_Ordinary Questions of Maxima and Minima._ In the common theory of _maxima_ and _minima_, it is proposed to discover, with reference to a given function of one or more variables, what particular values must be assigned to these variables, in order that the corresponding value of the proposed function may be a _maximum_ or a _minimum_ with respect to those values which immediately precede and follow it; that is, properly speaking, we seek to know at what instant the function ceases to increase and commences to decrease, or reciprocally. The differential calculus is perfectly sufficient, as we know, for the general resolution of this class of questions, by showing that the values of the different variables, which suit either the maximum or minimum, must always reduce to zero the different first derivatives of the given function, taken separately with reference to each independent variable, and by indicating, moreover, a suitable characteristic for distinguishing the maximum from the minimum; consisting, in the case of a function of a single variable, for example, in the derived function of the second order taking a negative value for the maximum, and a positive value for the minimum. Such are the well-known fundamental conditions belonging to the greatest number of cases. _A new Class of Questions._ The construction of this general theory having necessarily destroyed the chief interest which questions of this kind had for geometers, they almost immediately rose to the consideration of a new order of problems, at once much more important and of much greater difficulty--those of _isoperimeters_. It is, then, no longer _the values of the variables_ belonging to the maximum or the minimum of a given function that it is required to determine. It is _the form of the function itself_ which is required to be discovered, from the condition of the maximum or of the minimum of a certain definite integral, merely indicated, which depends upon that function. _Solid of least Resistance._ The oldest question of this nature is that of _the solid of least resistance_, treated by Newton in the second book of the Principia, in which he determines what ought to be the meridian curve of a solid of revolution, in order that the resistance experienced by that body in the direction of its axis may be the least possible. But the course pursued by Newton, from the nature of his special method of transcendental analysis, had not a character sufficiently simple, sufficiently general, and especially sufficiently analytical, to attract geometers to this new order of problems. To effect this, the application of the infinitesimal method was needed; and this was done, in 1695, by John Bernouilli, in proposing the celebrated problem of the _Brachystochrone_.