The greatest purposes of notation seem to be answered when the reader or learner can tell what is meant, first, with the utmost cer tainty, secondly, with sufficient facility ; it being always understood that the second must be abandoned when it clashes with the first. Too much abbreviation may create confusion and doubt as to the meaning ; too little may give the meaning with certainty, but not with more certainty than might have been more easily attained. Thus the old algebraists, in using A quadratum for A multiplied by A, in their transition from words at length to simple notation, used ten symbols where two only are requisite; and those who first adopted the symbol A A lost no certainty, and gained materially in simplicity. The suc cessors of these, again, who employed A A, AAA, A A AA, &c., to stand for the successive powers of A, were surpassed in the same manner by those who adopted &c. Beyond this it is obvious the notation cannot go in simplicity. The symbol which is to represent " a AB multiplied together" must suggest all three components of the preceding phrase—namely, a and A, and multiplied together. In A *, the 71 and A are obvious, and the position of the letters is the symbol of multiplication ; but, on the other hand, those who teach the beginner to signify by the square described on the line A, purchase simplicity at the expense of certainty. The same mathematical phrase with them stands for two different things, connected indeed, but of more dangerous consequences from that very connection ; for where similarities exist, the reader should not be made to genvert them into identities. It is of as much importance to impress the distinction of the things signified as the analogy of their properties.
Certainty, then, and the greatest facility of obtaining it, seem to be the main points of good notation ; and this is true with respect to the learner of all that has gone before. Grant that the mathematical sciences are never to advance further, and many alterations might be made, and many new practices adopted, which would give facility in acquiring time past, without any introduction of obscurity. But the future must also be thought of ; and no scheme will merit approbation which enlightens one end of the avenue at tho expense of the other. Notation influences discovery by the suggestions which it makes : hence it is desirable that its suggestions should be as many, as plain, and as true, as it is possible. Here we are on quite a different ground : reason is the builder amid settler, but imagination is the discoverer; and it might turn out that a notation which suggests many and obvious new ideas, though some of them should be fallacious, would be prefer able in its consequences to another of less suggesting power, but more honest in its indications. And while we speak of positive suggestion, it must not be forgotten that a notation may be faulty in occupying the part of the symbol which properly belongs to the extension of another notation. The latter is thus deprived of its natural direction
of growth, and must find its way elsewhere, to the injury perhaps of some other part of the symbol. In throwing together a few rules, pre viously to a little description of the present state of mathematical notation, we do not pretend to have exhausted the list of cautions which the eubjeet requires. It is to be remembered that the language of the exact sciences, instead of being, as should be the case, a separate subject, is hardly ever treated at all, and then only in connection with some isolated parts of the system. With the exception of an article by Mr. Babbage, in the ' Edinburgh Encyclopiedia,' we do not know of anything written in modern times on notation in general. Much may be collected, having notation for its specific object, from the writings of Arbogast, Babbage, Carnet, Cauchy, J. Herschel, and Peacock, writers who all have considered it necessary, when proposing a new symbol or modification of a symbol, to assign some reason for the pro posal. In general, however, it is the practice to adopt or reject notation without giving any justification of the course pursued. If it Could be rendered necessary, by the force of opinion, that every author should, in making a new symbol, explain the grounds, firstly, of his departure from established uasge,--secondly, of his choice from among the different methods which would most obviously present themselves,— two distinct advantages would result. In the first place, we should in most cases retain that which exists, until something was to be gained by altering it ; in the second, research and ingenuity would have a call into action which does not now exist. We hardly need mention a thing so well known to the mathematician as that the progress of his science now depends more than at any previous time upon the pro tection of established notation, when good, and the introduction of nothing which is of an opposite character. We should rather say the rate of progress ; for, however bad may be the immediate consequences of narrow and ignorant views in this respect, they cannot be permanent. The language of the exact sciences is in a continual state of wholesome fermentation, which throws up and rejects all that is incongruous, obstructive, and even useless. Had it been otherwise, it is impossible that the joint labours of three centuries and many countries, of men differing in language, views, studies, and habits, could have produced so compact and consistent a whole, as, with some defects (though no two persons agree precisely what they are), the present structure of mathematical language must be admitted to present.
The following rules and cautions, with respect to notation, are drawn from observation of the present state just alluded to.