"super" as in "over the top"

I've never developed any lasting fidelities to professional sports teams. It's true though that I've had an interest in the football Giants going back to the glory years of Frank Gifford, Sam Huff, Roosevelt Brown, Andy Robustelli, Alex Webster, and, ultimately, the great old man, Yelberton Abraham Tittle. And so I'll be watching the Super Bowl this Sunday. And so also I've been reading Mike Tanier's blog postings in the NYT, hoping, as he does, that fortune will favor the New Yorkers while expecting that the Patriots will pull off a win.

Yesterday, Mike wrote about the transformation of the Super Bowl's host city. The post has an eye-popping typo in its headline which alerts you to the fact that this is a true weblog and not an article in the Grey Lady herself. The blog post is Everything Indy: The Invastion of the Roman Numerals in
The Fifth Down (New York Times N.F.L. Blog), January 30, 2012.*

Mike says "The N.F.L.’s fascination with Roman numerals is one of the league’s most bombastic traits, but the commitment to classical pretense pays off when the league builds on an epic scale. Park a giant '46' in the middle of a city, and passersby scratch their heads. Erect a hulking 'XLVI' into town central, and watch everyone stop to take pictures in front of it. ... While monolithic and omnipresent, the N.F.L.’s makeover is obviously temporary upon close inspection. The enormous numerals are made of fabric fastened over mesh, the sunburst pattern on the sides of the numerals resembling plywood when viewed from a distance."

{This image is in the public domain. Mike's blog shows one that's similar but copyright protected.}

The giant numerals remind me of another extravagant urban makeover, one that celebrated a naval victory over the Spanish Empire in 1898: the Dewey Arch.

Wikipedia explains:

The Dewey Arch was a triumphal arch that stood from 1899 to 1901 at Madison Square in Manhattan, New York. It had been erected for the parade in honor of Admiral George Dewey to celebrate his victory in the Battle of Manila Bay at the Philippines in 1898.

In spring 1899, planning for the parade, which was scheduled for September, began. Architect Charles R. Lamb found support for his idea of building a triumphal arch amongst the members of the National Sculpture Society, of which he also was a member. A committee of the society, comprising Lamb, Karl Bitter, Frederick W. Ruckstull, John Quincy Adams Ward, and John De Witt Warner, proposed the construction of an arch to the city of New York, which approved these plans in July 1899.

With only about two months left, it was decided to build the arch and its colonnade in staff, a material that had been used for the temporary buildings of several World's Fairs. Modeled after the Arch of Titus in Rome, the Dewey Arch was decorated with the works of 28 sculptors and featured a large quadriga (done by Ward) on top that showed four horses drawing a ship. At night, the arch was illuminated with electric light bulbs.

After the parade on September 30, 1899, the arch quickly began to deteriorate. An attempt to raise money to have the arch rebuilt with more durable materials (as had been done for the arch in Washington Square Park) failed, and thus the arch was demolished in 1901. The larger sculptures were sent to Charleston for an exhibit, and were destroyed afterwards.

Here is a photo of the Dewey Arch taken an employee of the Detroit Publishing Co. and found in collections of the Prints and Photos Division of the Library of Congress. It is a photochrom, that is to say a lithographic print made via an early type of colorizing that was generally used to make postcards.

{Caption: The Naval Arch at Madison Square, New York City, ca. 1900}

Here's the original from which the color version was made. It comes from a large-format glass plate negative.

{Caption: Madison Square and Dewey Arch, New York, N.Y., 1900, 1 negative, glass, 8 x 10 in.}

The photo just below is the right half of a panoramic view of Madison Square Park taken in 1905. It shows what the park and the intersection of Fifth Avenue and Broadway looked like without the Dewey Arch. See my blog post, Madison Park 1905, for details from the photo-pair. (I've written frequently about the park and its part of the city. To find the posts, search 'Madison Park' in this blog.)

{Caption: Madison Square from the Flat-Iron [i.e. Flatiron] Building, New York, 1905, 2 glass negatives, 8 x 10 in.}

Here are some details from the Dewey Arch photo.

1. You can tell that it's 20 past noon on a warm day. The park is full of people enjoying the shady peace it offers. It seems to be a work day; in other photos there are prams and children, but not this one. As usual the streets have lots of pedestrians and you can see bicyclists, two in this case, one of whom you see here, framed by the massive arch.


2. The wagon catches your eye, but also the crumbling imitation rock (called "staff") out of which the arch is constructed.


3. The contrast between Corinthian capitals and self-absorbed bench sitters is pointed, as they say.


3. A nice contrast here, as well, between imaginary heroic figures and a a very real New Yorker, pedestrian, literally, but I think not figuratively.


4. A contrast here between mundane and angelic with our interest falling more naturally with the former over the latter.


5. The great tower atop the old Madison Square Garden shows the only advertising on view in this photo.


-----------
*Update: the headline has been corrected: Everything Indy: The Invasion of the Roman Numerals

men holding rules

Shakespeare puts words in Cleopatra's mouth that tell more about class attitudes in his time than in hers:

                                      Mechanic slaves
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)

She'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.[1]

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.[2]

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.

{Illustration of surveyors with their equipment from The English Improver Improved or the Survey of Husbandry Surveyed by Walter Blith (London, 1652); source: MSU.edu hst425}

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.

{A navigator and his instruments from the late sixteenth century; source: Os oitavos na História blog}

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. [3]

{From: The sector on a quadrant, or, a treatise containing the description and use of three several quadrants Accommodated for dyalling, for the resolving of all proportions instrumentally, and for the ready finding the hour and azimuth; with large cuts of each quadrant, printed from the original plates graved by Henry Sutton, by John Collins (printed by J.Macock, to be sold by George Hurlock, William Fisher and Henry Sutton mathematical instrument maker, 1658)}

This is Sutton's quadrant, made from a plate he engraved for Collins' book.

{source: Royal Museums, Greenwich}

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.[4]

{This reads: "Of the Mathematical and Optical Instrument, and Spectacle-Maker.

"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.

{an Account of John Collins's Mathematical Works Reprinted from the Appendix to Sir E. Sherburne's Translation of Bk. 1 of the Astronomicon of Manilius. Edited by N. Brook (Nathanael Brook: London, 1675?)}

------

Some sources:

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

-------

Notes:

[1] 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).

[2] The making of the English middle class; business, society, and family life in London, 1660-1730 by Peter Earle (University of California Press, 1989)

[3] 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)

[4] 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)

Note on reproductions: Where not in the public domain, reproductions made in the blog post appear in accordance with fair use provisions of US copyright law. If you believe I have abused the privilege of fair use, please let me know.

mathematical practitioners

In his criticism of the universities for teaching subjects that have little practical value, John Webster asks (rhetorically) "What is Grammar, Lodgick, Rhetorick, Poesie, Politicks, Ethicks, Oeconomicks, nay Metaphysicks? if they serve to no other use than bare and fruitless speculation?" In arguing that the universities should emphasize mathematics and the empirical sciences which have some practical use he tacitly acknowledges that scholars can learn some mathematics at Oxford or Cambridge, but he says this math is the wrong kind. He asks "Can the Mathematical Sciences, the most noble, useful, and of the greatest certitude of all the rest, serve for no more profitable end, than speculatively and abstractively to be considered of?"[1]

He says, in other words, the math that's taught should be useful. Regarding this more profitable mathematics he asks, "How could the life of man be happily led, nay how could men in a manner consist without it? Truly I may justly say of it as Cicero of Philosophy, it hath taught men to build houses, to live in Cities and walled Towns; it hath taught men to measure and divide the Earth; more facilely to negotiate and trade one with another: From whence was found out and ordered the art of Navigation, the art of War, Engins, Fortifications, all mechanick operations, were not all these and innumerable others the progeny of this never-sufficiently praised Science?"

Webster was a preacher and not a particularly astute scholar. As I pointed out in my last post, he joined many of his contemporaries in believing magic, alchemy, and astrology to be subjects of equal weight with mathematics and natural philosophy (as what we call simply science was then called). In doing so he echoes a man, John Dee, who lived half a century before him and whom he calls a "myrror of manifold learning."[2] In a well-known preface to the first English translation of Euclid's Elements, Dee, like Webster, praises math and science as topics for university study and, just as much, magic, alchemy, and astrology.[3] And, like Webster, he says the application of mathematics is at least as important as is abstract speculation. In his words, "the very chief perfection (almost) of Numbers Practicall use" can be attained by the "mixtyng of speculation and practise."

In the preface Dee catalogs many of math's practical uses — from merchants' reliance on arithmetic, to the tangible uses of algebra[4], and to the many uses of geometry made by surveyors, military commanders, navigators, builders, excise men, and others. With regret Dee says he's been writing against a deadline ("the Printer, hath looked for this Præface, a day or two") and tells us his subject "is so ample and wonderfull, that, an whole yeare long, one might finde fruitfull matter therin, to speake of: and also in practise, is a Threasure endeles."

Despite the passion he shows for his subject in the preface, Dee's life was devoted more to the intangible aspects of math than the material ones. It's true he used Euclidian geometry to solve problems of navigation and trained the crews of ships so they could find their way across the Atlantic in early voyages to North America, but he believed his life's mission to be the uncovering of the spiritual forms underlying the material world. To him math was a language for use in speaking with angels. He associated its abstract beauty with mystical powers of divination which he claimed to possess.

Of the angelical beauty of mathematics he wrote: "All thinges ... do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator. ... By Numbers propertie ... we may ... ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible, mortall and immortall, Corporall and Spirituall."[5]

The only son of a minor member of the royal court, he had a brilliant career at university and possessed both inclination and sufficient means to extend his education after graduation through extensive travel in Europe. Not himself wealthy, he was able to make himself useful to wealthy members of the aristocracy of England and the European continent. There appears to have been no snobbishness in him however. At a time when "gentles" treated unlettered artisans with contempt, scorn, or — at best — indifference, and when dramatists could be sure to draw laughs by poking fun at men whom they characterized as "rude mechanicals"[6], Dee was unusual in the sympathetic recognition he gave to the emerging class of "Common Artificer."

He closes the Preface by citing advantages of instruction — not in the Latin of the universities but in the English of the shop and street — made available to London tradesmen, many of whom were the first of their families to have acquired the ability to read. Of the book in which the Preface appears — Billingsley's translation of Euclid's Elements (which, as I say, was the first version to be published in English) — he writes: "[H]ow many a Common Artificer, is there, in these Realmes of England and Ireland, that dealeth with Numbers, Rule, & Cumpasse: Who, with their owne Skill and experience, already had, will be hable (by these good helpes and informations) to finde out, and devise, new workes, straunge Engines, and Instrumentes: for sundry purposes in the Common Wealth? or for private pleasure? and for the better maintayning of their owne estate?"

This title page to Billingsley's Euclid depicts some of the practical applications of mathematics to which Dee refers. (It's also, you'll notice, not prudish in depicting naked bodies.)

{The title page of Henry Billingsley's translation of Euclid's Elements (1570), with preface by John Dee; source: wikipedia}

It's generally thought that Dee's Preface and all of Billingsley's Euclid helped set in motion a gradual shift in attitudes toward mathematics and the book's readers appear to have fostered the growth of applied mathematics outside the universities. As a partial result, it's pretty clear that in the century following its appearance mathematical practitioners — the men who employed mathematics in their work as well as the printers and authors of practical math texts, the makers of technical instruments, and the technical advisors who assisted university-trained natural philosophers — became more prosperous and grew somewhat in social standing.[6]

----

Some sources:

The Mathematical Practitioners of Tudor and Stuart England by E.G.R. Taylor (Cambridge, 1954)

John Dee's "Mathematicall Praeface": A Sixteenth Century Classification of the Mathematical Arts and Sciences by Charles St. Clair (pdf)

The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara by John Dee from Sir Henry Billingsley's first English version of Euclid's Elements, 1570

"John Dee" by Thompson Cooper in Dictionary of national biography ed. by Leslie Stephen and Sidney Lee (Smith, Elder, & co., 1888)

"John Dee and His Supplication to Queen Mary" by P. Evans Lewin, Woolwich Public Libraries in The Library world, Vol. 5 (Library Supply Co., 1903) Extract: 'Whilst at Cambridge he only slept four hours every night, and spent eighteen hours of the day in study. So great was his knowledge, that his acquaintance was eagerly sought by such men as Gemma Frisius, Mercator, and Gaspar a Mirca, all of whom he visited in his twentyfirst year. Even at this period he was looked on askance, for he relates that in 1547 he "sett forth" at Trinity College a Greek comedy of Aristophanes, "with the performance of the Scarabaeus, his flying up to Jupiter's palace with a man and his basket of victuals on her back, whereat was great wondering and many vain reports spread about." This, probably, was only a piece of stage mechanism suitable to the crude ideas of the time and in keeping with Greene's instructions in "Tamburlaine" — "exit Venus; or if you can conveniently let a chair come down from the top of the stage and draw her up."'

"The Mistaking of 'the Mathematicks' for Magic in Tudor and Stuart England" by J. Peter Zetterberg in The Sixteenth Century Journal, Vol. 11, No. 1 (Spring, 1980), pp. 83-97. Stable URL: http://www.jstor.org/stable/2539477

"Science and Education in the Seventeenth Century: The Webster-Ward Debate" by G. Allen, reviewed by Theodore M. Brown in Isis, Vol. 64, No. 3 (Sep., 1973), pp. 422-424. (The University of Chicago Press) Stable URL: http://www.jstor.org/stable/2297

A general dictionary: historical and critical, in which a new and accurate translation of that of the celebrated Mr. Bayle, with the corrections and observations printed in the late edition at Paris, is included; and interspersed with several thousand lives never before published. The whole containing the history of the most illustrious persons of all ages and nations particularly those of Great Britain and Ireland, distinguished by their rank, actions, learning and other accomplishments. With reflections on such passages of Bayle, as seem to favor scepticism and the Manichee system, Volume 10 by Pierre Bayle, John Peter Bernard, Thomas Birch, John Lockman, George Sale, Alexis Gaudin, Anthelme Tricaud, Pierre Desmaizeaux (Printed by J. Bettenham, 1741)

Billingsley Euclid in Mathematical Treasures by Frank J. Swetz and Victor J. Katz

Shakespeare from the margins by Patricia A. Parker (University of Chicago Press, 1996)

---------

Notes:

[1] My quotes from John Webster come from his Academiarum Examen: Academiarum examen, or the examination of academies wherein is discussed and examined the matter, method and customes of academick and scholastick learning by John Webster (Calvert, 1654).

[2] My quotes from John Dee come from his The Mathematicall Praeface to Elements of Geometrie of Euclid of Megara by John Dee from Sir Henry Billingsley's first English version of Euclid's Elements, 1570. I have modernized Dee's use of the letter "u" where we would put "v" and given the "long s" (ſ) as "s".

[3] Dee's passion for mathematics leads him to claim (quoting Boetius) that "All thinges (which from the very first originall being of thinges, have bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator." And further: "By Numbers propertie therefore, of us, by all possible meanes, (to the perfection of the Science) learned, we may both winde and draw our selves into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and Formes: And also, farder, arise, clime, ascend, and mount up (with Speculatiue winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and invisible, mortall and immortall, Corporall and Spirituall."

[4] Of algebra he says: "This Rule, and Arithmetike of Algiebra, is so profound, so generall and so (in maner) conteyneth the whole power of Numbers Application practicall: that mans witt, can deale with nothyng, more proffitable about numbers: nor match, with a thyng, more mete for the divine force of the Soule, (in humane Studies, affaires, or exercises) to be tryed in."

[5] Here's the full quote: "All thinges (which from the very first originall being of thinges, have bene framed and made) do appeare to be Formed by the reason of Numbers. For this was the principall example or patterne in the minde of the Creator. O comfortable allurement, O ravishing perswasion, to deale with a Science, whose Subiect, is so Auncient, so pure, so excellent, so surmounting all creatures, so used of the Almighty and incomprehensible wisdome of the Creator, in the distinct creation of all creatures: in all their distinct partes, properties, natures, and vertues, by order, and most absolute number, brought, from Nothing, to the Formalitie of their being and state. By Numbers propertie therefore, of us, by all possible meanes, (to the perfection of the Science) learned, we may both winde and draw our selves into the inward and deepe search and vew, of all creatures distinct vertues, natures, properties, and Formes: And also, farder, arise, clime, ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible, mortall and immortall, Corporall and Spirituall."

[6] A Midsummer Night's Dream: Act 3, Scene 2

PUCK
6 My mistress with a monster is in love.
7 Near to her close and consecrated bower,
8 While she was in her dull and sleeping hour,
9 A crew of patches, rude mechanicals,
10 That work for bread upon Athenian stalls,
11 Were met together to rehearse a play
12 Intended for great Theseus' nuptial-day.
13 The shallowest thick-skin of that barren sort,
14 Who Pyramus presented, in their sport
15 Forsook his scene and enter'd in a brake
16 When I did him at this advantage take,
17 An ass's nole I fixed on his head:
18 Anon his Thisby must be answered,
19 And forth my mimic comes. When they him spy,
20 As wild geese that the creeping fowler eye,
21 Or russet-pated choughs, many in sort,
22 Rising and cawing at the gun's report,
23 Sever themselves and madly sweep the sky,
24 So, at his sight, away his fellows fly;
25 And, at our stamp, here o'er and o'er one falls;
26 He murder cries and help from Athens calls.

[7] Stephen Johnston makes this point. See The identity of the mathematical practitioner in 16th-century England from 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. See also: The making of the English middle class; business, society, and family life in London, 1660-1730 by Peter Earle (University of California Press, 1989)