63 



SYMMETRY, SYMMETRICAL. 



SYMPTOM. 



small letter, not a capital. The latter species is generally used for 

 functions of small letters. 



2. The letters rf, A, 8, and D, are appropriated for operations of the 

 differential calculus, and should hardly ever be used in any other sense. 



::. When co-ordinates ore used, the letters r, //, --, must be reserved 

 to signify them ; .r, y, ;, and {. n, (, may be \\aed if different species be 

 n quired, and if at ', y', /, Ac., or x,, y lt z p Ac., should not be judged con- 

 venient. 



4. When functional symbols are wanted, the letters <t>,^/,x, F, /, 

 +, V, should be first reserved for them ; afterwards *, u, T, sometimes 

 r, {, /i, . 



6. The letter w is, by universal consent, appropriated to 3-] 41. 19 

 .... and < (by the French e) to 271828 . . . . ; r to the functional 

 symbol for 1 .2.8. n. 



(i. When many operations of differentiation occur, superfixed accents 

 should be avoided in any other sense than that of differentiation. 



7. When exponents are wanted to aid in signifying operations, the 



id In- carefully distinguished. Thus, in a process in which 



sin"" 1 * is very much used for the angle whose sine is x, the square 



cube, Ac., of siux should not be sin-'x, sin'*, Ac., but (siiur) 2 , (sin*) 3 , 



-..me writers would have the latter notation employed in all 



cases ; but this is, we think, asking a little too much. 



8. Greek letters are generally used for angles, and Italic letters for 

 in., -. in geometry. To this rule it is desirable to adhere as far as 

 possible, but it cannot be mode universal 



9. Suffixed numerals are generally the particular values of some 

 function. Thus a, means a function of v, of which the values for 

 t;=0, r=l, Ac. are BO , 0| , Ac. 



10. As to the radical sign, Vi \/ "> * c -> do not generally mean any 

 one of the square roots, cube roots, Ac., of a, but the simple arith- 

 metical root. The indeterminate root is usually denoted by the 

 exponent. Thus ; V* Buy be necessary, but a + M has a super- 

 fluity. 



11. The same letters should be used, so far as possible, in the same 

 sense throughout any one work; and some preceding good writer 

 should be followed. As a general rule, those only are entitled to 

 invent new symbols who cannot express the results of their own 

 investigations without them. 



The writer who is most universally acknowledged to be a good guide 

 in the matter of notation is Lagrange. This subject is of great 

 importance; but fortunately it is pretty certain that no really bad 

 symbol, or system of symbols, can permanently prevail. Mathematical 

 language, as already observed, is, and always has been, in a state of 

 gentle fermentation, which throws up and rejects all that cannot 

 assimilate with the rest. A received system may check, but cannot 

 ultimately hinder, discovery : the latter, when it comes, points out 

 from what symbolic error it was so long iu arriving, and suggests the 

 proper remedy. 



For the progress of mathematical language, see MATHEMATICS, 

 RECEST TEHMINOLOOT is ; and TIIA.NSCKXDKNTAL : see also SYMMKTHY. 



SYMMETRY, SYMMETRICAL (Mathematics). These terms are 

 now applied to order and regularity of any kind, but this is not their 

 mathematical meaning. Euclid first used the word "summetros" 

 (avufttrpos) to signify commensurable, and this well-known Latin word 

 is in fact merely the literal translation of the Greek : two magnitudes 

 then were symmetrical which admitted of a common measure. In later 

 times, and those comparatively recent, the word was adopted both in 

 geometry and algebra in different senses. 



Since symmetrical applies in its etymology to two magnitudes which 

 can be measured together (by the same magnitude), the term would, 

 as to space-mil guitudes, uaturally apply to those which may be made 

 to coincide. But the term equal had occupied this ground ; and when, 

 in Euclid, the word equal, which was originally defined in the manner 

 just expressed, had degenerated into signifying equality of area only, 

 the term SIMILAR entered to express sameness of form, so that figures 

 having perfect capability of coincidence, or the same both in size and 

 form, were called equal and similar. The word symmetrical was 

 therefore not wanted, and was finally introduced to signify that 

 obvious relation of equal and similar figures which refers to their 

 position merely, and consists in then: corresponding portions being 

 similarly placed on diferent tides of the same straight line ; so that 

 coincidence counot be procured without turning one t'gare round that 

 straight line. Suppose, for instance, the front of a building to be 

 rymmetrical : draw a vertical line through the middle of the elevation, 

 and the two lateral portions are equal and similar, as Euclid uses those 

 words. But they are more than equal and similar ; they are sym- 

 metrical : the right-hand side stands in the right-hand portion of space, 

 with respect to the dividing line, and in exactly the same manner as 

 the left-band side stands in the left-hand portion of space. It the 

 architect were to preserve equality and similarity, without symmetry, 

 be would make two left sides, or two right sides, to his building, but 

 not one right and one left. In the letter w there is a want of nyin- 

 . but not in o : to make w symmetrical, both the inner lines 

 should Ire thin, and Ixith the outer ones thick. 



Kuclid assumes the power of turning a plane round, so as to apply 



it of two figures to one another, in such manner that, after the 



application, the spectator must be supposed to see through the paper 



or other imaginary substance of which his plane is the surface. He 



has then no occasion to consider symmetry; that is, figures being 

 equal and similar, no cases can arise in which it makes any difference 

 of demonstration whether they be symmetrical or not. When lie 

 comes to solid figures, he assumes a postulate in the garb of a dclini- 

 tion, which dispenses him from the consideration of synu 

 namely, that solid figures consisting of the same number of equal 

 planes, similarly placed, are equal. He seems to imagine tli ," 

 solids must evidently be capable of being made to occupy tli 

 space, which, though true as to quantity of space, is not true as to its 

 disposition. Two.solids may be equal in every respect, and yet it may 

 be impossible (and precisely on account of their symm. -ti \) t> make 

 one occupy the space previously occupied by the other. The two 

 hands furnish an instance : they give the idea of equality (ot 

 similarity (of form), and symmetry (of disposition). Yet they 

 be made to occupy the some space, so as, for instance, to fit exactly the 

 same glove ; and a sculptor who should cast both hands from the some 

 mould, would be detected immediat. ug given his figure two 



right bauds or two left handy. Again, suppose two solids, in 

 nice, composed of planes similar and equal, i- 



one to one of the other. Let coincidence be attern pted ge< 1 1 1 1 t r '.( 1 1 y : 

 the two bases must of course be made to coincide. If, the n, the two 

 vertices fall on the same side of the common base, the figui < 

 coincide altogether ; but if the two vertices fall on opposite sides of the 

 bases, absolute coincidence is impossible. Legendre proposed to call 

 such solids by the name of .- in doing which lie introduced 



the term of common life in an appropriate manner. 



In algebra, a function is said to be symmetrical with respect to any 

 two letters when it would undergo no change if these letter 

 interchanged, or if each were made to take the place of the other. 

 Thus, 



is symmetrical with respect to a and 6 ; interchange would give 



the same as before. But this expression is not symmetrical with respect 

 to a and x, for interchange would here give 



An expression is symmetrical with respect to any number of letters 

 when any two of them whatsoever may be interchanged without 

 tion of the function. Thus d*b + alt" + d 2 c + ar + Ifc + lie- is symmetri- 

 cal with respect to a, b, and c. It is not sufficient that certain contem- 

 poraneous changes should be practicable without producing alteration : 

 must be interchangeable, the rest remaining. Thus ''/) + 

 b-c + c-a is unaltered if a become b, b become c, and c become a, at the 

 same time, but it is not symmetrical ; for if a and b only be inter- 

 changed, it becomes b-a + a-c + c-b, or is altered. t 



Attention to symmetry is of the utmost consequence in mathematical 

 notation. Here 'the word means that quantities which in any manner 

 have a common relation should have something common in the symbols 

 of notation; and analogy is perhaps a better word than symmetry. 

 Suppose, for instance, we had taken, for the equation of a Slid 



OND DEGREE, ax 2 + by'' + cz 2 + dxy + cxz +fifz + ijj~ 4 // 

 Our formuhc would have been confused masses of letters, n 

 which would have presented any similarity, or have . ued in 



the memory. But iu the article cited there is no set of Winnl ..... f 

 which more than one need be remembered ; the others must be sug- 

 gested by it. 



SYMMETRY, CENTRE OF. [CENTRE, col. 734.] 



SYMPATHETIC INK. [COBALT.] 



SYMPHONY (aw, willt, and <pwr/i, sound), a term very differently 

 understood at different periods of musical history. Some v, 

 according to Zarlino (Parte 3za, cap. Ixxix.), have considered it as an 

 instrument of the lyre kind. Others have thought it a sort of ilium. 

 If an instrument, that it was used as an accompaniment mn .-i 

 bably to the voice the word in its original signification leads us 

 naturally to conjecture. 



With the moderns, Symphony, or x;,,fi,ti!a, signifies a musical 

 position for a full band of instruments, and, up to the latter p 

 the last century, the word was synonymous with overture; sym- 

 , and among these several of Haydn's early ones, having been 

 called overtures. Even at the present day the overture in tin- 

 poser's score of an Italian opera is usually termed >'/</',)/. The 

 modern symphony generally consists of four movements : a brilliant 

 allegro, which is commonly preceded by a slow introduction ; an ex- 

 pressive adagio or andante ; a minuet with its trio ; and a 

 Instead of the minuet, what is called a .V/ur,-o, a short, anii 

 sportive movement, is sometimes substituted. But d ire nut, 



restricted by any rule regarding the number of movements. Mozart's 

 second symphony in u has but three, besides the slow introduction ; 

 while Beethoven's Pastoral and Choral symphonies may be said to 

 comprise six or more. 



Symphony is a term also applied to the instrumental introdu- 

 ctions, &c., of vocal compositions; and these are SOUK 

 called ritornels, from the French rittn'i-i>'tt<, or the Italian 



SYMPIESOMETER. [BAKOMETKB.] 



SYMPTOM (ffu/xu-TGyia, an incident, or coincidence) is any change in 

 the appearance or functions of, the body differeii from those which 



