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 ?
]]>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!
]]>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
]]>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...
]]>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 ???
]]>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 ?
]]>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
]]>Bob
]]>Try this
http://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
How good is your Greek translation ?
Bob
]]>