
Euclid's Elements (1997) - nilsocket
http://aleph0.clarku.edu/~djoyce/elements/toc.html
======
wyc
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.

~~~
elzr
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.

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aphextron
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.

~~~
umanwizard
Why is this specific to Euclidean geometry as opposed to any other
mathematical writing?

If you pick up a modern calculus book, one advantage over Euclid is that the
proofs will actually be correct ;)

~~~
claytonjy
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.

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kqr2
For a nicely illustrated and colored text of Euclid's Elements see: _Byrne:
Six Books of Euclid_ :

[https://smile.amazon.com/gp/product/3836559382](https://smile.amazon.com/gp/product/3836559382)

~~~
reggieband
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.

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davidmr
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.

~~~
gglitch
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.

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leephillips
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.

~~~
thaumasiotes
> 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.

~~~
leephillips
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.

~~~
thaumasiotes
> 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.

~~~
leephillips
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.

~~~
thaumasiotes
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.

~~~
wahern
Both of you make persuasive arguments.

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.

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adrianratnapala
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.

It's not like there was anything else to do.

~~~
Koshkin
> _society is no longer a slave to classicism_

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.

~~~
adrianratnapala
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.

~~~
Koshkin
Still, Aristotle (or Kongzi) would be much more preferable to some of the old
books that are almost universally studied today.

------
sevensor
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.](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.

~~~
JadeNB
A lot of the applets are still available on the pages for individual
constructions. (I'm using it in a class right now.)

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nathell
Shouldn't the (1997) in the submission title read (300 BC)?

~~~
adamnemecek
It would have to be in ancient Greek.

~~~
jacobolus
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/](https://amzn.com/1888009187/) (note: this
doesn’t include all of Heath’s critical commentary)

------
jackfoxy
Excellent! I have Heath's edition and read through it 20 years ago. Prompted
me to take it off the shelf and bookmarking this site.

------
mjfl
would it be possible to rewrite Euclid's Elements with a proof assistent like
Coq?

~~~
thaumasiotes
We don't even understand some of the things Euclid was trying to say.

One of his postulates is "all right angles are equal". How do you recognize
two angles as "right" other than by measuring their equality?

~~~
schoen
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.)

~~~
thaumasiotes
> [...] 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.

~~~
schoen
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.

~~~
thaumasiotes
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.

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sillysaurus3
A lot of people don't know this, but in the same way Euler is pronounced
Oiler, Euclid is pronounced Oiklid.

~~~
jlos
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.

~~~
sillysaurus3
Just a little joke. It's nice for trolling your friends though. They go the
whole day thinking it's pronounced Oiklidian geometry.

