The social life of precision instruments: artisans’ trials in early-modern England, 1550–1700

ABSTRACT This paper examines the role of mathematical instrument makers in establishing a public culture of precision measurement in early-modern England. I argue that this culture was promoted through trials and demonstrations, in the context of which artisans held a privileged position. The trials described here cover land surveying, the measurement of magnetic variation, and standards of measurement for customs and excise. These trials were decisive moments in the ‘cultural biographies’ of precision instruments. I ask how it was that instrument makers were able to assume positions of authority, and what this means for our understanding of the socio-material system of precision measurement in the early-modern period, and the contemporary rise of ‘experimental’ and ‘philosophical’ trials. Because practical mathematics was a self-consciously economic activity – motivated by trade, commerce, exploration and colonization – direct connections to natural philosophy are significant, a point explored in my conclusion.

itself. 3 In the decades following Wright's death in 1797, collective endeavours in the pursuit of science were to fragment and move into the laboratory, field, lecture hall and learned society. 4'Collective empiricism' remained the paradigm, but the audience of what William Whewell called 'spectators and amateurs' feeling themselves 'on a level' with 'profound thinkers' was replaced with highly trained research workersno less concerned with norms of experimental practice, but occupying entirely different physical places and social spaces. 5More recently, access to equipment has come to dominate the physical sciences, and in a 'post-genomic' age the same is now true of biology. 6Technologies of remote sensing and the management of disparate networks of observers are also recent trends, albeit with histories that go back to the Baconian programme. 7Knowledge, we are to understand, is a thing that is created when certain groups of people place their trust in certain kinds of devices. 8n this image of science, instruments themselves too often 'simply are', to borrow Igor Kopytoff's phrase, used by him to describe the treatment of commodities by economic theorists. 9One reason for the apparent self-sufficiency of instruments is that instrumental techniques can often look like they succeeded 'naturally', by dint of the superiority of measurement over other ways of knowingfor instance traditional methods of surveying by using landmarks as boundaries, or mapmaking with scales of significance rather than for mathematical proportion, both of which were replaced with forms of measurement in the early modern period. 10Because of this, the social structures of legitimation, meaning, and use are too often implicitly or explicitly portrayed as separate from the function of specific instruments.This problem is particularly acute for precision instrumentation, because precision is taken to be an objective measure of performance of objects rather than a result of social processes. 11opytoff's response to the problem of reification was to propose that we use 'cultural biographies' of objects as they move into and out of commoditization, in order to build up a picture of the 'social life of things'. 12A 'theoretically aware' form of biography will, for Kopytoff, be based on 'the range of biographical possibilities that the society in question offers'. 13To consider the 'range of biographical possibilities' afforded to instruments is to place them into narratives of justification and polemics of performance.This approach has already been promoted in studies by Stephen Johnston, Jim Bennett and others.They have argued that to understand the adoption of specific instruments we need sophisticated accounts of why techniques that seem obvious to us were in fact taken up: Whose opinions were trusted?How were new 7 Lorraine Daston, 'On Scientific Observation', Isis, 99 (2008), 97-110.8   Noting that books can function as instruments as well; see Boris Jardine, 'The Book as Instrument: Craft and Technique in Early Modern Practical Mathematics', BJHS Themes, 5 (2020), 111-29.9 Igor Kopytoff, 'The Cultural Life of Things: Commoditization as a Process', in The Social Life of Things: Commodities in Cultural Perspective, ed.Arjun Appadurai (Cambridge: Cambridge University Press, 1986), pp.64-91, see p. 64. 10 Often the 'naturalness' argument is presented negatively, in the form 'such instruments or techniques were not yet available', as if problems would be solved immediately upon the arrival of said instrument.Older progressivist histories of technology are not hard to find so I will avoid singling out a specific instance of this kind of argument. 11An exception that proves the rule: optical instruments differ in important ways from precision/mathematical instrumentsyet two of the crucial episodes in the history of the development of the telescope and microscope are, first, the invention of micrometry, and, second, the precision measurement of lens systems themselves.Typically these are understood to be far more significant than, for example, changes in the construction of the bodies/stands of optical instruments, modifications of observational technique, improvements in microscope slide preparation, and so on.For the classical statement of the progressivist view of lens manufacture see Gerard L'E.Turner, Essays on the History of the Microscope (Oxford: Senecio Publishing, 1980), Chapter 9, ''The microscope as a technical frontier in science'. 12Kopytoff, 'The Cultural Life of Things', pp.66-8. 13Ibid., p. 66.
techniques advocated and demonstrated? 14In this paper, I explore public trials or demonstrations as decisive moments in cultural biographies of precision instrumentation.Trials established the credibility of instrumental techniques, and form a neglected part of the genealogy of the experimental method.My concern is with the ways in which instruments were adopted, promoted and trusted: what is the moral economy that stands behind the apparently objective economy of instrumental practice?
Of course, for the early-modern period we have a very well known narrative about collective observation and its relation to the sociomaterial dynamics of trust: the case of the Royal Society, gentlemanly norms, and 'virtual witnessing'. 15Steven Shapin discusses the 'literary technology' used by Boyle to define the boundaries of a new community of experimentalists investigating 'matters of fact'. 16The central components of Boyle's literary technology are: a separation of fact from speculation; the shifting of blame and credit onto instruments and machines; the creation of a public space for free discussion about facts; limiting that, a moral code for conduct.Perhaps centrally, there is an insider/outsider criterion for the assessment of a valid witness.As Shapin puts it: 'Oxford professors were accounted more reliable witnesses than Oxfordshire peasants'. 17Instrument makers, goldsmiths, surveyors and carpenters certainly were not peasantsbut nor were they professors.They possessed a meagre but important 'literary technology', as we will see.The spaces they inhabited were public, but in a different sense to the public space of the savants.We will encounter open fields, taverns, guild halls and workshops in what follows.What little we know of the moral conduct of artisans is gleaned mainly from guild regulations, records of transgressions and occasional disputes.We can only infer the moral codes that bound particular groups and conferred trustworthinessbut we can, I argue, recover the kinds of practical reasoning that mediated between the thing itself (the instrument) and the context of its justification (the trial).This is a version of the 'rise of experimental philosophy' in which authority in social settings lies with artisans rather than gentleman-practitioners.What kinds of authority and expertise might craftsmen bring to the table?How were conventions of trial, demonstration and collective observation established and maintained?How might we recover these occasions, given that many of those involved did not have the necessary skills or opportunities to provide a written record of their activities?Would it alter our reading of Wright's two masterpieces if we were to imagine that the philosophers had also constructed the orrery and the air pump? 18 lot depends, as we will see, on sources.Shapin has elsewhere written of the importance of 'invisible technicians'but curiously it is the visibility of the artisans in my story that is at issue. 19These were well known, worldy and successful individuals.It is precisely in their social network that we see their success.An invaluable source for the group analysed herethe mathematical practitioners remains Eva Taylor's monumental survey of that title, published in the 1950s and '60s. 20Taylor's attention to the revealing anecdote and her method of drawing cross-references via co-publication, collaboration and correspondence are, in my view, highly prescient.The evidence used in this paper has been hiding in plain sight for many decades.
This paper covers the period leading up to Wright's two famous paintings.My earliest examples date from the sixteenth century, but instrumental trials and demonstrations were so enmeshed in the promotion of new devices that the pattern described here continued well past my last (late seventeenth century) examples. 21I describe trials that relate first to the use of precision or 'mathematical' instruments, and later to patterns of observing, manipulating and measuring that are more recognizably part of the Baconian empiricist programme.The relation between older mathematical and newer 'optical' and 'philosophical' forms of instruments has remained reasonably obscure, mainly owing to an overly strict demarcation between instrument typesitself in part a product of modern historiography, museum classifications and collecting habits. 22If we look at social practices, identities, and forms of legitimation, there is a clear continuity between what went beforeartisans' observations, trials and demonstrationsand what came after, namely the experimental philosophy of the Royal Society.With the growing trade in mathematical instruments as a backdrop, my examples come from surveying, magnetism, and gauging (the measurement of the contents of vessels, usually barrels).I begin by defining the kinds of trials that were undertaken in the early-modern period, before looking in detail at specific instances in each of the areas just named.In conclusion, I turn to the broader question of community formation, the relationship between artisans' trials and the nascent Royal Society, and to the potentially far-reaching implications of taking artisanal know-how seriously.

Trials, real and idealized
Instruments and trials are not strictly speaking separable: technical devices require exposition, or trial by demonstration.Instrument texts and manuals are textual records of the teaching practices of those who offered lessons in practical mathematics. 23The act of using instruments is itself, through repetition and learning, a form of personal trialling. 24In light of this, we should not be surprised to find that one of the very earliest English discussions of practical mathematics involves a trial.This is reported in a letter of June 1542, from Sir John Wallop to Sir Thomas Cheneythat is, from the Captain of Guînes Castle to the Treasurer of the Royal Household.Wallop writes that four gentlemen from Kent had spent time with him at Guînes, and that they had amply demonstrated their skill aswell in gemetrie as thinges concernyng navigacon and the dissernyng of altitude as longetude, and as for the arte belonging to gonners [gunners] I have sein none such, in so muche that all thois that rekenethe theam selffes connyng on this sides of the see gevithe place vnto them 25 Wallop is keen to stress that these were not only verbal but also practical demonstrations: in thois things that thei haue reasenyd of here thei haue shewed themselfes marvelous wittely and thei cowde not be confowndyd by any that reasoned with them […] as well in argumentes of ther syences as in ther doing, whiche I haue bothe harde and seyn 26 These must have been instrumental demonstrations: one of the party, Leonard Digges, was to become the pre-eminent author on instrumental techniques in the middle years of the sixteenth century, and the mention of navigation and gunnery can only refer to angular measurement, probably with simple tools like the gunners level, quadrant or possibly cross-staff (Figure 2).These practitioners, Wallop emphasizes, were 'gentilmen', and not artisans, and he goes on to recommend to Cheney that the King employ such expert practitioners throughout the realm.Yet the promotion of the mathematical project quickly descended the social scale, with Digges himself writing books for the use of 'Surveyers, Landmeters, Joyners, Carpenters, and Masons'. 27he pattern of the development of the instrument trade shows conclusively that instruments were used by a diverse group with a range of financial means and practical aims.By the 1580s, for example, Edward Worsop could include the following advertisement in his treatise on surveying A Discoverie of Sundrie Errours: Scales, compasses and sundry sorts of Geometricall instruments in metall, are to be had in the house of Humfrey Cole, neere unto the North dore of Paules, and at the house of John Bull at the Exchange gate; in wood, at Iohn Reades in Hosier Lane, at James Lockersons dwelling neere the Conduite at Dowsegate, and at Iohn Reynolds at Tower Hill. 28me 28 instruments from the workshop of the goldsmith Humfrey Cole (d.1591) surviveall of them finely crafted in brassyet not a single instrument by any of the other makers listed here survives. 29The order of the list likely preserves the order of the quality of the wares, and wooden instruments were clearly cheaper than brass: the absence of surviving cheap Elizabethan instruments must not be taken as evidence that they were not made and used.
In addition to giving us a glimpse of the diversity of the instrument trade and, by extension, its clientele, Worsop's text on surveying also presents us with a very full account of an instrumental trial, this time involving a more variegated group than were present at Guînes in the early 1540s, though in some respects the real and imagined events are markedly similar.
Worsop presents a dialogue on methods of surveying, in which he is also an interlocutor, between two gentlemen (Master Peter and Master Watkins) a clothier called Johnson and a servant called Steven. 30Early on in the discussion, Johnson complains about the use of obscure words in mathematical practice, and Worsop himself gives a reasonably lengthy defence ('There is not any doctrine, or science, but hath of necessitie his peculiar termes'). 31The servant Steven also joins the argument, on Worsop's side, offering a vivid description of a scene he witnessed at his master's house: One time […] there was great talke, and arguing by the way, whether it were possible to tel how far one place in view is distant from an other: & how much one place is higher then an other, except it were first measured.Master Morgan answered that it was possible, & very easie to be done.Then Master Allen asked him if he could doe it by any of those instruments we carried with vs.He answered that he could.Wherevpon M. Allen suddenly staying, sayd: I pray you tel me how farre it is from the place where I stand to yonder oke. 32f course 'Master Morgan' is able to achieve the task: I wil, sayd he, and immediatly he piched one of his instrumentes, and looked thorowe a fine knacke, or Iig, and measured a good pretie way from him, not towards the marke, but sidewise: and at the corner where his measuring ended, he looked againe through his Iig, and casting a little with his pen, he tolde iustly almost how many perches it was from his foote to the oke: for he missed not a perch in a length that was aboue fiue furlonges. 33e significance of this moment is the esteem that it establishes for the mathematical practitioner Morgan, who is then able to discourse on the utility of mathematics for the state, giving a neat summary of arguments previously made in print by John Dee, Leonard and Thomas Digges and others.But this is only the first of Morgan's trials, and having earned his spurs by his use of (presumably) trigonometry to measure distances, he is then tasked with a surveying competition.Morgan is pitted against 'certaine lawyeres, surueyors, and countrye measurers'.'Such a stur', comments the servant Steven, 'as I neuer sawe amongst wise men': Some would haue the lande measured one way, some another: some brought long poles, some lines that had a knot at the ende of euerie perche, some lines that were sodden in rosen and waxe, M. Morgane had a line of wyers.They measured the poles, and lines with two foote rulers, & yardes, wherof some differed from other, halfe an inche, which made great variance, for euery man iustified his owne ruler. 34tting takes place, and Steven regrets that he did not have the wherewithal to back Master Morgan.This leads Stephen to contrast financial gain with good reasoning: Maister Morgane layed little or nothing, but alwayes as he sayde, so it was agreed vpon: he could alwayes giue such reasons, and so well proue his doings. 35e procedure of the trial is reasonably complex, and inevitably all parties disagree in their answers.Nevertheless, a simple conclusion is reached: Hereupon rose great contention and wagering, but at last all gaue place to Maister Morgane his measure. 36 have to take a step back here, and remember that this is not a purely factual report but a literary performance by Worsop: what can we conclude about the relationship between instrumental techniques, trust and demonstration?
First, Worsop has the innocent Steven tell this story: it is a story reported by someone who does not understand what he has seen, introducing further 33 Ibid., sig C v . 34Ibid., C2 r . 35Ibid. 36Ibid., C2 v .
characters about whom we know nothingthis lends the events recounted a 'ring of truth'.Then there is the two-stage trial.Morgan is able to rise to a challenge, to provide an explanation of how he has achieved his miraculous measurement, and then to discourse on the social utility of mathematical practice: these three things help to establish his credibility.When he is further tested in a competitive trial of surveying, he does not take part in the betting, and he benefits from the confusion of the different measures brought by each surveyor.So, while showing his disinterestedness and faith in his instruments, Morgan wins the day by being able to give something more than simple, variable measure.He gives an explanation of how his geometrical rules bring him to his figure for the meadow.This is the ideal type of the practical demonstration, where the force of instrumental and geometrical demonstrationcombined with arguments for the betterment of the commonwealth and the justice of tradewin the day for the new methods of survey. 37nstrumental trials, then, concerned the accuracy of new methods of measurement, and were part of the general phenomenon of mathematical projecting in the Tudor period.Gentlemen like Leonard Digges were early and crucial figures, but alongside them were printers, engravers, instrument makers, and practitioners in the various 'mathematical arts'. 38Over the course of the sixteenth century the trialas a learned, practitioner-led form of activitywas one tool amongst others in the promotion of instruments and geometrical techniques.

Trials and theorics: magnetism
By the early seventeenth century the range and extent of the mathematical artsespecially as a component of commercial and everyday lifehad developed significantly.This is perhaps best demonstrated by the large growth of the instrument trade: Worsop's advertisement had listed five makers active in 1582, and this, as far as we know, is almost the entire cohort. 39By the 1620s there were at least thirty instrument makers active in London, and probably many more amateur practitioners and journeyman artisans able to make instruments.The trade was also now, to a limited extent, a provincial affair, with makers active in Bristol and perhaps one or two other cities. 40 The broader community of practitionersincluding navigators, gunners, surveyors and architectsperhaps extended into the hundreds, though here questions of identity and expertise become complex. 41n terms of the sociability surrounding instruments, one decisive shift had occurred: authority for instrumental demonstration was now divested to practitioners and artisans themselves.Instrument makers' shops became lively sites for the exchange of expertise and information on new devices and techniques, and craftsmen could be expected to demonstrate their wares to potential customers.New kinds of gatherings also appeared alongside the kinds of demonstrations discussed above, and these included instrumental trials and collective observations.
One way to mark this shift is to look briefly at a case in which similar kinds of observation had been made at different times, to see how cultures of empiricism and instrumental practice changed.Instructive for this purpose is the sequence of measurements of the variation of the compass from true North, made between the 1580s and 1630s. 42In the late sixteenth century consensus had been reached that in London the magnetic compass deviated from true North by about 11°15' East.This value is recorded in the works of Thomas Digges, can be found on surviving sundials, and was checked and confirmed by William Borough in 1580.In 1622 the mathematical scholar and author Edmund Gunter observed that this figure appeared to have decreased to 5°5 5' East, though he assigned no cause to the discrepancy between his and his predecessors' observations.A little over a decade later a further set of observations found that the figure had decreased still further, to 4°10' East.At this point, Henry Gellibrandone of the group who had made the new observationswent into print with these observations and, notably, the conclusion that this change constituted a 'variation of the variation' or 'secular variation', that is, a natural phenomenon in which the location of magnetic North was itself shifting over time.
The episode is well known and has been extensively analysed by Stephen Pumfrey and others. 43For my purposes it illustrates the way in which a subject investigated by a socially variegated community of practitioners can contribute to the increased authority of craftspeople.The decisive member of the 1630s group was the compass needle-maker John Marr, who had been present in the early 1620s at Edmund Gunter's first test of the Elizabethan values.At the earlier trials Gunter, Marr and others had been content to conclude merely that previous authors had been mistaken: their confidence in their own observations was just about on a par with their conservatism with regard to magnetical theory.The 'discovery' of secular variation a decade later came only with Marr's return to the same set of measurements.Finding a lower value for variation than he himself had observed a decade prior, his new set of collaboratorsincluding the 'discoverer' Gellibrandfelt emboldened to conduct a series of observations, and in light of these a change to the theory of terrestrial magnetism itself.
Pumfrey rightly points out that it was the existence of a coherent and closeknit group of London-based practitioners that was central to the establishment of new matters of factas opposed to the critique of previous poor measurements.But why exactly did this matter?I propose that it mattered because it is an indication of Marr's status as a well connected practitioner, and his practical grasp of a properly 'philosophical' matter.At the time of Marr's measurements with Gunter, we were considering only two individuals, and their ability to critique earlier experimentalists.By the 1630s, through Marr himself, we can now connect Gunter to Gellibrand and an extended group.In addition to these two, the naval storehouse keeper John Wells was also involved.Wells was known to be a talented sundial maker, though this interest was personal and scholarly rather than commercial.Note that the only expertise in priming magnetic needles was Marr's.
Nor should we be surprised by Marr's elevated status: as Mario Biagioli has pointed out, mathematicians in this period had many opportunities for social mobility; 44 in Marr's specific case, we know that he was in the household of the Prince of Wales (as clerk of the kitchen), was tutor to John Collins in mathematics, wrote a treatise on the construction of sundials for King Charles I, and (with Elias Allen) supplied needles in 1631 to Captain Thomas James for his Arctic voyage. 45e might pause here and consider the kinds of expertise that artisans could bring with them.A knowledge of materials and how to work with them was obviously paramount: instrument makers were often also metallurgists, 44 Mario Biagioli, 'The Social Status of Italian Mathematicians, 1450-1600', History of Science, 27 (1989), 41-95. 45 especially the goldsmiths who were prominent in the early trade. 46Through Guild membership, artisans, craftsmen and merchants were exposed to a particularly important form of trial: 'the search', conducted in the course of enforcing standards of quality manufacture (though here of course it was the artisan's own work that was directly under consideration, their authority in question). 47Whatever material they worked in, instrument makers knew how to divide scales accurately, and probably also the rudiments or tricks of practical geometry, including the projection of the sphere, and of the hour lines necessary for making sundials.Instrument makers could also calibrate one instrument against another, and devised tests to check their own working. 48But instrument makers were not the only practitioners whose expertise was valued: artisans and merchants working in architecture, the cloth trade, carpentry, gunnery and allied fields were increasingly expected to have some facility with geometry and arithmetic, and to be able to use, craft and even modify the precision tools of their trade.For example, following the work of Nicolo Tartaglia and, more locally, Thomas Digges and William Bourne, English gunners relied increasingly on mathematical and instrumental techniques through the early seventeenth century.Yet this reliance was not simply one of trust in new techniques dictated from on high: gunners like Bourne, William Eldred (1562-1646) and Nathaniel Nye (1624-1647?) made their own original investigations of the trajectories of projectiles. 49n important question here concerns competency: could instrument makers in fact explain the principles on which their products were based?If not, then their authority must lie solely in their status as a master craftsman and, potentially, their collaboration with a learned inventor or virtuoso.If they could explain the grounds of their work, however, then their authority was two-fold: as maker and collaborator, and as a source of explanation and justification.There is plenty of evidence that contemporary scholars believed that artisans had a deep knowledge of geometry.For example, John Dee used the term 'mathematical mecanician', to refer to craftspeople who were also 'speculative'that is, who could grasp the geometry of their inventions.In using this term Dee had specific individuals in mind: certainly the navigator-craftsman Richard Chancellor (c.1521-1556), and possibly also the Louvain cosmographer-instrument maker Gerard Mercator (1512-1594) and the London engraver-instrument maker Thomas Gemini (d.1562).Gabriel Harvey borrowed Dee's term and used it to describe Humfrey Cole, the leading instrument maker of his generation (mentioned by Worsop), so we can be fairly confident that Cole and perhaps others were known not only to invent and make, but also to demonstrate and discourse upon instruments.Worsop, for his part, insists that anyone using instruments.
is not to be accounted therefore a sufficient Landmeater, except he can also prooue his instruments, and measurements, by true Geometricall Demonstrations 50 These 'Demonstrations' were probably quite limited, but there is no reason to doubt that artisans had them to hand.This may be taken literally.Robert Norman claimed in 1581 that notwithstanding the learned in those sciences, beeyng in their studies amongest their bookes, can imagine greate matters […] there are in this land diuers Mechanicians, that in their seuerall faculties and professions, have the vse of those artes in their fingers endes 51 This important claim sets us an interpretative challenge: can we approach the kinds of knowledge possessed by artisans and communicated in trials and demonstrations, or is this inaccessible owing to a lack of sources or even its tacit nature?
I do not propose to solve this problem, but will instead suggest that the answer lies in a concept explored by Jim Bennett, namely 'theorics'.'Theoric' was originally a term of art in astronomy, referring to the mathematically described course of a planet.The term became more general in the earlymodern period.As Bennett explains, theorics were rules intended to contain and generalize measurements across space or across time, often employed mathematical techniques close to those involved in instrumentation, and could take an instrumental form. 52other useful gloss is given by Laura Georgescu: Theorics combine observational results with geometrical considerations in a construction meant to be used as a predictive tool. 53 course, recognizably modern proofs were also available to artisans like Norman who had read their Euclid.But in my view the kinds of 'geometrical considerations' in play were not always so extensive, and may have rested on intuitive and graphical approaches to geometry, and even on the explanation 50 Worsop, A Discoverie of Sundrie Errours, sig.E3 r-v . 51Robert Norman, The newe Attractive […] (London, 1581), sig.I v . 52 of the construction and use of an instrument: just as a theoric could be embodied in a device, the theoric of an instrument might involve only as much geometry was involved in its construction.Stephen Johnston has shown that geometrical constructionsespecially involved in the spherical trigonometry of diallingwere often made using techniques of transposition and 'folding', rather than by calculating diameters or linear measurements. 54or Marr the case must be slightly different: he was a diallist and a compass-maker, with extensive experience of both.What he contributed to Gellibrand's group was a confidence in the quality and constancy of his compass needles, and a detailed knowledge of finding both true and magnetic north.As Johnston and others have shown, magnetism itself was understood in terms of theorics.This is the thrust of Norman's remarkable Newe Attractive, and surely also the way in which his successor Marr understood his work.Norman was an important predecessor in another sense: in his announcement of magnetic dip, Norman, like Marr, had been able to establish on his own authority that an unexpected magnetic phenomenon was a property of nature rather than a defect of manufacture. 55Both men had, in Norman's words 'the vse of those artes in their fingers endes'.

Trials and theorics: barrel gauging
In addition to contributing to scholarly enterprises like Gellibrand's, instrument makers also pursued their own objectives.They had different (commercial) motives, to be sure, but they employed the same means: private and public demonstrations, trials, and collective observations.The most prominent instrument maker active in London in the first half of the seventeenth century was Elias Allen (c.1588-1653), whose workshop on the Strand was to become a meeting houseeven a 'clearing house'for mathematical expertise and information. 56Allen's deepest engagement with the world of learning came through his collaborations with the mathematician William Oughtred  (1575-1660).Although Oughtred publicly scorned the use of instruments as a distraction from proper mathematical training, he nevertheless delighted in devising new kinds of instruments, and adapting and improving old ones. 57Over time a fruitful and mutually beneficial relationship grew between Oughtred and Allen, with the latter on hand to make newly invented devices, while simultaneously benefiting from the increased stock and burnished reputation brought about by association with the well known scholar.
In 1633 Oughtred published an account of a newly invented 'gauging rod' for measuring the contents of barrels: an important commercial task, as barrels were the means of transporting not only liquids like wine, beer and spirits, but also foodstuffs like fish and grain. 58Oughtred's book contains a lively account of the circumstances surrounding his invention.He claims that he was visited by 'an old man (that said he was) the Gauger of London', who defended the rules for gauging given previously by the mathematician Edmund Gunter. 59Oughtred argues the case and gives an example showing that the 'Gauger of London' (and by extension Gunter) are using a faulty procedure for estimating the volume of a spheroid from measures of its central and terminal diameters.This dispute was then, says Oughtred, conveyed to the instrument maker Elias Allen, who found himself in company with the Master of the Worshipful Company of Vintners and related it once again.At this point, the Master of the Vintners, George Ethredge, asked if an instrument could be devised that would cut out the need for any calculation at all, and this is the instrument described in the text.
The convoluted nature of this portrayal is not accidental to the question of authority and trust.It is important to Oughtred's own position as a disinterested scholar that Allen should negotiate with the Vintners.This complex relationship of scholar-craftsman-guildsman is carried over into Oughtred's report of the subsequent trial of his new gauging rod: For when Mr. Elias Allen had finished up one of those my instruments or gauging rods, and had brought it to their Hall, they presently deputed certaine of their society to see the experience and performance thereof, at the Taverne by Leadenhall under the signe of the Kings-head: and they tooke the paines to examine the truth of it in many and sundry kinds of wine vessels; where, as I have beene told (for I was not there present my selfe) beyond all expectation they found such an exact agreement with the measure of water they filled in by gallons.Whereupon they agreed with Mr. Allen for a price, and bespake of him threescore of the same my rods or instruments. 60his is obviously a less polished literary performance than Worsop's, and again we simply have to take the success of the demonstration at face value.But there is still plenty to analyse here.First, Oughtred claims that he was not even aware of the trial: this guarantees his own distance from proceedings, while also placing extra emphasis on the trustworthiness of Allen.Then there is the venue: the instruments are produced at the Vintners' Hall, but the group then moves to the public space of a tavern to conduct the experiment.Success is marked by commercial opportunity for Allen: the vintners acquire the Oughtred-designed instrument.For Oughtred this is more than a commercial point, however: as he goes on to explain at length, the success of the trial offers proof that barrels are truncated spheroidsthis is the principle on which his instrument is based, and the point of his conflict with Gunter and othersso the assumption underlying his instrument is verified by the success of the instrument in use.
The only question that remains is whether Allen himself could explain the principle of the instrument, in addition to demonstrating its use.At the end of Oughtred's book there is an advertisement that states: These Instruments are made in brasse by Elias Allen over against St. Clements Church without Temple-barre: where also those who are desirous may bee instructed in the practicall use thereof: and such as shall have occasion may have vessels gauged. 61r Allen himself we have relatively little evidence for his learning.He was certainly literate, and appears to have been able to construct quite complicated instruments from written descriptions aloneyet Oughtred on a couple of occasions had cause to comment critically on Allen's understanding and correct his working. 62Like Cole in the previous generation, Allen's standing as a City man was clearly high, and it was surely this, in combination with a small amount of mathematical skill, that allowed him not only to make but also to demonstrate his instruments.
Again we can understand Allen's learning in terms of 'theorics'.Oughtred in his text gives a general description of the instrument, and then four rules for taking measurements.Unlike the Worsop/Morgan survey above, in which the correct functioning of the instrument resulted in an accurate depiction of the surveyed land, or the magnetic trial by Gellibrand, in which trust was placed in Marr's grasp of the theoric of magnetism, here the trial itself is not sufficient to constitute a theoric, which must be reserved for the construction of the scales on the instrument itself.Oughtred is explicit: 61  the maner of computing the Gauge-divisions I have concealed: both because that speculation is impertinent to the managing and hand-working therewith: and also that because unto men of art by comparing the rule with the performance, it will not bee difficult to find out the reason: but especially because I intend and wish the benefit of making and fabricating this instrument, unto Mr. Elias Allen, who gave the occasion of it, and at whose request I invented it. 63 might be contended that this doesn't prove Allen's own knowledge of the 'maner of computing the Gauge-divisions'.But here we are fortunate in having a letter from Oughtred to Allen giving detailed instructions on scale division of a closely related instrument with a number of divided scales, so we have no reason to doubt that Oughtred was willing and able to communicate the method of scale division to Allen (rather than, for example, providing him with a mere template). 64We will never know whether or not Allen could make the further step of explaining (with proofs) why the divisions were placed at their exact locations; but for our purposes this does not matter.Allen possessed the secret knowledge of how to make the scales, and as the trial was a success this gave a special kind of authority.
In spite of the success of Oughtred's instrument, the problems of barrel gauging were only to grow in subsequent years, and here again we have evidence of artisans' involvement in settling matters of fact.In 1643, in order to raise funds during the Civil Wars, the Government of England and Wales introduced excise duty on a range of goods, including beer and wine. 65This put immediate pressure on the accuracy of gauging, because now a far greater quantity of goods was subject to taxation.Now it was important to discover not only the true rules of gauging but the quality of standards in use, and again an informal committee was formed, to go to the Guildhall and try the measures there. 66his group consisted of John Wybard, Edmund Wingate, John Reynolds and Baptist Sutton.Of this group Wybard is the most obscurehe was a London physician about whom little is known.Reynolds is much better documented: he was trained as a goldsmith and worked at the Royal Mint.Wingate is very well known for his innovations in logarithmic scaleshe was also a successful legal and mathematical author.Baptist Sutton was rather an extraordinary character: he was a church window maker and painter who also specialized in painted sundials and wrote a learned treatise on logarithms. 67Of this group, it was Reynolds who seems to have been the main motivator: as early as 1641 he had been involved in testing official standards. 68ybard gives quite an extensive account of the group's experiments at the Guildhall.They made multiple trials of the standards, which they examined using their own precision-made wooden measures.In the manner of later Royal Society members he gives plentiful detail on the exact kind of sealant to use on wood, the care taken over the handling of liquids, the procedures they put in place to be certain of what they were finding out. 69With this equipment the group found that the Guildhall measures were well short of the accepted standard.This might seem like a disastrous result: the structure of English excise duty was based on false measuresand this was a cry echoed in a pamphlet that was issued protesting at the new tax. 70But just as Wybard's book brought the whole matter into question, it also promoted both his own experimental programme, and the work of the mathematical practitioners.
If the Guildhall standards were not ultimately usable as a reference for specific measuring jugs or instruments, then how was the whole practice of gauging, and with it the levying of taxes, to be carried on?One part of the answer is given by an extraordinary survival at the Whipple Museum of the History of Science in Cambridge.Figures 3 and 4 show a gauging rule designed by Reynolds, of which this is the only known example.The rule was never published, and aside from the rule itself there is no account of its design and use.But from its survival we know that Reynolds gave his personal authority to the use of the unofficial standards. 71So the gauging trials were a mix of success and failure: the original critique of Gunter's rules led to the development of new instruments, and business opportunities for people like Elias Allen and Henry Sutton, maker of Reynolds' rule.The failure of the Guildhall standards in fact vindicated both the experiments of Wybard and his group, and gave additional validity to the use of instruments in maintaining impromptu standards.

The cultures and spaces of instrumental life
What conclusions can we draw from these mathematical and experimental trials?One important fact is that they continued into the period of the Royal Society, and overlapped with the work of its members. 72Across all of these examples -Worsop's surveying trial, the various gauging trials and the famous example of secular variationwe can see continuities.In the later trials, the presence of the craftsman is decisive: Elias Allen's demonstration wins him the commission; the diallist Baptist Sutton and the metallurgist John Reynolds vouch for Wybard's observations; the compass-maker John Marr's observations open up the pathway to Gellibrand's celebrated discovery.Just as astronomy was a crucible of mathematicized natural philosophy, and the use of instruments formed another bridge between the practical and the experimental sciences, trials were a 'social technology' that shows clear continuity across the sixteenth and seventeenth centuries.
If we recall Shapin's important discussion of Robert Boyle, we will notice a series of strong parallels between the trials I have described and the 'technology of producing knowledge' that had to be 'built, exemplified and defended against attack' by the members of the early Royal Society.Shapin argues that  the establishment of matters of fact utilized three technologies: a material technology embedded in the construction and operation of the air-pump; a literary technology by means of which the phenomena produced by the pump were made known to those who were not direct witnesses; and a social technology which laid down the conventions natural philosophers should employ in dealing with each other and considering knowledge-claim. 73idently the material technology is ever-present in instrument trials.The literary technology, meanwhile, consists in the descriptions of trials that proliferated in practical mathematics books after Worsop.We can be quite precise about the role of these accounts: they were 'lively descriptions' in the tradition of enargia (and more specifically pragmatographia), which were intended to justify the laborious grounding in theoretical principles that occupied the majority of these texts themselves.The social technology, as I have said, is the trial itself, and the 'theorical' way of knowing that instrument makers possessed and communicated.
If the number of trials I have brought forward here might seem rather small, this can be countered with the considerable evidence for a vibrant community of mathematical practice first brought forward by E.G.R. Taylor, and subsequently developed by Stephen Johnston, Hester Higton, Lesley Cormack, Deborah Harkness, Philip Beeley and others. 74The point here is not merely that trials formed a singular if important part of the 'cultural biography' of a new invention or modified instrument, but also that there was a vast community of artisan-practitioners involved in the development, advocacy and use of these instruments, and that within this community trials and demonstrations were commonplace. 75n order to draw connections between mathematical and philosophical trials, specific individuals and communities can be pursued over time.For example, the Royal Society's 'Magnetics Committee' was in part a response to the researches of Gellibrand and his group, and subsequent practitioners such as Henry Bond, a dockyard mathematics teacher who accurately predicted 73 75 There has been a lively debate over the use of the term 'mathematical practitioner'.Taylor's collective biography indeed brings too many individuals with diverse aims and careers under that heading, and there is a risk that the term 'mathematical practitioner' either suggests a completely coherent and self contained national (Anglophone) culture or a flattened out pan-European culture.In either case problems abound.To clarify my position in this paper: I am focused on a reasonably specific group (artisans) and a definite sphere of practice (the use of mathematical instruments).John Dee's authority in trials and demonstrations is not at issue, for instance, but Elias Allen's is.
changes in secular variation in the 1650s. 76The Committee sought to collect data from a wide range of practitioners, resulting in figures that were to end up playing a role in Newton's Principia. 77f we seek individuals who worked across disciplines and social strata we can point to the silk-weaver Robert Anderson (fl.1661-d.before 1710), who occupied a central role between instrument makers, publishers and other enthusiasts. 78Early in the 1660s, we find Anderson co-ordinating astronomical observations with his fellow weaver Thomas Streete, the instrument makers Henry Sutton and Joseph Moxon, and the optician Richard Reeve.Subsequently Anderson was to become prominent for his experiments in ballistics, which were conducted first on Wimbledon Heath, and then in a public demonstration in 1674 on Blackheath.Another set of Galilean experiments was conducted at Cripplegate steeple on 24 June 1686, with the clockmaker Thomas Tompion, who supplied a watch that could read to quarter seconds.Anderson was able to design and conduct his own experiments, style himself as a natural philosopher and collaborate and go into print in collaboration with other members of his artisanal community.
At the outset, I proposed Kopytoff's notion of cultural biography as a framework for understanding artisans trials.The importance of this for my analysis can be brought out by contrast with anotherperhaps superficially more apt historiographical notion, namely the 'trading zone'. 79Briefly, trading zones are sites (sometimes literal buildings or areas, but shading into social groups and even intellectual or practical traditions of work) in which forms of communication are developed that can allow 'sharply different global meanings' can 'come to […] coordination'. 80This concept was developed by Peter Galison (drawing heavily on linguistic anthropology) in the context of modern highly instrumentalized physics.It is relevant in the present context because it has been extended to describe early-modern relationships between artisans and university-trained individualsin fact, instrument makers' shops are specifically cited as trading zones. 81A virtue of thinking about trials as trading zones is that it draws our attention to the communicative practices involved, and here I will again recall Bennett's use of 'theoric' as a period-specific lingua franca of the protocol-based use of instruments, diagrams and other practical aids to understanding.But this is also the limit of adopting the idea of the trading zone: reconstructing the discursive space of the trial is an almost impossible taskperhaps achievable for some specific examples, but hardly generalizable.The payoff would be a truly anthropological approach to the past, but this is not my aim here.
For my purpose, I find the object-biography method more useful: what the trials offer is an idealized account of how instruments should function in a given social context.We are concerned with the objects themselves, and the status of those involved, rather than communicative practice.Cultural biography allows us to understand theoric not just as a means of communication but as a source of authority, hence the trial was a type of event that could be carried over from one context to another (gentlemanly to shop-floor demonstration; mathematical to experimental practice).
What both the 'trading zone' and 'cultural biography' have in common is an economic motive.For the former, Galison was drawing directly on Michael Taussig's well known anthropology of the capitalism's unruly 'edges', and the translation of economic concepts into ritual practices. 82Kopytoff, meanwhile, is motivated in his analysis by the 'drive to commoditization'. 83Here we may gain additional insight into the adoption of mathematical instruments.In Kopytoff's terms, the mathematical practitioners were creating a small but significant 'new exchange technology'i.e. the dividing up of space, time and objects into mathematically discrete units by instrumental means.Each of the trials mentioned above has an important economic context: military (state) power; land management; navigation; the measurement of commodities.Because trials were carried over into experimental inquiry, the trials and demonstrations I have described establish a clear relationship between the origins of capitalism and natural philosophy.Gellibrand is perhaps the ideal example of what I mean by this.In the attempt to reform the explicitly commercial practice of navigation and the more subtly commercial practice of time-telling, the compass maker Marr and diallist Wells played an important role in what was to be claimed later for Gellibrand as an experimental demonstration of a natural phenomenon, namely the variation of the variation.A philosophical matter, first expounded in its canonical form by William Gilbert at the beginning of the seventeenth century, was decisively settled instrumentally in fourth decade.The pattern was set: trials, demonstrations and artisanal know-how formed the basis of the new philosophya process that has been profitably explored in Simon Schaffer's papers on the 'information order' of Newtonian physics. 84Instruments were commodities, though they had a special status in that they could not be used by just anyone.Worsop went so far as to argue that new institutions needed to be set up so that they could be correctly adopted into surveying practice. 85Instruments were also part of a system whose purpose was explicitly to provide universal rules for the measurement and management of quantity, space and time.This was not simply a matter of aiding those who wished to conduct practical tasks like measuring the height of a tower or the depth of a well (two classic examples from practical textbooks).Rather, these measurements were to form an integrated whole that relied on geometrical principles and could be applied to a very wide range of problems, and to combine with arithmetic to form a monetary-numerical system in which large or small problems alike could be solved by translating between measurement and money, money and measurement. 86In an ambitious mode, John Dee could advocate for the reform of navigation itself as an aid to the realm through saving lives, improving the efficiency of transport of goods and discovering and settling new colonies. 87More homely problems could be posed and solved by a practical teacher like Edmund Gunter, who explained how to measure a plot of land, rent it at a certain rate, and then calculate interest over a set period of time using a numerical scale. 88Separated by a generation or two and differences of education and status, Dee and Gunter would nevertheless have agreed on the importance of a grounding in geometry and arithmetic, the centrality of instrumental techniques and the need for well made tools.Trials demonstrated the efficacy of particular devices, and established the status and learning of the instrument's user.Instruments were emblematic of the mathematical arts, and artisans' trials provided the social basis for their acceptance, and for the flourishing of the instrument trade itself.

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Figure 1 .
Figure 1.Mezzotint after Joseph Wright of Derby, A Philosopher giving that Lecture on the Orrery in which a lamp is put in place of the Sun (1766).Mezzotint by William Pether, 1768.Wh.1602, Whipple Museum, Cambridge, UK.Reproduced with permission.

Figure 2 .
Figure 2.An instrumental demonstration in Leonard and Thomas Digges' Pantometria (London, 1591; first edition 1571).Note the social niceties of the situation, with the practitioner literally below the lady and gentleman.Classmark 58:15, Whipple Library, Cambridge, UK.Reproduced with permission.
Taylor records Marr as a Scotsman (fl.1614-47) 'who served both James I and Charles I as compass-maker, diallist and dialmaker, an excellent mathematician and geometrician, according to William Lilly,' continuing: 'The famous astrologer recalls that Marr was familiar with both Henry Briggs and Napier of Merchiston and was present in 1614 at their first meeting in Edinburgh.In 1631 Marr and Elias Allen supplied magnetic needles of special quality to Captain Thomas James for his Arctic voyage […] He was subsequently visited at Richmond by John Pell who was following up Gellibrand's work.The stone dials in the Royal Garden at Hampton Court were no doubt erected by Marr who was the author of a "Description and Use" of the lines and circles upon one of them, a MS.dedicated to Charles I and now in the British Museum.John Marr was succeeded by William Marr, presumably a son or nephew.' Taylor, Mathematical Practitioners of Tudor and Stuart England, pp.203-4.See also Philip Beeley, 'Practical Mathematicians and Mathematical Practice in Later Seventeenth-Century London', The British Journal for the History of Science, 52 (2019), 225-248 (p.235).