Quadrature (mathematics)

In mathematics, quadrature is a historic term for the computation of areas and is thus used for computation of integrals. The word comes from the Latin quadratus, meaning "square," because for Ancient Greek mathematicians, finding an area meant constructing a square of the same size. This idea is why the term squaring is sometimes used today.

One famous example is the quadrature of the circle, also known as squaring the circle, which was a famous problem that was proven impossible to solve with the tools available to the Ancient Greeks. Later, in the 17th century, integral calculus provided a general way to calculate areas, and the term quadrature began to refer to the calculation of any integral.

Even today, quadrature is still used in numerical analysis to distinguish the calculation of integrals from solving differential equations or differential systems. This helps scientists and mathematicians solve complex problems by breaking them down into simpler parts that can be measured as areas.

History

Greek mathematicians thought about finding the area of shapes by drawing a square with the same size. This is why they called it quadrature. They could find the area of some curved shapes like the lune of Hippocrates and the parabola, but not all shapes, like a circle, using just a compass and straightedge.

Later, mathematicians used new methods to find areas. For example, Archimedes discovered that the area of a sphere’s surface is four times the area of its biggest circle, and that a part of a parabola can be measured using a special triangle. These ideas helped lead to the development of integral calculus, which gives a universal way to calculate areas today.