»«ci. ..] DISSERTATION SECOND. 27 



these problems, a particular provision seemed to be made in 

 the new geometry. 



When any function becomes either the greatest or the 

 least, it does so by the velocity of its increase or of its de- 

 crease ceasing entirely, or, in the language of algebra, be- 

 coming equal to nothing. But when the velocity with which 

 the function varies becomes nothing, the fluxion which is 

 proportional to that velocity must become nothing also. 

 Therefore, it is only necessary to take the fluxion of the 

 given function, and by supposing it equal to nothing, an equa- 

 tion will be obtained in finite terms (for the fluxion will en- 

 tirely disappear), expressing the relation of the quantities 

 when the function assigned is the greatest or the least pos- 

 sible. 



Another kind of maximum or minimum, abounding also in 

 interesting problems, is more difficult by far than the preced- 

 ing, and, when taken generally, seems to be only accessible 

 to the new analysis. Such cases occur when the function of 

 the variable quantities which is to be the greatest or the least 

 is not given, but is itself the thing to be found ; as when it is 

 proposed to determine the line by which a heavy body can 

 descend in the least time from one point to another. Here 

 the equation between the co-ordinates of the curve to be 

 found is, of course, unknown, and the function of those 

 co-ordinates which denotes the time of descent cannot 

 therefore be algebraically expressed, so that its fluxion 

 cannot be taken in the ordinary way, and thus put equal 

 to nothing. The former rule, then, is not applicable in 

 such cases, and it is by no means obvious in what man- 

 ner this difficulty is to be overcome. The general pro- 

 blem exercised the ingenuity of both the Bemoullis, as it 

 has since done of many other mathematicians of the greatest 

 name. As there are in such problems always two condi- 

 tions, according to the first of which, a certain property is to 



