Engineering

"Bernoulli writes... to Euler... Sept. 1743 [and] extends his principle of the 'vis viva potentialis laminae elasticae' to laminae of unequal elasticity, in which case \int E ds/R^2 is to be made a minimum. The... letter...in... April or May 1744... expresses his pleasure that Euler's results on the oscillations of laminae agree with his own."

"[In an Oct. 20, 1742 letter, Daniel Bernoulli] suggests for Euler's consideration the case of a beam with clamped ends, but states that the only manner in which he has himself found a solution of this "idea generalissima elasticarum" is "per methodum isoperimetricorum." He assumes the "vis viva potentialis laminae elasticae insita" must be a minimum, and thus obtains a differential equation of the fourth order, which he has not solved, and so cannot yet shew that this "aequatio ordinaria elasticae" is general.Ew. reflectiren ein wenig darauf ob man nicht konne sine interventu vectis die curvaturam immediate ex principiis mechanicis deduciren. Sonsten exprimire ich die vim vivam potentialem laminae elasticae naturaliter rectae et incurvatae durch \int ds/R^2, sumendo elementum ds pro constante et indicando radium osculi per R. Da Niemand die methodum isoperimetricorum so weit perfectionniret als Sie, werden Sic dieses problema, quo requiritur ut \int ds/R^2 faciat minimum, gar leicht solviren. [Ew. reflect a little on whether one can not deduce the curvature of the bar directly from the principles of mechanics. In the first place I express the actual elastic laminar potential, naturally right and yet curving, by \int ds/R^2, summing the element ds per constant radius of curvature R. Since no one has perfected the isoperimetric method as much as You, So this problem, which requires that \int ds/R^2 be minimum, might be easily solved.]"

"Galilei's problem had determined the direction of later researches... while James Bernoulli solved the problem of the elastic curve his nephew Daniel first obtained a differential equation which really does present itself in the consideration of the transverse vibrations of a bar."

"The direct impulse to investigate elastic problems... came to Euler from the Bernoullis."

"Gravesande in his Physices Elementa Mathematica [Vol.1; Vol. 2], 1720, explains elasticity by Newtonian attractions and repulsions. The... chapter... entitled De legibus elasticitatis [The laws of elasticity].... is of opinion that within the limits of elasticity, the force required to produce any extension is a subject for experiment only. ...he considers elastic cords, laminae and spheres (supposed built up of laminae), and finds the deflection of the beam in Galilei's problem proportional to the weight. He makes... no attempt to discuss the elastic curve."

"The importance of Riccati's work lies not in his practical results, which are valueless, but in his statement of method, and his desire to replace by a dynamical theory semi-metaphysical hypotheses. ...[H]is writings remind us... of Bacon, who in like fashion failed to obtain valuable results, although he was capable of discovering a new method. Euler's return to the semi-metaphysical hypothesis... is a distinct retrogression on Riccati's attempt, which had to wait till George Green's day before it was again broached."

"Riccati... tells us that the forza viva must be measured by the square of the velocity. The consideration of the impact of bodies is more suggestive; the forza viva existing before impact is converted at the moment into forza morta and this re-converted into forza viva partly in the motion of either body as a whole, and partly in the vibratory motion of their parts, which we perceive in the sound vibrations they give rise to in the air."

"This paragraph... unit[es] the old theologico-mathematical standpoint, with the first struggling towards the modern conception of the . It is this principle of energy which la mia novella sentenza endeavours so vaguely to express, namely that the mechanical work stored up in a state of strain, must be equivalent to the energy spent in producing that state."

"Riccati states la mia novella sentenza [my new sentence]... Every deformation is produced by forza viva and this force is proportional to the deformation produced. ...The forza viva spent in producing a deformation remains in the strained body in the form of forza morta; it is stored up in the compressed fibres. Riccati comes to this conclusion after asking whether the forza viva so applied could be destroyed? That... he denies, making use strangely enough of the argument from design, a metaphysical conception such as he has told us ought not to be introduced into physics!La Natura anderebbe successivamente languendo, e la materia diverrebbe col lungo girare de' secoli una massa pigra, ed informe fornita soltanto d' impenetrabilità, e d' inerzia, e spogliata passo passo di quella forza (conciossiachè in ogni tempo una notabil porzione se ne distrugge) la quale in quantità, ed in misura era stata dal sommo Facitore sin dall' origine delle cose ad essa addostata per ridurre il presente Universo ad un ben concertato Sistema. [Nature would then be languishing, and matter would become a lazy, unformed mass with the long passage of centuries, and only provided impenetrability, and inertia, and stripped step by step of that force (because at any time a notable portion destroys it) which in quantity, and to an extent had been from the supreme Authority since the origin of the things, subjected to, in order to reduce the present Universe to a well-organized System.]"

"Riccati... will not enter into these disputes [as to current hypotheses as to the nature of elasticity]... For [in] his own theory he will not call to his assistance the aether of Descartes or the attractions of Newton. ...[H]e ...seems ignorant of and quotes Gravesande [Physices elementa mathematica experimentis confirmata, 1720.] to shew that the relation of extension to force is quite unknown... curious as he elsewhere cites Hooke..."

"Riccati gives a characteristic paragraph with regard to the English theorists:...Non ci ha fenomeno in Natura, ch' eglino non ascrivano alle favorite attrazioni, da cui derivano la durezza, la fluidità, ed altre proprietà de' composti, e spezialmente la forza elastica ... [There is no phenomenon in Nature, which is not ascribed to the favorite attractions, from which is derived hardness, fluidity, and other properties of the compounds, and especially the elastic force. ...]"

"Riccati... sketches briefly some [current] theories... Descartes... supposed [that] elasticity to be produced by a subtle matter (aether) which penetrates the pores of bodies and keeps the particles at due distances; this aether is driven out by a compressing force and rushes in again with great energy on the removal of the compression. ...John Bernoulli... supposes the aether enclosed in cells in the elastic body and unable to escape. In this captive aether float other larger aether atoms describing orbits. When a compressing force is applied the cells become smaller, and the orbits of these atoms are restricted, hence their centrifugal force is increased; when the compressing force is removed the cells increase and the centrifugal forces diminish. Such is... how the forza viva [live force] absorbed by an elastic body can be retained for a time as forza morta [dead force]. (This theory of captive aether was at a later date adopted by Euler although in a slightly more reasonable form...)"

"[Riccatti states that] the physicists of his time had troubled themselves much with the consideration of elasticity: E si può dire, che tante sono le teste, quante le opinioni, fra cui qual sia la vera non si sa, se pure non son tutte false, e quale la più verisimile, tuttavia con calore si disputa. [And it can be said that so many are the thinkers, how many opinions, among which the true is not known, even if they are not all false, and which is the most verisimilar, nevertheless, it is hotly disputed.]"

"Riccati... attempt[s]... a general explanation of the character of elasticity... in his Sistema dell' Universo [System of the Universe]... written before 1754... [and] first published in the Opere del Oonte Jacopo Riccati... 1761. [Two chapters] are respectively entitled: Delle forze elastiche and Da quali primi principi derivi la forza elastica... display... dislike... of any semi-metaphysical hypothesis introduced into physics; and desire to discover a purely dynamical theory for physical phenomena."

"Riccati... has no clear conception of , nor does [his] theory... of acoustic experiments lead him to discover that law. In his third canon he states that the 'sounds' of a given length of stretched string are in the sub-duplicate ratios of the stretching weights. The 'sounds' are to be measured by the inverse times of oscillation. ...from this ...he deduces ...that, if u be a weight which stretches a string to length x and u receive a small increment \partial u corresponding to an increment \partial x of x, then the law of elastic force is that \frac{\partial u}{u} is proportional to \frac{\partial x}{x^2}. Hence according to Riccati we should have instead of Hooke's Law: \boldsymbol{u = Ce^{-\frac{1}{x}}}, where C is constant. For compression the law is obtained by changing the sign of x. Riccati points out that James Bernoulli's statements... do not agree with this result...He notes that the equation du/u = \pm dx/x^2 has been obtained by Taylor and Varignon for the determination of the density of an elastic fluid compressed by its own weight"

"[T]he state of physical investigation with regard to elasticity in Riccati's time... [is indicated by the] remark of Bernoulli... in the corollary to his third lemma: "Au reste, il est probable que cette courbe" (ligne de tension et de compression) "est différente de différens corps, à cause de la différente structure de leurs fibres." [Moreover, it is probable that this curve (line of tension and compression) is different for different bodies, because of the different structure of their fibers.] It struck Riccati... to consider the acoustic properties of bodies. For, he remarks, the harmonic properties of vibrating bodies are well known and must undoubtedly be connected with the elastic properties—("canoni virium elasticarum" canon elastic forces])."

". ...first is a memoir entitled Verae et germanae virium elasticarum leges ex phaeiwmenis demonstratae, 1731... printed in the De Bononiensi scientiarum Academia Commentarii, 1747. ...[I]t marks the first attempt since Hooke to ascertain by experiment the laws which govern elastic bodies."

"[On] the question of extension and compression of the fibres of the beam under flexure... [Bülfinger] cites the two theories... that of Mariotte, that the neutral line is the 'middle fibre' of the beam, and that of Bernoulli that its position is indifferent. He... rejects both theories, and gives... sufficient reasons... [N]ot having accepted Hooke's principle... he holds that till the laws of compression are formulated, the position of the neutral line must be found by experiment."

"Varignon: De la Résistance des Solides en général pour tout ce qu'on peut faire d'hypothèses touchant la force ou la ténacité des Fibres des Corps à rompre; Et en particulier pour les hypothèses de Galiée & de M. Mariotte. Memoires de l'Académie, Paris 1702... considers that it is possible to state a general formula which will include the hypotheses of both Galilei and Mariotte, but... it will [in most practical cases] be necessary to assume some definite relation between the extension and resistance of the fibres. ...Varignon's method ...[is] generally adopted by later writers (although in conjunction with either Galilei's or the Mariotte-Leibniz hypothesis), we shall briefly consider it here ..."

"G. W. Leibniz: Demonstrationes novae de Resistentiâ solidorum. Acta Eruditorum Lipsiae July 1684. The stir created by Mariotte's experiments... seem to have brought the German philosopher into the field. He treats the subject in a rather ex cathedrâ fashion, as if his opinion would finally settle the matter. He examines the hypotheses of Galilei and Mariotte, and finding that there is always flexure before rupture, he concludes that the fibres are really extensible. Their resistance is, he states, in proportion to their extension. ...[i.e.,] he applies " Hooke's Law" to the individual fibres. As to the application of his results to special problems, he will leave that to those who have leisure for such matters. The hypothesis... is usually termed by the writers of this period the Mariotte-Leibniz theory."

"Mariotte seems to have been the earliest investigator who applied anything corresponding to the elasticity of Hooke to the fibres of the beam in Galilei's problem. ...[H]is Traité du mouvement des eaux, Paris 1686... shows that Galilei's theory does not accord with experience. He remarks that some of the fibres of the beam extend before rupture, while others again are compressed. He assumes however without the least attempt at proof ("on peut concevoir" [we can conceive]) that half the fibres are compressed, half extended."

"The modern expression of the six components of stress as linear functions of the strain components may perhaps he physically regarded as a generalised form of ."

"By 'spring' Hooke does not merely denote a spiral wire, or a bent rod of metal or wood, but any "springy body" whatever. Thus after describing his experiments he writes: From all which it is very evident that the Rule or Law of Nature in every springing body is, that the force or power thereof to restore it self to its natural position is always proportionate to the Distance or space it is removed therefrom, whether it be by rarefaction, or separation of its parts the one from the other, or by a Condensation, or crowding of those parts nearer together. Nor is it observable in these bodies only, but in all other springy bodies whatsoever, whether Metal, Wood, Stones, baked Earths, Hair, Horns, Silk, Bones, Sinews, Glass and the like. Respect being had to the particular figures of the bodies bended, and to the advantageous or disadvantageous ways of bending them."

"The discovery apparently of the modern conception of elasticity seems due to Robert Hooke, who in his work De potentiâ restitutiva, London 1678, states that 18 years before... he had first found out the theory of springs, but had omitted to publish it because he was anxious to obtain a patent for a particular application of it. He continues:— About three years since His Majesty was pleased to see the Experiment that made out this theory tried at White-Hall, as also my Spring Watch. About two years since I printed this Theory in an Anagram at the end of my Book of the Descriptions of s, viz. ceiiinosssttuu, id est, Ut Tensio sic vis; That is, The Power of any spring is in the same proportion with the Tension thereof."

"A beam (ABCD) being built horizontally into a wall (at AB) and strained by its own or an applied weight (E), to find the breaking force upon a section perpendicular to its axis. This problem is always associated... with Galilei's name, and we shall call it... Galilei's Problem. The 'base of fracture' being defined as the section of the beam where it is built into the wall; we have the following results :— (i) The resistances of the bases of fracture of similar prismatic beams are as the squares of their corresponding dimensions. In this case the beams are supposed loaded at the free end till the base of fracture is ruptured; the weights of the beams are neglected. (ii) Among an infinite number of homogeneous and similar beams there is only one, of which the weight is exactly in equilibrium with the resistance of the base of fracture. All others, if of a greater length will break,—if of a less length will have a superfluous resistance in their base of fracture."

"Galilei gives 17 propositions with regard to the fracture of rods, beams and hollow cylinders. ...[H]e supposed the fibres of a strained beam to be inextensible. There are two problems... discussed... which form the starting points of many later memoirs. They are the following:"

"Galileo Galilei['s] second dialogue of the Discorsi e Dimostrazioni matematiche, Leiden 1638... both from its contents and form is of great historical interest. It not only gave the impulse but determined the direction of all the inquiries concerning the rupture and strength of beams, with which the physicists and mathematicians for the next century principally busied themselves."

"The modern theory of elasticity may be considered to have its birth in 1821, when Navier first gave the equations for the equilibrium and motion of elastic solids, but some of the problems which belong to this theory had previously been solved or discussed on special principles, and to understand the growth of our modern conceptions it is needful to investigate the work of the seventeenth and eighteenth centuries."

"My colleague, Professor A. B. W. Kennedy, has continually placed at my disposal the results not only of special experiments, but of his wide practical experience. The curves figured in the Appendix, as well as a variety of practical and technical remarks scattered throughout the volume I owe entirely to him; beyond this it is difficult for me to fitly acknowledge what I have learnt from mere contact with a mind so thoroughly imbued with the concepts of physical and technical elasticity."

"To the late M. Barré de Saint-Venant I am indebted for the loan of several works, for a variety of references and facts bearing on the history of elasticity, as well as for a revision..."

"There has been far too much invention of 'solvable problems' by the mathematical elastician; far too much neglect of the physical and technical problems which have been crying out for solution. Much of the ingenuity which has been spent on the ideal body of 'perfect' elasticity ideally loaded, might I believe have wrought miracles in the fields of physical and technical elasticity, where pressing practical problems remain in abundance unsolved. I have endeavoured... to abrogate this divorce between mathematical elasticity on the one hand, and physical and technical elasticity on the other. With this aim in view I have introduced the general conclusions of a considerable body of physical and technical memoirs, in the hope that by doing so I may bring the mathematician closer to the physicist and both to the practical engineer. I trust that in doing so I have rendered this History of value to a wider range of readers, and so increased the usefulness of Dr Todhunter's many years of patient historical research on the more purely mathematical side of elasticity. In this matter I have kept before me the labours of M. de Saint-Venant as a true guide to the functions of the ideal elastician."

"My greatest difficulty arose with regard to the rigid line which Dr Todhunter had attempted to draw between mathematical and physical memoirs. Thus while including an account of Clausius' memoir of 1849, he had omitted Weber's of 1835, yet the consideration of the former demands the inclusion of the latter..."

"I have inserted... memoirs of mathematical value, omitted [by Todhunter] apparently by pure accident. For example all the memoirs of F. E. Neumann, the second memoir of Duhamel, those of Blanchet etc. I cannot hope that the work is complete in this respect even now, but I trust that nothing of equal importance has escaped..."

"With a similar aim I have introduced throughout the volume a number of memoirs having purely historical value which had escaped Dr Todhunter's notice."

"The labours of Lagrange and Riccati also required some recognition... [of] interest, whether judged from the special standpoint of the elastician or from the wider footing of insight into the growth of human ideas."

"Taking the first chapter of this History the author has discussed the important memoirs of James Bernoulli and some of those due to Euler. The whole early history of our subject is however so intimately connected with the names of Galilei, Hooke, Mariotte and Leibniz, that I have introduced some account of their work."

"Keeping the twofold object of this work in view I have endeavoured to give it completeness (1) as a history of developement, (2) as a guide to what has been accomplished."

"It would be a great aid to science, if, at any rate, the innumerable mathematical journals could be to a great extent specialised, so that we might look to any one of them for a special class of memoir. ...the would-be researcher either wastes much time in learning the history of his subject, or else works away regardless of earlier investigators. The latter course has been singularly prevalent with even some firstclass British and French mathematicians."

"The use of a work of this kind is twofold. It forms on the one hand the history of a peculiar phase of intellectual development, worth studying for the many side lights it throws on general human progress. On the other hand it serves as a guide to the investigator in what has been done, and what ought to be done. In this latter respect the individualism of modern science has not infrequently led to a great waste of power; the same bit of work has been repeated in different countries at different times, owing to the absence of such histories as Dr Todhunter set himself to write. ...the various Jahrbücher and Fortschritte now reduce the possibility of this repetition, but besides their frequent insufficiency they are at best but indices to the work of the last few years; an enormous amount of matter is practically stored out of sight in the Transactions and Journals of the last century and of the first half of the present century."

"The principle which has guided me throughout the additions I have made has been to make the work, so far as it lay in my power, a standard work of reference for its own branch of science."

"I may appear to have exceeded the duty of an editor. For all the Articles in this volume whose numbers are enclosed in square brackets I am alone responsible, as well as for the corresponding footnotes, and the Appendix with which the volume concludes."

"I... regret that I have not devoted special chapters to such elasticians as Hodgkinson, [Guillaume] Wertheim and F. E. Neumann; in the latter case the regret is deepened by the recent publication of his lectures on elasticity."

"The changes I have made in that manuscript are of the following character; the introduction of a uniform terminology and notation, the correction of clerical and other obvious errors, the insertion of cross-references, the occasional introduction of a remark or of a footnote. The remarks are inclosed in square brackets. With this exception any article in this volume the number of which is not included in square brackets is due entirely to Dr Todhunter."

"Dr Todhunter's manuscript consists of two distinct parts, the first contains a purely mathematical treatise on the theory of the 'perfect' elastic solid; the second a history of the theory of elasticity. The treatise based principally on the works of Lamé, Saint-Venant and Clebsch is yet to a great extent historical, [i.e.,] many paragraphs are composed of analyses of important memoirs."

"The symbols and terms used in the manuscript are occasionally those of the original memoirs, occasionally those of Lamé or of Saint-Venant... the memoirs being of historical rather than scientific interest, and their language often the most characteristic part of their historical value."

"[C]onsistency in [notation and terminology] will be found after the middle of the chapter devoted to Poisson."

"[T]he notation and terminology will be found fully discussed in Notes B—D of the Appendix, which I would ask the reader to examine before passing to the text."

"[I]t was not till I had advanced... into the work that I felt convinced that... the... writer's terminology and notation must be abandoned and a uniform terminology and notation adopted for the whole book... to be available for easy reference, and not merely of interest to the historical student."

"In the summer of 1884... the Syndics... placed in my hands the manuscript of the late Dr Todhunter's History of Elasticity, in order that it might be edited and completed..."

"G. B. Bülfinger: De solidorum Resistentia Specimen, Commentarii Academiae Petropolitanae... is a memoir of August 1729... first published... 1735. [It] commences with a reference to the labours of Galilei, Leibniz, Wurtz, Mariotte, Varignon, James Bernoulli and Parent... [His following] sections are concerned with the breaking force on a beam when it is applied longitudinally and transversally. Galilei's and the Mariotte-Leibniz hypotheses are considered. It is shewn that the latter is the more consonant with... fact, but... is not exact because it neglects the compression (i.e. places the neutral line in the lowest horizontal fibre of the beam)."