The fact is, that owing to the very great complexity of the mathe matical part of the subject, the part of the calculus of variations which relates to the maxima and minima of integral forms is in a very incom plete state : and it is found impossible to introduce what has been done into elementary works. How long it will be before the mere vanishing of a differential or variation will cease, in elementary works, to be taken as the conclusive evidence of a maximum or minimum, depends on the degree in which mathematics will be studied as a dis cipline, and not solely as an instrument of physical inquiry.
The history of a large part cf the calculus of variational is simply that of dynamics from the time when D'Alembert proposed his cele brated principle (1743). But long before this, the questions of maxima and minima which ultimately came to occupy the greater part of pro fessed works on the calculus of variations, took their rise in the researches of the two Bernoullis, and led to their celebrated quarrel. in Moo. Div.] The first problem, namely, to find the curve of shortest descent between two given points, proposed by John Bernoulli, was quickly followed by others of the same kind, proposed by James Bernoulli, in which the curve to be found was required to bo of a given length. The prevalence of problems in which this last con dition was contained, led to the name of the Sobatien of lsoperimetrical problems, by which the calculus in question was long distinguished. But it must be noted that the first who solved any such problem aa has since been referred to the calculus of variations, whatever may have been his method, was Newton, who, in the Scholium to the 34th proposition of his second book, gives, without demonstration, the con struction requisite for finding the solid of least resistance. [Pacscrsia, cols. 739-40.] The subject was successively taken up by Brook Taylor, Euler, Simpson, Emerson, and Maclaurin, the second of whom first gave the general equation which determines the nature of the function required, independently of the limits of integration ; and his ' 3lethodua inveniendi lineal curves proprietate maximi minimive gaudentes,' published in 1744, being the last of his efforts on this subject which was made before Lagrange came into the field, is an epoch in its history. Lagrange's first change in existing methods was the intro
duction of the specific symbol S to stand for the variation of x (which suggested to Euler the name of the Calculus of Variations) and of the formation of all that part of 6f vdx which is free from the integral sign. Furnished with such an apparatus, ho undertook problems of a much more complicated etass than any of his predecessors, and stamped upon the subject the form which it has never since lost, at the same time that he gave it an extension which it cannot be said to have since greatly exceeded. Lagrange's memoirs were contained in the first and fourth volumes of the' Miscellanea Taurinensia,' published in 1760 and 1773. The M6canique Analytique ' of Lagrange (first edition, 1793) must also he regarded as the first work in which the calculus of varia tions was fully applied to problems of statics and dynamics, in the manner since universally followed. A complete and most excellent history of the rise and progress of the branch of this calculus which treats of the maxima and minima of undetermined integrals is con. Mined in, and forms the substance of, Woodhouse Treatise on Iso perimetrical Problems,' Cambridge, 1810. This work carries on the history to the end of the last century, and is worthily succeeded by Mr. Todhunter's recently published ' History of the Calculus of tions; Cambridge, 1861, 8vo, which describes what has been done by the successors of Lagrange down to the present time. Accordingly, there is now no branch of mathematics of which all the history is an well written as the calculus of variations. The only complete and separate elementary work on the subject is Jellett's ' Calculus of Variations,' Dublin, 1850, an able, elaborate, and, the subject con sidered, intelligible work.