6. False analogies should never be introduced ; and, above all, the incorrect analogies which custom and idiom produce in language should not be perpetuated in notation. It is becoming rather common to make editions of Euclid which are called symbolical, and which supply signs in the place of many words. To this, if properly done, there cannot be any objection in point of correctness : nor can we take any serious exception to the use of An to stand for the square on AD, to II for parallel, < fbr angle, for perpendicular, &c. But when we come to no for the rectangle on AB and B c, for the square on a s, we feel the case to be entirely altered. These are already arithmetical symbols : it is bad enough that the word square should have both an arithmetical and a geometrical meaning, and causes plenty of confusion : a good notation, if it cannot help in avoiding this con fusion, should at least not make it worse. At the same time, with regard to symbolic geometry, we feel some repugnance to introduce it into the elements, from observing that all the best writers seem to feel with one accord that pure reasoning is best expressed in words at length. If it be desirable that a student should be trained to drop reasoning, except as connecting process with process, and to think of process alone in the intervening time, it is also most requisite that he should have a corrective of certain bad habits which the greatest caution will hardly hinder from springing up while he is thus engaged. Arithmetic and algebra amply answer the first end ; and geometry, in the manner of Euclid, is the correcting process. Will symbolic geometry do as well d We will not answer positively, but we must say we much doubt it.
7. Notation may be modified for mere work in a manner which cannot be admitted in the expression of results which are to be reflected upon. The mathematical inquirer must learn to substitute, for his own private and momentary use, abbreviations which could not be tolerated in the final expression of results. Work may sometimes be made much shorter, and the tendency to error materially diminished, by attention to this suggestion. • For example, the complexity of the symbols, d: d=z dy d.r dy greatly impedes the operations connected with problems in solid geometry : the letters p, q, r, s, t, which are often substituted for them, make us lose sight of the connection which exists between the meanings. But the symbols Z,, Z, Zey, Zyy are not long nor complicated enough to partake much of the disad vantage of the complete symbols, while they are entirely free from that of the isolated letters.
8. In preparing mathematical writings for the press, some attention should be paid to the saving of room. In formula which stand out from the text, this is not of so much consequence ; but in the text itself a great deal of space is often unnecessarily lost. For example, it is indispensable in formula to write a fraction, such as in the manner in which it here appears ; but if this be done in the text, a line is lost ; and, generally speaking, a : b, or a±b, would do as well in mere explanation. Also, in printing, redundancies which are
tolerated in writing, should be avoided, such as J7, where ,/7 would do as well.
9. Strange and unusual symbols should be avoided, unless there be necessity for a very unusual number of symbols. The use of script letters, such as ,,CV, &c., or old English letters, as (a, 13, b, &c., except in very peculiar circumstances, is barbarous. A little attention to the development of the resources of established notation will prevent the necessity of having recourse to such alphabets. Nor is it wise to adopt those distinctions in print which are not easily 't• copied in writing, or which it is then difficult to preserve : such as the use of A and A, &c., in different senses ; even the distinction of Roman and Italic small letters, a and a, &c., should be sparingly intro duced.
10. Among the worst of barbarisms is that of introducing symbols which arc quite new in mathematical but perfectly understood in common language. Writers have borrowed from the Germans the abbreviation n ! to signify 1 . 2 . 3 ... —1)n, which gives their pa.g,es the appearance of expressing surprise and admiration that 2, 3, 4, &c., should be found in mathematical results.
The subject of mathematical printing has never been methodically treated, and many details are left to the compositor which should be attended to by the mathematician. Until some mathematician shall turn printer, or some printer mathematician, it is hardly to be hoped that this subject will be properly treated.
The elements of mathematical notation arc as follows : 1. The capitals of the Roman alphabet, and the small letters of the Italic. The small Roman letters and the Italic capitals are rarely used, and should be kept in reserve for rare occasions.
2. The small letters of the Greek alphabet and such capitals as are distinguishable from the corresponding Roman ones, as A, +, T.
3. The Arabic numerals, and occasionally the Roman ones.
Of all these there should be three different sizes in a good mathe matical press, and the different sorts should bear a much better proportion to one another than is usual. The Greek letters seldom set properly with the Roman ones, and few indeed are the instances in which such symbols as an" are, as they ought to be, good copies of the manner in which they are written. The handwriting of a bad writer is frequently more intel ligible to the mathematical eye than the product of the press. Among the faults to which the compositor is naturally subject, and which frequently remain uncorrected by the author, is that of placing bla, ks or spaces in the manner iu which he would• do in ordinary matter, by which he is allowed to separate symbols which are in such close ecn neetion that absolute junction would not be undesirable. For instance, cos 0 for cos0, (a b + c d) for (ab+cd). As a general rule, the maim script should be imitated.