395 ERRAT A. 149, 28, for √a+√±b, and √a-√ ±b, read /a+ √±b, and ya—√ ± b. 13, for observing, read and observing. 8, for (a-B)-3, read (α-B)√-3. 2, 3, 4, &c. in the denominators, read 218, 1. 14, for number, read member; and in the note, for differential, read différentiel. ALGEBRA. (A) ALGEBRA is the science which treats of a general method of performing calculations, and resolving mathematical problems, by means of the letters of the alphabet (a). Its leading rules are the same as those of arithmetic; and the operations to be performed are denoted by the following characters: + plus, or more, the sign of addition; signifying, that the quantities between which it is placed are to be added together, Thus, a+b shows that the number, or quantity, represented by b, is to be added to that represented by a; and is read a plus b. minus, or less, the sign of subtraction; signifying, that the latter of the two quantities be 3 (a): The science here treated of, which is called, by Newton and others, Universal Arithmetic, differs from that of numbers chiefly in this; that in the latter, every figure has a determinate and individual value peculiar to itself; whereas the algebraic characters, being general, or independent of any particular or partial signification, represent all sorts of numbers, or quantities, according to the nature of the question to which they are applied. Hence, when any of the operations of addition, multiplication, &c. are to be performed upon numbers, or other magnitudes, which are represented by the letters a, b, c, &c. it is obvious that the results, so obtained, will be general; and that any particular case, of a similar kind, may be readily derived from them R tween which it is placed is to be taken from the former. Thus, a-b shows that the quantity represented by b is to be taken from that represented by a; and is read a minus b. Also, a b represents the difference of the two quantities a and b, when it is not known which of them is the greater. x into, the sign of multiplication; signifying that the quantities between which it is placed are to be multiplied together. Thus, axb shows that the quantity represented by a is to be multiplied by that represented by b; and is read a into b. The multiplication of simple quantities is also frequently denoted by a point; or by joining the letters together in the form of a word. Thus, a × b, a.b, and ab, all signify the product of a and b: also, 3 × a, or 3a, is the product of 3 and a; and is read 3 times a. by, the sign of division; signifying that the former of the two quantities between which it is placed is to be divided by the latter. by barely substituting for every letter its real numeral value, and then computing the amount accordingly. Another advantage also, which arises from this general mode of notation, is, that, while the figures employed in common arithmetic disappear in the course of the operation, the characters used in algebra always retain their original form, so as to show the dependence they have upon each other, in every part of the process; which circumstance, together with that of representing the operations of addition, subtraction, &c. by certain signs, tenders the language of this science extremely simple and commodious. |