I S O 



327 



I S O 



toff**'- integral sign,) at the limit* between which the integral 

 "i 1 is taken. Now since the law, according to which )y 



^ ^ varies, is arbitrary, we may suppose 3_y =: i. if (x), i be- 

 ing an infinitely small given quantity, and 9 (x)_a func- 

 tion of i of a form absolutely arbitrary, and which may 

 even be snl-je.-t to no analytical law. If we suppose 

 X,, x,, JP B . y,. to represent the values of x, y, at the re- 

 jject . and *'(*) ?"(*) &c - the differential 



co-efficients of 9 (x), the part A, A,, will be of the 

 form, 



m,9 (*.) +>, 9' GO 4- v, ' CO + &c. 7 . 

 -o 9 (*.) - J. ' (*o) y. f GO &c. p 

 This, therefore, if determinate quantity, when the 

 extreme values of 9 (f), and its differential co-efficients 

 are any how determined ; but the other part,y*B )ydx, 

 or i. y B. f (x) <f x, being dependent on the whole ex- 

 tent of the function 9 (x), from one limit to the other, 

 ii not determined by those extreme values, because the 

 assigning any number of particular values of an arbi- 

 trary function, and its differential co-efficienU, is not 

 sufficient to determine its form. The condition of the 

 maximum, however, requires that the aggregate of these 

 two parts should vani-h ; which is therefore impossible, 

 unless the nature of the curve be such, that thi* inde- 

 terminate integral shall disappear from the variation, 

 that is, unless B = 0. This equation determines, there- 

 fore, the nature of the curve in general ; but there re- 

 main* to be satisfied the equation A ' A=0, in which 

 no quantities are involved but those which relate to the 

 limits of the integral, and which, therefore, determines 

 those conditions to which the limits must be subject, 

 that among all the curves which have their general na- 

 ture represented by B=0, the particular species and in. 

 <!r.ulual shall be iclected, which renders the proposed 

 integral the greatert or least possible ; in other words, 

 the equation at the limits determine* the arbitrary con. 

 Cants which enter into the integral of B=0. 



The quantity B, when V is a function of x, y, p, 

 f, Sec. such that 



d\' = M/x + N'/.y 

 Is no other than 



+ oic, 



\_ 



1 



from which Euler's general equation results at once ; 

 and in the more complicated case*, where V contains 

 indeterminate integrals, &c. the value of B, deduced by 

 the process for finding the variation of 'f V d x in such 

 CMC*, and pat equal to zero, will, in like manner, afford 

 the general equation expressive of the nature of the 

 curve nought. 



The processes of Euler and the Bernoulli* enabled 

 them merely to discover, in a general way, the nature 

 of the curve possessed of a given property of maxima 

 or minima ; as, for instance, that it is a cycloid, or an 

 ellipse. But the question immediately and naturally 

 arose, of all such curve*, What particular position of the 

 cycloid, or what position and eccentricity of the ellipse, 

 will give the integral in question a greater or less value 

 than it will have for any other cycloid or ellipse, &c. 

 lMnm*i drawn > To determine this, a separate and 

 independent jnoctas, of the nature of an ordinary ques- 

 tion of maiima and minima may be instituted, and 



Problems. 

 Itotonic 

 Sy-K-ni of 

 Tempera- 

 ment. 



would be found sufficient ; but no general formulae for 

 the purpose had been given, nor was it even suspected 



that the two problems might be resolved together by Iioperimt 

 one analysis, till Lagrange, by means of the definite lncal 

 part of his formulae, effected it at once, and indeed ren- 

 dered it an indispensable part of the solution. The dis- 

 cussion of this point belongs, however, more properly 

 to the calculus of variations, and can scarcely be sepa- 

 rated from it. Indeed, since the discovery of that cal- 

 culus, isopcrimetrical problems can no longer be re- 

 garded as forming a separate branch of mathematical 

 investigation ; though, as a point of scientific history, 

 they must ever continue to maintain a degree of inte- 

 rest proportional to the importance of the discovery to 

 which they gave rise. Like the theory of the tangents, 

 evolutes, &c. of curves, they take their place in our ele. 

 mentary treatises, on account of the geometrical illus- 

 tration and relief they give to the abstract theory ; but 

 much of their intrinsic importance is lost, when all they 

 can afford is exhausted, and they have ceased to present 

 a field of unexplored research. 



For further information, we must refer to the me- 

 moirs, &c. above cited ; to a treatise, by Mr. Wood- 

 house, on isoperimetriful problems ; to the various ex- 

 positions which have appeared of the calculus of varia- 

 tion* ; and to the article VARIATIONS in the present 

 work. (j. r. w. HEKSCHEL.) 



ISOTONIC "SYSTEM or TEMPERAMENT, or the equal 

 temperament of the musical scale, consists of 12 equal 



'* /~ 

 semitone*, of the value l-s-^2, SlZ + f-j- i/ t m. 



This system derives it* chief consequence from the great 

 number of writers, and the respectability of several of 

 those who have appeared as its advocates ; among those 

 who have fallen under our notice, we remember the 

 name* of D'Alembert, Broadwood, Cevallo, Chladni, 

 Couperin, Crotch, Davis, Des Cartes, Emerson, Euler, 

 Kirnberger, Kollmann, Marpurg, Merrick, Mersennus, 

 Rameau, Riccio, Scrogs, Sorge, Sulzer, Vogler, &c. 



It i* plain from the account given by most of the wri- 

 ters alluded to, that they had neither submitted this 

 system to the test of experiment, or thoroughly calcu- 

 lated and considered the harmonic effects of its grossly 

 tempered chords, the thirds and the sixths especially ; 

 while many of them -were utterly unacquainted with 

 the true nature and limits of the musical scale, as ap- 

 pear* from their statements, and as to what could or 

 could not be done, owing to the immutable relations 

 which any one of the tempered chords has with several 

 others, and of the whole combined, in a regular dou- 

 zeave. 



In page 273 of our ninth volume, we have inserted 

 a Table of the full particulars of a system, very care- 

 fully and minutely calculated, which Mr. Farey disco- 

 vered in 1807, and first announced in the Philosophi- 

 cal Magazine, vol. xxviii. p. 65, not for the purpose of 

 recommending or advocating the Uotonic system, of 

 which we are now treating, a* being adapted to use ; 

 but for the purpose of shewing a practicable mode of 

 exhibiting a new system, so indefinitely near to the 

 true isotonic, that all its merits and defects might there- 

 by be shewn, and the controversy so long subsisting re- 

 garding this system, ended by an appeal to actual and 

 indisputable experiment. 



In pursuing the same object, Mr. Farey hns very re- 

 cently recommended, in the periodical work above 

 quoted, vol. xlix. p. 447, the undertaking of an experi- 

 mental enharmonic organ, on a scale sufficiently exten- 

 ded to admit of exhibiting two or more octaves, of the 

 great scale of intervals, 612 in the octave, which i* 

 given under our article INTERVAL, so contrived, that the 



