MATHEMATICAL SIGNS AND SYM BOLS, in mathematics, a symbol employed to denote an operation to be performed, to show the nature of a result of some previous opera tion, or to indicate the sense in which an indicated quantity is to be considered. The present mathematical symbolism is due to the labors of many men — men having different habits of thought, men living in different ages, men speaking different languages. A physicist will employ sine a and sine b to calculate the angle of refraction of light. Now certainly there is neither sine a nor sine b in reality; yet there are most certainly relations of reality which are accurately described in these ex pressions, sine a and sine b. Such is the func tion of all signs, symbols or characters. When ever the eye can be brought to the aid of one's imaging faculty, a success in grasping a thought may often be gained which would otherwise be a failure. This is in particular true in mathematics where the subiject matter, while objective, is non-sensible. n this way signs, and symbols, stand for the emphatic presenta tion of mathematical ideas, often very subtle even when symbolized; and with such signs these ideas become, as it were, easy to exhibit. that is, in thought; and the relations of these ideas become thinkable, even very often to the whole complex train of ideas in which they occur.
One very important property for symbolism to possess is that it should be concise, so as to be visible at a glance of the eye and to be rapidly written. A sign should if possible al ways represent the same object, and the same object should always be represented by the same sign. If a new sign be advisable, per manently or even temporarily, it should carry with it always some mark of distinction from that which is already in use, unless it be a demonstrable extension of the latter (De Mor gan). The importance of notation is recog nized when it is remembered, for example, how great an advance toward the solution of the famous problem of the three bodies was made when special attention was given to it, and a special symbolism chosen for it. And the invention of the symbol by Gauss affords a fine example of the advantage which may be derived from a good notation; it, without ex aggeration, marks an epoch in the development of the science of arithmetic. The language of analysis, says Laplace, being in itself a power ful instrument of discoveries, its notations, especially when they are necessarily and happily conceived, are so many germs of new calculi.
Nowadays we have a symbol for each mathe matical operation. Sometimes, even a choice of symbols. And most of the letters of the Eng lish alphabet are now engaged for special mathe matical purposes. Thus: a signifies sometimes a finite quantity; at other times a known number, the side of a triangle opposite A; also an intercept on the axis of x, and, finally, altitude.
b like a signifies a known number, and also a side of a triangle, the one opposite B ; it also stands for base; and lastly, for an inter cept on the axis of y.
c signifies constant.
e signifies the base of the Napierian loga rithms.
A considerable inroad has been made, also, into the Greek alphabet: Y signifies the inclination of the axis of x; ? stands for the ratio 3.14159; e is used for sum of tens similarly obtained; and el indicates the standard deviation in the theory of measurements.
Some examples of reading notation may be given: a + b, a— b, a + b, a X b, and a.... b are read a plus b, a minus b, a divided by b, a multiplied by b, and the difference between a and b.
Further:—a>b, a < b,a=b, b, and amb are to be read a greater than b, a less than b, a approximating to b, and (Gauss' symbol) a ml with b.
Our present day symbols of operation, + (plus) and — (minus), appear to be among the oldest. Both are found in Widmann's arithmetic published in 1489, at Leipzig. In the time of Widmann, the symbols ( ), X, +, and < were unknown. Rudolf had already begun to employ the radical sign. Buts had not yet appeared. In those days almost every thing was expressed by words, or mere abbrevi ations. Yet even then .both cubic and biquad ratic equations had already been solved; the methods even were published. Oughtred used the term aseparatrix° in sense of a mark be tween the integral and fractional parts of a number written decimally. His symbol for a separatrix was L • Stevens had already used a figure a circle over or under each decimal place to indicate its order. And of the various separatrixes that have been em ployed by mathematicians, four are still em ployed : (a) A vertical line is still employed to separate cents from dollars in ledgers, etc. Such a separatrix appears already in 1613, em ployed in a work by Richard Witt. Napier also used a vertical line for the same purpose in his in 1617; (b) a period, still employed as a separatrix, is so used as early as 1612 in the trigonometry of Pitiscus, a Ger man. Napier, in his speaks of so employing a period or comma. The period has always been the prevailing form of the decimal point in America; (c) the Greek colon, a dot above the line, was advocated as a separatrix by no one less than Sir Isaac New ton. His desire was to prevent it from being confounded with a period used as a mark of punctuation. This form of the decimal point is now commonly used in England; (d) Pi tiscus is the author of an Italian work on trigonometry. He, in this work, published in 1608, uses a comma as a decimal point. Kep ler, in 1616, seems to have introduced this mark, the comma, into Germany for the same purpose. Briggs likewise used a comma in his loga rithmic tables, in 1624; and several other early English writers generally employed the comma as a decimal point. But to-day the comma is customary form of the decimal point, not in England, but in countries upon the mainland of the European continent.