]] from 1929, although by then he was not making machines. It is actually a Badenia (manufactured by the German Company Math. Bauerle in the Black Forest) which Hertzstark was by then re-badging (not long before he died). It features an electric motor, still in good working order, and with a keyboard instead of sliders for input. The features of it were not only that it had a keyboard for input, but also a control mechanism consisting of a column of keys which enabled a number to be entered and then added through 0 to 8 repeats representing multiplication by one to nine.
Whilst the above was an obvious innovation, the clumsiest approach in all the calculating devices - from the first arithmometer through to the Millionaire was division, which could only be done by a process along the lines of that done in long division. That is, the number to be divided (the dividend) is considered sequentially from the highest power of ten, and thus decomposed into a series of partial products of the successive parts of the dividend with the divisor. This required a process of guessing each partial product and then trying it out, the result being subtracted from the dividend to give a remainder, exactly as is done in pen and paper long division.
As well as Haman (and the [[site.Mercedes291923|Mercedes-Euklid]]) mentioned earlier, H.W. Egli also began the chain of innovation to mechanise this process, but only by abandonning the sophistication of the Millionaire and like Haman returning to the principles of the arithmometer. The result was the MADAS calculator, first produced in manual form, which could advance the carriage automatically one place at a time as division and multiplication were carried out. With progressive further modifications, including the insertion of an electric motor, the MADAS became the highly effective electric mechanical calculating machine whose use continued right into the 1960s. The late [[site.MADAS1950-60s|MADAS 20BTG calculator]] which is part of this collection represents a classic exemplar of the late stages of this evolutionary development. Its mechanism was also adopted in its essentials, then with many innovations (but with less of the Swiss robust construction of the Madas), in the US Frieden calculators. As it turned out, this final class of machines represented the pinacle of achievement in motorised mechanical calculation.
!! The Vanishing Point (1972)
Further innovation in mechanical calculation however was thwarted by a completely new approach, utlising the rapidly developing technology of electronics, becoming possible. Here the trend set in place by the return to simple basics and increasing speed through electrical processes was paradigmatically transformed, and all the prior approaches simultaneously reduced to a historical curiosity, once the application of integrated circuitry created the modern high speed and ultra-minituarised 'abacus' which combined with the capacity to make programmable conditional loops could rapidly move through the most complicated calculations, as exemplified by the [[site.HP-35-1972|HP-35]] pocket scientific calculator, at the mechanical calculation vanishing point of 1972.
24 July 2013
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What follows on this page is a first layer story - that of a developing solution to the problem of calculation. I should point out at once that this story is both a common theme amongst collectors' websites, and from another point of view rather superficial. After all the "problem" of how to calculate does not arise from nowhere. Facilitating calculation is a problem to the extent that calculation is proving necessary in people's lives, or institutions' operations. This has been a changing story too over the centuries in which these have developed, moving as it has from human pre-history through to the dynamic period of invention, construction and intensifying trade which is increasingly visible from the 1600s. It is also but a shallow account to consider calculators and calculation outside the challenges arising through the evolution of mathematical thinking. Some examination of that, over the last 10,000 years, is sketched in the section dealing with [[Site.Introduction|the context of the history of calculating technology]] (including the evolving mathematical context). Still, it can be helpful to start with the objects and some simple-minded thoughts about 'why' they have developed.
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What follows on this page is a first layer story - that of a developing solution to the problem of calculation. I should point out at once that this story is both a common theme amongst collectors' websites, and from another point of view rather superficial. After all the "problem" of how to calculate does not arise from nowhere. Facilitating calculation is a problem to the extent that calculation is proving necessary in people's lives, or institutions' operations. This has been a changing story too over the centuries in which these have developed, moving as it has from human pre-history through to the dynamic period of invention, construction and intensifying trade which is increasingly visible from the 1600s. It is also but a shallow account to consider calculators and calculation outside the challenges arising through the evolution of mathematical thinking. Some examination of that, over the last 10,000 years, is sketched in the section dealing with the context of the history of calculating technology (including the evolving mathematical context). Still, it can be helpful to start with the objects and some simple-minded thoughts about 'why' they have developed.
19 July 2013
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!! Collection
As with many others I have discovered (belatedly) that the act of forming a collection carries with it an insidious if not addictive desire. Not the least that is because there is always one more thing you could add which would make it all so much better, and maybe a wonderful opportunity to do so lurks just around the corner... This aspect needs some reflection - so I will write about it later. But putting that awkward fact aside, at least explicitly the aim of this collection has been to bring together some nice examples which focus primarily on manual mechanical means of calculation. A couple of electric motorised calculators have been acquired also as examples of the evolutionary direction of innovation during the transition period between manual machines, and the advent of electronic machines based on integrated circuitry.
Rather than seek to be comprehensive,[^For a comprehensive listing see http://www.rechenmaschinen-illustrated.com/^] which in any case turns out to be impossible because of the wide range of devices that were built over the last few centuries, the items in this collection have been acquired with the idea of illustrating the dynamics and trajectory of development. I have tried to acquire interesting machines which were both significant in their time, work as they are supposed to, and act as "bookends" to form examples of directions in which development was evolving. So, above all else, these objects are here because they please me![^Most, but not every acquisition was a success. However, even [[Fakes and False Promises|fakes]] can be interesting.^]
What follows on this page is a first layer story - that of a developing solution to the problem of calculation. I should point out at once that this story is both a common theme amongst collectors' websites, and from another point of view rather superficial. After all the "problem" of how to calculate does not arise from nowhere. Facilitating calculation is a problem to the extent that calculation is proving necessary in people's lives, or institutions' operations. This has been a changing story too over the centuries in which these have developed, moving as it has from human pre-history through to the dynamic period of invention, construction and intensifying trade which is increasingly visible from the 1600s. It is also but a shallow account to consider calculators and calculation outside the challenges arising through the evolution of mathematical thinking. Some examination of that, over the last 10,000 years, is sketched in the section dealing with [[Site.Introduction|the context of the history of calculating technology]] (including the evolving mathematical context). Still, it can be helpful to start with the objects and some simple-minded thoughts about 'why' they have developed.
!! Abacus (Pre-Historic - C21)
The first item may seem prosaic, but the abacus has the most venerable history of all, and is believed to have been introduced into Sumeria during the Akkadian occupation of ~2100-2000 BCE. The demand to compute all four operations (or as is often referred to in discussions of calculators - "four functions" or "four species"), being addition, subtraction, multiplication and division, has been long-standing, especially as out of these all computable functions can in principle be constructed. For reliable computation it is of course attractive to represent the numbers digitally (whether in cogs or beads or some other mechanism) and to be able to manipulate each pair of numbers precisely. The earliest (and highly effective) mechanism for achieving this is the abacus - which was used in ancient Rome, and analogically lies at the conceptual heart of all of most of the calculating machines (including the computer).
In this collection there is a [[site.abacus|contemporary Chinese abacus]]. There were of course different styles of such machines, each with their own particular capabilities and one day there should be added to this collection an early abacus (or Japanese Soroban). Beyond these the collection includes bookends to a series of branching devices designed to make these functions more simply utilised (if not necessarily any faster).
!! Sectors (C16-20)
From the beginning a more general central problem, has been how to add, subtract, multiply and divide quickly. Considering the technical aspects of this (rather than why the problem mattered) we can note that addition and its inverse, subtraction, seem relatively simple compared with the task of multiplication and division. True, multiplication and division can be performed through repeated addition and subtraction, but this is often very laborious. An early methodology which could be used for a more direct form of multiplication and division, invented in the sixteenth century was embodied in the sector. This instrument utilised the principle that the ratio of the base to side of two similar triangles is the same, to make multiplication or division easy. The device consisted of a hinged pair of rules with a variety of scales and was in use, particularly in gunnery, architecture and navigation, up to the early twentieth century. In this collection there are two sectors.
The first from c. 1700 was made by [[site.ButterfieldSector1700|Michael Butterfied]] (1635-1715 [or 1724 depending on source]) an English instrument maker who settled in Paris c. 1675. It is a gunnery sector of French design with additional scales for canon priming and orientation. Other very similar Butterfield sectors are held by the Metropolitan Museum of Art, the British Museum, and the National Museum of Scotland.
The second sector in this collection is an [[site.DoublettSector1830|English architect's sector]] made in c. 1830 from oxbone by the well known instrument makers T. and H. Doublett, London.
!! Early Logarithms (C17)
The discovery of logarithms which transform the problem of multiplication to that of addition (and division to subtraction) was a particularly helpful development for a time. The principle had been noted by the ancient Greeks but use of the principle required an easy means of evaluating the logarithms and their inverses. This was achieved with the first publication of logarithmic tables by Napier in 1614 (only nine years after the unsuccessful gunpowder plot against King James I in 1605).
The earliest bookend here is a set of logarithm and other tables published 12 years after Napier by [[site.Henrion1626|Henrion]] (a French mathematician) in 1626. This was the first French set of logarithm tables (but using the more practical decimal form than Napier's which had been published in 1624 by Briggs). It also contains [[site.Henrion1662|the design of the Gunter calculating rule]], published in England in 1624 (of which, more later).
The Henrion logarithm tables were the first to be introduced in French into Europe. The process of presenting tables of logarithms and associated functions was beset by the problem of errors that would easily creep in. In addition, different levels of accuracy were required in different situations.
Other publications of logarithms, with improvements, and always some errors, followed over subsequent centuries.
In addition to the very early logarithm tables by [[site.Henrion1662|Henrion (1626)]] this collection also contains a set of [[site.Gardiner1783|tables by Gardiner]] (with improvements by Callett) one and a half centuries later in England in 1783.
!! Mechanisation of multiplication, Napier's Rods and Schickard's calculator (C17)
Napier had also tackled the question of how to multiply and divide by creating in 1617 his "rabdologiae" a clever way of encoding multiplication tables onto 9 rods (or as they became known "[[Site.EncyclopediaBritannica1797|Napier's bones]]"). The moment (of 1617) is also therefore near what is regarded by some as a turning point.
Only 6 years after the publication of radbologiae, in 1623 Wilhelm Schickard constructed his "Calculating Clock" which is regarded as the first arithmetical machine since the abacus. It embodied Napier's rabdology but with the rods incorporated in a way that the operations including the necessary additions and subtractions could be carried out with the help of the machine. Schickard's machine was notable for not only its sophisticated use of Napier's rods through a set of sliders set in a rack, but also a register for storing intermediate numbers and a machine using wheels and cogs to enable carry forward to add the partial products obtained from the rods.
The Schickard approach would later be further developed in the [[site.BambergerOmega1904|Bamberger Omega]] (see later) which enjoyed a brief commercial appearance in an art nouveau design, almost three centuries later (1903-5). Unfortunately we only know of the Schickard machine through his correspondence with his friend Kepler (for whom he says he constructed a machine only for it to be destroyed in a fire.) All that is left of the machine is a rough drawing and description of what it does.
Another approach by Gaspar Schott (1668) turned the rods into cylinders and fixed them in a box ("cistula") so that the right one could be got simply with a turn of the knob. To this he had added a table of addition and subtraction to the inside cover of the cistula.[^"Habits of Knowing" pp 182-3.^] This approach is not unlike the Napier rod part of the Schickard.
In 1957 mathematician Bruno Baron von Freytag Löringhoff constructed a working replica from the Schickard drawings and description. A small number of other replicas now exist, and a reconstruction of [[site.Schickard1623|Schickard's Calculating Clock]] based on his sketch and description in a letter to Keppler is in this collection. (This reproduction Schickard was manufactured in a small set in Germany through the collaboration of three public, and one private, museums in Europe).
!! Slide rules (C17-C21)
The mechanical means of adding together logarithms (as first expounded by Gunter) in 1624 represented the logarithms as lengths on an appropriately calibrated scale (the so called "Gunter Line" or "Gunter Rule"). The [[site.Henrion1662|graphical calculation of the Gunter scale]] accompanies the [[site.Henrion1662|Henrion tables]] of 1626 in this collection.
The term "sliding rule" was coined by Everard who developed an instrument based on Gunter's representation of logarithms, but incorporated a slide to allow ease of calculation by adding lengths directly (rather than with dividers). The instrument was designed to support the work of excise officers as they sought to work out the tax payable on different containers of spirits. Logarithmic as well as trigonometric scales were used to calculate quantities in various shaped containers. The earliest such bookend here (beyond Henrion's graphical design for a slide rule based on the work of Gunter, already mentioned) is an [[site.Everard1759|early "sliding rule" gauger based on Everard's design]], but made by Edward Roberts (snr) of Jewery Lane, London, between 1759-69.
Slide rules of course developed in many ways - accuracy, the incorporation of a sliding cursor, the range of computable functions, use of different materials, minituarisation and other design innovations, represented here by a little series through a [[site.KeuffelandEsser1908|Keuffel and Esser (1908)]] to the widely regarded finest slide rule - the [[site.FaberCastell1967|Faber and Castell 2/83N Novo Duplex]].
There was much experimentation with how to get the most computational value out of different shapes and arrangements. In this collection there is a series of circular and cylindrical slide rules - the [[site.Thacher1911|Thacher Calculation Instrument]] from 1911 (the most accurate slide rule ever made, equivalent to a 60 foot slide rule), and then on the path to minituarisation the Fuller (40 foot equivalent), [[site.OtisKing1960|Otis King]], and the watch style pocket circular slide rules - the [[site.KL11960s|Russian KL-1]] and the [[site.Fowler1948|Fowler Jubilee Long Scale]] (1948).
!! Conversion devices
The bane of earlier centuries, and even still, is the fact that quantities were often measured in different incompatible units. Special purpose devices, basically just forms of mechanical look-up tables were devised to deal with this sort of issue, since performing the calculation each time was time consuming, and many did not have the skills in any case to do it. The need to be able to measure and compare occurred extensively in shipping and commerce rather early. In this collection we have a [[site.CHAIX1790|"Convertisseur"]] from (1780-1810) designed by Gabrielle Chaix in Paris, to assist merchants and artisans to convert between the old measure of distance (the "Aune") and the new measure introduced by the Revolutionary Government of the metre in 1791.
!! The first cogged calculation - Pascale and Morland (C17)
Following Schickard's embodiment of Napier's radibology into a machine in 1623 mathematical prodigy Blaise Pascale in 1642 invented a much more robust calculating device focussed on addition. The young Pascale embarked on this to assist his father in adding taxes. There exists a letter (of which I have only the text) from Pascale which explains how this device is used. Nine versions of the machine remain in existence in museums and other collections. Several replica have been constructed of which one such replica "[[Site.Pascaline1652|Pascaline]]" is in this collection.
Around 1666 Sir Samuel Morland invented two cogged machines - one for doing additions and the other for multiplication. In each case the machines were designed to make basic arithmetical operations more accessible for those not versed in them. He published descriptions of the machines in what is regarded as (arguably with Pascale's letters) the first calculating machine (or even "computer") manual. A copy of the [[site.Morland1672|first edition of Morland's book published in 1672]], the book itself being exceptionally rare, is in this collection.
!! Adders (C17-C20)
Other simple mechanisms for adding and subtracting were developed in a series of devices whose basic mechanism using metal strips or cogs which can be used by means of a stylis can be moved linearly or circularly the distance displaced standing for added numbers. This type of device, first developed in the 17th Century by inventors such as Claude Perrot (1613-88) was popularised by J. Louis Troncet, a French inventor who created his Arithmographe,[^for more, including illustrations from his patents see http://www.ami19.org/Troncet/Troncet.html.^] later to be known as the [[site.Addiator1920|Addiator]] (of which there is one in this collection).
Other devices used variations of this principle start with the [[site.Webb1869|Webb Patent Adder]] (1869) and [[site.Stevenson1890|Stevens Adder]] (1890s), and then proceeding through the [[site.GoldenGem1915|Golden Gem]], [[site.Addiator1920|Addiator]], and [[site.Lightning1946|Lightning]] (1940s) - just illustrations of course of the wide range of different types and shapes of such devices that were sold. All of these work by adding movements (of cogs, chains, or straps) imprinted on which are the equivalent numbers. One using rods - a [[site.comptator1910|Comptator]] from 1910 is in this collection.
In a sense the above devices are linear digital sliding rules. One of the key innovations achieved by most of these devices (but not by the 'Troncets') was to develop a mechanical means of 'carrying' (interestingly already solved by Shickard) when a sum of two numbers in the same 'column' was greater than 9. A particularly simple approach to the issue of carrying was addressed in the [[Site.Adall1910|Addall]] from 1910, a circular proportional adder which simply used successive rotations of a wheel to wind on successive numbers whilst a ball bearing, moving in a spiral groove, tracked the successive rotations.
Such adders, whilst simple in concept and cheap to manufacture, are slow and virtually useless for efficient multiplication and division. Early experiments with keyboards can be seen in the [[site.Adix1903|Adix]] adding machine of 1903, but as the one in this collection shows, it can only add and is very hard even to zero.
!! The first commercial calculators (C19-C20)
More elaborate machines were developed in the nineteenth century each aiming to allow all four operations to be successfully carried out. Early examples in this collection include a Felt and Tarrant [[site.Comptometer"Woodie"1896|"woodie" Comptometer]] (1896) - one of the first 40 known to still exist; a highly sought after [[site.ThomasDeColmar1884|Thomas de Colmar arithmometer]] (1884) - also in the first 180 of these known to still exist, and a very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster pinwheel]] (1896) calculator.
Each represents a different mechanical approach to digitally carrying out the four functions, with an easy input, and a digital readout. Each then was improved through a series of innovations, first by the original inventors, and then as more players across the world competed to produce and market better variants.%0a %0aThomas de Colmar was the first person to produce a commercially useable calculator based on Leibnitz's stepped drum principle, and he named his machine the "arithmometer". The Thomas design was copied and improved by a number of other engineers and marketed from different countries.
As well as an [[site.ThomasDeColmar1884|original Thomas de Colmar]], this collection has a [[site.TIM1909|TIM ("Time is Money" )]] from 1909 which was by now designed as a more robust arithmometer, mounted now in an all steel frame. Burkhardt in Austria produced the first Austrian version of the Thomas machine, and later improvements were marketed in association with Bunzel.
In this collection there is a rare and probably unique [[site.BunzelPrototype1913|Bunzel-Delton arithmometer]] (1913) - the only known one existing with a forward crank, and from this fact, and the expert way in which the modification has been made, almost certainly a prototype created in the Bunzel workshops to match a patent, also reproduced int his site.
The pinwheel calculator was developed as an improvement on the arithmometer, utilising a clever 'counting gear' in which the number of teeth could be adjusted by sliders (and later a push-down keyboard). In this collection there is the already mentioned very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster]] pinwheel calculator from 1896 (an Ohdner Brunsviga which had been re-badged by Schuster who on-sold them). There is also a later [[Site.OriginalOhdner1938|Ohdner]] from 1938 and a [[site.Facit1945|Facit calculator]] from around 1945, and then from near the end of production of such machines, from the 1950s a very nice [[site.waltherDemonstrator1957|demonstration Walther 160]] calculator, showing its mechanism, complemented by a complete and [[site.Walther1957|fully operational Walther]] of the same model.
A further approach to the problem of the four functions is found in the American invention by Felt, of a device that he called the comptometer and patented in 1887. It performs the task of addition by a system of keys and levers. Notably no crank handle needs to be turned in order to perform the addition. Addition of a number automatically occurs whenever a corresponding key is pressed. This is fast. But the difficulty is that subtraction must be carried out by a process of addition of complementary numbers. Carrying of "tens" must be done by addition also. And the process of multiplication is done by multiple presses of the number. As already mentioned, in this collection we have one of the first models of the comptomoter - [[site.Comptometer"Woodie"1896|one from 1896 cased in wood]] and one of the 40 oldest known to still be in existence, as well as a later [[site.SumlockDemonstrator1955|demonstration machine from Bell Punch]], and then its embodiment in a fully working [[site.SumlockComptometer1950s|Bell Punch Sumlock]] (from the 1950s).
!! Mechanical minituarisation at its best - the Curta Calculator (1947-1970)
Finally there is the last and most beautifully mintuarised of the four function manual mechanical calculators buit by Curt Hertzstark, the son of Samuel Hertzstark (mentioned earlier) which he developed when a prisoner in Buchenwald concentration camp. The first model (the Model 1) began production in 1947. A (rare) example of the [[site.Curta-1-1948|Model I Curta ( with pin sliders which were soon improved upon)]] in mint condition coming from the first 5500 made in ~July 1948, is in this collection along with [[site.Curta-1-1967|another Model I (complete with its packing box and instructions)]] from 1967, and a [[site.Curta-2-1963|Model II Curta]] from 1962 (with an 'extra' leather carrying case). These machines constitute the pinacle reached in the development of the personal hand-operated mechanical digital calculator - able to carry out all four functions by means of a crank operated machine of exquisite minituarised workmanship and design. Production ceased in November 1970 although sales continued through 1973.
!! Building on Napier's rods: the search to automate multiplication and division
An entirely different evolutionary path which has already been mentioned was attempted to solving the problem of mechanising the four arithmetic operations (+, -, x, /), with emphasis on finding ways to directly perform the more difficult two operations of multiplication and division. This developed from the Napier's rods - or "bones" (developed by John Napier (1550-1617) to which we have already referred.
As already mentioned calculational approaches were designed around them by Wilhelm Schickard in 1623, Charles Cotterel in 1667, Gaspard Schott in 1668, and Samuel Morland between 1625 and 1695. A late and unique expression of these in this collection is [[site.BambergerOmega1904|Justin Bamberger's Omega Calculating Machine (1903-6)]]. It combines a stylus operated adding machine (complete with provision for automatic carry) with a sophisticated Napier bones based machine for discovering the partial products of two multiplied numbers (a bit like that in [[[site.Schickard1623|Schickard's much earlier device]]), which can then be added in the adding machine to find corresponding products and quotients. I am working on an English language set of instructions for its use have been devised and are available. For that, watch this space.
One helpful development in design was the development of a //proportional rack// which moved cogs through a series of increments depending on the length of a toothed rack with which they engaged. Different lengths thus represented different numbers to be added. In 1906 Christel Haman founded the Mercedes-Euklid company who adopted this principle to create a machine that after a process of development eventually could automate the process of division. There is both a [[site.Mercedes29Demonstrator1923|demonstration Mercedes-Euklid (model 29 from 1934)]] and then a [[site.Mercedes291923|fully working Mercedes-Euklid 29]] in this collection.
An arithmometer, the MADAS (standing for "Multiplication, Addition, Division - Automatically, Substraction") was developed by Erwin Janz and manufactured by Egli in Zurich, starting in 1913. As the acronym suggests, this machine achieved an entirely automatic division process - the first ever for an arithmometer style calculator. In this collection there is a [[site.Madas-IX-Maxima|MADAS IX Maxima calculator]] (produced in 1917) which could display 16 numbers in its results register.
A much heavier and complex mechanical approach was also explored. First it was embodied in Léon Bollée's calculating machine which won a gold medal at the Paris Exposition of 1889. One surviving example of this bulky but beautiful machine can be seen at the Musée des Arts et Métiers in Paris. This collection has only an article on this [[site.Bollee1890|"New Calculating Machine of very General Applicability"]] from the //Manufacturer and Builder// of 1890. Similar principles were however utilised by Otto Steiger who patented a rather more practical "Millionaire calculating machine" which had a simple enough mechanism to enable production on a commercial scale. The Millionaire calculating machine combines the idea of the physical embodiment of multiplication tables with that of the proportional rack. It is able to interrogate a multiplication table represented by metal rods, and in a single crank of the handle multiplies the multiplicand by a number set between 1 and 9 then advancing its internal carriage one place ready for the next multiplier to be set and applied. Manufactured by H.W. Egli, some 4,655 of these were sold. The [[site.Millionaire1912|Millionaire calculating machine]] in this collection is from 1912, and is rather rare since it is capable of greater accuracy - 10 column - than the more common 8 column ones.
!! One step backward, many steps forward: applying the electric motor (C20)
The electric motor marked the beginning of the end for all forms of mechanism more ingenious than those depending on the simple minded operation of addition and its inverse, subtraction. The greatest gains in efficiency could be obtained by simply increasing the speed with which these operations were repeated and controlled. Speed gains followed from simpler rather than more complex basic mechanisms. The control mechanisms that utilised these simple repeated basic operations, however did become more complex in the interests of using them to produce more complex and accurate outputs.
In this collection there is an [[site.Hertstark1929|arithmometer branded by Samuel Hertzstark]] from 1929, although by then he was not making machines. It is actually a Badenia (manufactured by the German Company Math. Bauerle in the Black Forest) which Hertzstark was by then re-badging (not long before he died). It features an electric motor, still in good working order, and with a keyboard instead of sliders for input. The features of it were not only that it had a keyboard for input, but also a control mechanism consisting of a column of keys which enabled a number to be entered and then added through 0 to 8 repeats representing multiplication by one to nine.
Whilst the above was an obvious innovation, the clumsiest approach in all the calculating devices - from the first arithmometer through to the Millionaire was division, which could only be done by a process along the lines of that done in long division. That is, the number to be divided (the dividend) is considered sequentially from the highest power of ten, and thus decomposed into a series of partial products of the successive parts of the dividend with the divisor. This required a process of guessing each partial product and then trying it out, the result being subtracted from the dividend to give a remainder, exactly as is done in pen and paper long division.
As well as Haman (and the [[site.Mercedes291923|Mercedes-Euklid]]) mentioned earlier, H.W. Egli also began the chain of innovation to mechanise this process, but only by abandonning the sophistication of the Millionaire and like Haman returning to the principles of the arithmometer. The result was the MADAS calculator, first produced in manual form, which could advance the carriage automatically one place at a time as division and multiplication were carried out. With progressive further modifications, including the insertion of an electric motor, the MADAS became the highly effective electric mechanical calculating machine whose use continued right into the 1960s. The late [[site.MADAS1950-60s|MADAS 20BTG calculator]] which is part of this collection represents a classic exemplar of the late stages of this evolutionary development. Its mechanism was also adopted in its essentials, then with many innovations (but with less of the Swiss robust construction of the Madas), in the US Frieden calculators. As it turned out, this final class of machines represented the pinacle of achievement in motorised mechanical calculation.
!! The Vanishing Point (1972)
Further innovation in mechanical calculation however was thwarted by a completely new approach, utlising the rapidly developing technology of electronics, becoming possible. Here the trend set in place by the return to simple basics and increasing speed through electrical processes was paradigmatically transformed, and all the prior approaches simultaneously reduced to a historical curiosity, once the application of integrated circuitry created the modern high speed and ultra-minituarised 'abacus' which combined with the capacity to make programmable conditional loops could rapidly move through the most complicated calculations, as exemplified by the [[site.HP-35-1972|HP-35]] pocket scientific calculator, at the mechanical calculation vanishing point of 1972.
As with many others I have discovered (belatedly) that the act of forming a collection carries with it an insidious if not addictive desire. Not the least that is because there is always one more thing you could add which would make it all so much better, and maybe a wonderful opportunity to do so lurks just around the corner... This aspect needs some reflection - so I will write about it later. But putting that awkward fact aside, at least explicitly the aim of this collection has been to bring together some nice examples which focus primarily on manual mechanical means of calculation. A couple of electric motorised calculators have been acquired also as examples of the evolutionary direction of innovation during the transition period between manual machines, and the advent of electronic machines based on integrated circuitry.
Rather than seek to be comprehensive,[^For a comprehensive listing see http://www.rechenmaschinen-illustrated.com/^] which in any case turns out to be impossible because of the wide range of devices that were built over the last few centuries, the items in this collection have been acquired with the idea of illustrating the dynamics and trajectory of development. I have tried to acquire interesting machines which were both significant in their time, work as they are supposed to, and act as "bookends" to form examples of directions in which development was evolving. So, above all else, these objects are here because they please me![^Most, but not every acquisition was a success. However, even [[Fakes and False Promises|fakes]] can be interesting.^]
What follows on this page is a first layer story - that of a developing solution to the problem of calculation. I should point out at once that this story is both a common theme amongst collectors' websites, and from another point of view rather superficial. After all the "problem" of how to calculate does not arise from nowhere. Facilitating calculation is a problem to the extent that calculation is proving necessary in people's lives, or institutions' operations. This has been a changing story too over the centuries in which these have developed, moving as it has from human pre-history through to the dynamic period of invention, construction and intensifying trade which is increasingly visible from the 1600s. It is also but a shallow account to consider calculators and calculation outside the challenges arising through the evolution of mathematical thinking. Some examination of that, over the last 10,000 years, is sketched in the section dealing with [[Site.Introduction|the context of the history of calculating technology]] (including the evolving mathematical context). Still, it can be helpful to start with the objects and some simple-minded thoughts about 'why' they have developed.
!! Abacus (Pre-Historic - C21)
The first item may seem prosaic, but the abacus has the most venerable history of all, and is believed to have been introduced into Sumeria during the Akkadian occupation of ~2100-2000 BCE. The demand to compute all four operations (or as is often referred to in discussions of calculators - "four functions" or "four species"), being addition, subtraction, multiplication and division, has been long-standing, especially as out of these all computable functions can in principle be constructed. For reliable computation it is of course attractive to represent the numbers digitally (whether in cogs or beads or some other mechanism) and to be able to manipulate each pair of numbers precisely. The earliest (and highly effective) mechanism for achieving this is the abacus - which was used in ancient Rome, and analogically lies at the conceptual heart of all of most of the calculating machines (including the computer).
In this collection there is a [[site.abacus|contemporary Chinese abacus]]. There were of course different styles of such machines, each with their own particular capabilities and one day there should be added to this collection an early abacus (or Japanese Soroban). Beyond these the collection includes bookends to a series of branching devices designed to make these functions more simply utilised (if not necessarily any faster).
!! Sectors (C16-20)
From the beginning a more general central problem, has been how to add, subtract, multiply and divide quickly. Considering the technical aspects of this (rather than why the problem mattered) we can note that addition and its inverse, subtraction, seem relatively simple compared with the task of multiplication and division. True, multiplication and division can be performed through repeated addition and subtraction, but this is often very laborious. An early methodology which could be used for a more direct form of multiplication and division, invented in the sixteenth century was embodied in the sector. This instrument utilised the principle that the ratio of the base to side of two similar triangles is the same, to make multiplication or division easy. The device consisted of a hinged pair of rules with a variety of scales and was in use, particularly in gunnery, architecture and navigation, up to the early twentieth century. In this collection there are two sectors.
The first from c. 1700 was made by [[site.ButterfieldSector1700|Michael Butterfied]] (1635-1715 [or 1724 depending on source]) an English instrument maker who settled in Paris c. 1675. It is a gunnery sector of French design with additional scales for canon priming and orientation. Other very similar Butterfield sectors are held by the Metropolitan Museum of Art, the British Museum, and the National Museum of Scotland.
The second sector in this collection is an [[site.DoublettSector1830|English architect's sector]] made in c. 1830 from oxbone by the well known instrument makers T. and H. Doublett, London.
!! Early Logarithms (C17)
The discovery of logarithms which transform the problem of multiplication to that of addition (and division to subtraction) was a particularly helpful development for a time. The principle had been noted by the ancient Greeks but use of the principle required an easy means of evaluating the logarithms and their inverses. This was achieved with the first publication of logarithmic tables by Napier in 1614 (only nine years after the unsuccessful gunpowder plot against King James I in 1605).
The earliest bookend here is a set of logarithm and other tables published 12 years after Napier by [[site.Henrion1626|Henrion]] (a French mathematician) in 1626. This was the first French set of logarithm tables (but using the more practical decimal form than Napier's which had been published in 1624 by Briggs). It also contains [[site.Henrion1662|the design of the Gunter calculating rule]], published in England in 1624 (of which, more later).
The Henrion logarithm tables were the first to be introduced in French into Europe. The process of presenting tables of logarithms and associated functions was beset by the problem of errors that would easily creep in. In addition, different levels of accuracy were required in different situations.
Other publications of logarithms, with improvements, and always some errors, followed over subsequent centuries.
In addition to the very early logarithm tables by [[site.Henrion1662|Henrion (1626)]] this collection also contains a set of [[site.Gardiner1783|tables by Gardiner]] (with improvements by Callett) one and a half centuries later in England in 1783.
!! Mechanisation of multiplication, Napier's Rods and Schickard's calculator (C17)
Napier had also tackled the question of how to multiply and divide by creating in 1617 his "rabdologiae" a clever way of encoding multiplication tables onto 9 rods (or as they became known "[[Site.EncyclopediaBritannica1797|Napier's bones]]"). The moment (of 1617) is also therefore near what is regarded by some as a turning point.
Only 6 years after the publication of radbologiae, in 1623 Wilhelm Schickard constructed his "Calculating Clock" which is regarded as the first arithmetical machine since the abacus. It embodied Napier's rabdology but with the rods incorporated in a way that the operations including the necessary additions and subtractions could be carried out with the help of the machine. Schickard's machine was notable for not only its sophisticated use of Napier's rods through a set of sliders set in a rack, but also a register for storing intermediate numbers and a machine using wheels and cogs to enable carry forward to add the partial products obtained from the rods.
The Schickard approach would later be further developed in the [[site.BambergerOmega1904|Bamberger Omega]] (see later) which enjoyed a brief commercial appearance in an art nouveau design, almost three centuries later (1903-5). Unfortunately we only know of the Schickard machine through his correspondence with his friend Kepler (for whom he says he constructed a machine only for it to be destroyed in a fire.) All that is left of the machine is a rough drawing and description of what it does.
Another approach by Gaspar Schott (1668) turned the rods into cylinders and fixed them in a box ("cistula") so that the right one could be got simply with a turn of the knob. To this he had added a table of addition and subtraction to the inside cover of the cistula.[^"Habits of Knowing" pp 182-3.^] This approach is not unlike the Napier rod part of the Schickard.
In 1957 mathematician Bruno Baron von Freytag Löringhoff constructed a working replica from the Schickard drawings and description. A small number of other replicas now exist, and a reconstruction of [[site.Schickard1623|Schickard's Calculating Clock]] based on his sketch and description in a letter to Keppler is in this collection. (This reproduction Schickard was manufactured in a small set in Germany through the collaboration of three public, and one private, museums in Europe).
!! Slide rules (C17-C21)
The mechanical means of adding together logarithms (as first expounded by Gunter) in 1624 represented the logarithms as lengths on an appropriately calibrated scale (the so called "Gunter Line" or "Gunter Rule"). The [[site.Henrion1662|graphical calculation of the Gunter scale]] accompanies the [[site.Henrion1662|Henrion tables]] of 1626 in this collection.
The term "sliding rule" was coined by Everard who developed an instrument based on Gunter's representation of logarithms, but incorporated a slide to allow ease of calculation by adding lengths directly (rather than with dividers). The instrument was designed to support the work of excise officers as they sought to work out the tax payable on different containers of spirits. Logarithmic as well as trigonometric scales were used to calculate quantities in various shaped containers. The earliest such bookend here (beyond Henrion's graphical design for a slide rule based on the work of Gunter, already mentioned) is an [[site.Everard1759|early "sliding rule" gauger based on Everard's design]], but made by Edward Roberts (snr) of Jewery Lane, London, between 1759-69.
Slide rules of course developed in many ways - accuracy, the incorporation of a sliding cursor, the range of computable functions, use of different materials, minituarisation and other design innovations, represented here by a little series through a [[site.KeuffelandEsser1908|Keuffel and Esser (1908)]] to the widely regarded finest slide rule - the [[site.FaberCastell1967|Faber and Castell 2/83N Novo Duplex]].
There was much experimentation with how to get the most computational value out of different shapes and arrangements. In this collection there is a series of circular and cylindrical slide rules - the [[site.Thacher1911|Thacher Calculation Instrument]] from 1911 (the most accurate slide rule ever made, equivalent to a 60 foot slide rule), and then on the path to minituarisation the Fuller (40 foot equivalent), [[site.OtisKing1960|Otis King]], and the watch style pocket circular slide rules - the [[site.KL11960s|Russian KL-1]] and the [[site.Fowler1948|Fowler Jubilee Long Scale]] (1948).
!! Conversion devices
The bane of earlier centuries, and even still, is the fact that quantities were often measured in different incompatible units. Special purpose devices, basically just forms of mechanical look-up tables were devised to deal with this sort of issue, since performing the calculation each time was time consuming, and many did not have the skills in any case to do it. The need to be able to measure and compare occurred extensively in shipping and commerce rather early. In this collection we have a [[site.CHAIX1790|"Convertisseur"]] from (1780-1810) designed by Gabrielle Chaix in Paris, to assist merchants and artisans to convert between the old measure of distance (the "Aune") and the new measure introduced by the Revolutionary Government of the metre in 1791.
!! The first cogged calculation - Pascale and Morland (C17)
Following Schickard's embodiment of Napier's radibology into a machine in 1623 mathematical prodigy Blaise Pascale in 1642 invented a much more robust calculating device focussed on addition. The young Pascale embarked on this to assist his father in adding taxes. There exists a letter (of which I have only the text) from Pascale which explains how this device is used. Nine versions of the machine remain in existence in museums and other collections. Several replica have been constructed of which one such replica "[[Site.Pascaline1652|Pascaline]]" is in this collection.
Around 1666 Sir Samuel Morland invented two cogged machines - one for doing additions and the other for multiplication. In each case the machines were designed to make basic arithmetical operations more accessible for those not versed in them. He published descriptions of the machines in what is regarded as (arguably with Pascale's letters) the first calculating machine (or even "computer") manual. A copy of the [[site.Morland1672|first edition of Morland's book published in 1672]], the book itself being exceptionally rare, is in this collection.
!! Adders (C17-C20)
Other simple mechanisms for adding and subtracting were developed in a series of devices whose basic mechanism using metal strips or cogs which can be used by means of a stylis can be moved linearly or circularly the distance displaced standing for added numbers. This type of device, first developed in the 17th Century by inventors such as Claude Perrot (1613-88) was popularised by J. Louis Troncet, a French inventor who created his Arithmographe,[^for more, including illustrations from his patents see http://www.ami19.org/Troncet/Troncet.html.^] later to be known as the [[site.Addiator1920|Addiator]] (of which there is one in this collection).
Other devices used variations of this principle start with the [[site.Webb1869|Webb Patent Adder]] (1869) and [[site.Stevenson1890|Stevens Adder]] (1890s), and then proceeding through the [[site.GoldenGem1915|Golden Gem]], [[site.Addiator1920|Addiator]], and [[site.Lightning1946|Lightning]] (1940s) - just illustrations of course of the wide range of different types and shapes of such devices that were sold. All of these work by adding movements (of cogs, chains, or straps) imprinted on which are the equivalent numbers. One using rods - a [[site.comptator1910|Comptator]] from 1910 is in this collection.
In a sense the above devices are linear digital sliding rules. One of the key innovations achieved by most of these devices (but not by the 'Troncets') was to develop a mechanical means of 'carrying' (interestingly already solved by Shickard) when a sum of two numbers in the same 'column' was greater than 9. A particularly simple approach to the issue of carrying was addressed in the [[Site.Adall1910|Addall]] from 1910, a circular proportional adder which simply used successive rotations of a wheel to wind on successive numbers whilst a ball bearing, moving in a spiral groove, tracked the successive rotations.
Such adders, whilst simple in concept and cheap to manufacture, are slow and virtually useless for efficient multiplication and division. Early experiments with keyboards can be seen in the [[site.Adix1903|Adix]] adding machine of 1903, but as the one in this collection shows, it can only add and is very hard even to zero.
!! The first commercial calculators (C19-C20)
More elaborate machines were developed in the nineteenth century each aiming to allow all four operations to be successfully carried out. Early examples in this collection include a Felt and Tarrant [[site.Comptometer"Woodie"1896|"woodie" Comptometer]] (1896) - one of the first 40 known to still exist; a highly sought after [[site.ThomasDeColmar1884|Thomas de Colmar arithmometer]] (1884) - also in the first 180 of these known to still exist, and a very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster pinwheel]] (1896) calculator.
Each represents a different mechanical approach to digitally carrying out the four functions, with an easy input, and a digital readout. Each then was improved through a series of innovations, first by the original inventors, and then as more players across the world competed to produce and market better variants.%0a %0aThomas de Colmar was the first person to produce a commercially useable calculator based on Leibnitz's stepped drum principle, and he named his machine the "arithmometer". The Thomas design was copied and improved by a number of other engineers and marketed from different countries.
As well as an [[site.ThomasDeColmar1884|original Thomas de Colmar]], this collection has a [[site.TIM1909|TIM ("Time is Money" )]] from 1909 which was by now designed as a more robust arithmometer, mounted now in an all steel frame. Burkhardt in Austria produced the first Austrian version of the Thomas machine, and later improvements were marketed in association with Bunzel.
In this collection there is a rare and probably unique [[site.BunzelPrototype1913|Bunzel-Delton arithmometer]] (1913) - the only known one existing with a forward crank, and from this fact, and the expert way in which the modification has been made, almost certainly a prototype created in the Bunzel workshops to match a patent, also reproduced int his site.
The pinwheel calculator was developed as an improvement on the arithmometer, utilising a clever 'counting gear' in which the number of teeth could be adjusted by sliders (and later a push-down keyboard). In this collection there is the already mentioned very early [[site.BrunsvigaSchuster1896|Brunsviga-Schuster]] pinwheel calculator from 1896 (an Ohdner Brunsviga which had been re-badged by Schuster who on-sold them). There is also a later [[Site.OriginalOhdner1938|Ohdner]] from 1938 and a [[site.Facit1945|Facit calculator]] from around 1945, and then from near the end of production of such machines, from the 1950s a very nice [[site.waltherDemonstrator1957|demonstration Walther 160]] calculator, showing its mechanism, complemented by a complete and [[site.Walther1957|fully operational Walther]] of the same model.
A further approach to the problem of the four functions is found in the American invention by Felt, of a device that he called the comptometer and patented in 1887. It performs the task of addition by a system of keys and levers. Notably no crank handle needs to be turned in order to perform the addition. Addition of a number automatically occurs whenever a corresponding key is pressed. This is fast. But the difficulty is that subtraction must be carried out by a process of addition of complementary numbers. Carrying of "tens" must be done by addition also. And the process of multiplication is done by multiple presses of the number. As already mentioned, in this collection we have one of the first models of the comptomoter - [[site.Comptometer"Woodie"1896|one from 1896 cased in wood]] and one of the 40 oldest known to still be in existence, as well as a later [[site.SumlockDemonstrator1955|demonstration machine from Bell Punch]], and then its embodiment in a fully working [[site.SumlockComptometer1950s|Bell Punch Sumlock]] (from the 1950s).
!! Mechanical minituarisation at its best - the Curta Calculator (1947-1970)
Finally there is the last and most beautifully mintuarised of the four function manual mechanical calculators buit by Curt Hertzstark, the son of Samuel Hertzstark (mentioned earlier) which he developed when a prisoner in Buchenwald concentration camp. The first model (the Model 1) began production in 1947. A (rare) example of the [[site.Curta-1-1948|Model I Curta ( with pin sliders which were soon improved upon)]] in mint condition coming from the first 5500 made in ~July 1948, is in this collection along with [[site.Curta-1-1967|another Model I (complete with its packing box and instructions)]] from 1967, and a [[site.Curta-2-1963|Model II Curta]] from 1962 (with an 'extra' leather carrying case). These machines constitute the pinacle reached in the development of the personal hand-operated mechanical digital calculator - able to carry out all four functions by means of a crank operated machine of exquisite minituarised workmanship and design. Production ceased in November 1970 although sales continued through 1973.
!! Building on Napier's rods: the search to automate multiplication and division
An entirely different evolutionary path which has already been mentioned was attempted to solving the problem of mechanising the four arithmetic operations (+, -, x, /), with emphasis on finding ways to directly perform the more difficult two operations of multiplication and division. This developed from the Napier's rods - or "bones" (developed by John Napier (1550-1617) to which we have already referred.
As already mentioned calculational approaches were designed around them by Wilhelm Schickard in 1623, Charles Cotterel in 1667, Gaspard Schott in 1668, and Samuel Morland between 1625 and 1695. A late and unique expression of these in this collection is [[site.BambergerOmega1904|Justin Bamberger's Omega Calculating Machine (1903-6)]]. It combines a stylus operated adding machine (complete with provision for automatic carry) with a sophisticated Napier bones based machine for discovering the partial products of two multiplied numbers (a bit like that in [[[site.Schickard1623|Schickard's much earlier device]]), which can then be added in the adding machine to find corresponding products and quotients. I am working on an English language set of instructions for its use have been devised and are available. For that, watch this space.
One helpful development in design was the development of a //proportional rack// which moved cogs through a series of increments depending on the length of a toothed rack with which they engaged. Different lengths thus represented different numbers to be added. In 1906 Christel Haman founded the Mercedes-Euklid company who adopted this principle to create a machine that after a process of development eventually could automate the process of division. There is both a [[site.Mercedes29Demonstrator1923|demonstration Mercedes-Euklid (model 29 from 1934)]] and then a [[site.Mercedes291923|fully working Mercedes-Euklid 29]] in this collection.
An arithmometer, the MADAS (standing for "Multiplication, Addition, Division - Automatically, Substraction") was developed by Erwin Janz and manufactured by Egli in Zurich, starting in 1913. As the acronym suggests, this machine achieved an entirely automatic division process - the first ever for an arithmometer style calculator. In this collection there is a [[site.Madas-IX-Maxima|MADAS IX Maxima calculator]] (produced in 1917) which could display 16 numbers in its results register.
A much heavier and complex mechanical approach was also explored. First it was embodied in Léon Bollée's calculating machine which won a gold medal at the Paris Exposition of 1889. One surviving example of this bulky but beautiful machine can be seen at the Musée des Arts et Métiers in Paris. This collection has only an article on this [[site.Bollee1890|"New Calculating Machine of very General Applicability"]] from the //Manufacturer and Builder// of 1890. Similar principles were however utilised by Otto Steiger who patented a rather more practical "Millionaire calculating machine" which had a simple enough mechanism to enable production on a commercial scale. The Millionaire calculating machine combines the idea of the physical embodiment of multiplication tables with that of the proportional rack. It is able to interrogate a multiplication table represented by metal rods, and in a single crank of the handle multiplies the multiplicand by a number set between 1 and 9 then advancing its internal carriage one place ready for the next multiplier to be set and applied. Manufactured by H.W. Egli, some 4,655 of these were sold. The [[site.Millionaire1912|Millionaire calculating machine]] in this collection is from 1912, and is rather rare since it is capable of greater accuracy - 10 column - than the more common 8 column ones.
!! One step backward, many steps forward: applying the electric motor (C20)
The electric motor marked the beginning of the end for all forms of mechanism more ingenious than those depending on the simple minded operation of addition and its inverse, subtraction. The greatest gains in efficiency could be obtained by simply increasing the speed with which these operations were repeated and controlled. Speed gains followed from simpler rather than more complex basic mechanisms. The control mechanisms that utilised these simple repeated basic operations, however did become more complex in the interests of using them to produce more complex and accurate outputs.
In this collection there is an [[site.Hertstark1929|arithmometer branded by Samuel Hertzstark]] from 1929, although by then he was not making machines. It is actually a Badenia (manufactured by the German Company Math. Bauerle in the Black Forest) which Hertzstark was by then re-badging (not long before he died). It features an electric motor, still in good working order, and with a keyboard instead of sliders for input. The features of it were not only that it had a keyboard for input, but also a control mechanism consisting of a column of keys which enabled a number to be entered and then added through 0 to 8 repeats representing multiplication by one to nine.
Whilst the above was an obvious innovation, the clumsiest approach in all the calculating devices - from the first arithmometer through to the Millionaire was division, which could only be done by a process along the lines of that done in long division. That is, the number to be divided (the dividend) is considered sequentially from the highest power of ten, and thus decomposed into a series of partial products of the successive parts of the dividend with the divisor. This required a process of guessing each partial product and then trying it out, the result being subtracted from the dividend to give a remainder, exactly as is done in pen and paper long division.
As well as Haman (and the [[site.Mercedes291923|Mercedes-Euklid]]) mentioned earlier, H.W. Egli also began the chain of innovation to mechanise this process, but only by abandonning the sophistication of the Millionaire and like Haman returning to the principles of the arithmometer. The result was the MADAS calculator, first produced in manual form, which could advance the carriage automatically one place at a time as division and multiplication were carried out. With progressive further modifications, including the insertion of an electric motor, the MADAS became the highly effective electric mechanical calculating machine whose use continued right into the 1960s. The late [[site.MADAS1950-60s|MADAS 20BTG calculator]] which is part of this collection represents a classic exemplar of the late stages of this evolutionary development. Its mechanism was also adopted in its essentials, then with many innovations (but with less of the Swiss robust construction of the Madas), in the US Frieden calculators. As it turned out, this final class of machines represented the pinacle of achievement in motorised mechanical calculation.
!! The Vanishing Point (1972)
Further innovation in mechanical calculation however was thwarted by a completely new approach, utlising the rapidly developing technology of electronics, becoming possible. Here the trend set in place by the return to simple basics and increasing speed through electrical processes was paradigmatically transformed, and all the prior approaches simultaneously reduced to a historical curiosity, once the application of integrated circuitry created the modern high speed and ultra-minituarised 'abacus' which combined with the capacity to make programmable conditional loops could rapidly move through the most complicated calculations, as exemplified by the [[site.HP-35-1972|HP-35]] pocket scientific calculator, at the mechanical calculation vanishing point of 1972.