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."
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.
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.
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.
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.
B., 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.
Now 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+..
By the exponential and multinomial theorems we obtain the results) 1,r -1 (E7r) !
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.
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.
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.
General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.
(iii.) There are important theorems as to the relative value of fractions; e.g.
Consideration 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.
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.
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.
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.
So 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+..
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.
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.
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.
(iii) Solids of revolution also form a special class, which can be conveniently treated by the two theorems of Pappus (§ 33).
These 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.
These theorems were discovered by Pappus of Alexandria (c. A.D.
They 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.
- 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.
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.
- +I I-x 2, which lead to theorems in the partition of numbers.
In 1851 Mr Spottiswoode published in the form of a pamphlet an account of some elementary theorems on the subject.
The work of WH may be summed up into two theorems: - (1) The text preserved in the later MSS.
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.
Euclid 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.
C. 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).
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.
Are worked out more fully and generally than they were in earlier treatises, and that a number of theorems in Book iii.
47, 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.
In 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.