Euclid's element (P14)

Al-Allo · 2014-06-10 13:24:46

Hi,
I was reading proposition 14

http://aleph0.clarku.edu/~djoyce/java/e … opI14.html

of Euclid's elements and there is only one thing which I find weird:

Why do we need postulate 4 to conclude that “the sum of the angles ∠CBA and ∠ABE equals the sum of the angles ∠CBA and ∠ABD.”

Why can't we just use common notion 1? It seems useless to me to use the postulate…

Thank you!

Bob · 2014-06-11 10:37:13

hi Al-Allo

Couldn't get that page.

In the version I have that is how this is proved.

Therefore, since the straight-line AB stands on the straight-line CBE, the (sum of the) angles ABC and ABE is thus equal to two right-angles [Prop. 1.13]. But
(the sum of) ABC and ABD is also equal to two right- angles. Thus, (the sum of angles) CBA and ABE is equal to (the sum of angles) CBA and ABD [C.N. 1]. Let (angle) CBA have been subtracted from both. Thus, the remainder ABE is equal to the remainder ABD [C.N. 3], the lesser to the greater. The very thing is impossible.
Thus, BE is not straight-on with respect to CB.

from:

http://farside.ph.utexas.edu/euclid/Elements.pdf

Just tried that link and it failed. I'll try again tomorrow.

Bob

Last edited by Bob (2014-06-11 10:38:49)

Al-Allo · 2014-06-11 12:28:17

bob bundy wrote:

hi Al-Allo
Couldn't get that page.
In the version I have that is how this is proved.
Therefore, since the straight-line AB stands on the straight-line CBE, the (sum of the) angles ABC and ABE is thus equal to two right-angles [Prop. 1.13]. But
(the sum of) ABC and ABD is also equal to two right- angles. Thus, (the sum of angles) CBA and ABE is equal to (the sum of angles) CBA and ABD [C.N. 1]. Let (angle) CBA have been subtracted from both. Thus, the remainder ABE is equal to the remainder ABD [C.N. 3], the lesser to the greater. The very thing is impossible.
Thus, BE is not straight-on with respect to CB.
from:
http://farside.ph.utexas.edu/euclid/Elements.pdf
Just tried that link and it failed. I'll try again tomorrow.
Bob

Weird, I just tried the link that I had posted and everything seems to work out fine. Anyway, the text you refer to doesn't seem to use nor common notion 1 or postulate 4.

Here's the text :

If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.

With any straight line AB, and at the point B on it, let the two straight lines BC and BD not lying on the same side make the sum of the adjacent angles ABC and ABD equal to two right angles.
I say that BD is in a straight line with CB.

If BD is not in a straight line with BC, then produce BE in a straight line with CB. (I.Post.2)

Since the straight line AB stands on the straight line CBE, therefore the sum of the angles ABC and ABE equals two right angles(I.13). But the sum of the angles ABC and ABD also equals two right angles, therefore the sum of the angles CBA and ABE equals the sum of the angles CBA and ABD.(I.Post.4 and C.N.1)
C.N.3
Subtract the angle CBA from each. Then the remaining angle ABE equals the remaining angle ABD, the less equals the greater, which is impossible.(C.N.3) Therefore BE is not in a straight line with CB.
Similarly we can prove that neither is any other straight line except BD. Therefore CB is in a straight line with BD.
Therefore if with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one another.
Q.E.D.

anonimnystefy · 2014-06-11 12:31:29

Well, I'm guessing it's because without the Postulate 4, you wouldn't actually be able to use Common Notion 1, because you don't know if two right angles is equal to two right angles.

Bob · 2014-06-11 19:54:01

Got yours using Chrome. IE wouldn't do it.

Try this

http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf

How good is your Greek translation ?

Bob

Al-Allo · 2014-06-12 00:23:29

We'll, I don't know how good it is but it has been translated by Heath, so I guess it's correct

Bob · 2014-06-12 01:04:59

As far as I can see your version and mine are identical in English apart from yours says postulate 4 as well. mine has the original Greek too, but I cannot get a decent translation. When I get a moment I'll try to work it out for myself and post back. Watch this space.

Bob

Bob · 2014-06-12 01:35:41

Stefy had this sorted in post 4 and we've both missed it.

the sum of the angles ABC and ABE equals two right angles(I.13). But the sum of the angles ABC and ABD also equals two right angles, therefore the sum of the angles CBA and ABE equals the sum of the angles CBA and ABD.

Post. 4 states that "all right angles are equal to one another". => two 90s = another two 90s

CN 1 states that "Things equal to the same thing are also equal to one another." => ABC + ABE = ABC + ABD

CN 3 states that "if equal things are subtracted from equal things then the remainders are equal". => ABE = ABD

But if E is one side of BD then ABE > ABD and if the other then ABE < ABD. Hence the conclusion.

So CN 1 is not enough on its own.

Bob

Al-Allo · 2014-06-12 02:20:44

Wait, to be sure I'm understanding,

We proved that the sum of two angles = 2*90 ( we don't know if these angles are right angles are not, we only know that they sum to two rights)
We assumed that the sum of two angles= 2*90

Are you saying we must prove that 2*90=2*90 (We must show that 2*90 is the same as 2*90)

So that we can after say : sum of two angles=sum of two angles (Regardless of what each angle is worth.)

So, basically, we're considering 2*90 and 2*90 like two separate ideas that needs to be connected together, right ?

anonimnystefy · 2014-06-12 02:58:27

That's how I'm understanding it.

Al-Allo · 2014-06-12 03:08:39

anonimnystefy wrote:

That's how I'm understanding it.

Well, I find it weird that we need to prove that two rights=two rights...I mean, isn't it obvious that we're talking of the same thing ???

anonimnystefy · 2014-06-12 03:28:30

Then, by the same logic, you would not even need Postulate 4.

Al-Allo · 2014-06-12 03:37:59

anonimnystefy wrote:

Then, by the same logic, you would not even need Postulate 4.

Mmmh. Good point. I think that the reason why I wasn't understanding the use of postulate 4 was because, even if I tried to search, I assumed already that the two right angles were equal to the two other right angles. In other words, I didn't see them as two separate perpendiculars but only as one, which caused me trouble. I'm not sure if you're understanding what I'm saying...

Last edited by Al-Allo (2014-06-12 03:38:28)

Bob · 2014-06-12 03:59:24

Yes, it does seem weird to be stating the obvious. But Euclid was the first (I think) to insist that everything must be stated as an axiom (ie. assumed to be true) or proved from those axioms. He defines his terms carefully, states his axioms (as postulates; another word for the same thing) or as common notions (which I think means axioms that can also be used in other disciplines).

All that follows are careful proofs using those starting points. Even things that seem obvious to us; such as if two lines meet at a point and the sum of their angles with a third point is 180 => the two lines are as one line, even such things must be proved. It is the starting point for all of modern maths; which is one reason for studying the Elements.

It can teach us something else too. Euclid's geometry doesn't actually exist in the 'real' world. There is no such thing as a point with no dimensions or a line with no thickness. If you draw a triangle and measure it's angles and total them, you won't get 180. You might get 179 - 181 if you're very careful. But you cannot really achieve the absolute accuracy of Euclidean geometry. But we are still able to build useful models of reality using his ideas and successfully land a man on the Moon. All maths is like that: we can make a model of the real world and provided we understand the limitations of the model, make use of it in practical situations. Use it badly and you can get silly results. We all know 1 + 1 = 2 but if you add one pile of sand to another pile of sand, how many piles of sand have you got ? Wrong model; that's all.

Bob

Al-Allo · 2014-06-12 04:45:12

bob bundy wrote:

Yes, it does seem weird to be stating the obvious. But Euclid was the first (I think) to insist that everything must be stated as an axiom (ie. assumed to be true) or proved from those axioms. He defines his terms carefully, states his axioms (as postulates; another word for the same thing) or as common notions (which I think means axioms that can also be used in other disciplines).
All that follows are careful proofs using those starting points. Even things that seem obvious to us; such as if two lines meet at a point and the sum of their angles with a third point is 180 => the two lines are as one line, even such things must be proved. It is the starting point for all of modern maths; which is one reason for studying the Elements.
It can teach us something else too. Euclid's geometry doesn't actually exist in the 'real' world. There is no such thing as a point with no dimensions or a line with no thickness. If you draw a triangle and measure it's angles and total them, you won't get 180. You might get 179 - 181 if you're very careful. But you cannot really achieve the absolute accuracy of Euclidean geometry. But we are still able to build useful models of reality using his ideas and successfully land a man on the Moon. All maths is like that: we can make a model of the real world and provided we understand the limitations of the model, make use of it in practical situations. Use it badly and you can get silly results. We all know 1 + 1 = 2 but if you add one pile of sand to another pile of sand, how many piles of sand have you got ? Wrong model; that's all.
Bob

Interesting read, thank you!

Al-Allo · 2014-06-24 12:25:37

Hey, I have another question. I was reading proposition 15 http://aleph0.clarku.edu/~djoyce/java/e … opI15.html and it also uses the same method to prove that 2 * 90 = 2 * 90

Now, I was wondering, shouldn't we use the common notion 2 which says : 2. If equals are added to equals, then the wholes are equal

Because we are dealing with a sum. So, we would have 90=90 and 90=90 by postulate 4 for both cases. After, we would use CN2 to prove that 2*90=2*90

or 90+(90)=90+(90)

What do you think about it ?

anonimnystefy · 2014-06-24 13:02:18

It amounts to the same thing, except with one more rule involved.

Al-Allo · 2014-06-24 13:15:25

Ahok, thank you for your answer

anonimnystefy · 2014-06-24 13:22:03

No problem.

Math Is Fun Forum

#1 2014-06-10 13:24:46

Euclid's element (P14)

#2 2014-06-11 10:37:13

Re: Euclid's element (P14)

#3 2014-06-11 12:28:17

Re: Euclid's element (P14)

#4 2014-06-11 12:31:29

Re: Euclid's element (P14)

#5 2014-06-11 19:54:01

Re: Euclid's element (P14)

#6 2014-06-12 00:23:29

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#7 2014-06-12 01:04:59

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#8 2014-06-12 01:35:41

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#9 2014-06-12 02:20:44

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#10 2014-06-12 02:58:27

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#11 2014-06-12 03:08:39

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#12 2014-06-12 03:28:30

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#13 2014-06-12 03:37:59

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#14 2014-06-12 03:59:24

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#15 2014-06-12 04:45:12

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#16 2014-06-24 12:25:37

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#17 2014-06-24 13:02:18

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#18 2014-06-24 13:15:25

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#19 2014-06-24 13:22:03

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