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The ancient Egyptians and Babylonians discovered abstract Geometry. They developed these ideas that were used to build pyramids and help with reestablishing land boundaries. While, the Babylonians used abstract geometry for measuring, construction buildings, and surveying. Abstract geometry uses postulates, rules, definitions and propositions before and up to the time of the Euclid.
Abstract geometry is deductive reasoning and axiomatic organization. Deductive reasoning deals with statements that have already been accepted. An example of deductive reasoning is proving the sum of the measures of the angles of a quadrilateral is 360 degrees. Another example of deductive reasoning is proving the sum of the angles of a trigon is equal to 180 degrees. From this we get, any quadrilateral can be divided into two trigons. Axioms, which are also called postulates, are statements that can be proved true by using deductive reasoning.
Measurement geometry contains theories that exist and can have supporting ideas to back them up, and cannot be disproved. Hyperbolic geometry and elliptic geometry are two examples of measurement geometry. Non-Euclidean geometry can be considered measurement geometry, since it is a branch in which the fifth postulate of Euclidean Geometry is replaced by one of the two alternative postulates. Mathematicians in the nineteenth century showed that it is possible to create consistent geometries with Euclid's postulates.
An example of the difference in the abstract geometry and the measurement geometry is the sum of the measures of the angles of a trigon. The sum of the measures of the angles of a trigon is 180 degrees in Euclidian geometry, less than 180 in hyperbolic, and more than 180 in elliptic geometry. The area of a trigon in hyperbolic geometry is proportional to the excess of its angle sum over 180 degrees. In Euclidean geometry all trigons have an angle sum of 180 without respect to its area. Which means similar trigons with different areas can exist in Euclidean geometry. It is not possible in hyperbolic or elliptic geometry. In two-dimensional geometries, lines that are perpendicular to the same given line are parallel in abstract geometry, are neither parallel nor intersecting in hyperbolic geometry, and intersect at the pole of the given line in elliptic geometry. The appearance of the lines as straight or curved depends on the postulates for the space.
At the present time mathematics is trying to figure out which of the three is the best representation of the universe.
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Alder, Irving, A New Look at Geometry, John Day Co. (1996)
Bake, H. F., Principles of Geometry, university Press (1922)
Euclid's Fifth Postulate, D.E.Joyce, Clark University