4. Accents, superfixed and suffixed, as in a', a,,. These are generally continued, when they become too numerous, by Roman numerals, in a„ a,,, a,„ as„ a, a,i, &c.
5. The signs + — x : -1, and the line which separates the numerator from the denominator. Of these there are generally not sizes enough, particularly as to the sign —. It frequently happens that such an expression as (x-1) (x —2).(x-3) &c. overruns a line very inconveniently, when the use of a shorter negative sign, as in (x-1) (x-2) x-3) would avoid such a circumstance altogether. Between the division line of a fraction and the numerator and denominator unsightly spaces very often occur, as in — a+ b instead of — c+d c d G. The integral sign/ . with its limits expressed, as in/ : the symbols of nothing and infinity, 0 and co.
7. Brackets, parentheses, &c. [ ], ( ),{ ' &c. These are often not properly accommodated to the size of the intervening expressions, particularly in thickness.
8. The signs of equality, &c., =, <,>.
9. Occasionally, but rarely, a bar or a dot is used over a letter, as a or d. In some works, accents and letters are placed on the left of a symbol, as in 'a, 'a, This however should be avoided, as it is difficult to tell to which letter the symbol belongs ; and there are ample means of expression in what has been already described.
There are no general rules laid down for the use of notation : a few hints however may be collected from the practice of the best writers of recent times.
I. When a letter is to be often used, it should be, if possible, a small letter, not a capital. The latter species is generally used for functions of small letters.
2. The letters d, o, 8, and D, are appropriated for operations of the differential calculus, and should hardly ever be used in any other sense.
3. When co-ordinates are used, the letters x, y, r, must be reserved to signify them ; x, y, and t, n, C, may bo used if different species be required, and if xi, z', &c., or x„ y,, z„ &e., should not be judged con venient.
4. When functional symbols are wanted, the letters p, ry, x, F, 4., 4+, should bo first reserved for them ; afterwards 7, w, r, sometimes 1. Pr P.
5. The letter 7 is, by universal consent, appropriated to 3'14159 , and e (by the French e) to 211828 ; r to the functional symbol for 1 .2.3. a.
6. When many operations of differentiation occur, superfixed accents should be avoided in any other sense than that of differentiation.
7. When exponents are wanted to aid in signifying operations, the powers should be carefully distinguished. Thus, in a process in which sin-1x is very much used for the angle whose sine is a', the square cube, fic., of sine should not be sin1x, aroma, &c., but (sha)', (sinx)°, &c. Some writers would have the latter notation employed in all cases; but this is, we think, asking a little too much.
8. Greek letters arc generally used for angles, and Italie letters for lines, in geometry. To this rule it is desirable to adhere as far as possible, but it cannot ho made universal.
9. Suffixed numerals are generally the particular values of some function. Thus a, means a function of r, of which the values for r=0, r=1, &c. are , , &c.
10. As to the radical sign, Va,'4/ a, &e., do not generally mean any one of the square roots, cube roots, &e., of a, but the simple arith metical root. The indeterminate root is usually denoted by the exponent. Thus a± may be necessary, but a441 has a super fluity.
11. The same letters should be used, so far as possible, in the same sense throughout any one work; and some preceding good writer should be followed. As a general rule, those only are entitled to invent new symbols who cannot express the results of their own investigations without them.
The writer who is most universally acknowledged to be a good guide in the matter of notation is Lagrange. This subject is of great importance; but 'fortunately it is pretty certain that no really bad symbol, or system of symbols, can permanently prevail Mathematical language, as already observed, is, and always has been, in a state of gentle fermentation, which throws up and rejects all that cannot assimilate with the rest. A received system may check, but cannot ultimately hinder, discovery : the latter, when it comes, points out from what symbolic error it was so long in arriving, and suggests the proper remedy.
For the progress of mathematical language, see MATHEMATICS, RECENT TERMINOLOOY Ix; and TRANSCENDENTAL: See also SYMMETRY.