devoted to the exposition of a method of solving problems relating to certain cases of motion, which was useful at an epoch previous to the introduction of D'Alembert's Principle. All that it contains relative to elastic bodies is reproduced in that part of the Methodus inveniendi lineas curvas which treats of the vibration of elastic curves; and Euler on page 283 of that work cites the present memoir. A few points may be noticed. Euler here, as elsewhere, is not very careful with respect to his notation. For instance, in his diagram g it will be seen that a and A denote certain points; in the corresponding text it will be found that a denotes also a certain length, and A a certain elastic force. 52. Euler assumes that the elasticity along a curve varies inversely as the radius of curvature (see his page 113); he gives only this brief reason, namely that the more the rod is bent the greater is the elastic force. His words are; "Cumque eadem vis elastica sit eo major, quo magis curvatur, erit vis elastica in M ut V divisum per radium osculi in M." The differential equation d'y dx seems unable to state at once the general form of the solution: (see his pages 116 and 1171). In the Methodus inveniendi, page 285, he gives the general form, namely, 53. Euler, 1744. The celebrated work of Euler relating to what we now call the Calculus of Variations appeared in 1744 under the title of Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. This is a quarto volume of 320 pages; of these pages 245-310 form an appendix called Additamentum I. De Curvis Elasticis, which we shall now examine. 1 The following extract from page 122 may interest those who study the history of music: "Ope hujus regulae inveni in instrumentis musicis, quae ad tonum choralem attemperata sunt, chordam infimam littera C notatam minuto secundo 118 edere vibrationes; summam vero, quae ễ signari solet, eodem tempore vibrationes 1888 absolvere (sic)." T. E. 3 [54] The memoir commences with a statement which is of extreme interest as shewing the theologico-metaphysical tendency which is so characteristic of mathematical investigations in the 17th and 18th centuries. It was assumed that the universe was the most perfect conceivable, and hence arose the conception that its processes involved no waste, its 'action' was always the least required to effect a given purpose. That the results obtained by such metaphysical reasoning would differ according to the method in which 'action' was measured, does not seem at first to have occurred to the mathematicians. Thus we find Maupertuis' extremely eccentric attempt at a principle of Least Action. On the whole it is however probable that physicists have to thank this theological tendency in great part for the discovery of the modern principles of Least Action, of Least Constraint, and perhaps even of the Conservation of Energy. The statement to which we refer is the following: Cum enim Mundi universi fabrica sit perfectissima atque à Creatore sapientissimo absoluta, nihil omnino in mundo contigit, in quo non maximi minimive ratio quaepiam eluceat, quamobrem dubium prorsus est nullum quin omnes mundi effectus ex causis finalibus, ope methodi maximorum et minimorum, aeque feliciter determinari queant, atque ex ipsis causis efficientibus. [55.] Euler then cites several examples of this natural principle and mentions the service of the Bernoullis in the same direction. He continues: Quanquam igitur, ob haec tam multa ac praeclara specimina, dubium nullum relinquitur quin in omnibus lineis curvis, quas Solutio Problematum physico-mathematicorum suppeditat, maximi minimive cujuspiam indoles locum obtineat; tamen saepenumero hoc ipsum maximum vel minimum difficillime perspicitur; etiamsi a priori Solutionem eruere licuisset. Then stating that Daniel Bernoulli (see Bernoulli's letter of Oct. 1742, Art. 46) had discovered in the course of his investigations that the vis potentialis represented by fds/R2 was a minimum for the elastic curve, Euler proceeds to discuss the inverse problem, namely: 56. To investigate the equation to a curve which satisfies the following conditions. The curve is to have a given length between two fixed points, to have given tangents at those points, and to render fds/R2 a minimum: see pages 247-250 of the book. No attempt is made to shew why this potential force should be a minimum in the case of the elastic curve. By the aid of the principles of his book Euler arrives at the following equations where a, a, B, y are constants, Ex quibus aequationibus consensus hujus curvae inventae cum curva elastica jam pridem eruta manifeste elucet. Quo autem iste consensus clarius ob oculos ponatur, naturam curvae elasticae a priori quoque investigabo; quod etsi jam a Viro summo Jacobo Bernoullio excellentissime est factum; tamen, hac idonea occasione oblata, nonnulla circa indolem curvarum elasticarum, earumque varias species et figuras adjiciam; quae ab aliis vel praetermissa, vel leviter tantum pertractata esse video. 57. Accordingly Euler gives on his page 250 his investigation of the elastic curve in what he has just called an a priori manner. But this method is far inferior to that of James Bernoulli; for Euler does not attempt to estimate the forces of elasticity, but assumes that the moment of them at any point is inversely proportional to the radius of curvature: thus he in fact writes down immediately an equation like (1) on page 606 of Poisson's Traité de Mécanique, Vol. I., without giving any of the reasoning by which Poisson obtains the equation'. 58. Euler starts with the supposition that the elastic curve is fixed at one point, and is bent by the application of a single force 1 On page 250 observe P is used as elsewhere in two senses, namely for a force, and for the position of a point. at some other point; and then he considers cases in which instead of his single force we have two or more equivalent forces. On his page 255 he supposes that the forces reduce to a couple, and he shews that the elastic curve then becomes a circle. Saint-Venant alludes to this in the Comptes Rendus, Vol. XIX. page 184, where he remarks: Déjà M. Wantzel, dans une communication faite le 29 juin à la Société Philomatique, a remarqué que la courbe à double courbure, affectée par une verge primitivement cylindrique, solicitée par un couple, est nécessairement une hélice. C'est une généralisation du résultat d'Euler, consistant en ce que lorsque la courbe provenant de la verge ainsi solicitée est plane, elle ne peut être qu'un arc de cercle. 59. Euler distinguishes the various species of curves included under the general differential equation of Art. 56; he finds them to be nine in number. The whole discussion is worthy of this great master of analysis; we may notice some of the points of interest which occur. The third species is that in which the differential equation reduces to dy = = x dx Na* see his page 261. This is not substan tially different from the particular case we have noticed in (Art. 24) our account of James Bernoulli. The curve touches the axis of a at the origin; and Euler calls it the rectangular elastic curve. In connection with the discussion of this species Euler introduces two quantities f and b which are thus defined: Quanquam autem hinc, neque b, neque ƒ per a accurate assignari potest; tamen alibi insignem relationem inter has quantitates locum habere demonstravi. Scilicet ostendi esse 4bf=τaa. I do not know where Euler has shewn this; however b and f can now be expressed in terms of the Gamma-function. For put (See Integral Calculus, page 291, Example 24.) 5a π Hence we obtain he finds approxi πα mately ƒ= X and still more closely f= x 1.1803206; 24/2 = x 1.1803206 instead of √/2 Thus he makes 834612, which is too is about. a 60. On his page 270 Euler observes that he has hitherto considered the elasticity constant, but he will now suppose that it is variable; he denotes it by S, which is supposed a function of the arc s;p is the radius of curvature. He proceeds to find the curve which makes fSds/p a minimum1; and by a complex investigation finds for the differential equation of the required curve a+Bx-vy=S/p, where a, B, y are constants. This he holds to be necessarily the correct result, by the same principle as in Art. 54; and he says on his page 272: Sic igitur non solum Celeb. Bernoullii observata proprietas Elasticae plenissime est evicta; sed etiam formularum mearum difficiliorum usus summus in hoc Exemplo est declaratus. [This 'principle' is again due to Daniel Bernoulli: see Art. 74. ED.] |