Euclid's Elements

BSHM Bulletin Volume 21 (2006), 2–25 ‘Much necessary for all sortes of men’: 450 years of Euclid’s Elements in English June Barrow-Green Open University This talk, given at the BSHM Textbooks Meeting in September 2005, is in two parts. First it looks at some of the English editions of the Elements published over the last 450 years. Then, to show how various editors differed in the way they approached the text, it looks at their treatments of Pythagoras’ Theorem (Book I, Proposition 47). Finally, it draws attention to some texts from the Fauvel Collection that either derive from the Elements or are closely connected with it. Introduction uclid’s Elements is the most famous mathematics textbook of all time, and so seems an appropriate subject to start off a meeting on mathematical E textbooks. I have never studied ancient Greek, however, and my Latin is of the schoolgirl variety, so in this paper I will restrict myself to some of the many editions of the Elements published in English. Rather than debate the finer points of the mathematics contained in them, I want to concentrate on some of their other features, such as printing, layout, authors’ notes, frontispieces, dedications, etc. In other words, I want to focus on those things that give you information about the general context in which a book was published. For the most part, I shall not be providing a detailed analysis or interpretation of these features but rather I shall be identifying them and indicating how they may be used to enrich our understanding of the milieu in which the texts were produced and used. Apart from the ubiquity of the Elements, there is another reason for wanting to include a talk on it at this meeting, and that relates to the Fauvel Collection, now owned by the BSHM and housed in the Open University Library. As many of you will know, John Fauvel had a very keen interest in the use of history in mathematics education and in connection with this had built up a substantial collection of editions of the Elements and related material. This meeting has provided an ideal opportunity to exhibit some items from the collection. (The catalogue of the full collection—sorted by author and by class mark—can be viewed or downloaded in pdf format from the BSHM website.) The talk itself will be in two parts. First I shall look at a number of editions of the Elements published over the last 450 years, taking them chronologically and pointing out their distinguishing features. Then, in order to show how various editors differed in the way they approached the mathematics, I shall look at their treatments of Pythagoras’ Theorem (Book I, Proposition 47). Finally, as a very short post script, I shall conclude by drawing attention to some texts from the Fauvel Collection that either derive from the Elements or are closely connected with it. BSHM Bulletin ISSN 1749–8430 print/ISSN 1749–8341 online ß 2006 Taylor & Francis http://www.tandf.co.uk/journals DOI: 10.1080/17498430600566527 Volume 21 (2006) 3 Figure 1. Title page of The pathway to knowledge 1551 Robert Recorde, 1551 Robert Recorde’s Pathway to knowledge, although largely derived from the first four books of the Elements, was not an actual edition. Rather, it was a rearrangement and simplification. It was published in 1551 and it was the first geometry book to be published in English. Recorde’s name does not appear on the title page—in fact his name does not appear until the end of a lengthy dedication some thirteen pages later. However, the title page (see Figure 1) did provide the source for the title of this talk. On the second page Recorde reveals his plan for the book: The argumentes of the foure bookes The first booke declareth the definitions of the termes and names used in Geometry, with certain of the chiefe grounds whereon the arte is founded. And then teacheth those conclusions, which may serve diversely in al workes Geometricall. The second booke doth sette forth the Theoremes, (whiche may be called approved truthes) servinge for the due knowledge and sure proofe of all conclusions and workes in Geometrye. The third booke intreateth of divers formes, and sundry protractions thereto belonging, with the use of certain conclusions. The fourth booke teacheth the right order of measuringe all platte formes [surfaces], and bodies also, by reson Geometricall. Despite Recorde’s ambition, only the first two parts, namely those on elementary geometry, were ever published, the third and fourth parts relating to practical mathematics never appeared.1 However, the evidence suggests that Recorde did get as far as producing manuscripts for the final two parts.2 1Later Recorde refers to the missing parts ‘...the other two books which shoulde have been sette forth with these two, yf misfortune had not hindered it ...’ Pathway, sign. z. iiv, the dedicatory Epistle to King Edward VI. 2John Bale, Index britanniae scriptorum, (ed R C Poole), Oxford, 1902. See Joy B Easton, ‘A Tudor Euclid’, Scripta Mathematica, 27 (1966), 339–355, 340. 4 BSHM Bulletin After the plan come two dedications: the first ‘To the gentle reader’ and the second ‘To the most noble and puissant prince Edwarde the sixte by the grace of God, of England, Fraunce and Ireland kynge, defendour of the faithe, and of the Churche of England and Irelande in earth the supreme head’. I am not going to discuss these dedications but would just draw your attention to the fact that the length of the first dedication is two pages while the length of the second is ten. I leave you to draw your own conclusions! Following the dedications there is an eight page preface ‘declaring briefely the commodities of Geometrye, and the necessitye thereof’ in which Geometry personified pleads her own case. She asserts her usefulness to the learned professions, although not only, as one might expect, to those pursuing logic and rhetoric, but also to those in medicine, law and divinity. But she provides no justification, relying instead on reference to authority. Her claim to be the friend of craftsmen of a myriad of types: seamen, merchants, carpenters, carvers, joiners, masons, painters, goldsmiths, tailors, shoemakers, weavers, clockmakers, etc. is more firmly founded and altogether more relevant to the content of the book. The Pathway, unlike Recorde’s other texts, is not written in dialogue, and could perhaps be more aptly described as a handbook rather than as a self-teaching text. It gives the definitions of terms, and then ‘teaches’ the conclusions, by demonstrating— not rigorously—the theorems. It is a book that, as one commentator put it, ‘steadily ignores proof’.3 Since at the time when Recorde was writing, the English language contained few technical terms, Recorde often had no choice but to create his own, which he did with great care, in an effort to minimise the difficulties for his readers, preferring common English words to Latin or Greek derivatives. Charming and evocative though his terms are—for example, prycke (point), dye (cube), touch line (tangent), threelike (equilateral triangle), likeside (parallelogram with all four sides equal), and likejamme (parallelogram with opposite sides equal—not many of them have survived. Uncharacteristically for Recorde, the Pathway has numerous printing and editorial errors (missing text, incorrectly labelled diagrams, etc), which are probably due to the fact that he was otherwise occupied while the text was actually being printed.4 It was reprinted in 1574 and again in 1602. Henry Billingsley, 1570 Billingsley’s Euclid, which was published in 1570 by John Day[e], was the first full edition of the Elements to appear in English.5 It includes, as part of the front matter, the celebrated Mathematical praeface by John Dee (of which more later) and as a consequence it has become one of the most famous of all English editions. It is monumental production of folio size, that is, roughly 8 Â 12 in., complete with notes extracted from all the most important commentaries from Proclus onwards, and at 3S Lilley, ‘Robert Recorde and the Idea of Progress’, Renaissance and Modern Studies, 2 (1958), 3–37, 26. 4It has been conjectured that these errors occurred because in May 1551, around the time of the book’s printing, Recorde was embroiled in controversy in connection with his current position as General Surveyor of Mines and Monies in Ireland, and so was unable to see the proofs of the book. See Easton, 1966. There is also confusion over the date of the book’s publication. The book itself is dated 28 Jan 1551 but at that time the New Year began on the 25 March. Thus by our system of dating (with the New Year beginning on 1 January) the date of publication would be 1552. 5For a more detailed discussion of Billingley’s Euclid, see R C Archibald ‘The First Translation of Euclid’s Elements into English and its Source’, American Mathematical Monthly, 57 (1950), 443–452. Volume 21 (2006) 5 Figure 2. Title page of Billingsley’s Euclid 928 pages long (excluding Dee’s preface) was clearly meant for patrons of substance (in contrast to Recorde’s work). Billingsley himself was a wealthy merchant who in 1596 became Lord Mayor of London.6 Apart from Dee’s preface, the edition is 6Henry Billingsley matriculated at St John’s College, Cambridge in 1550 and became a scholar there in 1551. He is also said to have studied at Oxford, although he did not take a degree at either University. He was afterwards apprenticed to a London haberdasher and rapidly became a wealthy merchant.

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