# Theorems Sentence Examples

Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence

**theorems**, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.These

**theorems**, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.If, however, we defined the strength of the source by the statement that the strength divided 1 The beginner is often puzzled by the constant appearance of the factor 47r in electrical

**theorems**. It arises from the manner in which the unit quantity of electricity is defined.Let us apply these

**theorems**to a portion of a tube of electric force.Mensuration involves the use of geometrical

**theorems**, but it is not concerned with problems of geometrical construction.AdvertisementThey are sometimes known as Guldinus's

**Theorems**, but are more properly described as the**Theorems**of Pappus.The

**theorems**are of use, not only for finding the volumes or areas of solids or surfaces of revolution, but also, conversely, for finding centroids or centres of gravity.These

**theorems**may prove useful in preliminary calculations where the pressure-curve is nearly straight; but, in the absence of any observable law, the area of the pressure-curve must be read off by a planimeter, or calculated by Simpson's rule, as an indicator diagram.In 1851 Mr Spottiswoode published in the form of a pamphlet an account of some elementary

**theorems**on the subject.The former of these

**theorems**has been generally accepted and may be taken as proved, but the second has been closely criticized and probably must be modified.AdvertisementEuclid devotes his third book entirely to

**theorems**and problems relating to the circle, and certain lines and angles, which he defines in introducing the propositions.To the former belong the

**theorems**(t), (2), and (3), and to the latter especially the theorem (4), and also, probably, his solution of the two practical problems. We infer, then, [t] that Thales must have known the theorem that the sum of the three angles of a triangle are equal to two right angles.He published a number of these

**theorems**without demonstration as a challenge to contemporary mathematicians.The preceding

**theorems**are purely kinematical.The full working out is in general difficult, the comparatively simple problem of three bodies, for instance, in gravitational astronomy being still unsolved, but some general

**theorems**can be formulated.AdvertisementThe validity of the recilirocal

**theorems**of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional forces of the kind here considered.These

**theorems**are too absolutely stated, and require much modification to adapt them to real life.But the truth of Ricardo's

**theorems**is now by his warmest admirers admitted to be hypothetical only.Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many

**theorems**concerning it.It occupies twenty-four octavo pages, and consists of four

**theorems**and seven problems, some of which are identical with some of the most important propositions of the second and third sections of the first book of the Principia.AdvertisementAnd some years ago I lent out a manuscript containing such

**theorems**; and having since met with some things copied out of it, I have on this occasion made it public, prefixing to it an introduction, and joining a Scholium concerning that method.In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find " da p s la geometrie plane, a chaque theoreme ii en repond necessairement un autre qui s'en deduit en echangeant simplement entre eux les deux mots points et droites; tandis que dans la geometrie de l'espace ce sont les mots points et plans qu'il faut echanger entre eux pour passer d'un theoreme a son correlatif "; and the plan is introduced of printing correlative

**theorems**, opposite to each other, in two columns.In these papers the subject was recast and enriched by new and important

**theorems**. through which the name of Jacobi is indissolubly associated with this branch of science.The general equations expressing the motion of a planet considered as a material particle round a centre of attraction lead to

**theorems**the more interesting of which will now be enunciated.Some valuable but isolated facts and

**theorems**had been previously discovered and proved, but it was he who first clearly grasped the idea of force as a mechanical agent, and extended to the external world the conception of the invariability of the relation between cause and effect.AdvertisementIn his Discorso intorno alle cose the stanno su l'acqua, published in 1612, he used the principle of virtual velocities to demonstrate the more important

**theorems**of hydrostatics, deducing from it the equilibrium of fluid in a siphon, and proved against the Aristotelians that the floating of solid bodies in a liquid depends not upon their form, but upon their specific gravities relative to such liquid.Hesse, "they are, like P. Fermat's

**theorems**, riddles to the present and future generations."Other

**theorems**were published in his Opera Varia, and in John Wallis's Commercium epistolicum (1658).The position assigned to logic by Kant is not, in all probability, one which can be defended; indeed, it is hard to see how Kant himself, in consistency with the critical doctrine of knowledge, could have retained many of the older logical

**theorems**, but the precision with which the position was stated, and the sharpness with which logic was marked off from cognate philosophic disciplines, prepared the way for the more thoughtful treatment of the whole question.According to Mises, all of the categories,

**theorems**, or laws of economics are implied in the action axiom.Obviously the converse is true of Individual B. Two

**Theorems**of welfare economics And now two**theorems**of welfare economics.They commonly treat the historical method with a sort of patronizing toleration as affording useful exemplifications or illustrations of their

**theorems**.Their theorem prover is written in the functional programming language lisp which is also the language in which

**theorems**are represented.Given these, we can prove some useful

**theorems**about the policy.We also implement a proof checker for SPL which derives

**theorems**in the HOL system from SPL proof scripts.Over all geometrical

**theorems**they would be in complete agreement, only interpreting the words in terms of their respective intuitions.We make use of the convolution

**theorems**for the Mellin transforms to produce analytic structures for part of the calculation.Fourteen years later the Academie Frangaise, in ignorance of Smith's work, set the demonstration and completion of Eisenstein's

**theorems**for five squares as the subject of their "Grand Prix des Sciences Mathematiques."The importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for

**theorems**which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers.The

**theorems**on the composition of forces in circular motion with which it concluded formed the true prelude to Newton's Principia, and would alone suffice to establish the claim of Huygens to the highest rank among mechanical inventors.General Arithmetical

**Theorems**. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric

**theorems**, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to algebra, more especially in the theory of equations.The

**theorems**of hydrostatics are thus true for all stationary fluids, however, viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the normality of fluid pressure is taken to hold when the fluid is in movement.The proof of these

**theorems**proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship.If, however, the unit point charge were defined to be that which produces a unit of electric flux through a circumscribing spherical surface or the electric force at distance r defined to be 1/47rr2, many

**theorems**would be enunciated in simpler forms.The general

**theorems**which enabled him to do this, after a start had been made, are A2n = 11A„A ' n (Snell's Cyclom.), P 2A„A' n - 2A' „AZ, Gre o A 2 ” - A n +A2n or A' n +A2„ (g r1') where A „, A'„ are the areas of the inscribed and the circumscribed regular n-gons respectively.Reciprocity

**theorems**relate different states occurring within the same domain.The singularity

**theorems**imply the universe had an infinite density at some time in the past.The course is also a vehicle for the introduction of

**theorems**in vector calculus that have wide application in physics.Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general

**theorems**so far as was required to prove the results for the special case of five squares.As a geometer he is classed by Eudemus, the greatest ancient authority, among those who "have enriched the science with original

**theorems**, and given it a really sound arrangement."In 1709 he entered the university of Glasgow, where he exhibited a decided genius for mathematics, more especially for geometry; it is said that before the end of his sixteenth year he had discovered many of the

**theorems**afterwards published in his Geometria organica.In it Maclaurin developed several

**theorems**due to Newton, and introduced the method of generating conics which bears his name, and showed that many curves of the third and fourth degrees can be described by the intersection of two movable angles.But the desire to obtain general enunciations of

**theorems**without exceptional cases has led mathematicians to employ entities of ever-ascending types of elaboration.All the world, including savages who cannot count beyond five, daily "apply"

**theorems**of number.Given that such observations at the surface of the sea, at intermediate levels and at the bottom are sufficiently numerous and are of a high degree of precision, general conclusions as to the movements of the ocean may be deduced from established

**theorems**in hydrodynamics.His mathematical bent, however, soon diverted him from legal studies, and the perusal of some of his earliest

**theorems**enabled Descartes to predict his future greatness.In the theory of numbers he furnished solutions of many of P. Fermat's

**theorems**, and added some of his own.Besides this most important contribution to the general fabric of dynamical science, we owe to Lagrange several minor

**theorems**of great elegance, - among which may be mentioned his theorem that the kinetic energy imparted by given impulses to a material system under given constraints is a maximum.Astronomy was also enriched by his investigations, and he was led to several remarkable

**theorems**on conics which bear his name.The long-sought cause of the "great inequality" of Jupiter and Saturn was found in the near approach to commensurability of their mean motions; it was demonstrated in two elegant

**theorems**, independently of any except the most general considerations as to mass, that the mutual action of the planets could never largely affect the eccentricities and inclinations of their orbits; and the singular peculiarities detected by him in the Jovian system were expressed in the so-called "laws of Laplace."**Theorems**and formulae are appropriated wholesale without acknowledgment, and a production which may be described as the organized result of a century of patient toil presents itself to the world as the offspring of a single brain.Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical

**theorems**and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to algebra.Fundamental

**theorems**in the theory of equations are to be found in the same work.At the same time, it delights the pure theorist by the simplicity of the logic with which the fundamental

**theorems**may be established, and by the elegance of its mathematical operations, insomuch that hydrostatics may be considered as the Euclidean pure geometry of mechanical science.