This is one of the best-written books ever! Most non-fiction works strive (knowingly or not) to reach such a fine form. Words are massaged into terms, sentences into propositions, and certain paragraphs into arguments. This work is the purest form of that, a true paragon with enviable succinctness. Even if you're not into math, try picking up a copy of Euclid's Elements to see how articulate thoughts _can_ be.
Have you read Mortimer J Adler's How to Read a Book? I just finished it a month ago and your mention of terms, propositions and arguments seems straight out of the book's theory of reading :)
Thanks for your review by the way, it motivated me to reread Euclid.
Taking a proofs based Euclidean geometry course was the single most useful class I've ever taken. It totally changes the way you think about mathematics, and even made me a better writer. The way it teaches you to begin with a premise and reach logical conclusions through concrete, connected steps is applicable to all fields of thought. Anyone unfamiliar with Euclid should remedy that immediately.
It's probably not _specific_ to Euclidean geometry, but in the american school system it's common to see proof-based geometry in high school, while you generally don't dwell much on the proofs of calculus unless you take a real analysis class at university. So it's more about timing/exposure than the field itself.
The US high school math curriculum relegates deductive proofs to a course in “geometry”, which typically ends up being a somewhat watered down version of Euclid, with a few extra topics tossed in at the end.
Arguably there are better subjects to use for teaching deductive reasoning, and better formalizations for teaching geometry. This particular approach has a lot of historical inertia though.
> The way it teaches you to begin with a premise and reach logical conclusions through concrete, connected steps is applicable to all fields of thought. Anyone unfamiliar with Euclid should remedy that immediately.
I majored in math and work in software. I feel comfortable with my ability to reach logical conclusions, regardless of the fact that I haven't read the Elements.
And no, proofs courses (I've taken a few) never changed the way I thought about mathematics.
I'm reeeeeally confused with what sort of maths major contains a few proofs courses. Mathematics is proofs upon proofs. If mathematical proofs don't change the way you think about mathematics, I'm really not sure what's left....
Yes, as has been discussed here before, thinking in terms of proofs is a huge part of mathematical maturity, as required by university undergraduate mathematics. See "The Book of Proof" etc.
You're assuming without evidence that I began by not thinking of math in terms of proofs. This is not true; I took pains to give proofs of the basic properties of exponents when asked to help my sister with her algebra.
> I'm reeeeeally confused with what sort of maths major contains a few proofs courses.
I was referring to courses for which the goal was "we'll teach you how to do proofs", not courses which required proofs.
Fair enough, I should have read your comment more charitably — as perhaps you could have of your parent's.
They were being somewhat hyperbolic but the sentiment was more along the lines of "Euclid is approachable for anyone, not just maths people" than "if you haven't read Euclid, you won't understand proofs."
Like your parent, I also took a Euclidean geometry course during undergrad and still cite it as one of my favourites. There's something beautiful about striping mathematics back its foundations and building it back up with just a ruler and a compass.
I've worked through at least the first book of this version and it is a very nice intro to the material.
One large annoyance I have is the mistakes. I was trying to follow the proofs closely and there are some errors both in the diagrams and in the text. Many of these are called out in the intro so I found myself flipping between the corrections section and the pages to make sure I had a very clear idea of each proof. This can make some of the later proofs a bit cumbersome to follow since they all build on one another. So if a later proof is built on proofs that contain mistakes it is distracting and a lot of page flipping.
Just saw this and came on to post the same thing. It's such a lovely book, I got it in hardback last year.
I've actually just switched over to HN from playing a Euclid game I've been totally addicted to this last week. Go on, I dare you.... :-) https://www.euclidea.xyz/game
I spent a year in a "Great Books" college, where there are no textbooks, no lectures, etc., just primary sources progressing roughly chronologically.
For first year math, you go through Elements almost entirely front-to-back. I have mixed feelings about the Great Books programs in general, but the Euclid class was remarkable. It tends not to be a math-heavy group of students, but even those who think they're bad at math can still follow (for the most part) the geometric proofs. It gave all of the students--those of us who were mathematically inclined and those who weren't--a shared vocabulary and methodology for talking about actual mathematics in a way that any of the other math classes I took later after transferring to another university would not have.
If anyone wants to go through them by yourself, do yourself a favor and get the Heath-annotated copies. Those annotations can be a life-saver when you get stuck.
I had exactly the same experience -- one year in a Great Books school (in my case, St. John's) -- and can say that doing the Elements was probably the single most transformative experience of my life until I had a kid. I ended up reconceiving almost everything I thought I knew about my intellectual constitution.
A couple of interesting tidbits about the Elements that I've run into:
It had a profound influence on Abraham Lincoln. He said (he was largely an autodidact) that the Elements taught him what it meant to actually know that something was true, or words to that effect.
It's an example of how profoundly the West is indebted to Arabian (sometimes called "Islamic") scholarship. For centuries, the version that Europeans actually studied was a Latin translation of an Arabic translation that Arab scholars made from the original. They rescued much of our Greek heritage this way.
> Arabian (sometimes called "Islamic") scholarship
"Islamic" seems like a much more accurate descriptor than "Arabian"? For example, looking up al-Khowarizmi, I see that the first sentence of his Wikipedia page is:
> Muḥammad ibn Mūsā al-Khwārizmī (Persian: محمد بن موسی خوارزمی, Arabic: محمد بن موسى الخوارزمی; c. 780 – c. 850), formerly Latinized as Algoritmi, was a Persian (modern Khiva, Uzbekistan) mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad.
Or, looking up ibn Battuta:
> All that is known about Ibn Battuta's life comes from the autobiographical information included in the account of his travels, which records that he was of Berber descent, born into a family of Islamic legal scholars in Tangier, Morocco, on 25 February 1304, during the reign of the Marinid dynasty.
There were a lot more subjects of Arabian empires than there were Arabs.
Why are you mentioning these particular eminent people? Were these the translators of the _Elements_?
al-Khwārizmī, of course, is very important in the history of mathematics. But that Battuta guy doesn't seem relevant. Your comment seems kind of random, but maybe you can help me understand your point.
> Why are you mentioning these particular eminent people?
They were the first two Islamic scholars who came into my head. I actually can't name many more.
I'm not commenting on the Elements; I'm commenting on your weird comment that "Islamic" scholarship would be better referred to as "Arabian". That isn't so.
Well, the translators of the _Elements_ were Arabs, and they translated it into Arabic, so I don't see what the controversy is.
I've noticed a tendency (a rather "weird" tendency, in fact) for people to refer to "Islamic" science, math, and other stuff. Sometimes even when the work in question predates Islam. I don't remember ever hearing people talk about Newton's "Christian" physics. Or describe calculus as "Christian" mathematics. That would be weird, wouldn't it? Even though he was a Christian, living in a Christian society. As was Leibniz. So I think it's weird to hear about "Islamic" mathematics. Maybe someone can explain to me why this is normal.
I agree that it would be unusual for people to refer to "Christian" science etc. However, that's not because it's unusual to want to specify what culture something came from -- it's because the conventional identifier is "Western". You shouldn't have much trouble finding discussions of "Western science".
> the translators of the _Elements_ were Arabs, and they translated it into Arabic
There was a translation into Arabic, yes. Do you actually know that it was done by Arabs? All wikipedia says it that this was done "under Harun al-Rashid", whose court was in Iraq or Syria (and seems to have been run mainly by Persians). The translation would have had to be done by people who understood Greek; this seems less likely for Arabs than for locals.
FWIW, in my [limited] understanding it was a strain of Sunni Islam that became enamored with learning and preserving Greek and Roman scholarship. This backfired, though, and a strain of theology which very intentionally rejected empiricism and logical deduction emerged to dominate Sunni religious scholarship. At that point the preservation and further development of Western thought decreased significantly; the works produced--those which hadn't been burned--wait for discovery by Europeans centuries later.
Which is why today there are, supposedly, marked differences in the way a society like Iran finances human and physical sciences domestically versus, say, Saudi Arabia where financing of science education is vulnerable to the ire of religious leaders who see it as more of a threat than they otherwise might, similar to many strains of Christian theology which perceive a conflict between scientific scholarship, and acceptance and adherence to their religious creed. Individual adherents can remain oblivious to these esoteric distinctions, but they can matter hugely in terms of the development and atrophy of institutional support within a society.
There is a conventional truth that Elements was a standard textbook for thousands of years, but now that society is no longer a slave to clacissism, nobody actually learns their plane geometry that way. And I had just assumed that I had learned mine some other way.
Indeed I had learned bits and pieces from my father and my teachers. But thinking back, I realized that the solid chunk of education I got, where I saw and undestood a good body of proofs, was from reading Elements while on a family holiday in Canberra.
True, that - unfortunately. We are no longer required to learn Latin and Greek or to get familiar with classical writings. Having thus moved away from the "good stuff", we (many of us, anyway) are still slaves of ancient prejudices. What we have lost with classicism and the critical thinking it taught us, we "gained" in falling for obscurantism, pseudo-science, political agenda, and advertisement.
Hmm, while I agree there is a trade-off, I am mostly in favour of the shift away from the classics. It's not that we shouldn't learn from the Greeks, it's just that we can only afford one corner of a modern education for them.
For one thing, languages are a big things that take a long time to learn. For people in most fields, the time spent learning ancient languages can be spent in better ways. Also, classical educations end up placing Plato and Aristotle on a pedestal, and those guys were wrong about most of the things they said (or rather, less right than their modern successors).
Finally, an education can no longer be well-rounded if it only looks at the classical West. Western civilisation has been preeminent since about the 1600's, but if we go back to the axial age, then thinkers from all over Eurasia have to be reckoned with. It's bad enough to distill the Greeks to just Socrates, Aristotle and Plato, but now we have to add Zoroaster, Kongzi, Mengzi, the Buddha, the Upanshads and more just to get a fragmentary and hackneyed overview of classical thought.
How much effort must have gone into the Geometry Applet? It's a shame that it's no longer available. (I get a 403 for http://aleph0.clarku.edu/~djoyce/Geometry/Geometry.html.) But even if it were, I don't think I've had a functioning Java browser plugin in half a dozen years.
This is Heath’s translation from 1908. For anyone who wants a physical copy, I highly recommend Green Lion Press’s version, which is a ridiculously well made book for the price: https://amzn.com/1888009187/ (note: this doesn’t include all of Heath’s critical commentary)
This is actually seriously non-trivial, because Euclid isn't rigorous enough for Coq. For example, you need definition of betweenness and Pasch's axiom at the very least.
Euclids rigor was remarkable for 300 BC, but its not really what we think of as rigorous now. Even the first theorm has several fairly obvious flaws (as the linked commentary points out). So you probably wouldn't have much Euclid left if you tried to make a machine checked version of Euclid.
Many people post-300BC have done more rigorous treatments of plane geometry. Most famously Hilbert, who had (IIRC) 21 axioms.
Euclid was working in a context of classical geometric construction with compass and straightedge.
If you have a line and a point not on the line, you can classically construct a perpendicular through the point. Use a compass centered on the given point, open to an arbitrary radius greater than the distance between the given point and the line. Draw a circle, which will intersect the line at two points A and B. Now draw a second circle centered on A and passing through B, and a third circle centered on B and passing through A. The second and third circles intersect in two points, C and D, one on each side of the given line. Draw a new line through C and D. This line is perpendicular to the original line and forms four right angles with it.
A way to check whether a given angle is right is to pick an arbitrary point on one of the lines and draw a circle centered there and passing through an arbitrary point on the other line. The circle will then pass through a second point on the other line. You can construct the perpendicular bisector of the segment formed by the two points in which the circle intersects the second line, and see whether the bisector is identical to the first line. (Although I'm not sure that's the right classical solution because I don't recall whether "noticing whether two constructed lines are identical" or "noticing whether a constructed line intersects a given point" are allowed tasks in classical construction.)
> [...] Draw a new line through C and D. This line is perpendicular to the original line and forms four right angles with it.
You could define a right angle this way, but that only makes sense if "right angles" were already interesting before you worked out how to construct them. That process is valuable because it produces right angles; right angles aren't valuable by virtue of being the result of that process.
If you believe that, you still need to ask why certain angles were labeled "right", bearing in mind that whatever the reason was, it cannot have implied that two "right" angles were necessarily equal to each other.
It seems like they have tons of meaning in terms of squares, parallel lines, and many parts of triangle geometry, including the Pythagorean theorem and the angle sum in a triangle being two rights angles.
I don't see how right angles are relevant to parallel lines, but sure, they're significant in many places, including all the others you mentioned.
But again, all of those things are built on an existing recognition of right angles. You could define a right angle as "an angle equal to half the angle sum of a triangle", but then "all right angles are equal" would be an easily provable theorem, not a postulate.
On the other hand, there is nothing wrong with leaving the property of an angle being 'right' undefined and postulating that all angles that possess this property are equal.
(The notion of acuteness requires the ability to compare angles; those 'smaller' than the right angle are 'acute'.)
> there is nothing wrong with leaving the property of an angle being 'right' undefined and postulating that all angles that possess this property are equal
Yes there is; it makes the postulate completely useless. Stating "all angles possessing a certain property which cannot be assessed are equal to each other" is useless, identical to declaring "any two equal angles are equal".
Actually -- are there proofs in the Elements which invoke that postulate?
Because then you have no way of identifying right angles. That leaves you with a claim you can make about right angles, and no angles you can ever apply the claim to. That is the same as not having the claim at all.
Consider this modified classic syllogism:
1. All men are mortal.
2. No one can ever know whether any particular object is or isn't a man.
3. Therefore, Socrates is mortal.
See the problem? It may still be possible to prove that Socrates is mortal, but you can't prove that or anything else with the postulate "all men are mortal", because that postulate, by hypothesis, doesn't apply to anything.
Of course, a theory that only includes the "equality axiom" would be incomplete, there would have to be other axioms that reference the (otherwise undefined) notion of the 'right angle'. Then it will be possible to prove theorems that involve this notion. For example, it will be possible to prove that "the particular object" that lies between the shorter sides of a triangle with side lengths three, four, and five, is, in fact, a right angle!
Until you have a definition of right angles, it is not possible to prove that anything, including the angle opposite the 5 side of a 3,4,5 triangle, is a right angle. Really, that's what a definition is.
It isn't necessary for the statement "all right angles are equal" to include a definition of right angles. But for the statement to have any meaning at all, the logical system that the statement is made in must include a definition of right angles. They cannot be left undefined.
What's your source? "Eu" is a very common Greek dipthong typically take as "yu",for the traditional Erasmian pronunciation, or "Ev" for modern reconstructed Greek.
