In case it is required to express a series continued to n terms, the nth term should be written, and not the words " to n terms" be attached to a few of the first. These different methods would appear thus : In this simple case, something is gained in point of brevity; and if deductions are to be made from it, much is gained in perspicuity. The justice of this observa tion will be more readily acknowledged if it is put into rather different language. lts object may be thus stated : /t is better to nzake any ezpression an apparent function of n, than to let it consist of operations n times repeated. Thus, if it is required to express the series —+ — n 2n Sn By means of a definite integral, its value vvould be d x fd x cl x x x 1—x [xx:°11 Where there are n integrations; or as it has been wri'. ten by Mr. Spence, d fd x lad x [x=0] x is Ji—x x=i Neither of these methods possess that clearness which ought instantly to convey to the mind the system of operations intended to he expressed ; for it does not ap pear quite certain whether the n means that there are /Id x n repetitions of /—, or that the whole number of in el z' tegrations is 71: this latter is the signification in which it is employed by Spence. A better method of express ing it would be, I n x n X of—x d xn, or (c1-1---x d xn To any one acquainted with the integral calculus, this could not convey an incorrect idea, although presented for the first time, and without explanation. The index n denoting the repetition of the quantity to which it is attached, gives ...fi x d x x I —x And it is clear that the ndxs must be assigned to the n integrations. This latter expression may indeed be employed as deviating still less from the received one, and as still less liable to misinterpretation.
Another principle, which is chiefly valuable as it con duces to brevity, is, That all notation should be so con trived as to have its parts callable of being employed separately. By observing this rule, we avoid being en cumbered with any other sig-ns than those actually ne cessary to express the properties wc are considering. This principle has been so universally followed, that it is difficult to produce examples of its neglect. Thus, although it has never been expressly proposed in terms, it has been virtually acted upon by analysts for the last two centuries.
It was in fact to be expected, before any genet al prin ciples of notation were ddivered, that it should be ex tended in exact proportion to the progress of the in quirer, and this progress proceeding from the simple to the more complicated, his Rotation would naturally increase by continued additions. Such being its origin, it will necessarily follow, that at any stage it might be used without reference to those additions with which subsequent considerations had obliged him to augment it.
It is a. valuable service rendered to any science, to embody into language those rules which perhaps insen sibly direct or govern the minds of those who improve it ; however apparently obvious some of them may be, or however well known amongst the instructed, a siderable time will always elapse before individual Letration will have formed for itself such guides ; nor the advantage be entirely confined to the student. How frequently, cloes it happen, even to the best in formed, that they prefer one thing and reject another, from some latent sense of their propriety or impropriety, without being immediately able to state the reasons on which such choice is founded; vet it cannot be doubter], when the selection appears to be the result of correct taste, that it is guided by unwritten rules, themselves the valued offspring of long experience. Any expla
nation of these is probably rendered unavailing at the instant, for want of having previously fixed them by lan guage.
Another reason \\filch appears to justify, or rather to oblige us to notice this principle, is, that although it has not hitherto been much infringed, it seems probable that, unless clue care be taken in the future formation of notation, some of the rules which have been proposed may themselves lead to its infraction. This observa tion more particularly applies to that which dirccts us " to contrive all notation, so that it shall be capable of expressing laws." In attempting to satisfy this condi tion, we are obliged to take wide and extensive view.s, and are therefore peculiarly liable to forget that Nvhich may, appear of minor importance. The two principles are by no means incompatible ; and it is very desirable that when WC have occasion to employ the latter, the formcr may always be borne in mind.
Having discussed the general principles on which notation should be formed, it now becomes necessary to explain several rules which are proposed for the pro per application of individual signs. Writers on alge bra have defined the meaning of the signs they use ; but something further than these definitions is requisite in the present state of the science, or, to speak more accurately, some limitations, as well as extensions of the meaning of several of them, are required.
There are in analysis two great divisions of symbols, —those which denote quantity, and those which indicate operations. These latter were all of them originially arbitrary marks, such as those employed to express ad dition, multiplication, Ere. At the present time, how ever, letters are frequently employed to signify opera tions, and hence arises occasionally a source, if not of error; at lust of inconvenience. The use of letters in two senses has been objected to the followers of Leib nitz, who employ the d to denote the differential of the quantity to Nvhich it is prefixed; and as differentials are much employed in the doctrine of series, it sometimes occurs that the fourth term of the series a + b x c x2 d x3 -1-, enters into calculations in Nvhich the d also represents an operation performed on the letter to which it is prefixed. This inconvenience has been avoided by M. Lacroix, who uses the Roman d for dif ferentiation, and tbe Italic d for quantity. Arbogast also, in his work on Derivations, employs the Roman d for thc same purpose. This completely removes the inconve nience ; but in order to prevent its recurrence in any other shape, and for the purpose of affording a wider choice in the selection of letters, we would propose the following general rule : .911 letters that denote quantity should be printed in Italics, but all those which indicate operations should be printed in a Roman character. Thus the letters x, y, z,. . a, b, c, . . a,f3, P, would re present quantity; and, d, f; A, B, C, would denote ope rations. The only inconvenience which Nvould attend this method, is of very minor impor(ance : it would be come necessary. to cast ncw types for a few of the Greek letters in common use, because the difference between those two modes of printing does not exist in that lan guage.