An Etymological Dictionary of Astronomy and Astrophysics
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Of or pertaining to Euclid, or his postulates. → Euclidean division, → Euclidean geometry, → Euclidean space, → non-Euclidean geometry.

After the Gk. geometrician and educator at Alexandria, around 300 B.C., who applied the deductive principles of logic to geometry, thereby deriving statements from clearly defined axioms.

In arithmetic, the conventional process of division of two → integers. For a → real number a divided by b > 0, there exists a unique integer q and a real number r, 0 ≤ r <b, such that a = qb + r.

→ Euclidean; → division.

The geometry based on the postulates or descriptions of Euclid. One of the critical assumptions of the Euclidean geometry is given in his fifth postulate: through a point not on a line, one and only one line be drawn parallel to the given line. See also → non-Euclidean geometry.

→ Euclidean; → geometry.

A space in which the → distance between any two points is given by the → Pythagorean theorem: d² = (Δx)² + (Δy)² + (Δz)², where d is distance and Δx, Δy, and Δz are differential → Cartesian coordinates. Euclidean n-space Rⁿ is the set of all column vectors with n real entries.

→ Euclidean; → space.

Any of several geometries which do not follow the postulates and results of Euclidean geometry. For example, in a non-Euclidean geometry through a point several lines can be drawn parallel to another line. Or, the sum of the interior angles of a triangle differs from 180 degrees. According to Einstein's general relativity theory, gravity distorts space into a non-Euclidean geometry.

→ non-; → Euclidean geometry.

A real vector space of dimension n having a symmetric bilinear form (x, y) such that in some basis e₁, ..., e_n, the quadratic form (x²) takes the form x₁² + ... + x_{n - 1}² - x_n². Such bases are called orthonormal.

→ pseudo-; → Euclidean; → space.