Rule a straight line from the centre of a circle so that it intersects and continues beyond the circumference. All four angles are right angles, but the ones outside the perimeter are not equal to the ones inside. Superposition will show that they're bigger, (if you can't see that at a glance).
Euclidean geometers however only recognise angles between straight lines.The concept of an angle between a straight line and a curve is effectively disallowed by redefining it as between the straight line and the straight line tangent to the curve. Only then can you say all the right angles are equal. An arbitrary ruling which if dispensed with might mark the long awaited departure from Euclid. (Those three right angles of a triangle bounded by great circles on a sphere are in fact angles between planar sections).
Note the axioms or postulates as set out in the Scientific American link. The first three are in the imperative grammatical form, as maybe the controversial 4th and 5th should be. They're not everlasting truths, or even meant to be, but instructions on how to do a certain kind of geometry.
Let me turn to Common Notion 1 of Euclid: "Things which are equal to the same thing are equal to each other". Ever been in an office where no-one can ever find the master for re-photocopying? You keep photocopying photocopies of photocopies and end up with a mess. Same with continually measuring off a length of say rope against the previous one you've measured off. Drift or error is bound to set in after a while if you don't do it against one template length. Practical instruction again, rather than inert truism.
Not all right angles are equal.
Rule a straight line from the centre of a circle so that it intersects and continues beyond the circumference. All four angles are right angles, but the ones outside the perimeter are not equal to the ones inside. Superposition will show that they're bigger, (if you can't see that at a glance).
Euclidean geometers however only recognise angles between straight lines.The concept of an angle between a straight line and a curve is effectively disallowed by redefining it as between the straight line and the straight line tangent to the curve. Only then can you say all the right angles are equal. An arbitrary ruling which if dispensed with might mark the long awaited departure from Euclid. (Those three right angles of a triangle bounded by great circles on a sphere are in fact angles between planar sections).
Note the axioms or postulates as set out in the Scientific American link. The first three are in the imperative grammatical form, as maybe the controversial 4th and 5th should be. They're not everlasting truths, or even meant to be, but instructions on how to do a certain kind of geometry.
Let me turn to Common Notion 1 of Euclid: "Things which are equal to the same thing are equal to each other". Ever been in an office where no-one can ever find the master for re-photocopying? You keep photocopying photocopies of photocopies and end up with a mess. Same with continually measuring off a length of say rope against the previous one you've measured off. Drift or error is bound to set in after a while if you don't do it against one template length. Practical instruction again, rather than inert truism.