The measurement of distance had been aided by dividers (or “compasses”) at least since the Roman era. Below is a Roman pair of dividers from the C1 to C3 AD.

http://meta-studies.net/pmwiki/uploads/RomanDividers.jpg|A pair of Roman 1st to 3rd century dividers (C1-C3 AD)
(collection Calculant)

From the C18 new instruments were developed building on that idea. A print of a drawing by Thomas Jefferys (~1710–1771) which shows a range of such instruments from the C18 is shown below(:if equal {Site.PrintBook$:PSW} "True":) (this figure)(:ifend:).

Change and the Modern Era

(:table align=center cellpadding=6 border=0 width=90% class=long id=SlideRules"Slide Rules":)

(:table align=center cellpadding=6 border=0 width=90% tableclass=long id=SlideRules"Slide Rules":)

(:table align=center width=80% cellpadding=6 border=0 tableclass=long id=SlideRules"Slide Rules":)

(:cell:)

(:cellnr:)

(:cell:)(All from collection Calculant)

(:cellnr:)(All from collection Calculant)

(All the above are from collection Calculant)

(:table align=center cellpadding=6 border=0 tableclass=long id=SlideRules"Slide Rules":) (:cellnr:) Note (:cell:)Date (:cell:) Maker (:cellnr:) (i) (:cell:)1626–1726 (:cell:)Jacob Leupold 3 designs (:cell:)http://meta-studies.net/pmwiki/uploads/SlideRuleDes1670W300.jpg (:cellnr:)(ii) (:cell:)1759–69 (:cell:)Edward Roberts Everard pattern gauger’s rule (:cell:) http://meta-studies.net/pmwiki/uploads/EverardW300.jpg (:cellnr:)(iii) (:cell:)1821–84 (:cell:)J. Long Alcohol gauger’s rule (:cell:) http://meta-studies.net/pmwiki/uploads/JosephLong1821W300.jpg (:cellnr:)(iv) (:cell:)1893–98 (:cell:)Tavernier Gravet Slide rule (:cell:) http://meta-studies.net/pmwiki/uploads/Tavernier1W300.jpg (:cellnr:)(v) (:cell:)~1928 (:cell:)K&E Slide rule (:cell:) http://meta-studies.net/pmwiki/uploads/KE1908W300.jpg (:cellnr:)(xii) (:cell:)1967–73 (:cell:)Faber Castell 2/83N Novo Duplex (:cell:) http://meta-studies.net/pmwiki/uploads/FaberCastellDuplexW300.jpg (:cell:) (:cell:) (:cell:) (:cellnr:)(All the above are from collection Calculant) (:tableend:)

(iv) Of those now acknowledged as the “inventor” each had not only great intellectual ability but also a wide ranging intellectual curiosity. Combined with this was personal motivation to seek to mechanise calculation. For Schickard it was an increasing interest in discovery and application of new knowledge, found in a dispersed, small, but communicating network of people interested in all manners of philosophy and theology. It included natural philosophers such as Kepler, who had an increasing need to utilise and overcome the drudgery of large numbers of calculation. Napier through his rods and logarithms, had provided means to greatly assist multiplication. But reducing the drudgery of associated additions and subtractions was emerging as something that would be valued. Pascal’s initial motivation was to assist his father in his extensive revenue collecting duties. But Pascal was also on a rapid rise as a natural philosopher and thinker in his own right, where the devising of a ground breaking mathematical instrument also stood to be valued by the network of other thinkers in which he and Schickard were participating.

(v) The network in which Schickard and Pascal engaged was could not be composed, in any case, of any people. They had to be well educated and with time to follow these pursuits. And that required that, almost without exception, they would be well connected to, or members of, the highest ranks in society, that is the nobility. From this point of view, the products of their work were likely to be intended to find favour with others of that rank.

This important relationship between inventor and artisan was subtle. As the above suggests, the guild that knew all about gear trains that could count seconds, hours, and even days and months, was the guild of clockmakers. Inevitably they would be called upon in the construction of different counting machines, and inevitably they would have insights to contribute. History would know primarily of the class of “inventors” who had the power and status to commission, purchase the skill of the artisans, and announce the outcome of their work. But this is a symptom more of what is visible in history than a faithful reflection of the process of invention at the time. Inevitably these came about as a partnership between artisan and intellectual through a process mostly obscured by the barriers of class at the time, and the obscuring passage of passing centuries.

(iii) Artisans, skilled particularly in the art of constructing clock mechanisms, existed with tools and workshops that could be turned to the task of constructing, similar, if differently configured and designed gear trains, dials and associated components. As already remarked, the importance of clocks is reflected in Pascal’s choice of lantern gears for his Pascaline. Even his famous sautoir, whilst highly innovative and different in form, is reminiscent of both the Verge escapement mechanism introduced into clocks from the late thirteenth century and even more in the remontoire gravity powered mechanism invented by Jost Burgi (1552–1631) in about 1595. In both of these, as with the Pascaline, a toothed mechanism was mechanically ‘wound up’ in a cycle and releasing at the correct moment in the cycle to control, and in the case of the Remontoire power by a falling weight, the motion of connected parts.¹

(iv) The relationship between inventor and artisan was subtle. As the above suggests, the guild that knew all about gear trains that could count seconds, hours, and even days and months, was the guild of clockmakers. Inevitably they would be called upon in the construction of different counting machines, and inevitably they would have insights to contribute. History would know primarily of the class of “inventors” who had the power and status to commission, purchase the skill of the artisans, and announce the outcome of their work. But this is a symptom more of what is visible in history than a faithful reflection of the process of invention at the time. Inevitably these came about as a partnership between artisan and intellectual through a process mostly obscured by the barriers of class at the time, and the obscuring passage of passing centuries.

(v) Of those now acknowledged as the “inventor” each had not only great intellectual ability but also a wide ranging intellectual curiosity. Combined with this was personal motivation to seek to mechanise calculation. For Schickard it was an increasing interest in discovery and application of new knowledge, found in a dispersed, small, but communicating network of people interested in all manners of philosophy and theology. It included natural philosophers such as Kepler, who had an increasing need to utilise and overcome the drudgery of large numbers of calculation. Napier through his rods and logarithms, had provided means to greatly assist multiplication. But reducing the drudgery of associated additions and subtractions was emerging as something that would be valued. Pascal’s initial motivation was to assist his father in his extensive revenue collecting duties. But Pascal was also on a rapid rise as a natural philosopher and thinker in his own right, where the devising of a ground breaking mathematical instrument also stood to be valued by the network of other thinkers in which he and Schickard were participating.

(vi) The network in which Schickard and Pascal engaged was could not be composed, in any case, of any people. They had to be well educated and with time to follow these pursuits. And that required that, almost without exception, they would be well connected to, or members of, the highest ranks in society, that is the nobility. From this point of view, the products of their work were likely to be intended to find favour with others of that rank.

(iii) Artisans, skilled particularly in the art of constructing clock mechanisms, existed with tools and workshops that could be turned to the task of constructing, similar, if differently configured and designed gear trains, dials and associated components. As already remarked, the importance of clocks is reflected in Pascal’s choice of lantern gears for his Pascaline. Even his famous sautoir, whilst highly innovative and different in form, is reminiscent of both the Verge escapement mechanism introduced into clocks from the late thirteenth century and even more in the Remontoire gravity powered mechanism invented by Jost Burgi (1552–1631) in about 1595. In both of these, as with the Pascaline, a toothed mechanism was mechanically ‘wound up’ in a cycle and releasing at the correct moment in the cycle to control, and in the case of the Remontoire power by a falling weight, the motion of connected parts.

The above provides some basis for understanding what followed: a series of developments and experiments in mechanical calculation, few of them seen abstractly providing much real advantage over traditional pen and jeton for doing arithmetic, but each embodied in beautifully worked prototypes, often frequently being found on the shelves or in the cabinets of curiosities of the nobility and others of standing, whether in Germany, France or England. Since details of these are available elsewhere³ we will rely on objects documented in this collection to simply act as signposts. In particular, two inventors following Pascal, Leibniz and Morland, will be briefly considered, each of which illustrates substantially the above contention.

As with Morland’s multiplication machine it was now possible to introduce a carriage which would be mechanically advanced (by means of a second crank handle at the end of a screw thread) to allow multiplication by more than single digit numbers to take place. The Pascaline had had no such mechanism and so the equivalent had had to be done by recording intermediate results with pen and paper. The deficiency in Leibniz’s machine, however, as already indicated, was that unlike the Pascaline, Leibniz had been unable to devise a robust carry mechanism able to handle either multiple carries across several output dials at once. As a result, the machine was far from perfected. It seems that only one was made in 1674 (by M. Olivier, a French clockmaker, working under instruction from Leibniz), that a few years later it was sent Herr Kastner in Göttingen (Lower Saxony) for repairs, and was later stored in an attic at Göttingen University where it was not recovered for 200 years. A diagram of the Leibniz calculator from 1901 is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure (:ifend:)below, and the actual surviving machine is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure also (:ifend:)below.

As to the innovation in mechanism there is no evidence that Pascal had any knowledge of Schickard’s invention. Nevertheless it was clear that he too was infected with the growing understanding that his invention could now be turned to reducing labour in new ways. His Pascaline turned out to be an adding machine, rather like that in the base of Schickard’s clock, but with two important improvements (and one significant deficiency). As with the Schickard’s machine, numbers were added by turning disks with a stylus which in turn turned interlinked gears. Pascal however experimented with ways to improve the practical functioning of this. Various details of the Pascaline mechanism, including diagrams in this collection from the famous Encyclopédie ou Dictionnaire raisonné of Diderot and d’Alembert in 1759 are shown in this table below.

First, drawing firmly on the history of clock design, Pascal introduced gears which were minaturised versions of the “lantern gears” used in tower clocks and mills. These could withstand very great stresses and still operate smoothly. Second, he sought to achieve a carry mechanism which could operate even if many numbers needed to be carried at once (e.g. for 9999999+1 to give 10000000). He finally achieved this with a system where as each gear passed from 0 to 9 a fork-shaped weighted arm (the “sautoir”) on a pivot was slowly rotated upward. As the gear passed from 9 to 0 this weighted arm was then released to drop thrusting an attached lever forward to rotate the gear ahead of it by one unit.

eight surviving Pascalines(:ifend:) can be found in museums and private collections. He also documented his machine in a short pamphlet,⁶ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.⁷

Having said that, the machines were expensive (about of a third of a year’s average wages of the time).¹⁰ Pascal in his surviving Advice on the use of the Pascaline promised that the machine could “perform without any effort whatsoever all the arithmetical operations that had so often worn out one's mind by means of the plume and the jetons”.¹¹ But in truth the use of it required a good knowledge of arithmetic to be used effectively (especially for division and multiplication, but even for subtraction). At best it was a mechanical substitute for what otherwise would have to be written down, save that it could perform addition with a little practice, and simple subtraction (where the result was not negative) with a little more. As Balthazaar Gerbier described the Pascaline in a letter to Samuel Hartlib in 1648:

Leibniz’s machine formed the final piece of a tryptage (with Pascal and Schickard) of foundation pioneering seventeenth century mechanical calculators. Together these constituted foundation stones in the subsequent development of a wide range of other calculational technologies over the next two centuries. Progressively these overcame many of the more apparent limitations in these foundation devices, yet they shared also one other characteristic. Despite hopes that may have been held by their inventors, none of them proved to be more than prototypes in the sense that their destiny would be as curiosities of great interest, perhaps prized by the few who might get hold of one, but for a variety of reasons (cost, capabilities, ease of use) of little broader practical importance. The reasons we have canvassed for constructing them, and possessing them, would remain pertinent, but the other possibility, that they might be cheap enough and be sufficiently intuitive in use to create a notable increase in mathematical facility in a widening group of users, would remain a dream, not to be realised, until Thomas de Colmar, two centuries after Leibniz, also utilised the step drum to open the door to the commercial production and widening public use of mechanical calculators.

(see calculating with the Pascaline)(:ifend:). What I (:if equal {Site.PrintBook$:PSW} "False":)argue elsewhere?(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)argue elsewhere(:ifend:) is a rather empty debate has at times erupted over whether Pascal or Schickard should be considered the more fitting candidate for ‘inventor of the modern calculator’.¹³ Nevertheless, it is Pascal’s machine which has been referred to with reverance down the following centuries, not the least because a significant number were produced, it was mechanically ingenious, and some survive. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure (:ifend:)below. (:if equal {Site.PrintBook$:PSW} "False":)

(see calculating with the Pascaline)(:ifend:). What I argue elsewhere is a rather empty debate has at times erupted over whether Pascal or Schickard should be considered the more fitting candidate for ‘inventor of the modern calculator’.¹⁴ Nevertheless, it is Pascal’s machine which has been referred to with reverance down the following centuries, not the least because a significant number were produced, it was mechanically ingenious, and some survive. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure (:ifend:)below. (:if equal {Site.PrintBook$:PSW} "False":)

(:title Early Evolution of the Modern Calculator:)

(:title Foundational Contributions to the Modern Calculator:)

(:title Foundations for the emergence of the Modern Calculator:)

Schickard’s invention is revealed in (:if equal {Site.PrintBook$:PSW} "False":)two letters to Kepler and note to artisans(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)two letters to Kepler(:ifend:), the first being in 1623, where he describes an “arithmeticum organum” (“arithmetical instrument”) that he has invented, but later as a Rechen Uhr (calculating clock) in a surviving note to the artisan constructing his machine.¹⁶ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although given he describes it operating, a prototype of the lower (adding machine) part was almost certainly built, and very probably a prototype of the whole was commissioned since he also ordered one for Kepler. Our knowledge is based on (:if equal {Site.PrintBook$:PSW} "False":)

notes to artisans on building the machine(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s notes to artisans on building the machine(:ifend:), and drawings (see below) as well as comments to Kepler (discussed later). However, in his second letter Schickard made clear that he was not giving a full description.¹⁸ Surviving sketches are shown in this table.

Schickard’s invention is revealed in (:if equal {Site.PrintBook$:PSW} "False":)two letters to Kepler(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)two letters to Kepler(:ifend:), the first being in 1623, where he describes an “arithmeticum organum” (“arithmetical instrument”) that he has invented, but which would later be described as a Rechenuhr (calculating clock) or Rechenmaschine (calculating machine).²⁰ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although given he describes it operating, a prototype of the lower (adding machine) part was almost certainly built, and probably a prototype of the whole was commissioned. Our knowledge is based on (:if equal {Site.PrintBook$:PSW} "False":) notes to artisans on building the machine(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s notes to artisans on building the machine(:ifend:), and drawings (see below) as well as comments to Kepler (discussed later). However, in his second letter Schickard made clear that he was not giving a full description.²¹ Surviving sketches are shown in this table.

Schickard’s invention is revealed in (:if equal {Site.PrintBook$:PSW} "False":)two letters to Kepler(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)two letters to Kepler(:ifend:), the first being in 1623, where he describes an “arithmeticum organum” (“arithmetical instrument”) that he has invented, but which would later be described as a Rechenuhr (calculating clock) or Rechenmaschine (calculating machine).²³ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although a prototype was probably commissioned based on (:if equal {Site.PrintBook$:PSW} "False":) notes to artisans on building the machine(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s notes to artisans on building the machine(:ifend:), and drawings (see below) as well as comments to Kepler (discussed later). Surviving sketches are shown in this table.

Blaise Pascal’s Pascaline.

Blaise Pascal (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.²⁶ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself with the skills to produce one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.²⁷ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure (:ifend:)below.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. A drawing of a set from 1797 is shown (:if equal {Site.PrintBook$:PSW} "True":)in this figure(:ifend:) below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae - a word he created from the Greek for rod (rabdos) and calculation (logos) - which he published in 1617.²⁹

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design before settling on a final design. Some (:if equal {Site.PrintBook$:PSW} "False":)

Schickard’s invention is revealed in two letters to Kepler, the first being in 1623, where he describes an “arithmeticum organum” (“arithmetical instrument”) that he has invented, but which would later be described as a Rechenuhr (calculating clock) or Rechenmaschine (calculating machine).³¹ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although a prototype was probably commissioned based on (:if equal {Site.PrintBook$:PSW} "False":)

Schickard’s invention is revealed in two letters to Kepler, the first being in 1623, where he describes an “arithmeticum organum” that he has invented, but which would later be described as a Rechenuhr (calculating clock) or Rechenmaschine (calculating machine).³³ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although a prototype was probably commissioned based on (:if equal {Site.PrintBook$:PSW} "False":)

The first replica was of Schickard’s device was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) this figure(:ifend:)(:if equal {Site.PrintBook$:PSW} "False":)replica(:ifend:) in this collection, shown below is more recent.³⁵

The first replica was of Schickard’s device was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) this figure(:ifend:)(:if equal {Site.PrintBook$:PSW} "False":)replica(:ifend:) in this collection, shown below is more recent.³⁷

Schickard’s invention is revealed in two letters to Kepler, the first being in 1623, where he describes an “arithmeticum organum” that he has invented, but which would later be described as a Rechenuhr (calculating clock) or Rechenmsaschine (calculating machine).³⁹ It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although a prototype was probably commissioned based on (:if equal {Site.PrintBook$:PSW} "False":)

Schickard’s invention is revealed in two letters to Kepler, the first being in 1623, where he describes an “arithmeticum organum” that he has invented (which would later be described as a Rechenuhr (calculating clock) or Rechenmsaschine (calculating machine).⁴⁰

 It is not known whether a complete
  (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:)
  was ever actually built, although a prototype was probably
  commissioned based on  (:if equal {Site.PrintBook$:PSW} "False":)

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810)

It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:) was ever actually built, although a prototype was probably commissioned based on (:if equal {Site.PrintBook$:PSW} "False":)

(:cell:)

(:cellnr:)

(:cell:)

(:cell:)

(:title Foundations of the Modern Calculator:)

(:title The Emergence of the Modern Calculator:)

(:table class=pictures width=100% align=left class=border cwidth=7:)

(:cell:)

(:cell:)

(:cell:)

(:cellnr:)

(:cell:)

(:cell:)

(:cell:)

(:cellnr:)

(:if equal {Site.PrintBook$:PSW} "False":) (:table class=pictures width=100% align=left class=border cwidth=10:)

(:cellnr:)

(:cell:)

(:cell:)

(:cell:)

(:cell:)

(:tableend:) (:ifend:) (:ifend:)

(:title Part 2. The Modern Era - The Emergence of the Modern Calculator:)

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810)// (collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenriMorinW400.jpg|French drawing instruments ~1880 (including proportional dividers- top right)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/NapiersBonesContempW600.jpg|Set of Napier’s Rods (of recent construction)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Bloch1W350.jpg|Bloch Schnellkalulator
~1924
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Zeitermittler2W350.jpg|Der Zeitermittler
~1947
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Schickard1W300_1.jpg|Working replica of Schickard’s Calculating Clock (1623)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardFrontPanelW600.jpg|Schickard front panel
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|Schickard rear view showing Napier cylinders
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline,⁴² style ~1650
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/PascalineH150.jpg|Wood engraved plate from 1901 depicting a Pascaline Calculator for accounting (~1642)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalcH400.jpg|Wood engraved plate from 1901 depicting the Leibniz Calculator (1673)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520 (collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenriMorinW400.jpg|French drawing instruments ~1880 (including proportional dividers- top right) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/NapiersRodsEnBr2W600.jpg|Depiction of Napier’s Rods, 1797.⁴⁴ (collection Calculant)

http://meta-studies.net/pmwiki/uploads/NapiersBonesContempW600.jpg|Set of Napier’s Rods (of recent construction) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion (collection Calculant)

http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Schickard1W300_1.jpg|Working replica of Schickard’s Calculating Clock (1623) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardFrontPanelW600.jpg|Schickard front panel (collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|Schickard rear view showing Napier cylinders (collection Calculant)

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline,⁴⁶ style ~1650 (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/PascalineH150.jpg|Wood engraved plate from 1901 depicting a Pascaline Calculator for accounting (~1642) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalcH400.jpg|Wood engraved plate from 1901 depicting the Leibniz Calculator (1673) (collection Calculant)

(:if equal {Site.PrintBook$:PSW} "False":)

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":)

(:ifend:)

http://meta-studies.net/pmwiki/uploads/JefferysPrintH500.jpg|1710–71: Compasses by T. Jefferys sculp (Collection Calculant)

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810) (collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenriMorinW400.jpg|French drawing instruments ~1880 (including proportional dividers- top right)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/NapiersRodsEnBr2W600.jpg|Depiction of Napier’s Rods, 1797.⁴⁸
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/NapiersBonesContempW600.jpg|Set of Napier’s Rods (of recent construction)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/nomogram_b=a+dH350.jpg|A simple nomograph for calculating the sum of two numbers (b=a+c)

http://meta-studies.net/pmwiki/uploads/Bloch1W350.jpg|Bloch Schnellkalulator
~1924 (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Zeitermittler2W350.jpg|Der Zeitermittler
~1947 (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Schickard1W300_1.jpg|Working replica of Schickard’s Calculating Clock (1623)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardFrontPanelW600.jpg|Schickard front panel
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|Schickard rear view showing Napier cylinders
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline,⁵⁰ style ~1650
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/PascalineH150.jpg|Wood engraved plate from 1901 depicting a Pascaline Calculator for accounting (~1642)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Morland1jW380.jpg|Morland Adding Machine
adapted to the then Italian currency
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

http://meta-studies.net/pmwiki/uploads/Morland2jW420.jpg|Morland Multiplying Instrument
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/StepDrum.jpeg|Twentieth Century step drum following the same principle as Leibniz’s conception
⁵²

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalcH400.jpg|Wood engraved plate from 1901 depicting the Leibniz Calculator (1673)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/LeibnitzRechenmaschW.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University. Reproduced with permission from the Gottfried Wilhelm Leibniz Bibliothek.

http://meta-studies.net/pmwiki/uploads/LeibnitzRechenmaschW.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University. Reproduced with permission from the Gottfried Wilhelm Leibniz Bibliothek.

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520

http://meta-studies.net/pmwiki/uploads/Misc/GregorReisch1504H400.jpg|Contested methods - Woodcut from 1503

A deficiency was that the carry mechanism did not allow the process of addition to be reversed. Instead when subtracting a (:if equal {Site.PrintBook$:PSW} "False":) rather clumsy system(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":) rather clumsy system(:ifend:) had to be used of adding ‘complementary numbers’ (where a number -x is represented by (10-x), with 10 subsequently being subtracted). This was assisted by a window shade which can be switched between showing numbers or their complements.

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design. Some (:if equal {Site.PrintBook$:PSW} "False":) eight surviving Pascalines(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":) eight surviving Pascalines(:ifend:) can be found in museums and private collections. He also documented his machine in a short pamphlet,⁵⁵ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.⁵⁶

Whilst Schickard’s machine could multiply directly (using its ingenious incorporation of Napier’s rods), Pascal’s machine (despite his claim that it could be used for all operations of arithmetic) was primarily useful for addition, and with some mental effort, subtraction (:if equal {Site.PrintBook$:PSW} "False":) (see calculating with the Pascaline)(:ifend:). Nevertheless, it is Pascal’s machine which has been referred to with reverance down the following centuries, not the least because a significant number were produced, it was mechanically ingenious, and some survive. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline woodcut (:ifend:)below. (:if equal {Site.PrintBook$:PSW} "False":)

It is worth pausing to consider the reasons that Schickard and Pascal had launched into their geared calculator projects, almost certainly with no knowledge of each other’s efforts, being the first known such projects in more than 1500 years. Notably, whilst Schickard included a rendition of Napier’s rods to aid multiplication, and Pascal worked to create a more effective carry mechanism, both in rather similar ways ultimately sought to mechanise addition and subtraction through a set of interconnected gears and dials. Of course there was no single reason why they embarked on such a similar task so close together in historical time. Rather a skein of factors were coming together to make such innovations appealing and therefore more likely to be addressed by people with the resources and ingenuity to do so.

Having said that, the machines were expensive (about of a third of a year’s average wages of the time).⁵⁹ Pascale in his surviving Advice on the use of the Pascaline promised that the machine could “perform without any effort whatsoever all the arithmetical operations that had so often worn out one's mind by means of the plume and the jetons”.⁶⁰ But in truth the use of it required a good knowledge of arithmetic to be used effectively (especially for division and multiplication, but even for subtraction). At best it was a mechanical substitute for what otherwise would have to be written down, save that it could perform addition with a little practice, and simple subtraction (where the result was not negative) with a little more. As Balthazaar Gerbier described the Pascaline in a letter to Samuel Hartlib in 1648:

Morland had already seen a Pascaline, probably when on a diplomatic mission to Queen Christina of Sweden, a supporter of the sciences who in 1649 had been presented with a Pascaline (similar in looks and identical in function to the replica in this collection).⁶³ He had also taken part in a diplomatic mission during which he stopped off at the court of Louis XIV for over a month, so he may well have learned more of the Pascaline, and made contact with scholars and associated mechanics at that time.⁶⁴ He was thus aware of the potential allure of such inventions. Over the next few years he devised three different calculator designs: an adding machine (see (:if equal {Site.PrintBook$:PSW} "True":)Morland adding machine (:ifend:)below), a multiplying device (see also (:if equal {Site.PrintBook$:PSW} "True":)Morland multiplying instrument (:ifend:)below), and an instrument with moveable arms for determining trigonometric relations. Several examples of Morland’s calculating devices are still in existence, in particular in museums in London and Italy. Not only did Morland design the machines but he wrote a book on the use of the instruments. Published in 1672 this can be considered the first known English language (and arguably the earliest surviving) ‘computer manual’. (A copy of this rare book, shown below(:if equal {Site.PrintBook$:PSW} "True":) in this table(:ifend:), is in this collection).

In essence the adding machine was a simplified adaptation of the Pascaline’s dials (turned by stylus), but without any carry mechanism (so that carrying had to be done by hand). Adding (or subtraction by an opposite rotation) was input through the larger wheels. Each larger wheel engaged with the small wheel above to register a rotation and thus accumulated a 1 to be carried. As with the Pascaline different input wheels were provided for different units (whether units, tens, hundreds, etc, or pounds, shillings and pence).

Gottfried Wilhelm Leibniz (1646–1716) had a life filled with so many ambitions, such high profile conflict, and so much capacity that his biography can seem overwhelming. He has been described as “…an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.”⁶⁸ At the same time his irascible personality was such that by the time he died, his contemporaries were said to feel a sense of relief and only his secretary attended his funeral.⁶⁹ His father was a professor of moral philosophy who died when Leibniz was only six years old, but he grew up in the surroundings of his father’s imposing library from which he derived much satisfaction. He gained his Doctorate of Laws (on De Casibus Perplexis, i.e. On Perplexing Cases) at the age of 20 from the University of Altdorf in Nürnberg.

The scope of Leibniz’s subsequent work was extraordinary encompassing poetry and literature, law, political diplomacy, work to unify all knowledge in part by bringing the scientific societies together, a similar desire to bridge the gap between the Lutheran and Catholic church in particular, and all churches in general, and more enduringly, his powerful initiatives in science and mathematics. The products of his work in mathematics included the development of binary arithmetic, methods of solving systems of linear equations, and either the discovery of calculus (priority in this was contested by Newton) or at least the modern notation used for it. Central to his work was a belief that if a logically defined (‘mechanical’) algebra of thought could be developed then truths could be automatically generated and proved. (This quest can be found stretching back to the Stoics, and forward to the later work of Gottlob Frege, George Boole, Bertrand Russell, Kurt Gödel and many others). Given this enticing objective it is not surprising that, having heard of Pascal’s calculating machine when he visited Paris in 1672 on a diplomatic mission (later to be elected to the French Academy of Sciences), Leibniz decided to turn his hand also to creating a calculating machine.⁷⁰

Almost certainly Leibniz did not have a chance to use a Pascaline or he would have discovered that an early idea that he had, to automate multiplication by placing a mechanism on top of the Pascaline to simultaneously move its input “star wheels”, would conflict with the machine’s internal mechanism. His second attempt was much more original. Although unlike Pascal he was never able to properly automate the carry system, he developed a machine which could more faithfully replicate the pen and paper methods not only of addition, but subtraction, multiplication, and with some ingenuity, division. The first and most enduring innovation was a new way to input numbers by setting an accumulating cog to engage with a “stepped drum”.

Several such accumulating cogs with their drums were put side by side, corresponding to units, tens, hundreds, etc, and once a number was set (e.g. 239) it could be multiplied (e.g. by 4) by turning the crank through a full rotation the corresponding number of times (thus adding 239 to itself 4 times to give 956). The capacity to add a multi digit number to itself repeatedly gave the machine the capacity to multiply by simply turning a handle, which was a considerable advance over the Pascaline. Further, the crank could be rotated in the opposite direction to produce a subtraction.

Multiplication tables and various tabluated functions (notably logarithms and trigonometric tables), and various physical embodiments of those notably in various forms of Napier’s rods, sectors and scales, would increase in use as need and access to education in their use broadened. But could a machine be constructed that would make the mathematical literacy required in the use of these, and traditional methods of arithmetic using pen, paper, and calculi obsolete? Clearly the answer was “not yet”.

Leibnitz’s machine formed the final piece of a tryptage (with Pascal and Schickard) of foundation pioneering seventeenth century mechanical calculators. Together these constituted foundation stones in the subsequent development of a wide range of other calculational technologies over the next two centuries. Progressively these overcame many of the more apparent limitations in these foundation devices, yet they shared also one other characteristic. Despite hopes that may have been held by their inventors, none of them proved to be more than prototypes in the sense that their destiny would be as curiosities of great interest, perhaps prized by the few who might get hold of one, but for a variety of reasons (cost, capabilities, ease of use) of little broader practical importance. The reasons we have canvassed for constructing them, and possessing them, would remain pertinent, but the other possibility, that they might be cheap enough and be sufficiently intuitive in use to create a notable increase in mathematical facility in a widening group of users, would remain a dream, not to be realised, until Thomas de Colmar, two centuries after Leibniz, also utilised the step drum to open the door to the commercial production and widening public use of mechanical calculators.

In 1683 (:if equal {Site.PrintBook$:PSW} "False":) Thomas Everard(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Thomas Everard(:ifend:) , an English Excise Officer (who is credited with introducing the term “sliding rule”), began promoting a new 1 inch square cross section slide rule with several slides for calculating excise.⁷⁴ Shown in (ii), below, is an English four sided Everard pattern sliding rule from 1759. It includes various gauging points and conversions to square and cube roots for calculating volumes.⁷⁵ In (iii) is a more modern looking slide rule shape, from 1821–84 by Joseph Long of London, also for use in gauging the amount of alcohol spirit in a container, and calculating the corresponding tax.

The slide rule evolved in what are early recognisable directions. The use of a hair line to read off multiple scales was suggested as early as 1675 by Sir Isaac Newton. Nevertheless, the introduction of a moveable cursor with this innovation included had to wait a century until a professor of mathematics, John Robertson, in 1775, added a mechanical cursor.⁷⁶

This simple nomograph is used to add two numbers (one in column a) and the other (in column c).⁷⁸ The properties of similar triangles give the result that a line drawn between the two numbers will cut the middle column (b) at the required sum. Since the outer scales may be logarithmic or trigonometric many more complicated expressions are able to be evaluated using this sort of approach.

This undermines the simple minded view of innovation as a linear process of invention and improvement. Rather, as in any transitional period, multiple strands of change were in motion. They were deflected or shaped along the way by different motivations and pressures. One of these pressures was simply intellectual conservatism. That included the usual suspicion of practical compromises by those privileged to be able to focus purely on intellectual pursuits. When Gunter applied for a chair in mathematics at Oxford he was rejected by the Warden of Merton College, Sir Henry Saville. The reason given was that his instruments were “mere tricks”.⁸¹ Rejected from Oxford Gunter took up the Gresham Chair at Cambridge University. As noted earlier, this position was dedicated to exposing mariners and others to the potential usefulness of mathematics. Somewhat later an Experimental Philosophy group would grow up at Oxford (and in 1672 would gain a Royal Charter from King Charles II and become the Royal Society).⁸²

There is a range of other factors that would have affected the spread and adoption of a new instrument such as the slide rule. For example, in order to manufacture slide rules for mass distribution, the skills and techniques for scribing logarithmic scales would need to be picked up by potential instrument makers. Until many were produced and means were found to cheapen the process, costs of new instruments might well have been at a premium. Perhaps more important would have been the need to learn how to use them. The naval profession was not considered a place for scholars. Rather training was on the job and at the hands of senior sailors and officers. In short, whether in marine environments, or on land amongst architects, builders and planners, skills deemed necessary for doing the job were taught from master to apprentice in the age old fashion of the guilds. This was a very suitable way of passing on stable and established best practice. It was not necessarily so receptive to or good at transmitting new fangled ideas of scholarly gentlemen living and working in the privileged seclusion of universities. The Gresham Chair was intended to break through that, but it was too large a job to be achieved by any one such establishment.

(ii) the practitioners of practical arts - whether sailor, cartographer, or clerk who might appreciate a tool that would ease their work. Complementing this there was a slowly growing demand for larger numbers of “calculators” - that is, people who could calculate. Given that this was not a widespread skill, as already noted, anything that might ease the learning and teaching of the skills, or replace the need for it with some device, could over time prove attractive.

As Laird and Roux note, “The establishment of mechanical philosophy in the 17th century was a slow and complex process, profoundly upsetting the traditional boundaries of knowledge. It encompassed changing view on the scope and nature of natural philosophy, an appreciation of “vulgar” mechanical knowledge and skills, and the gradual replacement of a casual physics with mathematical explanations.”⁸⁴ It was this slow and complex process that would eventually give rise to mechanised calculation. But as suggested above, the developments would be complex, many stranded, and over time increasingly tangle together the two worlds of the philosophical and practical arts.

The Roman emphasis on the importance of transport for the state had led also to the use of measuring machines including the odometer of Heron of Alexandria - a device used to measure distance traversed by a rotating carriage wheel by coupling it to a system of incrementing gears. As described by Vitruvius (~15 BC),⁸⁷ the rotation of the carriage wheel was measured by this device in units, tens, hundreds, thousands and tens of thousands of paces.⁸⁸ This sort of counting mechanism could form a natural basis for the important “carry” mechanism (e.g. where 9+1 is converted to 10) of an adding machine. Still, for that potential to be capitalised upon there had to be a recollection or re-discovery of the sort of mechanism, and perhaps more importantly, an interest in constructing one. As already mentioned, the budding interest in mechanisation in the early C17 (building in particular on the widely understood appeal of the success of clocks, clockwork, and new mechanical insight about nature) provided an appropriately fertile base for that interest.

Given the increasingly multi-stranded stress on the importance of calculation, it is not surprising that at least some natural philosophers in the England and Europe took up the challenge of how best to facilitate it. Even though often distant from mundane economic or practical need there were intellectuals of the day who shared not only an enthusiasm for discovery, but also a growing enthusiasm for invention. It was only a matter of time before a growing interest in mechanisation would intersect with desire to simplify the process of calculating solutions to a variety of mathematical problems.

It is not known whether a complete (:if equal {Site.PrintBook$:PSW} "False":)Schickard’s “calculating clock”(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s “calculating clock”(:ifend:)

was ever actually built, although a prototype was probably
 commissioned based on  (:if equal {Site.PrintBook$:PSW} "False":)

notes to artisans on building the machine(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Schickard’s notes to artisans on building the machine(:ifend:), and drawings (see below) as well as comments to Kepler (discussed later). Surviving sketches are shown in this table

The first replica was of Schickard’s device was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) Schickard replica(:ifend:) in this collection, shown below is more recent.⁹⁰

Kepler was greatly excited by the publication of logarithms which he utilised to greatly reduce the enormous calculational effort in his computations of The Rudolphine Tables and later works. He dedicated his Ephemerides to Napier in 1617.⁹⁴ Professor Briggs (astronomer and geometrician) now at Gresham College, Cambridge, as mentioned earlier, was a close colleague of Napier and had written of his tables “…I never saw a book which pleased me better or made me more wonder”.⁹⁵ He would later take Napier’s work further forward producing new tables of logarithms to base 10. In 1609 Briggs was also impatiently awaiting Kepler’s exposition on ellipses.⁹⁶

The most often noted deficiency of Schickard’s machine was in its “carry mechanism” (the mechanism that moves the displayed result from 09 to 10 when 1 is added). In Schickard’s design it seems that the “carry” was achieved every time an accumulator wheel rotated through a complete turn, by a single tooth catching on an intermediate wheel causing the next highest digit in the accumulator to be increased by one.

Carrying a number required extra rotational force to be applied (since more than one wheel had to be moved simultaneously). If, for example, 1 was to be added to a number like 99999 to give 100000, all of the wheels bearing a 9 would have to be moved by the force applied to the right hand wheel as it moved from 9 to 0. Unfortunately that would require so much force as to break the machine. In short, the adding part of the machine would jam if too many numbers had to be carried simultaneously.

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁹⁹ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself with the skills to produce one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.¹⁰⁰ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline replica (:ifend:)below.

As to the innovation in mechanism there is no evidence that Pascale had any knowledge of Schickard’s invention. Nevertheless it was clear that he too was infected with the growing understanding that his invention could now be turned to reducing labour in new ways. His Pascaline turned out to be an adding machine, rather like that in the base of Schickard’s clock, but with two important improvements (and one significant deficiency). As with the Schickard’s machine, numbers were added by turning disks with a stylus which in turn turned interlinked gears. Pascale however experimented with ways to improve the practical functioning of this. Various details of the Pascaline mechanism, including diagrams in this collection from the famous Encyclopédie ou Dictionnaire raisonné of Diderot and d’Alembert in 1759 are shown in this table below.

A series of designs followed. Three of these appear in (:if equal {Site.PrintBook$:PSW} "False":)Jacob Leupold’s(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Jacob Leupold’s(:ifend:) book Theatrum Arithmetico-Geometricum¹⁰³ of which Table XII (page 241), held in this collection, is shown (:if equal {Site.PrintBook$:PSW} "False":) in (i) in the Table below(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":) in (i) in Early Slide Rules(:ifend:). (The top drawing is a design for a Gunter scale, and the two below are early designs for slide rules).¹⁰⁴

Gunter rules were usually equipped with both Gunter’s combination of a logarithmic and a linear scale. Often they also included a range of other scales (notably trigonometric scales for navigational tasks, such as required to work from elapsed time, speed and changes to compass bearing to distances travelled). In addition, important constants could be marked on them as “gauge marks”. Over the seventeenth century Gunter rules gained increasing acceptance and were used right through into the late nineteenth century. Some of the scales of a two foot long mid-nineteenth century navigational Gunter rule in this collection are shown (:if equal {Site.PrintBook$:PSW} "True":)in The Gunter Rule(:ifend:) below.

A series of designs followed. Three of these appear in (:if equal {Site.PrintBook$:PSW} "False":)Jacob Leupold’s(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)Jacob Leupold’s(:ifend:) book Theatrum Arithmetico-Geometricum¹⁰⁷ of which Table XII (page 241), held in this collection, is shown (:if equal {Site.PrintBook$:PSW} "False":) in (i) in the Table below(:ifend:)(:(:if equal {Site.PrintBook$:PSW} "True":) in (i) in Early Slide Rules. (The top drawing is a design for a Gunter scale, and the two below are early designs for slide rules).¹⁰⁸

Tables of logarithms, improved in various ways over time, were published either by themselves (:if equal {Site.PrintBook$:PSW} "False":)(e.g. Tables by Gardiner, 1783)(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)the Tables by Gardiner from 1783(:ifend:), or in the context of “ready reckoners” (e.g. (:if equal {Site.PrintBook$:PSW} "False":)Ropp’s ready reckoner, 1892(:ifend:) (:if equal {Site.PrintBook$:PSW} "True":)Ropp’s Ready Reckoner from 1892(:ifend:), in this collection). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

Tables of logarithms, improved in various ways over time, were published either by themselves (:if equal {Site.PrintBook$:PSW} "False":)(e.g. Tables by Gardiner, 1783)(:ifend:) (:if equal {Site.PrintBook$:PSW} "True":)the Tables by Gardiner from 1783(:ifend:), or in the context of “ready reckoners” (e.g. (:if equal {Site.PrintBook$:PSW} "False":)Ropp’s ready reckoner, 1892(:ifend:) (:if equal {Site.PrintBook$:PSW} "True":)Ropp’s Ready Reckoner from 1892(:ifend:), in this collection). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

Tables of logarithms, improved in various ways over time, were published either by themselves (:if equal {Site.PrintBook$:PSW} "False”:)(e.g. Tables by Gardiner, 1783)(:ifend:) (:if equal {Site.PrintBook$:PSW} "True”:)the Tables by Gardiner from 1783(:ifend:), or in the context of “ready reckoners” (e.g. (:if equal {Site.PrintBook$:PSW} "False”:)Ropp’s ready reckoner, 1892(:ifend:) (:if equal {Site.PrintBook$:PSW} "True”:)Ropp’s Ready Reckoner from 1892(:ifend:), in this collection). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

Tables of logarithms, improved in various ways over time, were published either by themselves (:if equal {Site.PrintBook$:PSW} "False”:)(e.g. Tables by Gardiner, 1783)(:ifend:)(:if equal {Site.PrintBook$:PSW} "True”:)the Tables by Gardiner from 1783(:ifend:), or in the context of “ready reckoners” (e.g. (:if equal {Site.PrintBook$:PSW} "False”:)Ropp’s ready reckoner, 1892(:ifend:)(:if equal {Site.PrintBook$:PSW} "True”:)Ropp’s Ready Reckoner from 1892(:ifend:), in this collection). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

The first developments in the technology of calculation in Early Modern Europe were perhaps as much cultural as economic in their motivation. In particular they reflected a growing fascination with the use of machinery. From the thirteenth century, windmills and the more reliable waterwheels (both of which had been known of since ancient Rome) began to be turned to an ever wider variety of productive purposes - from the traditional role of crushing grain, to cutting stone, blowing bellows in metal work, sharpening knives and weapons, sawing planks, printing ribbons, dressing leather, rolling copper plate, and much else.¹¹⁰ The productive value of machinery was becoming more widely recognised, as perhaps also was the enjoyment of the way it could extend human capacity and power. Indeed there is no reason to think that it would have been any less rewarding an experience to show off one’s new adaption of a water mill than it is to show off a new car or electronic device now. A quote from Walker (if set in pre-feminist language) sums up the way in which innovation is a product of a whole cultural as well as economic and technological system:

In this drawing, at the top can be seen the emerging shape of various compasses, constructed from brass and steel. Below,(:if equal {Site.PrintBook$:PSW} "True":) shown in Pair of dividers(:ifend:) from this collection, is a typical pair of such dividers, also from the eighteenth century.

At the bottom left of this Jeffries print can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.¹¹² Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below(:if equal {Site.PrintBook$:PSW} "True":) in French drawing instruments(:ifend:), is from ~1880.

Others (including Galileo Galilei) have claims to the invention of the sector.¹¹⁶ Hood first described his sector in 1598. Also known as a proportional compass the sector consisted of two rulers of equal length joined by a hinge and inscribed with various scales, to facilitate, in particular multiplication and division (but which could also be used to assist in problems of proportion, trigonometry, and calculations of square roots). It utilises the geometric principle, articulated by Euclid, that the like sides of similar triangles are in the same proportion. By forming an equilateral triangle with side and base in a particular ratio, multiplications in the same ratio could be read off for any other length of side allowed by the instrument.

Hood’s sector was constructed by Charles Whitwell, a fine instrument maker.¹¹⁷ However, the process of adoption was far from immediate. It was helped when, in 1607, Edmund Gunter, a young mathematician not long from his studies at Oxford, began to circulate his hand-written notes on The Description and Use of the Sector showing in particular how well it could be applied to the problems of navigation by Mercartor’s charts. In this he found support from Henry Briggs. In 1597, Briggs had been appointed as the first Gresham Professor of Geometry at Cambridge University with an aim of bringing geometry to within reach of the people of London.¹¹⁸ The very creation of this Chair indicated that there was a growing understanding of the potential utility of mathematical thinking. (However, the Chair was kept within the bounds of propriety by focussing on geometry which had less association with the “dark arts”). Briggs discharged his duty in popular education, amongst other things, by giving his lectures not only in the obligatory learned language of Latin, but repeating them in English in the afternoons. With Brigg’s support Gunter’s Use of The Sector was issued in print in 1623.

Sectors were in common use right through to the early twentieth century. Two sectors (from collection Calculant) are shown in Two Sectors below. The first is a (:if equal {Site.PrintBook$:PSW} "False":)Brass French Gunnery Sector from about 1700(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)brass gunnery sector(:ifend:) by Michael Butterfield, Paris. Michael Butterfield, and English clock maker was born in 1635 and worked in Paris from ~1680 to 1724. The second is an (:if equal {Site.PrintBook$:PSW} "False":)Oxbone Architect’s Sector by T. and H. Doublett(:ifend:)(:if equal {Site.PrintBook$:PSW} "True":)architect’s sector by T. and H. Doublett(:ifend:) who practiced their craft in London around 1830.

First, for reasons already mentioned, the learned designers of these early calculational aids seldom had the skills necessary to make them. Artisans, such as Michael Butterfield, who did have the highest relevant skills were frequently found amongst the members of the watch and clock makers guilds. Later specialist mathematical and scientific instrument makers (such as T. and H. Doublett) began to emerge.

John Napier (1550–1617), Eighth Lord of Merchiston, was an imposing intellectual of his time. He pursued interests in astrology, theology, magic, physics and astronomy, methods of agriculture, and mathematics.¹²⁶ If these seem a strange set of areas, it is to be remembered, as Stephen Snobelen reminds us also of Newton, that few people now: “…have an understanding of what an intellectual cross-road the early modern period was. In fact, we now know that Newton was in many ways a Renaissance man, working in theology, prophecy and alchemy, as well as mathematics, optics and physics. In short, neither Napier nor Newton who came after him was “a scientist in the modern sense.”¹²⁷

Napier was an ardent Protestant. He wrote a stinging attack on the Papacy in what he would have regarded as his most important work.¹²⁸ It was a great success, and was translated into several languages by European reformers.¹²⁹ He also devoted considerable time to seeking to predict, on the basis of the Book of Revelation in the Bible, the likely timing of what was believed to be the coming apocalypse, which he concluded would come in 1688 or 1700. (At the time the “apocalypse” was not taken to mean so much the end of the world as the “temporary social disintegration and moral chaos, which is in turn mirrored in the devastation of nature.”.¹³⁰) In any case, Napier did not live long enough to be confronted with the prediction’s failure. (Nor indeed did he need to confront the implications of Sir Isaac Newton’s musings, in 1704, based on similar numerological investigation of the Bible, that the apocalypse would be no earlier than 2060).¹³¹ Napier did however, live to see himself celebrated as the prodigiously gifted mathematician, and ingenious inventor, that he was.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. A drawing of a set from 1797 is shown (:if equal {Site.PrintBook$:PSW} "True":)in Napier’s Rods(:ifend:) below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae, published in 1617.¹³²

Tables of logarithms, improved in various ways over time, were published either by themselves (:if equal {Site.PrintBook$:PSW} "False”:)(e.g. Tables by Gardiner, 1783) (:ifend:)(:if equal {Site.PrintBook$:PSW} "True”:)the Tables by Gardiner from 1783(:ifend:), or in the context of “ready reckoners” (e.g. (:if equal {Site.PrintBook$:PSW} "False”:)Ropp’s ready reckoner, 1892(:ifend:)(:if equal {Site.PrintBook$:PSW} "True”:)Ropp’s Ready Reckoner from 1892(:ifend:), in this collection). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

In addition, despite theological concern associated with the rekindling of scientific interest, “the idea of progress” was finding particular favour with the increasingly powerful merchant class. An underlying promise here was that, rather than awaiting one’s rewards for the afterlife, technical and industrial development could increasingly be relied on to satisfy needs in the here and now.¹³⁵ “The idea of progress” stressed a claimed progressive character to science. Over time, seized on by the emerging entrepreneurial class it came to be part of an argument for the new order, built around the market, to be given greater freedom to act, and corresponding political standing. As a consequence, as Bury puts it, during this period increasingly “Self-confidence was restored to human reason, and life on this planet was recognised as possessing a value independent of any hopes or fears connected with a life beyond the grave.”¹³⁶

The idea of progress was an ideology which could give legitimacy to the claims of a particular emerging class. But it was based on developments in thinking that were led by a comparatively small set of intellectuals. The questions they worked on often would have seemed quite divorced from everyday life. These mathematical and other scientific pioneers were frequently drawn from the aristocracy or church, or at least were gentlemen of considerable independent means. The motivations for doing this work might be scattered across a spectrum. It could include: a delight in learning and discovery, a desire to build prestige amongst peers, a hope for economic return from practical applications, or a desire to find favour with a rich or royal patron. Over time an increasing number of kings, queens, and other nobles began to enjoy being seen as a supporter of progress, or became interested in the work of intellectual pioneers.

A gulf still stood between the discoveries by mathematicians and other intellectuals, and the many others to whom this work could be of practical assistance. On the one side of that gulf, reflecting upon these early intellectual innovators, was the long-standing idea that a man of elevated (or aristocratic) heritage - a “gentleman” or in France “un honnête homme” - would consider it demeaning (as would an ancient Greek or Roman of standing some 1500 years before) to lower himself to associate himself with practical work. On the other side, amongst those whose life was devoted to practical work (for example, artisans) a parallel image was common, of the impractical nature of the gentleman mathematician and the products of mathematical thinking. This gulf would have to be surmounted before these the technical insights of these “gentlemen” could be turned to widespread practical purpose.

The second half of the fifteenth century through the sixteenth century was a time of such dramatic European exploration by sea that it is often referred to as “the age of exploration”. Notable amongst the European achievements were the charting of sea routes to India, Africa and the Americas. (Christopher Columbus reached America in 1492, whilst Sir Francis Drake claimed San Francisco Bay for Queen Elizabeth in 1579.) As a consequence, there was a large flow of gold and silver, amongst many other commodities, from the Americas to Europe. This was the source of a powerful inrush of wealth and thus investment and purchasing power for those who gained possession of it.¹³⁹ Increasingly complex financial techniques were needed to take advantage of long-distance trade.¹⁴⁰ This was but an early contribution to the increasingly complex financial flows, instruments and organisations which would be developed in support of, and in order to gain advantage, in the increasingly complex market capitalist economy. This economy would develop over the next several centuries creating ever greater demands for an ever more distributed capacity for efficient calculation.

The early seventeenth century was a turbulent time comprising widespread conflict and upheaval across Europe. Indeed it was so turbulent as to comprise what some historians have referred to as “the General Crisis”,¹⁴⁴ a time of confrontation, and in some places overthrow, of the legitimacy of the existing order, and more broadly, the relationships between state and society.¹⁴⁵ Whether it was more profound than the Hundred Years War between France and England (1348–1453) or the Black Death which almost halved England’s population (1348–9)¹⁴⁶ is hardly the point. It was a tumultuous time, and the tumult was widespread.

In England, just as an example, religious and political turmoil included: the schism between Rome and England under King Henry VIII between 1533–40, repression of “Papists” under Queen Elizabeth I (who established the English Protestant Church in 1559 and was declared a heretic by the Pope in 1570), further suppression under King James I, the Gunpowder Plot of 1605 in reaction to the treatment of Catholics, struggle in England between those in the House of Commons and King Charles I which ended in his execution in 1649 following the two English Civil Wars (1642–5 and 1648–9), rule by Oliver Cromwell as Lord Protector from 1653–8, and the subsequent restoration of the monarchy under King Charles II in 1660. But also across Europe, parts of the New World, and even beyond, it was a century of major wars and revolutionary upsurges, fertile ground for furious political machination and contest, and as it would turn out, innovation.

The increasingly widespread use of gun and cannon, in particular, provided a growing practical need to be able to calculate the trajectory of cannon balls and other projectiles. There was thus a demand for improved and more widely accessible methods of estimating all the relevant parameters (for example, matching quantity of powder required to wind, inclination and target, and projectile weight and type). By the early seventeenth century the search to solve these sorts of problems began to result in the development, explication, elaboration, popularisation, and with time, greater use ,of various helpful approaches to making the necessary calculations.

As already mentioned the Modern era was characterised by increasing flows of trade and finance between and within nations. Corresponding to this was the growth of cities, the increasing power of the state and merchants, and the growth across Europe of expanding bureaucracies as states attempted to regulate, control, and facilitate the powerful trends underway. Military conflict added to the pressure to wield collective force across kingdoms. Rulers in turn needed to plan, command, and control the collected forces. Consequently, as the Modern era developed, there emerged a virtual army of clerks, customs officials, excise officers, inspectors, quantity surveyors, architects, builders, and then technicians. These were assembled to form the apparatus of states as they sought to shape, manage, and control an ever more complex world. At the same time in the ever more complex organisations of commerce, a similar virtual army of employees was constructed to assist in the achievement of profit.¹⁴⁸

Whether in the state, or the commercial sector, the need for calculation and the spread of the capacity to calculate became greater. Eventually that need would in part be met by the development of a host of calculational aids. But the pattern of change would not simply be one of invention following developing need. Rather those early insights and inventions aimed at aiding calculation, and even their deployment in practice, initially acted more as a prelude to deployment. It was only over a considerable period of time that the society began to understand that these tools of calculation could play a potentially vital role in the emerging work of state and corporation.

The system of “pebble” or “ocular arithmetic” using counters stretched back seemingly into the mists of pre-history. But in the middle of the sixteenth century a process of dramatic change was beginning. It heralded the re-growth of political power in Europe and Britain. Over the next two centuries this would be accompanied by a dynamic development in knowledge, trade and innovation. By the beginning of the eighteenth century the industrial revolution in the “West” would be well underway. Drawing from earlier innovation in the “East” this now outpaced it in terms of economic and political power and technological innovation. It was a time of increasing fascination with new knowledge and mechanisation, continuing rise in scale and economic importance of cities, growth of what became industrial capitalism. Increasing dominance of institutions of the market (including the modern corporation) shaped also new products produced by increasingly sophisticated technological means. In support of this, availability of paper and literacy spread. Roman numerals were increasingly replaced by their Arab-Indian counterparts in every-day use, and the reliance on “ocular arithmetic” with counters was on the decline. Developments in mathematics and calculation, although important, were just one component of this remarkable transition.

By the late eighteenth century and into the nineteenth new technologies were shaping commerce. These included the telegraph, steam power, the railway, and the development of the factory system now powered with such technologies. The feudal system became increasingly submerged by the ever more dynamic, productive and powerful force of the industrial revolution.¹⁵⁰ Trade was increasing not only in volume but also reach. New technologies of navigation and shipping were resonating with new means of production, forms of transportation, and ways of transmitting information. Whilst in 1750 it had taken as long to travel or send information from one place to another as in the ancient Greek or Roman empires, by the end of the following century travel by railway across great distances was becoming vastly faster, and information could be sent by telegraph nearly instantaneously. Factories, trade and cities all expanded as the needs of the new system were met and fed. The celebration of technology, and use of it for all aspects of this transition to industrial production, was becoming a central tenet of Modern life.

The desire to predict the motion of heavenly bodies in itself provided demand for not only more powerful mathematical insights, but also aids to calculation. Astronomical investigation was based on observations of planets and stars where the calculated paths were redicted from a theory which involved not circular cycles, but also epicycles and ellipses. Making such predictions required a vast number of calculations involving repetitive additions and multiplications. New ways would soon be developed which could help with this.

The beginnings of the renewal of mathematical and scientific learning thus did not amount to a neat picture. Old ways of doing things lay not just with the aristocracy. As Spencer Jones points out,¹⁵² even into the seventeenth century whilst learned men began to press forward mathematics, navigation remained “a practical art, in which successes depended upon experience, common sense and good seamanship. The navigator had for his use the compass, the log, and some sort of cross staff” with which, together with his estimate of wind speed and currents, he would estimate his position by a “crude method of dead reckoning”.

Innovation requires multiple inventions. Once they are applied in practice this creates new opportunities for innovation creating a dynamic system of change. Increased navigation meant increased trade, requiring increased naval protection of trading routes, requiring improved navigation. The improvement of navigation depended as much on the capacity to print, which Gutenberg had pioneered in 1449, as on new forms of calculation. For example, in sixteenth century England, an early innovation was to replace the oral instruction and reliance on memory which had characterised British navigation at sea, with books of charts, tables and sailing practices, an approach that the Dutch had already pioneered with the Spiegel der Zeevaert published in two parts over 1584–5. In England, when a copy was displayed in the Privy Council a decision was made to translate the document and modify it for use in England, with it duly appearing as The Mariners Mirrour in 1588.¹⁵⁶ With an acceptance that seafaring could be assisted by printed aides it was only a matter of time for the desire to improve them to create a further demand for more accurate calculation of more useful navigational tables.

One mathematician - Thomas Hood - a Doctor of Physic from Cambridge University, took as part of his teaching task the design and documentation of potentially useful instruments. One instrument which he describes emerged naturally from the concepts of dividers, and proportional scaling inherent in the proportional compass. This was the sector - a set of dividers, but with flattened legs upon which could be marked various scales.¹⁵⁸ It represented a considerable advance in aids to calculation.

Hood’s sector was constructed by Charles Whitwell, a fine instrument maker.¹⁶¹ However, the process of adoption was far from immediate. It was helped when, in 1607, Edmund Gunter, a young mathematician not long from his studies at Oxford, began to circulate his hand-written notes on The Description and Use of the Sector showing in particular how well it could be applied to the problems of navigation by Mercartor’s charts. In this he found support from Henry Briggs. In 1597, Briggs had been appointed as the first Gresham Professor of Geometry at Cambridge University with an aim of bringing geometry to within reach of the people of London.¹⁶². The very creation of this Chair indicated that there was a growing understanding of the potential utility of mathematical thinking. (However, the Chair was kept within the bounds of propriety by focussing on geometry which had less association with the “dark arts”). Briggs discharged his duty in popular education, amongst other things, by giving his lectures not only in the obligatory learned language of Latin, but repeating them in English in the afternoons. With Brigg’s support Gunter’s Use of The Sector was issued in print in 1623.

The initial insight was that the logarithms of two numbers could be added by sliding two Gunter scales against each other. There is however debate about who was the first to realise this.¹⁶⁴ It was William Oughtred who published his design for a slide rule in 1632.

The first European ‘scientific journal’ (Le Journal des sçavans - later Le Journal des savants) published its first edition on 5 January 1665. Once established, in 1666, it soon became the written forum for the Paris Academy of Sciences.¹⁶⁷ (In England, the Philosophical Transactions of the Royal Society began publishing only three months later in March 1666.) Prior to that (indeed after, in many places) news of discoveries was spread by letters sent to trusted figures in the small ‘invisible college’ of thinkers. Thus, for example, William Oughtred (first to publish about the sliding part of the slide rule) was one of the key contact points in England. Others would learn of developments in the popular seminars he delivered at his home.¹⁶⁸ The network was thus characterised by a shared excitement and objectives, a developing communication system, the stimulus of possible positive outcomes, and consequently a level of competition (as revealed in the often quite tense and hard fought battles beginning to rage over who had primacy in any particular insight or invention).

The scope of Leibniz’s subsequent work was extraordinary encompassing poetry and literature, law, political diplomacy, work to unify all knowledge in part by bringing the scientific societies together, a similar desire to bridge the gap between the Lutheran and Catholic church in particular, and all churches in general, and more enduringly, his powerful initiatives in science and mathematics. The products of his work in mathematics included the development of binary arithmetic, methods of solving systems linear equations, and either the discovery of calculus (priority in this was contested by Newton) or at least the modern notation used for it. Central to his work was a belief that if a logically defined (‘mechanical’) algebra of thought could be developed then truths could be automatically generated and proved. (This quest can be found stretching back to the Stoics, and forward to the later work of Gottlob Frege, George Boole, Bertrand Russell, Kurt Gödel and many others). Given this enticing objective It is not surprising then, that having heard of Pascal’s calculating machine when he visited Paris in 1672 on a diplomatic mission (later to be elected to the French Academy of Sciences), Leibniz decided to turn his hand also to creating a calculating machine.¹⁷²

As to the innovation in mechanism there is no evidence that Pascale had any knowledge of Schickard’s invention. Nevertheless it was clear that he too was infected with the growing understanding that his invention could now be turned to reducing labour in new ways. His Pascaline turned out to be an adding machine, rather like that in the base of Schickard’s clock, but with two important improvements (and one significant deficiency). As with the Schickard’s machine numbers were added by turning disks with a stylus which in turn turned interlinked gears. Pascale however experimented with ways to improve the practical functioning of this. Various details of the Pascaline mechanism, including diagrams in this collection from the famous Encyclopédie ou Dictionnaire raisonné of Diderot & d’Alembert in 1759 are shown in this table below.

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).¹⁷⁵ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut (:if equal {Site.PrintBook$:PSW} "True":)(Contested methods) (:ifend:)from 1503,¹⁷⁶ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

For such counting boards the horizontal lines represent rows of multiples of 1, 10, 100, 1000 (or in Roman numerals I, X, C, M), whilst the mid point between those lines represent the half-way mark of 5 (V), 15 (XV), 50 (L), 500 (D) and 1500 (MD). For addition the number on the left could be progressively added to the number on the right by rows. In the woodcut, Boethius is adding the number MCCXXXXI (1,241) on his left (our right), to the number LXXXII (82) on his right. (Multiplication could be done by repeated additions.¹⁸⁰)

As the use of calculi reached its zenith they became important in other ways. The example minted in the Netherlands in 1577 shown below (this table) demonstrates that by then, counters were in such general use that they were being used to convey messages to the public - becoming the equivalent of official illustrated pamphlets.¹⁸¹ The face on the left (“CONCORDIA. 1577. CVM PIETATE”) translates (loosely) to “Agreement. 1577. With Piety” and seems to carry the visual message that we are joined together (with hearts and hands) under our ruler (the crown) and God (represented by a host). The opposite face (“CALCVLI.ORDINVM.BELGII. Port Salu”) shown on the right) translates to “Calculi. The United Provinces of the Netherlands, Port Salu” and shows a ship (perhaps the ship of state)¹⁸² coming into Port. (The term “port salu” at the time often simply meant “safe harbour”). It was not a calm time. It was in the midst of the European wars of religion (~1524–1648), and in particular the Eighty Years’ War in the Low Countries (1568–1648). It was only 28 years after Holy Roman Emperor Charles V had proclaimed the Pragmatic Sanction of 1549 which established “the Seventeen Provinces” as a separate entity. (This comprised what with minor exception is now the Netherlands, Belgium and Luxembourg.) Given that there had been a revolt in 1569 (leading to the Southern provinces becoming subject to Spain in 1579), only two years after this calculi was minted, we may presume that the messages of widespread agreement, safe harbour and rule from God might have been intended as a calming political message.

The feudal system of land had been controlled through a system of manors by feudal secular and religious lords (and more elevated nobles). Church and state supported a view of the feudal order as natural and immutable - one in which lords and serfs performed enduring roles within a system of mutual obligation. As agriculture became more efficient increasing numbers of “free men” with greater social mobility began to challenge the entrenched ways and power of the feudal system. Freed from the obligation of labouring on the land, as early as the eleventh and twelfth centuries, these freemen began to find new work as merchants or in other productive occupations first in towns, and then large industrial towns. Unshackled also from the manorial system their allegiance was more directly to kings (queens and princes) rather than lords. As merchants became more numerous they increasingly gained the concessions from the Royal courts necessary to carry out ever more sophisticated forms of commerce.¹⁸⁴

By the late eighteenth century and into the nineteenth new technologies were shaping commerce. These included the telegraph, steam power, the railway, and the development of the factory system now powered with such technologie. The feudal system became increasingly submerged by the ever more dynamic, productive and powerful force of the industrial revolution.¹⁸⁶ Trade was increasing not only in volume but also reach. New technologies of navigation and shipping were resonating with new means of production, forms of transportation, and ways of transmitting information. Whilst in 1750 it had taken as long to travel or send information from one place to another as in the ancient Greek or Roman empires, by the end of the following century travel by railway across great distances was becoming vastly faster, and information could be sent by telegraph nearly instantaneously. Factories, trade and cities all expanded as the needs of the new system were met and fed. The celebration of technology, and use of it for all aspects of this transition to industrial production, was becoming a central tenet of Modern life.

It has been estimated that in 1668 in England “the temporal and spiritual lords, baronets, knights, esquires, gentlemen, and persons in offices, sciences, and liberal arts” together represented about 4% of the population¹⁹⁰, (although enjoying about 23% of the income).¹⁹¹ Clearly those associated with the sciences in general, and mathematical work in particular, constituted only a tiny fraction of this. It may have been an initially small group of people involved, but as inevitably it was drawn from those who could afford the ‘time out’ to devote themselves in this work, drawn as they often were from aristocratic families, they had the capacity to convey their excitement at new insights amongst those of standing, and not least to reach the ears of royalty.

The role of the emerging practice of science became particularly confrontational for religious authorities (and particularly the powerful Catholic Church). Astronomical observation and theory had long played an important role in human life. Apart from its traditional role in astrological prognostication, astronomy could be turned to the prediction of seasonal changes such as tides, and the fixing of time and position. However, religious orthodoxy was the Ptolemaic concept that the Earth lay at the centre of the universe with the planets, sun and stars revolving around it in concentric spheres. The seminal work of Nicolaus Copernicus De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) was published in 1543 (just before his death) providing a systematic justification of a view of the solar system in which the planets, including the Earth, revolved around the sun. This sparked a major theological and scientific controversy. Whilst Copernicus had the planets moving in circles, it remained for the German mathematician, astrologer and astronomer, Johannes Kepler, in 1609, to publish mathematical arguments showing (amongst other important insights) that a much simpler explanation was that the planets move in ellipses.¹⁹² Steadily the convergence of observation and mathematical insight was bringing astronomy from an adjunct to philosophical speculation and theological dogma, to a science of the motion of the heavenly bodies. It was not however, until 1758, that the Catholic Pope of the time removed Copernicus’s book from the index of forbidden reading.

In addition, despite theological concern associated with the rekindling of scientific interest, “the idea of progress” was finding particular favour with the increasingly powerful merchant class. An underlying promise here was that, rather than awaiting one’s rewards for the afterlife, technical and industrial development could increasingly be relied on to satisfy needs in the here and now.¹⁹⁵ In this way “the idea of progress” and the accompanying claims for the value of science, came to be part of the argument for the new order, built around the market, to be given greater freedoms and political standing. As a consequence, as Bury puts it, during this period increasingly “Self-confidence was restored to human reason, and life on this planet was recognised as possessing a value independent of any hopes or fears connected with a life beyond the grave.”¹⁹⁶

The beginnings of the renewal of mathematical and scientific learning thus did not amount to a neat picture. Old ways of doing things lay not just with the aristocracy. As Spencer Jones points out,¹⁹⁸ even into the seventeenth century whilst learned men began to press forward mathematics, navigation remained “a practical art, in which successes depended upon experience, common sense and good seamanship. The navigator had for his use the compass, the log, and some sort of cross staff” with which, together with his estimate of wind speed and currents, he would estimate his position by a “crude method of dead reckoning”.

Even prior to the Modern era, the increasing importance of mercantile and thus also naval shipping was reshaping the understanding of needs especially at the level of government. Early developments around navigation and the construction of ships enabled more reliable open sea transport and naval expeditions. By the mid fifteenth century Portuguese ships had rounded the Cape Bojadar on the West Coast of Africa (in 1434), and the quadrant and a few decades later the astrolabe (in about 1480) had come into use.²⁰⁵

The second half of the fifteenth century through the sixteenth century was a time of such dramatic European exploration by sea that it is often referred to as “the age of exploration”. Notable amongst the European achievements were the charting of sea routes to India, Africa and the Americas. (Christopher Columbus reached America in 1492, whilst Sir Francis Drake claimed San Francisco Bay for Queen Elizabeth in 1579.) As a consequence, there was a large flow of gold and silver, amongst many other commodities, from the Americas to Europe. This was the source of a powerful inrush of wealth and thus investment and purchasing power for those who gained possession of it.²⁰⁶ Increasingly complex financial techniques were needed to take advantage of long-distance trade.²⁰⁷ This was but an early contribution to the increasingly complex financial flows, instruments and organisations which would be developed in support of, and in order the gain advantage, in the increasingly complex market capitalist economy. This economy would develop over the next several centuries creating ever greater demands for an ever more distributed capacity for efficient calculation.

Innovation requires multiple inventions. Once they are applied in practice this creates new opportunities for innovation creating a dynamic system of change. Increased navigation meant increased trade, requiring increased naval protection of trading routes, requiring improved navigation. The improvement of navigation depended as much on the capacity to print, which Gutenberg had pioneered in 1449, as on new forms of calculation. For example, in sixteenth century England, an early innovation was to replace the oral instruction and reliance on memory which had characterised British navigation at sea, with books of charts, tables and sailing practices, an approach that the Dutch had already pioneered with the Spiegel der Zeevaert published in two parts over 1584–5. In England, when a copy was displayed in the Privy Council a decision was made to translate the document and modify it for use in England, with it duly appearing as The Mariners Mirrour in 1588.²⁰⁸ With an acceptance that seafaring could be assisted by printed aides it was only a matter of time for the desire to improve them to create a further demand for more accurate calculation of more useful navigational tables.

The need for more accurate maps added to the demand for simpler ways of carrying out calculations. The defeat of the Armada at Gravelines during the (undeclared) Anglo-Spanish War of 1585–1604 provided a telling lesson in the sixteenth century of the importance of not only the economic power of merchant shipping, but also of the military importance of naval power. In particular it reinforced the need for manoeuvrable naval ships effectively utilising the best available gunnery.²¹⁴

The use of cannon, muskets and pistols in warfare both on land and sea, had a history stretching back several centuries. But it had become a recognised feature of warfare by the mid-sixteenth century. So much was this so that King Henry VIII found himself troubled by shortage of gunpowder in his invasion of France in 1544 AD and had to import it.²¹⁵

The early seventeenth century was a turbulent time comprising widespread conflict and upheaval across Europe. Indeed it was so turbulent as to comprise what some historians have referred to as “the General Crisis”²¹⁶ a time of confrontation, and in some places overthrow, of the legitimacy of the existing order, and more broadly, the relationships between state and society.²¹⁷ Whether it was more profound than the Hundred Years War between France and England (1348–1453) or the Black Death which almost halved England’s population (1348–9)²¹⁸ is hardly the point. It was a tumultuous time, and the tumult was widespread.

As already mentioned the Modern era was characterised by increasing flows of trade and finance between and within nations. Corresponding to this was the growth of cities, the increasing power of the state and merchants, and the growth across Europe of expanding bureaucracies as states attempted to regulate, control, and facilitate the powerful trends underway. Military conflict added to the pressure to wield collective force across kingdoms. They in turn needed to plan, commandl, and control the collected forces. Consequently, as the Modern era developed, there emerged a virtual army of clerks, customs officials, excise officers, inspectors, quantity surveyors, architects, builders, and then technicians. These were assembled to form the apparatus of states as they sought to shape, manage, and control an ever more complex world. At the same time in the ever more complex organisations of commerce, a similar virtual army of employees was constructed to assist in the achievement of profit.²²⁰

The first developments in the technology of calculation in Early Modern Europe were perhaps as much cultural as economic in their motivation. In particular they reflected a growing fascination with the use of machinery. From the thirteenth century, windmills and the more reliable waterwheels (both of which had been known of since ancient Rome) began to be turned to an ever wider variety of productive purposes - from the traditional role of crushing grain, to cutting stone, blowing bellows in metal work, sharpening knives and weapons, sawing planks, printing ribbons, dressing leather, rolling copper plate, and much else.²²³ The productive value of machinery was becoming more widely recognised, as perhaps also was the enjoyment of the way it can extend human capacity and power. Indeed there is no reason to think that it would have been any less rewarding an experience to show off one’s new adaption of a water mill than it is to show off a new car or electronic device now. A quote from Walker (if set in pre-feminist language) sums up the way in which innovation is a product of a whole cultural as well as economic and technological system:

At the bottom left of the Jeffries print (above) can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.²²⁶ Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below(:if equal {Site.PrintBook$:PSW} "True":) in French drawing instruments(:ifend:), is from ~1880.

One mathematician - Thomas Hood - a Doctor of Physic from Cambridge University, took as part of his teaching task the design and documentation of potentially useful instruments. One instrument which he describes emerged naturally from the concepts of dividers, and proportional scaling inherent in the proportional compass. This was the sector - a set of dividers, but with flattened legs upon which could be marked various scales.²²⁸ It represented a considerable advance in aids to calculation.

Hood’s sector was constructed by Charles Whitwell, a fine instrument maker.²³¹ However, the process of adoption was far from immediate. It was helped when, in 1607, Edmund Gunter, a young mathematician not long from his studies at Oxford, began to circulate his hand-written notes on The Description and Use of the Sector showing in particular how well it could be applied to the problems of navigation by Mercartor’s charts. In this he found support from Henry Briggs. In 1597, Briggs had been appointed as the first Gresham Professor of Geometry at Cambridge University with an aim of bringing geometry to within reach of the people of London.²³². The very creation of this Chair indicated that there was a growing understanding of the potential utility of mathematical thinking. (However, the Chair was kept within the bounds of propriety by focussing on geometry which had less association with the “dark arts”). Briggs discharged his duty in popular education, amongst other things, by giving his lectures not only in the obligatory learned language of Latin, but repeating them in English in the afternoons. With Brigg’s support Gunter’s Use of The Sector was issued in print in 1623.

John Robertson, in 1755, described the method thus: “To take a diſtance between the points of the compaſſes. Hold the compaſſes upright, ſet one point on one end of the dſtance to be taken, there let it reſt; and (as before ſhewn) extend the other point to the other end.”²³⁴

John Robertson, in 1755, described the method thus: “To take a diftance between the points of the compaffes. Hold the compaffes upright, fet one point on one end of the diftance to be taken, there let it reft; and (as before fhewn) extend the other point to the other end.”²³⁶

Using the same dividers, following the above, it was necessary to measure off the distance between the two “legs” of the sector at the required point along them.²³⁸ As can be seen from the dividers in this collection, the accuracy of calculations using sectors was limited by the fineness with which their scales were rendered and the precision with which the points of the dividers could be applied to the task of measuring them. The process was thus slow, inherently inaccurate, and required considerable dexterity and practice to achieve a credible result.

John Napier (1550–1617), Eighth Lord of Merchiston, was an imposing intellectual of his time. He pursued interests in astrology, theology, magic, physics and astronomy, methods of agriculture, and mathematics.²⁴⁶ If these seem a strange set of areas, it is to be remembered, as Stephen Snobelen reminds us of Newton also that few people now: “…have an understanding of what an intellectual cross-road the early modern period was. In fact, we now know that Newton was in many ways a Renaissance man, working in theology, prophecy and alchemy, as well as mathematics, optics and physics. In short, neither Napier nor Newton who came after him was “a scientist in the modern sense.”²⁴⁷

Napier was an ardent Protestant. He wrote a stinging attack on the Papacy in what he would have regarded as his most important work.²⁴⁸ It was a great success, and translated into several languages by European reformers.²⁴⁹ He also devoted considerable time to seeking to predict, on the basis of the Book of Revelation in the Bible, the likely timing of what was believed to be the coming apocalypse, which he concluded would come in 1688 or 1700. (At the time the “apocalypse” was not taken to mean so much the end of the world as the “temporary social disintegration and moral chaos, which is in turn mirrored in the devastation of nature.”.²⁵⁰) In any case, Napier did not live long enough to be confronted with the prediction’s failure. (Nor indeed did he need to confront the implications of Sir Isaac Newton’s musings, in 1704, based on similar numerological investigation of the Bible, that the apocalypse would be no earlier than 2060).²⁵¹ Napier did however, live to see himself celebrated as the prodigiously gifted mathematician, and ingenious inventor, that he was.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. An drawing of a set from 1797 is shown (:if equal {Site.PrintBook$:PSW} "True":)in Napier’s Rods(:ifend:) below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae, published in 1617.²⁵²

More importantly, after 20 years of work Napier also devised and had printed in 1614, a set of tables, Mirifici Logarithmorum Canonis Descriptio (“Description of the Marvellous Rule of Logarithms”). These enabled direct multiplication to be carried out through simple addition. It is from these that modern tables of logarithms (with some additional conceptual improvements) are derived. These are based on the fact that the powers of numbers add when the numbers are multiplied (i.e. 2³x2⁴=2⁷). This fact had been known since the time of Archimedes. But it was Napier who thought to use this sort of property to tabulate numbers in terms of the power of some “base” (2 in this example). In fact, Napier did not achieve his tables quite in this way. Rather it is believed he started from an earlier approach using a property from trigonometry.²⁵⁵ This led him to a geometric argument based on the theory of proportions. Using this he constructed his functions.²⁵⁶

Henry Briggs, the first Gresham Professor of Geometry at Cambridge (already mentioned) had travelled to meet Napier in Edinburgh in 1615. There it is reported the two men gazed in admiration at each other for a full quarter hour before finding words to speak.²⁵⁹ Briggs had suggested in a prior letter that it would be good to develop tables of logarithms to base 10. Napier who had had a similar idea was unable to do the work because of ill-health. He was however very pleased that Briggs might carry the work through.

Briggs did develop such tables of “common logarithms” the first of which gave the logarithms from 1 to 1000. It was published as a 16 page leaflet Logarithmorum Chilias Prima in 1617. His colleague Edmund Gunter at Gresham College published a more complete set from 1 to 20,000, in 1620. It was accurate to 14 decimal places.²⁶⁰ The usefulness of such tables for serious calculations involving multiplication, division, and powers of numbers to high levels of accuracy was clear to those who became familiar with them. That knowledge and the tables themselves, usually supplemented by corresponding tables for trigonometric functions, spread rapidly.

Not only had Brigg’s colleague, Professor Edmund Gunter, published his Canon triangulorum in 1629, which contained logarithmic sines and tangents for every minute of arc in the quadrant to seven decimal places. In 1624 Gunter followed this with a collection of his mathematical works entitled The description and use of sector, the cross-staffe, and other instruments for such as are studious of mathematical practise. This work contained, amongst other things the detail of “Gunter’s scale” (or “Gunter’s rule”) which was a logarithmically divided scale able to be used for multiplication and division by measuring off lengths. It was thus the predecessor to the slide rule.²⁶²

The initial insight was that the logarithms of two numbers could be added by sliding two Gunter scales against each other. There is however debate about who was the first to realise this.²⁶⁶ It was William Oughtred who published his design for a slide rule in 1632.

A series of designs followed. Three of these appear in Jacob Leupold’s book Theatrum Arithmetico-Geometricum²⁶⁷ of which Table XII (page 241), held in this collection, is shown in (i) in the Table below. (The top drawing is a design for a Gunter scale, and the two below are early designs for slide rules).²⁶⁸

One consequence of Britain’s increasing strength in shipping and maritime trade was that trade became an obvious target for revenue raising. During the C17 taxation was aggressively applied to offshore trade. The income raised was in part invested in the increased naval capacity and colonial infrastructure required to protect shipping. Tax was applied to commodities as diverse as glass, paper, soap, vinegar, famously tea, and of course alcohol in wine, ale and spirits (the taxation of which began in 1643). One consequence of this was that the quantities of these in diverse containers needed to be audited.²⁷⁵ This created a rapidly growing need for “gaugers” who could apply the mathematics of “stereometry” to estimating volumes of fluid held, and the corresponding alcoholic content. These measurements needed to be applied not only to barrels (whether on their side or standing), but also butts, pipes, tuns, firkins, puncheons and long-breakers (amongst other now long forgotten containers).²⁷⁶ Given the lack of widespread mathematical literacy, it was essential to have aids to enable gaugers to do this. Extensive manuals, tables and guides were published, but even so, it was clear to practitioners that they really needed something easier to use.

In 1683 Thomas Everard, an English Excise Officer (who is credited with introducing the term “sliding rule”), began promoting a new 1 inch square cross section slide rule with several slides for calculating excise.²⁷⁷ Shown in (ii), below, is an English four sided Everard pattern sliding rule from 1759. It includes various gauging points and conversions to square and cube roots for calculating volumes.²⁷⁸. In (iii) is a more modern looking slide rule shape, from 1821–84 by Joseph Long of London, also for use in gauging the amount of alcohol spirit in a container, and calculating the corresponding tax.

The slide rule evolved in what are early recognisable directions. The use of a hair line to read off multiple scales was suggested as early as 1675 by Sir Isaac Newton. Nevertheless, the introduction of a moveable cursor with this innovation included had to wait a century until a professor of mathematics, John Robertson, in 1775 added a mechanical cursor.²⁷⁹

It was Victor Mayer Amédée Mannheim, a Colonel in the French artillery and professor of geometry in Paris, who introduced the now familiar scale system, combined with a now fully functional cursor. This effectively brought together the key elements of what would become the modern slide-rule. He described it in a pamphlet published in 1851²⁸⁰ An early slide rule made in 1893–98 by Tavernier-Gravet, but based on a pattern devised by Lenoir in 1814 (iv) is also shown, now fitted with a brass cursor. In (v) is a slide rule from about 1928 by the firm Keuffel and Esser with cursor and familiar scales. The Faber Castell 2/83N Novo Duplex slide rule (xii) with its multiple scales on both sides is often spoken of as the high point reached in elegance and utility of the straight slide rules. Production of it lasted until 1973 when electronic calculators rendered it obsolete. A progression of such slide rule designs is shown in Early Slide Rules below.

For more complex equations with multiple variables it was either train workers to evaluate the equations from first principles or find some other way of solving them. Nomography provided an approach to achieving this. This methodology was invented by Maurice d'Ocagne (1862–1938) in 1880 and was used through to the 1970s. Its principles are described in several good references.²⁸² A simple example of the approach is given in (:if equal {Site.PrintBook$:PSW} "True":)A simple nomograph (:ifend:)below:

The above is a nomograph for adding two numbers (one in column a) and the other (in column c).²⁸⁴ The properties of similar triangles give the result that a line drawn between the two numbers will cut the middle column (b) at the required sum. Since the outer scales may be logarithmic or trigonometric many more complicated expressions are able to be evaluated using this sort of approach.

As the above suggests, multiple solutions were emerging to multiple formulations of the “problem of multiplication”. Each solution had its limitations, whether it be ease of use in different practical circumstances, or accuracy. Thus, although slide rules had apparent advantages over Gunter rules, and Gunter rules over sectors, none of these simply vanished once the other had been invented. As Robertson noted in 1775, Gunter rules had simply been added to sectors as available approaches.²⁸⁷ Indeed, as shown by the objects in this collection, Gunter rules and sectors continued to be used right up into the nineteenth century. Even to this day, nomographs, often represented now in the form of computer graphics, continue to be used for particular applications.²⁸⁸

Laird and Roux note, “The establishment of mechanical philosophy in the 17th century was a slow and complex process, profoundly upsetting the traditional boundaries of knowledge. It encompassed changing view on the scope and nature of natural philosophy, an appreciation of “vulgar” mechanical knowledge and skills, and the gradual replacement of a casual physics with mathematical explanations.”²⁹⁰ It was this slow and complex process that would eventually give rise to mechanised calculation. But as suggested above, the developments would be complex, many stranded, and over time increasingly tangle together the two worlds of the philosophical and practical arts.

The Roman emphasis on the importance of transport for the state had led also to the use of measuring machines including the odometer of Heron of Alexandria - a device used to measure distance traversed by a rotating carriage wheel by means of a coupled system of incrementing gears. As described by Vitruvius (~15 BC),²⁹³ the rotation of the carriage wheel was measured by this device in units, tens, hundreds, thousands and tens of thousands of paces.²⁹⁴ This sort of counting mechanism could form a natural basis for the important “carry” mechanism (e.g. where 9+1 is converted to 10) of an adding machine. Still, for that potential to be capitalised upon there had to be a recollection or re-discovery of the sort of mechanism, and perhaps more importantly, an interest in constructing one. As already mentioned, the budding interest in mechanisation in the early C17 (building in particular on the widely understood appeal of the success of clocks, clockwork, and new mechanical insight about nature) provided an appropriately fertile base for that interest.

In the “Problemata” for one week of his course, Ciermans noted slightly obscurely (loosely translated) that while many seek savings in multiplying and dividing the outcomes usually require more effort to do so than from first principles. However, he wrote, there is a method with “rotuli” (normally rolls of parchment for writing upon, but could also intend the more modern diminutive for rota i.e. little wheels) with indicators (or pointers), which enables multiplication and division to be done “with a little twist” so the work is shown without error.²⁹⁶

It is worth remembering the smallness of the circle of people who had the necessary skills, interests and resources to participate in what we now know of as science. The astronomer Johannes Kepler, who was instrumental in demonstrating that the planets might most simply be considered to move in ellipses around the sun, and who provided other crucial insights that would later inform the work of Newton, was a graduate of the University of Tübingen where he had studied under Magister Michael Maestlin, one of the leading astronomers of the time.³⁰⁰ It was through Maestlin that Kepler came to know of the still suppressed work of Copernicus for who’s solar-centric theory he became a strong supporter.³⁰¹ His first book, Mysterium Cosmographicum, was published in Tübingen and it was presumably on one of his visits to Maestlin there that he was introduced to Schickard. They had much in common - both being mathematically gifted and knowledgeable, and both also sharing strong Lutheran theological interests.³⁰²

The first European ‘scientific journal’ (/’Le Journal des sçavans/’ - later Le Journal des savants) published its first edition on 5 January 1665. Once established, in 1666, it soon became the written forum for the Paris Academy of Sciences.³¹¹ (In England, the Philosophical Transactions of the Royal Society began publishing only three months later in March 1666.) Prior to that (indeed after, in many places) news of discoveries was spread by letters sent to trusted figures in the small ‘invisible college’ of thinkers. Thus, for example, William Oughtred (first to publish about the sliding part of the slide rule) was one of the key contact points in England. Others would learn of developments in the popular seminars he delivered at his home.³¹² The network was thus characterised by a shared excitement and objectives, a developing communication system, the stimulus of possible positive outcomes, and consequently a level of competition (as revealed in the often quite tense and hard fought battles beginning to rage over who had primacy in any particular insight or invention).

Kepler was greatly excited by the publication of logarithms which he utilised to greatly reduce the enormous calculational effort in his computations of The Rudolphine Tables and later works. He dedicated his Ephemerides to Napier in 1617.³¹³ Professor Briggs (astronomer and geometrician) now at Gresham College, Cambridge, as mentioned earlier was a close colleague of Napier and had written of his tables “…I never saw a book which pleased me better or made me more wonder”.³¹⁴ He would later take Napier’s work further forward producing new tables of logarithms to base 10. In 1609 Briggs was also impatiently awaiting Kepler’s exposition on ellipses.³¹⁵

Finally, of Schickard, Kepler wrote admiringly that he has “a fine mind and is a great friend of mathematics; … he is a very diligent mechanic and at the same time an expert on oriental languages.”³¹⁶ It is known that Kepler and Schickard had discussed applications to astronomical calculation by Kepler of Napier’s logarithms and rods as early as 1617. This may well have inspired Schickard to find a mechanical embodiment of the rods.

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.³²¹ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.³²² In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline replica (:ifend:)below.

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design. Some eight surviving Pascalines can be found in museums and private collections. He also documented his machine in a short pamphlet,³²⁷ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.³²⁸ Further, even though Schickard’s machine in principle could multiply (using its ingenious incorporation of Napier’s rods), whilst Pascal’s machine (despite his claim that it could be used for all operations of arithmetic) was primarily useful for addition, and with some mental effort, subtraction, (see calculating with the Pascaline) it is Pascal’s machine which was referred to with reverance down the following centuries. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline woodcut (:ifend:)below. (:if equal {Site.PrintBook$:PSW} "False":)

(i) Perhaps most subtly, this was a time when philosophical inquiry, and the emerging practice of what would more commonly become known as scientific inquiry, were taking a more practical turn. There was a growing realisation that investigation which engaged with the natural world though exploration of how it behaved, could yield rich results. Notable in leading this idea was Francis Bacon, who in 1620 had written his Novum Organum, a strong argument that systematic empirical engagement of this type, could not but result in “an improvement in man’s estate and an enlargement of his power over nature.”³³¹ Implicit in this was a narrowing of the gap between science and technology, new ideas and application for betterment, and intellectual investigation, tools and technique. It was no less than a launch of “the idea of progress” which, as mentioned earlier, over subsequent centuries was to act as a reinforcing ideology for merchants and entrepreneurs, eventually helping sweep before them and the market much of the religious and customary authority of the aristocracy.³³²

In a fascinating thesis,³⁴⁰ and subsequent published book chapter,³⁴¹ Jean-François Gauvin develops a multi-stranded analysis of the role of scientific instruments, including the Pascaline, and their creators and use in the seventeenth century. Key to this are conflicts and resonances between continuities in cultural habit, and social, philosophical and ideological challenges to them that were beginning to gain force at the time.

Challenges such as Bacon’s Novum Organum reflected the beginnings of a discomfort with knowledge founded on the verities of ancient Greek knowledge, church orthodoxy and aristocratic tradition. Of course both Schickard and Pascal, sought and enjoyed the indulgence and support of aristocratic and religious sponsors and Pascal in particular would use his machines as much as a way of attracting that as for any commercial gain. But the transformation reflected in Bacon’s call was to progressively reposition the man of substance (a “gentleman”, or in France “l'honnête homme”)³⁴² in relation to some tools such as these. Building on the existing prestige of clockwork and the growing understanding that intellectual advance was tied to mechanical engagement with the natural world, these devices could be seen to potentially restore the separation of the intellectual world of l'honnête homme from the drudgery of the ‘mechanical’ act of routine calculation.

Having said that, the machines were expensive (about of a third of a year’s average wages of the time),³⁴³. Pascale in his surviving “Advice” on the use of the Pascaline promised that the machine could “perform without any effort whatsoever all the arithmetical operations that had so often worn out one's mind by means of the plume and the jetons”.³⁴⁴ But in truth the use of it required a good knowledge of arithmetic to be used effectively (especially for division and multiplication, but even for subtraction). At best it was a mechanical substitute for what otherwise would have to be written down, save that it could perform addition with a little practice, and simple subtraction (where the result was not negative) with a little more. As Balthazaar Gerbier described the Pascaline in a letter to Samuel Hartlib in 1648:

But like rare books need not be read to add luster to the shelf, the machines needed not necessarily be used to add something to the owner. Basic mathematical skills were far from universally held, even amongst the nobility, and certainly amongst gentlemen. From this point of view, even if one could not add and subtract reliably in one’s head, let alone multiply using jeton (calculi) and pen, the potential allure of such a calculating machine was that by owning it, one could come to at least be seen to have an intimacy with such literacy. In these senses, as Gauvin argues, “Even though Pascal invented the machine to alleviate his father’s headaches as a royal tax collector, financiers and merchants, who tallied large amount of numbers, were not especially in Pascal’s mind. The pascaline was more than a mechanical contraption useful for business: used properly, it could bestow l'honnête.”³⁴⁶

Samuel Morland (1625–95) - son of an English clergyman - had a complex life in a difficult time. At the age of 24 (the year he matriculated from Cambridge) he experienced the English revolution with the execution of King Charles I. Then he began work for Cromwell as a courtier-inventor a year later primarily providing intelligence through methods of postal espionage (intercepting, opening, decrypting and interpreting, and re-sealing mail). In the course of this, he was almost killed by Cromwell on suspicion of overhearing a plot to lure to England and kill the exiled Charles II,³⁵³ son of the executed King Charles I. Indeed Morland had overheard the plot and subsequently reported it to Charles II’s supporters. After Cromwell’s death (in 1658) Morland was able to manage the delicate transition to service under the newly restored King Charles II and was knighted by him in 1660 and made a Baronet soon after.³⁵⁴

In the course of these events Morland, whilst certainly comfortably provided for (not the least after he married a Baroness), complained of the failure of his positions to yield wealth saying: 'Now finding myself disappoynted of all preferment and of any real estate, I betook myself too the Mathematicks, and Experiments such as I found pleased the King's Fancy.'³⁵⁵ On the basis of his mastery of engineering (in particular, fluid systems) Morland was eventually granted a newly created position of “Master of Mechanicks” to the King, and later was made a gentleman of his Majesty’s privy chamber.³⁵⁶ His inventions, thus played a two-fold role, first in his search to establish a role for himself in the context of the new Court, and second to supplement his wealth by seeking commercial success. In particular, he hoped for a lucrative outcome to devising mathematical instruments and selling them.

Morland had already seen a Pascaline, probably when on a diplomatic mission to Queen Christina of Sweden, a supporter of the sciences who in 1649 had been presented with a Pascaline (similar in looks and identical in function to the replica in this collection).³⁵⁷ He had also taken part in a diplomatic mission during which he stopped off at the court of Louis XIV for over a month, so he may well have learned more of the Pascaline, and made contact with scholars and associated mechanics at that time.³⁵⁸ He was thus aware of the potential allure of such inventions. Over the next few years he devised three different calculator designs: an adding machine (see (:if equal {Site.PrintBook$:PSW} "True":)Morland adding machine (:ifend:)below), a multiplying device (see also (:if equal {Site.PrintBook$:PSW} "True":)Morland multiplying instrument (:ifend:)below), and an instrument with moveable arms for determining trigonometric relations. Several examples of Morland’s calculating devices are still in existence, in particular in museums in London and Italy. Not only did Morland design the machines but he wrote a book on the use of the instruments. Published in 1672 this can be considered the first known English Language (and arguably the earliest surviving) ‘computer manual’. (A copy of this rare book, shown below(:if equal {Site.PrintBook$:PSW} "True":) in this table(:ifend:), is in this collection).

These machines were variously received as “those incomparable Instruments”(Sir Jonas Moore),³⁶⁵ “not very useful” (Henri Justel),³⁶⁶ or “very silly” (Robert Hook).³⁶⁷ But in terms of obtaining patronage on the one hand (not only in England but also from the Medici in Italy), and at least some sales to those men and women with wealth but not much knowledge of addition or the multiplication tables, the instruments served at least some of the needs of their inventor. That being so, they perhaps provided more reward to both maker and purchaser in terms of status than they returned financial benefit for the former, or enhanced arithmetic capability for the latter.

Gottfried Wilhelm Leibniz (1646–1716) had a life filled with so many ambitions, such high profile conflict, and so much capacity that his biography can seem overwhelming. He has been described as “…an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.”³⁶⁸ At the same time his irascible personality was such that by the time he died, his contemporaries were said to feel a sense of relief and only his secretary attended his funeral.³⁶⁹ His father was a professor of moral philosophy who died when Leibniz was only six years old, but he grew up in the surroundings of his father’s imposing library from which he derived much satisfaction. He gained his Doctorate of Laws (on De Casibus Perplexis, i.e. ‘On Perplexing Cases’) at the age of 20 from the University of Altdorf in Nürnberg.

The scope of Leibniz’s subsequent work was extraordinary encompassing poetry and literature, law, political diplomacy, work to unify all knowledge in part by bringing the scientific societies together, a similar desire to bridge the gap between the Lutheran and Catholic church in particular, and all churches in general, and more enduringly, his powerful initiatives in science and mathematics. The products of his work in mathematics included the development of binary arithmetic, methods of solving systems linear equations, and either the discovery of calculus (priority in this was contested by Newton) or at least the modern notation used for it. Central to his work was a belief that if a logically defined (‘mechanical’) algebra of thought could be developed then truths could be automatically generated and proved. (This quest can be found stretching back to the Stoics, and forward to the later work of Gottlob Frege, George Boole, Bertrand Russell, Kurt Gödel and many others). Given this enticing objective It is not surprising then, that having heard of Pascal’s calculating machine when he visited Paris in 1672 on a diplomatic mission (later to be elected to the French Academy of Sciences), Leibniz decided to turn his hand also to creating a calculating machine.³⁷⁰

http://meta-studies.net/pmwiki/uploads/NapiersBonesContempW600.jpg|Set of Napier’s Rods (of recent construction)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/NapiersBonesContempW600.jpg|Set of Napier’s Rods (of recent construction)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Schickard1W300_1.jpg|Working replica of Schickard’s Calculating Clock (1623)
(collection Calculant)

The first replica was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) Schickard replica(:ifend:) in this collection, shown below is more recent.³⁷²

The first replica was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) FIG(Schickard replica)(:ifend:) in this collection, shown below is more recent.³⁷⁴

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810) (collection Calculant)

The first replica was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The (:if equal {Site.PrintBook$:PSW} "True":) Schickard replica(:ifend:) in this collection, shown below is more recent.³⁷⁶

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|Schickard rear view showing Napier cylinders
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardFrontPanelW600.jpg|Schickard front panel
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|Schickard rear view showing Napier cylinders
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardBackViewW400.jpg|
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/SchickardFrontPanelW600.jpg|
(collection Calculant)

The vertical section at the top was a mechanical cylindrical embodiment of Napier’s bones (published six years earlier) to aid multiplication.

The use of cannon, muskets and pistols in warfare both on land and sea, had a history stretching back several centuries. But it had become a recognised feature of warfare by the mid-sixteenth century. So much was this so that King Henry VIII found himself troubled by shortage of gunpowder in his invasion of France in 1544 AD and had to import it.³⁷⁸

The Roman emphasis on the importance of transport for the state had led also to the use of measuring machines including the odometer of Heron of Alexandria - a device used to measure distance traversed by a rotating carriage wheel by means of a coupled system of incrementing gears. As described by Vitruvius (~15 BC),³⁸¹ the rotation of the carriage wheel was measured by this device in units, tens, hundreds, thousands and tens of thousands of paces.³⁸² This sort of counting mechanism could form a natural basis for the important “carry” mechanism (e.g. where 9+1 is converted to 10) of an adding machine. Still, for that potential to be capitalised upon there had to be a recollection or re-discovery of the sort of mechanism, and perhaps more importantly, an interest in constructing one. As already mentioned, the budding interest in mechanisation in the early C17 (building in particular on the widely understood appeal of the success of clocks, clockwork, and new mechanical insight about nature) provided an appropriately fertile base for that interest.

http://meta-studies.net/pmwiki/uploads/LeibnitzRechenmaschW.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University. Reproduced with permission from the Gottfried Wilhelm Leibniz Bibliothek.

http://meta-studies.net/pmwiki/uploads/LeibnitzRechenmaschW.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University. Reproduced with permission from the Gottfried Wilhelm Leibniz Bibliothek.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. An drawing of a set from 1797 is shown (:if equal {Site.PrintBook$:PSW} "True":)in Napier’s Rods(:ifend:) below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae, published in 1617.³⁸⁴

In the course of these events Morland, whilst certainly comfortably provided for (not the least after he married a Baroness), complained of the failure of his positions to yield wealth saying: 'Now finding myself disappoynted of all preferment and of any real estate, I betook myself too the Mathematicks, and Experiments such as I found pleased the King's Fancy.'³⁸⁷ On the basis of his mastery of engineering (in particular, fluid systems) Morland was eventually granted a newly created position of “Master of Mechanicks” to the King, and later was made a gentleman of his Majesty’s privy chamber.³⁸⁸ His inventions, thus played a two-fold role, first in his search to establish a role for himself in the context of the new Court, and second to supplement his wealth by seeking commercial success. In particular, he hoped for a lucrative outcome to devising mathematical instruments and selling them.

The early seventeenth century was a turbulent time comprising widespread conflict and upheaval across Europe. Indeed it was so turbulent as to comprise what some historians have referred to as “the General Crisis”³⁹² a time of confrontation, and in some places overthrow, of the legitimacy of the existing order, and more broadly, the relationships between state and society.³⁹³ Whether it was more profound than the Hundred Years War between France and England (1348–1453) or the Black Death which almost halved England’s population (1348–9)³⁹⁴ is hardly the point. It was a tumultuous time, and the tumult was widespread.

Napier was an ardent Protestant. He wrote a stinging attack on the Papacy in what he would have regarded as his most important work.³⁹⁹ It was a great success, and translated into several languages by European reformers.⁴⁰⁰ He also devoted considerable time to seeking to predict, on the basis of the Book of Revelation in the Bible, the likely timing of what was believed to be the coming apocalypse, which he concluded would come in 1688 or 1700. (At the time the “apocalypse” was not taken to mean so much the end of the world as the “temporary social disintegration and moral chaos, which is in turn mirrored in the devastation of nature.”.⁴⁰¹) In any case, Napier did not live long enough to be confronted with the prediction’s failure. (Nor indeed did he need to confront the implications of Sir Isaac Newton’s musings, in 1704, based on similar numerological investigation of the Bible, that the apocalypse would be no earlier than 2060).⁴⁰² Napier did however, live to see himself celebrated as the prodigiously gifted mathematician, and ingenious inventor, that he was.

Gottfried Wilhelm Leibniz (1646–1716) had a life filled with so many ambitions, such high profile conflict, and so much capacity that his biography can seem overwhelming. He has been described as “…an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.”⁴⁰⁵ At the same time his irascible personality was such that by the time he died, his contemporaries were said to feel a sense of relief and only his secretary attended his funeral.⁴⁰⁶ His father was a professor of moral philosophy who died when Leibniz was only six years old, but he grew up in the surroundings of his father’s imposing library from which he derived much satisfaction. He gained his Doctorate of Laws (on De Casibus Perplexis, i.e. ‘On Perplexing Cases’) at the age of 20 from the University of Altdorf in Nürnberg.

Even by the late sixteenth century in practice methods of calculation had not changed much since the Roman Empire. Indicative of this “calculi” (Latin: “calculus” for “pebble” (or “limestone”) - the origin of the word “calculation”) were still being struck (as they had been for centuries) to facilitate counting and arithmetic especially in many fields of commerce. They were referred to differently in different places. One such - a French “jeton” from 1480–1520 is shown (:if equal {Site.PrintBook$:PSW} "True":)in French jeton (:ifend:)below. It is made to look like a coin, but its inscription is meaningless.

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁴⁰⁹ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut (:if equal {Site.PrintBook$:PSW} "True":)(Contested methods) (:ifend:)from 1503,⁴¹⁰ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

The measurement of distance had been aided by dividers (or “compasses”) at least since the Roman era and now instruments were developed building on that idea. A print of a drawing by Thomas Jefferys (~1710–1771) which shows a range of such instruments from the C18 is shown below(:if equal {Site.PrintBook$:PSW} "True":) (Compasses)(:ifend:).

In this drawing, at the top can be seen the emerging shape of various compasses, constructed from brass and steel. Below,(:if equal {Site.PrintBook$:PSW} "True":) shown in Pair of dividers(:ifend:) from this collection, is a typical pair of such dividers, also from the eighteenth century:

At the bottom left of the Jeffries print (above) can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.⁴¹² Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below(:if equal {Site.PrintBook$:PSW} "True":) in French drawing instruments(:ifend:), is from ~1880.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. An drawing of a set from 1797 is shown (:if equal {Site.PrintBook$:PSW} "True":)in Napier’s Rods(:ifend:) below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae//, published in 1617.⁴¹⁴

In 1625 Wingate published a French edition of Brigg’s latest tables. A year later in 1626, Dennis Henrion published his tables “Traicté de logarithms” (of which there is a copy in this collection, see (:if equal {Site.PrintBook$:PSW} "True":)Logarithms by Henrion(:ifend:) below). Together with Wingate’s, these tables made available the power of logarithms across Europe.

Tables of logarithms, improved in various ways over time, were published either by themselves (e.g. Tables by Gardiner, 1783) or in the context of “ready reckoners” (e.g. Ropp’s ready reckoner, 1892). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown (:if equal {Site.PrintBook$:PSW} "True":)in Logarithms by Gardiner (:ifend:)below.

In a second section (see below), the book details the design of a logarithmic proportional rule (derived from Gunter’s 1624 publication), along with additional explanations, charts and other elaborations (see (:if equal {Site.PrintBook$:PSW} "True":)Graphical Construction of Gunter Scales (:ifend:)below). The proportional rule could be used directly by means of a pair of dividers to measure off lengths corresponding to logarithms and thus to evaluate multiplications and divisions.

Gunter rules were usually equipped with both Gunter’s combination of a logarithmic and a linear scale. Often they also ncluded a range of other scales (notably trigonometric scales for navigational tasks, such as required to work from elapsed time, speed and changes to compass bearing to distances travelled). In addition, important constants could be marked on them as “gauge marks”. Over the seventeenth century Gunter rules gained increasing acceptance and were used right through into the late nineteenth century. Some of the scales of a two foot long mid-nineteenth century navigational Gunter rule in this collection are shown (:if equal {Site.PrintBook$:PSW} "True":)in The Gunter Rule(:ifend:) below.

For more complex equations with multiple variables it was either train workers to evaluate the equations from first principles or find some other way of solving them. Nomography provided an approach to achieving this. This methodology was invented by Maurice d'Ocagne (1862–1938) in 1880 and was used through to the 1970s. Its principles are described in several good references.⁴¹⁶ A simple example of the approach is given in (:if equal {Site.PrintBook$:PSW} "True":)A simple nomograph (:ifend:)below:

Two devices which utilise these nomographic principles are shown below. The first (:if equal {Site.PrintBook$:PSW} "True":)(Bloch Schnellkalulator) (:ifend:)is a Bloch Schnellkalulator from ~1924. The second (:if equal {Site.PrintBook$:PSW} "True":)(Der Zeitermittler) (:ifend:)is a Zeitermittler from ~1947. In both cases these are nomographic devices for calculating parameters required to cut metal with a lathe. The Bloch Schnellkalulator is notable for its use of a linked mechanism which enables the results of one calculation to be fed as the input to another.

The first replica was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The one in this collection, shown below(:if equal {Site.PrintBook$:PSW} "True":) in Schickard replica(:ifend:) is more recent.⁴¹⁸

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁴²¹ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.⁴²² In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline replica (:ifend:)below.

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design. Some eight surviving Pascalines can be found in museums and private collections. He also documented his machine in a short pamphlet,⁴²⁵ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.⁴²⁶ Further, even though Schickard’s machine in principle could multiply (using its ingenious incorporation of Napier’s rods), whilst Pascal’s machine (despite his claim that it could be used for all operations of arithmetic) was primarily useful for addition, and with some mental effort, subtraction, (see calculating with the Pascaline) it is Pascal’s machine which was referred to with reverance down the following centuries. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown (:if equal {Site.PrintBook$:PSW} "True":)in Pascaline woodcut (:ifend:)below. (:if equal {Site.PrintBook$:PSW} "False":)

Morland had already seen a Pascaline, probably when on a diplomatic mission to Queen Christina of Sweden, a supporter of the sciences who in 1649 had been presented with a Pascaline (similar in looks and identical in function to the replica in this collection).⁴²⁹ He had also taken part in a diplomatic mission during which he stopped off at the court of Louis XIV for over a month, so he may well have learned more of the Pascaline, and made contact with scholars and associated mechanics at that time.⁴³⁰ He was thus aware of the potential allure of such inventions. Over the next few years he devised three different calculator designs: an adding machine (see (:if equal {Site.PrintBook$:PSW} "True":)Morland adding machine (:ifend:)below), a multiplying device (see also (:if equal {Site.PrintBook$:PSW} "True":)Morland multiplying instrument (:ifend:)below), and an instrument with moveable arms for determining trigonometric relations. Several examples of Morland’s calculating devices are still in existence, in particular in museums in London and Italy. Not only did Morland design the machines but he wrote a book on the use of the instruments. Published in 1672 this can be considered the first known English Language (and arguably the earliest surviving) ‘computer manual’. (A copy of this rare book, shown below(:if equal {Site.PrintBook$:PSW} "True":) in this table(:ifend:), is in this collection).

The drum had 9 radial splines of incrementally increasing length and could be turned with a crank handle. An accumulating cog could be slid along an axle parallel to the drum and when the drum turned through a full rotation depending on where the cog engaged with the splines of the drum,the cog would be turned by anything from 0 to 9 of the drum’s splines thus accumulating 0 to 9 units of rotation. An example from a C20 calculator is shown (:if equal {Site.PrintBook$:PSW} "True":)in Step drum (:ifend:)below.

As with Moreland’s multiplication machine it was now possible to introduce a carriage which would be mechanically advanced (by means of a second crank handle at the end of a screw thread) to allow multiplication by more than single digit numbers to take place. The Pascaline had had no such mechanism and so the equivalent had had to be done by recording intermediate results with pen and paper. The deficiency in Leibniz’s machine, however, as already indicated, was that unlike the Pascaline, Leibniz had been unable to devise a robust carry mechanism able to handle either multiple carries across several output dials at once. As a result, the machine was far from perfected. It seems that only one was made in 1674 (by M. Olivier, a French clockmaker, working under instruction from Leibniz), that a few years later it was sent Herr Kastner in Göttingen (Lower Saxony) for repairs, and was later stored in an attic at Göttingen University where it was not recovered for 200 years. A diagram of the Leibniz calculator from 1901 is shown (:if equal {Site.PrintBook$:PSW} "True":)in Leibniz wood cut (:ifend:)below, and the actual surviving machine is shown (:if equal {Site.PrintBook$:PSW} "True":)in Surviving Leibniz machine also (:ifend:)below.

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H300.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

As with Moreland’s multiplication machine it was now possible to introduce a carriage which would be mechanically advanced (by means of a second crank handle at the end of a screw thread) to allow multiplication by more than single digit numbers to take place. The Pascaline had had no such mechanism and so the equivalent had had to be done by recording intermediate results with pen and paper. The deficiency in Leibniz’s machine, however, as already indicated, was that unlike the Pascaline, Leibniz had been unable to devise a robust carry mechanism able to handle either multiple carries across several output dials at once. As a result, the machine was far from perfected. It seems that only one was made in 1674 (by M. Olivier, a French clockmaker, working under instruction from Leibniz), that a few years later it was sent Herr Kastner in Göttingen (Lower Saxony) for repairs, and was later stored in an attic at Göttingen University where it was not recovered for 200 years. A diagram of the Leibniz calculator from 1901 is shown in Leibniz wood cut, and the actual surviving machine is shown in Leibniz wood cut below.

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalcH400.jpg|Wood engraved plate from 1901 depicting the Leibniz Calculator (1673)
(collection Calculant)

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁴³⁴ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.⁴³⁵ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown in Pascaline replica below.

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline,⁴³⁶ style ~1650
(collection Calculant)

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁴⁴⁰ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.⁴⁴¹ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany,⁴⁴² and acquired for this collection in 2013, is shown in Pascaline replica below.

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline, style ~1650
(collection Calculant)

The system of “pebble” or “ocular arithmetic” using counters stretched back seemingly into the mists of pre-history. But in the middle of the sixteenth century a process of dramatic change was beginning. It heralded the re-growth of political power in Europe and Britain. Over the next two centuries this would be accompanied by a dynamic development in knowledge, trade and innovation. By the beginning of the eighteenth century the industrial revolution in the “West” would be well underway. Drawing from earlier innovation in the “East” this now outpaced it in terms of economic and political power and technological innovation. It was a time of increasing fascination with new knowledge and mechanisation, continuing rise in scale and economic importance of cities, growth of what became industrial capitalism. Increasing dominance of institutions of the market (including the modern corporation) shaped also new products produced by increasingly sophisticated technological means. In support of this availability of paper and literacy spread. Roman numerals were increasingly replaced by their Arab-Indian counterparts in every-day use, and the reliance on “ocular arithmetic” with counters was on the decline. Developments in mathematics and calculation, although important, were just one component of this remarkable transition.

At the bottom left of the Jeffries print (above) can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.⁴⁴⁴ Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below in French drawing instruments, is from ~1880.

One mathematician - Thomas Hood - a Doctor of Physic from Cambridge University, took as part of his teaching task the design and documentation of potentially useful instruments. One instrument which he describes emerged naturally from the concepts of dividers, and proportional scaling inherent in the proportional compass. This was the sector - a set of dividers, but with flattened legs upon which could be marked various scales.⁴⁴⁶ It represented a considerable advance in aids to calculation.

The use of these rods can be illustrated by, say, multiplying 4 x 89. To do this read the 4 row of the last two rods (rod 8 and 9). Reading from right to left the right most digit in the result is 6 and 3 is carried. Moving left the next digit of the result is 2 to which the carried 3 must be added to give 5, with 3 to be carried. This then gave 356 as the result. This is no more than a simple way of replacing the multiplication tables that otherwise must be memorised. Pen and paper were still required especially if the multiplier was composed of more than one digit (e.g. 34 x 89), so that the process needed to be repeated for each of these (3×89 x10 + 4×89) then adding together these resulting “partial products”.

Over the late nineteenth and first half of the twentieth century a two-way innovation race began to achieve accuracy and capacity on the one hand, and compactness and versatility on the other. Greater accuracy required longer slide rules so they could be more finely divided. To overcome the practical difficulties American bridge engineer Edwin Thacher effectively chopped two 10 metre slide rules into 20 segments and set them side by side around in a cylindrical manner to form an open “cage” with forty scales and an equivalent length of 20 metres. The cage of scales could rotate around a fixed inner cylinder which bore corresponding scales. Compacted in this way the instrument (vii), which was patented in 1881, remained very bulky, but capable of multiplications and divisions to an accuracy of 4 to 5 decimal places. Professor Fuller subsequently developed a cylindrical slide rule (viii) which wrapped the scale around the cylinder in a spiral pattern giving a more compact instrument with a scale of equivalent length 12.7 metres, able to calculate to 3 to 4 decimal places. This was sufficiently practical to go into widespread use.

The above is notable for the extent to which the slide rule, in its multiple variants was able to be shaped into a tool of trade in multiple emerging and growing professions. Its advantage over logarithm tables was its speed of use at the expense of complete accuracy. As noted above, where equivalent accuracy was created the instrument became very large and clumsy.

Through twenty-first century eyes the principle of the machine was simple enough. It was primarily intended for addition, subtraction and multiplication. (Division was possible but difficult with this device).

A deficiency was that the carry mechanism did not allow the process of addition to be reversed. Instead when subtracting a rather clumsy system had to be used of adding ‘complementary numbers’ (where a number -x is represented by (10-x), with 10 subsequently being subtracted). This was assisted by a window shade which can be switched between showing numbers or their complements.

Multiplication tables and various tabluated functions (notably logarithms and trigonometric tables), and various physical embodiments of those notably in various forms of Napier’s rods, sectors and scales would increase in use as need and access to education in them broadened. But could a machine be constructed that would make the mathematical literacy required in the use of these, and traditional methods of arithmetic using pen, paper, and calculi obsolete? Clearly the answer was “not yet”.

The drum had 9 radial splines of incrementally increasing length and could be turned with a crank handle. An accumulating cog could be slid along an axle parallel to the drum and when the drum turned through a full rotation depending on where the cog engaged with the splines of the drum,the cog would be turned by anything from 0 to 9 of the drum’s splines thus accumulating 0 to 9 units of rotation. An example from a C20 calculator is shown in Step drum below.

As the use of calculi reached its zenith they became important in other ways. The example minted in the Netherlands in 1577 shown below (this table) demonstrates that by then, counters were in such general use that they were being used to convey messages to the public - becoming the equivalent of official illustrated pamphlets.⁴⁴⁹ The face on the left (“CONCORDIA. 1577. CVM PIETATE”) translates (loosely) to “Agreement. 1577. With Piety” and seems to carry the visual message that we are joined together (with hearts and hands) under our ruler (the crown) and God (represented by a host). The opposite face (“CALCVLI.ORDINVM.BELGII. Port Salu”) shown on the right) translates to “Calculi. The United Provinces of the Netherlands, Port Salu” and shows a ship (perhaps the ship of state)⁴⁵⁰ coming into Port. (The term “port salu” at the time often simply meant “safe harbour”). It was not a calm time. It was in the midst of the European wars of religion (~1524–1648), and in particular the Eighty Years’ War in the Low Countries (1568–1648). It was only 28 years after Holy Roman Emperor Charles V had proclaimed the Pragmatic Sanction of 1549 which established “the Seventeen Provinces” as a separate entity. (This comprised what with minor exception is now the Netherlands, Belgium and Luxembourg.) Given that there had been a revolt in 1569 (leading to the Southern provinces becoming subject to Spain in 1579), only two years after this calculi was minted, we may presume that the messages of widespread agreement, safe harbour and rule from God might have been intended as a calming political message.

In this drawing, at the top can be seen the emerging shape of various compasses, constructed from brass and steel. Below, shown in Pair of dividers from this collection, is a typical pair of such dividers, also from the eighteenth century:

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810) (collection Calculant)

At the bottom left of the Jeffries print (above) can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.⁴⁵² Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below in French drawing instruments, is from ~1880.

http://meta-studies.net/pmwiki/uploads/HenriMorinW400.jpg|French drawing instruments ~1880 (including proportional dividers- top right)
(collection Calculant)

Sectors were in common use right through to the early twentieth century. Two sectors (from collection Calculant) are shown in Two Sectors below. The first is a Brass French Gunnery Sector from about 1700 by Michael Butterfield, Paris. Michael Butterfield, and English clock maker was born in 1635 and worked in Paris from ~1680 to 1724. The second is an Oxbone Architect’s Sector by T. and H. Doublett who practiced their craft in London around 1830.

It is worth making a couple of observations about these sectors.

In the course of his mathematical ‘hobby’ Napier invented an ingenious system of rods (now known as “Napier’s Rods” or “Napier’s Bones”) which could be manipulated to enable two numbers to be multiplied together with little mental effort, although some manipulation of the rods. An drawing of a set from 1797 is shown in Napier’s Rods below. The mathematical principle had been described by Al-Khwarizmi in the ninth century and had later been brought to Europe by Fibionacci. It involved using a lattice of numbers for multiplication (essentially a way of writing down a multiplication table). But by breaking the columns of the lattice into 10 rods sitting neatly on a board Napier created a calculational aid which was considerably easier (although still somewhat tedious) to use. Napier described this invention in his book Rabdologiae//, published in 1617.⁴⁵⁵

http://meta-studies.net/pmwiki/uploads/Misc/NapiersRodsEnBr2W600.jpg|Depiction of Napier’s Rods, 1797.⁴⁵⁶
(collection Calculant)

In 1625 Wingate published a French edition of Brigg’s latest tables. A year later in 1626, Dennis Henrion published his tables “Traicté de logarithms” (of which there is a copy in this collection, see Logarithms by Henrion below). Together with Wingate’s, these tables made available the power of logarithms across Europe.

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)

Tables of logarithms, improved in various ways over time, were published either by themselves (e.g. Tables by Gardiner, 1783) or in the context of “ready reckoners” (e.g. Ropp’s ready reckoner, 1892). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget. Tables of logarithms by Gardiner in 1783 are shown in Logarithms by Gardiner below.

http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner
(collection Calculant)

In a second section (see below), the book details the design of a logarithmic proportional rule (derived from Gunter’s 1624 publication), along with additional explanations, charts and other elaborations (see Graphical Construction of Gunter Scales below). The proportional rule could be used directly by means of a pair of dividers to measure off lengths corresponding to logarithms and thus to evaluate multiplications and divisions.

Gunter rules were usually equipped with both Gunter’s combination of a logarithmic and a linear scale. Often they also ncluded a range of other scales (notably trigonometric scales for navigational tasks, such as required to work from elapsed time, speed and changes to compass bearing to distances travelled). In addition, important constants could be marked on them as “gauge marks”. Over the seventeenth century Gunter rules gained increasing acceptance and were used right through into the late nineteenth century. Some of the scales of a two foot long mid-nineteenth century navigational Gunter rule in this collection are shown in The Gunter Rule below.

It was Victor Mayer Amédée Mannheim, a Colonel in the French artillery and professor of geometry in Paris, who introduced the now familiar scale system, combined with a now fully functional cursor. This effectively brought together the key elements of what would become the modern slide-rule. He described it in a pamphlet published in 1851⁴⁵⁸ An early slide rule made in 1893–98 by Tavernier-Gravet, but based on a pattern devised by Lenoir in 1814 (iv) is also shown, now fitted with a brass cursor. In (v) is a slide rule from about 1928 by the firm Keuffel and Esser with cursor and familiar scales. The Faber Castell 2/83N Novo Duplex slide rule (xii) with its multiple scales on both sides is often spoken of as the high point reached in elegance and utility of the straight slide rules. Production of it lasted until 1973 when electronic calculators rendered it obsolete. A progression of such slide rule designs is shown in Early Slide Rules below.

The search for compactness and versatility (at the cost of accuracy) continued resulting in a wide variety of circular pocket slide rules. Examples here are the the very first circular slide rule to be introduced in America, Palmer’s Computing Scale (vi), the Supremathic (ix), and the Fowler (x) - which combined a pocket watch style system with multiple scales. The Otis King pocket cylindrical slide rule (xi) was a particularly charming slide rule with its neat spiral scale design but of course less accurate than its bulkier predecessors, being equivalent to 1.7 metres in length. These are shown in Circular Scales below.

For more complex equations with multiple variables it was either train workers to evaluate the equations from first principles or find some other way of solving them. Nomography provided an approach to achieving this. This methodology was invented by Maurice d'Ocagne (1862–1938) in 1880 and was used through to the 1970s. Its principles are described in several good references.⁴⁶¹ A simple example of the approach is given in A simple nomograph below:

http://meta-studies.net/pmwiki/uploads/Misc/nomogram_b=a+dH350.jpg|A simple nomograph for calculating the sum of two numbers (b=a+c)

The above is a nomograph for adding two numbers (one in column a) and the other (in column c).⁴⁶² The properties of similar triangles give the result that a line drawn between the two numbers will cut the middle column (b) at the required sum. Since the outer scales may be logarithmic or trigonometric many more complicated expressions are able to be evaluated using this sort of approach.

Two devices which utilise these nomographic principles are shown below. The first (Bloch Schnellkalulator) is a Bloch Schnellkalulator from ~1924. The second (Der Zeitermittler) is a Zeitermittler from ~1947. In both cases these are nomographic devices for calculating parameters required to cut metal with a lathe. The Bloch Schnellkalulator is notable for its use of a linked mechanism which enables the results of one calculation to be fed as the input to another.

http://meta-studies.net/pmwiki/uploads/Bloch1W350.jpg|Bloch Schnellkalulator
~1924 (collection Calculant)

http://meta-studies.net/pmwiki/uploads/Zeitermittler2W350.jpg|Der Zeitermittler
~1947 (collection Calculant)

It is not known whether a complete Schickard’s “calculating clock” was ever actually built, although a prototype was probably commissioned based on notes to artisans on building the machine, and drawings (see below) as well as comments to Kepler (discussed later). Surviving sketches are shown in this table

The first replica was constructed in 1960 by Professor Bruno, Baron von Freytag-Löringhoff of the University of Tübingen. The one in this collection, in Schickard replica is more recent.⁴⁶⁴

http://meta-studies.net/pmwiki/uploads/Schickard1W300.jpg|Working replica of Schickard’s Calculating Clock (1623)
(collection Calculant)

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁴⁶⁸ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.⁴⁶⁹ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship. A fully working replica of a Pascaline, crafted by Jan Meyer in Germany, and acquired for this collection in 2013, is shown in Pascaline replica below.

http://meta-studies.net/pmwiki/uploads/PA_1W500.jpg|Working replica of a Pascaline,⁴⁷⁰ style ~1650
(collection Calculant)

As to the innovation in mechanism there is no evidence that Pascale had any knowledge of Schickard’s invention. Nevertheless it was clear that he too was infected with the growing understanding that his invention could now be turned to reducing labour in new ways. His Pascaline turned out to be an adding machine, rather like that in the base of Schickard’s clock, but with two important improvements (and one significant deficiency). As with the Schickard’s machine numbers were added by turning disks with a stylus which in turn turned interlinked gears. Pascale however experimented with ways to improve the practical functioning of this. Various details of the Pascaline mechanism, including diagrams in this collection from the famous Encyclopédie ou Dictionnaire raisonn&#233 of Diderot & d’Alembert in 1759 are shown in this table below.

First, drawing firmly on the history of clock design, Pascale introduced gears which were minaturised versions of the “lantern gears” used in tower clocks and mills. These could withstand very great stresses and still operate smoothly. Second, he sought to achieve a carry mechanism which could operate even if many numbers needed to be carried at once (e.g. for 9999999+1 to give 10000000). He finally achieved this with a system where as each gear passed from 0 to 9 a fork-shaped weighted arm (the “sautoir”) on a pivot was slowly rotated upward. As the gear passed from 9 to 0 this weighted arm was then released to drop thrusting an attached lever forward to rotate the gear ahead of it by one unit.

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design. Some eight surviving Pascalines can be found in museums and private collections. He also documented his machine in a short pamphlet,⁴⁷³ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.⁴⁷⁴ Further, even though Schickard’s machine in principle could multiply (using its ingenious incorporation of Napier’s rods), whilst Pascal’s machine (despite his claim that it could be used for all operations of arithmetic) was primarily useful for addition, and with some mental effort, subtraction, (see calculating with the Pascaline) it is Pascal’s machine which was referred to with reverance down the following centuries. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown in Pascaline woodcut below. (:if equal {Site.PrintBook$:PSW} "False":) Other details of the mechanism design are shown here.(:ifend:)

http://meta-studies.net/pmwiki/uploads/Misc/PascalineH150.jpg|Wood engraved plate from 1901 depicting a Pascaline Calculator for accounting (~1642)
(collection Calculant)

It is worth pausing to consider the reasons that Schickard and Pascal had launched into their geared calculator projects, almost certainly with no knowledge of each other’s efforts, being the first known such projects in more than 1500 years. Notably, whilst Schickard included a rendition of Napier’s rods to aid multiplication, and Pascal worked to create a more effective carry mechanism, both in rather similar ways ultimately sought to mechanise addition and subtraction through a set of interconnected gears and dials. Of course there was no single reason, but rather a skein of factors were coming together to make such innovations appealing and therefore more likely.

Morland had already seen a Pascaline, probably when on a diplomatic mission to Queen Christina of Sweden, a supporter of the sciences who in 1649 had been presented with a Pascaline (similar in looks and identical in function to the replica in this collection).⁴⁷⁷ He had also taken part in a diplomatic mission during which he stopped off at the court of Louis XIV for over a month, so he may well have learned more of the Pascaline, and made contact with scholars and associated mechanics at that time.⁴⁷⁸ He was thus aware of the potential allure of such inventions. Over the next few years he devised three different calculator designs: an adding machine (see Morland adding machine below), a multiplying device (see Morland multiplying instrument below), and an instrument with moveable arms for determining trigonometric relations. Several examples of Morland’s calculating devices are still in existence, in particular in museums in London and Italy. Not only did Morland design the machines but he wrote a book on the use of the instruments. Published in 1672 this can be considered the first known English Language (and arguably the earliest surviving) ‘computer manual’. (A copy of this rare book, shown below in this table, is in this collection).

http://meta-studies.net/pmwiki/uploads/Morland1jW380.jpg|Morland Adding Machine
adapted to the then Italian currency
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

http://meta-studies.net/pmwiki/uploads/Morland2jW420.jpg|Morland Multiplying Instrument
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

In essence the adding machine was a simplified adaptation of the Pascaline’s dials (turned by stylus), but without any carry mechanism (so that carrying had to be done by hand). Adding (or subtraction by an opposite rotation) was input through the larger wheels. Each larger wheel engaged with the small wheel above to register a rotation and thus accumulated a 1 to be carried. As with the Pascaline different input wheels were provided for different units (whether units, tens, hundreds, et, or pounds, shillings and pence).

The drum had 9 radial splines of incrementally increasing length and could be turned with a crank handle. An accumulating cog could be slid along an axle parallel to the drum and when the drum turned through a full rotation depending on where the cog engaged with the splines of the drum,the cog would be turned by anything from 0 to 9 of the drum’s splines thus accumulating 0 to 9 units of rotation. An example from a C20 calculator is shown in FIG(step drum) below.

http://meta-studies.net/pmwiki/uploads/Misc/StepDrum.jpeg|Twentieth Century step drum following the same principle as Leibniz’s conception
⁴⁸⁰

Several such accumulating cogs with their drums were put side by side, corresponding to units, tens, hundreds, etc, and once a number was set (e.g. 239) it could be multiplied (e.g. 3) by turning the crank through a full rotation the corresponding number (eg 3) of times (adding 239 to itself 3 times to give 717). The capacity to add a multi digit number to itself multiple times gave the machine the capacity to multiply, which was a considerable advance over the Pascaline. Further, the crank could be rotated in the opposite direction to produce a subtraction.

As with Moreland’s multiplication machine it was now possible to introduce a carriage which would be mechanically advanced (by means of a second crank handle at the end of a screw thread) to allow multiplication by more than single digit numbers to take place. The Pascaline had had no such mechanism and so the equivalent had had to be done by recording intermediate results with pen and paper. The deficiency in Leibniz’s machine, however, as already indicated, was that unlike the Pascaline, Leibniz had been unable to devise a robust carry mechanism able to handle either multiple carries across several output dials at once. As a result, the machine was far from perfected. It seems that only one was made in 1674 (by M. Olivier, a French clockmaker, working under instruction from Leibniz), that a few years later it was sent Herr Kastner in Göttingen (Lower Saxony) for repairs, and was later stored in an attic at Göttingen University where it was not recovered for 200 years. A diagram of the Leibniz calculator from 1901 is shown in Leibniz wood cut, and the actual surviving machine is shown in Leibniz wood cut below.

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalcH400.jpg|Wood engraved plate from 1901 depicting the Leibniz Calculator (1673)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H300.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

Even by the late sixteenth century in practice methods of calculation had not changed much since the Roman Empire. Indicative of this “calculi” (Latin: “calculus” for “pebble” (or “limestone”) - the origin of the word “calculation”) were still being struck (as they had been for centuries) to facilitate counting and arithmetic especially in many fields of commerce. They were referred to differently in different places. One such - a French “jeton” from 1480–1520 is shown in French jeton below. It is made to look like a coin, but its inscription is meaningless.

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁴⁸³ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut (Contested methods) from 1503,⁴⁸⁴ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

http://meta-studies.net/pmwiki/uploads/Misc/GregorReisch1504H400.jpg|Contested methods - Woodcut from 1503

As the use of calculi reached its zenith they became important in other ways. The example minted in the Netherlands in 1577 shown below (this figure) demonstrates that by then, counters were in such general use that they were being used to convey messages to the public - becoming the equivalent of official illustrated pamphlets.⁴⁸⁷ The face on the left (“CONCORDIA. 1577. CVM PIETATE”) translates (loosely) to “Agreement. 1577. With Piety” and seems to carry the visual message that we are joined together (with hearts and hands) under our ruler (the crown) and God (represented by a host). The opposite face (“CALCVLI.ORDINVM.BELGII. Port Salu”) shown on the right) translates to “Calculi. The United Provinces of the Netherlands, Port Salu” and shows a ship (perhaps the ship of state)⁴⁸⁸ coming into Port. (The term “port salu” at the time often simply meant “safe harbour”). It was not a calm time. It was in the midst of the European wars of religion (~1524–1648), and in particular the Eighty Years’ War in the Low Countries (1568–1648). It was only 28 years after Holy Roman Emperor Charles V had proclaimed the Pragmatic Sanction of 1549 which established “the Seventeen Provinces” as a separate entity. (This comprised what with minor exception is now the Netherlands, Belgium and Luxembourg.) Given that there had been a revolt in 1569 (leading to the Southern provinces becoming subject to Spain in 1579), only two years after this calculi was minted, we may presume that the messages of widespread agreement, safe harbour and rule from God might have been intended as a calming political message.

(:table align=center cellpadding=5 border=0 id=Calculi"Calculi":)

The measurement of distance had been aided by dividers (or “compasses”) at least since the Roman era and now instruments were developed building on that idea. Compasses below is a print of a drawing by Thomas Jefferys (~1710–1771) which shows a range of such instruments from the C18.

http://meta-studies.net/pmwiki/uploads/JefferysPrintH500.jpg|1710–71: Compasses by T. Jefferys sculp (Collection Calculant)

Kepler was greatly excited by the publication of logarithms which he utilised to greatly reduce the enormous calculational effort in his computations of The Rudolphine Tables and later works. He dedicated his Ephemerides to Napier in 1617.⁴⁹² Professor Briggs (astronomer and geometrician) now at Gresham College, Cambridge, as mentioned earlier was a close colleague of Napier and had written of his tables “…I never saw a book which pleased me better or made me more wonder”.⁴⁹³ He would later take Napier’s work further forward producing new tables of logarithms to base 10. In 1609 Briggs was also impatiently awaiting Kepler’s exposition on ellipses.⁴⁹⁴

Kepler was greatly excited by the publication of logarithms which he utilised to greatly reduce the enormous calculational effort in his computations of The Rudolphine Tables and later works. He dedicated his Ephemerides to Napier in 1617.⁴⁹⁸ Professor Briggs (astronomer and geometrician) now at Gresham College, Cambridge, as mentioned earlier was a close colleague of Napier and had written of his tables “…I never saw a book which pleased me better or made me more wonder”.⁴⁹⁹ He would later take Napier’s work further forward producing new tables of logarithms to base 10. In 1609 Briggs was also impatiently awaiting Kepler’s exposition on ellipses.⁵⁰⁰

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁵⁰³ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut from 1503,⁵⁰⁴ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁵⁰⁷ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut from 1503,⁵⁰⁸ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁵¹¹ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut from 1503, below,⁵¹² where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁵¹⁵ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut from 1503, below,⁵¹⁶ where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

http://meta-studies.net/pmwiki/uploads/Misc/GregorReisch1504H400.jpg|Contested methods - Woodcut from 1503

One group of such intellectuals were the Jesuit theologians who were now emerging as mathematical thinkers. By 1650 some 50 mathematical chairs had emerged in Jesuit colleges across Europe.⁵¹⁸ It was Joannes Ciermans (1602–1648), a Flemish Jesuit, who in 1641 published one of the most comprehensive surviving courses covering geometry, arithmetic and optics. These were presented in a practical way, designed for his students who were mostly expected to become military officers.

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H300.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H300.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University

(:if equal {Site.PrintBook$:PSW} "False":)

(:ifend:)

(:if equal {Site.PrintBook$:PSW} "True":) http://meta-studies.net/pmwiki/uploads/Misc/LeibnizCalc2H200.jpg|Surviving Leibniz Calculator recovered from the attic of Göttingen University (:ifend:)

In a second section (see below), the book details the design of a logarithmic proportional rule (derived from Gunter’s 1624 publication), along with additional explanations, charts and other elaborations. The proportional rule could be used directly by means of a pair of dividers to measure off lengths corresponding to logarithms and thus to evaluate multiplications and divisions.

Gunter rules were usually equipped with both Gunter’s combination of a logarithmic and a linear scale. Often they also ncluded a range of other scales (notably trigonometric scales for navigational tasks, such as required to work from elapsed time, speed and changes to compass bearing to distances travelled). In addition, important constants could be marked on them as “gauge marks”. Over the seventeenth century Gunter rules gained increasing acceptance and were used right through into the late nineteenth century. Some of the scales of a two foot long mid-nineteenth century navigational Gunter rule in this collection are shown below.

The initial insight was that the logarithms of two numbers could be added by sliding two Gunter scales against each other. There is however debate about who was the first to realise this.⁵²² It was William Oughtred who published his design for a slide rule in 1632.

A series of designs followed. Three of these appear in Jacob Leupold’s book Theatrum Arithmetico-Geometricum⁵²³ of which Table XII (page 241), held in this collection, is shown in (i) in the Table below. (The top drawing is a design for a Gunter scale, and the two below are early designs for slide rules).⁵²⁴

With its multiple scales (including the vital logarithmic scale) the Gunter rule was a lot more flexible in use than a Sector. It was much easier to make a rough calculation using it. All that was necessary was to measure off and add with dividers lengths against the various scales. This was much easier than having to write down the intermediate results. It was not long however before it was realised that instead of using dividers to add these lengths, the same thing could be achieved more easily by sliding two scales against each other.

John Robertson, in 1755, described the method thus: “To take a diftance between the points of the compaffes. Hold the compaffes upright, fet one point on one end of the diftance to be taken, there let it reft; and (as before fhewn) extend the other point to the other end.”⁵²⁶

John Robertson, in 1755, described the method thus: “To take a diſtance between the points of the compaſſes. Hold the compaſſes upright, ſet one point on one end of the diſtance to be taken, there let it reſt; and (as before ſhewn) extend the other point to the other end.”⁵²⁸

(:if equal {Site.PrintBook$:PSW} "False":) John Robertson, in 1755, described the method thus: “To take a diſtance between the points of the compaſſes. Hold the compaſſes upright, ſet one point on one end of the dſtance to be taken, there let it reſt; and (as before ſhewn) extend the other point to the other end.”⁵²⁹ (:ifend:)

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John Robertson, in 1755, described the method thus: “To take a di∫tance between the points of the compa∫∫es. Hold the compa∫∫es upright, ∫et one point on one end of the di∫tance to be taken, there let it re∫t; and (as before ∫hewn) extend the other point to the other end.”⁵³¹

John Robertson, in 1755, described the method thus: “To take a diſtance between the points of the compaſſes. Hold the compaſſes upright, ſet one point on one end of the diſtance to be taken, there let it reſt; and (as before ſhewn) extend the other point to the other end.”⁵³³

John Robertson, in 1755, described the method thus: “To take a diſtance between the points of the compaſſes. Hold the compaſſes upright, ſet one point on one end of the diſtance to be taken, there let it reſt; and (as before ſhewn) extend the other point to the other end.”⁵³⁵

(:ifend:)

A gulf still stood between the discoveries by mathematicians and other intellectuals, and the many others to whom this work could be of practical assistance. On the one side of that gulf, reflecting upon these early intellectual innovators, was the long-standing idea that a man of elevated (or aristocratic) heritage - a “gentleman” or in France “un honnête homme” - would consider it demeaning (as would an ancient Greek or Roman of standing some 1500 years before) to lower himself to associate himself with practical work. On the other side, amongst those whose life was devoted to practical work (for example, artisans) a parallel image was common, of the impractical nature of the gentleman mathematician and the products of mathematical thinking. This gulf was a significant obstacle to the new technical insights being utilised.

The need for more accurate maps added to the demand for simpler ways of carrying out calculations. The defeat of the Armada at Gravelines during the (undeclared) Anglo-Spanish War of 1585–1604 provided a telling lesson in the sixteenth century of the importance of not only the economic power of merchant shipping, but also of the military importance of naval power. In particular it reinforced the need for manoeuvrable naval ships effectively utilising the best available gunnery.⁵³⁷

John Napier (1550–1617), Eighth Lord of Merchiston, was an imposing intellectual of his time. He pursued interests in astrology, theology, magic, physics and astronomy, methods of agriculture, and mathematics.⁵⁴⁰ If these seem a strange set of areas, it is to be remembered, as Stephen Snobelen reminds us of Newton also that few people now: “…have an understanding of what an intellectual cross-road the early modern period was. In fact, we now know that Newton was in many ways a Renaissance man, working in theology, prophecy and alchemy, as well as mathematics, optics and physics. In short, neither Napier nor Newton who came after him was “a scientist in the modern sense.”⁵⁴¹

More importantly, after 20 years of work Napier also devised and had printed in 1614, a set of tables, Mirifici Logarithmorum Canonis Descriptio (“Description of the Marvellous Rule of Logarithms”). These enabled direct multiplication to be carried out through simple addition. It is from these that modern tables of logarithms (with some additional conceptual improvements) are derived. These are based on the fact that the powers of numbers add when the numbers are multiplied (i.e. 2³x2⁴=2⁷). This fact had been known since the time of Archimedes. But it was Napier who thought to use this sort of property to tabulate numbers in terms of the power of some “base” (2 in this example). In fact, Napier did not achieve his tables quite in this way. Rather it is believed he started from an earlier approach using a property from trigonometry.⁵⁴⁵ This led him to a geometric argument based on the theory of proportions. Using this he constructed his functions.⁵⁴⁶

Napier’s tables were welcomed by mathematically minded scholars across Europe. Indeed the astronomer Keppler (1571–1630) noted that his ground breaking calculations in relation to Tycho Brahe’s astronomical observations would have been impossible without the use of the tables.⁵⁴⁷

It was Victor Mayer Amédée Mannheim, a Colonel in the French artillery and professor of geometry in Paris, who introduced the now familiar scale system, combined with a now fully functional cursor. This effectively brought together the key elements of what would become the modern slide-rule. He described it in a pamphlet published in 1851⁵⁴⁹ An early slide rule made in 1893–98 by Tavernier-Gravet, but based on a pattern devised by Lenoir in 1814 (iv) is also shown, now fitted with a brass cursor. In (v) is a slide rule from about 1928 by the firm Keuffel and Esser with cursor and familiar scales. The Faber Castell 2/83N Novo Duplex slide rule (xii) with its multiple scales on both sides is often spoken of as the high point reached in elegance and utility of the straight slide rules. Production of it lasted until 1973 when electronic calculators rendered it obsolete.

For more complex equations with multiple variables it was either train workers to evaluate the equations from first principles or find some other way of solving them. Nomography provided an approach to achieving this. This methodology was invented by Maurice d'Ocagne (1862–1938) in 1880 and was used through to the 1970s. Its principles are described in several good references.⁵⁵¹ A simple example of the approach is given in (i) below:

The first European ‘scientific journal’ (Le Journal des sçavans - later Le Journal des savants) published its first edition on 5 January 1665. Once established, in 1666, it soon became the written forum for the Paris Academy of Sciences.⁵⁵⁶ (In England, the Philosophical Transactions of the Royal Society began publishing only three months later in March 1666.) Prior to that (indeed after, in many places) news of discoveries was spread by letters sent to trusted figures in the small ‘invisible college’ of thinkers. Thus, for example, William Oughtred (first to publish about the sliding part of the slide rule) was one of the key contact points in England. Others would learn of developments in the popular seminars he delivered at his home.⁵⁵⁷ The network was thus characterised by a shared excitement and objectives, a developing communication system, the stimulus of possible positive outcomes, and consequently a level of competition (as revealed in the often quite tense and hard fought battles beginning to rage over who had primacy in any particular insight or invention).

Blaise Pascale (1623–1662) had been only a 13 years old, when Schickard died but had already showed both a strong interest and flair for mathematics. His father Etienne, was a reasonably good mathematician in his own right, of noble birth and reasonably well to do, and when Blaise was 19 years old, with an appointment by Cardinal Richelieu to “Commissioner deputed by His Majesty for the Imposition and Collection of Taxes in Upper Normandy”.⁵⁶⁰ The pressure of work involved was enormous and Blaise was needed to help out in the extraordinary effort involved in adding seemingly endless numbers as taxes were calculated, collected, paid and audited. As a consequence he turned what would prove to be his brilliant mathematical mind to the more practical task of constructing a mechanism to assist in addition. One of his heirs (chevalier Durant Pascal) would later claim that finding no artisan clever enough, Pascal himself trained himself produced one of the machines with a few tools. However, this is very likely false. The clock-maker’s guild was well established in Rouen, clock-makers had all the necessary tools, and further had an exclusive right to make any machine like a clock.⁵⁶¹ In any case, Pascal definitely did seek to wring the very best out the local artisans, and in this he is considered to have been very successful, producing machines which were not only useable, but works of of the finest craftsmanship.

Unlike Schickard’s machine which is known only by documents, and which in reality would have had practical problems with the carry mechanism, Pascal sought to find profit from his invention. According to him he created some 50 of his calculators of varying design. Some eight surviving Pascalines can be found in museums and private collections. He also documented his machine in a short pamphlet,⁵⁶⁶ achieved what in modern terms would be called a patent for his invention - a right awarded by the monarch to exclusive production of the invention (thereby over-riding the clock-maker guild’s claim to this) - and commissioned an agent (Prof. Gilles de Roberval) to sell it.⁵⁶⁷ Further, even though Schickard’s machine in principle could multiply (using its ingenious incorporation of Napier’s rods), whilst Pascal’s machine (despite his claim that it could be used for all operations of arithmetic) was primarily useful for addition, and with some mental effort, subtraction, (see calculating with the Pascaline) it is Pascal’s machine which was referred to with reverance down the following centuries. A woodcut from 1901 of an accounting Pascaline, similar to that in the Léon Parcé collection, is shown below. Other details of the mechanism design are shown here.

Consistent with this, many of Pascal’s machines would end up, not in the hands of practitioners of mathematically intense duties, but as curiosities on the shelves and in the cabinets of persons of eminence. The names indicating the provenance of some of the surviving Pascalines - Queen of Sweden, Chancelier Séguier, Queen of Poland, Chevalier Durant-Pascal - are consistent with this. But perhaps equally so is the beautiful workmanship and decorative working of materials which are characteristic of these instruments.

In a fascinating thesis,⁵⁷⁵ and subsequent published book chapter,⁵⁷⁶ Jean-François Gauvin develops a multi-stranded analysis of the role of scientific instruments, including the Pascaline, and their creators and use in the seventeenth century. Key to this are conflicts and resonances between continuities in cultural habit, and social, philosophical and ideological challenges to them that were beginning to gain force at the time.

Challenges such as Bacon’s Novum Organum reflected the beginnings of a discomfort with knowledge founded on the verities of ancient Greek knowledge, church orthodoxy and aristocratic tradition. Of course both Schickard and Pascal, sought and enjoyed the indulgence and support of aristocratic and religious sponsors and Pascal in particular would use his machines as much as a way of attracting that as for any commercial gain. But the transformation reflected in Bacon’s call was to progressively reposition the man of substance (a “gentleman”, or in France “l'honnête homme”)⁵⁷⁷ in relation to some tools such as these. Building on the existing prestige of clockwork and the growing understanding that intellectual advance was tied to mechanical engagement with the natural world, these devices could be seen to potentially restore the separation of the intellectual world of l'honnête homme from the drudgery of the ‘mechanical’ act of routine calculation.

Having said that, the machines were expensive (about of a third of a year’s average wages of the time),⁵⁷⁸. Pascale in his surviving “Advice” on the use of the Pascaline promised that the machine could “perform without any effort whatsoever all the arithmetical operations that had so often worn out one's mind by means of the plume and the jetons”.⁵⁷⁹ But in truth the use of it required a good knowledge of arithmetic to be used effectively (especially for division and multiplication, but even for subtraction). At best it was a mechanical substitute for what otherwise would have to be written down, save that it could perform addition with a little practice, and simple subtraction (where the result was not negative) with a little more. As Balthazaar Gerbier described the Pascaline in a letter to Samuel Hartlib in 1648:

But like rare books need not be read to add luster to the shelf, the machines needed not necessarily be used to add something to the owner. Basic mathematical skills were far from universally held, even amongst the nobility, and certainly amongst gentlemen. From this point of view, even if one could not add and subtract reliably in one’s head, let alone multiply using jeton (calculi) and pen, the potential allure of such a calculating machine was that by owning it, one could come to at least be seen to have an intimacy with such literacy. In these senses, as Gauvin argues, “Even though Pascal invented the machine to alleviate his father’s headaches as a royal tax collector, financiers and merchants, who tallied large amount of numbers, were not especially in Pascal’s mind. The pascaline was more than a mechanical contraption useful for business: used properly, it could bestow l'honnête.”⁵⁸¹

In the course of these events Morland, whilst certainly comfortably provided for (not the least after he married a Baroness), complained of the failure of his positions to yield wealth saying: 'Now finding myself disappoynted of all preferment and of any real estate, I betook myself too the Mathematicks, and Experiments such as I found pleased the King's Fancy.'⁵⁸⁴ On the basis of his mastery of engineering (in particular, fluid systems) Morland was eventually granted a newly created position of “Master of Mechanicks” to his the King, and later was made a gentleman of his Majesty’s privy chamber.⁵⁸⁵ His inventions, thus played a two-fold role, first in his search to establish a role for himself in the context of the new Court, and second to supplement his wealth by seeking commercial success. In particular, he hoped for a lucrative outcome to devising mathematical instruments and selling them.

http://meta-studies.net/pmwiki/uploads/Misc/StepDrum.jpeg|Twentieth Century step drum following the same principle as Leibniz’s conception
⁵⁸⁷

(:ifend:)

Sometimes these calculi (however named, and in whatever form they took) would be “cast” on a “counting board”. Sometimes they would be simply piled up to aid the process of accounting or calculation. Their use petered out in England over the C18 and similarly in France ending with the revolution (1789).⁵⁸⁹ There was in particular a contest between the incoming technology of pen and paper using arabic numerals, and calculi and counting board based on Roman numerals. This contest is shown in the following woodcut from 1503 where, presided over by Dame Arithmetic, on the left Pythagoras is using the new technology, whilst on the right the ancient Roman mathematician Boethius is calculating with counting board and calculi.

The system of “pebble” or “ocular arithmetic” using counters stretched back seemingly into the mists of pre-history. But in the middle of the sixteenth century a process of dramatic change was beginning. It heralded the re-growth of political power in Europe and Britain. Over the next two centuries this would be accompanied by a dynamic development in knowledge, trade and innovation. By the beginning of the eighteenth century the industrial revolution in the “West” would be well underway. Drawing from earlier innovation in the “East” this now outpaced it in terms of economic and political power and technological innovation. It was a time of increasing fascination with new knowledge and mechanisation, continuing rise in scale and economic importance of cities, growth of what became industrial capitalism. Increasing dominance of institutions of the market (including the modern corporation) shaped also new products produced by increasingly sophisticated technological means. In support of this availability of paper and literacy spread. Roman numerals were increasingly replaced by their Arab-Indian counterparts in every-day use, and the reliance on “ocular arithmetic” with counters was on the decline. Developments in mathematics and calculation, although important, were just one component of this remarkable transition.

At the bottom left of the Jeffries print (above) can be seen another device - a pair of dividers with a moveable central pivot. This instrument created a means of drawing distances to a particular scale. With one end marking an actual distance, the other could be reduced by a desired ratio by moving the position of the central pivot. An early reference to such a “proportional compass” was made by architect Daniel Speckle, as early as 1589.⁵⁹¹ Dividers provided just part of the spectrum of mathematical tools that were under steady development. By the C18 such instruments were being produced in sets which included dividers, protractors, increasingly precisely scribed rulers, finer drawing pens, and other instruments, such as the proportional compass. The set of instruments, below, is from ~1880.

One mathematician - Thomas Hood - a Doctor of Physic from Cambridge University, took as part of his teaching task the design and documentation of potentially useful instruments. One instrument which he describes emerged naturally from the concepts of dividers, and proportional scaling inherent in the proportional compass. This was the sector - a set of dividers, but with flattened legs upon which could be marked various scales.⁵⁹³ It represented a considerable advance in aids to calculation.

It is worth making a couple of observations about these sectors.

The use of these rods can be illustrated by, say, multiplying 4 x 89. To do this read the 4 row of the last two rods (rod 8 and 9). Reading from right to left the right most digit in the result is 6 and 3 is carried. Moving left the next digit of the result is 2 to which the carried 3 must be added to give 5, with 3 to be carried. This then gave 356 as the result. This is no more than a simple way of replacing the multiplication tables that otherwise must be memorised. Pen and paper were still required especially if the multiplier was composed of more than one digit (e.g. 34 x 89), so that the process needed to be repeated for each of these (3×89 x10 + 4×89) then adding together these resulting “partial products”.

Tables of logarithms, improved in various ways over time, were published either by themselves (e.g. Tables by Gardiner, 1783) or in the context of “ready reckoners” (e.g. Ropp’s ready reckoner, 1892). Even up until the late C20 most school children were expected to have a passing understanding of how to use such logarithms prior to graduating to adult work. Extraordinarily useful as tables of logarithms were, a certain level of skill was required to use them. As many later in life would find, that skill also was not hard to forget.

Gunter rules were usually equipped with both Gunter’s combination of a logarithmic and a linear scale. Often they also ncluded a range of other scales (notably trigonometric scales for navigational tasks, such as required to work from elapsed time, speed and changes to compass bearing to distances travelled). In addition, important constants could be marked on them as “gauge marks”. Over the seventeenth century Gunter rules gained increasing acceptance and were used right through into the late nineteenth century. Some of the scales of a two foot long mid-nineteenth century navigational Gunter rule in this collection are shown below.

With its multiple scales (including the vital logarithmic scale) the Gunter rule was a lot more flexible in use than a Sector. It was much easier to make a rough calculation using it. All that was necessary was to measure off and add with dividers lengths against the various scales. This was much easier than having to write down the intermediate results. It was not long however before it was realised that instead of using dividers to add these lengths, the same thing could be achieved more easily by sliding two scales against each other.

Over the late nineteenth and first half of the twentieth century a two-way innovation race began to achieve accuracy and capacity on the one hand, and compactness and versatility on the other. Greater accuracy required longer slide rules so they could be more finely divided. To overcome the practical difficulties American bridge engineer Edwin Thacher effectively chopped two 10 metre slide rules into 20 segments and set them side by side around in a cylindrical manner to form an open “cage” with forty scales and an equivalent length of 20 metres. The cage of scales could rotate around a fixed inner cylinder which bore corresponding scales. Compacted in this way the instrument (vii), which was patented in 1881, remained very bulky, but capable of multiplications and divisions to an accuracy of 4 to 5 decimal places. Professor Fuller subsequently developed a cylindrical slide rule (viii) which wrapped the scale around the cylinder in a spiral pattern giving a more compact instrument with a scale of equivalent length 12.7 metres, able to calculate to 3 to 4 decimal places. This was sufficiently practical to go into widespread use.

The above is notable for the extent to which the slide rule, in its multiple variants was able to be shaped into a tool of trade in multiple emerging and growing professions. Its advantage over logarithm tables was its speed of use at the expense of complete accuracy. As noted above, where equivalent accuracy was created the instrument became very large and clumsy.

The above is a nomograph for adding two numbers (one in column a) and the other (in column c).⁵⁹⁵ The properties of similar triangles give the result that a line drawn between the two numbers will cut the middle column (b) at the required sum. Since the outer scales may be logarithmic or trigonometric many more complicated expressions are able to be evaluated using this sort of approach.

Through twenty-first century eyes the principle of the machine was simple enough. It was primarily intended for addition, subtraction and multiplication. (Division was possible but difficult with this device).

A deficiency was that the carry mechanism did not allow the process of addition to be reversed. Instead when subtracting a rather clumsy system had to be used of adding ‘complementary numbers’ (where a number -x is represented by (10-x), with 10 subsequently being subtracted). This was assisted by a window shade which can be switched between showing numbers or their complements.

It is worth pausing to consider the reasons that Schickard and Pascal had launched into their geared calculator projects, almost certainly with no knowledge of each other’s efforts, being the first known such projects in more than 1500 years. Notably, whilst Schickard included a rendition of Napier’s rods to aid multiplication, and Pascal worked to create a more effective carry mechanism, both in rather similar ways ultimately sought to mechanise addition and subtraction through a set of interconnected gears and dials. Of course there was no single reason, but rather a skein of factors were coming together to make such innovations appealing and therefore more likely.

In essence the adding machine was a simplified adaptation of the Pascaline’s dials (turned by stylus), but without any carry mechanism (so that carrying had to be done by hand). Adding (or subtraction by an opposite rotation) was input through the larger wheels. Each larger wheel engaged with the small wheel above to register a rotation and thus accumulated a 1 to be carried. As with the Pascaline different input wheels were provided for different units (whether units, tens, hundreds, et, or pounds, shillings and pence).

Several such accumulating cogs with their drums were put side by side, corresponding to units, tens, hundreds, etc, and once a number was set (e.g. 239) it could be multiplied (e.g. 3) by turning the crank through a full rotation the corresponding number (eg 3) of times (adding 239 to itself 3 times to give 717). The capacity to add a multi digit number to itself multiple times gave the machine the capacity to multiply, which was a considerable advance over the Pascaline. Further, the crank could be rotated in the opposite direction to produce a subtraction.

Multiplication tables and various tabluated functions (notably logarithms and trigonometric tables), and various physical embodiments of those notably in various forms of Napier’s rods, sectors and scales would increase in use as need and access to education in them broadened. But could a machine be constructed that would make the mathematical literacy required in the use of these, and traditional methods of arithmetic using pen, paper, and calculi obsolete? Clearly the answer was “not yet”.

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It was Victor Mayer Amédée Mannheim, a Colonel in the French artillery and professor of geometry in Paris, who introduced the now familiar scale system, combined with a now fully functional cursor. This effectively brought together the key elements of what would become the modern slide-rule. He described it in a pamphlet published in 1851⁵⁹⁷ An early slide rule made in 1893–98 by Tavernier-Gravet, but based on a pattern devised by Lenoir in 1814 (iv) is also shown, now fitted with a brass cursor. In (v) is a slide rule from about 1928 by the firm Keuffel and Esser with cursor and familiar scales. The Faber Castell 2/83N Novo Duplex slide rule (xii) with its multiple scales on both sides is often spoken of as the high point reached in elegance and utility of the straight slide rules. Production of it lasted until 1973 when electronic calculators rendered it obsolete.

(xii)||1967–73||Faber Castell
2/83N
Novo
Duplex
|| http://meta-studies.net/pmwiki/uploads/FaberCastellDuplexW300.jpg
(All the above are from collection Calculant)||

Over the late nineteenth and first half of the twentieth century a two-way innovation race began to achieve accuracy and capacity on the one hand, and compactness and versatility on the other. Greater accuracy required longer slide rules so they could be more finely divided. To overcome the practical difficulties American bridge engineer Edwin Thacher effectively chopped two 10 metre slide rules into 20 segments and set them side by side around in a cylindrical manner to form an open “cage” with forty scales and an equivalent length of 20 metres. The cage of scales could rotate around a fixed inner cylinder which bore corresponding scales. Compacted in this way the instrument (vii), which was patented in 1881, remained very bulky, but capable of multiplications and divisions to an accuracy of 4 to 5 decimal places. Professor Fuller subsequently developed a cylindrical slide rule (viii) which wrapped the scale around the cylinder in a spiral pattern giving a more compact instrument with a scale of equivalent length 12.7 metres, able to calculate to 3 to 4 decimal places. This was sufficiently practical to go into widespread use.

The search for compactness and versatility (at the cost of accuracy) continued resulting in a wide variety of circular pocket slide rules. Examples here are the the very first circular slide rule to be introduced in America, Palmer’s Computing Scale (vi), the Supremathic (ix), and the Fowler (x) - which combined a pocket watch style system with multiple scales. The Otis King pocket cylindrical slide rule (xi) was a particularly charming slide rule with its neat spiral scale design but of course less accurate than its bulkier predecessors, being equivalent to 1.7 metres in length.

http://meta-studies.net/pmwiki/uploads/K&E1908W300.jpg||

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810)
(collection Calculant)

//

//

http://meta-studies.net/pmwiki/uploads/Morland2jW420.jpg|Morland Multiplying Instrument
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

http://meta-studies.net/pmwiki/uploads/Morland1jW380.jpg|Morland Adding Machine
adapted to the then Italian currency
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

width=420px% http://meta-studies.net/pmwiki/uploads/Morland2jW420.jpg|Morland Multiplying Instrument
(Istituto e Museo di Storia della Scienza, Florence - Photos by Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)

In 1625 Wingate published a French edition of Brigg’s latest tables. A year later in 1626, Dennis Henrion published his tables “Traicté de logarithms” (of which there is a copy in this collection). Together with Wingate’s, these tables made available the power of logarithms across Europe.

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)
http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/HenrionH300.jpg|1626: Traicté de logarithms by Dennis Henrion
(collection Calculant)
http://meta-studies.net/pmwiki/uploads/GardinerH300.jpg|1783: Tables Portatives de Logarithms by Gardiner
(collection Calculant)

(:ifend:)

http://meta-studies.net/pmwiki/uploads/HenriMorinW400.jpg|French drawing instruments ~1880 (including proportional dividers- top right)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Jeton1480H200.jpg|Sides 1 & 2, French jeton 1480–1520
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/GregorReisch1504H400.jpg|Contested methods - Woodcut from 1503⁶¹⁷

(:cellnr:) http://meta-studies.net/pmwiki/uploads/Calculi2H200.jpg (:cell:) http://meta-studies.net/pmwiki/uploads/Calculi1H200.jpg

http://meta-studies.net/pmwiki/uploads/DividersW200.jpg|Two pairs of English eighteenth century dividers (~1740–1810)
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/Misc/NapiersRodsEnBr2W600.jpg|Depiction of Napier’s Rods, 1797.⁶¹⁹
(collection Calculant)

http://meta-studies.net/pmwiki/uploads/GunterHenrionW500.jpg|Graphical construction of Gunter scale (1624)