# 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. - ALGEBRAIC FORMS. The subject-matter of algebraic forms is to a large extent connected with the linear transformation of
**algebraical**polynomials which involve two or more variables. - The 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. - Arithmetical and
**Algebraical**Treatment of Equations.-The following will illustrate the passage from arithmetical to**algebraical**reasoning. - Arithmetical 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. - - The calculation of the values of simple
**algebraical**expressions for particular values of letters involved is a useful exercise, but its tediousness is apt to make the subject repulsive. - (ii.) These
**algebraical**formulae involve not only the distributive law and the law of signs, but also the commutative law. - 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. **Algebraical**division therefore has no definite meaning unless dividend and divisor are rational integral functions of some expression such as x which we regard as the root of the notation (ï¿½ 28 (iv.)), and are arranged in descending or ascending powers of x.- -Fp n by x-a, according to
**algebraical**division, the remainder is R= po an +pla? - 55ï¿½
**Algebraical**Division. - If, moreover, we examine the process of
**algebraical**division as illustrated in ï¿½ 50, we shall find that, just as arithmetical division is really the solution of an equation (ï¿½ 14), and involves the tacit use of a symbol to denote an unknown quantity or number, so**algebraical**division by a multinomial really implies the use of undetermined coefficients (ï¿½ 42). - 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. - In this sense, the laws mentioned in ï¿½ 54 apply also to
**algebraical**division. - (i.) By
**algebraical**division, I _ I +0.X+0.x2ï¿½...+0.xr+l I+ x - 1+x = I- x-f-x2-...--(-)rxrd-(-)'+1 + x (24). - (v.) The further extension to fractional values (positive or negative) of n depends in the first instance on the establishment of a method of
**algebraical**evolution which bears the same relation to arithmetical evolution (calculation of a surd) that**algebraical**division bears to arithmetical division. - From this point of view, the function which, by
**algebraical**operations on i+o.x+o.x2+..., produces the series, is called its generating function. - Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with
**algebraical**expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)). - 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. - And therefore T, are measured by the
**algebraical**sum of their individual moments with respect to the axis. - (B) Mensuration of Graphs of
**Algebraical**Functions. - A rational integral
**algebraical**function) of x, or of x and y, of a degree which is known. - 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. - This is a particular case of a general theorem, due to Gauss, that, if u is an
**algebraical**function of x of degree 2p or 2p + I, the area can be expressed in terms of p -}- i ordinates taken in suitable positions. - - Since all points on any ordinate are at an equal distance from the axis of u, it is easily shown that the first moment (with regard to this axis) of a trapezette whose ordinate is u is equal to the area of a trapezette whose ordinate is xu; and this area can be found by the methods of the preceding sections in cases where u is an
**algebraical**function of x. - 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. - The 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. - - We have next to consider the extension of the preceding methods to cases in which u is not necessarily an
**algebraical**function of x or of x and y. - 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. - - The extension of this method consists in dividing the trapezette into minor trapezettes, each consisting of two or more strips, and replacing each of these minor trapezettes by a new figure, whose ordinate v is an
**algebraical**function of x; this function being c h osen so that the new figure shall coincide with the original figure so far as the given ordinates are concerned. - 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. - (8) A parallel projection of a curve, or of a surface of a given
**algebraical**order, is a curve or a surface of the same order. - 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.