ISOPERIMETRICAL PROBLEMS. 



321 



1:.' 1- 



pertod, obtained over the mind of this singular man. 

 No sooner had he identified the bracbystochrone, on the 

 supposition of uniform gravity, with the cycloid, whkb 

 bad previously been identified with the Uutochrone, on 

 the same hypothesis^, titan he celebrates it as a wise 

 dispensation of Providence, or, at least, as a wonderful 

 instance of the frugality of nature in her operations, 

 thus to make one curve servo two purposes, ob-ervinp, 

 that if"* could not have happened, were not Galileo's 

 by pathetic of uniform gravity agreeable to nature, al- 

 though Newton's discoveries had long ago demon- 

 strated iu talaehood, an<l Bernoulli well knew the fact 

 to be to. 



It would not be difficult to clear Bernoulli's solution 

 from any objection on the score of rigour ; and, in fact, 

 he *peedi!y obtained a more direct one, upon which, 

 in June 1696, he proposed his problem in the Leipsic 

 sets, under the title, Proklema riorum, ad cujut tolulio- 

 nem Mathrmaiui ini-itantur, allowing six months for the 

 solution, which, however, at the request of Leibnitz, 

 who, as well a his brother James Bernoulli, had resol- 

 ved the problem, was prolonged to a .year. At the ex- 

 piration of this period, a multitude of the chief mathe- 

 maticians of the age were found to have been success- 

 ful, but the only direct analysis which appeared was 

 that of James Bernoulli. The principle on which this 

 analysis turns is of very extensive application, and it 

 once reduces this, and other problems of the same na- 

 ture, to questions of ordinary maxima. It is this that 

 thf maximum or minimum properly, mhick belongt to the 

 n-hiAe curve, belongt also to eetry rlemmtary, or infinite. 

 ly rmall partt'.n <* / It is true, this principle is not ab- 

 ohitely general, and therefore roust be verified before 

 it ic applied in any particular case. In the present it 

 is easily shewn to hold good. 



Let ABCDE be the curve (Fig. 1.) required, BCD 



any element, and, if pouible, let B c D be sn arc of some 

 other nmre, the tip u which is less than 



that of describing ' -ice the velocity n' 



the same, whether thr body fall down A Be D r A BCD, 

 the time through DF, will br the same on cither sup- 

 position, and tin rrtVirr the whole time through AB c DE 

 less than orrtra hrpothesin. 



Suppose, now, the curve ABODE, according to the 

 spirit of the differential calculu*, resolvd into an infi- 

 nite number of rectilinear dements, of which H( . ( I), 

 are two, corresponding to two element* I equal 



to each other.) irf the vertical abscissa AP (T). the arc 

 AB IH Ti2 called i, and the ordinate PB, y On tliii 

 n, the velocity must bo regarded as uniform 

 ich of the elements in d if we call 



D the velocity of describing BC, then v + d v (or v') 



rot. xii. 



will be the consecutive velocity with which the arc 

 CD = d t -f- d * * (= ds') is described. Now, to dis- 

 cover what function y is of j, we must endeavour to 

 establish some relation between the differentials d t and 

 dy ords, by means of the proposed minimum property ; 

 and to this effect, regarding the points B, D, as fixed, 

 and C moveable along the line QC, we must inquire 

 what must be the position of C upon that line, (or what 

 relation FC and BF must bear to each i ther,) that the 

 time through BC, with the uniform velocity v, pint the 

 tim through CD, with the uniform velocity v', shall be 

 minimum. Now, this is evidently an ordinary ques- 

 tion of maxima and minima ; it is, in fact, identical witli 

 Fermat's problem concerning refraction above mention- 

 ed, and the solution is precisely similar ; BCD will be 

 described in a minimum of time, when the two lines 

 BC, CD, make angles with the vertical, whose sines arc 

 to each oilier as the velocities v and v' with which they 

 are described. This gives at once 



' sin. ICD 



or 



- sin. HCB' sin. ICD ~ sin. HCB ' 



that is, (since sin. HCB = ^ , and sin ICD = 



. at u t 



v'dS vdt 



. _ 



or, since - ,- is the consecutive value of 



vdt 



whence we get 



vd* 



as constant = a. 



This is the relation required between the differen- 

 tials, or the differential equation of the curve, for v ii 

 given in functions of x. On the supposition of uniform 

 gravity, we have t>=v/I, nd writing y^ for a to make 

 both sides homogenous, 



the equation of cycloid, whose base is the horizontal 

 line AK. Such is the solution of James Bernoulli, 

 cleared of the geometrical form which embarrasses and 

 obscures it, and expressed, as he probably would have 

 expressed it, in the present state of symbolic reason- 

 ing. 



There is one peculiarity, however, in the question of 

 ordinary maxima and minima, to which the problem is 

 here reduced, v'z. that the quantity to be made a maxi- 

 mum or minimum is itself a differential expression, or 

 infinitely small magnitude. This does not at all affect 

 the truth of the conclusion, which is independent of 

 the absolute magnitude of the lines ami velocities con- 

 cemed, but it does the manner of treating it In pro- 

 blems of maxima and minima, the quantities concerned 

 are supposed to vary by increments infinitely smaller 

 than the quantities themselve*. Now, the variation of 

 the length of FC, or the differential of FC (dy), on the 

 supposition that C changes its place on QC. cannot be 

 expressed by d dy, because it would thus be confound- 

 is 



triad 

 Problems. 



