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Among other precious items they preserved are some results and the general approach of Pythagoras (Pythagoreans convinced themselves that all things are, or owe their relationships to, numbers.

The doctrine gave mathematics supreme importance in the investigation and understanding of the world.

A right triangle made at random, however, is very unlikely to have all its sides measurable by the same unit—that is, every side a whole-number multiple of some common unit of measurement.

By the late 19th century the hegemony of Euclidean geometry had been challenged by non-Euclidean geometry and projective geometry.

The first notable attempt to reorganize the study of geometry was made by the German mathematician Felix Klein and published at Erlangen in 1872. Analytic geometry was initiated by the French mathematician René Descartes (1596–1650), who introduced rectangular coordinates to locate points and to enable lines and curves to be represented with algebraic equations.

The German mathematician Carl Friedrich Gauss (1777–1855), in connection with practical problems of surveying and geodesy, initiated the field of differential geometry.

Using differential calculus, he characterized the intrinsic properties of curves and surfaces.

One way that they could have employed a rope to construct right triangles was to mark a looped rope with knots so that, when held at the knots and pulled tight, the rope must form a right triangle.

The simplest way to perform the trick is to take a rope that is 12 units long, make a knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop, as shown in the animation.

Ancient builders and surveyors needed to be able to construct right angles in the field on demand.

The method employed by the Egyptians earned them the name “rope pullers” in Greece, apparently because they employed a rope for laying out their construction guidelines.

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