# Algebraical Sentence Examples

Such is the basis of the

**algebraical**or modern analytical geometry.Thus, if x= horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as

**algebraical**symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers.The only other

**algebraical**symbol is A for minus; plus being expressed by merely writing terms one after another.In his youth he went to the continent and taught mathematics at Paris, where he published or edited, between the years 1612 and 1619, various geometrical and

**algebraical**tracts, which are conspicuous for their ingenuity and elegance.His earliest publications, beginning with A Syllabus of Plane

**Algebraical**Geometry (1860) and The Formulae of Plane Trigonometry (1861), were exclusively mathematical; but late in the year 1865 he published, under the pseudonym of "Lewis Carroll," Alice's Adventures in Wonderland, a work that was the outcome of his keen sympathy with the imagination of children and their sense of fun.AdvertisementThe present article is merely concerned with

**algebraical**linear transformation.The leading idea of this work was contained in a paper published in the Berlin Memoirs for 1772.5 Its object was the elimination of the, to some minds, unsatisfactory conception of the infinite from the metaphysics of the higher mathematics, and the substitution for the differential and integral calculus of an analogous method depending wholly on the serial development of

**algebraical**functions.In algebra he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving

**algebraical**equations of every degree.The distinction between

**algebraical**and arithmetical reasoning then lies mainly in the fact that the former is in a more condensed form than the latter; an unknown quantity being represented by a special symbol, and other symbols being used as a kind of shorthand for verbal expressions.The number by which an

**algebraical**expression is to be multiplied is called its coefficient.AdvertisementArithmetical and

**Algebraical**Treatment of Equations.-The following will illustrate the passage from arithmetical to**algebraical**reasoning.This is an

**algebraical**process.In the same way, the transition from (x 2 +4x+4) - 4= 21 to x 2 +4x+4 = 25, or from (5+2) 2 =25 to x+2= 1 /25, is arithmetical; but the transition from 5 2 + 45+4= 25 to (5+2) 2 = 25 is

**algebraical**, since it involves a change of the number we are thinking about.In

**algebraical**transformations, however, such as (x-a)2 = x 2 - 2ax+a 2, the arithmetical rule of signs enables us to combine the sign-with a number and to treat the result as a whole, subject to its own laws of operation.We therefore define

**algebraical**division by means of**algebraical**multiplication, and say that, if P and M are multinomials, the statement " P/M = Q " means that Q is a multinomial such that MQ (or QM) and P are identical.AdvertisementFrom this point of view, the function which, by

**algebraical**operations on i+o.x+o.x2+..., produces the series, is called its generating function.His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both algebra and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and

**algebraical**.It includes the properties of numbers; extraction of roots of arithmetical and

**algebraical**quantities, solutions of simple and quadratic equations, and a fairly complete account of surds.So far the development of algebra and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of

**algebraical**problems. Certain minds had long suspected the advantages which would accrue from the unrestricted application of algebra to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected.A number of cases are worked out in the American Journal of athematics (1907), in which the motion is made

**algebraical**by the se of the pseudo-elliptic integral.AdvertisementIn his Discourse on the "Residual Analysis," he proposes to avoid the metaphysical difficulties of the method of fluxions by a purely

**algebraical**method.Generally, if the area of a trapezette for which u is an

**algebraical**function of x of degree 2n is given correctly by an expression which is a linear function of values of u representing ordinates placed symmetrically about the mid-ordinate of the trapezette (with or without this mid-ordinate), the same expression will give the area of a trapezette for which u is an**algebraical**function of x of degree 2n + 1.If u is an

**algebraical**function of x of degree not exceeding p, and if the area of a trapezette, for which the ordinate v is of degree not exceeding p+q, may be expressed by a formula Aovo-1--yivi+..To extend these methods to a briquette, where the ordinate u is an

**algebraical**function of x and y, the axes of x and of y being parallel to the sides of the base, we consider that the area of a section at distance x from the plane x = o is expressed in terms of the ordinates in which it intersects the series of planes, parallel to y=o, through the given ordinates of the briquette (§ 44); and that the area of the section is then represented by the ordinate of a trapezette.This ordinate will be an

**algebraical**function of x, and we can again apply a suitable formula.AdvertisementThe result of performipg both operations, in order to obtain the volume, is the result of the operation denoted by the product of these two expressions; and in this product the powers of E and of E' may be dealt with according to

**algebraical**laws.If, as is usually the case, the ordinate throughout each strip of the trapezette can be expressed approximately as an

**algebraical**function of the abscissa, the application of the integral calculus gives the area of the figure.The first, which is the best known but is of limited application, consists in replacing each successive portion of the figure by another figure whose ordinate is an

**algebraical**function of x or of x and y, and expressing the area or volume of this latter figure (exactly or approximately) in terms of the given ordinates.It is also clearly impossible to express u as an

**algebraical**function of x and y if some value of du/dx or duldy is to be infinite.According to these experiments, the resistance of the air can be represented by no simple

**algebraical**law over a large range of velocity.Thus log x is the integral function of 1/x, and it can be shown that log x is a genuinely new transcendent, not expressible in finite terms by means of functions such as

**algebraical**or circular functions.In various systems of triangular co-ordinates the equations to circles specially related to the triangle of reference assume comparatively simple forms; consequently they provide elegant

**algebraical**demonstrations of properties concerning a triangle and the circles intimately associated with its geometry.In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational

**algebraical**equation of any degree.3 To prove the same proposition regarding 7r is to prove that a Euclidean construction for circle-quadrature is impossible.At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for

**algebraical**equations up to and including the fifth degree.During the later years of his life he resided in London, devoting himself to the construction of machines capable of performing arithmetical and even

**algebraical**calculations.Cramer, that, when a certain number of the intersections of two

**algebraical**curves are given, the rest are thereby determined.The work falls into two parts, which treat of the asymptotes and singularities of

**algebraical**curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches.In 1707 William Whiston published the

**algebraical**lectures which Newton had delivered at Cambridge, under the title of Arithmetica Universalis, sive de Compositione et Resolutione Arithmetica Liber.When the unit is not determined, the reasoning is

**algebraical**rather than arithmetical.If, for instance, three terms of a proportion are given, the fourth can be obtained by the relation given at the end of § 57, this relation being then called the Rule of Three; but this is equivalent to the use of an

**algebraical**formula.More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of

**algebraical**formulae.We obtain from the equation the notion of an

**algebraical**as opposed to a transcendental curve, viz.The most important are those relating to

**algebraical**curves and surfaces, especially the short paper Allgemeine Eigenschaften algebraischer Curven.Apart from the interest attaching to these manuscripts as the work of Napier, they possess an independent value as affording evidence of the exact state of his

**algebraical**knowledge at the time when logarithms were invented.He published, first in 1830, and then in an enlarged form in 1842, a Treatise on Algebra, in which he applied his philosophical ideas concerning

**algebraical**analysis to the elucidation of its elements.The great development of all branches of mathematics in the two centuries following Descartes has led to the term algebra being used to cover a great variety of subjects, many of which are really only ramifications of arithmetic, dealt with by

**algebraical**methods, while others, such as the theory of numbers and the general theory of series, are outgrowths of the application of algebra to arithmetic, which involve such special ideas that they must properly be regarded as distinct subjects.To transpose a term which is not the last term on either side we must first use the commutative law, which involves an

**algebraical**transformation.The first difficulty to be overcome was the

**algebraical**solution of cubic equations, the " irreducible case" (see Equation).With the values above of u, v, w, u', v', w', the equations become of the form p x + 4 7rpAx -Fax -{-hy-}-gz =o, - p - dy+ 4?pBy + hx+ay+fz =o, P d p + TpCZ +f y + yz = o, and integrating p p 1+27rp(Ax2+By2+CZ2) +z ('ax e +ay e + yz2 2 f yz + 2gzx + 2 hx y) = const., (14) so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by

**algebraical**reduction; and 4c2 (c 2 - a2)?The fundamental principles of his system (see Scholasticism) are that "Essentia non sunt multiplicanda praeter necessitatem" ("Occam's Razor"), that nouns, like

**algebraical**symbols, are merely denotative terms whose meaning is conventionally agreed upon (suppositio), and that the destructive effect of these principles in theological matters does not in any way destroy faith (see the Centilogium Theologicum, Lyons, 1495, and Tractatus de Sacramento Altaris).Allied to the matter just mentioned was Plucker's discovery of the six equations connecting the numbers of singularities in

**algebraical**curves (see Curve).