Studies in History and Philosophy of Science 43 (2012) 470–479

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Studies in History and Philosophy of Science

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John Dee on geometry: Texts, teaching and the Euclidean tradition

Stephen JohnstonMuseum of the History of Science, University of Oxford, Oxford, UK

a r t i c l e i n f o

Article history:Available online 10 January 2012

Keywords:John DeeEuclidGeometryTyrocinium mathematicumThomas DiggesPetrus Ramus

0039-3681/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.shpsa.2011.12.005

E-mail address: stephen.johnston@mhs.ox.ac.uk1 Marshall Clagett’s edition and translation of Dee’s w

the motivation for the text was not purely geometrica

a b s t r a c t

John Dee’s mathematical interests have principally been studied through his Mathematicall praeface toHenry Billingsley’s 1570 translation of Euclid’s Elements. The focus here is broadened to include the noteshe added to Books X–XIII of the Elements. I argue that this additional material drew on a manuscript text,the Tyrocinium mathematicum, that Dee wrote a decade earlier, probably as tutor to the youthful ThomasDigges. Using new evidence for this now-lost work, as well as his notes on Euclid, makes it possible toclarify Dee’s approach to geometry. The contrasting positions adopted by his Parisian acquaintance PetrusRamus also illuminate Dee’s geometrical choices and values. Unlike Ramus, Dee was not a pugnaciousadvocate of radical reform, yet he did look beyond the limits of Euclid’s geometry towards deeper disci-plinary visions of knowledge. The first published work of his pupil Thomas Digges not only suggests howDee shaped the younger man’s work but also reflects fresh light back on Dee’s own programme for a‘more general art Mathematical.’

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When citing this paper, please use the full journal title Studies in History and Philosophy of Science

1. Introduction

In his Mathematicall praeface to the 1570 English Euclid, JohnDee places great stress on the fundamental role of arithmetic andgeometry. They lead upwards to the contemplations of pure intel-lect and downwards to productive engagement with the naturalworld, serving as the source and root of a whole spectrum of ‘artsmathematical derivative.’ Strangely, given the scholarship that hasbeen devoted to the Mathematicall praeface and to Dee’s active pur-suit of various practical mathematical arts, remarkably little atten-tion has been given to his treatment of these two principalmathematical sciences.1

In this paper I want to focus on Dee’s approach to geometry,both to identify his intellectual position and to assess his signifi-cance and impact. Twenty years on from its publication, my prin-cipal point of reference is Nicholas Clulee’s John Dee’s naturalphilosophy (1988). Faced with the terrifyingly broad field of Dee’sintellectual formation and disciplinary ambition, Clulee’s work

ll rights reserved.

ork on the geometry of the parabol: it was part of Dee’s larger work o

represents a model of scholarship—integrating biography and thepursuit of patronage with the careful analysis of intellectual devel-opment and affiliation. He always seeks to pinpoint Dee’s sourceswith precision rather than constantly invoking a single overarchingexplanatory theme (whether Neoplatonism, Renaissance magic,Hermeticism, or any other all- encompassing interpretative term).

Clulee’s attention was centred on Dee’s natural philosophy andhis position within the history of science. Although the Mathemat-icall praeface appears as one of his key texts, Dee’s mathematics en-ters the account primarily for the light it sheds on these centralissues. Here I attempt a similar style of analysis, but take Dee’sgeometry as worthy of study in its own right. This means givingclose and serious attention to his writings, both surviving and lost,and establishing them within the chronological framework of Dee’scareer, while also being alert to wider resonances and responses.

Textbook histories of mathematics have typically focused onthe innovations of algebra as the central development of theRenaissance. Judged against transhistorical (or even ‘eternal’)

la is a notable exception; Clagett (1980). However, as Clulee (1988), p. 68 emphasises,n burning mirrors.

S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479 471

criteria of mathematical significance, sixteenth-century geometryhas been considered a static or even stagnant field.2 Bennett(2002) has responded to ‘the baneful influence of timelessness’ inthe history of Renaissance mathematics by shifting attention to prac-tical geometry, to the vigorous world of instruments and practice. Afull account of Dee’s geometry would have to address that broadfield, to encompass his inventions and devices for astronomy, navi-gation, geography and much else. But the realm of texts and scholar-ship need not be entirely abandoned: far from the cliché ofstagnation, geometry was in reality culturally dynamic and diverse.Ancient authorities were not only textually recovered and renewed,but also reworked, supplemented and even attacked. Rather than aplacid backwater, geometry was a field pockmarked by active debateand controversy, most famously on the quadrature of the circle.3 Deewas familiar with most of these cross-currents, both through hisextraordinarily rich library and his personal acquaintance with manyof the leading mathematicians of Europe. By locating him within thiscontemporary mathematical landscape we can recover what was atstake in the pursuit and teaching of geometry, and see how his math-ematical values encouraged the work of others.

Rather than dwelling exclusively on the familiar terrain of Dee’sMathematicall praeface, I want to take as my point of departure thenotes he added to Euclid’s Elements. In the Compendious rehearsal’slisting of the printed and manuscript works produced during hisfifty years of study, Dee gave these additions a separate recordfrom the entry for the Mathematicall praeface. He described themas ‘divers & many Annotations, and Inventions Mathematicall,added in sundry places of the foresaid English Euclide, after thetenth Booke of the same.’4 They are scattered throughout Books Xto XIII and, while not adding up to a connected work, they do repre-sent a significant investment of time.5 I argue that these commentsdrew on a manuscript text, the Tyrocinium mathematicum, that Deewrote a decade earlier, probably as tutor to the youthful ThomasDigges. Using new evidence for this now-lost work together withhis notes on Euclid makes it possible to clarify Dee’s approach togeometry, particularly when illuminated by the contrasting posi-tions adopted by his Parisian acquaintance Petrus Ramus. Unlike Ra-mus, Dee was not a pugnacious advocate of radical reform, yet he didlook beyond the limits of Euclid’s geometry towards deeper disci-plinary visions of knowledge. The first published work of his pupilThomas Digges not only suggests how Dee shaped the youngerman’s work but also reflects fresh light back on Dee’s own pro-gramme for a ‘more general art Mathematical.’ Situating Dee in hiscontemporary context provides vital insight, but I conclude that inmany respects Dee was continuing well- established traditions ofmedieval geometry, just as he did in natural philosophy.

2. Printing and writing

The hurried and harassed tone of Dee’s Mathematicall praefacehas often been noted. Dee refers several times to being ‘pinchedwith straightnes of tyme’: ‘the Printer, hath looked for thisPraeface, a day or two’ and ‘still the Printer awayting, for my penstaying.’6 Dee was not exaggerating the urgency of his composition.The preface was completed on 9 February 1570, a few days after he

2 For a neat encapsulation, see Cajori (1991 [1893]), p. 141. Note that major recent worparticularly Bos (2001).

3 Recent work on circle squaring controversies includes Leitão (2009) for Fine and Nun4 Dee (1726), p. 525. I have reformed the use of i/j and u/v in all quotations but otherw5 John Heilbron’s essay in Dee (1978) provided a first examination of Dee’s contributi

Mandosio (2003) classifies the variety of Dee’s additions.6 Dee (1570), sigs. A.iiij.v, c.iij.v and d.iiij.v.7 Euclid (1570), fol. 391v. As with his diary entries in printed ephemerides, Dee’s dates are

1 January.8 There is a substantial literature on Day and his workshop, including estimates of his r9 Dee (1726), p. 526.

had finished the striking tabular ‘Groundplat’ on the 3rd. As pub-lished, the latter includes a marginal note that it was printed on25 February, the date possibly reflecting the printer’s relief thatthe work was finally complete. Although perhaps less pressured thanhis work on the Praeface, Dee’s additions to the later books were notmuch more relaxed. He complains in similar terms that ‘utterly lei-sure is taken from me’ (Euclid, 1570, fol. 362r) and that ‘if more ley-sor had happened, many more straunge matters Mathematicall had,(according to my purpose generall) bene presently published to yourknowledge’ (ibid., fol. 371v). He even provides the date of 18 Decem-ber 1569 for two theorems on mean and extreme proportion addednear the beginning of book XIII.7

The sequence of Dee’s writing corresponds to the standard prac-tice of an Elizabethan printing house. A work’s dedication, prefaceand other preliminary materials normally went to press at the endof the production process. Dee was working on the Praeface up tothe last minute and the printer’s anxious enquiries show thatDee was holding up the completion of the job. Fortunately, HenryBillingsley had entrusted his translation to the safe hands of JohnDay, Elizabethan England’s most successful printer. Day was accus-tomed to the financial and technical demands of large and complextexts—at the same time as the Elements he was also supervising theenormous second edition (2300 folio pages in two volumes) ofFoxe’s Book of martyrs. Though not as large, Billingsley’s Euclidwas still a very substantial folio of nearly a thousand pages. Theprofusion of figures provided its own set of challenges and thecomposition and typography were varied and elaborate, makingthis easily the most lavish and prestigious English mathematicalpublication of the period. The typesetting and printing of the com-plete volume would have extended over many months in Day’sworkshop.8

The dated theorems that Dee contributed to Book XIII, as well ashis complaints of lack of time, strongly suggest that his additions toEuclid’s text were supplied during printing rather than well in ad-vance. The restriction of his comments to Euclid’s last four booksmay simply mean that the preceding books were already printedby the time Dee joined the editorial process. (He would certainlyhave had things to say about the earlier books; he had, for example,lectured on Books I and II of the Elements at Paris in 1550.9) If Deewas indeed working under severe time constraints, this raises ques-tions about the composition of the material which he did manage toadd. Few have appreciated just how sustained are some of the sec-tions he authored. Certainly, many of his additions are brief com-ments and corollaries interspersed with the main text. But there isalso, for example, a sequence of 17 folio pages at the end of BookXII which provides ‘Theoremes and Problemes (whose use is mani-folde, in Spheres, Cones, Cylinders, and other solides)’ (Euclid,1570, fols. 381v–389v). A substantial insertion such as this, pre-sented in the formal style of classical geometry, could not have beendashed out, with the printer hovering at his shoulder.

Rather than composing all of his Euclidean additions fromscratch, might Dee have drawn on material that he had alreadycompiled? The latter suggestion has not only a practical plausibil-ity but also a neat symmetry with the composition of the Mathe-maticall praeface. The retrospective character of the Praeface is

k in the conceptual history of mathematics offers a far more sophisticated view, see

es and Hogendijk (2010) for Scaliger and Van Ceulen.ise left the orthography unchanged.on, listing the additions (p. 22, n. 74) and working through some of the problems.

here evidently in astronomical rather than civil style, with the new year beginning on

ate of production; see Pettegree (2004) and, most recently, Evenden (2008).

472 S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479

now widely recognised, since Dee primarily depended on booksthat had been in his library before 1560.10 I suggest that in assem-bling his additions to Euclid he would have worked in a similar way,except in this case it was his own work from the 1550s that served asa source.

3. Tyrocinium mathematicum

Billingsley’s Euclid in fact gives us a candidate for the role ofDee’s source text. It appears in a note that must at least have beenauthorised if not actually authored by Dee himself. The commentappears deep in the theory of irrational magnitudes at Book X,53, and follows a corollary translated from the 1566 Latin editionof François Foix de Candale (Flussas). It is accompanied by a mar-ginal note, ‘M. Dee his booke called Tyrocinium Mathematicum’,and although it has been cited before, is worth quoting in full:

Although I here note unto you this Corollary out of Flussas, yet,in very conscience and of gratefull minde, I am enforced to cer-tifie you, that, many yeares, before the travailes of Flussas (uponEuclides Geometricall Elementes) were published, the orderhow to devide, not onely the 6 Binomiall lines into their names,but also to adde to the 6 Residuals their due partes: and farther-more to devide all the other irrational lines (of this tenth booke)into the partes distinct, of which they are composed: with manyother straunge conclusions Mathematicall, to the better under-standing of this tenth booke and other Mathematicall bookes,most necessary, were by M. Iohn Dee invented and demon-strated: as in his booke, whose title is Tyrocinium Mathematicum(dedicated to Petrus Nonnius, An. 1559) may at large appeare.Where also is one new arte, with sundry particular pointes,whereby the Mathematicall Sciences, greatly may be enriched.Which his booke, I hope, God will one day allowe him opportu-nitie to publishe: with divers other his Mathematicall andMetaphysicall labours and inventions.11

Given that Dee does not mention this lost Tyrocinium mathem-aticum in the later lists of his writings, this evidence requires care-ful consideration. Billingsley had evidently seen and used themanuscript, and it was presumably in a reasonably complete stateif it could be consulted and its publication envisaged.

As befits its location in Book X, the note reveals that a signifi-cant part of the Tyrocinium mathematicum dealt with the classifica-tion and manipulation of irrational magnitudes. Book X is thelongest and, for many of its readers, the most difficult of the Ele-ments. Dee presumably provided some form of synopsis to smooththe way to an understanding of its confusing mass of medial, bino-mial, apotome and other lines.12 That his purpose was expository issuggested by the title of the work. In a later age, ‘Tyrocinium math-ematicum’ could have been appropriately translated as ‘Mathemat-ics for beginners’, and the title certainly matches the pedagogictenor of Dee’s additions to Euclid. A typical note is ‘M. Dee his deuise,to helpe the imagination to young students in Geometry’ (Euclid,1570, fol. 380r). But although Dee’s Tyrocinium mathematicum mayhave included an overview of Book X, most contemporaries wouldnot have judged this a topic for real beginners. Many early editionsof the Elements include only the first six books, and these were con-sidered a sufficient introduction to geometry for the generality of

10 Clulee (1988), p. 147 and Roberts & Watson (1990), p. 10.11 Euclid (1570), fol. 268r–v. The note was referred to by Calder (1952), Ch. 6.12 There are some fragmentary notes by Dee on incommensurable lines in British Librar13 It may be worth noting here that, despite the coincidence of titles, there was evidentl

Sinclair (1661).14 For a detailed study of this relationship, see Almeida (2011) in this issue.

students. While explicating the Elements, Dee was therefore aimingbeyond the entirely elementary.13

Billingsley’s note makes clear that the Tyrocinium mathemati-cum was not concerned exclusively with Book X of the Elements,but that it helped with other books too. Whether these were withinthe Elements or elsewhere is left unclear. Nor did Dee’s text offerguidance only to existing mathematical works. Without givingany indication of its character, Billingsley says that the Tyrociniummathematicum contained a new mathematical art. The formalinvention of new arts was something of a speciality for Dee. Ashe was later to recall, the Mathematicall praeface contained ‘manyArts, of me, wholy invented (by name, definition, propriety anduse,) more then either the Graecian, or Roman Mathematiciens,have left to our knowledge’ (Dee, 1726, p. 525). Only the mostcharitable and sympathetic of Dee’s modern readers would grantthat his many neologisms and novel topics (from anthropographieto zographie, hypogeiodie to trochilike) represent a significant con-tribution to the mathematical sciences, and most of his new arts(or new names) immediately sank without trace. But it would betoo cynical to see this invention of arts as purely self-aggrandise-ment: it was also a way of magnifying and extending the realmof the mathematical sciences themselves, especially when pre-sented in the tabular form of the ‘groundplat’ to the Mathematicallpraeface.

We will return to Dee’s ambitions for a new and more generalmathematical art than could be found in Euclid, as well as to thecontemporary context for a detailed interest in Book X of the Ele-ments. There is just one more point to draw out here from Billings-ley’s note. He reports that the Tyrocinium mathematicum waswritten in 1559 and dedicated to the Portuguese mathematicianand cosmographer Pedro Nunes. This date and dedication fit clo-sely with Dee’s contemporary interests. Judging by the lists of hisown writings, Dee was most active as a mathematical author inthe later 1550s. Pedro Nunes also assumed a special importancefor him at that time. In the letter to Gerhard Mercator which pref-aces his Propaedeumata aphoristica of 1558, Dee names Nunes ashis mathematical executor (Dee, 1978, pp. 114–115). The dedica-tion of the Tyrocinium mathematicum to Nunes in 1559 underlinesthe close intellectual relationship which Dee evidently felt with hiscolleague.14 Moreover, the Letter apologeticall mentions anotherwork dedicated to Nunes from 1560, this time on the area of trian-gles (Dee, 1599, sig. B[1]r).

Intriguing as are these closely contemporary references to Nun-es, they leave obscure both Dee’s reason for dedicating an intro-ductory work to an expert mathematician and also the morebasic question of why Dee wrote the Tyrocinium mathematicum atall. I suggest that the Tyrocinium mathematicum was composedspecifically for the instruction of the youthful Thomas Digges. Dig-ges, now best known for his Copernican Perfit description of thecelestial orbes (1576), was a well- born gentleman. In addition tohis position as a prominent mathematician, he became a Memberof Parliament, technical advisor and military administrator (John-ston, 2004). But his public career was in the distant future whenhis father, the mathematical author Leonard Digges, died in about1559. Thomas, then probably aged thirteen, had already receivedsome mathematical instruction from his father and was now en-trusted to the tuition of Dee (Johnston, 2006). The Tyrocinium

y, MS Cotton Vitellius C. VII fols. 274–279v which may relate to his text.y no connection between Dee’s work and that of the Scottish mathematician George

A E

D

C

B

Fig. 1. Reconstruction of a diagram for proposition I, 12 of John Dee’s Tyrociniummathematicum.

S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479 473

mathematicum would then be the plan or product of Dee’s mathe-matical instruction.

Dee was certainly capable of transforming a singular experienceinto a more general treatise. His 1592 listing of works includes atract on underground surveying from this period (‘De itinere sub-terraneo – libri – 2 – Anno – 1560’).15 The Mathematical praefacementions this text in its entry on ‘hypogeiodie’ where Dee recordsthat the ‘occasion of my Inuenting this Arte’ was a dispute overmining rights between two gentlemen (Dee, 1570, sig. dj.v). Broughtin to determine under whose land the profitable lode lay, Dee notonly produced a Latin treatise but even dignified his effort as a wholenew art.

Dee’s sudden responsibility for the mathematical education ofDigges coincides with the character and date of the Tyrociniummathematicum. It requires no great stretch of the imagination toconceive his tuition of Digges as both the motivation and the basisfor an avowedly didactic mathematical text. I have stressed else-where the significance of the relationship between Dee and Digges,arguing not only that Digges’s early mathematical career recapitu-lated the role that Dee had occupied twenty or so years earlier, butalso that Digges’s first publication owes much to Dee’s interestsand teaching (Johnston, 2006, p. 67). Once equipped with a bettersense of the character of the Tyrocinium mathematicum and of Dee’splace in contemporary mathematical culture, I will return at theend of this paper to a deeper examination of Digges’s first workand its debt to Dee.

Billingsley’s brief account of the Tyrocinium mathematicum isrichly revealing but gives at best a partial indication of its scope.It can now be supplemented by some new evidence which signif-icantly extends our sense of Dee’s 1559 treatise. This does not sim-ply slot into our picture, like the last piece in a jigsaw puzzle, butraises some fresh questions and bibliographic possibilities. It isfound in Edinburgh University Library amongst a collection ofmathematical books bequeathed in 1635 by James Douglas, secre-tary to James VI. Many of the books came from the library of theScottish mathematician and royal physician John Craig.16 Shelf-mark Dh.5.195 forms part of Douglas’s bequest and is a copy ofFrancesco Maurolyco’s Opuscula mathematica (Venice, 1575). Boundin as a leaf preceding the title page is a manuscript letter in a neatitalic hand from one Gratianus Niger to John Craig:

Your singular humanity both towards me, and towards myespecial teacher, that noble Briton, Master John Dee, is alreadysplendidly evident enough. I willingly offer this letter as wit-ness, so that the friendship we have already begun will be richlyrewarding to each of us. And meanwhile see for yourself (mostlearned Craig) that which you seek: theorem 12 of the first bookof my teacher’s volume entitled Tyrocinium mathematicum,which he completed 18 years ago.Theorem 12The solid which is made from the product of the semiperimeterof any right-lined triangle and the radius of the circle inscrib-able within it, together with the same radius, is equal to thesolid which is made from the three residues left by subtracting

15 Dee (1726), p. 527. This was also inserted in Dee’s list of future publications idescribed as a modest single book in 1568, it had grown to two books by 1592.

16 Douglas’s benefaction is noted in Guild & Law (1982), p. 48. The list drawn up i17 ‘Perspecta iam satis luculenter singulari vestra humanitate cum erga me, tum

Lubenter polliceor, & hoc scripto testatum relinquo hanc ita initam amicitiam, nobisrequiris), primi libri, Theorema 12m ex eo volumine, cui (ante 18 annos absoluto) titualicuius Trianguli [word erased] rectilinei, in semidiametrum Circuli eidem inscriptibex tribus illis residuis, quae ex singulorum laterum a Semiperimetro eiusdem TriangOderam.’Expanded contractions are indicated by italics. My thanks to Edinburgh Un

18 Friedlaender (1887–1891) includes two sixteenth-century individuals called Geormatch with a Gratianus.

19 Halliwell (1842), p. 5; and Dee (1726), p. 522.

n the

n 163erga

vtrisqlum p

ilis, etuli subiversitgius N

each side from the semiperimeter of the same triangle.Your friend, Gratianus Niger, 18 December 1578, at Frankfurt ander Oder.17

Gratianus Niger, the letter’s author, has not so far been traced.18

However, the recipient John Craig is well known: he had matricu-lated at Frankfurt an der Oder in 1573 and became professor ofmathematics and logic there before graduating MD from the Univer-sity of Basel in 1580 and then returning to Scotland in about 1582 topursue a successful medical career (Henry, 2004).

But how did a specific theorem from Dee’s now-lost manuscriptcome to be quoted in a letter written in Frankfurt in 1578? Thedate gives the clue. The letter was written exactly one week afterDee arrived in Frankfurt while on his 100 day journey ‘undertakenand performed to consult with the learned physitians and philoso-phers beyond the seas for her Majesties health recovering and pre-serving.’19 Dee must have taken part in mathematical as well asmedical discussions while in Frankfurt. Presumably he had theTyrocinium mathematicum with him and made it available to satisfythis specific request from Craig.

Given what we know of the Tyrocinium mathematicum from Bill-ingsley’s Euclid, Dee’s theorem is surprising. Rather than irrationalmagnitudes, it deals with volumes constructed from dimensions ina plane figure. Dee works with a triangle, its semi-perimeter andthe circle inscribed within the triangle. This provides a strong indi-cation of the mathematical context from which the theorem arises.

Let the triangle be ABC; its semi-perimeter is ½ (AB + BC + CA)(Fig. 1). By Euclid IV, 4 a circle can be inscribed in the triangle so asto touch each side in one point. If the semi-perimeter is s and theradius DE of the inscribed circle is r, then Dee’s theorem states thats r2 = (s � AB)(s � BC)(s � CA).

There can be no doubt that the result is closely related to Her-on’s theorem, which expresses the area of the triangle as

p{s (s �

AB)(s � BC)(s � CA)}. Dee’s proposition can be readily derived fromHeron’s by subdividing the triangle into three smaller trianglesABD, BCD and CAD, each with a side as their base and the centreD as their apex. Then the area of the smaller triangle ABD is ½AB ED = ½ AB r, and likewise for the triangles BCD and CAD. Thus,if the total area of the triangle is T,

second edition of Propaedeumata aphoristica (Dee, 1568, sig. Aiiij.r). Whereas it was

5 is at EUL Da.1.29/7.vnicum praeceptorem meum, Nobilem virum, Dominum Joannem Dee Brytannum,

ue gratissimam fore. Atque en tibi interim (Doctissime Craige) praeceptoris mei (quodraefixit, Tyrocinium Mathematicum / Theor. 12m / Solidum quod fit ex semiperimetroex eo quod inde procreatur, in eandem semidiametrum: Aequale est illi Solido quod fitstractione fiunt.Amicus vester, Gratianus Niger Ao 1578. Decemb. 18Francofurti iuxtay Library for permission to reproduce the text of the letter.iger, one an Italian. There are also many entries under Schwarz, but none that seem to

474 S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479

T ¼ 12

AB rþ 12

BC rþ 12

CA r

so T ¼ 12ðABþ BCþ CAÞ r

or T ¼ s r

Therefore, by Heron’s theorem,

s r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifs ðs� ABÞðs� BCÞðs� CAÞ

pg

so s2 r2 ¼ s ðs� ABÞðs� BCÞðs� CAÞ

and s r2 ¼ ðs� ABÞðs� BCÞðs� CAÞ; as required:

This derivation is not classically geometrical, since not onlydoes it multiply magnitudes but it even goes beyond three dimen-sions in squaring an area. But we should not be too squeamish:Heron’s theorem itself requires the square root of the product offour magnitudes and yet was accepted by medieval geometers. Itwas not until Petrus Ramus’s Scholae mathematicae (1569) thatdimensional scruples were raised against Heron. Previous mathe-maticians who had seen the possibility of an in-principle objectionhad responded pragmatically by treating magnitudes as numbers,for which dimensional considerations were irrelevant.20

With no more than the enunciation of this single theorem avail-able to us, it is impossible to say whether Dee responded to suchconcerns in the Tyrocinium mathematicum. The formulation of thetheorem in terms of solids suggests a wish to remain close toEuclidean style, but this can scarcely be considered a firm conclu-sion. Given the connection with Heron’s theorem, we can inferwith greater confidence that the Tyrocinium mathematicum mustalso have dealt with the areas of triangles. Gratianus Niger alsotells us that Dee’s work had been written eighteen years before,thus dating from about 1560. If we only had Niger’s letter andnot Billingsley’s note, it would be natural to jump to the conclusionthat the Tyrocinium mathematicum should be identified with thepreviously mentioned work listed in Dee’s Letter apologeticall as‘De Triangulorum rectilineorum Areis – libri – 3 – demonstrati:ad excellentissimum Mathematicum Petrum Nonium conscripti –Anno – 1560.’21 Yet such a treatise on the areas of triangles seemsan unlikely home for the material on irrational magnitudes that Bill-ingsley identified as a key component of Tyrocinium mathematicum.

One way in which the testimonies of Billingsley, Niger and Deehimself may be reconciled is to imagine that Tyrocinium mathemat-icum was composed as a large work, from which a specific sectionon the areas of triangles was subsequently extracted. Niger’s letterreveals that the text was arranged in more than one book. A broadranging text would certainly make sense as a guide for the teen-aged Thomas Digges. But in the absence of additional evidence, fur-ther speculation seems fruitless. What does seem clear is that by1560 Dee had written a didactic work which dealt, inter alia, withirrational magnitudes as expounded in Euclid’s Elements Book X. Aspart of the same work (or a related one), he had written on solidgeometry and the areas of plane triangles. Building on these mod-est conclusions, and making use of Dee’s additions to Euclid (whichquite likely reflect the content of the Tyrocinium mathematicum),we can now move on to a wider consideration of his geometricalwork.

20 For the history of Heron’s theorem and Ramus’s objections, see de Wreede (2007), Ch21 Dee (1599), sig. B[1]r. Note that the Compendious Rehearsal (Dee, 1726, p. 526) gives a

omitting both the dedication to Nunes and the restriction of topic to right-lined triangles22 Book X had already been the object of other important new work, such as that of Stif23 Ramus (1569), pp. 257–258: ‘nihil enim unquam tam confusum vel involutum legi vel

more generally, resonates strongly with the account of Book X given by Knorr (1983). Boththe heuristic practice that created it.

24 For the variety of responses to Book X, from Ramus onwards, see de Wreede (2007), p25 Euclid (1570), fol. 371r, in a passage marked as ‘M. Dee his chiefe purpose in his addi

4. The challenge of Ramus

One area in which we can be confident of continuity betweenthe Tyrocinium mathematicum and Dee’s additions to Euclid is hiswork on Book X of the Elements. This sustained interest may in it-self seem unremarkable, particularly given the notorious difficultyof the book. But Dee’s attention came at a very specific moment,just when Book X became the starting point for a vigorous and divi-sive programme of mathematical reform.22 Dee was not just a wit-ness to a developing controversy, but responded to it.

The principal agent of reform was Petrus Ramus, whom Dee hadmet after arriving in Paris in the summer of 1550 (Dee, 1726, p.504). The two subsequently remained in contact, and Dee’s librarywould come to include many of Ramus’s works. While in Paris, Deewas also close to the mathematician Petrus Montaureus (PierreMondoré) (Dee, 1570, sig. ⁄ij.v). Montaureus produced an editionand commentary on Book X, first issued in 1551, as well as actingas Ramus’s guide to the book.

In the Scholae mathematicae (1569), Ramus would subsequentlyexcoriate Book X for its wilful obscurity: ‘I have never read or heardanything so confused or intricate.’23 Goulding (2006) has describedRamus’s engagement with Book X in the 1550s as a moment of epis-temological crisis, when he realised that mathematics—at least inthe form in which it had been passed down—was not plain and nat-ural but could be as difficult and involved as the worst excesses ofthe much-derided scholastic logic. His application of method was in-tended to reform the material of the book, but there was no way tomake it more practical. Ramus’s pugnacious intervention had apolarising effect on the next generation of mathematicians, whoseattitude to Book X often serves as a touchstone for their divergentphilosophical and methodological stances.24

Dee’s contact with Ramus while in Paris exposed him to thevery beginnings of this strident assault on the authority of Euclid.But he himself did not seek to dethrone Euclid: rather than acerbicattack, Dee offered a more sympathetic reading. While Euclid’s textmight need clarification and occasional correction, the model ofdemonstrative rigour which it embodied was not challenged. ThusDee’s comments on Book X are meant to assist the student ratherthan confront the author. There is indeed some suggestion thatthey may reflect his actual teaching practice. After X, 62 Billingsleyadds the note ‘Here follow certaine annotations by M. Dee, madeupon three places in the demonstration, which were not very evi-dent to yong beginners’ (Euclid, 1570, fol. 275r).

More generally, where Ramus proposed radical reform, Deenoted that his intent was ‘not to amend Euclides Method, (whichnedeth little adding or none at all).’25 Much more specifically, Deeappears to reproach his turbulent Parisian associate directly (but dis-creetly) in an unusually topical comment in Book XIII. In its receivedform, the text of Euclid breaks off after the fifth proposition and,rather than proceeding directly onwards, introduces a short sectionin which the first five theorems of the book are recapitulated in adouble process of resolution and then composition (analysis andsynthesis). This is a surprising turn in the text: there is no earlier hintin the Elements of the problem-solving technique of analysing backto established principles and then demonstratively synthesizing inthe opposite direction to prove the result.

. 7.briefer listing of the same book: ‘De triangulorum areis libri demonstrati 3.—A. 1560’,(as opposed, for example, to spherical triangles).

el (1544), book II.audivi.’ Interestingly, Ramus’s interpretation of Book X in particular, and the Elementssee it as a philosophically-motivated demonstrative system which deliberately hides

p. 192–205.tions.’

S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479 475

Immediately after the sequence, Billingsley’s edition includes an‘Advise’ by Dee, which notes that this material is evidently a laterinterpolation. Dee thereby means to clear Euclid of any blame forthis sudden interruption to his normal methodical progression,and for the unannounced introduction of a procedure alien to therest of the Elements. He goes on to give thanks to the ancient com-mentator for revealing so useful a method. Dee particularly em-braces the method of resolution in his own mathematical worksince ‘to do other wayes, were to me a confusion, and an unme-thodicall heaping of matter together’ (ibid., fol. 397v). He signsoff with a parting shot explaining the occasion of his ‘friendely ad-vise’: ‘some, of late, have inveyed against Euclide, or Theon in thisplace, otherwise than I would wish they had.’ The target for thiscriticism is surely Ramus, whose recently published Scholae math-ematicae attributed the addition on analysis and synthesis toTheon. Rather than admiring the technique, Ramus hit out at it:it was a source of needless duplication and redundancy, and brokehis rules of proper method. Altogether, he judged it to be ‘contrivedand utterly trifling.’26

Compared to the violent controversies to which Ramus wasaccustomed, Dee’s mild mannered rebuke would scarcely have reg-istered on his radar. But it is indicative of the way that Dee delib-erately positioned his geometrical endeavours in contemporarycontext. While some have seen the tabular arrangement of Dee’s‘groundplat’ to the Mathematicall praeface as a distinctively Ramistcontribution (Roberts & Watson, 1990, p. 10), it is clear that Ra-mus’s reforms held little attraction for either Dee’s Proclus-in-spired Neoplatonist philosophy of mathematics or his pedagogicpractice of geometry. Did Dee intend a deliberate additional digat the method-obsessed Ramus when he said that, without theanalysis that Ramus dismissed, mathematical composition wouldresult in an ‘unmethodicall heaping of matter together’? Certainly,on points such as Ramus’s constant view that demonstration wasan unnecessary encumbrance to mathematics, Dee was consis-tently opposed:

I would wish all Mathematiciens, as well of verities easy, as ofverities rare and obscure, to seeke the causes demonstrative,the finall fruite thereof, is perfection in this art. (Euclid, 1570,fol. 385r)

5. Beyond Euclid

If Dee acted against Ramus as a loyal defender of Euclid and an-cient geometry, he was not a slavish adherent. As has been recog-nised before, Dee looked beyond the limits and prescriptions whichare embodied in Euclid’s Elements (Dee, 1978, p. 27). We might in-deed see Dee’s career as a succession of attempts to overcome con-ventional demarcations so as to reach a more profound anduniversal insight than provided by ordinary arts. Geometry has aspecial status in this process since it provided a model of successfultranscendence which legitimated similar attempts elsewhere.

In common with several contemporary Euclidean commenta-tors and editors, Dee blurred the classical boundaries betweennumber and magnitude.27 In the Praeface he recognises and repeatsthe definitions which establish them as entirely distinct types of

26 Ramus (1569), p. 306: ‘commentitium est, & valde nugatorium.’ Dee owned a copy ofwould have seen it soon after publication since Ramus ensured that Dee immediately receivdated by Dee 20 February 1567, with a note that it came via Sir William Pickering. PickerinEuclid [1661], p. 608), since it also has Ramus’s dedication, ‘J. Deeo Londonensi amico singuis reproduced in Sherman (1995), p. 105.

27 For a general account of this contemporary development, including reference to Billin28 de Wreede (2010), p. 385 and Bockstaele (2009). More generally on the coming toget29 Dee (1570), Groundplat. For the character and sources of archemastrie, see Clulee (198

classical problems as the doubling of the cube. While he did not mistake his mechanicalmaterial is strikingly unusual for the canonical presentation of a prestigious ancient scien

intellectual object. Thus numbers are initially considered exclusivelyas groups of indivisible units, while magnitudes are infinitely divis-ible lines, planes and solids. In practice however, Dee treated magni-tudes in unorthodox ways, employing operations outside theEuclidean canon such as the multiplication of proportions. The scopeof numbers is also greatly expanded when Dee moves from arithme-tic in its pure or principal capacity to its ‘vulgar’ use as a derivativeart. Rather than only positive integers, vulgar arithmetic includes aricher realm of fractions, roots and algebraic operations. These pro-vide a bridge between number and magnitude because roots can ex-actly represent even incommensurable magnitudes. ‘Practise hathled Numbers farder, and hath framed them, to take upon them, theshew of Magnitudes propertie: Which is Incommensurabilitie and Irra-tionalitie’ (Dee, 1570, sig. ⁄ij.r). For Dee, the historical development ofalgebra had delivered a way in which the older formal definitions atthe foundation of arithmetic and geometry could be surpassed.

Dee seems to hint that the interconnection of the two sciencesheralds a more powerful and unified art, though he nowhereclearly spells out his conception. Billingsley’s comment that theTyrocinium mathematicum contained ‘one new arte, with sundryparticular pointes, whereby the Mathematicall Sciences, greatlymay be enriched’ may be an early version of this ambition. Explain-ing his additions to the Elements, Dee himself says that,

my desire is somwhat to furnish you, toward a more general artMathematical then Euclides Elements, (remayning in the termesin which they are written) can sufficiently helpe you unto. Andthough Euclides Elementes with my Additions, run not in oneMethodicall race toward my marke: yet in the meane spacemy Additions either geve light, where they are annexed to Euc-lides matter, or geve some ready ayde, and shew the way todilate your discourses Mathematicall; or to invent and practisethinges Mechanically. (Euclid, 1570, fol. 371r)

If Dee’s ‘more general art Mathematical’ had been an algebraicapproach to magnitude it might have presaged similarly-framedambitions towards a mathesis universalis among the next genera-tion of mathematicians. This ideal was most concretely articulatedby Adrianus Romanus, who sought an algebraically-inspired foun-dation for all mathematics, but it is also relates to the work ofFrançois Viète.28 But Dee’s vision seems larger and more elusive.His inclusion of the practical, mechanical realisation of mathematicalconclusions under the umbrella of this general art suggests a possi-ble assimilation to his master science of ‘archemastrie’, which ‘teach-eth to bring to actuall experience sensible, all worthy conclusions, byall the Artes Mathematicall purposed.’29

Dee’s aspirations towards a general art may never haveachieved the methodical form that he envisaged. The idea mayhave been more of a moving horizon of possibility than a set ofdoctrines, as Dee successively elaborated his notions of hiero-glyphic writing in the Monas hieroglyphica, archemastrie in theMathematicall praeface, and the Adamic Enochian language in theangelic conversations. Although a far more limited disciplinaryunification, the linking of arithmetic and geometry through alge-braic work with roots had an exemplary significance. It showedboth the possibility and the value of transcending conventionalboundaries, and the powerful techniques that could result. Little

the 1569 Scholae mathematicae but it has not been located. There is a good chance heed the earlier Prooemium mathematicum. One of Dee’s copies of the latter is signed andg evidently acted as a mail box or courier (cf. Dee’s 1563 instructions to Commandino;lari P. Ramis dono misit’: Roberts and Watson (1990), no. 805; the inscribed title page

gsley and Dee, see Malet (2006).her of arithmetic, geometry, algebra and analysis, see Bos (2001), Ch. 6.8), pp. 170–176. In the Praeface Dee outlined mechanical procedures to carry out suchrecommendations for demonstrative conclusions, the inclusion of such unorthodox

ce.

476 S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479

wonder that Dee is so rhapsodic in praise of algebra. In the Praefacehe says that it,

is so profound, so generall and so (in maner) conteyneth thewhole power of Numbers Application practicall: that mans witt,can deale with nothyng more proffitable about numbers: normatch, with a thyng, more mete for the divine force of the Soule,(in humane Studies, affaires, or exercises) to be tryed in. (Ibid.,sig. ⁄ij.v)

There is moreover an intriguing linguistic aspect to this generalpower of algebra, which appears in Dee’s justification of his finaland potentially most profound art, that of scrying. Dee openedthe angelic conversations with the hope that they would providea new route through which to pursue the ambitions of his previousphilosophical studies, so as ‘to fynde or get some ynkling, glimpse,or beame, of such aforesaid radicall truthes.’30 This may be a longway from geometry and arithmetic, but the wording echoes his ear-lier studies in those more familiar arts, since mathematics indeedsupplied Dee with general and radical truths. Algebra was, for Dee,quite literally a radical art because it depended so much on theextraction and manipulation of roots. He chose his words carefullywhen describing such numbers:

(beyng desirous to deliver the student from error and Cavilla-tion) I do give to this Practise, the name of the Arithmetike ofRadicall numbers: Not, of Irrationall or Surd Numbers. (Dee,1570, sig. ⁄.ij.r)

After referring to Book X of the Elements as justification for histerminology, he burst out with a sense of wonder at their extraor-dinary power:

Therfore to call them, generally, Radicall Numbers, (by reason ofthe signe

p. prefixed,) is a sure way: and a sufficient generall

distinction from all other ordryng and using of Numbers: Andyet (beside all this) Consider: the infinite desire of knowledge,and incredible power of mans Search and Capacitye: how, they,joyntly have waded farder (by mixtyng of speculation and prac-tise) and have found out, and atteyned to the very chief perfec-tion (almost) of Numbers Practicall vse. Which thing, is well tobe perceived in that great Arithmeticall Arte of Æquation: com-monly called the Rule of Coss. or Algebra. (Ibid., sig. ⁄.ij.r–v)

Dee here uses the word radical in a self-conscious way ratherthan as a merely conventional term. Its later appearance in the an-gelic conversations may carry over some of the same freight, inaddition to the rest of its metaphorical range.31 Beyond such sug-gestive connections, we can more securely recognise how algebra—despite being a practical art not sanctioned by ancient geometry—could nevertheless get at the root of things, and point to a deeperand more universal perfection. This successful transcendence ofgeometry and arithmetic provided Dee with a warrant for the feasi-bility and desirability of similar moves in other areas of knowledge.

6. Dee and Digges

From the exploration of the further frontiers and more elusiveambitions of Dee’s enterprise, I want to come back down to a morereadily documented dimension of his contribution to the culture ofgeometry. I argued above that the Tyrocinium mathematicum hadbeen prepared as part of Dee’s tuition of Thomas Digges. Havingidentified the connections between that lost work and Dee’s addi-

30 Cited from the manuscript Liber mysteriorum, sextus et sanctus (1583) in Clulee (1988)31 Dee also employs a more horticultural or agricultural sense of ‘root’ in referring to the P

the angelic conversations even include – in an inadvertently comic passage – a mathema32 Mehl (2003), pp. 442–443; Kepler (1997), pp. 11–14.

tions to Euclid, and recovered something of Dee’s place within thecontemporary field of mathematics, we can now return to the rela-tionship between master and pupil—particularly through the prismof Digges’s first publication.

Digges issued his Mathematicall discourse of geometrical solids in1571 as an appendix to his posthumous edition of his father’s Pan-tometria. The latter is a practical geometry which has been muchcited by historians of surveying, mensuration and mathematicalinstruments. If the publication of Pantometria was the dutiful actof a son in memory of his father, then I suggest that the Mathemat-icall Discourse recognises the debt he owed to Dee as his ‘reveredsecond mathematical father’ (Digges, 1573, sig. A2r).

Digges’s work provides an advanced and novel treatment of thefive regular solids and their transformation, taking its startingpoint from Euclid and yet going beyond the Elements both in itscontent and its style. Before looking at it more closely, we shouldnote the significance of its subject matter. The five Platonic sol-ids—tetrahedron, cube, octahedron, icosahedron and dodecahe-dron—are presented right at the end of Book XIII. Each solid hasits own proposition in which it is geometrically constructed and in-scribed within a sphere. There is then a final proposition whichcompares the sides of the solids and a proof that no other regularpolyhedra are possible. These five solids were traditionally seenas the aim and culmination of Euclid’s work. Proclus provided anancient statement of the purpose of the Elements: the treatisewas meant ‘both to furnish the learner with an introduction tothe science [of geometry] as a whole and to present the construc-tion of the several cosmic figures’, by which Proclus meant the fiveregular Platonic solids (Proclus, 1970, p. 59). But by the time of Dig-ges’s work, this was no longer universally accepted. Ever the icon-oclast, Petrus Ramus took a diametrically opposed view anddisputed the value of studying the regular solids—prompting therighteous indignation of Kepler.32 Despite Ramus’s attempt toundermine the traditional doctrine, Dee inclined strongly to Pro-clus’s interpretation. Commenting on the last six propositions ofBook XIII in his notes to the English edition, Dee described them as‘the ende, Scope, and principall purpose, to which all the premissesof the 12. bookes, and the rest of this thirtenth, are directed and or-dered’ (Euclid, 1570, fol. 397v).

This match between Dee’s endorsement of the special signifi-cance of the Platonic solids and Digges’s choice of subject for hisfirst publication is a promising start. To go further requires a quickdetour through sixteenth-century understandings of the Elements.For the modern reader, most likely approaching Euclid throughthe English edition of Sir Thomas Heath, it is taken for granted thatthe Elements consists of Euclid’s thirteen books. However, bothauthorship and extent were much debated in the sixteenth cen-tury. Many scholars considered only the enunciations of the prop-ositions to be by Euclid and the proofs to have been supplied byTheon and other later commentators. The text itself came withnot thirteen but fifteen books, the last two being considered now(and by Billingsley) as later additions. These so-called Books XIVand XV continue the treatment of regular solids, the first derivingfurther results on their dimensions and the second inscribing themwithin each other, rather than only within a sphere.

Attentive sixteenth-century readers of the Elements would havenoticed that the text of these two books was far from stable. Thiswould have been particularly obvious to anyone using one of theseveral editions which conveniently juxtaposed the popularmedieval version by Campanus of Novara with Zamberti’s new

, p. 209.ropaedeumata aphoristica (Clulee, 1988, pp. 22, 27). Apart from their ‘radical’ character,tical square root (Casaubon, 1659, p. 80; Sz}onyi, 2004, p. 200).

S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479 477

translation from the Greek. (Such a dual edition was first issued atParis in 1516; Billingsley worked from a later issue.33) Zamberti’snotionally more pristine versions of Books XIV and XV containedfar fewer propositions than Campanus’s Arabic-based text. If thesetwo books were not the polished products of antiquity but hadgrown over time, might they still be in need of further extensionand improvement? The project of enhancing and ‘completing’ theElements was pursued particularly vigorously in the 1566 Latin edi-tion of the Elements by François Foix de Candale (the same mathema-tician and Hermeticist who prompted Billingsley’s comment on theTyrocinium mathematicum).34 Foix de Candale not only added furtherpropositions to books XIV and XV, but appended an entirely newBook XVI on the dimensions of the mutually inscribed regular solids.For good measure, he then added a short tract on a pair of semi-reg-ular solids, apparently unaware that they were just two of the thir-teen semi-regular polyhedra identified by Archimedes. Foix deCandale’s labours were well known in England, where Billingsleyprovided double versions of Books XIV and XV, one according tothe Greek version and another following Foix de Candale (with duereference to the intermediate contribution of Campanus). Nor didBillingsley stop there. Both Book XVI and the tract on semi-regularsolids were added so that ‘the reader should want nothing conducingto the perfection of Euclides Elements’ (Euclid, 1570, fol. 445v). (Cla-vius would likewise add Foix de Candale’s Book XVI to his 1574 ver-sion of the Elements, but both he and Billingsley were spared theadditional Books XVII and XVIII which Foix de Candale appendedto his 1578 Euclid edition.)

Thomas Digges came to mathematical maturity at just thismoment of Euclidean enthusiasm. He does not mention Foix deCandale, but must have been familiar with his work either inLatin or English for, as well as establishing many results on theregular solids, he provides his own treatment of five (different)semi-regular solids. His independent and critical stance is alsoevident in the treatment of the mutual inscription of the Platonicsolids, where he prefers a more narrowly defined notion ofinscription to the all-inclusive version used by Foix de Candale.35

More remarkable than these particular divergences is Digges’schoice of mathematical style. Foix de Candale (and Billingsley) ad-hered fully to the classical form of Euclidean geometrical demon-stration, constructing the solids and proving theorems on theirlengths, areas and volumes. Digges both supplemented and com-plemented this approach, giving greatest priority to providing rulesof calculation for the many magnitudes involved. Rather thanproofs, he provides a mass of numerical results which, in order toexpress the irrational magnitudes involved, make liberal use ofnested roots. For his more advanced problems it is true that he alsoprovides a purely geometrical construction (e.g. Digges, 1571, sig.Aa1r). But the majority of the work proceeds, in a striking phrase,‘geometrically by algebraical calculations’ (ibid., sig. S4v), so thatdimensions can be computed numerically rather than left asabstract magnitudes and proportions. The title, a Mathematicalldiscourse of geometrical solids, was therefore carefully chosen: whilethe solids Digges discusses are geometrical in origin, he treats themin a more generally mathematical way.

But although the presentation is very different from Euclid, Dig-ges cannot here be characterised as a problem-solving practitionerinsensitive to the qualities and intellectual values embodied in the

33 Archibald (1950) discusses Billingsley’s annotated copy of the 1558 Basel edition, now34 On Foix de Candale, see Harrie (1978).35 Billingsley moved Foix de Candale’s definitions of inscription and circumscription from

analogous definitions for plane figures of Book IV. Digges’s tighter definition of inscriptionsome additional detail on Digges’s Mathematicall Discourse, see Johnston (1994), pp. 64–74

36 Digges (1571), sig. Vj.r. Digges recognises the practical utility of approximation, as ha37 In Ramist vein, Snellius considered that Book X should be excluded from the teaching38 Euclid (1661), p. 606. This English Euclid contains a complete translation of Command

Elements. He clearly distinguished approximate arithmetical meansfrom his exact work with roots:

Nowe bicause long working with irrationall numbers, may bre-ede confusion in such as are not perfect in the rules of Algebra,ye may by the rule of proportion (supposing the Spheres dime-tient 1000) reduce surd numbers to integers, although not soexactly as the subtilitie of geometricall demonstration requireth(considering these cordes cannot precisely be expressed inrationall numbers) yet for any Mechanicall operation or manualmensuration, the difference shall not be sensible.36

Moreover, although not bound by the format of the Elements,Digges nevertheless respected and made use of Euclid’s work. Hisalgebraic approach could have completely ignored the Euclideanclassification of irrational lines that makes up a major part of BookX, and which Dee had set out to clarify. But Digges went out of hisway to employ the language of medials and major and minor lines,along with the various orders of apotomes and binomials, to iden-tify the magnitudes that arise in his study.

Together with the basic judgement that the regular solids wereworthy of study, this commitment to the classification of irrationallines represents a real intellectual position, highlighted by contrastwith Ramus’s contemporary critique of Book X and his dismissal ofthe regular solids as irrelevant. Subsequent critics of Book X wouldconsider that the obscurity of its purpose was even more frustrat-ing than its laborious and repetitive proofs. What was its pointwhen, as Willebrord Snellius would later observe, its contents arenot used elsewhere in more advanced ancient Greek mathemat-ics?37 Digges’s use of Book X in his deeper study of solids could serveas an apology for Euclid, and a reproof to those who saw such mat-ters as uselessly subtle.

Digges’s work and approach are strikingly congruent with Dee’sposition, as revealed by the Tyrocinium mathematicum and his con-tributions to the 1570 Euclid. Dee represented an open and moder-ate stance between rigid adherence to Euclidean tradition andoverzealous Ramist rejection. He endorsed the significance of thePlatonic solids, and his glowing account of algebra magnified itas a science of high intellectual stature rather than merely the vul-gar workings of commercial arithmetic. Because ‘radical numbers’could capture ‘the show of magnitude’s property’, traditionalgeometry could be supplemented by just the kind of algebraicinvestigation of solids that Digges explored so thoroughly. The for-mulation of Digges’s problems—establishing a sought dimensionon the basis of a given one—is also closely related to the type of‘data problems’ that Dee elaborated in his text on the parabola asa burning mirror (Clagett, 1980, p. 493).

If congruence can never quite prove influence, we can note thatDee elsewhere had a history of actively encouraging other mathe-maticians to pursue particular topics. Visiting Italy in 1563, he gaveFederico Commandino a manuscript on the division of figures thatDee thought might be ultimately attributable to Euclid. As Dee hadhoped, Commandino published this, adding his own generalisedtreatment of the subject. Dee’s letter to Commandino had hopedfor even greater things, specifically encouraging him to turn fromthe division of plane figures by lines to the division of solids byplanes.38 Dee’s friendly exhortation may not have diverted the moresenior Commandino, but his wish to foster more advanced work in

in Princeton.

Book XI to XV (Euclid, 1570, fol. 431v) and commented on how they differ from theresults in a more restricted set of possible inscriptions; Digges (1571), sig. Tiij.r. For.

d Regiomontanus (Bos, 2001, 136–138), but maintains the higher value of exactness.of geometry; see de Wreede (2010), p. 390; Bos (2001), pp. 141–142.ino (1570).

478 S. Johnston / Studies in History and Philosophy of Science 43 (2012) 470–479

solid geometry closely resonates with the actual endeavours of hispupil and ‘most worthy mathematical heir’, Digges.39

7. Conclusion

The Tyrocinium mathematicum of 1559 dates from an intenseperiod in Dee’s mathematical career. In the same year that he pub-lished the Propaedeumata aphoristica of 1558, he wrote the De spec-ulis comburentibus, part of which survives as a study of thegeometry of the parabola. It was in 1559 that he deciphered andmade the fresh copy of the medieval Latin translation of Machom-etus Bagdedinus’s On the division of figures that he would give toCommandino in 1563 (Rose, 1972). We have seen that the texton the area of triangles from 1560 was either part of or closely re-lated to the Tyrocinium mathematicum. It is also striking that Deenot only renewed his Euclidean interests at the time but was eventhinking about how geometry should best be presented: he ac-quired a Greek and Latin edition of Euclid in August 1558, and in1559 added a note that the misleadingly earth-bound term ‘geom-etry’ would be better replaced by ‘megathoscopica.’40 We thereforeshould not be fooled by the publication date of the English Euclid:the central period for Dee’s concern with geometry came a decadebefore 1570.

The detailed reconstruction of Dee’s career, and its chronologi-cal and intellectual disentanglement, is one of the principalachievements of Clulee (1988). My focus on Dee’s geometry is in-tended to offer a small supplement to that account. However, inshifting the centre of gravity of Dee’s mathematics from 1570 tothe later 1550s I have considered only Dee’s texts (rather thanhis instrumental and practical innovations). My account is there-fore incomplete, though it does makes good sense within Clulee’snarrative of the overall trajectory of Dee’s career. A fuller analysiswould move beyond what Dee classed in the ‘groundplat’ as theprincipal and vulgar aspects of geometry, so as to examine the fullrange of derivative arts which forms so prominent a part of thePraeface and its case for the utility of mathematics.

The absence of that broader scope here is simply a practicalmatter of available space; I certainly do not want to suggest a divi-sion of mathematics into separate spheres of theory and practice, adivision for which Dee’s work offers no support. Nor is my atten-tion to Dee’s texts an effort to elevate him to the intellectual pan-theon of the history of mathematics. Dee becomes interesting,significant and influential only when he is placed within a broadercultural history of mathematics.

I have set Dee within a contemporary context: the importanceof the various sixteenth-century editions of Euclid, the ideologicalclashes provoked by Ramus, and the inspiration and encourage-ment Dee evidently provided for Thomas Digges. To conclude, itis worth taking one more leaf from Clulee’s book, and looking be-yond the sixteenth century and the modern category of the Renais-sance. Clulee shows the depth of Dee’s indebtedness to medievalnatural philosophy and magic, particularly as developed in RogerBacon’s mathematical programme for natural philosophy. Thecharacteristically medieval treatment of geometry also helps tomake sense of Dee’s endeavours.

Rommevaux (2003) has outlined medieval interests in geome-try which were, above all else, centred on the Elements. Euclidwas the object of much didactic clarification, with corollaries, com-ments and meta-mathematical reflections frequently added to thetext. New and creative material was attached and embedded in a

39 Dee (1573), sig. A.ij.v describes Digges as ‘charissimus mihi Iuvenis, Mathematicusque40 Clulee (1988), p. 281 n. 5. In the Mathematicall Praeface Dee opted for ‘megethologia’41 This link to medieval practices provides a more fully historical version of the tens

Renaissance humanist interests and (particularly Dee’s) commitment to a more transcend42 Compare Clulee (1988), p. 84 on the relationship of the traditional exoteric discipline

wide range of editions, but this work was oriented towards trans-mission and pedagogy rather than innovation for its own sake.Moreover, much of medieval mathematics was pursued primarilyfor the insights it could bring to issues of theology and naturalphilosophy.

This perspective can be fruitfully juxtaposed with Dee’s practiceas a geometer. He was a commentator on Euclid and a pedagogue,supplementing and revising the text, and improving its accessibil-ity to students. Serving as a link in the chain of traditio was a virtuein its own right and there was no shame for Dee in the orderly pre-sentation and didactic compilation of geometry. But improvementwas also possible, and Dee had no hesitation in proclaiming thenovelty and importance of his own contributions, even if they in-volved no more than the restitution of a Euclidean diagram (Euclid,1570, fol. 380r–v).41 He was likewise open to the value of more sig-nificant historical change, such as the emergence of algebra. Butwhile he celebrated the ‘mervaylous newtralitie’ of ‘thinges Mathe-maticall’, Dee was ultimately more interested in their

straunge participation betwene thinges supernaturall, immor-tall, intellectual, simple and indivisible: and thynges naturall,mortall, sensible, compounded and divisible. (Dee, 1570, sig.

.iiij.v).

It was the light they shed on supercelestial and natural philos-ophy that provided their strongest justification.

For the modern reader, Dee’s career seems to embody both lim-itless ambition and also a curious conservatism. Unlike Ramus’sprojects of reform, Dee’s transformations of the arts transcendedthe individual disciplines which, though subsumed, were left lar-gely unchanged.42 Hence the coexistence of apparently disparateintellectual endeavours at different stages of his life. However, Dee’saim was increasingly directed towards the recovery of a profoundoriginal wisdom and a form of power that was both spiritual andworldly. Specific new results in the individual arts and sciences werenot to be dismissed but paled in significance beside the deepertruths Dee sought. If this seems disappointing from the perspectiveof a modern progressive vision of the sciences, it only serves tounderline how distant Dee was from such a world.

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