# Euclid's Elements

A Site About Euclid's Elements

Herbert M Sauro: August, 2023

"He was 40 years old before he looked on Geometry; which happened accidentally. Being in a Gentleman’s Library, Euclid’s Elements lay open, and ’twas the 47 El. Libri 1. He read the proposition. By God, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is impossible! So he reads the Demonstration of it, which referred him back to such a Proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that trueth. This made him in love with Geometry."

Thomas Hobbes by John Aubrey (1626-1697). Quoted in O L Dick, Brief Lives (Oxford 1960)

Euclid's Elements is probably one of the most famous books in the world. It's certainly one of the most published, with over 1000 different editions. It's a collection of 13 `books' (today, we might call them chapters) that lay out the foundation for geometry, number theory, and many core concepts of mathematics and logic still important today. Its popularity is highlighted by the fact it was still used in schools and colleges well into the 20th century as a textbook on geometry, though more modern treatments have now supplanted it. There is, however, continuing interest in using it as an approach to teaching deductive reasoning.

The entire collection comprises definitions, postulates, and a large number of mathematical proofs, many of which are related to geometric constructions. The 13 books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines.

I recently put together a new color rendering of Book I which might be of interest.

You can find a wide range of links to Euclid related pages at List of Web Links

A list of classical editions at archive and google at the bottom of the page.

Contents: Euclid's Elements

BOOK I Triangles, parallels, and area

BOOK II Geometric algebra

BOOK III Circles

BOOK IV Constructions for inscribed and circumscribed figures

BOOK V Theory of proportions

BOOK VI Similar figures and proportions

BOOK VII Fundamentals of number theory

BOOK VIII Continued proportions in number theory

BOOK IX Number theory

BOOK X Classification of incommensurables

BOOK XI Solid geometry

BOOK XII Measurement of figures

BOOK XIII Regular solids

EUCLID

Very little is known about Euclid. What we do know is that he (we assume he was a he) lived in Alexandra about 300 BC. This is based on a passage from Proclus' "Commentary on the First Book of Euclid's Elements.", copies which you can still purchase today, Proclus was a philosopher who lived from 412 to 485. Note that he was writing 700 years after Euclid. Here is a quote from Proclus who describes Euclid:

"All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures."

This passage tells us that Euclid collated earlier work as well as contributing his own work. It also tells us roughly when Euclid lived as well as an anecdote about his character. The passage also tells us that Proclus didn't have a precise date for when he lived. We also know that Euclid wrote other books on mathematics, a number of which we still have and a number which, sadly, have been lost to history.

By far, the best book to learn more about Euclid is the series of volumes by Heath, which is still readily available today in print and online:

Heath, Sir Thomas Little (1861-1940)

The thirteen books of Euclid's Elements translated from the text of Heiberg with introduction and commentary. Three volumes. University Press, Cambridge, 1908. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publ., New York, 1956.

The Wikipedia page on Euclid is also an excellent source of information on Euclid.

THE BOOK

The main material in Euclid' Elements are the proofs, but each book usually has at the beginning one or more definitions, postulates, and axioms. There are a huge number of editions available online as well as copies that can be purchased from sites like Amazon, abebooks, Etsy, or eBay. However, the editions by Todhunter and by Hall & Stevens were specifically written for High School students and the new reader might find these a good entry to Euclid. However, no matter what editions you use, Euclid requires sustained effort to master. However, there is a nice shortcut today.... YouTube.

On YouTube, you'll find numerous videos on Euclid's Elements. I can recommend three channels: one by Sandy Bultena, which covers Book I to Book VII, and part of Book VIII. A second is by Euler's Academy, which covers Book I and part of Book II; and a third by mathematicsonline which covers Book I. There are lots of other smaller channels that go over some of the material. Euclid is a substantial piece of work, and it is no surprise that there isn't a complete set of videos for every Book in Euclid.

If you're after a completely new fresh copy of Euclid, there are a couple to choose from. The most well-known current copy of Euclid is by Dana Densmore. Other than a short introduction, the book is pure Euclid and comes in at 527 pages, This gives you some idea of the volume of material.

If you're looking for an edition with lots of commentary, then you should get hold of the books by Heath (mentioned above). Because of the amount of commentary provided by Heath, this edition comes in three separate volumes.

The Proofs

The Greeks had particular restrictions on how to do geometric proofs and constructions. The first is that any proof or construction could only use a straight-rule and a compass. This is one reason why Propositions 2 and 3 in Book I appear to be a bit odd when read today. In fact, for those coming to Euclid for the first time, they will find the first preposition easy to digest but might be in for a bit of a shock when they read propositions 2 and 3 which seem totally unrelated to the kind of geometry one might be used to. After propostion 3 the book settles down again to what we might think of as geometry. I have more to say about this in the Book I page.

The other interesting point about Euclid is that actual measurement was not allowed even though the Greeks did have standards of measurement. Instead, magnitudes are compared, such that they might be equal, or that one magnitude is larger than another. You may also notice that the book never talks about measuring angles in terms of degrees even though in astronomy the ancient Greeks used degrees in their measurements. In Euclid, everything is in terms of right angles which partly accounts for the strangeness of Postulate 4: "That all right angles equal one another.".

The proofs also have a particular structure which was used in other ancient mathematical books although the details can vary. Proclus gave a detailed description as follows:

Enunciation: This states the result with possible reference to a figure.

Setting-Out: A statement on how we will start out on the proof.

Demonstration: This is the proof itself.

Conclusion: A statement of the result with reference to the enunciation.

Let's look at the first proposition of Book I to give you an idea of what Euclid looks like. This describes the construction of an equilateral triangle. That is a triangle with three sides of equal length. It should be noted that many theorems are not constructions but proofs related to some geometric theorem.

First I will present Proposition 1 using more modern language which might be easier for you to read. This is from my book on A rendering of Book I. Changing Euclid is, of course, committing sacrilege but this is the only place I do this.

Hall & Stevens 1898

Let's now look at some genuine Euclid editions to see how Proposition 1 is described. What follows is a series of screenshots from various editions (in no particular order) of Euclid that highlights some of the variations you'll find. Most of the time the variations are quite minor but sometimes there are significant deviations especially among editions from 1000 years ago when finding unadulterated copies of Euclid was difficult.

I'll first give a rendering of Proposition 1 from an edition by Hall & Stevens, a book used by High Schools in the last century. One thing you'll notice in the following examples is that in a number of places you'll find references to a postulate or definition next to a statement. For example, Post. 3 or often in square or round brackets such as [Post 3.] This is actually a new development, since there is no evidence that this is what Euclid actually did and editions up to the 16th century show no such referencing to definitions, postulates, axioms or earlier propositions in a proof.

Dana Densmore

The second example of Proposition 1 is a rendering from Dana Densmore, an edition of Euclid published in the last 20 years, but is essentially the proof show in Heath, 1920:

Book I: A new rendering

The third is my own, based on Richard Fitzpatrick's edition, which is a re-rendering of Heiberg's Greek edition in Greek and English. but which is very similar to Heath's edition. I have also colored-coded the proposition to make it easier to see as hunting for the letters in the figure can be a nuisance.

Oliver Byrne: 1847

The fourth is the version presented by Oliver Byrne who wrote the graphical edition of Euclid. The screenshot is from a copy held at archive.org but several people have republished this edition in the last few years. I won't give links since there are a number of editions but do a search on Amazon. Note the letters that look like 'f' are actually the old way to write the letter 's'.

Adelard of Bath

Here is a version of Proposition 1 from Adelard of Bath, probably written between 1126 and 1130 (The first Latin translation of Euclid's Elements commonly ascribed to Adelard of Bath, Busard, 1983, pp20). Adelard was an English natural philosopher (likely Anglo-Saxon) who did much to help restart learning in England and beyond in medieval Europe. His edition of Euclid is based on an earlier Arabic edition, probably by al-Hajjaj, so it should come as no surprise to find that it is very different from current versions. First, I will give the Latin version I obtained from "The first Latin translation of Euclid's Elements commonly ascribed to Adelard of Bath" by the great late scholar Busard, 1983. Not having a classical education in Latin, I called upon Chatgpt to translate Latin to English, which you can see below the Latin copy.

What follows is a computer-generated Latin translation of the above to English of Book I, Proposition 1 by Adelard:

Now we have to show how to make the surface of a triangle of equal sides over a straight line of assigned size.

Let the line be assigned ab. Let the center be placed above a occupying the space that is between a and b circle, above which gdb. Another circle is placed above the center of b occupying the space between a and b, above which gah.

Let them proceed from the point g above which the intersection of two circles is made by straight lines to the point a and to the point b Let him/them be called ga and gb. I say that here/behold we have made/constructed a triangle of equal sides above the assigned line ab.

Reason: Because the point a became the center of the circle gdb, the line ag became equal to the line ab. And since the point b is the center of the circle gah, the line bg is equal to the line ba. Thus, each of the lines ga and gb is equal to the line ab. But each thing is equal to one thing, and each thing is equal to another. Therefore, the three lines ag and ab and bg are equal to each other. A triangle of equal sides abg is therefore made above the line assigned to ab. And this is what we intend to demonstrate in this figure.

This is an image of Proposition 1 (and by the looks of it maybe part of Prop 2) from the Harley MS 5266. Image taken from the British Library site. This is an early 14th century copy of Adelard I. The oldest copy MS 47 in Trinity college is not available online :(

D'Orvile Edition 888 AD

Last but no means least here is a translation provided by the Clay Institute of Proposition 1 from the oldest known edition of Euclid's Elements, the MS D’Orville 301 at the Bodleian in Oxford. Written in Constantinople in 888AD.

"On a given finite straight line to construct an equilateral triangle. Let AB be the given finite straight line.

Thus it is required to construct an equilateral triangle on the straight line AB.

With centre A and distance AB let the circle BCD be described; [Post. 3] again, with centre B and distance BA let the circle ACE be described; [Post. 3] and from the point C, in which the circles cut one another, to the points A, B let the straight lines CA, CB be joined. [Post. 1] Now, since the point A is the centre of the circle CDB, AC is equal to AB. [Def. 15] Again, since the point B is the centre of the circle CAE, BC is equal to BA. [Def. 15] But CA was also proved equal to AB; therefore each of the straight lines CA, CB is equal to AB. And things which are equal to the same thing are also equal to one another; [C.N. 1] therefore CA is also equal to CB. Therefore the three straight lines CA, AB, BC are equal to one another.

Therefore the triangle ABC is equilateral; and it has been constructed on the given finite straight line AB. "

The Bodleian D'Orville 301 manuscript, written in Constantinople in 888AD. Opened to the page showing Proposition 1 on the right.