"THE WATERFALL. From the German, by Geffner. Is this the vale, whofe fhadowy flood Plung'd lightly from the wood-crown'd height? A frozen column meets my eye, Sufpended from the beetling brow. Ye rocks that crown the vallies deep, No cares fhall there my bofom pain, No fearful thoughts my heart alarm; From hill, from grove, and flowery plain Shall fweetly steal a foothing charm. And wherefore envy those that shine And bask in Fortune's tranfient beam? While, with my flask of jovial wine, While fweet fuccefs may crown my lays, Amid these cool delicious bow'rs: And future ages learn to praife The paftime of my harnlefs hours." P. 16. "FROM THE DANISH. IN EVALES FISKERNE. From high the feaman's wearied fight, O fweetest O fweeteft pledge of love and pleasure, That, if beneath that garb of beauty, I ftrait may feek the waters wide." P. 34. It would far exceed our regular limits, to enumerate every thing with which we are gratified, in this truly pleafing publication: our readers will, however, not object to the following additional proof of Mr. Herbert's poetical abilities. "FROM CATULLUS III. With mournful voice and faultering tongue, Begin ye loves the funeral fong! The bird, my fair one's joy is gone! The bird the nurfed with anxious care, The little darling wifh'd to ftray. Now perch'd upon her graceful head He fweetly tuned his artless note, The furies of your cruel pow'r! Behold my fair one's fwollen eyes With tears of never-ceafing grief! 'Tis ye that cause that breaft to heave; Your hands have oped the fource of woe, And doom'd my lovely nymph to grieve." P. 37. Mr. Herbert's name is already familiar to the literary world, and we anticipate with pleasure, the refult of thofe hours which he may hereafter dedicate to the public infruction or entertainment. We must obferve, however, that this is not the Mr. Herbert who publifhed Sir Reginalde, and other poems, reviewed by us in vol. xxiii. p. 198. ART. V. Analytical Inftitutions, in Four Books, &c. (Concluded from vol. xxiv. p. 600.) Sect. 11. Of the Rules of Integration having Recourfe on infinite Series. AGNESI here firft lays down three rules for reducing fractional and radical quantities into infinite feries, the last of which is the famous binomial theorem of Sir Ifaac Newton; thefe rules fhe alfo illuftrates by examples, of which one is, to raise a multinomial, or infinite feries, to any power denoted By one or other of thefe reductions, fluxionary expreffions, containing quantities of the above defcription, are prepared for integration, by the rules delivered in the preceding Section. She then refers the reader to James Bernoulli's tracts de feriebus infinitis, for fome properties of feries, of which it was no part of her defign to treat. Laftly, the gives a very general, and extenfively ufeful, feries for computing fluents, when the quantity under the vinculum confifts of two terms, together with the investigation of it, and an example of its application. This fublime and widely extended branch of the mathematics has been confiderably increased since Agnefi's time. Sect. III. The Rules of the foregoing Sections applied to the Rectification of Curve-lines, the Quadrature of Curvilinear Spaces, the Complanation of Curve Superficies, and the Cubature of their Solids. This is a large and very valuable Section; containing, firft, the investigations of general formula for the purposes mentioned in the title of it, and then an illuftration of those formula, by forty-feven examples, the particulars of which our limits will not permit us to enumerate; but we obferve among them, not only the quadrature and rectification of the circle, ellipfis, and the reft of the conic fections; and the cubature, and complanation of the furfaces, of the folids generated by these curves; and many others which are now frequently found in treatifes of fluxions; but alfo the curious inveftigations of the folidity and fuperficies of ungulas of various kinds; and several other ufeful matters in menfuration, which of late years have appeared in other books, but (as is ufual with fome bookmakers) without any acknowledgment of the ftores from which they were taken. L BRIT. CRIT. VOL. XXV. FEB. 1805. If If the illuftration of rules by a proper number of wellchofen examples be the best way of inftruction, as has been the opinion of the most eminent teachers, and particularly of Sir Ifaac Newton*, who had many years experience as Profeffor of Mathematics in the Univerfity of Cambridge, then is Agnefi entitled to much commendation for what fhe has done in this Section. Sect. IV. The Calculus of Logarithmic and Exponential Quantities. Here exponential quantities are defined, and their several degrees defcribed; and, as thefe quantities are beft managed by logarithms, the learner is firft taught how to find the fluxion of a logarithm, of any power of a ogarithm, and of the logarithm of a logarithm; then, how to find the fluxion of any exponential quantity; and, laftly, how to find the fluents of fluxionary expreffions, in which there are logarithmic or exponential quantities. In fhort, this curious and difficult fubject is treated by the Italian lady with great ability and elegance, and with it the ends the third Book of these Inftitutions. Book IV. The Inverfe Method of langents. We are now arrived at that part of the work which affords the greatest scope for genius, and accordingly it is here difplayed. The learned lady begins this Book, in her ufual way, with a definition, which is as follows: "As, when any curve is given, the manner of finding its tangent, fubtangent, perpendicular, or any line of that kind, is called the direct method of tangents; fo, when the tangent, fubtargent, perpendicular, or any fuch line is given, or when the rectification or area is given, to find the curve to which fuch properties belong, is called the inverfe method of tangents." Agnefi then illuftrates her definition by three examples, of which the firft is this: "Let the curve be required of which the fubtangent is double to the abfcifs. Calling the abfcifs x, and the ordinate y, the formula of Sir Ifaac fays, in his Arithmetica Univerfalis, "Artes exemplis faciliùs quàm præceptis addifcuntur." See Bishop Horfley's edition of his works, vol. I. p. 63. She She then gives a further explanation of the matter in these words: "The equations which arife by proceeding after this manner will always have (as is eafy to perceive) the indeterminates and differentials intermixed and blended with each other, fo that at prefent they cannot be managed, in order to proceed to their integration, fo as to obtain the curves required; and much more if they contain differentials of the fecond, third, and higher degrees. For, in the third Section aforegoing, the differe tial formulæ have always been fuppofed to be compounded of one indeterminate only, with its difference or fluxion. Therefore other expedients are neceffary, to try to reduce fuch equa tions to integration or quadratures, which is called the conftruction of differential equations, of the firft, fecond, &c. degrees: and as to the construction of thofe of the first degree, we may proceed two ways; one is, to pafs immediately to integrations or quadratures, without any previous feparation of the indeterminares and their differentials; the other is, firit to feparate the indeterminates, and fo to make the equation fit for integration or quadrature. "I fhall proceed to show feveral particular methods for both ways, by which we may attain our purpose in moft equations. But very often we shall meet with others, which will be found fo ftubborn, as not to fubmit to any methods hitherto difcovered, or which have not the univerfality that is neceffary." P. 248. We observe, in tranfitu, that what is, by moft foreign mathematicians, called the conftruction of differential equations, is, by the English, called the refolution of fluxional equations; that is, the finding of their fluents; and that the English notation is ufed throughout this tranflation. Sect. 1. Of the Conftruction of Differential Equations of the first Degree, without any previous Separation of the Inde berminates. This Section contains (in eight pages) the conftructions (that is, refolutions) of many fluxional equations, in which are two variable quantities, fome by algebraic expreffions, fome by logarithms, and fome by expreffions of both thefe kinds; plying or dividing the given equation by fome power or product of the variable quantities, and fome of them by both thefe operations and a tranfpofition of the terms. But we cannot convey to our readers an adequate idea of the ingenuity here. difplayed without a tranfcript of this Section, which our plan will not admit. It will, however, be only justice, both to them and the author, to remark, that among the easiest of the operations here performed, are complete and general folu L 2 tions |