A Conversation for Curved Space and the Fate of the Universe

Is #4 needed?

Post 1

Apollyon - Grammar Fascist

Why is # stated as a postulate? One would expect all right angles to be equal by definition, since a right angle is defined as being one of 90 degrees. Can you have a right angle of 89deg? Exactly.

Wouldn't it have been equally valid to state #4 as 'All 37deg angles are equal to each other' or 'all angles of e radians are equal to each other'?

Is #4 needed?

Post 2

Gnomon - time to move on

You're assuming that angles work the way that they intuitively should work. You assume that there's the same "amount of angle" around any point (360 degrees) no matter where the point is. It's a reasonable assumption, so Euclid just lists it explicitly. But instead of talking in terms of number of degrees, he starts with what he considers the most basic angle, the right angle.

He then goes on to show that if right angles are all equal, then other portions of angles are also equal, so it is valid to divide up the right angle into degrees.

Euclid incidentally defines a right angle as that angle such that two of them togeher make a straight line.

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