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 MSc 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 principal 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 aicending application of the more complicated of the former to the more complicated of the latter. But 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 formuhr. For example, if there be two words of more frequent occurrence than any others, they are numerator* and denominator; the parts of a fraction cannot be described under nine syllables. A mathematician will have occasion to write and speak these words ten thousand times, for every occasion on which he will have to use the word reap, of four letters. A, comparatively rare idea, used in an isolated subject, can be 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 power to avoid the same sort of inversion in his notation.
3. Pictorial or is preferable to any other, when it can be obtained by simple symbols. Many instances occur in astro.
nomy, and the use of the initial letters of words may be cited as a class of examples : as in f for force, r for velocity, &c.
4. Legitimate associations which have become permanent must not be destroyed, even to gain an advantage. The reason is, that the loss of facility in reading established works generally more than compensates for the advantage of the proposed notation ; besides which, it seldom happens that the desired object, absolutely requires an invasion of established forms. For instance, perhaps the most uniform of all the notations of the higher mathematics is the use of the letter d to signify an increment which is either infinitely small, or may be made as nearly so as we please. A few Cambridge writers, some years ago, chose to make a purely arbitrary change, and to signify by dy, dz, &c., not increments, but limiting ratios of increments : and students trained in their works must learn a new language before they can read Euler, Lagrange, Laplace, and a host of others. Thus d,y has been made to stand for dy :dx, and the old association connected with dy has (in the works spoken of) been destroyed. Now if the letter D had been employed instead, the only harm would have been that the student would have had to learn a new language before he communicated with the greatest mathematicians; as it is, many will have to form a new language out of the materials of the old one, which is a much harder task. This injudicious innovation is now extinct.
5. Analogies should not be destroyed, unless false : for true analogy has been frequently the parent of discovery, and always of clearness. Thus the real analogy of Icpxtix and jcpxdx was lost to the eye by the use of Lepx to signify the latter ; an innovation which preceded the one last-mentioned, and has obtained more approbation in this country, though now almost extinct. The notation used by Fourier to express a definite will certainly prevent the spread of the one just alluded to; though this last itself is chargeable with breach of analogy forrfxd.r!, f &c., ought to represent the successive integrals of fp.rdx. Fortunately, however, the symbols (fdr);;Ox &c., may represent these successive integrals; and thus the two notations may be combined. For instance, (f:d.r)Cpx represents the fourth integral of isx, each integration being made from 0 to x.