THE BOOK OF WONDERS
The Many Lives of Euclid’s Elements
Benjamin Wardhaugh

Copyright

William Collins

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This eBook first published in Great Britain by William Collins in 2020

Copyright © Benjamin Wardhaugh 2020

Cover images: God, the Divine Architect © Bridgeman Images; The Friday Mosque © Gerard Degeorge / Bridgeman Images

Benjamin Wardhaugh asserts the moral right to be identified as the author of this work

A catalogue record for this book is available from the British Library

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Source ISBN: 9780008299903

Ebook Edition © August 2020 ISBN: 9780008299927

Version: 2020-07-10

Dedication

For my parents

Contents

Cover

Title Page

Copyright

Dedication

Prologue

I: AUTHOR

Alexandria: The geometer and the king

Elephantine: Pot shards

Hypsicles: The fourteenth book

Theon of Alexandria: Editing the Elements

Stephanos the scribe: Euclid in Byzantium

Al-Hajjaj: Euclid in Baghdad

Adelard: The Latin Euclid

Erhard Ratdolt: Printing the Elements

Marget Seymer her hand: Owning the Elements

Edward Bernard: Minerva in Oxford

Interlude

II: SAGE

Plato: The philosopher and the slave

Proclus Diadochus: Minerva in Athens

Hroswitha of Gandersheim: Wisdom and her daughters

Rabbi Levi ben Gershom: Euclid in Hebrew

Christoph Clavius: The Jesuit Elements

Xu Guangqi: Euclid in China

Blame not our author: Geometry on stage

Baruch Spinoza: The geometrical manner

Anne Lister: Improving the mind

Interlude

III: HERO

Petechonsis: Taxing and overtaxing

Dividing the monochord

Hyginus: Surveying the land

Muhammad Abu al-Wafa al-Buzjani: Dividing the square

Lady Geometria: Depicting the liberal arts

Piero della Francesca: Seeing in perspective

Euclid Speidell: Teaching and learning

Isaac Newton: Mathematical principles

Interlude

IV: SHADOW AND MASK

Mary Fairfax: Euclid and the straitjacket

François Peyrard: Manuscript 190

Nicolai Ivanovich Lobachevskii: Parallels

Maggie and Tom: The torture of the mind

Simson in Urdu: The Euclidean empire

His modern rivals

Thomas Little Heath: The true con amore spirit

Max Ernst: Euclid’s mask

Euclidean designs

Lambda: Curved space, dark energy

Epilogue

Notes on Sources

Select Bibliography

Index

Acknowledgements

About the Author

Also by Benjamin Wardhaugh

About the Publisher

Prologue

Alexandria. Alexándreia. The reign of Ptolemy, first of the Alexandrians. Perhaps the tenth year of his reign: a little after 300 BC.

Arrive by sea, the Egyptian sun on the water. Cross the harbour; enter the city. Building upon building. Through the Gate of the Moon and up the boulevard, the Canopic Way. White marble; dust and hammering: building work everywhere. The grandest of cities. You can drive your carriage up the streets if you want to. Paved streets and white-faced buildings. The sea murmuring.

To the crossroads; turn left onto the Street of the Sema: long and cool, catching the wind. Into the palace quarter. Temples; museum; library.

One of the workers in the famous cultural quarter is a man named Euclid. One of his books, the Elements of Geometry. When Ptolemy’s glorious Alexandria is dust, it will live.

For twenty-three centuries the Elements of Geometry has been changing the world. A compendium of facts about space and its properties – lines and shapes, numbers and ratios – it has drawn countless readers into its limitless world of abstract beauties and pure ideas. And it has been on the most incredible journey. Few artefacts survive the collapse of the culture that produced them; few texts survive the abandonment of the language in which they are written. The Elements has survived both. Indeed, it seems to have positively thrived on its transplantation into a series of fantastically diverse situations. Its very austerity seems to have enabled readers to find in it qualities that made it interesting to them, characteristics that gave it meaning in their time and place.

The sculptors of the west facade of Chartres Cathedral depicted Euclid, and the scholars of Abbasid Baghdad translated his book. An American artist turned his diagrams into art, and an Athenian philosopher wrote a commentary on them. The Elements was implicated in the Scientific Revolution: the fateful decision to read the book of nature as if it was written in the language of mathematics.

In Beijing, between August 1606 and April the following year, the scholar Xu Guangqi and the Italian Jesuit Matteo Ricci worked to translate the Elements, one of the books Ricci had brought with him from the far west, into the language of the Mandarin scholar. They struggled with terminology, with the structure of the text, with the very different assumptions each brought to its subject matter. They revised the text three times before they were happy to release it for publication.

From May to November 1817, on the other side of the world, Anne Lister set aside her mornings for – in equal proportions – arithmetic and Euclid. By the autumn she had worked through more of the Elements than most university graduates.

A thousand years before, at Gandersheim in Lower Saxony, the canoness Hroswitha wrote into one of her plays Euclid’s definition of perfect numbers. It formed part of Wisdom’s mockery of the emperor Hadrian, who was seeking to torture her and her daughters.

Again and again, new generations in new places have encountered the Elements and done new things with it. It has travelled through worlds that could never have been imagined by the Greeks who first wrote and read the text.

What does it mean for a book to live 2,000 years and more? To survive the wreck of the civilisation that produced it? To find readers again and again, in place after place and time after time? How vast a range of meanings do readers have to find in it? How vast a range of readers does it have to find?

Come on a journey and find out.

I

Alexandria
The geometer and the king

Alexandria, around 300 BC.

A dinner, say: a symposium, in the palace quarter, perhaps in the Museum. Ptolemy himself in attendance: general, hero, king, deity. And the talk touches on geometry: why so hard? Why is there no easier way? The geometer – a dusty man, but pert – answers: Majesty, there is no royal road to geometry.

Ptolemy snubbed is one of the irresistible stories. The man had been a childhood friend of Alexander the Great; one of his bodyguard. Maybe his illegitimate half-brother. He was a trusted general (the name is said to mean ‘warlike’): level-headed but capable of the grand gesture; a man of no nonsense.

He was one of the great survivors, too. In the twenty years of chaos that followed Alexander’s death, when many abler men died, Ptolemy played and won. Of all the successors who eventually carved up Alexander’s brief, continent-spanning empire, he founded the longest dynasty, the stablest country. He chose Egypt, and he never risked it for a larger realm. Fourteen Ptolemaic rulers followed him, until Cleopatra lost it all at the Battle of Actium 250 years later. The first king, then, of the last Egyptian dynasty. Basileus to the Greeks, pharaoh to the Egyptians. Inheritor of 3,000 years of Egyptian kingship and, yes, a god too. In 306 BC he defeated an attack on Rhodes so soundly that altars were erected to him and he acquired the title ‘saviour’. By 278 BC there were Ptolemaic games in his honour: four-yearly, like the Olympics.

Ptolemy I Soter.

Silver stater of Ptolemy I, 305–285 BC. Cleveland Museum of Art 1916.994. (Creative Commons/Public Domain, CC0 1.0)

The geometer, by contrast – a man named Euclid; Eukleídēs – is utterly obscure, historically speaking. The story about the ‘royal road’ is moonshine, sadly; it is also told about another geometer (Menaechmus) and another king (Alexander), and there is little reason to suppose it really happened. Even his dates – sometime around 300 BC – are the mere surmise of authors writing centuries after him. Unlike the exceptionally well-documented Ptolemy, Euclid left no biographical traces whatever. He founded no dynasty, built no palaces. His legacy was solely intellectual.

But what a legacy. The line of his students at Alexandria outlasted his life. His book outlasted his civilisation.

What was this city, that produced such a man and such a book? Alexandria was the right setting for the Elements of Geometry. It was Ptolemy’s greatest achievement. Alexander himself decreed the city, on the site of what had been a village and, like a dozen others, it bore his name. He never saw a stone erected, but Ptolemy took it for his capital, moving the Egyptian royal seat from Memphis. It was a Greek polis in a profoundly un-Greek world, a new foundation in a land where cities were 2,000 years old. Ptolemy did everything to make it splendid; it had an assembly, a council, its own coinage and its own laws. There were broad boulevards, colonnades, avenues of trees, street lamps. In 322 BC he kidnapped Alexander’s corpse and put it on display in his new royal city.

It was indeed the most splendid site for a city, at the meeting place of two continents, just to the west of the mouths of the Nile. It would be a major commercial port for centuries, a strategic military site until the Second World War. Ptolemy laid the foundations of the famous lighthouse: fortress and beacon, it would be one of the seven wonders of the world when it was done, 400 feet high and topped by a statue of Zeus. It stood for 1,500 years. Blessed with such a city, people immigrated from all over the Greek world, and Alexandria became not just notoriously big and splendid but notoriously crowded and cosmopolitan, with Greeks, Macedonians, Egyptians, Jews, Syrians and more, its streets teeming like ant heaps. Within a few generations it would have more than a million inhabitants.

As well as town planning, and the sheer erection of building upon building, Ptolemy involved himself in cultural policy: and with his characteristic effectiveness. To make himself a convincing Egyptian pharaoh, he indulged in sculpture to match, and devised a new cult of ‘Serapis’, a blatantly invented figure with a hybrid iconography. Like all his achievements, it endured: the temple, the Serapeum at Alexandria, stood for 600 years.

And to please the Greek mind and heart there were pageants, festivals, and a palace with tapestries the gods would prize. As one contemporary put it, Alexandria had ‘wealth, wrestling schools, power, tranquillity, fame, spectacles, philosophers, gold, youths, the sanctuary of the sibling gods … the Museum, wine, every good thing [one] could desire’.

All of which was invaluable in projecting Greek power and an idea of Greekness in a profoundly alien environment; in saying, this is what we do in the great Greek world. This is our right to rule.

And so, the Museum. The Mouseion: the shrine of the Muses. It was royally subsidised, with scholars in discipline after discipline. Their head was a priest of the Muses, and they included poets, grammarians, historians, philosophers, doctors, natural philosophers, geographers, machine-builders, astronomers, and of course geometers. It was partly Ptolemy’s own impulse, partly the creation of Demetrios of Phaleron, a famous pupil of Aristotle brought over from Athens to oversee the creation of the new institution. It had courts, covered walks and gardens, a dining hall and an observatory. The staff numbered perhaps forty, and they spent their time researching, writing and sometimes teaching. They held learned symposia, some attended by the king. They were a remarkable collection of people, sometimes compared a little sourly with the collection of animals that Ptolemy also founded: ‘well-fed bookworms, arguing endlessly in the Muses’ birdcage’. If there were bookworms, there were certainly books, too; the library at Alexandria would become the most famous in the world, though it seems to have been organised a little later, under Ptolemy’s son.

All of which is how the famous Greek mathematician ended up working in Egypt. Was Euclid another part of Ptolemy’s collection, someone brought in to swell the ranks of the Museum? It is not certain whether he was native to Alexandria or an immigrant, although at this early date in the city’s life the latter is much the more likely. An immigrant from where? His austere prose gives no hint of an accent: unlike Archimedes, in the next generation, who wrote in the Doric dialect of Syracuse.

What came to Alexandria in Euclid’s person (and perhaps in the persons of other mathematicians; it is not clear whether or not he was the only one) was a well-established tradition of Greek geometry. Greeks liked to have things to think about. They liked hobbies. Some Greeks raced chariots, some talked about philosophy, some occupied themselves with politics. From perhaps the late fifth century BC onwards, some did geometry.

What was it like? Perhaps it is clearest to think of Greek geometry as an outgrowth of the Greek love of talk, of disputation. For its geometry was nothing if it was not a performance.

Draw a line, a square, a circle. Reason out loud as you draw; play to the inevitable audience. From such beginnings the long-lived game of geometrical reasoning took shape. The figure of the geometer drawing in the sand remains part of the image of ancient Greek mathematics to this day: raking in the ‘learned dust’, as the Roman orator Cicero put it. He memorably evoked Archimedes as a man of ‘dust and drawing-stick’. (Though have you ever tried to draw a detailed diagram in dry sand? Pieces of clay, wax tablets or, for showing to a larger audience, wooden boards seem more probable.)

The number of Greek mathematicians was never very large, and they had to write their ideas down to preserve what they had found out about their lines and circles; there weren’t enough of them, it seems, for a purely oral transmission to stand a chance. So a genre was born: a special style of mathematical writing. That style would come to define mathematics in the West for more than two millennia, as constraining as any poetic metre, and as long-lived. Its components were the statement (of something to be proved); the diagram, with its points labelled with letters; and a chain of reasoning from things already known to things newly proved. That chain ended with the proposed, intended result, and the section – the ‘proposition’ – was rounded off with the note that ‘this is what was to be proved’: hóper édei deîxai; quod erat demonstrandum; QED. In some cases it was ‘what was to be drawn’. Here is an example:

How to draw an equilateral triangle

Start with any straight line; call its ends A and B.

Now draw two circles, each with its radius equal to that line: one centred at A, the other centred at B.

The two circles will cross at two points. Pick one of them and call it C. Now join up A, B and C. They make an equilateral triangle.

Why?

Because of the way the distance from A to B was used to find C. C is that same distance from A, and C is also that same distance from B. In other words, all three of the triangle’s sides – AB, BC and CA – are the same length. So it’s an equilateral triangle. Which is what was to be drawn.

The same ancient sources that report when Euclid lived also report that there were written collections of geometrical learning in Greek by perhaps 400 BC: a century before him. They are quite informative about the subjects involved, and even some of the specific results and techniques. The written matter itself has not survived, though, throwing it all into doubt. There is a strong temptation to make up genealogies for mathematical ideas when real evidence is lacking. So yes, maybe the study of the circle was done by the Pythagoreans; likewise the work on numbers and their properties. Very probably the work on ratios was done by a geometer called Eudoxus early in the fourth century BC. Some work on the regular solids originated with another, named Theaetetus. The claim that there were, before Euclid, well-developed books called Elements of Geometry seems distinctly more doubtful.

Constructing an equilateral triangle.

Christoph Clavius, Euclidis elementorum libri XV (Rome, 1574), fol. 21v. (Author’s collection. Image © Benjamin Wardhaugh)

What was Euclid’s role, then? He took all the easier material known to the Greek geometers of his day and put it all together in a single book. He organised it, both on the large scale and the small. Certainly he added some new matter of his own, though no one can now confidently identify what was new and what was not. Historians continue to argue – and they always will – about how much was compilation and how much composition in Euclid’s book. His was a work of collection, like Ptolemy’s; Euclid the Museum artefact became himself a curator, the Elements a smaller museum.

Yet this museum contained a world. It displayed the geometrical style of prose, in a relentless, ritualistic tramp of proposition after proposition: 400 of them, arranged in thirteen ‘books’ or chapters. Every verb was perfect, imperative and passive: ‘let a circle have been drawn’. There was something hypnotic about it; something infinitely calm. The book began with definitions: what was meant by a line? A point? A circle? It continued with the simplest manipulations of lines and shapes in two dimensions: how to draw different kinds of triangle; how to divide a line or an angle in two. The fact that, in a triangle, any two sides add up to more than the third. The Epicurean philosophers thought that last piece of geometrical information was ‘evident even to an ass’ since ‘if straw is placed at one extremity of the sides, an ass in quest of provender will make his way along the one side and not by way of the two others’.

Euclid did not care how obvious any of it was. He arranged and exemplified a toolbox of the basic techniques and results that he had inherited: ways of arguing, ways of proving; facts that geometers commonly assumed or used but seldom proved in full. At the end of the first book he placed ‘Pythagoras’ theorem’. Draw a triangle, with one of its angles a right angle. Using its shortest side as a base, draw a square, whose side is the same length as that side of the triangle. Repeat the procedure with the triangle’s two longer sides, so that you end up with three squares of different sizes, resting flush against the three sides of the triangle. Now, it turns out, the areas of the two smaller squares will add up to that of the larger one: a startling fact, not evident to any ass, which Euclid proved in his characteristically meticulous manner.

And so the ideas and the diagrams became harder and more complicated throughout the book. There were purely geometrical sections: a description of how to draw a regular pentagon or hexagon inside a given circle, for example. Parts of the book dealt not with geometry but with numbers and ratios, ranging from the most basic facts (‘if an odd number is multiplied by an odd number, then the product is odd’) to a procedure for finding the mysterious ‘perfect’ numbers, equal to the sum of their divisors.

Finally, Euclid turned to three-dimensional shapes. The last three books of the Elements – books 11, 12 and 13 – were concerned with spheres, cones and cylinders, with cubes and cuboids and with the regular polyhedra. These were beautiful solids whose faces were all the same regular polygon: triangles, squares or pentagons. There were just five regular polyhedra: the tetrahedron (four triangular faces), the cube, the octahedron (eight triangles), the dodecahedron (twelve pentagons) and the icosahedron (twenty triangles). He showed how to make such shapes, starting from, say, a given triangle or a given circle; he showed how to find their surface areas and their volumes. Euclid’s explorations in these final books were frequently ingenious, and sometimes applied an almost unbelievable amount of lateral thinking. Despite its gentle beginnings, and its incorporation of a lot that anyone could understand, the Elements as a whole was a virtuoso performance; a road that only the keenest of geometrical minds could follow all the way.

It added up to more than 20,000 lines of text in Greek. Euclid was careful, but he was no superman, and both joins and slips were sometimes visible. A few definitions (oblong, rhombus, rhomboid) seemed to have been carried across from older sources but never actually used. Quite a number of terms were used that were, conversely, not defined earlier on. Some words were ambiguous. A surprising amount was taken for granted about the properties of points and lines that Euclid never explicitly set out in his assumptions. Some propositions were merely special cases of others; some propositions were, strictly speaking, unnecessary because they were the mere logical consequences of others. But despite such wrinkles, the Elements was a fine, even an awe-inspiring monument to all that had been done in Greek geometry so far.

Euclid was by no means the author of just one book. The order of events is not clear, but it is certain that he wrote more. There were perhaps four other books on special topics in elementary geometry, and there were books on the applications of mathematics – to music, astronomy, optics and more – as well. In total nearly a dozen books are mentioned in the early evidence; eight actual texts survive, though most are disputed by historians.

Back in seething Alexandria, where the building work is still going on and the streets are ever more crowded. By the end of Euclid’s life the great lighthouse at Pharos has been built (did the architects consult Euclid? It would be intriguing to think so); the library and Museum are nearing completion and the palace complex is grander than ever. The Elements is finished: thirteen rolls of papyrus covered with neat columns of text and diagrams. And Euclid is still teaching, still taking on new students.

One beginner is impatient, like the king before him. After he has understood the first proposition he bursts out, ‘What is my profit now that I have learned that?’

A glance of contempt, or perhaps of pity. Euclid calls for a servant. ‘Give him threepence, since he must always make a profit out of what he learns.’

Another romantic legend, perhaps: one that circulated later in Greece, during its period of Roman domination. Like the ‘royal road’ story, it helped to protect Euclid from the whiff of servility, of sycophancy that hung over anyone connected with Ptolemaic Alexandria and its institutions. It preserved and dramatised the idea that geometry was a leisured, cultured pursuit, part of the life of the mind. Not a profitable trade, but pure, truthful and beautiful for its own perfect sake.

Some 350 geometrical propositions in the driest of styles. It’s an odd thing, to have become one of the most enduring cultural artefacts of the Greek world. Ptolemaic Alexandria is largely dust today; a few wrecked statues are dug from the ground or pulled from the sea from time to time, but the splendour is all gone. Ptolemy’s dynasty died with Cleopatra. The library is scattered. But the books – the Elements among them – the books lived.

Elephantine
Pot shards

Elephantine island, Upper Egypt, in the reign of Ptolemy III (r.246–221 BC; grandson of Ptolemy I). A Greek garrison at the end of the world. One man, off duty, is writing.

He grabs the nearest thing to hand to write on: a few broken pieces of pot. Scratch, scribble. A quick diagram, a few lines of text. His hands are confident; only slightly less confident is his memory of the mathematics. Is this how it worked? Or this? Ah, that’s right.

And, his mind and hand refreshed, the potsherds, the cheapest available writing material – perhaps the cheapest possible – go into the rubbish heap where they belong.

Euclid’s original manuscripts do not survive, nor anything like them. The papyrus on which he wrote is durable enough, in the right conditions. Scrolls hundreds of years old were not terribly unusual in the ancient world, and they could remain smooth, pliable and legible for much longer. A story is told of a museum curator who used to display the strength and flexibility of papyrus by blithely rolling and unrolling an Egyptian sheet 3,000 years old (this was in the 1930s, when attitudes to museum artefacts were perhaps less reverent than today).

In the right conditions, that is. Most conditions are not right. If it gets too wet, papyrus rots; too dry and it crumbles. Insect larvae like papyrus, and the worms destroyed many a literary reputation in the ancient world. So did the rats. Plus, the long rolls tore easily and were thrown away when they did. The upshot is that large or complete papyri surviving from the ancient world are extremely rare. What more often survive are fragments: discarded rolls, pieces reused to make mummy cases, pieces recovered from rubbish dumps or ruined houses. Rough, dark and brittle with age, nearly all are from provincial locations in Middle and Upper Egypt, where the dry conditions preserved them. Finds have come from cemeteries along the Nile Valley and in the Faiyum Oasis, and from certain villages. From the big towns, by contrast, there is next to nothing: Alexandria itself, having a high water table, has no preserved papyri at all.

For all that, there are a lot of papyrus fragments. People have been systematically digging them out of the ground since the mid-nineteenth century, and hundreds of thousands are now amassed. And, yes, some of them contain fragments of Euclid’s Elements. Seven, in fact, totalling about sixty complete lines of the text and another sixty fragmentary lines.

What parts of the Elements do they preserve? They include, written around 100 BC: three propositions from book 1, with one summary proof (these come as citations in a philosophical treatise preserved – carbonised – in Herculaneum by the eruption of Vesuvius in AD 79: an exception to the normal generalisations about papyrus survival). An enunciation from book 2, with a rough figure, written in the Egyptian city of Oxyrhynchus around AD 100. Parts of two more propositions from book 1, written at Arsinoë (modern Faiyum) in the second half of the second century AD. A second-century copy of three figures and enunciations from book 1, carefully written with ruled diagrams. And a schoolteacher’s or pupil’s copy of the ten opening definitions, made in the third century AD.

It is not much: these are small pieces from the easy parts of the book, in one case from its very beginning. But they do reveal something about the way the Elements was spreading. It did not just stay in Alexandria: already, by the first few centuries after its composition, it – or parts of it – was being copied out by people hundreds of miles away around the Greek-speaking world. It was moving out from the cultural centre to the provinces.

Euclid’s Elements will have been published in the ancient sense: sent to a scribal copying house which produced multiple copies for sale. But most of the papyrus fragments are not from those copies; only the Faiyum fragment looks like the work of a professional scribe. Instead, they bear witness to the activity of individuals copying out parts of the text for their own use, teaching or learning.

So, the writers of these papyrus fragments represent the ‘public’ for Greek geometry: a tiny minority, in a world in which the literate themselves were already a minority. These were people who understood geometry, who accepted and shared its conventions, who knew enough of the basics and the methods to comprehend Euclid’s book. Their needs surely shaped what was written and how it was written. The very packaging of mathematics in a self-contained written form already presumes that they existed. But nothing more is known about them.

And this evidence can tell only about those places where it was dry enough to preserve papyrus fragments: for the rest of the Greek world – the islands and the mainland north of the Mediterranean, for instance – the lack of evidence reveals nothing, positive or negative. Surely the Elements went to Athens, for example: but it is centuries before there is evidence for that.

As well as papyrus there was a cheaper writing surface still: ostraka or pot shards. Literally, broken pieces of pot: waste, and therefore free. Ostraka were used in Egypt before the Ptolemies and up to the end of antiquity, in Athens from the seventh century BC: they were written on in ink or simply scratched, to form pictures or writing in Hieratic, Demotic, Greek, Coptic or Arabic as the case might be. Schoolboys, soldiers, priests and tax collectors all used them. (They were also used as voting tokens: if the word sounds familiar it is because ‘ostracism’ was a procedure for expelling a man from the country for ten years on suspicion of disloyalty: the votes were written on ostraka. It happened at Athens for most of the fifth century BC, and in other Greek cities too.)

A Euclidean ostrakon.

Ostrakon, Elephantine, third century BC. Berliner Papyrusdatenbank P. 11999. (© bpk-Bildagentur)

A set of ostraka bearing geometrical writing, written at Elephantine, were preserved by the random sieve of history. They were dug up by German archaeologist Otto Rubensohn in the winters of 1906 and 1907, and they are now held with the papyrus collection at Berlin; their contents were transcribed and published in the 1930s. They are the oldest surviving physical evidence for any part of Euclid’s Elements.

Elephantine is more than 500 miles south of Alexandria: an island in the Nile at the northern end of the first cataract. In the third quarter of the third century BC, when the ostraka were written, it was the frontier of the Ptolemaic kingdom. Traditionally the ‘ivory island’ or ‘elephant island’, whose settlement stretched back into prehistory, it was the capital of the first Upper Egyptian administrative district or ‘nome’, controlling trade with the quarries of the cataract region and the trade route to Nubia. It was a garrison town, far removed from the centres of Greek culture, under threat from brigands.

It had temples, priests, fairly elaborate housing and a degree of bustle; by the Byzantine period the town had a public camel yard. But the documents that have survived – papyri again, mainly – are dominated by the characteristic anxieties of soldiers settled far from their widely diverse homelands. In the third century BC there were men at Elephantine from Greek cities and islands as far afield as Crete and Rhodes as well as Alexandria and the mainland at Euboea and Phocis: a veritable Homeric catalogue of soldiers. They kept aloof from the native Egyptians, referring to their town as ‘the fortress’, and the papyri show them making wills, marrying, appointing guardians, providing accounts to their superiors or petitioning them for justice. It seems a most unlikely setting for the earliest Euclidean evidence.

The ostraka – six of them, one clearly broken on all sides – bear a text concerned with constructing a regular polyhedron. It relates to propositions 10 and 16 in book 13 of the Elements: that is, from very near the end of the book. In those propositions a pentagon, hexagon and decagon are employed to build up an icosahedron: a regular solid with twenty sides, all of them equilateral triangles. The inevitable diagram is clearly present on one of the ostraka, with its letter labels, and the shards give a clear sense of someone doing what every Greek geometer did: draw a picture and tell a story about it.

The propositions in the Elements depend on one another in an elaborate, tree-like structure, each one referring implicitly to many that have come before. To be confident in something from this late in the book, a person would need to have studied much of what came before it: but the ostraka provide no direct evidence for that. The text is in a confident, flowing handwriting, too: that of an experienced writer whose grammar and spelling were unhesitating and correct.

Who was the writer? It is most frustrating not to know. Serafina Cuomo, historian of ancient mathematics, remarks of these ostraka that ‘while their contents denote a high level of education, both the humble material and the location (a remote outpost in the heart of “Egyptian” Egypt) seem to jar with that conclusion’. Priest, teacher, soldier or camp follower? It will never be known whose were the hands that scratched out this earliest surviving piece of Euclideana.

There is a further twist. The matter discussed on the ostraka came straight from Elements book 13, but their text is not the text that has been transmitted as part of the book. It is the same diagram, and the same ideas, but the words are not exactly the same. That is also true, albeit to a lesser degree, of the other early fragments that survive from the Elements, on papyrus: their versions of the text do not exactly match what has been preserved in later, more complete versions of the book.

This reinforces the sense that Greek geometry was essentially a performance, consisting of drawing a diagram and talking about it, to oneself or to an audience. What was written down was a transcript of the performance, an aide-memoire, a skeleton or a set of prompts, together with the finished, static version of a diagram that, in use, had been dynamic and growing. Those written traces could be of use for private study or a help for a teacher, who would need to perform the same proof several times. They could also, as in a book like the Elements, transmit the idea of the proof or construction to people far away in time or place.

And, as a result, reading a geometric proof is not like reading a novel or a poem. You can only really follow it by recreating the original live performance; by picking up a pen and some scrap paper, papyrus, or a shard of pot, and constructing the diagram as you read, so as to see how it grows.

So the written forms of Greek geometric propositions were not so much something one would learn and copy slavishly as prompts that said: here is something interesting; try it yourself. The Elements was not a dead repository of facts but a support for learning and practice; an invitation to perform for oneself, in the same way that rhetoric textbooks aimed to prepare students for rhetorical performance. With that in mind it becomes less surprising that the early fragments of the Elements surviving on pot and papyrus have quite ‘wild’ versions of the text. This set of ostraka, in particular, should probably be read in this way: as an attempt to recreate something the writer had read or seen performed.

It is fitting that the earliest evidence for the Euclidean Elements is so enigmatic, so shakily related to the Elements themselves, and so, literally, fragmented. The text and its ideas would travel about as widely as it is possible for a cultural artefact to travel, but they would be much changed by the journey; and, what is more, it is not clear that they were ever simple, single and stable, even at the very beginning. Euclid was not a master but a muse, an inspiration: he did not just reveal facts but offered a set of tasks. His readers knew they could always go deeper and create more, because although the Elements had already done everything, everything was still to be done.

Hypsicles
The fourteenth book

Back in Alexandria, and still under Ptolemy III. Another legend, this one reported by Galen, the Roman physician.

Ptolemy III’s interest in collecting ancient books mounts to an obsession, a mania, and he has ships arriving at the harbour searched for texts that might be of interest to the Alexandrian library. On one occasion he receives, on loan from Athens, the precious scrolls containing the plays of Aeschylus, Sophocles and Euripides, depositing the fantastic sum of fifteen silver talents as security on the understanding that he will have copies made and return the originals.

He has copies made, indeed, and on the best papyrus. But it is those copies that he sends back to Athens, retaining the originals for his library. The Athenians can do nothing; they had accepted the silver on the condition that it would be theirs if the scrolls were not returned. They keep the copies, and they keep the money.

Ptolemy I (who learned there was no royal road to geometry) had abdicated in 284 BC in favour of his son, but the cultural institutions he started continued to grow, as did the great, the fabulous city of Alexandria. Its library became the largest collection of books in the world, an unequalled symbol of literary culture, a vast stocktaking of Greekness that surpassed anything to be found in Greece itself. The Ptolemies’ hunger for books became a legend; so did the lengths to which they would go to secure good copies of important works: as in the fabulous story quoted above, where Ptolemy III in the next generation paid a king’s ransom for the original Athenian texts of the great tragedies. As well as plays, histories and epics, though, the library had books on cookery, magic and fishing: no subject was excluded.

Meanwhile in mathematics, Euclid’s Elements remained the acknowledged toolbox of standard techniques and results, in which later Greek geometers used to discover all kinds of new things. Where Euclid had dealt with points, straight lines and circles, they dealt with the shapes made by the intersections of cones and planes, and with the solids formed when those intersections were themselves rotated about an axis: weird forms, surely close to the limit of what the human mind can conceive without the aid of algebra or digital visualisation. Yet they used Euclid’s terminology and relied on his propositions constantly. As well, they extended the Greek geometrical style itself in new ways that reflected the cultural preoccupations of Alexandria, with twists and turns, elements of surprise and sudden flashes of intellectual light.

Perhaps a generation or less after Euclid, Archimedes (c.287–212 BC) earned a reputation as the most brilliant of the Greek geometers, the most skilled at suspense, surprise, and showy unexpected results. One of his best-known surprises was that if you take a sphere and a cylinder with the same height and the same diameter, their volumes are related in the ratio of 2:3.

The great geometer of the next generation was Apollonius (c.262–c.190 BC); he shared his name, confusingly, with an Alexandrian poet who was head of the library. One ancient report says he studied in Alexandria with the pupils of Euclid (the passage is in fact the same one – the only one – that places Euclid himself in Alexandria). He systematised the study of conic sections – curves derived from slicing up cones – and adopted, too, the Archimedean manner in which beautiful results emerge as surprises at the end of long arguments.

The tradition continued. In 235 BC Eratosthenes, an astronomer and geometer, became head of the library (in succession to Apollonius the poet, in fact): an important moment for the integration of geometry into the culture of the Museum. Eratosthenes would become famous for deducing, correctly, the size of the earth from astronomical observations.

The icosahedron and dodecahedron, in a Renaissance woodcut.

Luca Pacioli, Divina proportione (Venice, 1509), plates XXII and XXVIII. Getty Research Institute, 47289. (Internet Archive/Public Domain)

One of the geometrical tricks and puzzles these geometers worked on was this. Take a dodecahedron and an icosahedron of the same size: that is, so that the two spheres into which the two solids would just fit are the same size. The dodecahedron’s faces are pentagons; the icosahedron’s are triangles. And it turns out – though you really wouldn’t expect it – that the pentagons and the triangles are the same size as each other: that is, the circle into which one of the pentagons would just fit is the same size as the circle into which one of the triangles would just fit. It’s a fun fact: not one that matters, but a good display of a geometer’s mental dexterity. It was admired enough to pass from hand to hand over several generations, and eventually to be annexed to the Elements.

It was a geometer named Aristaeus who first made the discovery, and published it in a book on the five regular solids which surely took its inspiration from the discussion that closed the Elements. Apollonius treated the problem in a work dealing solely with the dodecahedron and the icosahedron. Next, one Basilides of Tyre came to Alexandria, discussed the matter with a colleague and found Apollonius’ account wanting. Apollonius issued a new version of his text, in which he achieved the remarkable feat of showing that the volumes of the equal-sized dodecahedron and icosahedron form the same ratio as do their surface areas. Finally a man named Hypsicles took up the subject and put together all of what he found valid in the previous accounts.

This was in the middle part of the second century BC, perhaps 150 years after Euclid. Much is uncertain, but it seems Hypsicles was an astronomer as well as a geometer; in addition to a now lost book on the harmony of the spheres, and possibly one on numbers, he wrote a work on the rising and setting of the stars. It was in the style of Euclid and showed how to find (approximately) the rising time of a given point in the heavens – say, of a particular star – at a given place on the earth’s surface. It was also notable for being the earliest Greek text to divide the circle into 360 parts, a practice that derived ultimately from Babylonia.

Hypsicles responded to Apollonius’ revised book with a text of his own on the comparison of the same-sized dodecahedron and icosahedron. He provided new proofs for the main results: that their faces fit into the same circle, and that their volumes have the same ratio as their areas (make a cube of the same size, by the way, and its side and the side of the icosahedron will make that same ratio again). It was a short book, with just five propositions plus three subsidiary results – ‘lemmas’ – and one alternative proof.

Little is certain about how such things circulated, but it is clear that the treatise of Hypsicles, despite or perhaps because of its recondite subject matter, acquired some currency and was read, cited and admired. But it would suffer a curious fate. A later, anonymous hand attached it to Euclid’s Elements; where Euclid had written thirteen books, the treatise of Hypsicles was labelled ‘book 14’ and came to be transmitted as part of the Elements. There was some editorial modification at the same time, changing how the ‘book’ started and ended and also adding an extra proposition to Euclid’s own book 13 to strengthen the link. Almost incredibly, the ‘fourteenth book’ would remain a widely accepted part of the Elements for a millennium and a half.

The fantastic collection of books and people at Alexandria, and the mathematical tradition that was becoming associated with the city, interacted in some strange ways and produced some unforeseeable results. The rooms full of boxes full of scrolls made for some unexpected juxtapositions, odd misattributions and unlikely amalgamations. The Elements was part of that process, becoming a site of research, controversy and improvement even as it was transmitted from generation to generation.

Theon of Alexandria
Editing the Elements

Alexandria in AD 370, or thereabouts. The city remains: Greek, proud and beautiful. But the world is changing: temples are closing, churches are being built. The library is gone, and the Museum’s final generation of scholars is at work.

Theon is one of the scholars; his daughter works with him. They teach in the city: geometry, astronomy and philosophy. To teach, you need books, and they prepare new versions of the classic texts: Claudius Ptolemy on astronomy; the Orphic hymns; works on divination; and on geometry, Euclid.

These are long books. The scratch of reed on papyrus is day-long; fingers ache.

Theon comes to Elements book 6, proposition 33, and finds that the proposition is not quite complete. He adds a section, extends the result. And so Euclid’s Elements becomes a little longer and – he hopes – a little better.

Alexandria at its height had been the largest, richest, grandest city of the Greek-speaking world, the Ptolemies the patrons of every art and science. But like all dynasties they had their troubles, and after the Battle of Actium, from 31 BC Egypt was ruled from Rome, made into a province on the Roman model. The city was still rich and splendid; architecturally, it remained perhaps the finest city of the Mediterranean world.

The library lived through it all. The Ptolemies’ bibliomania gave it a size that contemporaries could only guess at. One suggested there were nearly half a million scrolls there: surely an impossible number; but whatever the arithmetic in miles of text or acres of papyrus, it was a storehouse of all that was Greek and all that could be amassed and translated into Greek. A most celebrated symbol of its cosmopolitanism was the translation of the Jewish scriptures into Greek during the third and second centuries BC.

But books are fragile, and libraries are fragile. Cultural institutions are not everyone’s priority when political instability sets in. Julius Caesar accidentally set the city on fire in 47 BC, and an enduring – and probably true – report says that some books were destroyed, perhaps in seafront warehouses. On the other hand, there was certainly still a functioning library a century later. And the Museum itself continued, even throve at first, under the patronage of the Roman emperors.

But the Roman period saw civil war and riot with at least the usual frequency for a city of such a size: the Alexandrians in fact acquired a reputation for mob violence during their first Roman centuries. In the late third century, civil war reduced the whole of the old palace quarter to ‘desert’, in the words of one witness, and there was most likely little left of library or Museum after that. It is impossible to be certain. In the evidence that survives, the last employee of the Museum was Theon, who taught mathematics from around 360 until perhaps the end of the century.

Whatever the details of its long demise, the tradition of scholarship at the library had been a crucially important one for the shaping and the sheer survival of Greek literature. Practically every Greek text of any length that survived at all passed through the Alexandrian library.

The library developed a specialism in textual criticism of older literature, studying questions like: what did the authors really write? Which books are authentic? Which lines have been garbled in transmission? It was here that there arose the long-lived metaphor of the corrupt text, the text that needed to be cured, purified of the dirt and disease it had acquired through contact with human beings.

The scholars at the library devised signs for recording which lines of a text were doubtful, and how and where they had emended it. They began to invent marks for section divisions; even line divisions. No punctuation proper; texts were still transmitted in ‘continuous writing’:

ALLINCAPITALSWITHNOWORDBREAKSACCENTSORPUNCTUA TIONOFANYKINDANDTHEREFOREREALLYHARDTOREAD.

These techniques and innovations were developed for the study and transmission of literary classics: Homer first, following a tradition of interpretation that went back to the Homeric reciters of classical Athens, and then the Athenian playwrights and others. The scholars at the library had less to say about prose, and less again about scientific texts; there is no evidence that Aristotle’s books, for instance, were worked on in this way at Alexandria. To work on mathematics required perhaps a slightly different approach, although the idea of textual purification was the same.

The text of Euclid’s Elements was already unstable in some ways during its first few centuries; the surviving evidence hints at more. It is quite possible that Euclid himself prepared more than one edition of the Elements, which would help to account for some of the variation the book shows in the evidence that survives. It is at least equally possible that zealous editors got to work on the text during the second and first centuries BC, producing versions that differed from the original and perhaps from one another.

What is certain is that by at least the first century AD, and continuing until at least the fifth century, there was a tradition in Greek – and often at Alexandria – of writing commentaries on the Elements. Was a definition missing? Was a term defined but not subsequently used? Was a proposition a ‘dead end’, never used to prove anything else? Did it come in the wrong place, after something it was supposed to help prove? All of those things could be pointed out in a commentary, to help teachers and students use the text, to help them avoid stumbling over such minor wrinkles. Commentary of this kind could hardly avoid suggesting changes to the Euclidean text itself, and so it provided raw material to the different but overlapping activity of editing the text: preparing a new version of it for the use of, say, one’s own students.

And so to Theon, and the last days of the Museum. He taught, and he dedicated books to his students. He wrote commentaries – most likely based on his lectures – on the second-century astronomical works of Claudius Ptolemy (no relation, as far as is known, of the royal house of the Ptolemies. On his so-called Handy Tables he even wrote two different commentaries: one for competent astronomers, and one for those who could not yet understand the reasoning and the mathematics the book contained. He also seems to have written on the interpretation of omens and on the Orphic hymns. A handful of his poems survive, in which he expressed his devotion to the perfect world of the heavens, the gods and the stars.

And he also prepared editions: consistent, improved versions of classic texts for use in his teaching. He certainly worked in this mode on the Elements and other works of Euclid, and on the works of Ptolemy such as the Almagest, doing what various other editors did to mathematical texts during this period. He cleaned up and smoothed out difficulties. He chose between variant versions of the texts, or combined the variants together to give a text with, sometimes, two alternative proofs for a single proposition. He filled in gaps, real or imagined. While the Homeric editors saw themselves standing outside the epic tradition, mathematicians of later Alexandria tended to behave as though Euclid was a colleague, and they themselves members of a still-living tradition. Instead of textual purity and fidelity (‘what did Euclid really write?’) they prized correctness, completeness and usability.

Most unusually, Theon actually left a record of one of his specific changes to the Elements. In another work – his commentary on Ptolemy – he mentioned it, a certain minor point that ‘we have proved in our edition of the Elements at the end of the sixth book’. Thus he recorded for posterity that a certain fragment of the text was his, and that that was the kind of thing he did.

The proposition in question says that if you take a slice out of a circle – like a slice of cake, with two straight lines meeting at the centre – then the length of its curved side is in proportion to the angle at the centre. Theon’s addition says that the area of the slice is also in proportion to the angle at the centre. It is easy to believe that it is true, and not at all hard to prove.

Theon’s throwaway comment has received an enormous amount of attention from scholars of the Elements over the years, since it seems to raise the tantalising possibility of seeing beneath the editorial accretions of the ancient world, to the real Euclid beneath. If the type of thing Theon did can be securely described, perhaps later scholars can recognise his modifications elsewhere and reverse them.

Unfortunately, attributing specific modifications to Theon other than this one proves to be a very doubtful business, although many have tried. One person’s logical gap is another person’s elegant brevity. One person’s Theonine interpolation is another person’s authentic Euclidean blunder. The evidence – most of it far, far later than Theon, and the subject of endless cross-contamination between Theon’s version and the version or versions that preceded him – does not allow anything like certainty.

History, meanwhile, has not been kind to Theon himself. Later ages would have different attitudes to text and authenticity, and would not be too impressed with the kind of modification in which he and his like engaged. Recent historians have characterised his editions as ‘trivial reworkings’, ‘mathematically banal’, and his scholarship as ‘completely unoriginal’. His biographer in the Dictionary of Scientific Biography is frankly devastating: ‘for a man of such mediocrity Theon was uncommonly influential’.

Perhaps it is not necessary to be quite so hard on him, particularly if he was doing his work with teaching in mind. To be sure, there are things in Euclid’s Elements that are ‘elementary’ in the sense of easy; quite a lot of them, ranging across simple constructions with lines and circles and triangles through the properties of numbers (even times even is even) and those of ratios. But the book also contains a great deal that is eye-wateringly hard by any standard whatever, and which requires a real talent for mathematics, and a lot of time spent on its study. An example is Euclid’s exhaustive classification – in book 10 of the Elements – of the different ways two lengths can make a ratio that is impossible to express using whole numbers: there are 115 propositions about the subject. Another is the construction of the tetrahedron, cube, octahedron, dodecahedron and icosahedron, and their relative sizes when set inside a sphere that just touches the corners of each of them. It is certainly possible to imagine students for whom Theon’s smoothing, explaining and organising would have been of real help.

Influential he certainly was. Although older texts of the Elements remained in circulation, Theon’s new edition was taken up enthusiastically, and nearly all of the complete manuscripts that have survived to the present contain what seems to be Theon’s version. Mediocrity or not, he had more impact on the Euclidean text than anyone else since Euclid.

The sharp-eyed will have noticed a reference to Theon’s daughter. Her name was Hypatia, she was born around 355, and early sources suggest her mathematical talents surpassed those of her father, although only the titles of her works survive. These seem to have included commentaries and possibly independent works in mathematics and astronomy.

Theon was living in a changing world. The emperor Constantine had converted – at least formally – to Christianity in 312, and removed the legal obstacles to Christian worship the following year. Churches were opening and – slowly – temples were closing across the Greek world. During Theon’s lifetime the temple of Serapis in Alexandria was destroyed: seat of Ptolemy’s novel cult, it had been closed since 325, but it was probably one of the main remaining libraries in the city.

The Alexandrian reputation for mob violence continued, and Hypatia herself was one of its victims. By the early 390s she was active as a teacher of philosophy, with an established circle of students. Her teaching included geometry, and she is therefore one of the last recorded individuals who continued the tradition of geometrical teaching in the city of the Elements’ birth: a well-known figure, even something of a celebrity in the city. Most of her identifiable students were – or later became – Christians, and her paganism does not seem to have been felt as a problem. But in the 410s her supporter the Roman prefect (Orestes, another Christian) came into violent conflict with the Alexandrian patriarch, and in March 415 Hypatia herself was murdered in something like reprisal: a victim, in the words of one ancient historian, of political jealousy. Since the nineteenth century the tragedy has been often – and often sensationally – fictionalised.

One of Theon’s astronomical commentaries bears an ambiguous heading referring to his daughter. It has been variously interpreted as implying that Hypatia checked or revised that part of the commentary (or more), or that she edited or revised part or all of the text on which Theon was commenting. Certainly it seems there was a period when she and Theon worked together on mathematical and astronomical material. But the surviving evidence does not link her with his edition of Euclid’s Elements. It is possible she did work on it with him, and tempting to speculate that the future teacher of geometry left her mark, somewhere, on the text of the Elements. But Theon’s career certainly began before his period of collaboration with his daughter, and most probably continued after it, meaning that he could equally well have edited Euclid independently of her. As with so much about the transmission of the Elements, it is impossible to be sure.

Stephanos the scribe
Euclid in Byzantium

Constantinople, in the 6,397th year of the world (AD 888). The scriptorium in one of the city’s monasteries.

The last page of vellum turns. The scribe flexes his weary fingers, stretches his aching back, and adds a final note:

Written in the hand of the clerk Stephanos in the month of September, the seventh indiction, the year of the world 6397. Purchased by Arethas of Patras, the price of the book 14 coins.

During the fifth century the Roman imperial system collapsed in the western part of the empire, from Britain to Portugal to the Balkans. By 480 there was no emperor. In the eastern part, which spoke Greek and had had its own separate emperor since 395, the empire flourished for another 1,000 years. The seat of its power was in Byzantium.

Byzántion. Constantinople. Another immense vanity project, it was founded in 324 as a new de facto Eastern capital, and designated the official capital from 330. A university of sorts opened in 425.

This empire was less prosperous, and less literate than its predecessor. Schools – the universities of the period – existed at Alexandria, Antioch, Athens, Beirut, Constantinople and Gaza. But they declined, and by the middle of the sixth century only those at Constantinople and Alexandria remained. The latter was conquered by Persia in 619, ruled by the caliphs from 641. The Greek-speaking world collapsed; Greek culture was now largely that of a single city, Constantinople.

If the barbarians had provided a solution of sorts to the problems of Rome’s unwieldy empire, the great contraction of culture provided a similar sort of solution to the unwieldy mass of Greek literature. The production of Greek books slowed dramatically, and what was not salvaged at Constantinople was, by and large, not salvaged at all. The worms got the rest, and the Greek world’s library was a manageable size once again.

For teaching purposes, the Byzantine scholars organised matters into a literary cycle and a scientific cycle, together making the ‘seven liberal arts’ that were also studied in the West. The list of seven subjects went back to classical Athens; it is not really certain when or where it was formalised into a curriculum, perhaps around 100 BC. On the literary side there were grammar, rhetoric and dialectic (logic), which amounted to three different ways of reading the favoured ancient authors. On the mathematical side there were arithmetic, geometry, astronomy and music. The authorities for these subjects ranged from late classical authors such as Aristoxenus on music, down to Greek authors of the Roman period such as Claudius Ptolemy on astronomy. On geometry, of course, the text to study was Euclid’s. For the first time, then, it is clear that knowledge of the Elements was part of a good education, mastery of it a requisite for those aspiring to join the cultural elite. Or at least, presumably, mastery of the easier parts.

Saving culture with limited resources, particularly when the priority was teaching tomorrow’s orators, did odd things to texts. It did them to the Elements. Scribes made mistakes; when they were copying mathematics that they did not always understand, they made particular kinds of mistakes, confusing numbers, abbreviated words and the names of geometrical points, all of which used the same set of Greek capital letters, sometimes in the same sentence.

Since readability rather than literality was their main value, scribes also made deliberate improvements, standardising notation, tidying up diagrams, even regularising the logical structure of proofs. Some compared the two versions of the Elements now in circulation – Theon’s edition and the version that preceded it – and adopted whichever text they preferred at any given point. Coherence and mathematical perfection resulted, perhaps, but at the expense of obscuring Euclid’s distinctive voice.

All this was taking place not on the papyrus rolls that Euclid used but on codices – books with pages – that had gradually supplanted them. A single codex could be large enough to hold the whole of the Elements, with commentary in the margins too: and that created further possibilities for confusion, when scribes copied pieces of commentary into the main text by mistake, or adopted as main-text readings what were proposed in the commentary as alternatives or improvements. Even when that did not happen, fragments of commentary became enduringly attached to the text as ‘scholia’, transmitted for generations and often completely detached from their original commentary context.

It was probably in the fifth or sixth century, too, that someone added Hypsicles’ treatise on the solids to the Elements, calling it book 14, and there it remained, enough scribes accepting the addition to give it a sort of authenticity of its own as a completion of what Euclid had supposedly left incomplete. Most likely in the same period, a fifteenth book was also added, addressing further questions in solid geometry. Although it was sometimes also associated with the name of Hypsicles, it was in reality a combination of material from at least three authors, of whom the latest was Isidorus of Miletus, the sixth-century architect of Hagia Sophia in Constantinople. It may indeed have been one of his pupils who assembled the fifteenth book and added it to the Elements.

The Byzantine literary world itself suffered serious disruption from around 550 to 850, with many of its institutions dormant or closed. The empire was at war with the caliphate (almost always) and the Bulgarian khanate (usually) and in conflict with the Roman Pope and the Carolingian empire (occasionally). And it was stirred internally by seemingly endless theological controversy and the hopeless muddle of the imperial succession. In the ninth century came a period of self-conscious revival and rediscovery; the imperial university reopened, and the names of certain teachers have been preserved: Leo the mathematician, Theodore the geometrician, Theodegius the astronomer, Cometas the literary scholar.

These scholars collected books. Leo, for instance, undoubtedly had copies of the works of Euclid; he also had copies of Apollonius’ book on conic sections and of works by Theon and by Proclus, as well as those of Archimedes and very likely of the astronomer Ptolemy. Books were now being copied in a new, ‘minuscule’ handwriting, smaller and easier to read than the old all-capital style, and better supplied with breath-marks and accents. And scribes still perhaps blundered, improved, and sifted the existing versions of texts for what struck them as best. Half the definitions and a third of the propositions in the Elements were by now affected by changes from a thousand years of commentary and editing.

Arethas of Patras was another of the scholars of this period, and evidently also a man who loved beautiful books. Born shortly after 850 in western Greece, he came to Constantinople and received a comprehensive classical education, reading widely in ancient literature. His career was in the church; he became a deacon by 895, and in 902 or 903 Archbishop of Caesarea, the most senior see dependent on Constantinople. He was an author and editor, and the writer of a commentary on the Apocalypse. A few of his letters survive, and a few of his epigrams. Along the way, he collected books.

He owned, by his death, twenty-four of the dialogues of Plato, and the main works of Aristotle. He had copies of the ancient authors Lucian, Aristides, Dio Chrysostom, Aurelius, and of Christian writers such as Justin, Athenagoras, Clement of Alexandria and the church historian Eusebius. The best of Arethas’ books were masterpieces of calligraphy on high-quality parchment. He paid twenty-one gold pieces for his copy of the works of Plato, when Civil Service salaries started at seventy-two per year. He commissioned his books from professional scribes, most of them monks; some were the finest calligraphers of their period. (A measure of the high-pressure perfectionism that ruled in the Byzantine scriptoria is that one of them had a specific penalty for breaking your pen in anger.)

And, of course, Arethas obtained a copy of the Elements. In fact, it seems to have been the first book he commissioned; perhaps in connection with the completion of his education in the seven liberal arts. Stephanos, its scribe, was one of the most accomplished of all Byzantine calligraphers. He worked for various patrons; copies survive of the Acts of the Apostles and the New Testament Epistles in his hand, and works by such authors as Ptolemy, Porphyry and Proclus in hands that may be his. He developed a characteristic decorative scheme in blue and gold, with cypresses, columns, lanterns, crosses and geometric figures such as rhombuses, circles, squares and rectangles. And he developed a spacious layout, with wide margins where notes and commentary could appear without making the page look messy.

The Elements in 888.

Oxford, Bodleian Library, MS D’Orville 301, fol. 268r. (The Picture Art Collection / Alamy Stock Photo)

All of this he brought to Arethas’ copy of the Elements in 888: 388 leaves of parchment in forty-eight gatherings, bound together to make a thick book seven inches by nine. At up to twenty-six lines to a page, there were nearly 20,000 lines of text all told, in a pleasing reddish-brown ink. Stephanos reported at the end of book thirteen that what he had copied was ‘Theon’s edition’; he went on to copy out books 14 and 15 as ‘Hypsicles’ addition’.

The diagrams were drawn – it is hard to be certain whether Stephanos did them himself – neatly, with rule and compass, and their labels added in capital letters. They were inserted into spaces in the main text that had been left blank on purpose: always on the right, and always a good legible size.

Arethas didn’t like his books just to look well on the shelves; they were not merely prestige objects to be owned for the sake of display. He read them, and annotated their margins in his own hand. In his Aristotle, he annotated a huge amount; in his Plato, little. Often he copied the remarks of others, but he certainly made contributions of his own too, albeit none of great importance (and his attempts at the correction of texts have generally been regretted by later scholars). He explained, commended, criticised, got angry: carried on a dialogue with the authors.

He did a great deal of this in his Elements. Evidently he wrote in commentary from other copies of the book, including fragments from ancient geometers and other accretions, some of them illustrated with their own diagrams. But about fifty of his additions appear in no other surviving copy of the Elements, and may well be Arethas’ own work. He corrected the scribe, added comments, or made brief remarks like ‘nice’, ‘note’, or just inserted a tiny ornament. On one page he added a comment about the addition and subtraction of fractions, which originated from a lecture by Leo the geometer. Near the start and end of the volume he wrote two epigrams about Euclid.

Arethas died sometime in the 930s. His was an impressive library but not a huge one; there were others collecting books, and his may well have been unremarkable in its own day. Nor did he always obtain the best texts of the works in which he was interested. And his reputation as a scholar has never at any time stood very high. But his library matters because, in part, it has survived to the present, providing some of the earliest known copies of some famous texts.

Eight of Arethas’ books survive, scattered today from Florence to Moscow. His copies of the works of Plato and Aristotle are important witnesses to the texts they contain, and the world owes to Arethas the only surviving copy of the Meditations of the emperor Marcus Aurelius, that long-lived favourite of later Stoic thought: Arethas had found an old manuscript, ‘not entirely fallen apart’, and had it recopied. His Elements is one of the two oldest complete copies of the text to survive, the other being an undated ninth-century book also from Byzantium.

It has fared no worse than many another artefact from the period. From about the tenth to the fourteenth century it was evidently much used and annotated. Owners and readers added to Arethas’ marginalia, sometimes in large, ugly handwriting and sometimes including large careless diagrams, which did something to spoil the look of the book. Someone added numbers to help understand one of the diagrams. Worse, a few early pages became detached and were lost, so that the first fourteen propositions of book 1 had to be replaced by a copy in a different hand.

The exact whereabouts of the manuscript during all this is uncertain; Greek books were coming to western Europe by various routes from the time of the Crusades onwards, but there is no sign of this particular volume in any records until it turned up in (probably) 1748, acquired for the collection of Jacques Philippe D’Orville, the Dutch-born son of a French family, who travelled in France, Italy and Germany before settling as a professor back in his native Amsterdam.

D’Orville had gradually acquired a large collection of manuscripts, among which Arethas’ Elements stood out as one of the oldest. On his death, the books passed to his son John and then his grandson; next they were sold to one J. Cleaver Banks, apparently a book dealer, and then in 1804 nearly all of the manuscripts were sold again to the Bodleian Library in Oxford, fetching £1,025. Recatalogued, the Elements became manuscript D’Orville 301.

The book remains in the Bodleian. Today, Arethas’ Elements is the oldest surviving manuscript of an ancient Greek author to bear a date. It is, indeed, the first dated manuscript in Greek minuscule writing, setting aside religious texts. Despite its age and its long journey – indeed, because of them – it is still beautiful. It is hard to look at it and not be moved.

Al-Hajjaj
Euclid in Baghdad

Baghdad, 14 Safar 204 AH (10 August 819). Dawn over the Round City. A little boy stands in a crowd.

I saw the caliph Ma’mun on his return from Khurasan. He had just left the Iron Gate and was on his way to Rusafa. The people were drawn up in two lines [to watch the caliph and his entourage go by] and my father held me up in his arms and said ‘That is Ma’mun and this is the year [two hundred] and four.’ I have remembered these words ever since: I was four at the time.

After six years of siege, Abu al-Abbas al-Ma’mun ibn Harun al-Rashid was returning to the city of his fathers. Heir to the caliphate, he was the inheritor of the spectacular conquests of the early Arab armies. Stretching across Egypt, the Fertile Crescent, Persia and India, it was almost the whole of what Alexander had taken a millennium before, and this empire was proving far more durable.

Of the three great heirs to the Roman Empire – the Greeks, the Arabs, the Latins – it was the Arabs who were the most dynamic, who most rapidly transformed a region beyond recognition. The vast world ruled by the caliphs stretched 4,000 miles from the Atlantic to the Oxus, funnelling goods, people and ideas through – and into – Iraq and Syria. Its wealth was incalculable.

Ma’mun’s dynasty, the Abbasids, came to power in 750. It would provide an unbroken line of caliphs for half a millennium, though its apogee came early, in the first century or two. Their values were universal and cosmopolitan, and they created a centralised Arabic, Islamic state around the caliph as commander, leader and lawmaker, with Turks and Iranians, Christians and Zoroastrians assimilated, employed and often powerful in their own right. Their story is the stuff of legend: Harun al-Rashid raiding against the Byzantines; the rise and fall of the Barmakid family; the execution of Ja’far, once the caliph’s right-hand man.

The Abbasids moved their capital – after a period of hesitation and experiment – to Baghdad in 762. It was the heartland of their political support and of the sources of their wealth: the rich alluvial plain of southern Iraq, watered by the Tigris and the Euphrates, growing wheat and barley, dates and other fruits.

Baghdad was close to the confluence of the two rivers: a great inland port as well as a centre of overland trade. Soon it became known as the Round City. The central government complex was a walled and ditched circle, built around a palace and a mosque, with wide arcades and generous courts. Suburbs and markets grew up around it; a separate palace – al-Khuld, the Palace of Eternity – was built beside the Tigris.

The city was a monument to the caliph who founded it, Abu Ja’far al-Mansur:

In the entire world, there has not been a city which could compare with Baghdad in size and splendour, or in the number of scholars and great personalities. The distinction of the notables and general populace serves to distinguish Baghdad from other cities, as does the vastness of its districts, the extent of its borders, and the great number of residences and palaces. Consider the numerous roads, thoroughfares, and localities, the markets and streets, the lanes, mosques and bathhouses, and the high roads and shops – all of these distinguish this city from all others, as does the pure air, the sweet water, and the cool shade. There is no place which is as temperate in summer and winter, and as salubrious in spring and autumn.

The palaces of Baghdad.

Francis Rawdon Chesney, Narrative of the Euphrates Expedition carried on by Order of the British Government during the years 1835, 1836, and 1837 (London, 1868), p. 406. (Archivac / Alamy Stock Photo)

The cultural mix in Iraq was an extraordinary one. There were Aramaic-speaking Christians and Jews in the country, Persian-speakers in the cities, and Arabs both Christian and Muslim. Meanwhile in Syria, Palestine and Egypt, Greek continued to be spoken alongside Arabic, as well as the Syriac and Hebrew into which quantities of Greek writing had now been translated. Along the lengthy border with Byzantium there were not just military skirmishes but cultural contact, bringing to the caliphate an awareness of just how much ancient Greek learning was still preserved.

Some Persian and Greek books had already been translated into Arabic, and now that the Abbasids were securely in power and their capital city was under construction, they turned to learning.

They had heard some mention of them by the bishops and priests among [their] Christian subjects, and man’s ability to think has (in any case) aspirations in the direction of the intellectual sciences. Abu Ja’far al-Mansur, therefore, sent to the Byzantine Emperor and asked him to send him translations of mathematical works. The Emperor sent him Euclid’s book and some works on physics. The Muslims read them and studied their contents. Their desire to obtain the rest of them grew.

In this first wave of translations are also mentioned works of Aristotle, Ptolemy and other Greek writers, as well as Persian and Syriac books and even those of Indian writers on astronomy.

The Arabic translator of Euclid was al-Hajjaj ibn Yusuf ibn Matar, the son of a scholarly Christian family who was present at the foundation of Baghdad. Nothing more is recorded of his personal life or his activities: just that he translated Euclid’s Elements and Claudius Ptolemy’s astronomical work the Almagest during the decades around 800. One source places the initial arrival of a Greek copy of the Elements as early as 775. The Arabic sources report that al-Hajjaj’s version of Euclid was made at the request of al-Mansur’s successor Harun al-Rashid; some say of his vizier.

What was it like, this Arabic Elements? Some of the scholars of this period seem to have translated the sense of their texts rather than working word for word, but others stayed as close as possible to their models. The latter is perhaps more likely for this classic mathematical book; in any event, it seems unlikely that al-Hajjaj substantially reorganised the Euclidean text. But his was quite a succinct version of the Elements: he appears to have worked from a version somewhat shorter than the text that Stephanos copied in Byzantium, lacking a proposition here and there and often missing some of the repetitious setting-out, specification and conclusion to be found in the longer Greek versions. It also quite probably lacked books 14 and 15. The ravages of time have left more than twenty manuscripts of the Elements in Arabic, but none seems to contain al-Hajjaj’s version in anything much like its pure form, making its detailed character largely a matter of conjecture.

The burgeoning translation movement – and the reception of Euclid’s Elements in Arabic – were severely disrupted about half a century after the foundation of Baghdad. A war of succession broke out in 196 AH (AD 811) between the brothers al-Amin and al-Ma’mun, sons of Harun al-Rashid, and for more than a year al-Amin was besieged in Baghdad by his enemy’s forces. The city was devastated; suburbs were demolished and trenches dug; palaces were dismantled and their materials sold. Al-Amin was murdered in 198/813 but the fighting continued for another six years, with pillage and banditry from soldiers and vigilantes. Cultural and intellectual life surely came to a standstill.

Al-Ma’mun’s return to Baghdad after it was all over marked a new beginning. In the midst of pacifying and rebuilding, he and his ruling circle developed a court culture of refinement and sophistication, with a lively interest in the translation movement and the scientific texts it was accumulating. More than ever, Baghdad now became a magnet for texts and ideas: alongside Chang’an in the distant Tang empire it was the most literate city in the world. Al-Ma’mun founded (or perhaps re-founded) a library, the ‘House of Wisdom’, to store the scientific and philosophical knowledge that was being collected.

Al-Ma’mun’s personal enthusiasm was critically important, and the court followed his lead; courtiers used the patronage of translators as a way to establish their own prestige and competed in the hunt for manuscripts. The well-off outside the court, and the military elite, also became patrons of intellectuals. There were new missions to Byzantium, and collections of manuscripts came back to Baghdad for translation. Large personal libraries were amassed, and at one point more than a hundred booksellers worked in Baghdad. At its peak the translation movement was supported by the whole elite of Abbasid society, from princes and courtiers to officials and generals; it cut across ethnic and religious groups, involving speakers of Arabic, Syriac and Persian, Muslims, Christians, Zoroastrians and pagans.

Perhaps the most famous sponsors of translation were the three sons of Musa ibn-Shakir, collectively known as the Banu Musa. Close to al-Ma’mun, they spent lavishly, supporting a circle of professional translators and book-finders with monthly salaries. They took a particular interest in the mathematical sciences, but they, like the rest of Abbasid society, were willing to range across the whole of science and philosophy: astrology, alchemy, metaphysics, ethics, physics, zoology, botany, logic, medicine, pharmacology, military science, falconry …

As translation became a profession, the translators’ collective knowledge of Greek was improving, and the accuracy and range of their scientific terminology were increasing, meaning that older translations went out of date or needed to be updated. The early Arabic translators of scientific texts were, indeed, faced with an extraordinary task. Arabic is a language quite unrelated to Greek, with entirely different ways of managing syntax and word formation. For many technical terms, new Arabic words had to be devised, by borrowing or modifying Greek (or Syriac) words or by inventing new terms. The translators eventually created a scientific vocabulary that would be used for centuries, from North Africa to China.

In this heady atmosphere al-Hajjaj saw that it would be to his advantage to prepare a new version of his Elements. Technical terminology was still unstable and there may have been value in updating his version for that reason; it may be that he had seen new Greek texts that he wished to take into account. What later Arabic historians reported was that he decided to seek the favour of al-Ma’mun (or perhaps was commissioned by his vizier) by correcting, clarifying and shortening his Elements, making a version for specialists in which anything superfluous was deleted and anything missing was added.

Al-Hajjaj, or if not he then someone else in this first period of the Arabic Elements, also started a habit of giving names to certain of its propositions. One was called ‘the Mamunian’ – al-ma’muni – after the caliph who was al-Hajjaj’s second patron. Another – the so-called Pythagoras’ theorem – was called ‘the one with two horns’, applying rather incongruously to its diagram a traditional Arabic surname for Alexander the Great. Later in the book came the ‘goose’s foot’, the ‘peacock’s tail’ and even the ‘devil’. Presumably the names made it easier to remember which proposition was which or to recall the general shape of some of the diagrams: foot, tail or horns.

The mere fact that there was an audience of specialists – trained, interested people – to read this new, leaner version of the Arabic Elements is an indication of what the translation movement was achieving. Over the course of two centuries at least eighty Greek authors would be translated, including the big names such as Aristotle and Plato. Almost everything of Greek science and philosophy that had survived the wreck of the Roman Empire was rendered into Arabic during this period. It was a renaissance of learning on the largest possible scale, with profound consequences for the formation of an Islamic intellectual culture, which assimilated and developed not just Greek but Indian and Persian learning too.

The continuing flow of manuscripts to Baghdad meant that the story of Euclid in Arabic did not end with al-Hajjaj. Later in the ninth century another translator produced a fresh version. It was afterwards revised by a third man: Thabit ibn Qurra, one of the Banu Musa’s protégés. The Elements was becoming part of the stock of Arabic mathematical learning, to be worked over and rethought as needed.

It had become, indeed, the most important mathematical text in the Islamic world, central to the teaching of geometry and a stimulus to all manner of new mathematical work. It was corrected, summarised, added to, excerpted and frequently commented upon. Perhaps fifty Arabic commentaries on the Elements survive today, as well as Arabic translations of ancient Greek commentaries. Omar Khayyam, better known in the West as a poet, wrote a treatise explaining the difficulties in Euclid’s postulates; Ibn Sina, called Avicenna in the medieval Latin world, made a summary of the Elements as the geometrical part of an encyclopaedic work. In the tenth century an early writer on fractions went by the nickname ‘al-Uqlidisi’: meaning, most likely, that he made his living copying manuscripts of Euclid (Uqlidis). The Round City of al-Mansur was a ruin by then, but the Arabic Elements to which it had given birth lived on.

Adelard
The Latin Euclid

Bath, in south-west England, early in the twelfth century. Two men and two books. One book is in Arabic, and one of the men reads from it, translating as he goes into – probably – Spanish. The other listens to the Spanish and writes it down in Latin in his book. Word after word, page after page. There are stumbles over vocabulary, discussions about terms. The Latin Euclid is, gradually, born.

In the medieval story of Euclid, the Arabic versions hold unparalleled importance. From them, the text was translated on into versions in Sanskrit, Persian, perhaps Syriac, and other languages, some of which survive only in fragments. And it was translated into Latin.

For around half a millennium, educated citizens of the Western Roman Empire read Greek geometry in Greek, if they read it at all. Then for half a millennium and more after the Western Empire collapsed, they were reduced to mostly proof-free summaries of Euclidean results in Latin (see this book’s chapter on Hyginus in Part III for a fuller story). It is not clear that the whole – or anything like the whole – of the Elements was translated into Latin before the year 1000.

But, from late in the eleventh century, the Christians reconquered Spain and the Normans dominated Sicily, creating new possibilities for books in Greek or Arabic to pass into the Latin-speaking world. The new situation also made it much more possible for people who could read and write Latin to come into contact with those who could read Greek or Arabic. (The contemporaneous crusades in the eastern Mediterranean perhaps did rather less for cultural exchange, although the sale or plunder of Islamic libraries was certainly a factor in the movement of texts in this period. Saladin auctioned the famous Fatimid library from AD 1171 onwards, for instance; and in 1204 Constantinople itself, summit of Byzantine manuscript culture, was taken by the Latins.)

What followed was a translation movement comparable in its overall scope to the translation movement at Baghdad two centuries earlier: but this was decentralised, driven not by court patronage so much as by a new breed of sometimes itinerant lay teachers, hungry for texts. Translators worked at Barcelona, Tarazona, Segovia, León, Pamplona, and also north of the Pyrenees in Toulouse, Béziers, Narbonne and Marseilles. An intense enthusiasm for translation into Latin would last at least a century. Toledo would become a particularly important centre for translation and scholarship. Much of Arabic science and learning was translated into Latin during the twelfth century, and the consequences were incalculable.

Naturally the Arabic Elements was a part of this. The book was translated into Latin several times during the twelfth century, by scholars whose origins ranged from Chester in northern England to Cremona in Lombardy. The very first was done by a man whose life would qualify as an adventurous one in almost any period: Adelard of Bath.

Fastred was one of the tenants of the Bishop of Wells; he held land at Wells, Yatton and Banwell, all in Somerset. His son Adelard – Athelardus – was born in about 1080: less than a generation after the Norman conquest of England. He was clever, perhaps extremely so, and the combination of his family’s means and perhaps patronage, and the new situation of a Norman-controlled England, combined to provide exciting possibilities for his education. He went to France, to Tours, studied the liberal arts and showed a particular interest in astronomy. He learned music and played – by invitation – before the queen. He wrote a dramatic dialogue in the manner of Boethius, exhorting youths to the study of philosophy.

Adelard of Bath.

Leiden University Libraries, SCA 1, fol. 2r. (Leiden University Libraries/Creative Commons Attribution International, CC-BY 4.0)

He could have become a career cleric or perhaps an administrator at court. But instead he went travelling. Leaving behind a nephew to concentrate on the tamer Gallic studies (the nephew, from Adelard’s own writings about his life, may possibly be a fiction), he set out in search of the novelty of the age: Arabic learning.

In an instance of what might well be called Crusade tourism, he made his way to the Norman kingdom of Antioch in Syria, taking in Salerno and perhaps Sicily and Cilicia along the way. He saw an earthquake in Syria, a pneumatic experiment in Italy, met a Greek philosopher and learned that light travels faster than sound. Adelard, in other words, appears to have travelled from England to the border with the Islamic empire – without any official protection or support – just to seek the knowledge of that other civilisation. The success of the first Crusade in 1098 had produced something of a vogue for visiting the Middle East, but it was far from usual for scholars to travel in that direction for the specific purpose of encountering Arabic science.

Adelard certainly succeeded in encountering Arabic scientific texts, including astrological books, astronomical tables and Euclid’s Elements. But the details become lamentably obscure at this point. His astronomical tables were calculated for use at Cordova, and the most natural place to find an Arabic Elements would also have been reconquered Spain. But Adelard, despite a tendency to show off about his travels, never said that he had been to the Iberian Peninsula. Thus, there is something of a mystery.

During the 1120s, Adelard returned to England. He turns up in legal and court records in the area of Bath, though his name was not quite unusual enough to be sure that every such report refers to the same man. This was a turbulent period for England, and it looks as though Adelard may have supported first King Stephen and later his rival Matilda; he dedicated a book to her son, the future Henry II. He was quite probably associated with a circle of learned clerics around the Bishop of Hereford and Walcher and Prior of Malvern. He wrote on astrology, translated the astronomical tables of al-Khwarizmi (head of the library of the House of Wisdom in Baghdad) and wrote a work of natural philosophy drawing in a general way on his knowledge of Arabic ideas. He taught, and wrote a little book about falconry. And he translated the Elements of Euclid from Arabic, probably shortly before 1130.

As one historian remarks, ‘nothing in his life so far has prepared us for this’. It is not altogether clear what had prepared Adelard for it, who nowhere stated that he could read Arabic. The writings of his earlier life betray no special knowledge of geometry nor particular interest in the subject, although he certainly knew some practical writings about surveying and may have seen a Latin summary of the Elements.

Prepared or not, it is remarkable to think of translating a book of the length and complexity of the Elements in the conditions of the time: the more so as Adelard presumably did the work after he was back in Bath, the better part of 1,000 miles from any location where Arabic was spoken. Yet it is likely that he did have some linguistic help.

Later reports show that Latin scholars in centres such as Toledo employed the help of local experts: there would be a text in Arabic, a local person who spoke both Arabic and Spanish, and a visiting or immigrant scholar who could understand Spanish and write Latin. Translation would be effectively a two-stage process, with the text passing through a spoken Spanish version along the way, and much of the intellectual work of understanding the text done by the arabophone ‘assistant’ rather than the Latin ‘scholar’.

It is quite conceivable that Adelard had an assistant, now lost to history, or something of this kind: or perhaps someone who helped him to render the Arabic directly into Latin. Indeed, it seems less than plausible that he made his way through the conceptual and terminological complexity of the Arabic Elements without some such help. At one stage, the learned circle in Hereford and Malvern had included Petrus Alfonsi from Huesca in Aragon, a converted Spanish Jew who became physician to King Henry. He worked on astronomical material, and may well have provided the base translation from which Adelard produced his version of the tables of al-Khwarizmi. Petrus moved to France too early, it seems, to have worked on Adelard’s Elements, but his example indicates the kind of collaboration that might have been involved, and the kind of person who may have stood, uncredited, behind Adelard’s work.

Be that as it may, the Latin Elements was a remarkable production. It clearly showed the linguistic stages it had been through, and the struggles of this first medieval translator to find the right Latin words for what he needed to say. In one place Adelard tried three different phrases in succession, all trying to express the idea of ‘a ratio repeated three times’. Elsewhere he hesitated between words meaning ‘difference’ or ‘remainder’ when talking about the result of taking a quantity away from another.

It is plain to see that he was not working from the Greek text. He used none of the Greek loan-words that would later become common in Latin, such as hypotenusa, parallelogrammum, ysosceles. Instead his Latin contained a wealth of terms that were simply Arabic: alkamud (perpendicular), alkaida (base), mutekefia (proportional), elmugecem (solid), elmugmez (pentagon). Even when he knew the equivalent Latin words, Adelard sometimes deliberately adopted an Arabic terminology: elkora for sphere and elmukaab for cube, for instance.

Also from Adelard’s Arabic source came the names for some of the propositions, of which he made some strange gibberish: elefuga, dulcarnon, thenep atoz, seqqlebiz. Some of them nevertheless stuck, notably dulcarnon, which went on to be mentioned by Chaucer and would later facilitate the popular pun ‘dull carnon’. He also took from the Arabic Elements a curious habit of introducing the different parts of each proposition with fixed, formal phrases: Now we must prove … So I say that … For the sake of argument … The reason why. These do not appear in any surviving Greek version, and were surely not invented by Adelard; they provide a glimpse of features of his Arabic source which are otherwise lost.

Of the two main Arabic versions of the Elements, the modern consensus is that Adelard’s version seems to be derived from the earlier, that of al-Hajjaj. The clearest piece of evidence is that the number of propositions in Adelard’s Euclid agrees with al-Hajjaj in ten of the fifteen books, whereas in four of those books the later Arabic version, by Thabit, has a different number of propositions. Similarities of wording support the conclusion, but several matters complicate it. Al-Hajjaj’s version is largely lost; Adelard’s survives only incompletely (book 9 and part of book 10 have been lost). No surviving Arabic manuscript is close enough to Adelard’s version to have been his actual source, and it is possible that what he worked from was not a ‘pure’ al-Hajjaj text but one with some modifications. For further intrigue, a fifteenth- or sixteenth-century manuscript containing much of book 1 in Syriac agrees almost word for word with Adelard; even the lettering on the diagrams matches. Does this mean that the Syriac translator and Adelard used the same, Arabic source, now lost? Certainly the last word has not yet been written on the subject.

Adelard’s was a remarkable achievement, and his name became celebrated, attached to Euclidean material by later scribes whether it was really his or not. His opinions on some details in the arithmetic of Boethius were carefully recorded by a student; one ‘Ocreatus’ dedicated an arithmetical work to him. His version of the Elements was a key moment when a mathematical tradition from the shores of the Mediterranean acquired a presence in northern Europe, from there to pass – a few generations later – into the chain of new universities stretching from Italy to England.

By about 1150 Adelard was dead, but the translation work of the twelfth-century scholars went on, and Latin attention to the Elements of Euclid gathered pace. The translators of the period worked with evident excitement on a mass of material to which their access was somewhat haphazard, and they worked in some isolation from one another. Texts were translated more than once, and tangles of different versions emerged as new Arabic texts were obtained or new ways were tried of amalgamating them. For the Elements alone, the cast of characters becomes quite large and the names of the translators and editors give a sense of the range of people interested in the book: Hermann of Carinthia, Robert of Chester, Gerard of Cremona, Ioannes of Tinemue. Gerard, in the mid-twelfth century, actually moved to Toledo, becoming the century’s most prolific translator from Arabic: he rendered Euclidean commentaries and Euclid’s Data as well as the Elements. In Sicily, where there was a tradition of bilingualism and trilingualism, someone translated the Elements directly from Greek into Latin, late in the twelfth century. A number of other reworkings have not yet been studied in full even today: some of them exist only in single manuscripts and represent complex mixtures of different versions.

Thus all of the early versions, including Adelard’s own, came to be eclipsed. First by the version of Robert of Chester, popular throughout the twelfth century and, ironically, often attributed in the manuscripts to Adelard himself (Robert may have been a student of Adelard, which does something to explain the confusion). And, second, by the work of Campanus of Novara. Writing in the 1250s, he put together what would become the Latin Euclid of the later Middle Ages. He used Robert’s version for the statements of each proposition: material that – much of it – went back to Adelard. But he composed new versions of the proofs, bringing together material from other versions and commentaries. And he added to the book, with new propositions and definitions from various sources: the impulse to modify and improve the Elements was proving as durable as the book itself.

A hundred manuscripts of Campanus’ Elements survive today: it was a bestseller in its day. For Adelard’s version there are just four, none of them complete. But through Campanus, much of what Adelard had done in rendering Euclid’s words into Latin survived and flourished for another three centuries, and passed on into the era of print.

Erhard Ratdolt
Printing the Elements

May 1482, in La Serenissima: the Republic of Venice. The print shop of Erhard Ratdolt.

In one room, compositors perch on high stools, slotting tiny pieces of lead type into little frames, called composing sticks. A stick holds a few lines of text (upside down and backwards), and when it is full the assembled letters are lifted out in a block, and added into the tray – the forme – in which a whole sheet of text is being assembled. A good compositor can set a few thousand characters – several hundred words – in an hour.

Once a full sheet is made up, the letters (and any pictures) are clamped together and carried into the next room, to the press itself. Out slides the bed from under the press, and the forme full of letters is placed on it and spread with ink. Two frames fold out to hold the paper; they fold back and the whole thing – paper, ink, letters – slides back under the press. Two pressmen pull a long lever to screw the press down, pushing the paper against the inked type and transferring ink onto the paper. They release the lever, slide out and unfold the frames and take out the paper. The printer’s boy (or ‘devil’) takes it away. Another sheet of paper follows, and another: hour after hour.

The workshop is busy, noisy and dangerous. The type is made of lead; the ink probably contains vitriol, and its manufacture involves burning pitch and boiling oil. The press itself is heavy and as powerful as a wine press, its close ancestor.

From one workshop in the first months of 1482 there come more copies of Euclid’s Elements than Europe has made in 1,000 years.

The owner and overseer of that workshop, Erhard Ratdolt, was part of the German diaspora that took Gutenberg’s invention across Europe in the second half of the fifteenth century. One printing press in the world used movable type in the 1450s; by 1470 such machines existed in fourteen cities, and by 1480 in over a hundred. Born in about 1447, Ratdolt grew up in Augsburg in southern Germany, and after an apprenticeship crossed the Alps by the Brenner Pass, setting up as a printer in the first and finest city he reached: La Serenissima.

Venice in the 1470s.

Werner Rolevinck, Fasciculum temporum (Venice, 1485), fol. 37v. (Digital Library@Villanova University, Creative Commons Attribution-ShareAlike 4.0 International, CC BY-SA 4.0)

A commercial and business hub with a fine natural harbour, Venice was at the height of its power and influence in the late fifteenth century: and of its splendour. In the previous half-century many of its famous landmarks had been erected: the facade of the Doge’s Palace, the Ca’ d’Oro, San Giovanni e Paolo, the Porta della Carta and the Ca’ Foscari. Giovanni Mocenigo occupied the doge’s throne. A French ambassador described the city:

I was conducted through the longest street, which they call the Grand Canal, and it is so wide that galleys frequently cross one another; indeed I have seen vessels of four hundred tons or more lie at anchor just by the houses. It is the fairest and best-built street, I think, in the world, and goes quite through the city. The houses are very large and lofty, and built of stone; the old ones are all painted; those of about a hundred years’ standing are faced with white marble from Istria, about one hundred miles off, and inlaid with porphyry and serpentine. Within they have, most of them, at least two chambers with gilt ceilings, rich chimney pieces, bedsteads of gold colour, their portals of the same, and exceedingly well furnished. In short, it is the most glorious city that I have ever seen, the most respectful to all ambassadors and strangers, governed with the greatest wisdom, and serving God with the most solemnity.

By 1476 Ratdolt was printing as a member of a three-man partnership, part of an immense expansion of print in the city: from nothing in 1469 to 150 print shops by the end of the century. The combination of convenient sources of paper in northern Italy and generous provision for the protection of texts and technical improvements made the city one of the centres for the new technology. Ratdolt and his partners printed calendars, histories and geographies, a total of eleven books in three years. They were some of the finest printed books ever produced, with ornamental title pages, careful woodcut illustrations, page borders and initial letters, and two-colour texts printed in black and red. Illustrations of eclipses in the calendars had colouring in yellow added by hand. The type, like that of other printers in Venice, was modelled on the handwriting of contemporary scholars; it is admired to this day for its harmony and elegance.

In 1478 the plague threatened to end it all. It afflicted Venice for four years, and at its height 1,500 people were dying every day. Many fled; half the printing houses closed, and Ratdolt’s partnership never printed again, although all three of its members survived. Ratdolt’s daughter Anna was born in June 1479 when the disaster was at its height.

By the following year he was beginning to rebuild his printing business, this time as sole owner and employer of a full team of compositors, pressmen, engravers, proofreaders, apprentices and assistants. He worked, not surprisingly in the circumstances, with a furious energy. In his first year on his own he produced eight books, and he took pains to reach a broad market, ranging across ecclesiastical works, histories and – a growing specialism of his – mathematical texts.

There had been plans to print the Elements before: the astronomer and printer Regiomontanus, under whom Ratdolt had served as apprentice, had proposed an edition in the 1470s but never carried it out. It fell to Ratdolt, then: and at his hands, ten years before Columbus crossed the Atlantic, Euclid made the momentous transition to print.

It was a vast project. Ratdolt chose the large ‘folio’ format, producing a book about eight inches by seven, and even so it filled 276 pages; nearly seventy sheets of paper had to go under the press, twice each, to produce each copy (each was then folded, so that when bound into the book it formed four pages). The compositors must have set well over half a million characters of type in all.

There were some problems no one would have expected. So many of Euclid’s propositions began with ‘If …’ (‘Si …’) or ‘Let there be …’ (‘Sit …’) that some pages needed up to thirteen decorated S’s: the set of decorated initials Ratdolt was using ran out and had to be supplemented by another set that did not match. In a similar way, the incessant reference to points in diagrams labelled A, B, C, D and so on, put pressure on the supply even of normal-sized capital letters. Four different sets of type had to be used in order to complete the book.

Technical challenges apart, Ratdolt took every care to make sure the book was of the highest quality. He seems to have obtained quite a good manuscript of the Elements, in Campanus’ Latin version: probably part of a celebrated ecclesiastical library deriving from the mid-century papal court, where there was a tradition of well-illustrated mathematical books. The decorated first page was put under the press three times: once for the text, once for the border and once for the title, which was printed in red. Pinholes in the paper were used to line up the separate layers of printing. The type was neat and strikingly accurate, and the decorated border – reused from one of Ratdolt’s earlier books – was a beautiful piece of design. The page layout was broadly similar to that of Gutenberg’s Bibles, with broad margins modelled ultimately on the layouts of fine medieval manuscripts.

And the diagrams? This was the first really geometrical book to be printed; the first for which a large number of accurate diagrams were needed. Printing diagrams was known to be hard; editions of Ptolemy and Vitruvius were appearing from other presses without diagrams, because they were too difficult to produce. And even in the world of manuscript, it had become common for diagrams to be poorly copied or even omitted because of the time and skill involved in their construction. The Elements needed 500 or more, and, as Ratdolt himself pointed out, nothing in geometry could be properly understood without them.

He was determined to get this right, and indeed his decision to print Euclid’s Elements seems to have depended on a breakthrough he achieved in the technique of printing geometrical diagrams. The usual way to insert illustrations in printed books was to carve them in relief in wood, clamping the block of wood into the frame with the type, inking it and printing from it in just the same way as the text. Good effects could be achieved, but there were limitations: particularly in creating fine lines of even thickness, keeping them straight when they were meant to be straight and circular when they were meant to be circular. It was particularly hard to make it work for geometrical diagrams.

Ratdolt boasted in a preface to his Elements that he had a special technique for printing the diagrams, but scholars do not agree about what it was. One suggestion is that he took strips of metal – zinc or copper most likely – bent them into the shapes required, and mounted them in plaster to hold them in place, together with pieces of type that would provide the labels – A, B, C, and so on – within the diagram. Another possibility is that he used cast pieces of metal clamped together much like the pieces of type that made up the words on the page. Or, again, he may have had special tools and techniques for making woodcuts of exceptionally fine quality.

In any case, the wooden or metal diagram could then be clamped into the set-up page of type – Ratdolt invariably placed the diagrams in the margin, which would simplify this part of the process – and its protruding edges inked and printed from just like the rest of the page. However it was done, Ratdolt achieved results of the highest quality, with fine, even lines that were curved or straight as required. He has deservedly been called the most inventive printer after Gutenberg. The result was a book of which Ratdolt could justly be proud, and which increased still further his already high reputation among Venetian printers. No surviving medieval manuscript of the Elements, in any language, has diagrams of a similar number and quality.

After a large number of copies had been printed, Ratdolt stopped his press and reset the first nine leaves of the book. He wanted to rearrange some of the diagrams, insert some new ones and generally improve the completeness and clarity of the pages. Such an obsessive care, a commitment to perfection, almost defies belief, even in the proud, skilled world of the early printers.

The Elements in 1482.

Preclarissimus liber elementorum Euclidis perspicacissimi (Venice, 1482), fol. 2r. (Wikimedia Commons/public domain)

It was a work fit for a king, and Ratdolt added to his book a separately printed leaf dedicating it to the doge. There’s no evidence that the doge actually commissioned or paid for the printing of the book (or wanted it), but Ratdolt’s dedicatory letter placed it unambiguously under his ruler’s protection, casting Giovanni Mocenigo in the role of patron and himself, Ratdolt, in the role of a creator offering up his work. It was a way to boost his own status – higher than a craftsman’s, and nearly as high as an author’s – as well as an opportunity to display to the world what he could do. For a few presentation copies, he printed on vellum instead of paper, and put gold dust into the ink instead of the usual lamp-black, so as to print the dedicatory letter in flamboyant gold letters. For the doge’s own copy, the first page was illuminated by hand to complete the effect.

The dedicatory letter itself spared no pains to emphasise Ratdolt’s own cleverness:

I used to wonder why it is that in your powerful and famous city there are many works of ancient and modern authors being published, but none or few and of little importance are books of mathematics … Until now, no one has found a way to make the geometric diagrams … I applied myself and with great effort I made the figures, so that the geometric figures are printed with the same ease as the verbal parts of the Elements.

Ratdolt probably printed over 1,000 copies of his remarkable edition of the Elements. He had set standards for mathematical typography as well as for the printing of geometrical diagrams; he had established a visual language for geometry within the new world of print, similar to the visual language of manuscript geometry but in subtle ways different.

There is sadly little information about where the copies went. A de luxe book is not necessarily a book that is read, although in the intellectual climate of fifteenth-century Italy it is reasonable to suppose that this, the first printed version of such a widely known ancient text, received its fair share of attention. And indeed, the surviving copies show that many a reader responded to the diagrams by working with pen or pencil in hand, re-copying them as the anonymous reader at Elephantine had done and using the blank space in Ratdolt’s generous margins as rough paper: continuing the ancient tradition of understanding and learning the ideas the words and diagrams embodied.

Before the century was out, there were a handful more editions of the Elements in a Latin abridgement, as well as a pirated reprint of Ratdolt’s edition that appeared at Vicenza in 1491. But the Elements really took flight during the next century, as print spread and cheapened and printed books decisively took over from manuscripts in many (though not all) situations. About forty substantially different versions of the Elements were produced between 1500 and 1600: some were printed several times; thus there was a new printing something like once a year on average. Most were in Latin, but Simon Grynäus edited the Greek text for publication in Basel in 1533, using manuscripts in Venice, Paris and Oxford. The first complete vernacular version to see print was the Italian translation by mathematician Niccolò Tartaglia, printed in 1543. By the end of the century the Elements would also be printed in German, French, Spanish, English and Arabic. It is not clear that any text except the Bible was being set up in type more frequently; that pattern would last another two centuries and more.

Ratdolt himself returned to his native Augsburg after ten brilliant years in Venice, there to print for another four decades. He developed a focus on liturgical books, though he also continued to work on scientific texts and histories. He didn’t use his novel system for printing geometrical diagrams again, and although he reprinted a number of his Venetian books, the Elements was not one of them.

Erhard Ratdolt died, wealthy and highly respected, in March 1528, at about the age of eighty-one. A tribute written by a fellow Venetian printer in 1482 – the year of his Elements – might stand as his epitaph:

The seven arts, abilities granted by the divine power, are amply bestowed on this German from Augsburg, Erhard Ratdolt, who is without peer as a master at composing type and printing books. May he enjoy fame, ever with the favour of the Sister Fates. Many satisfied readers can confirm this wish.

Marget Seymer her hand
Owning the Elements

A handwritten inscription in a 1543 printed edition of Euclid’s Elements, now in the National Library of Wales: ‘Marget Seymer her hand’.

Putting the Elements into print sent it into the wild, making it common property in a way it had never been before. Printed editions varied very widely, and reflected not just the range of medieval versions of the text but also the new agendas of the Renaissance world, and the commercial imperatives of its printers and publishers.

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.

There was much more to it than that, though. People – a lot of them – have read the Elements not only for its own pure sake, but for what it could do for them: as a source of wisdom, or a spur to personal transformation. A different story would make the Elements a philosophy and Euclid a sage. It would show people reading the book because it put them in touch with eternal truths.

Back to Greece, then, to tell a different story this time.

II

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