Maligned for mathematics: Sir Thomas Urquhart and his Trissotetras

ABSTRACT

Thomas Urquhart (1611–1660), celebrated for his English translation of Rabelais’ Gargantua et Pantagruel, has earned some notoriety for his eccentric, putatively incomprehensible early book on trigonometry The Trissotetras (1645). The Trissotetras was too impractical to succeed in its own day as a textbook, since it lacked both trigonometric tables and sample calculations. But its current bad reputation is based on literary authors’ amplifications of the verdict prefaced to its 19th century reprinting by one mathematician, William Wallace, who lacked the background to appreciate the book’s historical context. Considering that context (including seventeenth century ‘copious’ prose, and medieval logic and ‘art of memory’), the bad reputation is undeserved: the book is mathematically clear, clever (e.g. in superimposing 16 problems into one diagram), and complete. The Trissotetras may thus be viewed as simply one more of Urquhart’s polymathic projects and involvements – which included education, rise of the middle class, religious and class conflicts, development of science and mathematics, search for patronage, universal language construction, and development of English prose – which serve to make him a lively and instructive intellectual Everyman for his time.

Acknowledgment

Deep thanks to Alan Rocke, Henry Eldridge Bourne Professor Emeritus and Distinguished University Professor Emeritus at Case Western Reserve University, for his extraordinary effort and kindness in guiding me through this long project. Thanks also to Prof. Glen Van Brummelen for his helpful comments on an earlier version of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Introduction to Thomas Urquhart, The Works of Sir Thomas Urquhart of Cromarty, Knight (New York, 1971, reprinted from the original editions, Edinburgh, 1834, xvi–xvii); also in Urquhart's only full-length biography (John Willcock, Sir Thomas Urquhart of Cromartie Knight (London, 1899, pp. 119–20)). Wallace retired from his professorship in 1838, ‘having been unable to perform his duties in person during the three previous seasons’ (Untitled anonymous obituary notice about William Wallace, Monthly Notices of the Royal Astronomical Society, 6 (1845), 31–41 (p. 37)). Thus (if ‘seasons’ are years, and printing was not delayed), his verdict on Urquhart printed in 1834 came only a year before that first year of incapacity. Possibly the onset of ill health contributed to his inability to give Urquhart a fair consideration.

2 Charles Whibley, Studies in Frankness, 1st ed. (Port Washington, NY, 1970 reissue of l898), pp. 227–62 calls Urquhart ‘the greatest translator of all time’ (244), and later elaborates (255):

… it is to his translation of Gargantua and Pantagruel that Urquhart owes his immortality, and surely no man better deserves the wreath of undying fame. His masterpiece shares the honour with our own Authorised Version of being the finest translation ever made from one language into another.

And in his introduction to François Rabelais, Gargantua and Pantagruel, trans. by Thomas Urquhart and Peter le Motteux (London: Tudor Translations, 1900; reprinted New York, 1967), p. lxxix, Whibley calls Urquhart's book ‘a translation, unique in its kind, which has no rival in profane letters’.

3 For information on the Maitland Club see Graeme Morton, ‘Historical Clubs and Societies’, in The Oxford Companion to Scottish History, ed. by Michael Lynch (Oxford, 2001), pp. 310–11; and Charles Sanford Terry, A Catalogue of the Publications of Scottish Historical and Kindred Clubs and Societies (Glasgow, 1909), pp. vii–x and 100–16. Founded in Glasgow in 1828 ‘to print works illustrative of the antiquities, history, and literature of Scotland’, the Maitland Club was one of the scores of historical clubs and societies that in the 1800s played an important role in sustaining Scottish culture and national identity. The works of Urquhart was #30 of the Maitland Club's 77 publications (which ceased in 1859); #47 was De arte logistica by John Napier.

4 The fullest biography of Wallace is the untitled anonymous obituary notice in the Monthly Notices of the Royal Astronomical Society (note 1), evidently written by a well-informed friend. A modern study is M. Panteki, ‘William Wallace and the Introduction of Continental Calculus to Britain: A Letter to George Peacock’, Historia Mathematica, 14 (1987), (pp. 119–32). Briefer biographies, which appear to draw most of their facts from the obituary notice, are T. A. A. Broadbent, ‘Wallace, William’, in Dictionary of Scientific Biography, ed. by Charles Coulston Gillispie (New York: Charles Scribner's Sons, 1976), vol. 14, 140–41; Robert Chambers, ‘Wallace, William’, in A Biographical Dictionary of Eminent Scotsmen (Hildesheim, New York: G. Olms, 1971 reprint of 1872 new ed.), vol. 3, pp. 489–90; Wallace, William, Encyclopaedia Britannica, 9th edn (1889), vol. 24, (1889) p. 348. J. C. Poggendorff, Biographisch-Literarisches Handwörterbuch zur Geschichte der exacten Wissenschaften, (Leipzig, 1863), vol. 2, pp. 1250–51; reprinted (Ann Arbor, Michigan: Edwards Bros., 1945); and George Stronach, ‘Wallace, William’, in Dictionary of National Biography: From the Earliest Times to 1900, ed. by Leslie Stephen and Sidney Lee, (London: Oxford University Press), (1921–1922 reprinting), vol. 20, pp. 572–73.

5 Note that this anecdote suggests Wallace never learned Greek, the language that fascinated Urquhart and was the source and explanation for so much of his Trissotetras’s intimidating new terminology like ‘loxogonospherical’.

6 Inventoried as PG 195 in Helen Smailes, The Concise Catalogue of the Scottish National Portrait Gallery (Edinburgh, 1990). Andrew Geddes (1783–1844) was a well-known Scottish portrait painter in Edinburgh and London: see Robin Simon, ‘Geddes, Andrew’, in Dictionary of Art, ed. by Jane Turner (London: Macmillan, 1996), vol. 12, p. 228; John Miller Gray, ‘Geddes, Andrew’, in Dictionary of National Biography, ed. by Leslie Stephen and Sidney Lee (London: Oxford University Press, 1921–1922), vol. 7, pp. 980–81; and Helen Smailes, ‘Andrew Geddes, 1783–1844: Painter-printmaker: ‘A Man of Pure Taste’ (catalogue of an exhibition held at the National Gallery of Scotland, Edinburgh, 15 February to 29 April, 2001) (Edinburgh: National Gallery of Scotland, 2001).

7 ‘ … he brought the maturity of his experience as a teacher, as well as his rich scientific acquirements as a mathematician, to a chair but too often filled with men unpractised in the common ways of life, and whose whole occupation is to muse and dream over a problem. Many of the scientific men of the present day can still remember, with gratitude, the efficiency with which Mr. Wallace discharged the duties of his professorship, and the impulse which his teaching imparted to their studies’. (Chambers (note 4), p. 489.

9 #1: See note 1.

#2: Willcock (note 1), 118 and 121.

#3: Rabelais (note 2), xcii.

#4: Thomas Seccombe, ‘Urquhart or Urchard, Sir Thomas’, in Dictionary of National Biography: From the Earliest Times to 1900, ed. by Leslie Stephen and Sidney Lee, (London, 1921–1922 reprinting), vol. 20, pp. 46–50 (p. 47).

#5: Morris Bishop, A Gallery of Eccentrics, or A Set of Twelve Originals & Extravagants from Elagabalus, the Waggish Emperor to Mr. Professor Porson, the Tippling Philologer, Designed to Serve, by Example, for the Correction of Manners & for the Edification of the Ingenious (New York, 1928), pp. 53–68 (p. 64). His conclusion: ‘So you have Sir Thomas Urquhart; a valiant warrior for doomed causes, a searcher-out of unwanted learning, a conceiver of fantastic inventions, the Scottish incarnation of Rabelais, and over and above all the greatest word-maker and word-lover in the records of our tongue. He was indeed a little mad; but he may be excused in the terms of Aristotle: Nullum magnum ingenium sine mixtura dementiae’. (67–68)

#6: Florian Cajori, A History of Mathematical Notations, Vol. II: Notations Mainly in Higher Mathematics (Chicago, 1929), p. 164.

#7: Thomas Urquhart, Selections from Sir Thomas Urquhart of Cromarty, ed. by John Purves (Edinburgh, 1942), p. 12.

#8: Harder, Kelsie B., ‘Sir Thomas Urquhart's Definition of Wit’, Notes and Queries, 1 (1954), 154–55; ‘Sir Thomas, Urquhart's Rimed Moral Discourse’, Notes and Queries, 3 (1956), 13–17; ‘Urquhart's “Lexicidion” and the O.E.D.’, Notes and Queries, 3 (1956), 70–72; ‘Sir Thomas Urquhart's “Trissotetras”’, Notes and Queries, 3 (1956), 237–39; ‘Sir Thomas Urquhart's Universal Language’, Notes and Queries, 3 (1956), 473–76; ‘Izaak Walton's Adventure and Urquhart's “Ekskubalauron”’, Notes and Queries, 4 (1957), 10.

#9: F. C. Roe, Sir Thomas Urquhart and Rabelais (London: Taylorian Lecture, 1957), p. 8.

#10: Edwin George Morgan, ‘Sir Thomas Urquhart’s, Encyclopaedia Britannica, 22 (1964), 902–03.

#11: ‘Urquhart, Sir Thomas’, Encyclopaedia Britannica, X (1974), 303.

#12: Thomas Urquhart, The Admirable Urquhart: Selected Writings Edited and Introduced by Richard Boston (London, 1975).

#13: Roger J. Craik, Sir Thomas Urquhart of Cromarty (1611–1660) Adventurer, Polymath, and Translator of Rabelais (Lampeter, 1993), discussing The Trissotetras on pp. 13–20.

#14: David Fergus, ‘Rhapsodies, Incoherent, Unintelligible and Extravagantly Absurd: The Works of Sir Thomas Urquhart’, Scottish Book Collector, 3 (1993), 7–11 (p. 7).

#15: John B. Corbett, University of Glasgow, English Language, John B. Corbett, MA, PhD . <http://www.gla.ac.uk/departments/englishlanguage/staff/johnbcorbett> [accessed 29 October 2010; ‘Prose Urquhart,’ Scottish Corpus of Texts and Speech, Document 28, (2004). <http://www.scottishcorpus.ac.uk/corpus/search/document.php?documentid=28> [accessed 28 August 2009]. ‘Lecture on Scottish Literature 2’, Scottish Corpus of Texts and Speech, Document 599, (2004). <http://www.scottishcorpus.ac.uk/corpus/search/document.php?documentid=599>. [accessed 28 August 2009]. In these lectures apparently not realizing that Urquhart is proving the Pythagorean theorem, Corbett seems to think he is giving an absurdly elaborated statement of the theorem. Urquhart's proof is not that easy to read because (as usual) there is no labeled figure to follow. Possibly Urquhart intended it merely as a reminder, assuming that anyone studying trigonometry would have already been through Euclid. For it is in fact a form of the simple proof–splitting the triangle to two similar smaller ones–which appears in Euclid VI.8 and 31 and has been, since at least the seventeenth century, the most common argument (Euclid, The Thirteen Books of Euclid's Elements, trans. by T. L. Heath, Great Books of the Western World, (Chicago, 1952), vol. 11, pp. 106, and 123–124; Eli Maor, The Pythagorean Theorem: A 4,000-year History (Princeton, 2007), pp. 41–42).

8 Biographical sources (see Table 1: Critical Opinion, below): Two excellent modern studies are Craik (#13) and Boston's introduction (#12). The only full-length biography of Urquhart is Willcock (#2). Briefer accounts are in Whibley’s introduction to Rabelais (#3), Purves's introduction to Urquhart (#7), and Roe (#9). Earlier sources are covered in Seccombe (#4). Selected literary aspects are discussed in Harder's articles (#8).

Accounts featuring Urquhart's eccentricity by Bishop (#5), MacDiarmid, and Powys appear to take their facts primarily from Urquhart’s own writings Logopandecteision (quoted in Craik, 2, Willcock, 21–25, and Boston, 10) and Ekskubalauron (quoted in Purves, 30, and in part in Craik, 27, Willcock, 154, and Boston, 27), and from Willcock. Hugh MacDiarmid, Scottish Eccentrics (New York, 1972 reprint of 1936 edition), pp. 26–56, highlights Urquhart's extravagant prose works, but does not mention The Trissotetras. Llewelyn Powys, Thirteen Worthies (London,1924), pp. 79–90 celebrates Urquhart with a deft touch – ‘the inimitable translator of Rabelais’ (79), in whose own works ‘buffoonery and wisdom stand so intermixed that whether they reach the height of super-subtle sagacity or the depths of fantastical folly remains still an open question’ (86) – and from The Trissotetras quotes just a number of its unusual made-up words (87).

For Urquhart's family history see Henrietta Tayler, History of the Family of Urquhart (Aberdeen, 1946). This is an over 300 page-long monograph presenting information collected over a 30-year period by the author and her late brother Alistair Tayler, who were specialists in Scottish family history; the ‘great Sir Thomas Urquhart’ is the centre and focus of the book. Thomas Innes of Learney, The Tartans of the Clans and Families of Scotland, 8th ed. (Edinburgh, 1971) gives authoritative one-page family histories, and illustrations of the badges, slogans, heraldic devices, and tartans, for over a hundred Scottish families, including Urquhart. Clan tartans began around the seventeenth century, but were systematized only in the nineteenth; the Urquhart tartan is a plaid of wide dark blue and narrow red stripes on a dark green background.

10 Urquhart, Logopandecteision (see note 8 above); amplified in Willlcock (note 1) pp. 9–29.

11 James Ernst, Roger Williams: New England Firebrand (New York, 1932), pp. 327–28. The letters of Cromwell (Wilbur Cortez Abbott, The Writings and Speeches of Oliver Cromwell (Cambridge, 1939)) include two petitions from Urquhart.

Did Urquhart ever interact with the man who would become the greatest literary figure of the age, John Milton (1608–1674)? Milton was a secretary to the council of state, and went totally blind, during the winter of 1651–1652 when Urquhart was in prison. There is no recorded contact with Urquhart in Milton's life records [J. Milton French, The Life Records of John Milton (New Jersey, 1954); for a clear precise chronology see also Gordon Campbell, A Milton Chronology (Houndmills, Bassingstoke, Hampshire, 1997)], or in Milton’s biography [David Masson, The Life of John Milton: Narrated in Connexion with the Political, Ecclesiastical, and Literary History of his Time (Gloucester, MA, 1965); for a fully narrated biography see also William Riley Parker, Milton: A Biography, vol. 2 (London, 1968)]. But Hugh C. H. Candy, ‘Milton, N.LL, and Sir Tho. Urquhart’, Library, 14 (1934), 470–76, found an intriguing clue in an anonymous tract, published in 1653, which he attributes to Urquhart, both on stylistic grounds (it is 29 pages long and supposedly written in one day!) and because he found one copy with an apparently genuine signature of Urquhart. This tract quotes extensively from a letter signed ‘N.LL’, which Candy interprets as from Milton to Marvell or Wall, suggesting some connection between Urquhart and Milton did occur. (An imagined meeting of the two men also forms the basis for Alasdair Gray's short story ‘Sir Thomas's Logopandocy’, Note 15 below.)

12 Distinguished ancestors include Methusalah, Noah, Narfesia (Queen of the Amazons), Termuth (Pharaoh's daughter who rescued Moses), Panthea (daughter of Deucalion), Hypermnestra (choicest of the 50 Danaids), Thymelica (daughter of Bacchus), and Nicolia (Queen of Sheba).

13 ‘ … you ar but as a yesterdayes grouen-up mushrome, mishapenly swolen to the greatnes quhairin … your most detestable, infamous, abominable, and viperis actings … you are a coward, a knave, a skelme, and a lier, iff ye stand not to it’ – Tayler (note 8) 55–58.

14 John Aubrey, Aubrey’s Brief Lives, ed. by Oliver Lawson Dick (London, 1949), p. 225. Willcock (note 1) 97–101 (see also Craik (Table 1 #13) 37) explains that the story, taken from Lucian (MAKPOBIOI (‘Octogenarians’) c.25 – doubtfully attributed to Lucian, in The Works of Lucian, trans. by A. M. Harmon, ((London: Loeb Classical Library, 1927), vol. 1, pp. 242–43) or Valerius Maximus, Memorable Doings and Sayings, ed., trans. by D. R. Shackleton Bailey, Loeb Classical Library, No. 493, (Cambridge, MA, 2000), vol. II, pp. ix. 12, 376–77), how Philemon died of laughter at seeing an ass eat the figs set out on the dinner table, is mentioned in Rabelais i 10, i 20, and iv 17. The same story is also, incidentally, told for the death of the philosopher Chrysippus, in Diogenes Laertius, Lives of Eminent Philosophers, trans. by R. D. Hicks, Loeb Classical Library, No. 185, (Cambridge, MA, 1979), vol. II, pp. 292–93.

15 Politics: At least seven Urquharts have been members of Parliament – Tayler (note 8) 15, 113, 139, 242, 252, and 261. Mathematics: Alasdair Urquhart (1945-), University of Toronto emeritus professor of philosophy, is an eminent researcher in logic, computational complexity theory, history of logic, and non-classical logics – http://philosophy.utoronto.ca/people/emeritus-faculty/alasdair-urquhart (printed 10/2/2009). Fiction: Alasdair Gray, ‘Sir Thomas's Logopandocy,’ in Unlikely Stories, Mostly, 134–92 (story); drawings representing Sir Thomas throughout the book, Edinburgh: Canongate Classics republication of 1983 Canongate edition, 1997; Andrew Drummond, A Hand-book of Volapük (Edinburgh, 2006).

16 Eric Temple Bell, Men of Mathematics (New York, 1937); David Eugene Smith, History of Mathematics, Vol. 1: General Survey of the History of Elementary Mathematics (New York, 1958); J. L. Heilbron, Galileo (Oxford, 2010), Chapter 1; Maria Rosa Antognazza, Leibniz: An Intellectual Biography (Cambridge, 2009).

17 Aubrey, note 14, pp. 150–51; 157–58 for relations with Galileo and Descartes (good) and Wallis and Boyle (poor).

18 It is unfortunate that this resource no longer exists today. Who would understand a comparison of Yeats's winding staircase to a Riemann surface?

19 John Donne, ‘A Valediction Forbidding Mourning’, in Robert M. Adams, ‘The Seventeenth Century’, in The Norton Anthology of English Literature, ed. by M. H. Abrams et al. (New York: W. W. Norton & Co.,1962), vol. 1, pp. 737–1125, pp. 767–68.

Our two souls therefore, which are one,/Though I must go, endure not yet/A breach, but an expansion, / Like gold to airy thinness beat.

If they be two, they are two so / As stiff twin compasses are two; / Thy soul, the fixed foot, makes no show/To move, but doth, if th' other do.

And though it in the centre sit, / Yet when the other far doth roam, /It leans and hearkens after it, /And grows erect, as that comes home.

Such wilt thou be to me, who must / Like th' other foot, obliquely run; /Thy firmness makes my circle just, /And makes me end where I begun.

20 Adams (note 19) pp. 1061–2; see also p. 752. For a biographical and critical study of Browne see Jonathan F. S. Post, Sir Thomas Browne, Twayne's English Authors Series No. 448 (Boston, 1987). Post mentions (8) that during his Oxford education Browne would have encountered geometry professor Henry Briggs, the inventor of common logarithms. Also, much as Urquhart delighted in Greek neologisms, Post points out that Browne ‘hatched an exotic species of Latinate English’ with a ‘brood of verbal oddities nearly unequaled in the seventeenth century’, including such words as: reminiscential, paradoxologie, ampliate, empuzzle, decollation, immoderancy, farraginous, indigitate, augurial, tripudary, fabulosities, sententiosity, desume, extispicious, mundification, and consectory (58).

21 Thomas Browne, ‘The Garden of Cyrus, or The Quincunciall, Lozenge or Net-work Plantations of the Ancients, Artificially, Naturally, Mystically Considered,’ in Urne Buriall and the Garden of Cyrus, ed. by J. Carter (Cambridge, 1958).

22 Ibid. 86.

23 Ibid. 114.

24 Desiderius Erasmus of Rotterdam, On Copia of Words and Ideas (De utraque verborum ac rerum copia), trans. by Donald B. King and H. David Rix, Mediaeval Philosophical Texts in Translation No. 12 (Milwaukee, 1963) – a partial edition, with an informative translator's introduction. For an overview of copia: Neil Rhodes, The Power of Eloquence and English Renaissance Literature (New York, 1992), pp. 41–63; for copia in more theoretical and Continental aspects: Terence Cave, The Cornucopian Text: Problems of Writing in the French Renaissance (Oxford, 1979), pp. 3–34; for shorter summaries: Marjorie Donker and George M. Muldrow, Dictionary of Literary-rhetorical Conventions of the English Renaissance (Westport, Connecticut, 1982), pp. 58–60 (copia), 33–36 (Ciceronian style), 187–90 (Senecan style).

25 T. W. Baldwin, William Shakspere’s Small Latine & Lesse Greek (Urbana, 1944) is a massive two-volume study of English grammar school education such as Shakespeare might have received in the late 1500s. Baldwin finds that this education – focused on the study of classical authors – was highly uniform across England, and changed little over three centuries (see e.g. 1: 98, 163, Chapt. VIII. The Movement toward Authorized Uniformity, and Chapt. XIII. Definitive Form Attained under Edward VI). The role of Erasmus is in Chapt. IV. Erasmus Laid the Egg; De Ratione Studii, Chapt. V. Erasmus Laid the Egg; His Textbooks and Approved Authors, Chapt. VI. The Egg which Erasmus Laid at Paul's, and Chapt. XXXVI. The Rhetorical Training of Shakspere: Erasmus, Copia. A confirmatory picture appears in books on the English grammar schools: Foster Watson, The Old Grammar Schools (Cambridge, 1916); W. A. L. Vincent, The Grammar Schools: Their Continuing Tradition 1660–1714 (London, 1969); Foster Watson, The English Grammar Schools to 1660: Their Curriculum and Practice (Cambridge, 1908); and Howard Staunton, The Great Schools of England, new edition (London, 1869).

26 Examples:

On receiving your letter I was carried away with joy … 

Immortal God! what great joy came to us from your letter? … 

What laughter, what applause, what exultant dancing your letter caused in me … 

Your pen has sated me with joys … 

What clover is to bees, what willow boughs are to goats, what honey is to the bear, your letter is to me … 

Shakespeare gleefully parodies Erasmian copia in the speech of the schoolmaster Holofernes in Love’s Labour’s Lost, Act IV, Scene 2:

Holofernes: The deer was, as you know, sanguis, – in blood; ripe as a pomewater, who now hangeth like a jewel in the ear of coelo, – the sky, the welkin, the heaven; and anon falleth like a crab on the face of terra, – the soil, the land, the earth.

(This Shakespeare reference and the others in this note are from Baldwin (note 25). By an old tradition Shakespeare had himself once been a country schoolmaster (Baldwin, Vol. 1, pp. 34–38), which might help account for the accuracy of his satire. As You Like It, V, 1, 52ff. has another parody of copia, while Taming of the Shrew, III, 1 has wooing under the guise of a lesson in Latin translation, and Merry Wives of Windsor, IV, 1 parodies a Latin grammar exam). Robert Burton (1577–1640) is another famous writer of seventeenth century copia:

How would our Democritus have bin affected, to see a wicked caitiffe, or foole, a very idiot, a funge, a golden asse, a monster of men, to have many good men, wise men, learned men to attend upon him … 

Robert Burton, The Anatomy of Melancholy, the Clarendon edition, ed. by Thomas C. Faulkner, Nicolas K. Kiessling, and Rhonda L. Blair (Oxford, 1989), vol. 1, p. 48. The italicized beginning of the passage is Burton's quotation from Utopia by Thomas More (who was a close personal friend of Erasmus). For stylistic overviews: Roger Pooley, English Prose of the Seventeenth Century, 1590–1700 (London, 1992), pp. 197–204; Michael O’Connell, Robert Burton, Twayne's English Authors Series 426 (Boston, 1986), pp. 75–91; Adams (note 19), pp. 752–53.

27 Otto Bird, Syllogistic and its Extensions (Englewood Cliffs, New Jersey, 1964) pp. 1–30, esp. 22–23.

28 Frances A. Yates, The Art of Memory (Chicago, 1966), pp. 1–128, and 266–86; Paolo Rossi, Logic and the Art of Memory: The Quest for a Universal Language, trans. by Stephen Clucas of the 1983 second edition of: Clavis universalis: Arti della memoria e logica combinatoria da Lullo a Leibniz, 2nd ed. (Chicago, 2000 translation of 1983) pp. xxi–xxviii and 1–28; and Mary J. Carruthers, The Book of Memory: A Study of Memory in Medieval Culture, Cambridge Studies in Medieval Literature 10 (Cambridge, 1990), pp. 2–8, 71–79,109–11, and 127–37.

29 Dard Hunter, Papermaking: The History and Technique of an Ancient Craft, 2nd ed. (New York, 1978 reprint of 1947) – a book presenting (vii) forty years of study; see map preceding Chapter 1.

30 Baldwin (note 25): System unaltered for 300 years: 1:142; memorization: 1:135, 136–37, 154 (see also 160), and 287; avoidance of ‘artificial’ memory techniques: 2:100–2. Books on the English grammar schools (note 25) also confirm the central role of memory work in the curriculum: Watson, The Old Grammar Schools, pp. 101–11; Vincent 58–92 (esp. 75–77); Watson, The English Grammar Schools, pp. 293–304 (esp. 293–94); and Staunton 24 (Eton) and 73 (Winchester).

31 The technique is apparently a natural one: The astonishing twentieth century mnemonist Sherashevsky whom the Russian psychologist A. R. Luria describes in his Mind of a Mnemonist (neither author nor subject seems aware of the classical art of memory) had reinvented the same technique, using the buildings of the street where he lived, or of his childhood hometown, as loci (A. R. Luria, The Mind of a Mnemonist: A Little Book about a Vast Memory, trans. by Lynn Solotaroff (Cambridge, MA, 1968)). Luria found that Sherashevsky's memory had no apparent limits in capacity or durability (e.g. the man could recall material perfectly when tested without warning sixteen years later) (11–12). The basis was synesthetic imagery, any sound immediately producing for him also sensations of light, colour, taste, and touch (21–29). Unfortunately he, unlike Aquinas, could not use his talent effectually. For the very intensity of his memory processing tended to overwhelm his capacities for abstract thought or planning, making him ponderous, timid, dull, awkward, and absent-minded (9,157).

32 Hugh Platt, The Jewell House of Art and Nature (London, 1594; reprinted Amsterdam, 1979, as facsimile Number 950 in The English Experience, pp. 81–85, section 98: ‘The Art of memorie which master Dickson the Scot did teach of late yeres in England’. See Yates (note 28) 266–86 for more on Dickson and his publications as mnemonist and controversialist.

33 This tactic agrees with modern neuropsychology, which has identified a specific brain locus, the amygdala, that functions to link emotion to memory: James L. McGaugh, ‘Affect, Neuromodulatory Systems, and Memory Storage’, pp. 245–68, and Joseph E. LeDoux, ‘Emotion as Memory: Anatomical Systems Underlying Indelible Neural Traces’, in The Handbook of Emotion and Memory: Research and Theory, ed. by Sven-Åke Christianson (Hillsdale, N.J.,1992), pp. 269–88; Eric R. Kandel, In Search of Memory: The Emergence of a New Science of Mind (New York, 2006), pp. 335–51 and 477–79. Other mnemonic techniques similarly aim to make their images striking, or funny, or obscene, or in some other way memorable. Peter of Ravenna attributed his fabulous memory to filling his memory-places with images of beautiful women (Rossi, note 28, p. 22). Another system used vivid multilingual puns: for instance, to remember the English battle at Berwick, one might think of a man holding an eel (‘anglia’) in one hand, and a mighty bear (‘Bere-wic’) in the other (Carruthers, note 28, p. 135). Alphabets offer ready-made ordered loci for memory work, and Carruthers (note 28) pp. 109–11,127–30) and Yates (note 28) pp. 91–101) suggest that medieval alphabets, bestiaries, and possibly other ‘grotesque’ art of the time may actually represent memory images.

34 ‘I had almost omitted to tell you, that for the more variety in the last two Figures of the Orthogonosphericals are set downe the two letters of Ch. and Shin, the first a Spanish, and the second an Hebrew letter. Now if to those helps for the memorie which in this Table I have afforded the Reader, both by the Alphabetical order of some Consonants, and homogeneity of others in their affections of sharpnesse, meannesse, obtusity, and duplicity, he joyne that artificiall aid in having every part of the Schemes locally in his mind (of all wayes both for facility in remembring, and stedfastnesse of retention, without doubt, the most expedite) or otherwise place the representatives of words, according to the method of the Art of memory, in the severall corners of a house (which, in regard of their paucity are containable within a Parlour or dining roome at most) he may with ease get them all by heart in lesse then the space of an houre: which is no great expence of time, though bestowed on matters of meaner consequence’. (16)

35 Richard Norwood, Trigonometrie, or, The Doctrine of Triangles (London, 1631; reprinted 1971 in facsimile as Number 404 in The English Experience by Theatrum Orbis Terrarum Ltd. (Amsterdam) and Da Capo Press (New York). Publication record: Louis Charles Karpinski, ‘Bibliographical Check List of all Works on Trigonometry Published up to 1700 A.D.’, Scripta Mathematica 12 (1946), 267–83, (p. 278). Norwood biography: Lettie S. Multhauf, ‘Norwood, Richard’ in, Dictionary of Scientific Biography, ed. by Charles Coulston Gillispie (New York: Charles Scribner's Sons, 1976), vol. 10, pp. 151–52; and E. G. R. Taylor, The Mathematical Practitioners of Tudor & Stuart England (Cambridge, 1967), pp. 74–77, 202, 347, 353, 363, 376.

William Oughtred, Trigonometrie, or, The Manner of Calculating the Sides and Angles of Triangles, by the Mathematical Canon, Demonstrated (London, 1657); Ann Arbor: University Microfilms, entry 0590. Biography: Florian Cajori, William Oughtred: A Great Seventeenth-century Teacher of Mathematics (Chicago and London, 1916).

36 … I would rather advise every man to commit to memory the foure Axiomes before going, and to ground his practise thereon (Norwood, note 35, p. 34).

37 Thomas Urquhart, The Trissotetras, or The Most Easy and Exact manner of Resolving all sorts of Triangles, whether Plain or Sphericall, Rectangular or Obliquangular; with greater facility, then ever hitherto hath been practised. Most necessary for all such as would attaine to the exact knowledg of Fortification, Dyaling, Navigation, Surveying, Architecture, the Art of Shadowing, taking of Heights, and Distances, the use of both the Globes, Perspective, the skil of making of Maps, the Theory of the Planets, the calculating of their motions, and of all other Astronomicall computations whatsoever. Lately invented, and perfected, explained, commented on, and, with all possible brevity, and perspicuity in the hiddest, and most re-searched mysteries, from the very first grounds of the Science itselfe, proved, and convincingly demonstrated (London, 1645). For the present study I have used a photocopy made by the Cleveland Public Library from their microfilm of the text (Early English books, 1641–1700; 1470:23; Ann Arbor, Mich.: University Microfilms International). This microfilm reproduces an original (London: Printed for William Hope, 1650; Wing; U138) in the Hunterian Museum. Another microfilm version (Wing (2nd ed.); U140; reproducing a 1645 original in the British Library), is available now electronically as part of Early English books online and is the source of this article's Figures 4 and 5.

38 For printers and the book trade see: Henry R. Plomer, A Dictionary of the Booksellers and Printers who were at Work in England, Scotland and Ireland from 1641 to 1667 (London, 1907), pp. 198–99; Paul G. Morrison, Index of Printers, Publishers and Booksellers in Donald Wing’s Short-title Catalogue of Books Printed in England, Scotland, Ireland, Wales, and British America and of English Books Printed in Other Countries 1641–1700 (Charlottesville, Virginia, for The Bibliographical Society of the University of Virginia, 1955), p. 217; H. S. Bennett, English Books and Readers, 1603 to 1640: Being a Study in the History of the Book Trade in the Reigns of James I and Charles I (London, 1970), p. 1.

39 ‘Trigonometrie. Or, The Doctrine of Triangles. Divided into two Bookes: The first shewing the mensuration of Right lined Triangles: The second of Sphericall: With the grounds and demonstrations thereof. Both performed by that late and excellent invention of Logarithmbs, after a more easie and compendious manner, than hath beene formerly taught. Whereunto is annexed (chiefly for the use of Seamen,) A Treatise of the application thereof in the three principall kindes of sailing. With certaine necessary Tables used in Navigation. By Richard Norwood, Reader of the Mathematicks in London.’

40 ‘To the right honourable Francis Earle of Bedford, Lord Russel, Baron Russel of Thornehavghe, Lord Lieutenant of the County of Devon: and Citty of Exeter …  And howsoever (being rudely composed) it may seeme unworthy the protection of one so eminent in place, and of such ripenesse and judgement in all kinde of learning: Yet I am bolde to present it to your Lordship, in confidence of your favourable acceptance, according to that noble respect you are accustomed to manifest towards all good endeavours … ’

41 ‘TO THE RIGHT HONOURABLE, And most noble LADY, My deare and loving Mother, the Lady DOWAGER of Cromartie  …  But in so much more especially, doe the most judicious of either sex admire the rare and sublime endowments, wherewith your Ladiship is qualified, that (as a patterne of perfection, worthy to be universally followed) other Ladies (of what dignity soever) are truly by them esteemed of the choiser merit, the nearer they draw to the Paragon proposed, and resemble your Ladiship; for that, by vertue of your beloved society, your neighbouring Countesses, and other greater Dames of your kindred and acquaintance, become the more illustrious in your imitation; amidst whom, as Cynthia amongst the obscurer Planets, your Ladiship shines, and darteth the Angelick rayes of your matchlesse example on the spirits of those, who by their good Genius have been brought into your favourable presence to be enlightned by them … ’

42 ‘But all that has been done these many hundred yeares, is not comparable to that which hath been effected in our times, by the Honourable Lord John Nepair Baron of Marchiston: who by an invention of Logarithmes takes away those difficulties that were in the practice thereof’.

43 ‘To write of Trigonometry, and not make mention of the illustrious Lord Neper of Marchiston, the inventer of Logarithms, were to be unmindfull of him that is our daily Benefactor  …  Pythagoras, all the seven wise men of Greece, Archimedes, Socrates, Plato, Euclid, and Aristotle, had (if coaevals) joyntly adored him, and unanimously concurred to the deifying of the revealer of so great a Mystery … ’ ‘ … the Philosophers stone is but trash to this invention, which will alwayes  …  be accounted of more worth to the Mathematicall world, then was the finding out of America, to the King of Spaine … ’

44 ‘ … this Mathematicall Tractate doth no lesse bespeak him a good Poet, and good Orator, then by his elaboured Poems he hath showne himselfe already a good Philosopher, and Mathematician’. (Expresse, p. 1)

45 ‘ … instead of three quarters of a yeere, usually by Professors allowed to their Schollers for the right conceiving of this Science … ’ (Expresse, p. 4)

46 ‘The novelty of these words I know will seeme strange to some, and to the eares of illiterate hearers sound like termes of Conjuration’. (12)

47 ‘If the novelty of this my invention be acceptable, as I doubt not but it will, to the most learned and judicious Mathematicians … ’ (‘finall Conclusion’)

48 Taylor (note 35) p. xi.

49 Ibid. 230, 360.

50 ‘ … [who] have rather contented themselves barely with Scale and Compasse, and other mechanick tooles and instruments … ’ (‘J.A.’, Expresse, p. 3)

51 For instance:

1. His ‘Case 1: The Angles and Base given: to finde the Perpendicular’, is a detailed working out, by three different approaches, of a single concrete problem in which the right triangle ADB has base AB of 768 paces and angle at D of 67°23'.

2. His summary table (Figure 3(a) above) is in fact an index to 12 worked examples.

3. An appendix details applications to sailing, such as how to calculate the change in latitude and longitude resulting from sailing 100 leagues in a fixed direction, or how to plot the great circle distance from the southernmost point in England, ‘The Lizard’, to the mouth of the Amazon. Surely linking the peril and excitement of sailing to the power of trigonometry would make Norwood's tenth example memorable to a young man in love with the sea:

   A Merchant man, being in the latitude of 43 degrees, falls into the hands of pyrats, who amongst other things take away his sea-compasse. But when he is gotten cleare, he sailes away as directly as he can, and after two dayes meetes with a man of war, who also had bin the day before in the latitude of 43° and had sailed thence sebs 37 leagues: He desirous to finde these pyrats, the Merchantman tells him, he left them lying to and fro where they tooke him, and he had sailed since at least 64 leagues, betweene the south and west: what course shall the man of warre shape to finde these pyrats?

52 ‘ … I doe not illustrate what I have written with variety of examples; seeing practically to treat of Triangulary calculations, in applying their doctrine to use, were to digresse from the purpose in hand, and incroach upon the subject of other Sciences; a priviledge, which I must decline, as repugnant to the scope proposed to my selfe, in keeping this book within the speculative bounds of Trigonometry … ’ (86)

‘ … for, if the Lord chamberlain of the Kings houshold should give me a Key, made to open all the doores of the Court, I could not but graciously accept of it, though he did not goe along with me to try how it might fit every lock. The application is so palpable, that, not minding to insist therein, I will here stop the current of my Pen … ’ (87)

53 In the 19-page ‘Lexicidion’ at the end of the book, which gives derivations for the book's technical terms, Urquhart is eager to share his invented world:

Being certainly perswaded, that a great many good spirits ply Trigonometry, that are not versed in the learned Tongues, I thought fit, for their encouragement, to subjoyne here the explication of the most important of those Greek, and Latin termes, which, for the more efficacy of expression, I have made use of in this Treatise. (88)

The Lexicidion then explains that ‘Diatyposis, is a briefe summary description, and delineation of a thing  …  for the instruction of the Reader  …  in a most neat and convenient order’. (95).

54 In between his Diatyposis and his Table Urquhart offers a fine summary-review of Euclidean plane geometry.

55 ‘ … I know not why Logick and Musick should be rather fitted with such helps then Trigonometrie, which for certitude of demonstration, hath been held inferior to no science, and for sublimity and variety of object, is the primell of the Mathematicks. This is the cause why I framed the Trissotetras … ’ (12)

56 ‘ … since the very infancie of learning, such inventions have beene made use of, and new words coyned, that the knowledge  …  might the more easily be retained in a memory susceptible of their impression … ’ (12)

57 The ‘Axiomes four’ preceding this Table will be discussed in the Appendix below.

58 Two changes from the previous name Upalem are:

1. The signs * and §, now divide the word into syllables, to focus attention on the critical information, the vowels: The first two or three vowels, up to the §, indicating the givens (‘Datas’, at the head of the column); the one after § the unknown (‘Quaesitas’);

2. The vowel U is written interchangeably with the consonant V. This is useful ‘to avoid vastnesse of gaping’ in pronouncing the ‘figures’ like Va*le or Ve*mane in the leftmost column of the table. Just as medieval logicians grouped several syllogistic moods into ‘figures’, Urquhart here groups his trigonometric problems, for instance Vb*em§a̳n and Vph*en§e̳r into Ve*ma̳ne̳. As the underlining indicates, the grouping is by the data (in both problems the hypotenuse and a side are given), with the ‘figure’ vowels after the * indicating both possible unknowns (an angle, or the third side).

59 This is the ‘artificial’ sine, that is, the logarithm of the ordinary (‘natural’) sine. The convention that one must use log values in additive forms of the formulas holds throughout both Norwood's and Urquhart's books.

60 Norwood omits Shenerolem, which is simply the combination of 9 (Xemenoro) and 8 (Danarele).

61 ‘Here endeth the doctrine of the right-Angled sphericalls, the whole diatyposis wherof is in the Equisolea or hippocrepidian diagram, whose most intricate amfractuosities, renvoys, various mixture of analogies, and perturbat situation of proportionall termes, cannot choose but be pervious to the understanding of any judicious Reader that hath perused this Comment aright. And therefore let him give me leave (if he think fit) for his memorie sake, to remit him to it, before he proceed any further’. (38)

62 ‘Equisolea, and Equisolearie, are said of the grand Orthogonosphericall Scheme; because of the resemblance it hath with a horse-shooe, and may in that sense be to this purpose applied with the same metaphoricall congruencie, whereby it is said, that the royall army at Edge-hill was imbatteld in a half-moon’. (96)

63 ‘OU>L’ following 3. nu perhaps instructs the reader to look for the next term 4.MIR outside and left of the crowded large triangle. The only other ‘OU>L’, in panel 13. Alamun, marks a similarly, though less extremely, displaced fourth term. This latter ‘OU>L’, which follows a 3.ma term, may alternatively be connected with the 3.ma at the extreme upper left of the diagram, which is the only term in the horseshoe not otherwise accounted for by the 16 formulas.

64 I transcribe it here for legibility, since it is noteworthy for both its content and its optimism:

By these 16 representatives 1. Lec. 2. Ter. 3. Ruc. 4. Cle. 5. Lu. 6. Tul. 7. Terc. 8. Tol. 9. Lec. 10. Ar. 11. Tul. 12. Clet. 13. Cret. 14. Tar. 15. Tur. 16. Le. (A. Signifying an oblique angle. E. the Perpendicular U. the Subtendent C. initiall, the Complement of a Side to the quadrant C. finall, the side Continued to the Radius or a Quadrant. L. Left. R. Right, and T. one of the top Triangles of the Scheme) it is evidenced in what part of the Diagram the Analogy of any of the 16. Moods begins, which being once knowne; the progressive sequence of the proportionable Sides & Angles is easily discerned out of the orderly Involutions of the Figure itself. Here it is to be observed, that as the Book explaineth the Trissotetral Table: so this Trigono-diatyposis unfoldeth all the intricate difficulties of the Book.

I do not understand the coding for cases 3 (Ruc), 4 (Cle), or 5 (Lu).

65 Urquhart's explanation in the ‘Animadversions’ appended to his ‘Explanation of the Trissotetras’ is:

In the letter T. I have been something large in the enumeration of severall Radiuses; for there being eleven made use of in the grand Scheme, whereof eight are Circumferentiall, and three Angularie, that they might be the better distinguished from one another, when falling in proportion we should have occasion to expresse them; I thought good to allot to every one of them its owne peculiar Character: all which I have done with the more exactnesse, that by the variety of the Radiuses amongst themselves, when any one of them in particular is pitched upon, we may the sooner know what part of the Diagram, by meanes thereof, is fittest for the resolving of any Orthogonosphericall problem … (14–15)

The ‘Explanation’ then gives these forms:

To. the Radius or total Sine, but in the Diagram it is taken for the left angularie Radius: Tω. the right angularie Radius in the Scheme proposed: Tol. the first hypotenusal Radius thereof. Tom. the second hyp. Radius. Ton. the third hyp. Rad. Tor. the fourth hyp. Rad. Tolb. the basiradius on the left hand. Torb. the basiradius on the right. Tolp. the Cathetorabdos, or Radius on the left. Torp. the Cathetoradius on the right. (14)

66 ‘The reason hereof will be manifest enough to the industrious Reader, if when by a peculiar Diagram, of whose equiangular Triangles the foresaid Sines and differences are made the constitutive sides, he hath evinced their Analogy to one another  …  and afterwards proceed in all the rest, according to the ordinary rules of Æquation, and Analogy, he cannot choose but extricat himselfe with ease forth of all the windings of this elaboured proposition.’ (40–41). A rigorous modern proof needs just a few lines of clever trigonometric algebra (published by Euler in 1753), based on a law of cosines for spherical triangles (p. 24) discovered by the great Arab astronomer Al-Battani (c. 858–929) – I.Todhunter and J. G. Leathem, Spherical Trigonometry (London, 1949), pp. 27–28.

67 Occasional later illustrations of Urquhart's trigonometer's mind might include the inventiveness and thoroughness of his genealogy in the Pantochronocanon (Urquhart, Works (note 1), pp. 155–72; reprinted in Tayler (note 8), pp. 69–97); his enhancement of Rabelais's already long lists of children's games and animal noises (Games: Rabelais (note 2), pp. 25–26; Admirable Urquhart (Table 1 #12) p. 56; Roe (Table 1 #9) p. 20. Noises: Rabelais (note 2) p. 152; Admirable Urquhart (Table 1 #12) pp. 56–57, Willcock (note 1) pp. 203–5, MacDiarmid (note 8) pp. 54–55; and the geometrical descriptions and imagery in his biography of the ideal Scots scholar and warrior Crichton (Ekskubalauron, Urquhart, Works (note 1), pp. 223, 240–41; cf. Admirable Urquhart (Table 1 #12) pp. 41–43).

68 Universal languages were a major intellectual preoccupation of Urquhart's time (useful general introductions are James Knowlson, Universal Language Schemes in England and France, 1600–1800 (Toronto, 1975); and Rhodri Lewis, Language, Mind and Nature: Artificial Languages in England from Bacon to Locke (Ideas in Context 80), (Cambridge, 2007); also available in paperback (2012)), and Urquhart's work shows affinity or influence from other plans, notably Mersenne's (Craik (Table 1 #13) pp. 99–108).

69 It is three pages of bitterly sarcastic ‘Epistle Dedicatory’ to Nobody, who has helped him so much in his troubles after the Battle of Worcester!

70 ‘Words are the signs for things … 

All things are either real or rational: and the real, either natural or artificial.

There ought to be a proportion betwixt the sign and thing signified; therefore should all things, whether real or rational, have their proper words assigned unto them.

Man is called a Microcosm, because he may by his conceptions and words contain within him the representatives of what in the whole world is comprehended.’

71 Numbers in square brackets are the numbered paragraphs (from a total of 137) in the first book Neaudethaumata of Logopandecteision.

72 Urquhart describes his principle for the logical ordering of words, then surveys ancient (Latin, Greek, Hebrew, Arabic, Caldean, Syriack, Aethiopian, and Samaritan) and modern (English, Irish, French, Spanish, Italian, and Dutch) languages for their strengths and weaknesses [12–21]. He combats with vigour and spirit the ‘adulterate sense of those pristinary lobcocks’ [65], these ‘archaemanetic coxcombs’ [68], who believed languages were miraculously infused from God into men at the time of the tower of Babel, and could not or should not be altered or improved [34–69], pointing out such immense subsequent progress as the syllogism of Aristotle, the sphere of Archimedes, gunpowder and printing of Swart and Gertudenburg, and logarithms of Neper [65], that hero of The Trissotetras. Urquhart also does some self-advertising: ‘I ascribe unto myself the invention of the Trissotetrail trigonometry’ [69], claiming also books on logic and philosophy [69], a treatise on arithmetic ready for the press [98], and ‘above a hundred other several books on different subjects’ [69].

73 ‘As all things of a single compleat being, by Aristotle into ten classes were divided; so may the words whereby those things are to be signified, be set apart in their several storehouses’. [6] The book of ‘Categories’ that begins Aristotle's Organon divides objects of thought into eight or ten categories of substance, quantity, quality, relation, place, time, etc., then subdivides them to primary or secondary substance, discrete or continuous quantity, and so forth (Aristotle, Categories, in The Works of Aristotle, ed. and trans. by W. D. Ross, reprinted in Great Books of the Western World, ed. by Robert Maynard Hutchins, 8: 3–21, (Chicago, 1952). As Urquhart's commentator Boston pointed out (Table 1 #12, pp. 37–38), this is exactly the same principle used in a modern Roget's thesaurus, which classifies words into eight classes (abstract relations, space, physics, matter, … ), each further divided (abstract relations = existence, relation, quantity, order, …) and subdivided (existence = being in the abstract, in the concrete, … ; etc.) to give 1,000 or so categories in all (Roget’s International Thesaurus, 3d ed. (New York, 1962), pp. xiii–xx.

74 ‘Every faculty, science, art, trade or discipline, requiring many words for expression of the knowledge thereof, hath each its respective root from whence all the words thereto belonging are derived.’ [91] Actually, several times more than 250 syllables are possible if one may place a consonant before, after, or on both sides of the vowel. Urquhart specifically affirms that words in his language will remain meaningful even when the letters are reversed [93].

75 This is probably the weak point in the plan–as Descartes had mentioned to Mersenne in their correspondence on universal languages (Knowlson, note 68, p. 65)–the difficulty being not in making words, but in classifying objects. For if the first syllable specifies categories of, say, plants vs. colours, then the second syllable must divide both plants and colours to 250 subcategories. But it seems unlikely that the same 250 subcategories could serve for both; one would, more likely, have to memorize a different set for each radical in the language. Memorizing the subcategorizations might turn out easier, though, than memorizing the uncategorized words of an ordinary language.

76 From a modern perspective one sees a mathematically exact relation for vocabulary size: since adding syllables multiplies the number of possible words, the vocabulary grows exponentially with the syllables (i.e. the logarithm of the vocabulary size is proportional to the number of syllables). Robert Haas, ‘This Sentence Would Pass the Spell Checker, or 168 Billion Puns’, Journal of Irreproducible Results, 42(6) (1997), 15–17 exploited this ‘logarithmic’ property of language for humorous effect, by tabulating 40 homonyms for each word of a seven-word sentence to generate 40⁷ potential variants.

77 ‘This world of words hath but two hundred and fifty prime radices, upon which all the rest are branched: for better understanding whereof, with all its dependant boughs, sprigs and ramelets, I have before my lexicon set down the division thereof (making use of another allegory) into so many cities, which are subdivided into streets, they again into lanes, those into houses, these into stories, whereof each room standeth for a word; and all these so methodically, that who observeth my precepts therein shall at the first hearing of a word know to what city it belongeth, and consequently not to be ignorant of some general signification thereof, till after a most exact prying into all its letters, finding the street, lane, house, story and room thereby denotated, he punctually hit upon the very proper thing it represents in its most specifical signification’. [73]

78 ‘Such as will harken to my instructions, if some strange word be proposed to them, whereof there are many thousands of millions deviseable by the wit of man which never hitherto by any breathing have been uttered, shall be able, although he know not the ultimate signification thereof, to declare what part of speech it is; or if a noun, unto what predicament or class it is to be reduced; whether it be the sign of a real or notional thing, or somewhat concerning mechanic trades in their tools, or terms; or if real whether natural or artificial, complete, or incomplete; for words here do suppone for the things which they signify … ’[72]

79 Features like the ending and stem vowel show the tense (present, imperfect, …  – 7 possibilities in all), voice (active, middle, or passive), mood (indicative, imperative, subjunctive, or optative), person (first, second, or third), and number (singular, dual, or plural), thus up to 7 × 3 × 4 × 3 × 3 = 756 possibilities, plus another 120 forms as a participle (verbal adjective) allowed 2 numbers, 3 genders, 4 cases, and 5 tenses (C. A. E. Luschnig, An Introduction to Ancient Greek: A Literary Approach, 2nd ed. (Indianapolis, 2007): verb forms explained 17–18 and tabulated 298–300; participles 135–39 and 288–89). The Commission on Enzymes of the International Union of Biochemistry similarly classified the thousands of enzymes catalyzing biochemical reactions by a four-number code giving the major class (oxidoreductase, transferase, hydrolase, etc.), sub- and sub-sub-classes, and finally the individual enzyme. Enzyme 1.11.1.6, for instance, is catalase, which decomposes hydrogen peroxide (Marcel Florkin and Elmer H. Stotz, eds., Enzyme Nomenclature, Comprehensive Biochemistry, 2nd ed., vol. 13 (Amsterdam, 1965)).

80 Urquhart seems to have anticipated, and forgiven, this outcome, saying in ‘The Finall Conclusion’ to The Trissotetras:

But as for such, who, either understanding it not, or vain-gloriously being accustomed to Criticise on the Works of others, will presume to carp therein at what they cannot amend, I pray God to illuminate their Judgments, and rectifie their Wils,* that they may know more, and censure lesse: for so by forbearing detraction, the venom whereof must needs reflect upon themselves, they will come to approve better of the endeavours of those, that wish them no harme.

*‘Wils’ is perhaps a variant spelling of ‘Wills’; the Maitland Club reprinting has ‘wits.’