Euclid's Definition of an Angle

Textkit is a learning community- introduce yourself here. Use the Open Board to introduce yourself, chat about off-topic issues and get to know each other.
Post Reply
PeterD
Textkit Enthusiast
Posts: 591
Joined: Sat Nov 15, 2003 6:54 pm
Location: Montreal, Canada

Euclid's Definition of an Angle

Post by PeterD » Sat Mar 04, 2006 2:22 am

Hi!

Def. 1.8. A plane angle is the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line.

What would you call two lines that meet at a common point but extend in opposite directions?

A straight line or a straight angle?

Is the term straight angle not an oxymoron?

Thank you.

~PeterD
Fanatical ranting is not just fine because it's eloquent. What if I ranted for the extermination of a people in an eloquent manner, would that make it fine? Rather, ranting, be it fanatical or otherwise, is fine if what is said is true and just. ---PeterD, in reply to IreneY and Annis

edonnelly
Administrator
Posts: 988
Joined: Sun Jan 16, 2005 2:47 am
Location: Music City, USA
Contact:

Re: Euclid's Definition of an Angle

Post by edonnelly » Sat Mar 04, 2006 12:45 pm

PeterD wrote: What would you call two lines that meet at a common point but extend in opposite directions?
Do you mean basically an "angle" of 180 degrees? Some use the term "straight angle" to describe this, but I think Euclid would not accept this as an angle according to his definition (because they do lie in a straight line).

I suspect that those who consider such a situation as an "angle" do it for the sake of completeness, because you can have an angle of 179.9 degrees and one of 180.1. Having 180 be undefined as an angle can cause an ugly discontinuity.
The lists:
G'Oogle and the Internet Pharrchive - 1100 or so free Latin and Greek books.
DownLOEBables - Free books from the Loeb Classical Library

PeterD
Textkit Enthusiast
Posts: 591
Joined: Sat Nov 15, 2003 6:54 pm
Location: Montreal, Canada

Post by PeterD » Sat Mar 04, 2006 8:23 pm

Thanks for replying, edonnelly.

I agree. It does not adhere to Euclid's definition. Yet, every geometry text I've encountered uses this term.

Here is another freaky term I have come across:

"unproved theorem"

(This term ranks up there with "half pregnant".)

Would you believe that the geometry textbook that contains this term is on the list of recommended school textbooks for both Texas and New Hampshire?! Texas I can understand (GWB's state). But the "Live Free or Die" state?

~PeterD
Fanatical ranting is not just fine because it's eloquent. What if I ranted for the extermination of a people in an eloquent manner, would that make it fine? Rather, ranting, be it fanatical or otherwise, is fine if what is said is true and just. ---PeterD, in reply to IreneY and Annis

ThomasGR
Textkit Enthusiast
Posts: 444
Joined: Mon Aug 23, 2004 8:49 pm

Post by ThomasGR » Sat Mar 04, 2006 10:26 pm

I do not see any problem on the "straight" angle. One may imagine easily two lines making an angle and spinning around until they make an angle of exactly 180 degrees. Still this angle and the lines follow Euclid’s definition, though the lines may overlap each other and merge into one line. In this case one cannot build the triangle, not anymore in the Euclid’s geometry, but I think in non-Euclidean geometry this is possible.

About the “unproved theorem?, my opinion is that we have here a problem of the language. It looks there is a paradox, for a theorem cannot be unproved. Today we distinguish between a conjuncture (=unproved theorem) and theorem, but for historical reasons people stick with “unproved theorem?.

edonnelly
Administrator
Posts: 988
Joined: Sun Jan 16, 2005 2:47 am
Location: Music City, USA
Contact:

Post by edonnelly » Sat Mar 04, 2006 10:47 pm

PeterD wrote: "unproved theorem"

(This term ranks up there with "half pregnant".)
I think it might depend upon the context. I know some books present a theorem and then present the proof. They also have a few theorems that are true, and have been proven, but they might require more advanced mathematics than the student has achieved. But they need to use the theorem for other proofs, and so it is presented, and stated to be true, but is unproved (from the student's perspective).

But there's another possibility, too. Technically, isn't a theorem something that is either true or, if unproved, something everyone believes to be true? Fermat's last theorem was one of these when I was in school (though I believe it has since been proven). Also, I don't think there is a proof of Euclid's theorem that given a line and a point not on the line that there is one and only one line through the point that is parallel to the line. Even if unproved this statement would be believed to be true by most people.

Some of this may really just be language and not mathematics, for there are surely unproven statements that are assumed to be true, whether you want to classify these as "theorems" or not seems to be more a question for the English classroom than the mathematics one.

EDIT: ThomasGR beat me on the language thing.
The lists:
G'Oogle and the Internet Pharrchive - 1100 or so free Latin and Greek books.
DownLOEBables - Free books from the Loeb Classical Library

Democritus
Textkit Fan
Posts: 331
Joined: Fri May 07, 2004 12:14 am
Location: California

Post by Democritus » Sun Mar 05, 2006 5:02 pm

PeterD wrote:"unproved theorem"
An "unproved theorem" is the one I left blank on the test because I didn't remember the proof. :)

edonnelly
Administrator
Posts: 988
Joined: Sun Jan 16, 2005 2:47 am
Location: Music City, USA
Contact:

Post by edonnelly » Sun Mar 05, 2006 7:00 pm

Democritus wrote: An "unproved theorem" is the one I left blank on the test because I didn't remember the proof. :)
Does that result in an "unpassed exam?"
The lists:
G'Oogle and the Internet Pharrchive - 1100 or so free Latin and Greek books.
DownLOEBables - Free books from the Loeb Classical Library

PeterD
Textkit Enthusiast
Posts: 591
Joined: Sat Nov 15, 2003 6:54 pm
Location: Montreal, Canada

Post by PeterD » Tue Mar 07, 2006 4:30 am

I e-mailed Mark Solomonovich, Ph D in theoretical and mathematical physics, regarding this. He was very kind to respond:

"It's not an oxymoron since it is a term for a certain type of a geometric figure, which is not an angle in regular sense. The words "straight angle" should be read as a single term, not as "which kind of angle". For example, we say "mountain lion", although it's not a lion that lives in the mountains but a cougar."

I don't know about you guys, but I am satisfied.

If, btw, anyone is interested in geometry/mathematics, check out the professor's site at www.solomonovich.com. The man is dedicated to his profession.

Thanks everyone,

PeterD
Fanatical ranting is not just fine because it's eloquent. What if I ranted for the extermination of a people in an eloquent manner, would that make it fine? Rather, ranting, be it fanatical or otherwise, is fine if what is said is true and just. ---PeterD, in reply to IreneY and Annis

Skylax
Textkit Enthusiast
Posts: 672
Joined: Fri Jun 06, 2003 8:18 am
Location: Belgium

Re: Euclid's Definition of an Angle

Post by Skylax » Tue Mar 07, 2006 7:13 pm

PeterD wrote: A straight line or a straight angle?
In French, they say un angle plat ("flat angle")

Post Reply