The Book of Wonders

On the one hand there was Euclid the great author, to be edited and translated with reverence and respect. To be involved with this tradition as editor or commentator was to display one’s credentials as a scholar, and the result tended to be large, heavy, expensive books meant for private or institutional libraries, well supplied with both space and money.

But there were also versions of the Elements for smaller pockets. Students at many universities were required to read specifically the first six books of the text, plus sometimes books 11 and 12: that is, the two-dimensional geometry, the theory of ratios, and the more basic three-dimensional geometry. The arithmetic or theory of numbers that made up books 7–9 was left out; so was the very tricky ratio theory in book 10 and the fairly tricky discussion of regular solids in books 13–15. There were, inevitably, printed editions catering to this particular need, containing only the prescribed eight books and usually in smaller formats, meant to be easier to carry around and cheaper to print and purchase.

There were slimmer Euclids still. Scholar and teacher Petrus Ramus in Paris did much to disseminate the belief, also aired by other early printers and editors, that the proofs in the Elements were not written by Euclid. Since it was quite true that the proofs in many printed Latin versions went back no earlier than Campanus in the thirteenth century, and since there was no direct evidence one way or the other about the authorship of the proofs to be found in Greek manuscripts of the Elements, the charge stuck easily, and Ramus and others were able to find a market for editions of the Elements containing just the statement of each geometrical proposition, leaving the provision of proofs and constructions to the preference of the individual teacher or the initiative of the individual reader: geometry on the do-it-yourself model.

Another world again were the editions of the Elements in European vernaculars that started to appear in the mid-sixteenth century. Some at least of these seem to have been aimed at the upwardly mobile merchant, anxious to demonstrate cultural sophistication but not anxious to claim a – perhaps non-existent – competence in Greek or Latin. That meant large formats, heavy price tags, copious annotation and the inclusion of learned prefaces and appendices.

Henry Billingsley’s English translation of 1570 followed this pattern: it was based on the printed Greek text but included comments and remarks selected from several Latin editions: ‘manifolde additions, Scholies, Annotations and Inventions … gathered out of the most famous and chiefe Mathematiciens, both of old time and in our age’. It contained a lengthy preface from the pen of Elizabethan mathematical celebrity John Dee, his remarks at various points in the text and a fold-out chart listing the different branches of mathematics and their relationships. The whole thing filled 1,000 pages, as against the forty-five of Ramus’ minimal Elements.

So, in the world of print, Euclid’s Elements was rapidly ceasing to be a single text, and becoming a wide tradition of different texts meant to be used in different ways by different kinds of people. It is immensely tempting to ask how they were actually used: who bought these books and what they really did with the Euclidean texts once they had them in their hands.

At the simplest level the answer is easy: they wrote all over them. There seems to have been a very long-lived convention that readers should write in mathematical books; by the later Middle Ages, owners were even writing on Stephanos’ priceless manuscript of the Elements. Print offered equally wide margins and less sense of defacing something unique; and in the world of print there was no tendency for marginal annotations to find their way by accident into subsequent copies of the text. Schools and universities seem to have positively encouraged the reading of mathematics with pen in hand, and the result is that up to three-quarters of surviving printed mathematics books from the sixteenth and seventeenth centuries bear readers’ annotations.

As a result, quite a lot of information is preserved about how people read Euclid. They read selectively, choosing a few propositions here and there, perhaps marking up the page to show which parts they had worked through or marking up the contents page or the index to the same effect. Surely teachers directed some of this, but surely, too, some of it was idiosyncratic: and there seems to be little evidence that the selections themselves were ever passed from person to person. For example: two surviving copies of the same late seventeenth-century edition of the Elements were marked up for use in English universities with selections from the first six books; yet the two sets of selections coincide hardly more than can be accounted for by chance.

Within the portions they chose to study, readers negotiated, sometimes aggressively, with the printed text, about what it should say. They corrected, added and emended. They noticed – and fixed – wrong labels in diagrams or missing lines in proofs. Some authors and editors frankly invited this kind of thing, by adding to the book at the last minute a list of ‘errata’ they had noticed while it was being printed: but most readers went far beyond any such authorised list in their corrections. Some added whole new propositions, cross-references to other books or portions from ancient or modern commentaries. Some collated more than one edition of the Elements, copying sections or details from one into another.

Finally, some readers used the margins of printed books simply to practise their own mathematics: writing out calculations, copying diagrams or repeating proofs until they were satisfied with their own performance. This kind of thing can give a most vivid impression: of learning mathematics as a sort of rehearsal; of mathematics itself as a special type of performance in the Renaissance world, just as it had been in antiquity.

But who were these readers, correctors, selectors and performers? In very many cases it is impossible to know. Some owners did sign their books, but it is not always possible to be sure that the hand of the signature was also responsible for the more interesting annotations found elsewhere in a book. And even when it is, most of the names are mere names, hardly capable of being located in time, place or society.

One such is the Marget Seymer who signed a copy, printed in Paris in 1543, of the Elementale geometricum of Johannes Voegelin; a fairly brief compendium drawn from Euclid that enjoyed a vogue in the sixteenth century. Women engaging with Euclid are elusive in this as in most periods, and it would be valuable to know more about her. But she seems to be virtually lost to history. She might be the Marget Seymer, daughter of Robert Seymer, who married Jerom Atwod at All Hallows, Honey Lane, in London on 11 May 1553 and was buried forty years later, in 1593. But she might as easily not be.

The same is true of many another Euclidean owner. Some can be connected with particular universities, schools or colleges; some signatures include a note of the date. Some readers provided other clues, such as by pasting in a grand bookplate with a coat of arms; and of course sometimes the book itself remains in situ in the library of a stately home or a university or college. More often, though, the books have moved, the records are lost and the trail is cold. When books were bought second-hand, some readers unhelpfully obliterated the signatures of previous owners, that might otherwise have helped to establish a trajectory. And the chronology is not always what it seems. One Sandie Hume remarked that Euclid’s Elements was ‘a Very Queen of a Buck’, and June Amelia Hume, presumably a relative, also signed and dated the copy. The book was printed in 1719, but she signed it more than a century later, in 1827.

Euclid in Renaissance Europe was thoroughly established as part of culture. It was more than ever – as it had been in the ancient and the medieval worlds – a site for annotation and engagement, for garlands of text draped around Euclid’s words themselves. But the very range of its travels makes some of those marks more tantalising than otherwise.

Edward Bernard

Minerva in Oxford

On the endless shelves of the Bodleian Library in Oxford there lives perhaps the most unusual copy of the Elements in the world. It has been put together from the separate sheets of four different printed versions of the text, in three different languages: Greek, Arabic and Latin. Its pages are dense with annotations made in the last two decades of the seventeenth century, in the hands of the then professors of geometry and astronomy at Oxford. Textual difficulties are wrestled with, mathematical problems are explained. Algebraic equivalents are provided for propositions about geometry or about ratios.

This polyglot Euclid is bound in two volumes, and it rarely sees the light of day. Its complex annotations are hard to interpret and often simply hard to read. But it is a monument to the kind of attention the text of Euclid’s Elements was beginning to attract by the years leading up to 1700.

The number of different printings of the Elements during the sixteenth and seventeenth centuries was staggering, approaching 300 by the end of that period. Even when there were scores of versions already available, printers and publishers showed no diminishment in their willingness to put the Euclidean text into print yet again: translated, rearranged, simplified, truncated or garbled as it might be. A new cohort of students every year did something to refresh the market, as did publishers’ invention of new markets by translating Euclid into modern vernaculars and by cutting, rearranging, annotating and re-presenting the book to appeal to new parts of the social scale.

By the second half of the seventeenth century an enthusiast with money to spend could amass quite a collection of Euclids. Robert Hooke, who, as well as being a founder member of the Royal Society of London and a celebrated experimenter and natural philosopher, was the lecturer on geometry at Gresham College in London, owned thirty-one different editions of the Elements, assembled at least in part for the sheer pleasure of collecting.

Yet for all the attention the Elements had received from printers and publishers over the two centuries since Ratdolt, it had remained strangely untouched by the kind of textual scholarship that other ancient books received. Version after version was put into print with scant regard for textual accuracy; the surviving Greek manuscripts were consulted only rarely, and the complete Greek text was put into print just once during 200 years (in 1533) – and that, badly.

An institution with a particular interest in ancient mathematical texts was the pair of mathematical professorships set up at Oxford in 1619 by Sir Henry Savile. The Savilian professors, who had their own rooms and a special collection of books within the Bodleian Library in central Oxford, amassed an impressive set of versions of the Elements. Savile himself donated not just printed books but manuscripts, and by the mid-seventeenth century Oxford was one of the few cities on earth to possess manuscripts of the Elements in Greek, Arabic and Latin.

Savile’s statutes for his two professorships aimed to promote a restoration of mathematical learning. They also aimed to refute – or facilitate the refutation of – some of the nonsense he felt was being talked on the European continent about mathematics and its ancient authors. Savile – who once described himself as passionately inflamed with love for geometry – felt a reverence for the Euclidean text that was worlds away from the quick-and-dirty printings it was receiving at the hands of some scholars, and he took particular exception to the notion that the book needed to be pruned or rearranged to make it suitable for the modern student. Polemicising against one cut-down version, he wrote that the Elements constituted a ‘perfect body’ of geometry: pure, shining and virtuous, fit to be admired, studied, and indeed loved. He wished his professors to undertake not just the teaching of the ancient mathematical authors but the study, editing and publication of their texts.

Good intentions lay dormant for a while, partly because in the mid-seventeenth century England had its mind on civil war rather than the editing of ancient mathematics. But late in the century Savile’s intentions began to be realised, with new, learned editions of certain texts: works on astronomy, on the mathematics of music and the like, newly edited from the rich manuscripts in Oxford’s libraries.

Into this situation came Edward Bernard: born in 1638 in the English Midlands, the elder child of a clergyman, and educated in London at the Merchant Taylors’ School and at St John’s College, Oxford. He learned the classical languages as well as Hebrew, and evidently had a talent in that direction; by early middle age he also knew some Arabic, Syriac and Coptic. He continued at St John’s College as a fellow and, later, a university proctor and college bursar.

Early in his time in Oxford, Bernard also studied mathematics. Evidently he had a flair for this subject too; in 1669 the professor of astronomy – Christopher Wren – appointed Bernard to deputise for him in Oxford when he was appointed surveyor of the royal works. Four years later, Wren resigned the professorship altogether and Bernard became in name as well as fact the Savilian Professor of Astronomy.

He lectured on ancient mathematics and astronomy, was elected a fellow of the Royal Society and acted as tutor to a handful of minor noblemen. He published minor works on ancient weights and measures, on ancient languages and on etymology, and a collection of prayers. He catalogued the manuscripts held by English and Irish libraries. He was something of a manuscript-hunter, travelling to Leiden on a couple of occasions to acquire rare items at auction. True to the intentions of Henry Savile, Bernard also edited ancient texts. He was involved with drawing up an ambitious proposal to edit the entirety of ancient mathematics, in twenty-one volumes. He actually began an edition of the ancient geometer Apollonius, but he never finished it. He also worked for years on an edition of the Jewish historian Josephus. But his real dream was a new edition of Euclid.

Looking at the surviving evidence for Bernard’s work on Euclid it becomes quickly clear why he was a man who published little and tended not to finish things. Bernard planned to compare the existing Greek editions with manuscripts, as well as correcting the best of the existing Latin translations against the Greek. And as well as a fresh, accurate version of the text, he intended to include the widest possible range of annotations. He planned notes on variant readings of the Greek text, cross-references to show the logical dependence of each proposition on others, and explanatory comments: all to be drawn from the riches of earlier printed editions and commentaries – Greek, Arabic, Persian and Latin – together with new information from Bernard himself and his contemporaries.

To this end, Bernard assembled a collection of printed copies of the Elements, as well as several manuscripts, and books containing the handwritten annotations of other scholars. On different printed copies of the text he wrote out parts of his carnival of Euclidean annotation, and he enlisted the help of his colleague, the Savilian Professor of Geometry John Wallis, to write and transcribe some of it.

Bernard was also interested in the Arabic versions of the Elements. There were manuscripts in Oxford, and he was able to draw on a printed source, too: a 1594 printing of a thirteenth-century Euclidean compendium attributed (wrongly) to the famous mathematician Nasir al-Din al-Tusi. Embodying the total of four centuries of study of the Euclidean text in Arabic, it was a storehouse of geometrical information and interpretation of the text, and Bernard chose it as the main basis for his study of the Arabic evidence.

It was perhaps inevitable that a project of such ambition would run into difficulties. There seems to have been some trouble coordinating the different layers of editing and commenting work, and Bernard’s collection of printed versions of the Elements came to bear a complex tangle of kinds and styles of annotation. Some of them directly contradicted one another. More than once a whole series of meticulous annotations had to be just as meticulously crossed out.

Having filled the margins of (at least) seven printed copies of the Elements with preliminary work, Bernard turned to the mind-bending labour of collating it all into a single unified text which might be capable of being printed. To that end, he obtained four more printed books, at what must have been considerable expense: copies of the 1533 Greek edition, the 1594 Arabic edition, a 1612 Latin edition and a 1620 partial edition in both Greek and Latin. He had all the pages taken out of their bindings, and he sorted the resulting mass of loose sheets together and had them re-bound so that he was left with, more or less, a single copy of the Elements in three languages, with the three texts running, as nearly as possible, concurrently on adjacent pages.

And he set to work to annotate this monstrous polyglot Euclid, putting in everything he had amassed so far: textual emendations and variant readings from the manuscripts, cross-references, endless detailed changes to wording, capitalisation, punctuation and even the placing of headings within the text, and of course commentary. Quickly the new copies themselves became something of a mess, with second thoughts and crossings-out, and several sections of the text never received certain of the layers of annotation at all.

The story now becomes a demoralising one. Bernard’s friends repeatedly tried to discourage him from persisting with a project that had taken leave if not of common sense then at least of the reality of the book trade. One wrote to him that there was neither publisher nor market for such an edition, since most mathematical scholars did not know Greek: ‘Men in this degenrous age wil not learne Greeke, or buy mathematicks att so great an expense of time and study and mony too.’ Even if he could persuade a printer to print the book, Bernard would be left to pay the bills when it failed to sell.

Nevertheless, Bernard several times had specimen sections of the text printed, including extensive sets of diagrams. But by doing so he merely reinforced the point that the costs of the edition were likely to be terribly high, and his friends became blunter still, urging him to ‘lay aside all thoughts of printing your Euclide’.

In the end, no publisher was found. Always in poor health, Bernard died of malnutrition and consumption in January 1697, his great Euclidean edition still no more than a mass of more or less disordered notes in the margins of printed books. He was buried in the chapel of St John’s College, and the Bodleian Library paid his widow £340 for a selection of his books and manuscripts. Three hundred years later they remain in the library, the manuscripts unpublished and unpublishable as ever.

The cautionary tale has a coda, however. Six years before his death, Bernard was succeeded as professor of astronomy by a man twenty years his junior. David Gregory, educated in Scotland and a former professor of mathematics at Edinburgh, was a likeable and energetic teacher: not a brilliant mathematician, perhaps, but a sound communicator and author. He worked successfully with Edmond Halley on an edition of Apollonius: something Bernard had failed to complete. Later, perhaps inevitably, his attention turned to Euclid.

By 1698 senior figures within the university were writing to the well-known London publisher Jacob Tonson in support of a proposal to print the works of Euclid in Greek, promising that Gregory would ‘take care of the Geometry & Reasoning’. The Greek text was to be overseen by one John Hudson, and there was further support from John Wallis, still in post as professor of geometry. Tonson had expertise in the printing of classical editions as well as being a founder of the notorious Kit-Kat Club.

No mention was made of Bernard, and the letters give the impression that this project was to be much scaled down compared with his ambitions. Tonson was, nevertheless, not impressed, and Euclid languished again. It was another five years before the Oxford University Press at last issued the complete works of Euclid in Greek and Latin. Gregory was credited as their editor, and in his preface he mentioned his debts to the manuscripts of Savile and the Bodleian Library, and the books of Edward Bernard. Some of Bernard’s specific corrections to the text can be clearly seen in Gregory’s edition, although the latter never acknowledged Bernard’s long efforts to bring the text to publication.

Euclid’s Elements had passed through a remarkable number of hands and been on a journey as long as that of any text, from the ancient world to the Renaissance. Around two millennia after its composition, and over two centuries after its first appearance in print – and after a multitude of more or less wayward printed versions – it had arrived at a point of rest, an edition with reasonable claims to be definitive: at least for the moment.

For all that, much of what Bernard had attempted to achieve was irretrievably lost in turning the edition into something that could practicably be printed and sold. There was no Arabic in this edition; there were no lengthy explanations, references to other authors, variant readings or discussions of what was in the manuscripts. Gregory’s Euclid was in most ways a pared-down edition, with little or no commentary and even some pruning of the text itself.

Athena in Oxford.

Euclidis quae supersunt omnia (Oxford, 1703), title page. (BEIC digital library/public domain)

One of its few adornments was an illustration on the title page, showing Minerva with shield, helmet and spear seated against an Oxford background (she was sitting in the middle of what was and is Broad Street, holding up the traffic in fine style). In the background could be seen the old Ashmolean Museum and the Sheldonian Theatre where the book was printed: and, fittingly, the Bodleian quadrangle, housing the Savilian books and the Savilian study, where Edward Bernard had laboured for so long.

Interlude

Wait. Stop. All of these encounters with Euclid tell only one of the many possible stories about the Elements. They tell a story in which the Elements was a work of Greek literature, that travelled where Greek literature did, and was transmitted and translated just like Homer and Hesiod, Aeschylus and Sophocles. It is a story in which Euclid’s was a text like other texts: beautiful, admittedly, but sharing a widespread tendency to become dirty, confused, corrupted under contact with human beings. It is a story, finally, in which Euclid is just an author like any other.