Two postulates in Euclid's Elements embody the ideal conception of the straightedge and compass inherited from Plato's realm of Ideas. The Alexandrian scholar built much of plane geometry—and the constructions he bequeathed to us—on these two postulates.
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The so-called inscribed angle theorem, still part of every middle-school student's mathematical toolkit today, was already known to ancient geometers. It can sometimes have unusual applications.
Geometric constructions are central to reasoning in Euclid's The Elements. The various methods of teaching geometry, right up to the present day, claim to follow this approach.
Some profound results, such as Pythagoras' theorem, predate any awareness of mathematics as a science. Rather, once theorized and proved, these results gave rise to this science.
Euclid's fifth postulate differs from the others: it seems provable. Yet its negation leads to other geometries, known as non-Euclidean geometries. They have their place within mathematics and have applications both in arithmetic, as Poincaré showed, and in general relativity.
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