Let rya, and 2x+2y=2a; this latter equation being deducible from the former, it involves no different supposition, nor requires any thing more for its truth, than that rya should be a just equation.

PROBLEMS WHICH PRODUCE SIMPLE

EQUATIONS.

From certain quantities which are known, to investigate others which have a given relation to them, is the business of Algebra.

When a question is proposed to be resolved, conditions. Then substituting for such unwe must first consider fully its meaning and known quantities as appear most convenient, termined, and we wished to try whether they we must proceed as if they were already dewould answer all the proposed conditions or not, till as many independent equations arise as we have assumed unknown quantities, which will always be the case if the question be properly limited; and by the solution of these equations, the quantities sought will be determined.

life. Afterwards algebra was applied to geometry, and all the other sciences in their turn. The application of algebra to geometry, is of two kinds; that which regards the plane or common geometry, and that which respects the higher geometry, or the nature of curve lines.

The first of these, or the application of algebra to common geometry, is concerned in the algebraical solution of geometrical problems, and finding out theorems in geometrical figures, by means of algebraical investigations or demonstrations. This kind of application has been made from the time of the most early writers on algebra, as Diophantus, Cardan, &c. &c. down to the present times. Some of the best precepts and exercises of this kind of application are to be met with in Sir I. Newton's “Universal Arithmetic," and in Thomas Simpson's "Algebra and Select Exercises." Geometrical Problems are commonly resolved more directly and easily by algebra, than

by the geometrical analysis, especially by young beginners; but then the synthesis, or construction and demonstration, is most elegant as deduced from the latter method. Now it commonly happens that the algebraical solution succeeds best in such problems as respect the sides and other lines in geometrical figures; and, on the contrary, those problems in which angles are concerned, are best effected by the geometrical analysis. Sir Isaac Newton gives these, among many other remarks on this branch. Having any problem proposed, compare together the quantities concerned in it; and, making no difference between the known and unknown quantities, consider how they depend, or are related to, one another; that we may perceive what quantities, if they are assumed, will, by proceeding synthetically, give the rest, and that in the simplest manner. And in this comparison, the geometrical figure is to be feigned and constructed at random, as if all the parts were actually known or given, and any other lines drawn that may appear to conduce to the easier and simpler solution of the problem. Having considered the method of computation, and drawn out the scheme, names are then to be given to the quantities entering into the computation, that is, to some few of them, both known and unknown, from which the rest may most naturally and simply be derived or expressed, by means of the geometrical properties of figures, till an equation be obtained, by which the value of the unknown quan

tity may be derived by the ordinary methods of reduction of equations, when only one unknown quantity is in the notation; or till as many equations are obtained as there are unknown letters in the notation.

For example: suppose it were required to inscribe a square in a given triangle. Let ABC, (Plate Miscellanies, fig. 1.) be the given triangle; and feign DEFG to be the required square; also draw the perpendicular BP of the triangle, which will be given, as well as all the sides of it. Then, considering that the triangles BAC, BEF are similar, it will be proper to make the notation as follows, viz. making the base ACb, the perpendicular BP = p, and the side of the square DE or EF=x. Hence then BQ=BP— ED=p-x; consequently, by the proportionality of the parts of those two similar triangles, viz. BP: AC:: BQ: EF, it is p:b::p-x:r; then, multiply extremes and means, &c. there arises p x= bp

bp-bx, or bx+px=bp, and x= 'b+p'

the side of the square sought; that is, a fourth proportional to the base and perpendicular, and the sum of the two, taking this sum for the first term, or AC + BP: BP :: AC: EF.

The other branch of the application of algebra to geometry, was introduced by Descartes, in his Geometry, which is the new or higher geometry, and respects the nature and properties of curve lines. In this branch, the nature of the curve is expressed or denoted by an algebraic equation, which is thus derived: A line is conceived to be drawn, as the diameter or some other principal line about the curve; and upon any indefinite points of this line other lines are erected perpendicularly, which are called ordinates, whilst the parts of the first line cut off by them are called abscisses. Then, calling any absciss x, and its corresponding ordinate y, by means of the known nature, or relations, of the other lines in the curve, an equation is derived, involving x and y, with other given quanti. ties in it. Hence, as x and y are common to every point in the primary line, that equation, so derived, will belong to every position or value of the absciss and ordinate, and so is properly considered as expressing the nature of the curve in all points of it; and is commonly called the equation of the

curve.

In this way it is found that any curve line has a peculiar form of equation belonging to it, and which is different from that of

every other curve, either as to the number of the terms, the powers of the unknown let ters x and y, or the signs or co-efficients of the terms of the equation. Thus, if the curve line HK, (fig. 2.) be a circle, of which HI is part of the diameter, and IK a perpendicular ordinate: then put HI=r, IK =y, and p= the diameter of the circle, the equation of the circle will be px-x2y2. But if HK be an ellipse, an hyperbola, or parabola, the equation of the curve will be different, and for all the four curves, will be respectively as follows: viz.

This way of expressing the nature of curve lines, by algebraic equations, has given occasion to the greatest improvement and extension of the geometry of curve lines; for thus, all the properties of algebraic equations, and their roots, are transferred and added to the curve lines, whose abscisses and ordinates have similar properties. Indeed the benefit of this sort of application is mutual and reciprocal, the known properties of equations being transferred to the curves they represent; and, on the contrary, the known properties of curves transferred to their representative equations.

Besides the use and application of the higher geometry, namely of curve lines, to detecting the nature and roots of equations, and to the finding the values of those roots by the geometrical construction of curve lines, even common geometry may be made subservient to the purposes of algebra. Thus, to take a very plain and simple instance, if it were required to square the binomial a+b; (fig. 3.) by forming a square, as in the figure, whose side is equal to a+b, or the two lines or parts added together denoted by the letters a and b; and then drawing two lines parallel to the sides, from the points where the two parts join, it will be immediately evident that the whole square of the compound quantity a+b is equal to the squares of both the parts, together with two rectangles under the two parts, or a2 and b2 and

2ab, that is, the square of a +b is equal to a2+b2+2 ab, as derived from a geometrical figure or construction. And in this very manner it was, that the Arabians, and the early European writers on algebra, derived and demonstrated the common rule for resolving compound quadratic equations. And thus also, in a similar way, it was, that Tartalea and Cardan derived and demonstrated all the rules for the resolution of cubic equations, using cubes and parallelopipedons instead of squares and rectangles. Many other instances might be given of the use and application of geometry in algebra.

ALGOL, the name of a fixed star of the third magnitude in the constellation Perseus, otherwise called Medusa's head. This star has been subject to singular variations, appearing at different times of different magnitudes, from the fourth to the second, which is its usual appearance. These variations have been noticed with great accuracy and the period of their return is determined to be 2d 20h 48' 56'. The cause of this variation, Mr. Goodricke, who has attended closely to the subject, conjectures, may be either owing to the interposition of a large body revolving round Algol, or to some motion of its own, in consequence of which, part of its body, covered with spots or some such like matter, is periodically turned towards the earth.

ALGORITHM, an Arabic term, not unfrequently used to denote the practical rules of algebra, and sometimes for the practice of common arithmetic; in which last sense it coincides with logistica numeralis, or the art of numbering truly and readily.

ALIEN, in law, a person born in a strange country, not within the king's allegiance, in contradistinction from a denizen or natural subject.

An alien is incapable of inheriting lands in England, till naturalized by an act of parliament. No alien is entitled to vote in the choice of members of parliament, has a right to enjoy offices, or can be returned on any jury, unless where an alien is party in a cause; and then the inquest of jurors shall be one half denizens and the other aliens.

Every alien neglecting the king's procla mation directing him to depart from the realm within a limited time, shall, on conviction, for the first offence be imprisoned for any time not exceeding one month, and