(61 



SYMBOLS AND NOTATION'. 



SYMBOLS AND NOTATION. 



982 



nomy, and the use of the initial letters of words may be cited as a class 

 of examples : as in / for force, r for velocity, Ac. 



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 : <ls, 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 

 talk. 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 2<ftrAx aaAjifucdx was lost to the eye by the 

 use otj^j:ia 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 integral, / <^d-f, will certainly prevent the spread of the one 

 just alluded to ; though this last itself is chargeable with breach of 



analogy : lurf'^^djftf^^jedi?, Ac., ought to represent the successive 

 integrals of ffxdx. Fortunately, however, the symbols (Jdjcffyx 

 (JdjcpQx, Ac., may represent these successive integrals ; and thus the 

 two notations may be combined. For instance, (J*dx)*$x represents 



the fourth integral of jv, each integration being made from to .r. 



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 tymbolical, 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 

 erious exception to the use of D A B to stand for the square on A B, 

 to H 1 'or parallel, < fbr angle, Jj for perpendicular, Ac. But when 

 we come to AB . BO for the rectangle on AB and BC, AB : for the square 

 on AB, we feel the case to be entirely altered. These are already 

 arithmetical symbols : it is bad enough that the word f/aare 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 element*, 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 

 procem 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 Kuclid, is the correcting process. Will symbolic 

 geometry do an well f We will not answer positively, but we must 

 lay 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, 



di d; iFz d-z dj^ 

 dJc' Ty' 1?' dTdi/' dy>' 



greatly impedes the operations connected with problems in solid 

 geometry : the letters p, q, r, , I, which are often substituted for 

 them, make us lose sight of the connection which exists between the 

 meanings. But the symbols 



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 

 ARTS ASD SCI. DIV. VOL. VII. 



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 formulae to write a fraction, such as r, iu the 



manner in which it here appears ; but if this be done in the text, a 

 line is lost ; and, generally speaking, a : ft, or -=-&, would do as well 

 in mere explanation. Also, in printing, redundancies which are 

 tolerated in writing, should be avoided, such as \/T, where V7 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 \S&, c$, Ac., or old English letters, as 9, 33, H, u, 



Ac., 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 

 copied in writing, or which it is then difficult to preserve : such as the 

 use of A and A, Ac., in different senses; even the distinction of 

 Roman and Italic small letters, a and a, Ac., should be sparingly intro- 

 duced. 



10. Among the worst of barbarisms is that of introducing symbols 

 which are quite new iu 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 

 pages the appearance of expressing surprise and admiration that 2, 3, 

 4, Ac., 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 are 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, 4>, 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 

 projierly with the Roman ones, and few indeed are the instances iu 

 which such symbols as 



a"" e (i+') s 



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 unconnected by the author, is that of placing bla. ks 

 or spaces in the manner in which he would do in ordinary matter, by 

 which he is allowed to separate symbols which are in such close cc n- 

 nection that absolute j unction would not be undesirable. For instanc e, 

 cos 9 for cos9, (a b + c d) for (ab + erf). Aa a general rule, the man; i- 

 script should be imitated. 



4. Accents, superfixed and suffixed, as in a", a u . These are generally 

 continued, when they become too numerous, by Roman numerals, as 

 in ,, a,,, a,,, a>,, a , a,;, Ac. 



5. The signs + x H- : *S t and the line which separates tht 

 numerator from the denominator. Of these there are generally not 

 sizes enough, particularly as to the sign . It frequently happens 

 that such an expression as (# 1) (jc 2) (z 3) Ac. overruns a line very 

 inconveniently, when the use of a shorter negative sign, as in (x-1) 

 (x-Z) x-3) would avoid such a circumstance altogether. Between 

 the division line of a fraction and the numerator and denominator 

 unsightly spaces very often occur, as in 



+* . a+b 



- instead of > 

 c+d c + d 



6. The integral sign /'. with its limits expressed, as in / : the 



symbols of nothing and infinity, and 



7. Brackets, parentheses, Ac. [],(), 

 properly accommodated to the size of 



7. Brackets, parentheses, Ac. [],(),{}, &e. These are often not 

 operly accommodated to the size of the. 

 particularly in thickness. 



he intervening expressions, 



8. The signs of equality, Ac., =, <,>. 



9. Occasionally, but rarely, a bar or a dot is used over a letter, as 

 a or d. In some works, accents and letters are placed on the left of a 

 symbol, as in 'a, 'a, ,p. This however should be avoided, as it is 

 difficult to tell to which letter the symbol belongs ; anil there are 

 ample means of expression in what has been already described. 



There are no general rules laid down for the use of notation : a few 

 hints however may be collected from the practice of the best writers 

 of recent times. 



1 When a letter is to bo often used, it should be, if possible, a 



