De Gua's theorem
Encyclopedia
De Gua's theorem is a three-dimensional analog of the Pythagorean theorem
and named for Jean Paul de Gua de Malves
.
If a tetrahedron
has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces.
The Pythagorean theorem
and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplices
with a right angle
corner.
Jean Paul de Gua de Malves (1713-1785) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Tinseau d'Amondans (1746-1818), as well. However the theorem had been known much earlier to Johann Faulhaber
(1580-1635) and René Descartes
(1596-1650).
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
and named for Jean Paul de Gua de Malves
Jean Paul de Gua de Malves
Jean Paul de Gua de Malves was a French mathematician who published in 1740 a work on analytical geometry in which he applied it, without the aid of differential calculus, to find the tangents, asymptotes, and various singular points of an algebraic curve.He further showed how singular points and...
.
If a tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces.
The Pythagorean theorem
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
and de Gua's theorem are special cases (n = 2, 3) of a general theorem about n-simplices
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
with a right angle
Right angle
In geometry and trigonometry, a right angle is an angle that bisects the angle formed by two halves of a straight line. More precisely, if a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles...
corner.
Jean Paul de Gua de Malves (1713-1785) published the theorem in 1783, but around the same time a slightly more general version was published by another French mathematician, Tinseau d'Amondans (1746-1818), as well. However the theorem had been known much earlier to Johann Faulhaber
Johann Faulhaber
Johann Faulhaber was a German mathematician.Born in Ulm, Faulhaber trained as a weaver and later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen...
(1580-1635) and René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
(1596-1650).