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SYMBOLS, Mathematical, signs or ab breviations used in mathematical operations for the sake of brevity and to facilitate expression. In arithmetic and algebra there are four general kinds of symbols used; namely, those of quan tity, operation, relation and abbreviation. Quantities are generally represented by letters. Known quantities are represented by the lead ing letters of the alphabet, or by the final letters with one or more accents, thus: x', y", etc. Unknown quantities are represented by the final letters of the alphabet, as a-, y, 2, etc. Besides the English letters, those of the Greek alpha bet are often made use of. Certain letters have come to represent certain quantities. Thus, re generally stands for the ratio of the diameter to the circumference of circle, or the number 3.1416; e denotes the base of the Naperian sys tem of logarithms, or the number 2.718281828; M denotes the modulus of any system of loga rithms. The symbol m denotes an infinitely great quantity.

Of the symbols of operation the sign +, plus, when written between two quantities, signifies that the second is to be added to the first; as, a + b. The sign —, minus, when placed between two quantities, denotes that the one on the right is to be subtracted from the one on the left; as, a— b. The sign X, when placed between two quantities, denotes that the one on the left is to be multiplied by the one on the right; as, a X b. Multiplication may be indicated by placing a point between the factors when they are both expressed by letters; as, a. b. This method is not applicable when the factors are numbers, because, in that case the indicated product would be confounded with a mixed decimal fraction; thus, 5.6, instead of being read, product of 5 by 6, would be read 5 and 6-tenths. There are cases, however, where the sign is used between numerical factors, as in series where the factors follow a law which it is desirable to keep before the eye: thus the general term of the binomial formula is m .(m — 1).(m — 2) . . . (m — n 1) 1 2 • 3• 4 n an The sign -+ , placed between two quantities, in dicates that the one on the left is to be divided by the one on the right; as, a χ b. Division may also be indicated by writing the quantities in place of the points; as, , or a: b, or a/b, 1 41 . .. . x — --= . The sign — denotes the difference between two quantities, without implying which is to be subtracted from the other; as, a -- b. The sign V is called the radical or evolution sign, and when placed over a quantity indicates that its root is to be taken; as, Va: the degree of the root is indicated by a number written over the sign, which is called the index of the root or radical; thus etc. The sign V indicates the square root. A vinculum —, bar J, brackets [], { , thesis (), etc., indicate that the quantities closed by them are to be regarded together; as, x, °Ix, etc. The symbol I denotes that

the algebraic sum of several quantities of the same nature as that to which the symbol is pre fixed is to be taken, thus, = Z 2-) Z --q- / n(n + P) P (Ps -F P)] is a formula, in which p being constant and and n arbitrary, signifies that of the algebraic sum of any number of terms deduced by at tributing values to q and n is equal to 1 — multi plied by the difference of the algebraic sum of the terms, which are deduced by attributing the same values to q and n in the expressions Of the symbols of relation f, F, p..

+ written before any quantity, or quantities, sepa rated by commas, as F(x), f(x, y), (x, y, s), etc., denotes quantities depending upon the quantity or quantities within the parenthesis, without designating the nature of the relation. The sign of equality, between two quan tities, denotes that those quantities are equal to each other. The sign of inequality, >, placed between two quantities, denotes that the one placed at the opening of the sign is greater than the one placed at the vertex of the sign; thus, a > b, a is greater than b, but a < b, is read: i a s less than b; also, a b, a is not greater than b; a b, a is not less than b. The sign is a negation of equality; as, a b, a is not equal to b. The signs of proportion :, placed between quantities, taken two and two, show that the quantities are in proportion; thus, a : b c : d, is read, a is to b as c is to d. The and third signs are signs of ratio, and the second the sign of equality, so that the above might be written b d a c Of the symbols of abbreviation the sign .'. stands for therefore or hence and •. • stands for since or because. Other algebraic and mathematical symbols are % for per cent or per thousand; f the symbol of .integra tion; the Decimal, as in 5.6 (America), 56 (Great Britain), 5,6 (Continental Europe), meaning 5 and 6-tenths; identity =--.

Geometry borrows most of its symbols from those of algebra just explained. Magnitudes are represented pictorially; the symbols 1.4 are pictorial representations of angle, angles. I means parallel to; I perpendicular to; A, A, represent pictorially the terms triangle and triangles respectively; while 10, a represent circle, circles; 0 , 0, square, squares; en, OD, rectangle, rectangles; and O. S., rep resent parallelogram and parallelograms. Arc is represented by the symbol and a designates radians; -' means congruent to and ... similar to. See NOTATION and consult Cantor, M. B., (Vorlesungen fiber Geschichte Mathe matik) (Leipzig 1910) ; (Vorschlip zur Verein heitlichtung der Mathematischen in Schilunter richt) in ten des deutschen Ausschusses fiir den mathematischen and naturwissenchaft lichen Unterricht' (ib. 1913). This contains an extended list of symbols.