SYMBOLS, MATHEMATICAL. The various signs and abbreviations used to facilitate mathe matical expression. They are of the following kinds: The question of the origin and development of mathematical symbols is a large one, and science has not yet given satisfactory answers at many points. The probable origin of the remarkable digits 1 . . 9, is discussed in the article on NIT MERALS. The origin of zero is unknown, there being no authentic record of its history before A.D. 400. The extension of the position system below unity is attributed to Stevin (15S5), who called tenths, hundredths, thousandths, . . . primes, sekondcs, tcrzes, and wrote subscripts to denote the orders, thus 4.628 was written 8 But Rudolf (1525) and Kepler (1571-1630) used the comma to set of the deci mal orders, and Biirgi (1552-1632) and Pitiscus (1612) in their tables used the decimal fraction in the form 0.32, 3.2. Although the early Egyp tians had symbols for addition and equality, and the Creeks, Hindus, and Arabs symbols for equal ity and for the unknown quantity, from earliest times mathematical processes were cumbersome for lack of proper symbols of operation. The expressions for such processes were either written out in full or denoted by word abbreviations. The later Creeks, the Hindus, and Jordanus in dicated addition by juxtaposition; the Italians usually denoted it by the letter P or p with a line drawn through it to distinguish it as an operation, but their symbols were not uniform. Paeioli, for example, sometimes used p and some times e, and Tartaglia commonly expressed the operation by p. The German and English alge braists introduced the sign +, but spoke of it as signiint addito•um and first employed it only to indicate excess. Subtraction was indicated by
Diophantus by the symbol Is. The Hindus used a dot, while the Italian algebraists denoted it by M or in with a line drawn through the letter.
The symbols i; and de were, however, used by Pacioli. The German and English algebraists were the first to use the present symbol and de scribed it as signum subtractorum. The symbols and — appeared first in print in an arith metic of Widmann (1489). The symbol X for 'times' is due to Ougiitred (1631). To Rahn (1659) is due the present sign for division; Harriot (1631) used a period to indicate multi plication, and Descartes (1637) used juxtapo sition. Leibnitz in 1688 employed the sign to denote multiplication and to denote di vision. Division among the Arabs was desig nated variously by a — b, but Clairaut (1760) made fandlbir the form : b. Descartes made popular the notation au for involution and Wallis defined the negative exponent. The sym bol of equality. =, is due to Recorde (1557), and the symbols >, <, for greater than and less than, orhtinated with Ilarriot (1631). \'ii•ta (1591) and Girard (16•29) introduced various symbols of aggregation. The symbol oo for in finity was first employed by Wallis in 1655. The symbols of differentiation dx and of integration used in calculus, are due to Leibnitz, as is also the symbol for similarity, as used in geometry. The symbolism p, f, F', as used in theory of functions, is due to Abel.
Consult Cantor, fiber Ceschichte dcr Mathematik, (2d ed., Leipzig, 1900).