ALGEBRA, Sped0718 or literal, or the new algebra, is that in which All the quanti ties, known and unknown, are express ed or represented by their species, or let ters of the alphabet 'Mere are instan ces of this method from Cardan, and others about his time ; but it was more generally introduced and used by Vieta. Dr. Wallis apprehends that the name of specious arithmetic, applied to algebra, is given to it with a reference to the sense in which the Civilians use the word species. Thus, they use the names Titus, Sempro nuns, Gains, and the like, to represent in definitely any person in such circumstan ces; and cases so propounded, they call species. Vieta, accustomed to the lan guage of the civil law, gave, as Wallis supposes, the name of species to the let ters, A, 11, C, which lie used to re present indefinitely any munher or quan tity so circumstanced, as the occasion required. This mode of expression frees the memory and imagination from that stress or effort, which isrequired to keep seventh matters, necessary for the disco very ()Nile truth investigated, present to the mind ; for which reason this art may be properly denominated metaphysical geometry. Specious algebra is not, like the numeral, confined to certain kitrds of problems ; but serves universally for the investigation or invention of theorems, as well as the solution and demonstration of all kinds Of problems, both arithmetical and geomethcal. The letters used in algebra do each of them, separately', re present either lines or numbers, as the problem is either arithmetical or geome trical ; and, together, they represent planes, solids, and powers, more or less high, as the letters are in a greater or less number. For instance, if there be two letters, a they represent a rectangle, whose two sides are expressed, one by the letter a, and the other by b ; so that by their mutual multiplication they pro duce the plane a b. Where the same let ter is repeated twice, as a a, they denote a square. Three letters, a b c, represent a solid, or a rectangular parallelopiped, whose three dimensions are expressed by the three letters a b c ; the length by a, the breadth by h, and the depth by c; so that by their mutual multiplication they produce the solid a b c. As the multipli cation of diniensions is expressed by the multiplication of letters, and as the 1111M ber of these may- be so great as to become incoinmodious, the method is only to write down the root, and on the right hand to write the index of the power, that is, the nuniber of' letters or which the quantity to be expressed consists ; as a', a3, a4, &e. the last of which signifies as much as a multiplied four times into itself; and so of the rest. But as it is necessary, before any progress can be made'in the science of algebra, to under stand the method of notation, we shall here give a general view of it. In alge bra, as we have already stated, every quantity, whether it be known or given, or unknown or required, is usually repre sented by some letter of the alphabet ; and the given quantities are commonly denoted by the initial letters, a, b, &c. and the unknown ones by the final letters, 71, 7(1, y, z. These quantities are connected together by certain signs or symbols, which seme to shew their mu tual relation, and at the same time to simplify the science, and to reduce its operations into a less compass. Accord ingly the sign -F, plus, or. more, signi fies that the quantity to which it is prefix ed is to be added, and it is called a posi tive or affirmative quantity. 'finis, a+b,
expresses the sum of the two quantities a and b, so that if a were 5, and b 3, a+b would be 5+3, or 8. if a quantity have no sign, 7,F, plus, is undentood, and the quantity is affirmative or positive.
The sign —, minus, or less, denotes that the quantity which it precedes is to be subtracted, and it is called a negative quantity. Thus tz--b expresses the dif ference of a and b ; so that a being 5, and b 3, a—b, or 5-3, would be equal to 2. If more quantities than two were con nected by these signs, the stun of those with the sign — must be subtracted from the sum of those with the sign Thus a+ b—c—d represents the quantity which would ren-min, when c and d are taken from a and b. So that if a were 7, b 6, c 5, and ti 3, a + — c — d, or 7 + 6 —5-3, or 13— 8, would be equal to 5. If two quantities are connected by the sign cr, as a b, this mode of expres sion represents the difference of a and b, when it is not knoun which of them is the greatest. The sign x signifies th'k the quantities between which it stands are to be multiplied together, or it repre sents their product. Thus, a x b ex presses the product of a and b; a Xbxc denotes the product of a, b, and c ; X c denotes the product of the compound quantity- a + b the simple quantity c ; and (a+b + c) X (a—b + c) X (a+ b) represents the product of the three cmn pound quantities, multiplied continually into one another ; so that if a were 5, b 4,, and c 3, then would (a + b 4- c) X (a — b + c) X (a+ c) be 12 x 4 x 8, or 384, The parenthesis used in the forego ing expressions indicate that the whole compound quantities are affected by the sign, and not simply the single terms be tween which it is placed. Quantities that are joined together without any interme diote sign form a product; thus a b is the same with a x b, and a b c the same with aXbX c. When a quantity is multi plied into itself, or raised to any power, the usual mode of expression is to draw a line over the quantity, and to place the number denoting the power at the end of it, which number is called the index or exponent. Thus, (a + b)' denotes the same as (a + b) X (a + b) or second power, or square, of a + b considered as one quantity ; and (a + b)3 denotes the same as (a + b) x (a + b) x (a + b), or the third power, or cube, of a+ b. In expressing the powers of quantities re presented by single letters, the line over the top is usually omitted : thus, a' is the same as a a or a x a, and b3 the same as bbb or bXbXb,anda'b3, the same as aa X bbb or axaxbxbX b. The full point . and the word into, .are sometimes used instead of x as the sign of multiplication. Thus, (a+b) . c), and a + b into a + c, signify the same thing as (a+ b) X (a + c), or the pro duct of a + 6 by a -1- c. The sign -4 is the sign of division, as it denotes that the quantity preceding it is to be divided by the succeeding quantity. Thus, c+b signifies that c is to be divided by b ; and (a + b) (a + c), that a + b is to he divided by a + c. The mark ) is some times used as a note of division ; thus a + b) a b denotes that a b is to be divi ded by a + b. But the division of alge braic quantities is most commonly ex pressed by placing the divisor under the dividend, with a line between them, like a vulgar fraction. Thus, represents the quantity arising by dividing c by b, or the quotient, and represents the quotient of a-f-b divided by a+c. Quan tities thus expressed are called algebraic fractions.