where K is the sum of the corrections introduced by the integrations.

If r = 0, or a have no variation, the first part of this formula will coincide with Euler's; for the latter part we are indebted to La Grange, who further deduced the value of the variation, when taken between two specified limits. Euler's formule will determine the nature of the curve or the relation of x to y; but other considerations sometimes occur in which they cannot be applied. For example: the curve of quickest descent between two given points, or between a given point and any other point in a right line or a curve, is a cycloid. This can be proved by the equation of Euler; but to determine the particular cycloid, down which the time is a minimum, or the angle in which the cycloid must cut the straight line or the curve, we must have recourse to the formula of La Grange.

The use of this formula, when adapted to the variation between two limits, is exemplified toward the close of the work in several cases, which may be considered as undetermined conditions, belonging to certain problems, in which the relation of to y had been previously determined. The defectiveness of Euler's equations has thus been supplied, and the solution of isoperimetrical problems may be considered as complete.

The remainder of this chapter relates principally to the second and third problems of La Grange; and the variation of svdx is determined according to different assumptions for the value of dv. These pages are so purely mathematical, that we despair of giving a very satisfactory idea of them without entering into detail; and we are unwilling to lengthen this already extended article by analytical disquisitions or the mere exhibition of formulæ. It may be sufficient to state, that among the expressions deduced is one adapted to the case, in which v contains an integral s, and that all the formulæ of solution to determine the nature of the curve were invented by Euler, with the exception of two; these two conclude the chapter.

The 7th chapter exhibits the general method of treating isoperimetrical problems, as given by La Grange in the "Leçons sur le Calcul des Fonctions." Mr. W. remarks of this method, that "it is distinguished rather by its mode of treating the question, than by any thing novel in its principles." (P. 100.) The

rules which it comprizes extend to all cases of maxima and minima, both absolute and relative.

The 8th chapter contains a variety of problems to exemplify the application of the principles already established. With a view to simplify and facilitate the solution of them, certain particular expressions are in the beginning of the chapter deduced from the general formula: they are for the most part easily remembered, and not difficult in application. The first eleven problems involve only one property, that of the maximum or minimum; "and therefore in strictness, as Mr. W. remarks, ought not to be classed amongst isoperimetrical problems, since they involve neither the isoperimetrical property, properly so called, nor any other equally affecting the theory and the analytical processes." (P. 121.) The remaining problems involve more than one property, and the author concludes his work with an illustration of the determinate formulæ of La Grange.

To those who are little conversant with mathematical studies, we are perfectly sensible that this account of Mr. Woodhouse's publication will at best appear somewhat obscure; and we are not quite certain that even all of our mathematical readers will follow out every part of it, unless their attention has been previously directed to this particular department of science. The subject certainly cannot be classed among such as are of very easy comprehension; but by those who have a taste for analytical pursuits, we think that the work before us will in general be read with pleasure.

To many the notation will be somewhat repulsive; and the question will probably be asked, why could not the author avail himself of the English notation instead of the foreign? where was the necessity for puzzling his readers by rejecting the language and the process, to which, if they understand the doctrine of fluxions, they are already accustomed, and involving his researches in the mists of ds and deltas? The advantages of a fresh notation ought unquestionably to be obvious, and such Mr. Woodhouse considers to be the case in the present instance*. Whether there was sufficient reason to justify the innovation we pretend not to decide; but we would certainly recommend to those who peruse the work to perform the differential operations in the differential language: habit will render the use of that language easy, uncouth and forbidding as it may at first sight

appear.

It is frequently urged as an objection to analytical disquisitions, that their authors are in too great haste to generalize; hence it

* See preface, p. 6.

happens that some of their reasonings appear hardly conclusive, and some of the results not perfectly satisfactory. The force of this objection must be sometimes admitted; but it must be further observed, that general reasoning is often better comprehended when we see it applied in particular instances; and, unless we deceive ourselves, the former chapters of this treatise will be more fully understood after the perusal of the last, in which the formulæ are applied to the solution of problems.

Should it after all be demanded, what is the immediate use of these enquiries, and what practical purpose are they likely to answer, we venture to reply, that though the first place is undoubtedly to be given to those works of science which can be converted to the purposes of life, yet no science is therefore to be rejected because its application is not at first perceptible. The seasons would doubtless have observed their appointed periods, and the enjoyments of life would have suffered little dimi nution, though problems on isoperimetry had never existed; but it would be a new and a barbarous rule, which would fetter the laudable exertions of genius, and without any respect for intellectual excellence or the general improvement of knowledge, would look at practical benefits as the sole test and standard of utility. To combine practice with theory is unquestionably the higher praise; such was the praise of Maclaurin: "His peculiar merit as a philosopher was, that all his studies were accommodated to general utility; and we find in many places of his works an application even of the most abstruse theories to the perfecting of mechanical arts." But he also must be considered as entitled to no mean commendation, whose labours are directed to improve the powers of the mind, and to extend the boundaries of liberal science.

In the prosecution of his enquiries Mr. Woodhouse has confined himself almost exclusively to the mathematicians of the continent. Among the reasons which induced him to pass over the researches of our own countrymen are these: 1. That he wished to arrive by the most regular process at the conclusions of La Grange. 2. That the chapters usually assigned to this subject in our treatises on fluxions are defective and inadequate. The chief notice which is taken of their labours we have in the following passage.

"The researches of Maclaurin, Emerson, and Simpson on this subject, may here be noticed. With regard to practical methods of solution, they do not extend so far as those of Euler, which we have been speaking of; and in point of perspicuity, if we except

• Life of Maclaurin prefixed to his Fluxious, p. xviii. second edition.

Maclaurin, the other two mathematicians are inferior to the learned foreigner.

"The methods of Maclaurin and Simpson (for Emerson's is plainly taken from that of the former,) extend to cases, in which more than one property is involved; but they are inapplicacle to the three cases, and the connected problems enumerated in p. 30. "Maclaurin's formula of solution is this: if x and z are functions of r, then if xds zdy be a minimum or maximum, xdy = zds. This result is included amongst Euler's. For since xds expresses one property, and dx = 2dx, ax, or since x is a function of x, we have by form 111, the quantity corresponding to P (see p. 41,) =d

dy

ds

(x ds), and for zdy expressing the other property, by form 11,

.

dz

the quantity, corresponding to P. dr; consequently the re

dy
ds

=

dx

sulting equation is d (x-1)= a. dz, and xdy = a zds, the same result in fact as Maclaurin's.

"Simpson's method is equally restricted with Maclaurin's; it rests too on the assumption of the principle, that the property of minimum or maximum, true for the whole curve, is true also for any portion of it. The want of generality, therefore, in this principle, would vitiate the method in its application to the excepted

cases.

"The methods just described solve not problems of greater depth and intricacy than those of the Bernoullis; although it must be remarked, they are invested with greater analytical neatness and compactness. They are not however more perspicuous; and even if they did possess greater extent and clearness, it would not suit the purpose of the present tract longer to insist on them, since they conduct us not towards that formula and algorithm, with which the researches on this subject have been closed." (P. 48.)

We must, however, be excused for thinking, that a little enlargement of the plan would have made this, at least in the estimation of Englishmen, a more perfect treatise. We would have recommended the addition of two chapters, each in fact independent of the plan, which the author has prescribed to himself. Of these the first should contain a distinct enunciation of the methods* proposed in the books of fluxions which have

* The following short account may give some idea of the methods adopted siuce the time of Maclaurin.

Emerson presents us with two rules. The first is deduced from the ordinates of a curve in arithmetic progression The determination of the equation depends upon the position of the inte mediate ordinate. The principle upon which his demonstration is founded is, that the maximum or minimum, which belongs to the whole curve, must belong to the part intercepted between the ordinates. Thus it

been published in this country, and a detail of the reasons why they are defective: the second should give the elements of the science, on geometrical principles. Many persons who shrink from the pursuit of abstruse enquiries, would read and understand the geometrical process, and make themselves masters of the elements. A very elegant and perspicuous chapter on this subject has recently appeared in the third volume of "A Course of Mathematics," by Dr. Hutton, who has reduced into system the chief propositions of L'huillier, Le Gendre, and Horsley, together with some additional propositions, which those geometers had not deduced. We should have recommended a geometrical chapter on the elements with the more earnestness, because with some writers it seems a settled principle, that geometry is never to be admitted, where analysis can by any contrivance supply its place.

If the length of this article appears to be disproportioned to the magnitude of the work under consideration, this circumstance

11

one given quantity = A+B+C+D+E+ &c. and another which is required to be a maximum = a+b+c+d+e &c. and all the quantities be supposed constant except two, which correspond, we have C + D constant, and c+d a maximum; hence Ċ + Ď = 0; and ¿ + d 0; and the parts C and D, c and d, being expressed in terms of the same variable quantities, we can from the solution of the equation determine the nature of the curve. His second rule applies to cases which are somewhat more complex. He supposes A - B to be a maximum or minimum, and proves Ax = B., A and B being functions of 1 or 2. This process merely reciprocates the functions A and B. The demonstration of Lyons is similar to Emerson's; the same remark applies to Mr. Vince's.

Simpson's theorem supposes that when fymu is equal to a given value, then

Sy'. u2±2 is a maximum or minimum; and his conclusion is that

27

is a constant quantity. Or more generally, if R and S

u2 + y2

must be a constant quantity. This expression is to be

Ꭱ

applied to particular cases, and the equation of the curve to be deduced by substitution. For the cases in which a new condition is introduced, and to which this

expression does not apply, he gives the equation

q are constant quantities.

The principle of uniformity is introduced by most writers in the solution of cases after the manner of John Bernoulli.

VOL. III. NO. V.

E