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 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. He was elected Lord Mayor of London in 1596 and he was knighted the following year. He was married five times and died in 1606. See: A McConnell, ‘Billingsley, Sir Henry (d. 1606)’, Oxford dictionary of national biography (ODNB), Oxford: Oxford University Press, 2004. 6 BSHM Bulletin remarkable for including what could be described (in modern parlance) as ‘pop-up diagrams’ to show the real form of solid figures. The figures are made of paper and pasted at the edges on to the page in the book so that they can be opened up to make actual models of the solid figures represented.7 The elaborate frontispiece (see Figure 2) is particularly interesting not least because it credits the contents to the wrong Euclid, namely Euclid of Megara (c.450–380 BC), a philosopher who lived some 100 years before the right Euclid. He was an associate of Socrates and a contemporary of Plato, and he was present at Socrates’ death. But Billingsley was not alone in making the mistake: confusing the two Euclids was a common error in the Middle Ages.8 The frontispiece is also full of all sorts of symbols, and symbolic and historical figures. The lower half contains the muses of the four quadrivium studies, with astronomy and music balancing geometry and arithmetic. And beneath them, lurking right at the bottom is Mercury, the god of learning, flanked by Gemini and Virgo. Floating up in the sky there is what appears to be slightly eclectic bunch of Greeks: Ptolemy, Hipparchus, Aratus, Strabo, and Polibius, the first two being rather more obviously mathematical than the other three. In fact the subject that links them all together is not mathematics but geography. Which begs the question as to why an edition of the Elements should be decorated with pictures of geographers. The answer is quite simple—economy. Such elaborate frontispieces were expensive to produce so it was to a publisher’s advantage if they could be reused. It turns out that this particular one was originally commissioned for William Cunningham’s Cosmographical glasse which was published in 1559, twelve years before Billingsley’s Euclid. The Cosmographical glasse contained the ‘principles of cosmographie, geographie, hydrographie and navigation’, which explains the presence of the constellation of geographers. Day[e] used the frontispiece for a third time in 1572 in Pandecte locorum communium (with an introduction by John Foxe) a commonplace book (a book with blank pages in which students jot down notes). That Day[e] chose to use this frontispiece for a commonplace book hints at the importance of mathematics and geography as areas of study at the time. Almost 400 years later the frontispiece was used again, but in a very different setting. In 1958 it appeared in a book on the history of German film which was published as part of the Rowohlts Deutsche Enzyklopa¨die.9 Billingsley’s Euclid also contains a notable colophon. Dated 1562, it is an excellent woodcut of the publisher, John Day[e], and is believed to be the earliest known portrait of a printer from this period.10 In the past it has been mistaken for a portrait of John Dee.11

7Two examples of ‘pop-up’ diagrams are reproduced at: http://www.math.ubc.ca/cass/Euclid/dee/ dee.html. 8Sir Thomas Heath, A history of Greek mathematics I, New York: Dover Publications, 1981, 355. 9Siegfried Kracauer, Von Caligari bis Hitler, Hamburg: Rowohlts, 1958. I am grateful to Volker Remmert of the University of Mainz for supplying me with this information. 10John Day[e] has been described as ‘one of the titans of the Elizabethan book world’ and ‘one of the London book trade’s most innovative and adept members’. See Andrew Pettegree, ‘Day, John (1521/2–1584)’, ODNB. 11Archibald (1950), 445. Volume 21 (2006) 7

Figure 3. First page of Dee’s Praeface and his Groundplat

Dee’s Mathematicall Praeface and Groundplat The extraordinary Mathematicall praeface by John Dee, the Cambridge educated astrologer and mathematician, is one of the most important British mathematical texts of the sixteenth century.12 Some 50 pages long, the Praeface surveys and classifies a wide range of mathematical arts and sciences. It was notably influential both in its encouragement of the belief in the use of mathematics in practical applications, and in its vindication of the publication of a classical text in the vernacular. At the end of his Praeface, Dee included a Groundplat, a large (14 20in) pull-out diagram summarising his classification of the mathematical arts (see Figure 3). Some of these subjects are familiar to us today, for example, perspective, astronomy, and architecture, but others are not so instantly recognisable, for example, ‘Trochilike’ (the study of circular motions), ‘Thaumaturgike (the production of marvels), and ‘Zographie’ (the study of painting and drawing).

Thomas Rudd, 1651 Captain Thomas Rudd’s edition, which contains the first reprinting of Dee’s preface, was published in 1651. Although it appeared more than 80 years after Billingsley’s edition, it was only the second edition to be published in English. The reprint to Dee’s preface was also sold separately.13 It is an altogether more modest volume than the Billingsley edition, being quarto rather than folio and being only 259 pages long.

12For a commentary on Dee’s Praeface and a reproduction of the original, see: John Dee ‘The Mathematical Praeface to the Elements of Geometrie of Euclid of Megara’ with an introduction by Allen G Debus, Science History Publications, 1975. 13E G R Taylor The mathematical practitioners of Tudor and Stuart England, Cambridge, 1967, 358. 8 BSHM Bulletin

It covers only the first six books of the Elements and those ‘In a compendious form, contracted and demonstrated ’. The only material beyond the mathematical text is an unsigned four page note ‘To The Reader’, which contains a rather curious admission. It begins, predictably enough, by stressing the dependence upon Euclid of all parts of mathematics, but then the writer informs the reader that due to the fact that the ‘author’ died before the text was completed, the text has had to be completed by a ‘revisor’. However, neither ‘author’ nor ‘revisor’ is actually named. Nevertheless, going on the contents of the title page, it would seem reasonable to suppose that the ‘author’ was Thomas Rudd. However, that assumption does not square with the fact that Rudd did not die until 1656.14 Thus it would appear that Rudd was actually the revisor rather than the author. Incidentally, Rudd had been the chief military engineer to Charles I.

Isaac Barrow, 1660 Isaac Barrow’s (1630–1677) first edition of the Elements was published in Latin in 1655, with his first English edition (translated from the Latin) following five years later. His editions were extremely successful and continued to be reprinted well into the eighteenth century. Barrow designed his edition with two aims in mind which he described in detail in the preface. The first aim was to make the (complete) proofs as brief as possible in order to make the volume ‘conveniently portable’. And portable it certainly is, especially when compared to Billingsley’s edition. Barrow’s edition measures 6½ 4½ inches and is only 369 pages long whereas Billingsley’s measures 8 12 inches and is 928 pages long. However, Barrow’s desire for brevity did not prevent him from including all fifteen books of the Elements (rather than, as editors such as Rudd had done, only those books that pertain most directly to geometry).15 The second aim, which worked in conjunction with first, was to produce an edition that would be attractive to those with a preference for symbolical rather than verbal proofs. He pointed out that such proofs, since they intermingle ‘words and signs at discretion’, have the advantage of avoiding ‘superfluous repetition’.

John Leeke and George Serle, 1661 The next edition in English to appear was published in 1661, the year after Barrow’s edition, and was the work of two obscure ‘students of mathematics’ John Leeke and George Serle. Unlike its predecessors it contains considerably more than just the Elements, including, not only the second reprinting of Dee’s preface, but also the first English edition of Euclid’s Data. In the nineteenth century various mathematicians, including Augustus De Morgan, described it as a second edition of Billingsley’s Euclid. However, as Archibald convincingly argues, this is incorrect.16 (In fact, Archibald shows how De Morgan’s account of Billingsley’s Euclid is mistaken in a number of ways.)

14Andrew Saunders, ‘Rudd, Thomas (1583/4–1656)’, ODNB. 15Books XIV and XV are continuations of Book XII (construction of the five regular solids) but are not by Euclid. Book XIV is the work of Hypsicles who lived in the second half of the second century BC; Book XV is much inferior to Book XIV and is attributed, in part, to Isidorus of Miletus 6th Century AD. 16See Archibald (1950), 450–451. Volume 21 (2006) 9

As the title pages show, the book was printed for two different booksellers. In fact the printer, Leybourn, was the same as the printer of Rudd’s edition of ten years earlier, and one of the booksellers, Richard Tomlins, was also the bookseller of Rudd’s edition. The other bookseller, George Sawbridge, was a partner in the King’s Printing House, treasurer of the Company of Stationers for much of his life, and Master of the Company in 1675. He died a wealthy man and was considered to be one of the greatest booksellers in England at the time.17 However, booksellers apart, the title pages of the two editions are identical and one could be forgiven for expecting the contents of the two books to be the same. But, there is a rather surprising difference. The Tomlins edition contains a ‘biography’ of Euclid while the Sawbridge edition contains a ‘portrait’ of Euclid. (Since nothing is known for certain about the life of Euclid, least of all what he looked like, the ‘biography’ and ‘portrait’ should be treated with caution!) Why the two editions should be different in this way is not clear but it is an interesting question.

Reeve Williams and William Halifax (from De Chales), 1685 In 1685 there was another case of two editions appearing at the same time, but this time, although they each derive from the same source, they are noticeably different. Both editions are translations of the French edition of 1672 by Claude Dechales,18 but each has a different editor and a different printer. One is edited by Reeve Williams ( fl. 1682–1793), an engraver and schoolmaster,19 and published in London, and the other is edited by William Halifax and published in Oxford. Since each of editors made their own translation from the French, there are minor textual differences between the two editions. But rather more significant than the textual variations, are the other differences between the volumes. Reeve Williams, who seems to have had his eye on the main chance, dedicated his edition to Samuel Pepys (1633–1703), who at that time was not only Secretary to the Admiralty but had been elected President of the Royal Society the previous year (1684). Halifax, on the other hand, did not include a dedication. Reeve Williams also incorporated a ‘portrait’ of a Euclid—completely different to the one included in the Leeke & Serle edition—below which the printer, Philip Lea, had included a small advertisement for his other wares.20 When Halifax’s edition was reprinted in 1700 (its third printing) it included an illustration of the well-known quotation from Vitruvius (from the preface to Book VI of De architectura, first century BC) about a shipwrecked philosopher (Aristippus) and his companions coming across geometrical diagrams in the sand. An illustration of the same story also appeared David Gregory’s Latin edition of The elements of 1703, published by the Oxford University Press. The Press reused the engraving in Halley’s 1710 edition of Apollonius’ Conics, and in Torelli’s 1792 edition of Archimedes,21 on each occasion replacing the diagrams in the sand with others more germane to the particular subject matter.

17EL Furdell ‘Sawbridge, George, the elder (b. in or before 1621, d. 1681)’ ODNB. 18Claude-Franc¸ ois Milliet Dechales, Huict livres des Ele´ments d’Euclide rendus plus faciles, Lyon, 1672. 19See E G R Taylor (1967), 278. 20Philip Lea ( fl. 1666–1700) was a notable cartographer, globe- and instrument-maker. See E G R Taylor (1965), 253. 21For reproductions of the Oxford University Press engravings, see J Fauvel, R Flood, R Wilson, Oxford figures: 800 years of Oxford mathematics, Oxford: Oxford University Press, 2000, 53, 129, 161 respectively. 10 BSHM Bulletin

Figure 4. Title pages of Geometry epitomiz’d and An epitome of geometry

William Alingham, 1695 and 1700 The next two editions, one of 1695 and one of 1700, were both the work of William Alingham (fl.1694–1710) who ran a mathematical school in Westminster,22 However, they differ in a number of ways, including in their titles (see Figure 4). The first, Geometry epitomiz’d, covers the first six books of Euclid, while the second, An epitome of geometry, includes the eleventh and twelfth books as well as a treatise on measuring and other practical subjects. The title page of the 1695 edition reveals that it can be bought from the author ‘over against the Rummur Tavern in Channel Row, Westminster’. However, in the 1700 edition, Alingham no longer mentions the tavern but simply states that the book can be bought from the author ‘in Channel Row, Westminster’; and he has added the phrase a ‘Teacher of the Mathematicks’ under his name. Alingham dedicated the first edition to ‘To the ingenious and my worthy Friend, William Lownds of Winslo in Buckinghamshire, Esq’, and it is probably no coincidence that the year in which the book was published was the same year as that in which William Lowndes (1652–1724) was appointed to a position in the Treasury. He dedicated the second edition ‘To John Earl of Bridgewater, J. Thompson, Ld. Haversham, Sir Robert Rich, Bar. Sir George Rook, Kt. Sir David Mitchel, Kt. Commissioners, for Executing the Office of Lord High Admiral of England’. Since the subjects taught by Alingham included ‘Arithmetic, Geometry, Navigation, Surveying, Measuring, Fortification, Throwing of Bombs ...’,23 and that he was in the business of making ‘Youth ...fit for any sort of Business, either by Sea

22See E G R Taylor (1967), 289. 23From an advertisement in W Alingham Thesaurarium mathematicae, by John Taylor, gent ...Enlarged by W. A., London, 1707. Volume 21 (2006) 11 or Land ...’, it would surely have been to his advantage to secure the patronage of the commissioners for the admiralty. Precisely why Alingham produced two such different editions within a relatively short time span, is so far unknown. Nevertheless, comparing the two editions does give an insight into some of the changes that must have taken place in Alingham’s life during the five year period between the dates of the publication of the respective editions.

Figure 5. William Whiston and the title page of his Euclid of 1714

William Whiston, 1714 William Whiston (1667–1752) published his first English edition of the Elements in 1714. He had previously published two editions in Latin, which is why the advertisement on the title page (see Figure 5) advertises it as the third edition. Altogether Whiston published six editions in Latin, the first appearing in 1703 and the last in 1795, and ten editions in English, the last of which appeared in 1792. There was an edition in Greek in 1805, and printings of his editions were made in Cambridge, London, Amsterdam, Venice and Vienna.24 As well as the Elements, the volume also contains ‘Select Theorems out of Archimedes by the Learned Andrew Tacquet’ and ‘Practical Corollaries, shewing the uses of many of the Propositions’. William Whiston (1667–1652) was Newton’s successor to the Lucasian chair at Cambridge, a post he held from 1702 until 1711 when he was expelled for heresy.25

24S D Snobelen, ‘Whiston, William (1667–1752)’, ODNB. 25Whiston had lectured as Newton’s deputy when Newton was called to the Mint and so was the natural successor to Newton’s chair. His conflict with the Church derived from his support for the anti-trinitarian doctrine. 12 BSHM Bulletin

Figure 6. Barrow’s Preface and Whiston’s ‘portrait’ of Euclid

Thus the 1714 edition appeared after Whiston had lost his professorship, despite the claim on the title page. To emphasize the value and significance of mathematics in general, Whiston opened the volume with a twelve page essay entitled ‘An historical account of the rise and progress of the mathematicks’, which, as one might expect in an edition of the Elements, concentrates mostly on the contribution of well-known Greek mathe- maticians. However, it does also contain a number of biblical references, particularly at the beginning, which gives a clue as to Whiston’s own set of beliefs. The historical account is followed by a general preface. However, rather than write his own, Whiston chose to use one previously written by Isaac Barrow whose works by this time were extremely popular (see Figure 6). Given Whiston’s religious convictions, Barrow’s preface, with its caption ‘God always acts Geometrically’ and opening words ‘How great a geometrician are thou O Lord?’, was a natural choice. Whiston’s edition also includes yet another image of Euclid. In this one Euclid is represented in profile with the source of the image—‘Taken from a Brass Coin in the Repository of the late Queen Christian of Sweden’—given as a caption.

Henry Hill, 1726 Henry Hill, as other editors had done before him, chose to include only the first six books and the eleventh and twelfth books of the Elements in his edition. But while Hill may have been traditional in his choice of material, he was certainly not traditional in the way he chose to deal with it. Using ‘a new, plain and easie method’ Hill’s take on his subject matter bears little resemblance to that of his predecessors. In the Preface he explains that the method is ‘wholly different from any that has hitherto been made choice of by any Interpreters of Euclid, and as such does not so much require Intent and Severe Thinking, as a bare and easie Inspection’. He goes on to say that he is ‘chiefly and for the most part’ going to use Algebra where he considers it ‘convenient to deviate from the original’, and that he will include Volume 21 (2006) 13

‘as much of Literal Arithmetic as necessary in the demonstrations’. The result is a version of the Elements which looks quite unlike any other so far considered (as will be shown in the following section on the treatment of Pythagoras’ Theorem). And it is perhaps worth making the point that there is little concern for the integrity of the original text. In common with many other books of the period, Hill’s edition included a list of subscribers. These lists, which provide a useful resource for information about the circulation of texts, and the economics of production, contain the names of all those who, prior to publication, have agreed to buy at least one copy of the book. Clearly, the more prestigious the subscriber, the better it is for the author from the point of view of future sales. Amongst Hill’s subscribers were the Duke of Somerset (Vice- Chancellor of the University of Cambridge) to whom the book was dedicated, the Marquis of Hertford, John Mather (Vice-Chancellor of the University of Oxford), plus several members of the universities of Oxford and Cambridge, and numerous members of the clergy. One subscriber, a certain Richard Heath signed up for nine copies. Altogether 91 subscribers are listed, including five women (three of whom were from the same family), and between them they subscribed for 103 books.

Robert Simson, 1756 Robert Simson (1687–1768) was a Scottish mathematician and professor at Glasgow University. His translation, which included Critical notes explaining how and why his account differed from the Greek text, ran to some thirty editions which went on appearing right until the middle of the nineteenth century, with only the first four being published before his death. Simson’s translation formed the basis for later versions by many different authors, including those of two of the most popular textbook writers of the nineteenth century, Robert Potts and Isaac Todhunter.

Dionysius Lardner, 1828 Dionysius Lardner (1793–1859), a graduate of Trinity College, Dublin, was the first holder of the chair of natural philosophy and astronomy at University College, London. He took up his position in London in 1828, the same year in which his edition of the Elements was published. Lardner’s edition consists of the first six books, together with a commentary and geometrical exercises, and, instead of Books XI and XII, he included a Treatise on solid geometry which was mostly based on Legendre. He also added a large number of corollaries and additional propositions (all in smaller print), as well as an Appendix on the theory of parallel lines in which he gave a brief history of attempts to get over the problem of the parallel postulate as far as Legendre. Lardner’s edition was very popular and by 1861 was being republished for the twelfth time.

Oliver Byrne, 1847 One of the most celebrated editions of the Elements ever published, and without doubt one of the prettiest, is the one edited by Oliver Byrne ( fl. 1835–1878). The presentation is entirely novel, for it is, as Byrne himself tells us, one in which ‘Coloured diagrams and symbols are used instead of letters for the greater ease of learners.’ And a quick glance at the title page (see Figure 7), which shows a diagram from Pythagoras’ theorem, makes it clear just how very different from other editions this one is going to be. 14 BSHM Bulletin

Figure 7. Title page of Byrne’s Euclid

In his introduction, which ran to eleven pages, Byrne explained the rationale behind his method, and showed how it worked. He was in no doubt as to the method’s pedagogical efficacy: for he claimed ‘such is the expedition of this enticing mode of communicating knowledge, that the Elements of Euclid can be acquired in less than one third the time usually employed, and the retention by the memory is much more permanent; these facts have been ascertained by numerous experiments made by the inventor, and several others who have adopted his plans.’ That the volume was a source of great pride to its printer, Charles Whittingham of the Chiswick Press, can be judged by the fact that was one of a very small number of British books displayed at the Great Exhibition of 1851. Although perhaps it was not merely a coincidence that Whittingham was on one of the Juries of the Exhibition.26

26For further information about Oliver Byrne and to view the entire book see the University of British Columbia website: http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/byrne.html. Volume 21 (2006) 15

Isaac Todhunter, 1862 (1878) Isaac Todhunter (1820–1884), one of the most prolific authors of mathematics textbooks in the nineteenth century, as well as the author of several histories, began his career as an schoolmaster, before going to Cambridge and graduating as senior wrangler. He went on to become a college lecturer, and then finally a private tutor, before devoting his energies to textbook writing. He was one of the staunchest defenders of Euclid against the rising tide of reformers who, inspired by the preface of James Wilson’s Elementary geometry (1868), advocated a move away from Euclid as a school textbook.27 Todhunter designed his edition—which was one of many based on the translation by Robert Simson—to maximise sales figures—it is of a very practical size and it was sold for a reasonable price of 3s 6d. The text is clearly laid out and there plenty of notes to help with its study. It was translated into Urdu and Japanese, and there was a special edition prepared for Indian schools. More than twenty editions were published, with the last one appearing in 1932 and a final reprint in 1955. Altogether more than half a million copies were sold. Todhunter, who went up to Cambridge as an impoverished student in 1844, made a considerable fortune out of writing textbooks. When he died in 1884 his estate was worth over £80,000, a large sum of money at the time.28

Charles Dodgson (Lewis Carroll), 1882 Charles Dodgson (1832–1898), who limited his edition to the first two books of the Elements, was a renowned champion of Euclidean geometry, not least through his five-act comedy Euclid and his modern rivals (1879) in which he showed his distaste (in common with Todhunter) for the movement away from the traditional methods of teaching classical geometry. He also shared with Todhunter a concern for the appearance of his text, and, in fact, was so impressed by Todhunter’s diagrams in the latter’s edition of the Elements that he obtained permission from him to use them for his edition.29 He illustrated the logical structure of the Books with diagrams showing how each of the propositions depend upon one another (see Figure 8).

Sir Thomas Heath (1908), 1956, 2002 The edition of Sir Thomas Heath’s (1861–1940), first published in 1908, was the first English translation of all thirteen books of the Elements since J Williamson’s edition of 1781–88 (all the intervening translations omitted Books VII–XXX and XIII, with several not going beyond the first six books) and it is still print today. It contains an extensive commentary. In the three volume Dover reprint, for example, the first proposition is not reached until page 241, the preceding 240 pages, more than half the volume, being taken up with an introductory discussion; and in the discussion of

27Isaac Todhunter ‘Elementary Geometry’, The conflict of studies, and other essays on subjects connected with education, Macmillan, 1879. For a discussion of Todhunter’s work see J Barrow-Green ‘Isaac Todhunter and his Mathematics Textbooks’, Teaching and Learning in 19th-Century Cambridge (ed. J Smith, C Stray), Boydell, 2001. 28By way of comparison, Henry Smith, Savilian Professor of Geometry at Oxford who died in 1883, was only worth about £3,000, and Arthur Cayley, Sadlerian Professor of Pure Mathematics at Cambridge, who died just over a decade later was worth about £23,000. 29Letter from Dodgson to Todhunter, 20 March 1876, Todhunter manuscripts (Notes I), St John’s College Archives, Cambridge. 16 BSHM Bulletin

Figure 8. Dodgson’s ‘Logical sequence’

Pythagoras’ theorem, the statement and proof run to just over a page, while the accompanying notes extend to eighteen pages. In addition to the full edition, Heath’s translation without the Notes was included as part of the Encyclopaedia Britannica Great Book Series which first appeared in 1952; and the translation without the Notes was republished by The Green Lion Press in 2002.30 Taken as a whole, Heath’s edition is an impressive work of scholarship and an extremely useful source of reference. However, it was produced almost a century ago and some of Heath’s interpretations have now been superseded by the work of more recent scholars such as Wilbur Knorr and David Fowler.31

David Joyce (Web edition), 1996 David Joyce’s web edition of the Elements is based on that of Heath’s but, by Joyce’s own admission, is slightly less literal so as to be more accessible. It contains the text of all thirteen books and the figures are illustrated using a Java applet (the Geometry Applet). This has the advantage that the diagrams can be dynamically changed so that the propositions can be seen to hold through all the possible ranges of the constructions. Joyce has not included Heath’s commentary but has provided his own guide through the text. (For those wishing to see Heath’s commentary, the website contains a link to the Perseus Project where Heath’s edition is available on-line.)

30Euclid, Euclid’s Elements, tr. Thomas L Heath (ed. D Densmore), Green Lion Press, 2002. 31See, for example, W Knorr ‘The wrong text of Euclid: on Heiburg’s text and its alternatives’, Centaurus, 38 (1996), 208–276; D Fowler, The mathematics of Plato’s Academy: a new reconstruction, Oxford: Clarendon Press, 1999 (second edition). Volume 21 (2006) 17

Treatments of Pythagoras’ Theorem The following is a selection from the above texts of treatments of Euclid I.47, otherwise known as Pythagoras’ Theorem. The idea is look at each one of these in a holistic way in order to get a sense of the pedagogical style of the editor concerned.

Figure 9. Recorde’s version of Elements I.47

Robert Recorde, 1551 Looking at the diagram (Figure 9) we can see that the theorem is being demonstrated by using the example of a 3, 4, 5 triangle—each of the three squares has been divided up into little squares, all of the same size. The square on the hypotenuse has been divided into 25 smaller squares, and the squares on the other two sides have been divided into 16 and 9 squares respectively. The accompanying text does not provide a proof of the theorem but shows why it is true in this particular instance. Notice how the text all runs together which makes the argument quite difficult to digest. (Incidentally, the diagram is incorrectly labelled—see the discussion on Recorde above.)

Henry Billingsley, 1570 In contrast to Recorde, Billingsley provides a complete proof of the theorem. But as in Recorde, the text is very dense and all runs together (see Figure 10). It is difficult to discern the structure and to follow the reliance on previous propositions, etc. The diagram, however, is nicely separated from the text and clearly labelled. 18 BSHM Bulletin

Figure 10. Billingsley’s version of Elements I.47

Figure 11. Rudd’s version of Elements I.47

Thomas Rudd, 1651 A quick glance at Figure 11 is enough to see that Rudd is not following Billingsley. Not only has he used an alternative proof—which can be seen immediately from the diagram—but he has made the text much easier to read by using indented paragraphs and an italic type face for the references to previous propositions. He has also taken care with the diagram. It is a good size and clearly positioned above the text. Moreover, it is repeated on a second page so that it is always visible to the reader. Volume 21 (2006) 19

Figure 12. Barrow’s version of Elements I.47

Isaac Barrow, 1660 Barrow’s treatment (see Figure 12), both in the statement of the proposition and in the proof, was markedly different from either Billingsley’s or Rudd’s. Although Barrow used an equivalent method of proof to Billingsley, as the similarity in their diagrams attests, he handled it altogether differently (to concord with the aims stated in his preface mentioned above). Where Barrow departs from his predecessors is in his use of notation and symbols. In the statement of the proposition he makes specific reference to letters on the diagram, which then enables him to be much more concise in the proof. In the latter he makes extensive use of symbols, particularly the ¼ sign, with the result that the entire proof is only ten lines long. (For comparison, Billingsley used 38 lines to convey the same information.) A further novelty of Barrow’s presentation is the placing of references to previous propositions etc. in the margin, which not only makes the references easier to see but also highlights the axiomatic structure of the whole enterprise. 20 BSHM Bulletin

Figure 13. Plate 3 from Williams and Halifax 1685

Reeve Williams and William Halifax, 1685 The distinguishing feature of Reeve Williams’ edition is the separation from the text of the diagrams. Instead of each diagram being incorporated into the text at the relevant point, the diagrams are printed in groups on separate sheets, with about a dozen diagrams on each sheet, and placed at the back of the volume (see Figure 13). This clearly was simpler (and undoubtedly cheaper) for the printer. Nevertheless, in doing this Reeve Williams was not following Dechales, the French editor, from whom he had made his translation. Halifax, on the other hand, did follow Dechales and included his diagrams in the body of text. Neither Reeve Williams nor Halifax followed Barrow’s example of placing references to previous propositions in the margin.

William Whiston, 1714 Whiston, like Reeve Williams, separated the diagrams from the text. But, unlike Reeve Williams, Whiston put all the diagrams together on one fold-out page which, since he included 92 diagrams, meant that each diagram had to be extremely small Volume 21 (2006) 21

Figure 14. Diagrams for Book I from Whiston 1714

and hence very difficult to see. However, Whiston did adopt the convention of placing references to previous propositions etc. in the margin. As well as the proposition and its proof, Whiston included a Scholium in which, characteristically, he did not waste the opportunity to emphasise his religious devotion, as the following quotation shows: ‘This Theorem ...is commonly call’d the Pythagoric Theorem, from Pythagoras the Inventor of it; who, as is attested by Proclus, Vitruvius and others, offer’d Sacrifices to the Muses, as supposing himself have been helped by them in so excellent an Invention; in which thing he shew’d himself to be ignorant of God, the Lord of Sciences, the true and only Author of all Wisdom; or certainly if he knew him, he glorified him not as God.’. The Scholium also included three problems and their solutions connected with the proposition, the last one of which showed, as a corollary, how tables of sines, tangents, and secants could be derived.

Henry Hill, 1726 Hill had promised in the Preface that his interpretation of Euclid would be ‘wholly different’ to any that had gone before, and a look at his treatment of Pythagoras’ Theorem shows that indeed it is (see Figure 15). Not only does the method of proof, which makes an unprecedented use of algebra, require a diagram unlike any used previously, but, for reasons of expediency, it also uses a special layout invented by John Pell which involves two ruled left-hand columns. With the diagram set up, Hill then uses algebra to dispose of the ‘demonstration’ of the truth of the theorem in a mere four lines. 22 BSHM Bulletin

Figure 15. Hill’s version of Elements I.47

Figure 16. Simson’s version of Elements I.47

Robert Simson, 1756 Robert Simson’s translation and layout provided a model which was used by many later editors right up until the end of the nineteenth century (see Figure 16).

Dionysius Lardner, 1828 Lardner’s edition was one of the most popular of the period, possibly because it was also crammed full of all sorts of extra information. It is not easy on the eye (see Figure 17)—every page is densely covered with text and diagrams are relatively small. In the case of Pythagoras’ Theorem, although the actual proof of the theorem is not very long and the diagrams are small, there are several extra corollaries etc., so that the theorem and its related material take up almost five full pages.

Oliver Byrne, 1847 Since Byrne’s interpretation of the Elements it is essentially visual (see Figure 18), with colours (red, yellow and blue) being used to distinguish between different lines, different angles, and different areas, there is only a minimum of text and no need for Volume 21 (2006) 23

Figure 17. Lardner’s version of Elements I.47

Figure 18. Byrne’s version of Elements I.47 letters on diagrams. His treatment of Pythagoras’ theorem is therefore remarkably short in comparison to that of most of the others considered here. It is also very clear and straightforward to follow. The presentation could hardly be in greater contrast to that of Lardner. 24 BSHM Bulletin

Figure 19. Todhunter’s version of Elements I.47

Isaac Todhunter, 1862 (1878) Todhunter placed a high value on overall presentation (see Figure 19) and in that it would appear he differed from Lardner whose text is certainly not easy on the eye. Particularly striking, is the amount of space Todhunter leaves around his diagrams so that they are not submerged in the text, and which makes the entire proof much easier to follow. In addition, he uses a new line for each new assertion, and he is liberal in his use of paragraphs. He deliberately placed all the notes and exercises together at the end of the text so that they did not clutter up the essential core of the mathematics.

Related texts in the Fauvel Collection Finally, I want to point out that as well as the numerous editions of the Elements published over the decades, there were also books produced that either relied on or that related to particular aspects of the Elements. The following are three such texts selected both because they represent a spectrum of activity and because they come from the Fauvel collection. First, B Donne’s rather obscure little book of 1826 (see Figure 20), the full title of which is ‘An Essay, on Mechanical Geometry, chiefly explanatory of a set of Schemes and Models, by which the knowledge of the most useful propositions of Euclid, and other celebrated Geometricians, may be clearly and expeditiously conveyed, even to youth of an early age’, is not listed in the British Library catalogue. It appears that it was written to accompany a set of apparatus, for the title page includes the statement ‘Price with the Apparatus, Two Guineas and a Half’. The title page also reveals that the author, B Donne, was the ‘Late Master in the Mechanics in Ordinary to His Majesty’; that the book was printed in Bristol for ‘M.A.D. Clayton (Daughter of the Author)’; and that this is a ‘new edition revised and corrected by S. Rootsey, F.S.L.’. The title page also includes a quotation from the Book of Job (which might hint at a religious connection). Although a quick internet search did not come up with any further information about or publications Volume 21 (2006) 25

Figure 20. The title page of Donne’s Essay by the author, it did come up with information on Samuel Rootsey (1788–1855), a Fellow of the Linean Society who lectured on botany at Bristol Medical School (1832–1854). The next two books are rather better known, so I will just mention their titles and then no more about them: Augustus De Morgan, The connexion of number and magnitude: an attempt to explain the fifth book of Euclid (1836) and John Ruskin, The elements of perspective. Arranged for the use in schools and intended to be read in connexion with the first three books of Euclid (1859).

Conclusion I hope this very quick spin through the Elements has alerted you to some of the things to look out for when studying a mathematics textbook from the past. I have of course only mentioned a small number of the editions of the Elements that were published in English, but I have tried to bring in as many from the Fauvel Collection as I could. At all events, I hope you have been able to see that a mathematics textbook has the potential to tell us much more than just how to tackle a particular branch of mathematics. In the time that I had available to prepare this talk, I was not able to study these texts in detail. However, the work I have done so far has made me think that in-depth study of Euclid in English would be a worthwhile addition to the literature, both from the pedagogical and the historical point of view.