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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.

11As 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."

11Under 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.

00So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt dz and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, (udx + vdy + wdz) = Du dx +u 1+..

00These 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.

00If, 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.

00Let us apply these theorems to a portion of a tube of electric force.

00Mensuration involves the use of geometrical theorems, but it is not concerned with problems of geometrical construction.

00(iii) Solids of revolution also form a special class, which can be conveniently treated by the two theorems of Pappus (§ 33).

00These two theorems may be stated as follows: (i) If any plane figure revolves about an external axis in its plane, the volume of the solid generated by the revolution is equal to the product of the area of the figure and the distance travelled by the centroid of the figure.

00These theorems were discovered by Pappus of Alexandria (c. A.D.

00They are sometimes known as Guldinus's Theorems, but are more properly described as the Theorems of Pappus.

00The 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.

00- Besides the work upon the geodetical operations connecting Paris and Greenwich, of which Legendre was one of the authors, he published in the Memoires de l'Academie for 1787 two papers on trigonometrical operations depending upon the figure of the earth, containing many theorems relating to this subject.

00These 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.

00- +I I-x 2, which lead to theorems in the partition of numbers.

00In 1851 Mr Spottiswoode published in the form of a pamphlet an account of some elementary theorems on the subject.

00The work of WH may be summed up into two theorems: - (1) The text preserved in the later MSS.

00The 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.

00Euclid 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.

00C. Huygens, in his De Circuli Magnitudine Inventa, 1654, proved the propositions of Snell, giving at the same time a number of other interesting theorems, for example, two inequalities which may be written as follows 8 - chd B }- 4 chd Bsin a (chd 0-sin >chd 8+3 (chd 0-sin 0).

00are worked out more fully and generally than they were in earlier treatises, and that a number of theorems in Book iii.

00In the same preface is included (a) the famous problem known by Pappus's name, often enunciated thus: Having given a number of straight lines, to find the geometric locus of a point such that the lengths of the perpendiculars upon, or (more generally) the lines drawn from it obliquely at given inclinations to, the given lines satisfy the condition that the product of certain of them may bear a constant ratio to the product of the remaining ones; (Pappus does not express it in this form but by means of composition of ratios, saying that if the ratio is given which is compounded of the ratios of pairs - one of one set and one of another - of the lines so drawn, and of the ratio of the odd one, if any, to a given straight line, the point will lie on a curve given in position), (b) the theorems which were rediscovered by and named after Paul Guldin, but appear to have been discovered by Pappus himself.

00To 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.

00In a scientific point of view: (a) we see, in the first place, that by his two theorems he founded the geometry of lines, which has ever since remained the principal part of geometry; (b) he may, in the second place, be fairly considered to have laid the foundation of algebra, for his first theorem establishes an equation in the true sense of the word, while the second institutes a proportion.'

00In a philosophic point of view: we see that in these two theorems of Thales the first type of a natural law, i.e.

00The most important are :- Euclid's Elements; Euclid's Data; Optical Lectures, read in the public school of Cambridge; Thirteen Geometrical Lectures; The Works of Archimedes, the Four Books of Apollonius's Conic Sections, and Theodosius's Spherics, explained in a New Method; A Lecture, in which Archimedes' Theorems of the Sphere and Cylinder are investigated and briefly demonstrated; Mathematical Lectures, read in the public schools of the university of Cambridge.

00He published a number of these theorems without demonstration as a challenge to contemporary mathematicians.

00The preceding theorems are purely kinematical.

00The 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.

00Various important reciprocal theorems formulated by H.

00The validity of the recilirocal theorems of Helmholtz and Lord Rayleigh, already referred to, is not affected by frictional forces of the kind here considered.

00For proofs of the theorems here stated and for applications to the more general indeterminate equation x 2 - Ny 2 = H the reader may consult Chrystal's Algebra or Serret's Cours d'Algbbre Superieure; he may also profitably consult a tract by T.

00These theorems are too absolutely stated, and require much modification to adapt them to real life.

00But the truth of Ricardo's theorems is now by his warmest admirers admitted to be hypothetical only.

00Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it.

00" I hope that it may not be considered as unpardonable vanity or presumption on my part, if, as my own taste has always led me to feel a greater interest in methods than in results, so it is by Methods, rather than by any Theorems, which can be separately quoted, that I desire and hope to be remembered.

00It 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.

00And 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.

00and iii., 1810-1813); and from the theorem we have the method of reciprocal polars for the transformation of geometrical theorems, used already by Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by various writers in the earlier volumes of Gergonne.

00In 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.

00by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (x, y, z.) instead of meaning point-co-ordinates pay, mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem.

00In 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.

00The 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.

00r 1.2 r1.3 r2.3 The theorems of motion just cited are expressed by seven integrals, or equations expressing a law that certain functions of the variables and of the time remain constant.

00Some 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.

00In 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.

00Hesse, "they are, like P. Fermat's theorems, riddles to the present and future generations."

00Eminent analysts succeeded in proving some of the theorems, but it was reserved to L.

00Other theorems were published in his Opera Varia, and in John Wallis's Commercium epistolicum (1658).

00The 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.

00According to Mises, all of the categories, theorems, or laws of economics are implied in the action axiom.

00convergence theorems on repeated averaging.

00convolution theorems for the Mellin transforms to produce analytic structures for part of the calculation.

00Obviously the converse is true of Individual B. Two Theorems of welfare economics And now two theorems of welfare economics.

00They commonly treat the historical method with a sort of patronizing toleration as affording useful exemplifications or illustrations of their theorems.

00limit analysis theorems and especially statically admissible force flows and their use for lower bounds to design.

00Their theorem prover is written in the functional programming language lisp which is also the language in which theorems are represented.

00reciprocity theorems relate different states occurring within the same domain.

00singularity theorems imply the universe had an infinite density at some time in the past.

00Given these, we can prove some useful theorems about the policy.

00We also implement a proof checker for SPL which derives theorems in the HOL system from SPL proof scripts.

00Over all geometrical theorems they would be in complete agreement, only interpreting the words in terms of their respective intuitions.

00We make use of the convolution theorems for the Mellin transforms to produce analytic structures for part of the calculation.

00Fourteen 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."

00The 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.

00b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,Ã¯¿½Ã¯¿½Ã¯¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.

00Now log (1+Ã¯¿½X1 +/22X2+/Ã¯¿½3X3 +Ã¯¿½Ã¯¿½Ã¯¿½) =E log (1+/2aix1+22aix2-1-/23ax3+...) whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of Ã¯¿½n gives (n) (-)v1+v2+v3+..

00The 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.

00General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.

00Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (Ã¯¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.

00In 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.

00The 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.

00The 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.

00If, 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.

00The 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.

0047, then follow various theorems on the circle, leading up to the problem of the construction of a circle which shall circumscribe three given circles, touching each other two and two.

00Most of the theorems concerning continued fractions can be thus proved simply from the properties of determinants (see T.

00The reason for publishing these two tracts in his Optics, from the subsequent editions of which they were omitted, is thus stated in the advertisement: " In a letter written to M Leibnitz in the year 1679, and published by Dr Wallis, I mentioned a method by which I had found some general theorems about squaring curvilinear figures on comparing them with the conic sections, or other the simplest figures with which they might be compared.

00Reciprocity theorems relate different states occurring within the same domain.

00The singularity theorems imply the universe had an infinite density at some time in the past.

00The course is also a vehicle for the introduction of theorems in vector calculus that have wide application in physics.

00In 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.

01In 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.

01But the desire to obtain general enunciations of theorems without exceptional cases has led mathematicians to employ entities of ever-ascending types of elaboration.

01All the world, including savages who cannot count beyond five, daily "apply" theorems of number.

01Given 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.

01b., we obtain by comparison with the middle series the symbolical representation of all symmetric functions in brackets () appertaining to the quantities p i, P2, P3,ï¿½ï¿½ï¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.

01Now log (1+ï¿½X1 +/22X2+/ï¿½3X3 +ï¿½ï¿½ï¿½) =E log (1+/2aix1+22aix2-1-/23ax3+...) whence, expanding by the exponential and multinomial theorems, a comparison of the coefficients of ï¿½n gives (n) (-)v1+v2+v3+..

01By the exponential and multinomial theorems we obtain the results) 1,r -1 (E7r) !

01His 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.

01In the theory of numbers he furnished solutions of many of P. Fermat's theorems, and added some of his own.

01Besides 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.

01Astronomy was also enriched by his investigations, and he was led to several remarkable theorems on conics which bear his name.

01The 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."

01Theorems 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.

01(iii.) There are important theorems as to the relative value of fractions; e.g.

01Consideration of the binomial theorem for fractional index, or of the continued fraction representing a surd, or of theorems such as Wallis's theorem (ï¿½ 64), shows that a sequence, every term of which is rational, may have as its limit an irrational number, i.e.

01Notwithstanding 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.

01Fundamental theorems in the theory of equations are to be found in the same work.

01At 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.

01

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