Evidence of the Elements | Benjamin Wardhaugh (2023)

The ancient Greek mathematician Euclid’s original manuscripts for Elements of Geometry—a compendium of facts about space and its properties, lines and shapes, numbers and ratios, written around 300 bc in Alexandria—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 three thousand years old (this was in the 1930s, when attitudes to museum artifacts 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, totaling 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 one, with one summary proof (these come as citations in a philosophical treatise preserved—carbonized—in Herculaneum by the eruption of Vesuvius in 79: an exception to the normal generalizations about papyrus survival). An enunciation from book two, with a rough figure, written in the Egyptian city of Oxyrhynchus around 100. Parts of two more propositions from book one, written at Arsinoë (modern Faiyum) in the second half of the second century. A second-century copy of three figures and enunciations from book one, carefully written with ruled diagrams. And a schoolteacher’s or pupil’s copy of the ten opening definitions, made in the third century.

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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 center 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: ostraca or pot shards. Literally, broken pieces of pot: waste, and therefore free. Ostraca 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 ostracismwas a procedure for expelling a man from the country for ten years on suspicion of disloyalty: the votes were written on ostraca. It happened at Athens for most of the fifth century bc, and in other Greek cities, too.)

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Evidence of the Elements | Benjamin Wardhaugh (1)

A set of ostraca bearing geometrical writing, written at Elephantine in Egypt, 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 five hundred 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 ostraca 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 centers 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 ostraca—six of them, one clearly broken on all sides—bear a text concerned with constructing a regular polyhedron. It relates to propositions ten and sixteen in book thirteen 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 ostraca, 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.

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The propositions in the Elements depend on one another in an elaborate, treelike 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 ostraca 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 ostraca 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.

Evidence of the Elements | Benjamin Wardhaugh (2)

There is a further twist. The matter discussed on the ostraca came straight from Elements book thirteen, 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-mémoire, 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 re-creating 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.

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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 ostraca, in particular, should probably be read in this way: as an attempt to re-create 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 artifact 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.

Excerpted from Encounters with Euclid: How an Ancient Greek Geometry Text Shaped the World by Benjamin Wardhaugh. Copyright © 2021 by Benjamin Wardhaugh. Reprinted by permission ofPrinceton University Press.

FAQs

How has Euclid's Elements survived? ›

It survived only by Arabic translatons. Phaenomena -- on spherical geometry, it is similar to the work by Autolycus. Optics -- an early work on perspective including optics, catoptrics, and dioptrics.

Who is the mother of geometry? ›

Euclid
Known forThe Elements Optics Data Various concepts Euclidean geometry Euclidean algorithm Euclid's theorem Euclidean relation Euclid's formula Numerous other namesakes
Scientific career
FieldsMathematics
InfluencesEudoxus, Hippocrates of Chios, Thales and Theaetetus
4 more rows

How do you prove Euclid's 5th postulate? ›

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

What was the purpose of Euclid's Elements? ›

Euclid's Elements (c. 300 bce), which presented a set of formal logical arguments based on a few basic terms and axioms, provided a systematic method of rational exploration that guided mathematicians, philosophers, and scientists well into the 19th century.

Who is the father of math? ›

The Father of Math is the great Greek mathematician and philosopher Archimedes.

Who is the father of Triangle? ›

Many have speculated that the claims of the 5th-century BC Greek mathematician Pythagoras being the first to deduce facts about right-angled triangles just did not add up. Some believe later commentaries indicate it was the collaborative work of his followers, the Pythagoreans.

Who first invented geometry? ›

Euclid was a great mathematician and often called the father of geometry. Learn more about Euclid and how some of our math concepts came about and how influential they have become.

Who are the mathematicians who tried to prove the 5th postulate? ›

The Persian mathematician, astronomer, philosopher, and poet Omar Khayyám (1050–1123), attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher (Aristotle), namely, "Two convergent straight lines intersect and it is impossible for ...

Is there an importance of proving the fifth postulates? ›

The fifth postulate is also known as the Parallel Postulate, because it can be used to prove properties of parallel lines. exterior angles equal to the interior and opposite angle and the sum of the interior angles on the same side equal to two right angles.

What are the 5 postulates of Euclid? ›

Euclid's postulates were : Postulate 1 : A straight line may be drawn from any one point to any other point. Postulate 2 :A terminated line can be produced indefinitely. Postulate 3 : A circle can be drawn with any centre and any radius. Postulate 4 : All right angles are equal to one another.

Where was Euclid's Elements found? ›

The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC.

Where is the original Euclid's Elements? ›

The manuscript now resides in the Bodleian Library, Oxford University.

What did Euclid write in the Elements? ›

Euclid of Alexandria (19th Century)

The thirteen volumes of Euclid's “Elements” contains 465 formulas and proofs, described in a clear, logical style using only a compass and a straight edge, it contains formulas for calculating the volumes of solids such as cones, pyramids and cylinders.

Who found zero? ›

About 773 AD the mathematician Mohammed ibn-Musa al-Khowarizmi was the first to work on equations that were equal to zero (now known as algebra), though he called it 'sifr'. By the ninth century the zero was part of the Arabic numeral system in a similar shape to the present day oval we now use.

Who found numbers 1 to 9? ›

They originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially al-Khwarizmi and al-Kindi, about the 12th century.

Who created algebra? ›

Muhammad ibn Musa al-Khwarizmi was a 9th-century Muslim mathematician and astronomer. He is known as the “father of algebra”, a word derived from the title of his book, Kitab al-Jabr. His pioneering work offered practical answers for land distribution, rules on inheritance and distributing salaries.

Who created shapes? ›

Early geometry. The earliest recorded beginnings of geometry can be traced to early peoples, such as the ancient Indus Valley (see Harappan mathematics) and ancient Babylonia (see Babylonian mathematics) from around 3000 BC.

Why is it called a triangle? ›

Triangle comes from the Latin word triangulus, "three-cornered" or "having three angles," from the roots tri-, "three," and angulus, "angle or corner."

Who invented calculus? ›

Today it is generally believed that calculus was discovered independently in the late 17th century by two great mathematicians: Isaac Newton and Gottfried Leibniz.

What was the first shape? ›

The first type of solid shapes to be discovered are known as Platonic solids, which include the cube, the tetrahedron (a 3D form made up of four triangular faces), the octahedron (a 3D form made up of eight triangles), the dodecahedron (a 3D form made up of 12 sides) and the icosahedron (a form made up of 20 triangular ...

Why is it called geometry? ›

Geometry comes from two Greek words, “ge” meaning “earth” and “metria” meaning “measuring.” The approach to Geometry developed by the Ancient Greeks has been used for over 2000 years as the basis of geometry.

What is a geometry proof? ›

Geometric proofs are given statements that prove a mathematical concept is true. In order for a proof to be proven true, it has to include multiple steps. These steps are made up of reasons and statements. There are many types of geometric proofs, including two-column proofs, paragraph proofs, and flowchart proofs.

How does Euclid affect us today? ›

He is most famous for his works in geometry, inventing many of the ways we conceive of space, time, and shapes. He wrote one of the most famous books that is still used today to teach mathematics, Elements, which was well received at its time and also is praised today for its thought and understanding.

How did Euclid impact the world? ›

The reason that Euclid was so influential is that his work is more than just an explanation of geometry or even of mathematics. The way in which he used logic and demanded proof for every theorem shaped the ideas of western philosophers right up until the present day.

What was Euclid's contribution to mathematics that's still in use today? ›

Euclid gave the proof of a fundamental theorem of arithmetic, i.e., 'every positive integer greater than 1 can be written as a prime number or is itself a prime number'. For example, 35= 5×7, etc. 2. He was the first one to state that 'There are infinitely many prime numbers, which is also known as Euclid's theorem.

What was Euclid's most important mathematical accomplishment? ›

Euclid's most famous work is his treatise on mathematics The Elements. The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years. Probably no results in The Elements were first proved by Euclid but the organisation of the material and its exposition are certainly due to him.

What was discovered by Euclid? ›

Among Euclid's extant works are the Optics, the first Greek treatise on perspective, and the Phaenomena, an introduction to mathematical astronomy. Those works are part of a corpus known as “the Little Astronomy” that also includes the Moving Sphere by Autolycus of Pitane.

Why is Euclid known as the father of geometry? ›

Euclid is called the father of geometry because he basically created the geometry that people do today. In his book "Elements," Euclid gathered up all of the known mathematics of his time, as well as a lot of his own, and then he subjected it all to logical, mathematic proofs.

What is an example of Euclidean geometry? ›

The two common examples of Euclidean geometry are angles and circles. Angles are said as the inclination of two straight lines. A circle is a plane figure, that has all the points at a constant distance (called the radius) from the center.

Who is the father of math? ›

The Father of Math is the great Greek mathematician and philosopher Archimedes.

How do you pronounce Euclid? ›

How to Pronounce Euclid? (CORRECTY) - YouTube

When did Euclid write the Elements? ›

The index below refers to the thirteen books of Euclid's Elements (ca. 300 BC), as they appear in the "Bodleian Euclid." This is MS D'Orville 301, copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD. The manuscript now resides in the Bodleian Library, Oxford University.

Who is the first mathematician in the world? ›

One of the earliest known mathematicians were Thales of Miletus (c. 624–c. 546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.

Who created shapes? ›

Early geometry. The earliest recorded beginnings of geometry can be traced to early peoples, such as the ancient Indus Valley (see Harappan mathematics) and ancient Babylonia (see Babylonian mathematics) from around 3000 BC.

What was the impact of Euclid's work? ›

Euclid's vital contribution was to gather, compile, organize, and rework the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry. In Euclid's method, deductions are made from premises or axioms.

What was the first shape invented? ›

The first type of solid shapes to be discovered are known as Platonic solids, which include the cube, the tetrahedron (a 3D form made up of four triangular faces), the octahedron (a 3D form made up of eight triangles), the dodecahedron (a 3D form made up of 12 sides) and the icosahedron (a form made up of 20 triangular ...

What is the history of Euclid geometry? ›

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these.

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