History Of Physics

"The ideas that prove to be of lasting interest are likely to build on the framework of the now standard world picture, the hot big bang model of the expanding universe. The full extent and richness of this picture is not as well understood as I think it ought to be, even among those making some of the most stimulating contributions to the flow of ideas."

"The most far-reaching implication of general relativity... is that the universe is not static, as in the orthodox view, but is dynamic, either contracting or expanding. Einstein, as visionary as he was, balked at the idea... One reason... was that, if the universe is currently expanding, then... it must have started from a single point. All space and time would have to be bound up in that "point," an infinitely dense, infinitely small "singularity." ...this struck Einstein as absurd. He therefore tried to sidestep the logic of his equations, and modified them by adding... a "cosmological constant." The term represented a force, of unknown nature, that would counteract the gravitational attraction of the mass of the universe. That is, the two forces would cancel... it is the kind of rabbit-out-of-the-hat idea that most scientists would label ad-hoc. ...Ironically, Einstein's approach contained a foolishly simple mistake: His universe would not be stable... like a pencil balanced on its point."

"If you think... the shattering of the bubble universe is... tragic... [W]hen the worst has happened our galaxy... will be left intact. ...not so bad a prospect."

"One of the few authors to have explicitly connected the physical issue of the expansion of the universe with the philosophical topic of the metaphysical status of space is Gerald James Whitrow."

"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just . This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational ... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. ). ..."

"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."

"The definition of inflation is extraordinarily simple: it is any period of the Universe's evolution during which the scale factor, describing the size of the Universe, is accelerating. This leads to a very rapid expansion of the Universe, though perhaps a better way of thinking of this is that the characteristic scale of the Universe, given by the Hubble length, is shrinking relative to any fixed scale caught up in the rapid expansion. In that sense, inflation is actually akin to zooming in on a small part of the initial Universe."

"A book, too, can be a star, explosive material, capable of stirring up fresh life endlessly, a living fire to lighten the darkness, leading out into the expanding universe."

"This model of the expanding universe I shall call the substratum. It achieves in the private Euclidean space of each fundamental observer the objects for which Einstein developed his closed spherical space. Although it is finite in volume, in the measures of any chosen observer, it has all the properties of an infinite space in that its boundary is forever inaccessible and its contents comprise an infinity of members. It is also homogeneous in the sense that each member stands in the same relation to the rest. This description of the substratum holds good in the scale of time in which the galaxies or fundamental particles are receding from one another with uniform velocities. This choice of the scale of time, together with the theory of equivalent time-keepers... makes possible the application of the Lorentz formulae to the private Euclidean spaces of the various observers. It thus brings the theory of the expanding universe into line with other branches of physics, which use the Lorentz formulæ and adopt Euclidean private spaces. ...[T]here is no more need to require a curvature for space itself in the field of cosmology than in any other department of physics. The observer at the origin is fully entitled to select a private Euclidean space in which to describe phenomena, and when he concedes a similar right to every other equivalent observer and imposes the condition of the same world-view of each observer, he is inevitably led to the model of the substratum which we have discussed."

"The cosmological constant['s]... most important consequence: the repulsive force, acting at cosmological distances, causes space to expand exponentially. There is nothing new about the universe expanding, but without a cosmological constant, the rate of expansion would gradually slow down. Indeed, it could even reverse itself and begin to contract, eventually imploding in a giant cosmic crunch. Instead, as a consequence of the cosmological constant, the universe appears to be doubling in size about every fifteen billion years, and all indications are that it will do so indefinitely."

"This circumstance of an expanding universe is irritating. ...To admit such possibilities seems senseless to me."

"If a distant galaxy is moving relative to us, its entire is Doppler-shifted in frequency. Its s are displaced relative to those of stationary light sources. Thanks to this effect, we know that distant galaxies recede from the solar system at speeds proportional to their distances from us. That's the effect that told us of the expanding universe, and of its birth, long ago, in the Big Bang."

"All kinds of questions remain. Many have to do with cosmology. How did the universe originate? How did the galaxies become distributed in space like the suds in the kitchen sink..? Why is the cosmological constant apparently very tiny but non-zero and has a peculiar value that leads the universe to expand more rapidly?"

"All of this picture of the expansion is exciting, pleasant, coherent, well in order. But what if the s are not to be interpreted by the Doppler-Fizeau law in the classical mechanical view, or general relativistically, by the fact that the ratio of the of a photon (as measured by a co-moving observer) to the space radius of curvature is independent of ? Not speaking of quasars, the first indications for non-Doppler redshifts for a galaxy have been provided... What if not all galaxies were formed at the dawn of the Big Bang; what if some are being formed now? Then, at least, the can be anything larger than the age of our own Galaxy..."

"Red-shifts are produced either in the nebulae, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulae, then red-shifts are probably velocity-shifts and the nebulae are receding. If the source lies in the intervening space, the explanation of red-shifts is unknown, but the nebulae are sensibly stationary."

"In Euler's work on the elastica the rod is thought of as a line of particles which resists bending. The theory of the flexure of beams of finite section was considered by Coulomb... [by investigating] the equation of equilibrium obtained by resolving horizontally the forces which act upon the part of the beam cut off by one of its normal sections, as well as of the equation of moments. He... thus... obtain[ed] the true position of the "neutral line," or axis of equilibrium, and he also made a correct calculation of the moment of the elastic forces. His theory of beams is the most exact of those [that assume] the stress in a bent beam arises wholly from the extension and contraction of its longitudinal filaments, and... ."

"The assumptions in the design of reinforced concrete beams are those of the ordinary beam theory, namely: the Bernoulli-Euler theory of flexure. The fundamental premise is that a plane section before bending, remains a plane section after bending, with the further assumption that , i.e., the stress is proportional to the strain, is true. Although the brilliant researches of Barre de St. Venant, have shown that plane sections do not remain plane during bending, the error becomes appreciable when the ratio of depth of beam to span exceeds one-fifth. Since for such ratios, stresses, other than those induced by , usually govern the required reinforcement and depth of beam e.g. the unit shear and adhesion, these assumptions of plane sections may be taken as valid, so long as the stresses induced by the bending moment govern the required depths and amounts of steel reinforcement. The concrete is assumed to take no tension."

"The first investigation of any importance is that of the elastic line or elastica by James Bernoulli in 1705, in which the resistance of a bent rod is assumed to arise from the extension and contraction of its longitudinal filaments, and the equation of the curve assumed by the axis is formed. This equation practically involves the result that the resistance to bending is a couple proportional to the of the rod when bent, a result which was assumed by Euler in his later treatment of the problems of the elastica, and of the vibrations of thin rods."

"We consider the case of a horizontal rod or beam slightly bent by vertical forces applied to it. The state of strain is no longer of the simple character appertaining to pure flexure; in particular there will be a relative shearing of adjacent cross-sections, and also a warping of the sections so that these do not remain accurately plane. We shall assume, however, that the additional strains thus introduced are on the whole negligible, and consequently that the bending moment is connected with the curvature of the axis..."

"Galileo does not attempt any theory to account for the flexure of the beam. This theory, supplied by , was applied by Mariotte, Leibnitz, De Lahire, and Varignon, but they neglect compression of the fibres, and so place the neutral in the lower face of Galileo's beam. The true position of the neutral plane was assigned by James Bernoulli 1695, who in his investigation of the simplest case of bent beam, was led to the consideration of the curve called the "elastica." This "elastica" curve speedily attracted the attention of the great Euler (1744), and must be considered to have directed his attention to the s. Probably the extraordinary divination which led Euler to the formula connecting the sum of two elliptic integrals, thus giving the fundamental theorem of the addition equation of s, was due to mechanical considerations concerning the "elastica" curve; a good illustration of the general principle that the pure mathematician will find the best materials for his work in the problems presented to him by natural and physical questions."

"The assumption that the varies as the curvature is the basis of the 'Euler-Bernoulli' theory of flexure. This was developed in memoirs by James Bernoulli (1705) D. Bernoulli (1742), L. Euler (1744)."

"[C]alculations... based on the simple theory of bending... are approximate only. While the simple (or Bernoulli-Euler) theory gives the deflections due to the bending moment with sufficient accuracy, the portion of the total deflection which is due to shearing cannot generally be estimated with equal accuracy from the distribution of shear stress... It becomes desirable, then, to check the results by those given in the more complex theory of St. Venant... if a very accurate estimate of shearing deflection is required. In a great number of practical cases, however, the deflection due to shearing is negligible in comparison with that caused by the bending moment."

"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)."

"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."

"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."

"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."

"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 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."

"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."

"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."

"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... 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."

"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."

"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."

"[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 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... 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...]"

"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. ... 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."

"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."

"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."

"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."

"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."

"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."

"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.]"

"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."

"[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 obtainds = \frac{a^2 dx}{\sqrt{a^4 - (\alpha + \beta x + \gamma x^2)^2}}"