yor y3

93 27

ys +r y3 +

27

--(-)

27

21 x

10— 64 —

54

441

y=

- 21 x +

4

21x+ 1 or x3 4

extracting the sq. root, x—

x=

4

21 15 =± 2

21±15

2 = 3 or 18.

By this process two values of x are found, but on trial it appears, that 18 does not answer the condition of the equation, if we suppose that (5 x + 10) represents the positive square root of 5x+ 10. The reason is, that 5 x + 10 is the ✔(5x+10) as well as of square of + (510); thus by squaring both sides of the equation ✓ (5x+10) 8 x, a new condition is introduced, and a new value of the unknown quantity corresponding to it, which had no place before. Here, 18 is the value which corresponds to the supposition that x — (5 +10) = 8.

therefore z-9, or 49.

3--3 2

*

Since the square of every quantity is positive, a negative quantity has no square root; the conclusion therefore shews that there are no such numbers as the question supposes. See BINOMIAL THEOREM; EQUATIONS, nature of; SERIES, SURDS, &c. &c.

ALGEBRA, application of to geometry.The first and principal applications of algebra were to arithmetical questions and computations, as being the first and most useful science in all the concerns of hu

man 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 andunknown 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 quantity 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 AC-b, 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: x; then, multiply extremes and means, &c. there arises pr=b p-b x, or b x+px b p =b and x= the side of the square

b+p

bp, 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 property 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, 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 quantities 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 letters 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=x, IK=y, and p = the diameter of the circle, the equation of the circle will be px - ry. 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.

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 geome try 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 a and b2 and 2 a b, that is, the square of a+b is equal to a+b2+2 a b, as derived from a geometrical figure or construction. And in this very manner it was, that the Arabians, and the earlier 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 va riations, appearing at different times on 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 nume

or the art of numbering truly and ly.

ALIEN, in law, a person born in a trange 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 proclamation, 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 not exceeding twelve months for the second; at the expiration of which, he shall depart within a time to be limited: and if such alien be found therein after such time so limited, he or she shall be transported for life.

ALIMENTARY duct, a name which some call the intestines, on account of the food's passing through them. See ANATOMY.

ALIMONY, alimonia, in law, denotes the maintenance sued for by a wife, in case of a separation from her husband, wherein she is neither chargeable with elopement nor adultery.

ALIQUANT parts, in arithmetic, those which will not divide or measure the whole number exactly. Thus, 7 is an aliquant part of 16, for twice 7 wants 2 of 16, and three times 7 exceeds 16 by 5.

ALIQUOT part, is such part of a number as will divide and measure it exactly, without any remainder. For instance, 2 is an aliquot part of 4, 3 of 9, and 4 of 16.

To find all the aliquot parts of a number, divide it by its least divisor, and the quotient by its least divisor, until you get a quotient not farther divisible, and you will have all the prime divisors or aliquot parts of that number. Thus, 60, divided by 2, gives the quotient 30, which divided by 2 gives 15, and 15 divided by 3 gives the indivisible quotient 5. Hence, the prime aliquot parts are 1, 2, 2, 3, 5; and by multiplying any two or three of these together, you will find the compound aliquot parts, viz. 4, 6, 10, 12, 15, 20, 30.

Aliquot parts must not be confounded with commensurable ones; for though the former be all commensurable, yet these are not always aliquot parts: thus,

4 is commensurable with 6, but is not an aliquot part of it.

ALISMA, great water plantain, in botany, a genus of the Hexandria Polyginia class of plants, the calyx of which is a perianthium, composed of three oval, hollow, permanent leaves; the corolla consists of three large, roundish, plane, and very patent petals; the fruit consists of capsules, arranged together in a roundish or trigonal form the seeds are single and small. There are nine species.

ALKAHEST, or ALCAHEST, among chemists, denotes a universal menstruum, capable of resolving all bodies into their ens primum, or first matter; and that without suffering any change, or diminution of force, by so doing. See ALCHEMY.

ALKALI, in chemistry, a word applied to all bodies that possess the following properties: they change vegetable blue colours, as that of an infusion of violets, to green: they have an acrid and peculiar taste: they serve as intermedia between oils and water: they are capable of combining with acids, and of destroying their acidity: they corrode woollen cloth, and, if the solution be sufficiently strong, reduce it to jelly and they are soluble in water. The alkalies at present known are three; viz. ammonia, potash, and soda: the two last are called fixed alkalies, because they require a red heat to volatilize them; the other is denominated volatile alkali, because it readily assumes a gaseous form, and is dissipated by a very moderate degree of heat. Barytes, strontian, lime, and magnesia, have been denominated alkalies by Fourcroy; but as they possess the striking character of earths in their fixity, this innovation does not seem entitled to general adoption.

Since writing the above, some discoveries of great importance, on the subject of alkalies, have been made known to the philosophical world by Mr. Davy, Professor of Chemistry at the Royal Institution. We shall in this place give a sketch of the two papers which he has just laid before the Royal Society, referring to some subsequent articles for further particulars. In a former discourse, read before this learned body, Mr. Davy, in speaking of the agencies of electricity, suggested the probability, that other bodies not then enumerated might be decomposed by the electric fluid. In the course of the last summer, this celebrated philosopher was employed in making a number of experiments with this particular view, and by means of very powerful galvanic troughs, consisting of a