The way I view Euclid's postulates are as follows:A line segment can be made between any two points on surface A.A line segment can be continued in its direction infinitely on surface A.Any line segment can form the diameter of a circle on surface A.The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.The way I view the five postulates is simple. Each postulate defines some quality a surface has.For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).I want to determine what "quality" 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be "categorized" under some combination of postulates, but that is irrelevant.I am merely asking:What two surfaces individually violate the first postulate and violate the fourth postulate?

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The way I view Euclid&#039;s postulates are as follows:A line segment can be made between any two points on surface A.A line segment can be continued in its direction infinitely on surface A.Any line segment can form the diameter of a circle on surface A.The result of an isometry upon a figure containing a right angle preserves the right angle as a right angle on surface A.If a straight line falling on two straight lines make the interior angles on the same side less than 180 degrees in total, the two straight lines will eventually intersect on the side where the sum of the angles is less than l80 degrees.The way I view the five postulates is simple. Each postulate defines some quality a surface has.For instance, 2 seems to define whether a surface is infinite/looped or finite/bounded, 3 seems to force a surface to be circular (or a union of circular subsets), and 5 I believe change the constant curvature of a surface (wether it is 0 or nonzero).I want to determine what &quot;quality&quot; 1 and 4 define in the context of the surface itself. 1 seems to imply discontinuity vs continuity, and I think 4 would imply non-constant curvature. However, I am unsure. Ultimately I would like to assign each of these a quality of a surface that they define such that all surfaces can be &quot;categorized&quot; under some combination of postulates, but that is irrelevant.I am merely asking:What two surfaces individually violate the first postulate and violate the fourth postulate?

zalitiaf · Accepted Answer

(1.) Two parallel planes in Euclidean space.(4.) It&#039;s not clear to me what Euclid means by equality of angles. But it would seem to imply (in any interpretation) that the total angle around any point is equal (to 4 right angles), so it would eliminate the possibility of a cone point on the surface?

The way I view Euclid's postulates are as follows: A line segment can be made between any two p

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