A History of the Theory of Elasticity and of the Strength of Materials

A History of the Theory of Elasticity and of the Strength of Materials: from Galilei to the Present Time is a two volume set edited and completed by Karl Pearson from notes written by. It was published by Cambridge at the University Press posthumously in Todhunter's name. Volume I. Galilei to Saint-Venant 1639-1850 was first published in 1886. Volume II. Saint-Venant to Lord Kelvin was first published in 1893.

Preface (June 23, 1886)

 * by Karl Pearson, editor and in parts, co-writer.


 * 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...


 * [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.


 * [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.


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


 * 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.


 * 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 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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.


 * 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...


 * 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...


 * 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.


 * 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...


 * 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.

Ch. 1. The Seventeenth and Eighteenth Centuries

 * 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.


 * 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.


 * 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:

(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.
 * 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 :—

"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."
 * 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:—

"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."
 * 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:


 * 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 .


 * 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.


 * 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.


 * 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 ...


 * Let $$ABCNML$$ be a beam built into a vertical wall at the section $$ABC$$, and supposed to consist of a number of parallel fibres perpendicular to the wall... and equal to $$AN$$ in length. Let $$H'$$ be a point on the 'base of fracture,' and $$H'E$$ [which is perpendicular to $$AC] = y, AE= x$$. Then if a weight $$Q$$ be attached by means of a pulley to the extremity of the beam, and be supposed to produce a uniform horizontal force over the whole section $$NML, \; Q = r \cdot \int ydx$$ where $$r$$ is the resistance of a fibre of unit sectional area and the integration is to extend over the whole base of fracture. $$Q$$ is by later writers termed the absolute resistance and is given by the above formula. Now suppose the beam to be acted upon at its extremity by a vertical force $$P$$ instead of the horizontal force $$Q$$. All the fibres in a horizontal line through $$H'$$ will have equal resistance, this may be measured by a line $$HK$$ drawn through $$H$$ in any fixed direction where $$H$$ is the point of intersection of the horizontal line through $$H$$ and the central vertical $$BD$$ of the base. As $$H$$ moves from $$B$$ to $$D, K$$ will trace out a curve $$GK$$ which gives the resistance of the corresponding fibres. Take moments for the equilibrium of the beam about $$AC$$ $$P \cdot l = \iint uydxdy$$ where $$l =$$ length of the beam $$DT$$ and $$u = HK$$.


 * This quantity $$\iint uydxdy$$ was termed the relative resistance of the beam or the resistance of the base of fracture. ...it is necessary to know $$u$$ before we can make use of it. He then proceeds to apply it to Galilei's and the Mariotte-Leibniz hypotheses.


 * In Galilei's hypothesis of inextensible fibres $$u$$ is supposed constant $$= r$$ and the resistance of the base of fracture becomes $$r \int ydxdy = \frac{r}{2} \cdot \int y^2dx$$. On the supposition that the fibres are extensible we ought to consider their extension by finding what is now termed the neutral line or surface. Varignon however, and he is followed by later writers, assumes that the fibres in the base $$ACLN$$ are not extended; and that the extension of the fibre through $$H'$$ varies as $$DH$$, in other words he makes the curve $$GK$$ a straight line passing through $$D$$. Hence if $$r'$$ be the resistance of the fibre at $$B$$, and $$DB = a$$, the resistance of the fibre at $$H = r'y/a$$ or the resistance of the base of fracture on this hypothesis becomes $$\frac{r'}{3a}\int y^3dx$$ This resistance in the case of a rectangular beam of breadth $$b$$ and height $$a$$ becomes on the two hypotheses $$\frac{ra^2b}{2}$$ and $$\frac{r'a^2b}{3}$$ ...his results are practically vitiated when applying the true ( Leibniz-Mariotte) theory by his assumption of the position of the neutral surface, but in this error he is followed by so great a mathematician as Euler himself.


 * The first work of genuine mathematical value on our subject is clue to James Bernoulli... Véritable hypothèse de la résistance des Solides, avec la démonstration de la Courbure des Corps qui font ressort... 12th of March 1705... begins by brief notices of what had been already done with respect to the problem by Galilei, Leibniz, and Mariotte; James Bernoulli claims for himself that he first introduced the consideration of the compression of parts of the body, whereas previous writers had paid attention to the extension alone.
 * Pearson's note: ...this [last] remark does not apply to Mariotte.

I. Des Fibres de même matière et de même largeur, ou épaisseur, tirées ou pressées par la même force, s'étendent ou se compriment proportionellement à leurs longueurs. [Fibers of the same material and of the same width, or thickness, drawn or pressed by the same force, extend or compress proportionally to their lengths.] II. Des Fibres homogènes et de même longueur, mais de différentes largeurs ou épaisseurs, s'étendent ou se compriment également par des forces proportionelles à leurs largeurs. [Fibers homogeneous and of the same length, but of different widths or thicknesses, extend or are also compressed by forces proportional to their widths.] III. Des Fibres homogènes de même longueur et largeur, mais chargées de différens poids, ne s'étendent ni se compriment pas proportionellement à ces poids; mais l'extension ou la compression causée par le plus grand poids, est à l'extension ou à la compression causée par le plus petit, en moindre raison que ce poids—là n'est à celui—ci. [Homogeneous fibers of the same length and width, but charged with different weights, neither extend nor compress proportionally to these weights; but the extension or the compression caused by the greatest weight, is to the extension or to the compression caused by the smaller, in less reason...]
 * Three Lemmas which present no difficulty are given and demonstrated [by James Bernoulli]:
 * Note: The third lemma points out the possibility of nonlinear behavior of the homogeneous fibers.


 * The fourth Lemma... may be readily understood by reference to Varignon's memoir. ...Varignon supposed the neutral surface to pass through... the so-called 'axis of equilibrium'... James Bernoulli... recognises the difficulty of determining the fibres which are neither extended nor compressed, but he comes to the conclusion that the same force applied at the extremity of the same lever will produce the same effect, whether all the fibres are extended, all compressed or part extended and part compressed about the axis of equilibrium. In other words the position of the axis of equilibrium is indifferent. This result is expressed by the fourth Lemma and is of course inadmissible.


 * Saint-Venant remarks in his memoir on the Flexure of Prisms in Liouville's Journal, 1856: On s'étonne de voir, vingt ans plus tard, un grand géomètre, auteur de la première théorie des courbes élastiques, Jacques Bernoulli tout en admettant aussi les compressions et présentant même leur considération comme étant de lui commettre sous une autre forme, précisement la même méprise du simple au double que Mariotte dans l'évaluation du moment des résistances ce qui le conduit même à affirmer que la position attribuée à l'axe de rotation est tout à fait indifférente. [It is surprising to see, twenty years later, a great geometer, author of the first theory of elastic curves, Jacques Bernoulli... commit precisely the same mistake of... Mariotte in the evaluation of moment of resistance which leads him... to assert that the position attributed to the axis of rotation is entirely indifferent.]


 * Bernoulli... rejects the Mariotte-Leibniz hypothesis or the application of Hooke's law to the extension of the fibres. He introduces rather an idle argument against [it], and quotes an experiment of his own which disagrees with Hooke's Ut tensio, sic vis.


 * James Bernoulli next takes a problem which he enunciates thus: "Trouver combien il faut plus de force pour rompre une poutre directement, c'est-à-dire en la tirant suivant sa longueur, que pour la rompre transversalement." [Find out how much more force is needed to break a beam directly... by pulling it along its length in order to break it transversely.] The investigation depends on the fourth Lemma, and is consequently not satisfactory.

Jacques Bernoulli a déterminé, le premier, la figure de la lame élastique en équilibre, d'après des considérations que nous allons développer,. . .[Jacques Bernoulli has determined, the first, the figure of the elastic blade in equilibrium, according to considerations that we will develop...]
 * The method of James Bernoulli with improvements, has been substantially adopted by other writers. The English reader may consult the earlier editions of Whewell's Mechanics. Poisson says in his Traité de Mécanique...

The parts of all homogeneal hard bodies, which fully touch one another, stick together very strongly. And for explaining how this may be, some have invented hooked atoms, which is begging the question; and others tell us, that bodies are glued together by Rest: that is, by an occult quality, or rather by nothing: and others, that they stick together by conspiring motions, that is by relative Rest among themselves. I had rather infer from their cohesion, that their particles attract one another by some force, which in immediate contact is exceeding strong, at small distances performs the chemical operations above-mentioned, and reaches not far from the particles with any sensible effect."
 * Sir Isaac Newton : Optics  or a Treatise of the Reflections, Refractions and Colours of Light. 1717. ...The Query [XXXIst, termed 'Elective Attractions,'] commences by suggesting that the attractive powers of small particles of bodies may be capable of producing the great part of the phenomena of nature:—"For it is well known that bodies act one upon another by the attractions of gravity, magnetism and electricity; and these instances shew the tenor and course of nature, and make it not improbable, but that there may be more attractive powers than these. For nature is very consonant and conformable to herself. ...


 * Newton supposes all bodies to be composed of hard particles, and these are heaped up together and scarce touch in more than a few points."And how such very hard particles, which are only laid together, and touch only in a few points can stick together, and that so firmly as they do, without the assistance of something which causes them to be attracted or pressed towards one another, is very difficult to conceive."


 * After using arguments from capillarity to confirm these remarks he continues:"Now the small particles of matter may cohere by the strongest attractions, and compose bigger particles of weaker virtue; and many of these may cohere and compose bigger particles, whose virtue is still weaker; and so on for divers successions, until the progression end in the biggest particles, on which the operations in chemistry, and the colours of natural bodies depend; and which by adhering, compose bodies of a sensible magnitude. If the body is compact, and bends or yields inward to pression without any sliding of its parts, it is Hard and Elastick, returning to its figure with a force rising from the mutual attractions of its parts."


 * The conception of repulsive forces is then introduced [by Newton] to explain the expansion of gases."Which vast contraction and expansion seems unintelligible, by feigning the particles of air to be springy and ramous, or rolled up like hoops, or by any other means than a Repulsive power. And thus Nature will be very conformable to herself, and very simple; performing all the great motions of the heavenly bodies by the attraction of gravity, which intercedes those bodies; and almost all the small ones of their particles, by some other Attractive and Repelling powers."


 * A suggestive paragraph... occurs... which is sometimes not sufficiently remembered when gravitation is spoken of as a cause :—"These principles—i.e. of attraction and repulsion—I consider not as occult qualities, supposed to result from the specifick forms of things, but as general laws of Nature, by which the things themselves are formed; their truth appearing to us by phenomena, though their causes be not yet discovered."


 * This seems to be Newton's only contribution to the subject of Elasticity, beyond the paragraph of the Principia on the collision of elastic bodies.


 * [W]hile the mathematicians were beginning to struggle with the problems of elasticity, a number of practical experiments were being made on the flexure and rupture of beams, the results of which were of material assistance to the theorists.


 * Petris van Musschenbroek: Introductio ad cohaerentiam corporum firmorum.... commences at of the author's Physicae experimentales et geometricae Dissertationes. Lugduni 1729. It was held in high repute even to the end of the 18th century. ...[The] historical preface,... has been largely drawn upon by Girard. [Van Musschenbroek] describes the various theories which have been started to explain cohesion, and rejects successively that of the pressure of the air and that of a subtle medium. ...He laughs at Bacon 's explanation of elasticity, and another metaphysical hypothesis he terms abracadabra.  ...[H]e falls back... upon Newton's thirty-first Query... and would explain the matter by vires internae [internal forces]. Musschenbroek assumes... we may determine them in each case by experiment. ...The source of elasticity is a vis interna attrahens... drawn directly from Newton's Optics.


 * Musschenbroek... treats of the extension (cohaerentia vel resistentia absoluta) and of the flexure (cohaerentia respectiva aut transversa) of beams, but does not seem to have considered their compression. His experiments are... on wood, with a few... on metals. ...Anything of value in his work is however reproduced by Girard.


 * Musschenbroek discovered by experiment that the resistance of beams compressed by forces parallel to their length is... in the inverse ratio of the squares of their lengths; a result afterwards deduced theoretically by Euler.

Pere '''Maziere finds a probable explanation of the physical cause of spring in that favorite hypothesis of a 'subtile matter' or étherée. ...Mazière ...applies the Cartesian theory of vortices to the aether''' ...
 * Pere Maziere: Les Loix du choc des corps à ressort parfait ou imparfait, déduites d'une explication probable de la cause physique du ressort. Paris, 1727... carried off the prize of the Acadimie Royale des Sciences... 1726. Pere Maziere, Pretre de I'Oratoire... brings out clearly the union of those theological and metaphysical tendencies of the time, which so checked the true or experimental basis of physical research. It shews us the evil as well as the good which the Cartesian ideas brought to science. It is startling to find the French Academy awarding their prize to an essay of this type, almost in the age of the Bernoullis and Euler. Finally it more than justifies Riccati' s remarks as to the absurdities of these metaphysical mathematicians.


 * 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).


 * Bülfinger... suggests a parabolic relation of the form tension &prop; (distance from the neutral line)m, where the [exponent m] power is a constant to be determined by experiment.


 * [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.


 * . ...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.


 * [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]).


 * 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


 * 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.


 * [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... 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...)


 * 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... 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 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.]"


 * 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... 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.


 * 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.


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


 * 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.


 * [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.]"


 * 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.


 * The celebrated work of Euler relating to... the Calculus of Variations appeared in 1744 under the title of Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. ...an appendix called Additamentum I. De Curvis Elasticis ...commences with a statement... shewing the theologico-metaphysical tendency... 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. ...Thus we find Maupertuis' extremely eccentric attempt at a principle of Least Action. ...[I]t is... 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.


 * [S]tating that Daniel Bernoulli... had discovered... that the vis potentialis represented by $$\int ds/R^2$$ was a minimum for the elastic curve, Euler proceeds to discuss the inverse problem... The curve is to have a given length between two fixed points, to have given tangents at those points, and to render $$\int ds/R^2$$ a minimum... No attempt is made to shew why... By the aid of the principles of his book Euler arrives at the following equations where $$a, \alpha, \beta, \gamma$$ are constants, $$dy = \frac{(\alpha + \beta x + \gamma x^2)}{\sqrt{a^4 - (\alpha + \beta x + \gamma x^2)^2}}$$ from this we obtain $$ds = \frac{a^2 dx}{\sqrt{a^4 - (\alpha + \beta x + \gamma x^2)^2}}$$


 * Euler gives... 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... writes... an equation like... Poisson's Traite de Mecanique, Vol. I., without giving any of the reasoning by which Poisson obtains the equation.


 * Euler... has hitherto considered the elasticity constant, but he will now suppose that it is variable... $$S$$, which is supposed a function of the arc $$s$$; $$\rho$$ is the radius of curvature. He proceeds to find the curve which makes $$\int S ds/\rho^2$$ a minimum; and... finds for the differential equation of the required curve $$\alpha + \beta x -\gamma y = S/\rho$$ where $$\alpha, \beta, \gamma$$ are constants.


 * Euler takes the case in which forces act at every point of the elastic curve; and he obtains an equation like the first volume of Poisson's Traiti de Mecanique.


 * Euler devotes his attention to the oscillations of an elastic lamina; the investigation is some what obscure for the science of dynamics had not yet been placed on the firm foundation of : nevertheless the results obtained by Euler will be found in substantial agreement with those in Poisson's Traite de Mecanique, Vol. II.


 * 1757. Sur la force des colonnes, Mémoires de l'Académie de Berlin, Tom. XIII. 1759... is one of Euler's most important contributions to the theory of elasticity. The problem... is the discovery of the least force which will suffice to give any the least curvature to a column, when applied at one extremity parallel to its axis, the other extremity being fixed. Euler finds that the force must be at least $$= \pi^2 \cdot \frac{Ek^2}{a^2}$$, where $$a$$ is the length of the column and $$Ek^2$$ is the 'moment of the spring' or the 'moment of stiffness of the column' (moment du ressort or moment de roideur).




 * If we consider a force $$F$$ perpendicular to the axis of a beam (or lamina) so as to displace it from the position $$AC$$ to $$AD$$, and $$\delta$$ be the projection of $$D$$ parallel to $$AC$$ on a line through $$C$$ perpendicular to $$AC$$, Euler finds by easy analysis $$D \delta = \frac{F\cdot a^3}{3\cdot Ek^2}$$, supposing the displacement to be small. This suggests to him a method of determining the 'moment of stiffness' $$Ek^2$$, and he makes various remarks on proposed experimental investigations. He then notes the curious distinction between forces acting parallel and perpendicular to a built-in rod at its free end; the latter, however small, produce a deflection, the former only when they exceed a certain magnitude. It is shewn that the force required to give curvature to a beam acting parallel to its axis would give it an immense deflection if acting perpendicularly.


 * Euler deduces the equation for the curve assumed by the beam $$AC$$ fixed but not built in at one end $$A$$ and acted upon by a force $$P$$ parallel to its axis. If $$RM$$ be perpendicular to $$AC$$ and $$y=RM, x = AM$$, he finds $$\frac{y}{\theta}\cdot \sqrt{\frac{P}{Ek^2}} = sin(x \sqrt{\frac{P}{Ek^2}})$$, where $$\theta = \angle RCM$$. Hence since $$y = 0$$, when $$x = a$$ the length of the beam, $$a \sqrt{\frac{P}{Ek^2}}$$ must at least $$= \pi$$, whence it follows that $$P$$ must be at least $$= \pi^2 \cdot \frac{Ek^2}{a^2}$$. This paradox Euler seems unable to explain.


 * If $$Q$$ be the total weight of the beam the differential equation $$Ek^2 ad^3 y + Pa (dx)^2 dy + Qx(dx)^2 dy = 0$$ is obtained... This is reduced by a simple transformation to a special case of Riccati's equation, which is then solved on the supposition that $$\frac{Q}{P}$$ is small. Euler obtains finally for the force $$P$$, for which the rod begins to bend, the expression $$P = \pi^2 \cdot Ek^2/a^2 - Q \cdot (\pi^2 - 8)/2\pi^2$$; which shews that the minimum force is slightly reduced by taking the weight of the beam into consideration.


 * Determinatio onerum quae columnae gestare valent. Examen insignis paradoxi in theoria columnarum occurrentis. De altitudine columnarum sub proprio pondere corruentium. [all in] Acta Academiae Petropolitanae [1778, 1780]. The first memoir... points out that vertical columns do not break under vertical pressure by mere crushing, but that flexure of the column will be found to precede rupture. ...[Euler] proposes to deduce a result which is now commonly in use... to find an expression connecting $$Ek^2$$ with the dimensions of the transverse section of the column. Euler finds $$Ek^2 = h \cdot \int x^2 ydx$$, where $$x$$ and $$y$$... Euler appears however to treat the unaltered fibre or 'neutral line' without remark as the extreme fibre on the concave side of the section of the column made by the central plane of flexure. Thus for a column of rectangular section of dimensions $$b$$ [with]in, and $$c$$ perpendicular to the plane of flexure, he finds... $$Ek^2 = \frac{1}{3} b^3 ch$$, and the like method is used in the case of a circular section.
 * Pearson's note: In the case of a beam or column bent by a longitudinal force it may be shewn theoretically that the neutral line does not necessarily lie in the material of the beam, its position and form vary with the amount of the deflecting force; in other words $$Ek^2$$ is not a constant, but a function of the force and of the flexure. The assumption that the 'moment of stiffness' is constant seems to me to vitiate the results not only of Euler and Lagrange but of many later writers on the subject. Prof. A. B. W. Kennedy['s]... experiments... conclusively prove that the position of the neutral line varies with the magnitude of the longitudinal force.


 * Euler... calculate[s] the flexure which may be produced in a column by its own weight. If $$y$$ be the horizontal displacement of a point on the column at a distance $$x$$ from its vertex, the equation $$Ek^2 \cdot \frac{d^2y}{dx^2} + b^2 \int_0^y xdy = 0$$ is found, where the weight of unit volume of the column is unity and its section a square of side $$b$$. ...[I]f $$a$$ be the altitude of the column and $$m = Ek^2/b^2$$, it is found that the least altitude for which the column will bend from its own weight is the least root of the equation, $$0 = \frac{1 \cdot a^3}{4! m} + \frac{1 \cdot 4 \cdot a^6}{7! m^2} - \frac{1 \cdot 4 \cdot 7 \cdot a^9}{10! m^2} + \frac{1 \cdot 4 \cdot 7 \cdot 10 \cdot a^{12}}{13! m^2} - \mathrm{etc.}$$ Euler finds that this equation has no real root, and thus arrives at the paradoxical result, that however high a column may be it cannot be ruptured by its own weight.


 * P. S. Girard. Traite Analytique de la Resistance des solides, et des solides d'e'gale Resistance, Auquel on a joint une suite de nouvelles Experiences sur la force, et Velasticite specifique des Bois de Chine et de Sapin. Paris, 1798. ...This work very fitly closes the labours of the 18th century. It is the first practical treatise on Elasticity; and one of the first attempts to make searching experiments on the elastic properties of beams. It is not only valuable as containing the total knowledge of that day on the subject, but also by reason of an admirable historical introduction... The work appears to have been begun in 1787 and portions of it presented to the Academie in 1792. Its final publication was delayed till the experiments on elastic bodies, the results of which are here tabulated, were concluded at Havre. ...We are... considering the period of the French Revolution.


 * The book... introduction is occupied with an historical retrospect of the work already accomplished in the field of elasticity... [and] concludes with an analysis of Girard's own work.


 * The first section of Girard's treatise is concerned with the resistance of solids according to the hypotheses of Galilei, Leibniz and Mariotte. He notes Bernoulli's objections to the Mariotte-Leibniz theory; but remarks that physicists and geometricians have accepted this theory... [H]e thinks it probable that Galilei's hypothesis of non-extension of the fibres may hold for some bodies—stones and minerals—while the Mariotte-Leibniz theory is true for sinews, wood and all vegetable matters (cf. p. 6). As to Bernoulli's doubt with regard to the position of the neutral surface, Girard accepts Bernoulli's statement that the position of the axis of equilibrium is indifferent, and supposes accordingly that all the fibres extend themselves about the axis $$AC$$...


 * [Girard's] book forms... a most characteristic picture of the state of mathematical knowledge on the subject of elasticity at the time and marks the arrival of an epoch when science was to free itself from the tendency to introduce theologico-metaphysical theory in the place of the physical axiom deduced from the results of organised experience.


 * General summary. As the general result of the work... previous to 1800... while a considerable number of particular problems had been solved by means of hypotheses more or less adapted to the individual case, there had as yet been no attempt to form general equations for the motion or equilibrium of an elastic solid. Of these problems the consideration of the elastic lamina by James Bernoulli, of the vibrating rod by Daniel Bernoulli and Euler, and of the equilibrium of springs and columns by Lagrange and Euler are the most important. The problem of a vibrating plate had been attempted, but with results which cannot be considered satisfactory.


 * A semi-metaphysical hypothesis as to the nature of Elasticity was started by Descartes and extended by John Bernoulli and Euler. It is extremely unsatisfactory, but the attempt to found a valid dynamical theory by did not lead to any more definite results.

Quotes about this work
The present volume is an indispensable hand-book of reference for the mathematician and the engineer, and in the editing and printing must be considered a very fitting tribute to the wonderful industry and application of its projector, the late Dr. Todhunter.
 * In the appendix Mr. Pearson has carefully analysed the conflicting notations of different writers, and proposed a very convenient terminology and notation, which would save great trouble if universally adopted. He has also given an account of experiments carried out by Prof. Kennedy in his mechanical laboratory, which have an important bearing on the limitations of the truth of Hooke's law, or in the language of elasticity, the constancy of the ratio of stress to corresponding strain.
 * A. G. Greenhill, Nature (Feb. 3, 1887) Review of A History of the Theory of Elasticity, Volume 35, pp. 313-314.


 * At Mr Webb's suggestion, the exposition of the theory is preceded by an historical sketch of its origin and development. Anything like an exhaustive history has been rendered unnecessary by the work of the late Dr Todhunter as edited by Prof. Karl Pearson, but it is hoped that the brief account given will at once facilitate the comprehension of the theory and add to its interest.
 * , A Treatise on the Mathematical Theory of Elasticity (1892) Vol. 1, Preface.