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exhibit the state of the subject at the close of the eighteenth century; the first chapter is therefore devoted to these works of Lagrange and Lacroix. The notice of the two works of Lagrange is very brief, for in fact both of them were accessible to Woodhouse, and he has given a good account of all that Lagrange accomplished. The notice of the work of Lacroix is fuller because the second edition of that work had not appeared when Woodhouse wrote; it was also necessary to indicate two important mistakes which occur in Lacroix on account of their influence on the history of the subject; see Arts. 27 and 39.

The second chapter contains an account of the treatises of Dirksen and Ohm.

The third chapter contains an account of a remarkable memoir by Gauss, which affords the earliest example of the discussion of a problem involving the variation of a double integral with variable limits of integration.

The fourth chapter contains an account of a memoir by Poisson on the Calculus of Variations. The great object of this memoir is to exhibit the variation of a double integral when the limits of integration are variable. The memoir is important in itself, and also from the fact that it may be considered to have led the way for those which were written by Ostrogradsky, Delaunay, Cauchy and Sarrus.

The fifth chapter contains an account of a memoir by Ostrogradsky; this memoir was suggested by Poisson's, and its object is to exhibit the variation of a multiple integral when the limits of the integration are variable.

The Academy of Sciences at Paris proposed for their mathematical prize subject for 1842, the Variation of Multiple Integrals. The prize was awarded to a memoir by Sarrus, and honourable mention was made of a memoir by Delaunay. The memoir of Delaunay is analysed in the sixth Chapter, and the memoir of

Sarrus in the eighth Chapter; the seventh chapter analyses a memoir by Cauchy, in which the results obtained by Sarrus are presented under a slightly different form.

Here that part of the present work terminates which treats of the variation of multiple integrals.

The next three chapters treat of another branch of the subject, namely, the criteria which distinguish a maximum from a minimum; these criteria were exhibited in a remarkable memoir published by Jacobi in 1837, which has given rise to a series of commentaries and developments. The method of Jacobi is founded upon one originally given by Legendre; accordingly the ninth chapter first explains what Legendre accomplished, and also what was added to his results by another mathematician, Brunacci, and then finishes with a translation of Jacobi's memoir. The tenth chapter contains an account of the commentaries and developments to which Jacobi's memoir gave rise. The eleventh chapter contains some miscellaneous articles which also bear upon Jacobi's memoir.

The twelfth chapter contains an account of various memoirs which illustrate special points in the Calculus of Variations. The thirteenth chapter contains an account of three comprehensive treatises which discuss the whole subject. The fourteenth chapter gives a brief notice of all the other treatises on the subject which have come to the writer's knowledge.

The fifteenth chapter notices various memoirs which have some slight connection with the subject. The sixteenth chapter notices various memoirs which relate principally to geometry, or differential equations, or mechanics, but the titles of which are suggestive of some relation to the Calculus of Variations.

The seventeenth chapter gives the history of the theory of the conditions of integrability.

The writer has endeavoured to be simple and clear, and he hopes that any student who has mastered the elements of the

subject will be able without difficulty to understand the whole of the work.

It may appear at first sight that great disproportion exists between the spaces devoted to the various treatises and memoirs which are analysed. The writer has not considered solely or chiefly the relative importance of these treatises and memoirs, but also the ease or difficulty of obtaining access to them; and thus a work of inferior absolute value may sometimes have obtained as long a notice as another of higher character when the latter could be procured far more readily than the former.

In citing an independent work the title has usually been given in the original language of the work, but in citing a memoir which forms part of a scientific journal it has generally been considered sufficient to give an English translation of the title. Sometimes a mathematician has been named in the history before an account of his contributions to the subject has been given; in such a case by the aid of the index of names at the end of the volume it will be easy to find the place which contains the account. Occasionally in the course of the translation of a passage from a foreign memoir the present writer has inserted a remark of his own; this remark will be known by being enclosed within square brackets.

The writer may perhaps be excused for stating that he has found the labour attendant on the production of this work far longer and heavier than he had anticipated. It would have been easy to have examined merely the introductions to the various treatises and memoirs, and thus to have compiled an account of what their respective authors proposed to effect; but the object of the present writer was more extensive. He wished to ascertain distinctly what had been effected, and to form some estimate of the manner in which it had been effected. Accordingly, unless the contrary is distinctly stated, it may be assumed that any treatise or memoir

relating to the Calculus of Variations which is described in the present work has undergone thorough examination and study. This remark does not, however, apply to all the productions which are noticed in the last two chapters of this work.

It will be found that in the course of the history numerous remarks, criticisms, and corrections are suggested relative to the various treatises and memoirs which are analysed. The writer trusts that it will not be supposed that he undervalues the labours of the eminent mathematicians in whose works he ventures occasionally to indicate inaccuracies or imperfections, but that his aim has been to remove difficulties which might perplex a student. In the course of his studies the writer frequently found that remarks which he intended to offer on various points had been already made by some author not usually consulted; for example, the considerations introduced in Art. 366 occurred to him at the commencement of his studies, and it was not until long afterwards that he found he had been anticipated by Legendre; see Art. 202.

The writer will not conceal his own opinion of the value of a history of any department of science when that history is presented with accuracy and completeness. It is of importance that those who wish to improve or extend any subject should be able to ascertain what results have already been obtained, and thus reserve their strength for difficulties which have not yet been overcome; and those who merely desire to ascertain the present state of a subject without any purpose of original investigation will often find that the study of the past history of that subject assists them materially in obtaining a sound and extensive knowledge of the position to which it has attained. How far the present work deserves attention must be left to competent judges to decide; should they consider that the objects proposed have been in some degree secured, the writer will be encouraged hereafter to undertake a similar survey of some other department of science.

The writer will receive most thankfully any suggestion or correction relating to the present work with which he may be favoured, and especially any information respecting those memoirs and treatises which may have escaped his observation, and those of which he has only been able to record the titles; see Arts. 394 and 420.

The writer takes this opportunity of returning his thanks to the Syndics of the University Press for their liberal contribution to the expenses of printing the work.


April 15, 1861.

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