For example: 1r21 would in America be written decimally as 1.50; in England as 1.50; and in Germany, France and in Italy as 1,50.
Symbols in general may now be mentioned: The sign of cancellation is simply drawn across the factors cancelled, like the one drawn across the zeros above. These with the ordinary Arabic and Roman numerals and the abbrevi ations for the names of things, like f, s, $, 0, ", constitute the main symbols of ordinary arithmetic. In algebra there are four kinds of symbols: symbols of quantity, of operation, of relation and of abbreviation. quantities are represented by the first letters of the alphabet; or by the final letters with one or more accents, as x', Be. sides letters of Greek and even of the Hebrew alphabet are employed. Thus M stands for ((modulus') of any system of logarithms. it has been spoken of.
Of symbols of operation, multiplication may be indicated by placing a point between the factors when both are expressed by letters, as a . b. In a series multiplication between fac tors expressed even by numbers, may be ex pressed by a point between such factors. When ". is used between two quantities, it denotes difference, but not which quantity is to be sub tracted from the other. Division has been sometimes indicated thus: a I b. The radical sign, V,when placed over a quantity indicates that its root is to be taken. Thus Va denotes I _ the square root of a. In the same manner Va and Va are read the cube root and the nth root of a, respectively. Here and n give the index of the radical. A vinculum, , a bar I, brackets [ ], and parentheses ( ) all indicate that the quantities enclosed by them are to be regarded together as wholes. In algebra, denotes that the algebraic sum of several quan tities of the same nature as that to which the sign is prefixed is to be taken. The letter f, F and 0, written before any quantity, or quan tities, separated by commas, as F (x), f (x, y), (x, y, a), et cetera, denote quantities depending upon the quantities, or the quantity, within the parenthesis, without designating the nature of the relation. The signs of proportion, : : :, when placed be tween quantities show them to be in proportion. R, r and P, and other symbols, denote radii of circles. L, I and X may denote latitude. The leading letters of the Greek alphabet are also sometimes used to denote known angles; and the final letters of the same alphabet to denote unknown angles. And when several quantities of the same kind are involved in an investiga tion, they may be designated by the same letter accented; thus: a' a" a'"; a, a, and as. An older usage of o denotes an infinitely small quantity; and only sometimes absolute zero. Clearer thinking has impelled moderns to use c (iota) or i for an infinitesimal, and to denote by o, absolute zero, and that only.
The symbol co, first employed by Wallis in the 17th century, has long been used both for a variable increasing without limit and for absolute infinity. Taylor, in 1898, introduced the symbol cV, a contraction of % for an in finite, the reciprocal of an infinitesimal. The sign denotes then or therefore; and the sign • since or because. The y = f(x) is a general sign, indicating that there is a general relation between y and x— that is, y and x are so con nected that x cannot change without y changing at the same time. The symbol F (x, y, z,) == 0 implies that there is a general relation between x, y and z, without specifying the relation. The symbols sin, cos, tan, co-tan, sec, co-sec, ver-sin, co-ver-sin, are abbreviations used, re spectively, for the words *sine,* gent," ((cotangent," °secant)) °co-secant,* gversed-sine and ((co-versed-sine?) When the arc is supposed, as in trigonometry, to depend for its value upon any of the trigonometric lines, the function is called the inverse tri gonometric function. The following symbols are used to denote this kind of a relation: cos—'y, tan—'y, cot—'y, sec—'y, co sec—'y, co-ver-sin—'y. ' These stand, respectively, for the arc upon the sine, cosine, tangent, co-tangent, secant, co secant, versed-sine and co-versed-sine,— is y. The principle of notation has been extended to all inverse functions; thus: logy, (x d x ), etc.
These stand, respectively, for the quantity whose logarithm is y', and the quantity whose differen tial is x d x, etc. The differential of a func tion, or independent variable, is denoted by d, thus: d If we suppose the form of a function to vary, the symbol employed to denote the variable is d, thus: Om, If both the form of the function and the inde pendent variable of the function vary together, the resulting variation is denoted by the symbol D, thus: D f (x,y) The differential is the difference between two consecutive states of the quantity differentiated. If it is desired to represent the difference be tween two states of a function which are not consecutive, the symbol , is employed. Succes sive finite differences are represented by the symbols: Au, L'u, Atu, A'u, etc.
We have already spoken of 2 as used in algebra, where it denotes an algebraical sum. Its use in the calculus is principally restricted to the denotation of the sum of the finite dif ferences of a function. The symbol f denotes an integration to be performed; while the symbol f a is used to denote a definite integral taken between the limits a and b. The symbol r + 1) stands for the integral fe The vector sign is U. in quaternions, should be read tensor of B.
In the foregoing list nearly all the symbols, commonly employed by American mathe maticians, have been enumerated; as well as some used abroad.