Mechanic slavesShe's telling an attendant how she feels about the prospect of being acclaimed by plebeian Romans. Though less pungently, Coriolanus said much the same. So did Puck, quoted in my last post.
With greasy aprons, rules, and hammers shall
Uplift us to the view. In their thick breaths,
Rank of gross diet, shall we be enclouded.
And forc'd to drink their vapor.
(AC. V. ii. 209-13)
It's not easy to generalize with any confidence about the attitudes of any class toward another. The attempt to locate the boundaries that separate the "us" of one class from the "them" of another can likewise be a daunting one. To Cleopatra the odiferous mass were all one. To themselves, they were many distinct groups.
I'm interested in the mechanic who's holding a rule. One can imagine him as a carpenter or builder, maybe a shipwright or gunner, or he might have been a surveyor, navigator, or excise officer. Or he might have been a man who made rules — a person who made tools for a living. Or, still, he might have been one who taught others how to make and use the instruments of the trade.
In Shakespeare's time, to many of those who were gently born, these mechanic slaves holding rules would have been men who were presumed not to amount to much simply because they worked with their hands. But although many surely did, as apprentices and journeymen whose daily drudge was limited to performing tasks the master assigned, approximate this description, other men — the masters themselves — also held rules in their hands. They were builders, architects, land surveyors, gunnery officers, and others who used mathematical tools — rules — in their work. This, for example, shows a gentlemanly-looking surveyor (at right) and equally-well-decked-out helper (left) with an early version of the theodolite (described as a "Semy Circle set upon a perfeck square"), some levels (one of them using water boxes), and a target stick.
This illustration shows a Portuguese navigator using a similar device (a forerunner of the sextant called a cosmolabe) to help draw a nautical chart. His instrument could not be used at sea, but more portable versions might be found on the distant ships, to be used with his charts when landmarks ashore were out of sight.
Another set of men holding rules were the men who made them. They were self-employed craftsmen or masters of small shops who possessed skill akin to goldsmiths or clock-makers or even, in those times, many plebeian artists, composers, and literary authors.
You can tell something of the skill possessed by one of these craftsmen, Henry Sutton, from this account by John Collins. Notice that Sutton needed no more than a verbal description in order to make a complex mathematical instrument. A mathematician "intimated his desires" and the craftsman, an engraver, "speedily found out the drawing" without further instruction. 
This is Sutton's quadrant, made from a plate he engraved for Collins' book.
Sutton was acclaimed one of the best, but others like him were skillful instrument makers, the technicians of their time. Here's one (quite late) description of their trade.
"THE Mathematical-Instrument-Maker makes all kind of Instruments constructed upon Mathematical Principles, and used in Philosophical Experiments: He makes Globes, Orrerys, Scales, Quadrants, Sectors, Sun-Dials of all Sorts and Dimensions, Air-Pumps, and the whole Apparatus belonging to Experimental Philosophy. He ought to have a Mathematically turned Head, and be acquainted with the Theory and Principles Upon which his several Instruments are constructed, as well as with the practical Use of them. He employs several different Hands, who are mere Mechanics, and know no more of the Use or Design of the Work they make, than the Engines with which the greatest Part of them are executed; therefore the Master must be a thorough Judge of Work in general.
"The Optical-Instrument-Maker is employed in making the various sorts of Telescopes, Microscopes of different Structures, Spectacles, and all other Instruments invented for the Help or Preservation of the Sight, and in which Glasses are used. He himself executes very little of the Work, except the grinding the Glasses: He grinds his Convex-Glasses in a Brass Concave Sphere, of a Diameter large in proportion to the Glass intended, and his Concave-Glasses upon a Convex Sphere of the same Metal: His Plane-Glasses he grinds upon a just Plane, in the same Manner as the common Glass-Grinder, mentioned Chap. XXXII. Sect. 4. He grinds them all with Sand and polishes them with Emery and Putty. The Cases and Machinery of his Instruments are made by different Workmen, according to their Nature, and he adjusts the Glasses to them.
"It is a very ingenious and profitable Business, and employs but a few Hands as Masters; The Journeymen earn, a Guinea a Week, and some more, according as they are accurate in their Trade. Such a Tradesman designed for a Master ought to have a pretty good Education, and a penetrating Judgment, to apprehend the Theory of the several Instruments he is obliged to make, and must be a thorough Judge of such Work as he employs others to execute. A Youth may be bound to either of these Trades any time between thirteen and fifteen Years of Age, and does not require, much Strength."}
Yet one more set of men with rules were teachers and authors of instructional texts who demonstrated the use of mathematical instruments.
John Collins himself was one of these. As an example, here is a list of his books on the use of mathematical instruments. It shows up in an "astronomical appendix" to a translation of The Sphere of Marcus Manilius by Sir Edward Sherburne, ca. 1675.
Navigation: The Mariner's Quadrant
Navigation, Smith Center for Cartographic Education, University of Southern Maine
The identity of the mathematical practitioner in 16th-century England by Stephen Johnston in the proceedings of a 1995 conference in Duisburg: Irmgarde Hantsche (ed.), Der “mathematicus”: Zur Entwicklung und Bedeutung einer neuen Berufsgruppe in der Zeit Gerhard Mercators, Duisburger Mercator-Studien, vol. 4 (Bochum: Brockmeyer, 1996), 93-120. It is closely based on material in the introduction to my thesis, and appears here by permission of Universitätsverlag Dr. N. Brockmeyer.
De republica anglorum, a discourse on the Commonwealth of England by Thomas Smith, ed. by L. Alston with a preface by F.W. Maitland (Cambridge, the University Press, 1906)
"Christs Teares over Ierusalem" by Thomas Nashe (1593) in Life in Shakespeare's England a book of Elizabethan prose comp. by John Dover Wilson (Cambridge University press, 1913) Extract: "In London, the rich disdain the poor. The courtier the citizen. The citizen the country man. One occupation disdaineth another. The merchant the retailer. The retailer the craftsman. The better sort of craftsmen the baser. The shoemaker the cobbler. The cobbler the carman. One nice dame disdains her next neighbour should have that furniture to her house, or dainty dish or device, which she wants. She will not go to church, because she disdains to mix herself with base company, and cannot have her close pew by herself. She disdains to wear that everyone wears, or hear that preacher which everyone hears. So did Jerusalem disdain God's prophets, because they came in the likeness of poor men. She disdained Amos, because he was a keeper of oxen, as also the rest, for they were of the dregs of the people. But their disdain prospered not with them. Their house, for their disdain, was left desolate unto them."
"Itinerary" by Fynes Moryson (1617) in Life in Shakespeare's England a book of Elizabethan prose comp. by John Dover Wilson (Cambridge University press, 1913)
The London tradesman; Being a compendious view of all the trades, professions, arts, both liberal and mechanic, now practised in the cities of London and Westminster. Calculated for the information of parents, and instruction of youth in their choice of business by R. Campbell, esq (London, printed by T. Gardner, 1747)
The making of the English middle class; business, society, and family life in London, 1660-1730 by Peter Earle (University of California Press, 1989)
Diary of Samuel Pepys, Tuesday 29 May 1660
The sphere of Marcus Manilius made an English poem with annotations and an astronomical appendix by Marcus Manilius, translated by Sir Edward Sherburne (London, printed for Nathanael Brooke, 1675)
"A Catalogue of Astronomers" in the work cited just above.
"The Institution of Residential Investment in Seventeenth-Century London" by William C. Baer, The Business History Review, Vol. 76, No. 3 (Autumn, 2002), pp. 515-551 (Harvard) Stable URL: http://www.jstor.org/stable/4127797
Biographies in John Aubrey's Brief Lives
The Mathematical Practitioners of Tudor and Stuart England by E.G.R. Taylor (Cambridge, 1954)
Shakespeare from the margins by Patricia A. Parker (University of Chicago Press, 1996)
Earthly Necessities: Economic Lives in Early Modern Britain by Keith Wrightson (Yale University Press, 2002)
Manilius and his intellectual background by Katharina Volk (Oxford University Press, 2009)
The English Improver Improved or the Survey of Husbandry Surveyed by Walter Blith (London, 1652)
Marcus Manilius on wikipedia
"The Search for the 'Middle Sort of People' in England, 1600-1800" by H. R. French, The Historical Journal, Vol. 43, No. 1 (Mar., 2000), pp. 277-293 (Cambridge University Press) Stable URL: http://www.jstor.org/stable/3021022
Stereographic projection in wikipedia
Hour angle in wikipedia
Azimuth in wikipedia
Solar azimuth angle in wikipedia
 Patricia Parker writes well about the disdain of gentles for their lesser bretheren. See Shakespeare from the margins by Patricia A. Parker (University of Chicago Press, 1996).
 The making of the English middle class; business, society, and family life in London, 1660-1730 by Peter Earle (University of California Press, 1989)
 Here is the relevant description: A limb is the outer edge of a sphere, in this case the earth.
Being in conference with my loving friend M. Thomas Harvie, he told me, that he had often drawn a Quadrant upon Paper pasteboard, &c. derived by himself, and never done by any man before, as to his knowledge, from the Stereographick Projection, which for a particular Latitude, would give the Hour in the equal Limb, and would also perform the Azimuth very well; and but that it was so particular, was very desirous to have one made in Brass for his own use by an Instrument Maker: whereto replying, that with the access of some other Lines to be used with Compasses, it might be rendered general for finding both the Hour and the Azimuth in the equal Limb: He thereupon intimated his desires to M. Sutton, promising within a fortnight after their conference, to draw up full directions for the making thereof. But M. Sutton having very good practise and experience in drawing Projections, speedily found out the drawing of that Projection, either in a Quadrant or a Semicircle, without the assistance of the promised directions, and accordingly, hath drawn the shape of it.
-- The sector on a quadrant, or, a treatise containing the description and use of three several quadrants by John Collins (printed by J.Macock, to be sold by George Hurlock, William Fisher and Henry Sutton mathematical instrument maker, 1658)
 This description comes from a book published a century after the period of which I'm writing. I couldn't find anything more closely contemporaneous. Although it does not contradict writers who published in the middle of the seventeenth century and seems to jibe with sources from the period, I can't really say it's as good a description of the one time as of the other. In the preceding chapter the author also describes the watch- and clock-making trades. My source is Chapter 55 in The London tradesman; Being a compendious view of all the trades, professions, arts, both liberal and mechanic, now practised in the cities of London and Westminster. Calculated for the information of parents, and instruction of youth in their choice of business by R. Campbell, esq (London, printed by T. Gardner, 1747)
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