• Such are, in my opinion, the defects of existing methods. Still, however, I have not composed a treatise on the subject, by merely remedying them ; that is, by inserting formulæ of sufficient extent, and by more fully explaining and illustrating their principles. But, on a novel plan, I have combined the historical progress with the scientific developenient of the subject ; and endeavoured to lay down and inculcate the principles of the calculus, whilst I traced its gradual and successive improvements. If this has been effected, which I think it has, in a compass not very wide of that which a strictly scientific treatise would have required, the only serious objection against the present plan is, in part, obviated. For, there is little doubt, the student's curiosity and attention will be more ex. cited and sustained, when he finds history blended with science, and the demonstration of formulæ accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject.' pp. iii, iv. Conformably with the plan Mr. Woodhouse has thus pre. scribed himself, he divides his work into eight chapters ; the principal subjects of which will appear from the subjoined analysis.-Chapter the 1st. relates to the problem of the curve of quickest descent, and contains a full developement of the principle of John Bernoulli's solution. In the 2nd chapter. Mr. Woodhouse announces the isoperimetrical problems proposed by James Bernoulli; describes the nature of the solution given by John Bernoulli; explains the distinction between his fundamental and specific equations; shews the application of them to the curve of quickest. descent, and of a given length ; describes Brook Taylor's solution of isoperimetrical problems; and points out the imperfections of the methods employed by him and the Bernoullis. The 3d chapter contains an account of Euler's first memoir on isoperimetrical problems, and of his very ingenious table of formulæ, with their application to the solution of some problems: it also contains a brief account of the methods of Maclaurin, Emerson, and Simpson, and points out their restrictions. The 4th chapter is employed in describing Euler's second memoir, (Comm. Petrop. tom. viii) his general formulæ of solution in that memoir, in tracing the characters of distinction between different problems, and in pointing out exceptions to the ge. neral formulæ: Mr. W. here sheirs in what manner the class of problems leads to the determination of the number of or. dinates that must vary, and the order, the number that must be introduced into the computation. Chapter the 5th is devoted principally to Euler's tract, intitled “ Methodus Inveniendi Lineas curvas," &c: in this the author explains the distribution of cases into absolute and relative maxima and mi. nima, exhibits rules for finding the increment of quantities dependent on their varied state, and more valuable formulæ of solution. The oth chapter relates to Lagrange's first me moir on the theory of variations: and here Mr. Woodhouse shews the uses of an appropriate symbol, such as d, to denote the variation of a quantity, traces the similarity between the differential calculus and that of variations, and deduces the principal rules for finding the variation in any proposed case; a new process is also exhibited of deducing Euler's formulæ, and several new formulæ are given, with their applications to some problems. In the 7th chapter Lagrange's general method of treating isoperimetrical problems is explained, and especially the nature and use of the equation of limits; and several useful remarks are added to shew the method of reducing cases of relative maxima and rninima to those of absolute. In the 8th and last chapter, the author has first shewn how to deduce several subordinate formulæ from Euler's gene. , ral formulæ; being such as, though they are more limited, materially expedite the solution of problems. He then presents a collection of thirty problems with their solutions. Of these some are very curious and interesting, especially those relating to the inquiry of the brachystochrone in all its varieties. From a work like the present, in which almost every page is so intimately connected with what precedes it, (either by the peculiarities of the notation, or the enchainment of logical method,) as scarcely to admit of any separation from it without becoming unintelligible, it is difficult to make any quotations. Perhaps, however, the following extracts may serve to communicate to the scientific reader, as well the spirit of the methods to which they relate, as the manner in which our author treats the respective subjects, • Euler reduced isuperimetrical problems of the second class to a de. pendance on two similar equations of the form Pbg (P + dPci=0, the determination of P depending on the proposed properties : for, if either the isoperimetrical property, or that from the maximum were 7.dx, TdT, T=fig), P would equal , or 2. dx. If the property were ST.dy, T=f(x), P would = or .dx. If the property were ST ds, T=f(x), P would equal d (1.4); and by observation on the resulting forms for P, Euler generalised his conclusions, and arranged them in a table, after the manner of the subjoined specimen. * Proprietates Valores Litteræ P propositae. respondentes. 1. ST.dx ... dT = Mdyt . . P = M.dx •* Comm. Acad. Petrop. tom. VI. p. 141. '+ dT= M dy, and M= Tor M is the differential coefficient of T, making II. ST.dy . . dT = Ndx ... P= N.dx &c. and of these forms he gave fifteen, by reference to which, any problemi belonging to the second class might be solved. For instance, suppose the curve to be required, which, amongst all others of the same length, should contain the greatest area. Here, the maximum property B, =s ydx, . the isoperimetrical A, =sds = a. By Form I. T=y; .: M=1, P= dx. By Form III. T =1 Pro le or R= dT in T, y to vary. Similarly, N= To is the differential coefficient making in T, x to vary. 'If T should contain both x and y, that is, if dT = Mdy + Ndr, then M and N would become partial differential coefficients. See Princ. Anal. Calc. p. 79. ** See Emerson's Fluxions, third edition, p. 187; also Simpson's Fluxions, p. 485. a.dx - = [c correction,] dxi + dy2) (3? -c, dy whence, dx = , an equation to the elastic curve ; V (a - [ye to c]?) and which in a particular case, when c 0, becomes y.dy dx = V(a' - y) and the curve in this case is called the rectangular elastic curve. * • As a third example, let the curve be required, which, amongst all others of the same length, shall have its center of gravity most remote from the axis. Here, (calling x the distance from the axis) B .. by Form III. (since s is a given quantity] P. again an equation to the catenary, • Ibis example could not have been solved by Euler's table, if the property had been any other than the isoperimetrical : for s, an integral. =/ dy *2); and Euler gives, in this memoir, no general met od of finding the resulting equation, such as P is in his tabl., when the ana. lytical expression of a property involves ini sils. See tom. VI. p 144. By means of this table, the practical solution of isoperimetrical problems, was, as it has been already said, very materially expedited. In a subsequent part of his memoir,t Euler increases his table by nine new forms: making the whole number twenty-four. And alinough this table is now superseded, yet its examination is not without interest. since we may discover in it the parcels of that general formula, which the author afterwards exhibited.' pp. 40–44. The subjoined quotation serves to explain an essential part of Lagrange's method, and is so simple as to need no parti. cular explanation. Whatever be the function V, if IV = Mdx + Ndy + Pdp + Qd.; + &c. then d V = Max + Nøy + Port log + &c. Since the processes for finding the differential and variation differ only in the symbois dy, dy, which are arbitrary ; it is plain, if both operations are to be performed on an analytical expression, that it is matter of indif * See Simpson's Fluxions, p. 486. where the solution is not general, t Comm. Petrop. tom, vi. p. 146. ference, which operation is performed first : or, if the symbols d, , meet together denoting operations, we may, at our pleasure, change their order : for instance ddy and ddy are alike significant ; for ddy means the firs term of two successive values of dy, or = d (y + dy) - dy = 8y + Sdy - dy= ddy: again, if V for instance be a function of y; then d? V V= dy, and do V = dy.dy ; .8V = ddv, or, in a particular instance, when V = yn, d iyn) = lst term of [(y+dy)n —yn] =nyn 1.dy n(n-1) yn-2.dy.dy, n(n- yn-2.dy dy; ..dd (yn) = dd (yn): And, by similar processes, d'} V = 8d V = 8ddV = dådV d8V = ddV = d8d2V=d&dV. • This rule is, in Lagrange's method, of the greatest importance ; it is an essential part of it Amongst other uses, it enables us when an integral is concerned, to introduce the symbol d within the symbol (s) of the integral : thus, since the symbols d and s indicate reverse operations, V=dSV; ...87=8df V = ddV. Hence, taking the integrals on each side 18 V =şdif V =:V ...[a] This result may be easily extended to double and treble integrals : for if V=SW, then ¿V= W=fW by [a]; ...f8V=S$&W : but fiv=jV= FF W, consequently offW= Sfå W' pp. 83_-85. It is now time for us to characterize this work, which we may do very shortly, by saying that we prefer it very much to any preceding performance of the same author. It is more methodical, more perspicuous, infinitely less affected, and will, we doubt not, be far more useful. In a few instances the links which connect one method with another in the history of discoveries, are not all supplied ; and two or three inadvertencies have escaped the author. But the chief things of which young mathematicians will complain, after they have read this treatise, will be ambiguities arising from the defects of the system of notation pursued by foreigners, and adopted, con amore, by Mr. Woodhouse. Thus in some cases, d, d, mark the extremi. ties of a line which is a variation of an ordinate, while, in others they are employed to designate, the former the differential, the latter the variation of a quantity ;—and then the reader needs to be told, (as at the note, p. 45.) that “dd has no con . |