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....
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.
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)
Readit News
Thanks for your review by the way, it motivated me to reread Euclid.
If you pick up a modern calculus book, one advantage over Euclid is that the proofs will actually be correct ;)
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.
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.
https://smile.amazon.com/gp/product/3836559382
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.
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
The text includes a lexicon which is surprisingly short.
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.
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.
"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.
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.
It's not like there was anything else to do.
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.
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.