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Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within the circle ABCD and outside ACK, which is absurd.
Therefore a circle does not touch a circle externally at more points than"

There's only one little detail which i'm not sure of. We are trying to prove that circles which touch one another will only touch at one point. Fine. I understood the first part which treats of a circle in another one. It's only the case where one circle touches another one from the outside. By using proposition 2 of book 3, we prove that the line AC will be inside both of circles since the two points are on each circumference of the two circles. Now, this is where I get lost. We say that "but it (line AC) fell within the circle ABCD and outside ACK" and we prove this by using definition 3 of book 3 (Circles are said to touch one another which meet one another but do not cut one another.) In other words, this definition says that circles which touch another do not cut one another. In our situation, we have two circles which touch one another and are not supposed to cut one another. This is where I don't understand, how does this justify this : "but it fell within the circle ABCD and outside ACK." How do we get that conclusion from the definition?

Thank you!

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII13.html

Last edited by Al-Allo (2015-06-16 14:11:37)

Bob

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Registered: 2010-06-20

Posts: 10,619

Re: Euclid's elements proposition 13 book 3

hi Al-Allo

I had hoped that there would be a definition somewhere to explain the word 'cuts'.  I searched all the way through books 1, 2 and 3 but Euclid seems to use the term without definition.  I suppose I would reason like this: 

When object A  cuts object B, there is a region that they share (the overlap) and a region that they do not share (what remains after the overlap is subtracted). If A does not cut B, then there is no overlap. If a line is in the overlap, then it is in both A and B.  If A does not cut B, then any line that is fully in A, cannot be in B.

Given how careful Euclid is about other proofs, I'm a bit surprised that this is omitted.  But all mathematical theories have this problem:  the writer may define some words, but cannot define all (for the words used in the definition would then have to be defined, and then those words,  ............).  He appears to take the meaning of 'cut' as obvious.  Wouldn't it be great if we could ask him. 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Euclid's elements proposition 13 book 3

hi Al-Allo

I had hoped that there would be a definition somewhere to explain the word 'cuts'.  I searched all the way through books 1, 2 and 3 but Euclid seems to use the term without definition.  I suppose I would reason like this: 

When object A  cuts object B, there is a region that they share (the overlap) and a region that they do not share (what remains after the overlap is subtracted). If A does not cut B, then there is no overlap. If a line is in the overlap, then it is in both A and B.  If A does not cut B, then any line that is fully in A, cannot be in B.

Given how careful Euclid is about other proofs, I'm a bit surprised that this is omitted.  But all mathematical theories have this problem:  the writer may define some words, but cannot define all (for the words used in the definition would then have to be defined, and then those words,  ............).  He appears to take the meaning of 'cut' as obvious.  Wouldn't it be great if we could ask him. 

Bob

So we are assuming that there exists two cases where the circles are crossing each other and where they aren’t crossing each others to be able to do the proof ? Also, the case where they don’t cross each other, is this where definition 3 comes in ? In other words, do we prove that “ If A does not cut B” by using definition 3 ? Just want to make sure I understand the use of definition 3. Also, to be able to do the proof, I assume that it is highly important that we prove first that both lines are in each circle so that when we use definition 3, we can say that a circle that don’t cut another one then have for consequence that the line can only be located in one of the circles and not in both of them. (Since we said earlier that they had to be in the two of them) Finally, the part where “then any line that is fully in A, cannot be in B.” is something that we assume. Shouldn’t it be proved or something ? Oh, I forgot another thing, are you sure your interpretation is a good one or maybe Euclid was thinking something else ?  I just feel bad when I skip something that I’m not sure of understanding. (It doesn’t quit my mind lol)
By the way, the link says that “There are logical flaws in this proof similar to those in the last two proofs.” I even found a document with comments by Heath if you want it. (The one who translated the elements)

Last edited by Al-Allo (2015-06-17 04:25:43)