Another instance of a breach of this rule is to be ob served in the Thicrie des -Vonzbres of Legendre, who, throughout that highly valuable work, employs the sign = in two different senses ; first, in its ordinary accepta tion ; and, secondly, he places it between two quantities, to denote, that when they are divided by the same quan tity, they will leave the same remainder. This relation of quantity is of frequent occurrence in the theory of numbers ; and the necessity of denoting it with brevity, induced the author of the Disquisitiones ..grithmeticx to invent the symbol =, and to adjoin (inclosed in brack ets) the quantity which is used as the divisor. A differ ent method has been adopted by.Mr. Barlow, who, in his Treatise on the Theory of Aiznz3crs, uses the double f (ff) placed in a horizontal position. These three me thods of denoting the same relation stand thus : L. an= — 1 G. 1 (mod. ft) B. an 'cz, pv-1.
The first of these sins unpardonably against therule we are endeavouring to enforce ; it is much more inconve nient than the use of dots by Lagrange, because, in the instance we are considering, the same symbol = is used in the same page, and even in the san:e line, in two differ ent senses. It has also the disadvantage of requiring the divisor to be expressed in words.
This innovation in the use of a well known sign pro bably arose from too strict an adherence to an admitted rule : Not to multiply the nunzber of mathematical sigps without necessity. We may, however, here be permitted to observe, that the necessity existed; and it is acknow ledged by Gauss, whose notation, although much pre ferable to the one we have been criticising, is decidedly inferior to that of Mr. Barlow. The preference we have given to this latter is also supported by another rule too evident to require much argument. When it is required to express new relations that are analogous to others for which signs are already contrived, we should employ a notation as nearly allied to those signs as 'WC conveniently can.
That analogy ought to be our guide in the formation of all new notations, is a truth, which, like tnany others, has been felt and acted upon, although it may not have been stated in express terms : and it was probably this feeling which induced Stifelius to inquire into the mean ing of negative exponents, the consequence of which was the establishment of the connection between the direct and reciprocal powers, a deduction which, when enlarged by the consideration of fractional exponents, was no mean addition to the state of algebra at the time it was suggested.
It has long been usual to denote known quantities by the first letters of the alphabet, and to represent un known ones by the last letters. In the sixteenth and seventeenth centuries, the practice used to be to employ the vowels for the unknown, and the consonants for the known quantities. This is in itself a matter perfectly
arbitrary, and, provided it is adhered to, either method of distinguishing known from unknown quantities is equally proper; but whenever one of these, suppose that now in use, is fixed ttpon for this purpose, if we wish to consider known and unknown functions, and to treat of their relations, it is no longer a matter of indifference how they are to be distinguished. The rule we are now illustrating must be attended to, and if WC have used the first letters of the alphabet for known quantities, it com pels us to employ the first letters of whatever other al phabet we may fix on to indicate functions for the known ones, and the latter letters of the same alphabet for the unknown ones. Thus, if a, b, c, Etc. are known quanti ties, and x, y, z, Sec. unknown ones, cz, p, sce. must denote known functions, and , x„ unknown ones. In another instance, the repetition of a letter xx has been marked thus .z.3. Analogy, therefore, would direct us when we wish to repeat an operation as 44.,, to contract it by writing it thus, 4,2, and similarly for others, as 4,3, We have already had occasion to remark, that we ought not to multiply the number of mathematical symbols without necessity. This is a maxim of so much import• ance that it deserves a fuller consideration.
The natural tendency of the science is to develop new relations and new combinations of those already known. When these new relations involve complicated combi nations of such as arc already received, or when they are of frequent °cent lence, it becomes necessary, if it were merely for the sake of brevity, that some new sym bol should be employed ; it is, however, a necessity that ought to be avoided as much as possible; and the rule is one which has suffered many infractions in modern times. Amongst the numberless instances may be re marked the case of a function consisting of the product of a number of terms in arithmetical progression. 'Phis species of function is of some itnportance, and has been frequently considered, yet scarcely any two authors have agreed in the notation they have employed. The first, in point of time, is that of Vandermoncle. (Acad. des Sciences, 1779.) He expressed the quantity, x . x — a . .r —2a . . . . — n—i a by the sign Le, , or when a = 1, more shortly, thus [x]n. Kramp, in his treatise on .4strononzical Refractions,' calls the functions in question fatal/era nutneriques, and denotes them thus X X . + a . . n—la. Arbogast has occa sion to treat of the same quantities,f when he represents thus Dn 14* 7I --. 1.