EUCLID 13 Books of The Elements, what is this?

[HSKLNG]

What exactly it is and How would you use it?

 

I only read that is related to Geometry, but Is it Geometry? What kind of Geometry?

 

Has anyone used it on these boards? How did you work it out?

 

Having some Geometry troubles and looking for soothing supplements.

 

Thanks for your time and inputs.:o

[Tina in Ouray]

This is the real thing, the original ancient text on geometry.

 

It's a very cool book, but it probably isn't a soothing supplement for geometry woes.

 

What text are you currently having trouble with?

 

Tina in Ouray, CO

 

 

http://www.greenlion.com/cgi-bin/SoftCart.100.exe/euclid.html?E+scstore

 

http://www.greenlion.com/Eu-I-1-7.pdf (sample pages)

[HSKLNG]

...a curriculum to learn Geometry or not?:confused:

 

What is it? How do you use it?

 

Tina, I saw the link, read the info, open the files and I still wonder how do you use these books (or even the all in one book)?:(

[KAR120C]

...a curriculum to learn Geometry or not?:confused:

 

What is it? How do you use it?

It's not really a curriculum in the way we usually think of it -- no lesson plans and teacher helps and homework and tests... It's more like a book of geometry proofs that, if one were to read through and understand them all, one could learn both geometry and the logic of proof from.

 

I actually like it as a geometry text, but not because it's "soothing" as much as because it's not ;) We have a second book by Benno Artmann that serves as a kind of guide to Euclid and interprets some of the complicated parts and makes comments on where geometry has gone since Euclid, etc. but again it's not really how one would think of a curriculum. The way we use it is to read a proposition or a group of related propositions (Artmann has them divided up rather neatly) and then actually try the constructions with a compass and straightedge and discuss why they work they way they work and the significance of each step.

 

Some of them are really obvious and some take a lot of thought. There is no extra explanation of why any one of them works beyond the plain instructions in Euclid and some commentary in Artmann -- nothing to help a struggling student. But that's why I like it myself... because we're in a situation where letting DS struggle with it is exactly what I want. I don't think I'd be as enthusiastic if we were under any time constraints or if I were teaching a child who had trouble with a more straightforward text.

 

If you're looking for something that would be helpful and not too overwhelming, I really like Jacobs Geometry. I've only used the 2nd edition (said to be "proofier" although I've not really looked at the 3rd to compare), and it's written in a very friendly and accessible way with good, challenging problems and nothing too frustrating.

 

Geometry can be a tricky subject for some kids (and parents!) because it is so much less math and so much more philosophy/ logic... I always loved it myself, but my twin sister (who went on to get a math degree and work as a programmer -- no lack of math or logic skills there!) found it overwhelming, and twenty years later still thinks she can't do it. I think if you can communicate the point that you're starting with a situation where there can be only one answer and then managing to find that answer and prove that it is the only one, you're "there". It's not about memorizing the various postulates and theorems, but understanding the process of proof.

 

Clear as mud? ;)

[Plaid Dad]

...a curriculum to learn Geometry or not?:confused:

 

What is it? How do you use it?

 

Tina, I saw the link, read the info, open the files and I still wonder how do you use these books (or even the all in one book)?:(

 

Here is a short article on Euclid that explains who he was. The Elements was originally written as a textbook for geometry and some still use it as such, but it is no more a "math curriculum" in the modern sense than Aristotle's Rhetoric is a writing curriculum. It's a Great Book that can be used to teach a particular subject.

 

Some classical schools and colleges use Euclid's Elements as the basis of their geometry courses. They have students read the book and work out the proofs that Euclid discusses. The teacher needs to have a very thorough knowledge of geometry to do this effectively, imo. There are also modern textbooks that follow Euclid more or less closely. HTH!

[Myrtle]

...a curriculum to learn Geometry or not?:confused:

 

What is it? How do you use it?

 

Tina, I saw the link, read the info, open the files and I still wonder how do you use these books (or even the all in one book)?:(

 

 

In the 1800s the teacher would begin by having the students memorize word by word all the definitions and axioms at the beginning of each book. They would then proceed on to the proofs and the kids would have to prove the question at hand. In other words, they would try to, on their own, come up with all the steps. If they couldn't do it they would ultimately memorize the steps and move on to the next proof. Your first problem would be, "to describe an equilateral triangle upon a given finite straight line." You will have to come up with a way of doing this as well as come up with a proof, or a chain of syllogisms, using the things that you "know" (given axioms and definitions) and show how your conclusion follows from those first principles.

 

 

If anyone is interested in how math was taught in the US during the 1800s, there is a very good book online that you can read by Florian Cajori in which he gives anecdotes of students in college courses and summarizes what was happening mathwise and how it was taught. Anyone looking fondly back on some Golden Age of Math education will not find it here! (I did just a blog entry on this).

 

If you are interested in what texts were used in high schools, especially classical education tracks, check out The Rise of the High School in Massachusetts. Some of the texts which they list out are found in Google Books.

 

The advantages of teaching geometry are that it allows the student to work within an axiomatic system and acquire knowledge through justification of their reasoning. The key piece to this training of the mind lies in the parsimonious set of axioms that are used which are slowly built up into proving quite sophisticated propositions. Today there are many modern geometry books, some do not teach proof at all and some claim to teach proofs, but give you an "infinitude of axioms" to work with which some might argue can create difficulties for the student.

[Charon]

...a curriculum to learn Geometry or not?:confused:

 

What is it? How do you use it?

 

Tina, I saw the link, read the info, open the files and I still wonder how do you use these books (or even the all in one book)?:(

 

I doubt the Elements were really meant as a textbook. They are more like a compilation of the mathematical knowledge of Euclid's day. All of the Elements together cover a lot more than just geometry. The high school course you would be interested in is the geometry of a plane. And, I think the main interest there should be to start from a few axioms (true/false statements that you can assume without havign to justify) and eventually work up to proving the Pythagorean Theorem. That's probably the best way to characterize it right there. (Of course, if you get through all of that then there is still plenty of material to take out of the elements to fill up a year or even all of high school.)

 

Mainly, I wanted to just point out, though, that Euclid wasn't some high school teacher from Ancient Greece. His Elements includes material that is taught to graduate math students right now today, open problems in mathematics, for that matter, and has been a source of inspiration and the very model of doing math for all sorts of great mathematicians in history. It's not just the geometry of the plane but also solid geometry, number theory, geometric sequences and series, the method of exhaustion (which is proto-calculus),....

 

I have come across people in my many travels that seem to think that Euclid is crap or something. Nothing could be further from the truth. And, incidentally, just to put it a little more in perspective, Euclid is probably largely an outgrowth of Platonist geometry (in case you might wonder how these things are all connected). In fact, if, say, you are totally into calculus (as some of the people I have seen disparage Euclid seem to be), then that Method of Exhaustion or, more particularly, what has now come to be called the Archimedean Property of the real numbers, was first articulated by one of Plato's students, Eudoxus. That Archimedean Property is probably one of the first and most fundamental things a student learns when they do calculus rigorously (i.e. correctly and usually in a much more advanced class called "Real Analysis").

 

That's probably a lot to digest if you didn't know of the Elements to begin with. I guess I'm just saying that this particular "rabbit hole" is infinitely deep, arguably deeper than any other one in all of academia or education. It doesn't just amount to some old high school class that is almost not even taught anymore.