Definition

1. Noun:
   • A mathematical process: In mathematics, quadrature historically refers to the process of determining the area of a geometric figure, especially by constructing a square with an area equal to that of the given figure.
   • A specific angular position: In astronomy, quadrature describes the position of a celestial body (especially the Moon or a planet) when its elongation from the Sun, as seen from Earth, is 90 degrees.

Usage and Examples

• Mathematical Context:

  • The ancient problem of the quadrature of the circle involved trying to construct a square of equal area using only a compass and straightedge.
  • Archimedes made significant contributions to the quadrature of the parabola.

• Astronomical Context:

  • The Moon is said to be in quadrature during its first and last quarter phases.
  • When Mars is in quadrature, it forms a right angle with the Sun and Earth.

Advanced Usage

• "Quadrature of the circle": A famous classical problem in geometry, now known to be impossible to solve under the constraint of using only a compass and an unmarked straightedge.

  • The proof that the quadrature of the circle is impossible was a major achievement in mathematics.

• Numerical Integration: In modern calculus and numerical analysis, the term quadrature is often used synonymously with numerical integration, which is the computation of an integral's approximate value.

  • Gaussian quadrature is a highly efficient method for numerical integration.

Variants and Related Words

• Quadrate (adj., archaic): Square or rectangular; having four equal sides and right angles.
• Quadrature amplitude modulation (QAM) (n): A method in telecommunications that conveys data by modulating the amplitude of two carrier waves, which are out of phase with each other by 90 degrees (in quadrature).

Synonyms

• Squaring (in the mathematical sense).
• Right angle configuration (in the astronomical sense).

Related Phrases and Technical Terms

• Phase quadrature: In signal processing, when two periodic waveforms are offset by one-quarter of a cycle (90 degrees).
  • The two signals were in phase quadrature, which is essential for the demodulation scheme.

Notes on Different Meanings

The mathematical and astronomical meanings are distinct but share the conceptual root of involving a 90-degree relationship—either in constructing an equal square (implying right angles) or in the angular separation between celestial bodies. The term is highly specialized and primarily used in technical, scientific, and historical contexts.

Noun

1. the construction of a square having the same area as some other figure