# theorem

## Definitions

### from The American Heritage® Dictionary of the English Language, 4th Edition

• n. An idea that has been demonstrated as true or is assumed to be so demonstrable.
• n. Mathematics A proposition that has been or is to be proved on the basis of explicit assumptions.

• n. A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. Theorems which are not very interesting in themselves but are an essential part of a bigger theorem's proof are called lemmas
• n. A mathematical statement that is expected to be true; as, Fermat's Last Theorem (as which it was known long before it was proved in the 1990s.)
• n. a syntactically correct expression that is deducible from the given axioms of a deductive system
• v. to formulate into a theorem

### from the GNU version of the Collaborative International Dictionary of English

• n. That which is considered and established as a principle; hence, sometimes, a rule.
• n. A statement of a principle to be demonstrated.
• transitive v. To formulate into a theorem.

### from The Century Dictionary and Cyclopedia

• To reduce to or formulate as a theorem.
• n. A universal demonstrable proposition.
• n. In geometry, a demonstrable theoretical proposition.
• n. The proposition that the velocity of a liquid flowing from a reservoir is equal to what it would have if it were to fall freely from the level in the reservoir; or, more generally, if p is the pressure, p the density, V the potential of the forces, q the resultant velocity, A a certain quantity constant along a streamline, then
• n. given by Daniel Bernoulli (1700–82) in 1738.
• n. The generalized multiplication theorem of determinants (1812).
• n. given by the eminent English mathematician George Boole (1815–64).
• n. The proposition that in the impact of inelastic bodies vis viva is always lost.
• n. The proposition that in explosions vis viva is always gained. These theorems are all due to the eminent mathematician General L. N. M. Carnot (1753–1823), who published in 1803 and and in 1786.
• n. The proposition that the ratio of the maximum mechanical effect to the whole heat expended in an expansive engine is a function solely of the two temperatures at which the heat is received and emitted: given in 1824 by Sadi Carnot (1796–1832): often called Carnot's principle.
• n. given by John Casey in 1866.
• n. The proposition that if the order of a group is divisible by a prime number, then it contains a group of the order of that prime. The extension of this—that if the order of a group is divisible by a power of a prime, it contains a group whose order is that power—is called Cauchy and Sylow's theorem, or simply Sylow's theorem, because proved by the Norwegian L. Sylow in 1872.
• n. The rule for the development of determinants according to binary products of a row and a column.
• n. The false proposition that the sum of a convergent series whose terms are all continuous functions of a variable is itself continuous.
• n. Certain other theorems are often referred to as Cauchy's, with or without further specification. All these propositions are due to the extraordinary French analyst, Baron A. L. Cauchy (1789–1857).
• n. given by E. Cesaro in 1885. It is an extension of Ceva's theorem.
• n. given by B. P. E. Clapeyron (1799–1868): otherwise called the theorem of three moments.
• n. given by L. Crocchi in 1880.
• n. given by Morgan W. Crofton in 1868. Certain symbolic expansions and a proposition in least squares are also so termed.
• n. Same as De Moivre's property of the circle (which see, under circle).
• n. A certain proposition in probabilities. All these are by Abraham De Moivre (1667–1754).
• n. The proposition that if two triangles ABC and A′ B′ C′ are so placed that the three straight lines through corresponding vertices meet in a point, then also the three points of intersection of corresponding sides (produced if necessary) lie in one straight line, and conversely. Both were discovered by Gérard Desargues (1593–1662).
• n. named from G. Dostor, by whom it was given in 1870. Certain corollaries from this in regard to the ellipse and hyperbola are also known as Dostor's theorems.
• n. so that in a synclastic surface ρ1 and ρ2 are the maximum and minimum radii of curvature, but in an anticlastic surface, where they have opposite signs, they are the two minima radii.
• n. The proposition that in every polyhedron (but it is not true for one which enwraps the center more than once) the number of edges increased by two equals the sum of the numbers of faces and of summits.
• n. One of a variety of theorems sometimes referred to as Euler's, with or without further specification; as, the theorem that (xd/dx + yd/dy)r f(x, y) = n f(x, y); the theorem, relating to the circle, called by Euler and others Fermat's geometrical theorem; the theorem on the law of formation of the approximations to a continued fraction; the theorem of the 2, 4, 8, and 16 squares; the theorem relating to the decomposition of a number into four positive cubes. All the above (except that of Fermat) are due to Leonhard Euler (1707–831.
• n. One of a number of arithmetical propositions which Fermat, owing to pressure of circumstances, could only jot down upon the margin of books or elsewhere, and the proofs of which remained unknown for the most part during two centuries, and which are still only partially understood—especially the following, called the last theorem of Fermat: the equation x + y = z, where n is an odd prime, has no solution in integers.
• n. The proposition that, if from the extremities A and B of the diameter of a circle lines AD and BE be drawn at right angles to the diameter, on the same side of it, each equal to the straight line AI or BI from A or B to the middle point of the are of the semicircle, and if through any point C in the circumference, on either side of the diameter AB, lines DCF, ECG be drawn from D and E to cut AB (produced if necessary) in F and G, then AG + BF = AB: distinguished as Fermat's geometrical theorem. This is shown in the figure by arcs from A as a center through G and from B as a center through F meeting at H on the circle.
• n. The proposition that light travels along the quickest path.
• n. given in 1820 by Sir J. F. W. Herschel (1792–1872).
• n. The proposition that forced vibrations follow the period of the exciting cause.
• n. given by the Rev. Hamnet Holditch (born 1800).
• n. The proposition that an equilibrium ellipsoid may have three unequal axes.
• n. One of a variety of other propositions relating to the transformation of Laplace's equation, to the partial determinants of an adjunct system, to infinite series whose exponents are contained in two quadratic forms, to Hamilton's equations, to distance-correspondences for quadric surfaces, etc. All are named from their author, K. G. J. Jacobi (1804–51).
• n. The proposition that the order of a group is divisible by that of every group it contains: also called the fundamental theorem of substitutions. Both by, J. L. Lagrange (1736–1813).
• n. A proposition relating to the apparent curvature of the geocentric path of a comet. Both are named from their author, J. H. Lambert (1728–77).
• n. where the modulus of x is comprised between R and R′ : given by P. A. Laurent (1813–54).
• n. is equal to the same after development of (Du + Dv) by the binomial theorem, where Du denotes differentiation as if u were constant, and Dv differentiation as if u were constant.
• n. given by S. A. J. Lhuilier (1750–1840).
• n. a monodromic function fz can always be found having for critical points α0, α1, … αn, etc., and such that
• n. φn being a function for which αn is not a critical point: given by G. Mittag-Leffler.
• n. The proposition that the three diagonals of a quadrilateral circumscribed about a circle are all bisected by one diameter of the circle.
• n. One of the two propositions that the surface of a solid of revolution is equal to the product of the perimeter of the generating plane figure by the length of the path described by the center of gravity, and that the volume of such a solid is equal to the area of the plane figure multiplied by the same length of path. Various other theorems contained in the collection of the Greek mathematician Pappus, of the third century, are sometimes called by his name.
• n. A certain proposition concerning uniform functions connected by an algebraic relation.
• n. The proposition that a quantity of the form R = √u + v cannot differ from αu + βv by more than R tan ½ε where α = cos (θ + ε)/cos ½ε, β = sin (θ + ε)/cos ½ε, ε = ½(Θθ), tan Θ ⟩ u/v ⟩ tan θ. Both were given by General J. V. Poncelet (1788–l877).
• n. The proposition that if a point be taken on each of the edges of any tetrahedron and a sphere be described through each vertex and the points assumed on the three adjacent edges, the four spheres will meet in a point: given by Samuel Roberts in 1881.
• n. where α, β, etc., are all the prime numbers one greater than the double of divisors of n: given in 1840 by K. G. C. von Staudt (1798–1867).
• n. given by James Stirling (1696–1770).
• n. The proposition that every quaternary cubic is the sum of the cubes of five linear forms.
• n. The proposition that if λ1, λ2, etc., are the latent roots of a matrix m, then
• n. given by the great algebraist J. J. Sylvester (born 1814).
• n. given by H. M. L. Tanner in 1879.
• n. where d represents the differential of the function u.
• n. named after the discoverer, John Wallis (1616–1703).
• n. where v is the velocity, r the radius vector of the point whose mass is m and its coördinates x, y, z, while X, Y, Z are the components of the force, f the force, and ⟩ the distance of two particles: given in 1872 by A. J. F. Yvon-Villarceau (1813–83). It much resembles the theorem of the virial.
• n. Synonyms See inference.

• n. an idea accepted as a demonstrable truth
• n. a proposition deducible from basic postulates

## Etymologies

Late Latin theōrēma, from Greek, from theōrein, to look at, from theōros, spectator; see theory.
(American Heritage® Dictionary of the English Language, Fourth Edition)
Via Late Latin theōrēma, from Ancient Greek θεώρημα (theōrēma, "speculation, proposition to be proved") (Euclid), from θεωρέω (theōreō, "I look at, view, consider, examine"), from θεωρός (theōros, "spectator"), from θέα (thea, "a view") + ὁράω (oraō, "I see, look"). See also theory, and theater. (Wiktionary)

## Examples

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