SVM iTATION. 



sVMii.n.s AND NOTATION. 



030 



II.-M found one of iU beat applications in the construction of 

 chinery iUelf. 

 We propose in this article to treat particularly of mftihamfttiotj 



notation, which, like language, boa grown up without much looking 

 t.>, it the dictates of convenience and with the sanction of tin- majority. 

 Resemblance, real or fancied, has been the first guide, and analogy has 

 : 



Signs are of two kinds : 1st, those which spring up anil arr found 

 in existence, but cannot be traced to their origin ; 2nd)y, those of 

 which we know either the origin, or the epoch of introduction, or 

 both. Those of the first kind pass into the second as inquiry advances. 

 I A i PIIABET.] In our present robject we have mostly to deal with the 

 second chvw. 



Mathematical marks or signs differ from those of written language in 

 being almost entirely of the purely abbreviative character, since it is 

 possible that any formula might be expressed in words at length. We 

 nay possible, because it is barely so, not meaning thereby to imply that 

 the mathematical sciences could ever have flourished under a system of 

 expressions in words. A well-understood collection of notions, how- 

 \tensive, becomes simple as a matter of conception by use and 

 habit, and thus becomes a convenient resting-point for the mind and a 

 suitable basis for new combinations of ideas. Now it is the charac- 

 teristic of the advance of human knowledge that the mind never 

 grapples at once with all that is contained in the notions under use for 

 the time being, but only with some abstraction derived from a previous 

 result, or some particular quality of that result. Hence no symbol 

 which should contain the representative of every idea which occurred 

 in the previous operations would ever be necessary ; and more than 

 this, it would even be pernicious from its complexity, as also from its 

 suggesting details which are not required. The generalisation, or 

 rather abstraction, which is the distinctive character of the civilised 

 language as compared with the savage (though the latter is not wholly 

 without it), must be the ruling process of mathematical notation, as it 

 is of the advance of spoken language ; and in this point of view the 

 connection of our subject with speech presents more analogies and 

 gives more instruction than its comparison with the written signs of 

 speech. The latter is a bounded subject. AVhen once it is agreed 

 how the different modifications of sound shall be represented, written 

 language follows immediately; nor do the infinite modes of using 

 words require any modification of the method of writing them. In 

 our modern works, for instance, it would be difficult to find many 

 artifices of notation with which to compare the never-ceasing varieties 

 of mathematical signs. In mentioning the marks of punctuation and 

 reference, the italics for emphatic words, and the varieties of print by 

 which notes are distinguished from text, &c., we have almost exhausted 

 the list. 



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 gitadratum 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, AAAA, &c., to stand 

 for the successive powers of A, were surpassed in the same manner by 

 those who adopted A S , A*, A 4 , &c. Beyond this it is obvious the 

 notation cannot go in simplicity. The symbol which is to represent 

 "HAS multiplied together " must suggest all three components, of the 

 preceding phrase namely, and A, and multiplied together. In A ", 

 the i 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 A S the square described on the line A, purchase 

 simplicity at the expense of certainty. The same mathematical phrase 

 witli 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 convert 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 the 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 the expense of the other. 

 Notation influences discovery by the suggestions which it makes : 

 hence it is desirable that its suggestions should be as many, us plain, 

 and as true, as it U ponmble. Here we are on quite a different ground : 

 reason a the builder and 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 nui.tt not be forgotten that a notation may he 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 rul< 

 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 subject requires. It is to be remembered that the language 

 of the exact sciences, instead of being, as should be the case, a *' 

 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 Encyclopaedia,' 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, Carnot, 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 co 

 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 usage, 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 consei | 

 of narrow and ignorant views in this respect, they cannot be perm 

 The language of the exact sciences is in a continual state of wholesome 

 fermentation, which throws up and rejects all that is in ogmoua, 

 obstructive, and even useless. Had it been otherwise, it is impos-qlile 

 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 struct 

 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. 



1. Distinctions must be such only as are necessary, and they must be 

 sufficient. For instance, in so simple a matter as the use of capitals or 

 small letters, whatever may guide the inquirer to adopt either in one 

 case should lead him to the same in another, unless some useful dis- 

 tinction can be made by the change. Thus a writer who in one instance 

 uses a capital letter to denote a complicated function of small letters 

 (which is a very desirable mode), will in another part of the same 

 question employ a small letter for a similar purpose, thus nullifying an 

 association of ideas which perspicuity would desire to be retained. If 

 such a course were necessary in the first case, it is still more so in the 

 second. It is not often that the second part of this rule is infringed ; 

 so small an addition makes a sufficient distinction, that the pri' 

 danger which arises is that of the same notative difference occurring 

 in too varied senses in different problems. 



The tendency to error is rather towards over-distinction than the 

 contrary. It is surprising how little practice enables the beginner in 

 mathematics to remember that so slight a difference as that of a and a 

 implies two totally different numbers, neither having any necessary 

 connection with the other. The older mathematicians [ACCENT] over- 

 did the use of distinctions by their uniform adoption of different and 

 unconnected letters ; and forgot resemblances. 



2. The simplicity of notative distinctions must bear some proportion 

 to that of the real differences they are meant to represent. Distinctions 

 of the first and easiest order of simplicity are comparatively few ; the 

 complications of ideas of which they are the elements of repre- 

 sentation are many, and varied to infinity. There is no better 

 proof of skill than the adaptation of simple forms to simple notions, 

 with a graduated and ascending application of the more complicated 

 of the former to the more complicated of the latter. 15ut some 

 writers remind us in their mathematical language of that awkward 

 mixture of long and short words to which the idiom of our language 

 frequently compels them in their written explanations of the f..rmnl;e. 

 For example, if there be two words of more frequent occurrein 



.any others, they are numerator* and titnumiiKitui'; the parts of a 

 fraction cannot be described under nine syllables. A mathematician 

 will have occasion to write and speak these words ten t!i"ii-aml times, 

 for every occasion on which he will have to use the wor.l rn.~,,. of four 

 letters. A comparatively rare idea, used in an isolated subject, can l.e 

 expressed in one syllable, while the never-ending notions of the parts 

 of a fraction require nine : this he cannot help; but it is in his pm, . r 

 to avoid the same sort of inversion in his notation. 



3. Pictorial or descriptive notation is preferable to any other, 

 it can be obtained by simple symbols. Many instances occur in 



* These worda might well be ihortcncd into numcr nml dcnomer. 



