Abstract
Ancient mathematics was motivated by very practical reasoning. What we now call land management and commerce were the overriding considerations, and calculational questions grew out of those transactions. As a result, many of the ideas considered involved meshing rectangles and triangles, their areas, and their relative proportions. Basic geometry and trigonometry grew out of these largely pragmatic considerations.
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Notes
- 1.
We cannot take the time to describe and discuss Euclid’s definitions. But they are fascinating, and indeed charming. For example, his definition of point is “that which has position but no mass.”
- 2.
“Congruent” means they coincide exactly when superimposed.
- 3.
In this discussion we use corresponding markings to indicate sides or angles that are equal. Thus if two sides are each marked with a single hash mark (or double or triple hash marks), then they are understood to be equal in length. If two angles are each marked with a single hash mark (or double or triple hash marks), then they are understood to have equal measure.
- 4.
Angles are supplementary if their measures add to 180∘.
- 5.
Angles measuring less than 90∘ are acute, while angles measuring more than 90∘, but less than 180∘, are obtuse.
References and Further Reading
Beltrami, E.: Saggio di interpretazione della geometria non-euclidea. Giornale di Mathematiche 6, 285–315 (1868)
Beltrami, E.: Teoria fondamentale degli spazii di curvatura costante. Annali di Matematica Pura ed Applicata 2(2), 232–255 (1868–1869)
Greenberg, M.J.: Euclidean and Non-Euclidean Geometries, 4th edn. W.H. Freeman, New York (2007)
Stahl, S.: The Poincaré Half-Plane: A Gateway to Modern Geometry. Jones & Bartlett Learning, Burlington (1993)
Wallace, E.C., West, S.F.: Roads to Geometry. Prentice-Hall, New York (1992)
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Krantz, S.G., Parks, H.R. (2014). Euclidean and Non-Euclidean Geometries. In: A Mathematical Odyssey. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_6
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