History of physics

"This history is intended mainly for the use of students and teachers of physics. The writer is convinced that some attention to the history of a science helps to make it attractive, and that the general view of the development of the human intellect, obtained by reading the history of science, is in itself stimulating and liberalizing."

"In the announcement of Ostwald's Klassiker der Exakten Wissenschaften [Classics of the Exact Sciences] is the following significant statement: "While, by the present methods of teaching, a knowledge of science in its present state of advancement is imparted very successfully, eminent and far-sighted men have repeatedly been obliged to point out a defect which too often attaches to the present scientific education of our youth. It is the absence of the historical sense and the want of knowledge of the great researches upon which the edifice of science rests." It is hoped that the survey of the progress of physics here presented may assist in remedying this defect so clearly pointed out by Professor Ostwald."

"In mathematics, metaphysics, literature, and art the Greeks displayed wonderful creative genius, but in natural science they achieved comparatively little. ...it is true that, as a rule, they were ignorant of the art of experimentation, and that many of their physical speculations were vague, trifling, and worthless."

"As compared with the vast amount of theoretical deduction about nature, the number of experiments known to have been performed by the Greeks is surprisingly small. Little or no attempt was made to verify speculation by experimental evidence."

"Mechanical subjects are treated in the writings of Aristotle. The great peripatetic had grasped the notion of the parallelogram of forces for the special case of the rectangle. He attempted the theory of the lever, stating that a force at a greater distance from the fulcrum moves a weight more easily because it describes a greater circle. He resolved the motion of a weight at the end of the lever into tangential and normal components. The tangential motion he calls according to nature; the normal motion contrary to nature. ...the expression contrary to nature applied to a natural phenomenon is inappropriate and confusing."

"Aristotle's views of falling bodies are very far from the truth. ...He says "That body is heavier than another which, in an equal bulk, moves downward quicker." In another place he teaches that bodies fall quicker in exact proportion to their weight. No statement could be further from the truth. ...If it had only occurred to him, while walking up and down the paths near his school in Athens, to pick up two stones of unequal weight and drop them together, he could easily have seen that the one of, say, ten times the weight, did not descend ten times faster."

"Immeasurably superior to Aristotle as a student of mechanics is Archimedes. He is the true originator of mechanics as a science. To him we owe the theory of the centre of gravity (centroid) and of the lever. In his Equiponderance of Planes [On the Equilibrium of Planes] he starts with the axiom that equal weights acting at equal distances on opposite sides of a pivot are in equilibrium, and then endeavours to establish the principle that "in the lever unequal weights are in equilibrium only when they are inversely proportional to the arms from which they are suspended.""

"In his Floating Bodies Archimedes established the important principle, known by his name, that the loss of weight of a body submerged in water is equal to the weight of the water displaced, and that a floating body displaces its own weight of water. Since the days of Archimedes able minds have drawn erroneous conclusions on liquid pressure. The expression "hydrostatic paradox" indicates the slippery nature of the subject. All the more must we admire the clearness of conception and almost perfect logical rigour which characterize the investigations of Archimedes."

"It is reported that he astonished the court of Hieron by moving heavy ships by aid of a collection of pulleys. To him is ascribed the invention of war engines, and the endless screw ("screw of Archimedes") which was used to drain the holds of ships."

"The Greeks invented the hydrometer, probably in the fourth century AD. ...It was first used in medicine to determine the quality of drinking water, hard water being at that time considered unwholesome. According to Desaguliers it was used for this purpose as late as the eighteenth century."

"Optics is one of the oldest branches of physics. A converging lens of rock crystal is said to have been found in the ruins of Nineveh."

"In Greece burning glasses seem to have been manufactured at an early date. Aristophanes in the comedy of The Clouds... introduces a conversation about "fine transparent stone (glass) with which fires are kindled," and by which, standing in the sun, one can, "though at a distance, melt all the writing" traced on a surface of wax."

"The Platonic school taught the rectilinear propagation of light and the equality of the angle of incidence to that of reflection."

"The astronomer Claudius Ptolemy... measured angles of incidence and of refraction and arranged them in tables."

"Metallic mirrors seem to have been manufactured in remote antiquity. Looking glasses are referred to in Exodus 38:8, and in Job 37:18; they have been found in graves of Egyptian mummies."

"Spherical and parabolic mirrors were known to the Greeks. To Euclid... is attributed a work on Catoptrics, dealing with phenomena of reflection. In it is found the earliest reference to the focus of a spherical mirror. In Theorem 30 it is stated that concave mirrors turned toward the sun will cause ignition. In the "fragmentum Bobiense," a document written, perhaps, by Anthemius of Tralles, the focal property of parabolic reflectors is demonstrated. Several Greek authors appear to have written on concave mirrors."

"The Greeks elaborated several theories of vision. According to the Pythagoreans, Democritus, and others vision is caused by the projection of particles from the object seen, into the pupil of the eye. On the other hand Empedocles, the Platonists, and Euclid held the strange doctrine of ocular beams, according to which the eye itself sends out something which causes sight as soon as it meets something else emanated by the object."

"Thales of Miletus... one of the "seven wise men" of early Greece, is credited with the knowledge that amber, when rubbed, will attract light bodies, and that a certain mineral, now called magnetite, or loadstone, possesses the power of attracting iron."

"Amber—a mineralized yellowish resin—was used in antiquity for decoration. In common with the bright shining silver-gold alloys, and gold itself, it was called "electron"; hence the word "electricity.""

"Theophrastus, in his treatise On Gems mentions another mineral which becomes electrified by friction. We know now that all bodies can be thus electrified."

"The polarity of magnets and the phenomenon of repulsion which may exist between electric charges or magnetic poles were unknown to Greek antiquity."

"It is in Athens that we find the oldest contrivance for observing the direction of the wind. There, in its essential parts standing to this day, is the "tower of the winds," built about 100 B.C. Upon an octagon of marble was a roof, the highest part of which carried a weather-vane in form of a triton."

"Among the Greeks meteorology can hardly be said to have risen to the dignity of a science."

"Theophrastus of Eresus... wrote a book On Winds and on Weather Signs, but like most other Greek philosophers, he was hardly the man to adopt patient and exact observation in place of dogmatic assertion and the teaching of authority."

"Aristotle makes a good observation on the formation of dew; viz. dew is formed only on clear and quiet nights."

"Aratus of Soli... wrote a book of Prognostics, giving predictions of the weather from observation of astronomical phenomena, and various accounts of the effect of weather on animals."

"In Newton's time only two kinds of force were available for quantitative investigation. One was the force of gravity; the other the forces of push and pull encountered in everyday life... Newton endeavored to construct a general theory of all forces, both those known in his time and those that might be discovered and investigated later. He intended his theory of gravitation to be one example that he himself could work out fully... Newton formulated his celebrated three laws: (1) In the absence of force, a body will continue at rest or in its present state of uniform rectilinear motion. (2) In the presence of force, a body will be accelerated in the direction of that force, the product of its mass by its acceleration being equal to the force (f = ma). (3) To every force there corresponds an equal counterforce, acting in a direction opposite to that of the force... According to the third law, then, each planet exerts an attractive counterforce to the sun, accelerating it toward the planet... a relatively small acceleration, because the mass of the sun so vastly exceeds... every planet..."

"However far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms. The argument is that simply by the word "experiment" we refer to a situation where we can tell others what we have done and what we have learned and that, therefore, the account of the experimental arrangement and of the results of the observations must be expressed in unambiguous language with suitable application of the terminology of classical physics."

"A law explains a set of observations; a theory explains a set of laws. The quintessential illustration of this jump in level is the way in which Newton’s theory of mechanics explained Kepler’s law of planetary motion. Basically, a law applies to observed phenomena in one domain (e.g., planetary bodies and their movements), while a theory is intended to unify phenomena in many domains. Thus, Newton’s theory of mechanics explained not only Kepler’s laws, but also Galileo’s findings about the motion of balls rolling down an inclined plane, as well as the pattern of oceanic tides. Unlike laws, theories often postulate unobservable objects as part of their explanatory mechanism. So, for instance, Freud’s theory of mind relies upon the unobservable ego, superego, and id, and in modern physics we have theories of elementary particles that postulate various types of quarks, all of which have yet to be observed."

"From the thick darkness of the middle ages man's struggling spirit emerged as in new birth; breaking out of the iron control of that period; growing strong and confident in the tug and din of succeeding conflict and revolution, it bounded forwards and upwards with restless vigour to the investigation of physical and moral truth; ascending height after height; sweeping afar over the earth, penetrating afar up into the heavens; increasing in endeavour, enlarging in endowment; every where boldly, earnestly out-stretching, til, in the AUTHOR of the PRINCIPIA, one arose, who, grasping the master-key of the universe and treading its celestial paths, opened up to the human intellect the stupendous realities of the material world, and, in the unrolling of its harmonies, gave to the human heart a new song to the goodness, wisdom, and majesty of the all-creating, all-sustaining, all-perfect God."

"Newton then elevates this approximate empirical discovery to the position of a rigorous principle, the principle of inertia, and states that absolutely free bodies hence will cover equal distances in equal times. ...It is the principle of inertia coupled with an understanding of spatial congruence that yields us a definition of congruent stretches of absolute time. ...The principle of inertia, together with the other fundamental principles of mechanics, enables us... to place mechanics on a rigorous mathematical basis, and rational mechanics is the result. ...science, in the case of mechanics, has followed the same course as in geometry. Initially our information is empirical and suffers from all the inaccuracies ...But this empirical information is idealised, then crystallised into axioms, postulates or principles susceptible of direct mathematical treatment. ...If peradventure further experiment were to prove that our mathematical deductions ...were not born out in the world of reality, we should have to modify our initial principles and postulates or else agree that nature is irrational. With mechanics, the necessity of modifying the fundamental principles became imperative when it was recognized that the mass of a body was not the constant magnitude we thought it to be; hence it was experiment that brought about the revolution. On the other hand, in the case of geometry, it was the mathematicians themselves who forsaw the possibility of various non-Euclidean doctrines, prior to any suggestion of this sort being demanded by experiment."

"... Inertia resists acceleration, but acceleration relative to what? Within the frame of classical mechanics the only answer is: Inertia resists acceleration relative to space. This is a physical prop­erty of space—space acts on objects, but objects do not act on space. Such is prob­ably the deeper meaning of Newton's assertion spatium est absolutum (space is absolute). But the idea disturbed some, in particular Leibnitz, who did not ascribe an independent existence to space but considered it merely a proper­ty of "things" (contiguity of physical objects)."

"Although many historians of the new millennium now take issue with the notion of a Scientific Revolution, it is generally agreed that Newton's work culminated the long development of European science, creating a synthesis that opened the way for the scientific culture of the modern age."

"I mentally conceive of some moveable [sphere] projected on a horizontal plane, all impediments being put aside. Now it is evident... that equable motion on this plane would be perpetual if the plane were of infinite extent, but if we assume it to be ended, and [situated] on high, the movable, driven to the end of this plane and going on further, adds on to its previous equable and indelible motion, that downward tendency which it has from its heaviness. Thus, there emerges a certain motion, compounded..."

"On the authority of Aristotle... motion in the planetary world was somehow directed by the more perfect motion in higher spheres, and so on, up to the outermost sphere of fixed stars, indistinguishable from the prime mover. This implied a refined animistic and pantheistic world view, incomparably more rational than the ancient world views of Babylonians and Egyptians, among others, but a world view, nonetheless, hardly compatible with the idea of "inertial motion" which is implied in Buridan's concept of "impetus"… a momentous breaking point... which was to bear fruit... in the hands, first of Copernicus and then of Newton."

"Wave functions, probabilities, quantum tunneling, the ceaseless roiling energy fluctuations of the vacuum, the smearing together of space and time, the relative nature of simultaneity, the warping of the spacetime fabric, black holes, the big bang. Who could have guessed that the intuitive, mechanical, clockwork Newtonian perspective would turn out to be so thoroughly parachial—that there would be a whole new mind-boggling world lying just beneath the surface of things as they are ordinarily experienced?"

"Newton's system was for a long time considered as final and the task... seemed simply to be an expansion.... From the theory of the motion of mass points one could go over to the mechanics of solid bodies, to rotatory motions, and one could treat the continuous motion of fluid or the vibrating motion of an elastic body. All these... were gradually developed... with the evolution of mathematics, especially of the differential calculus... checked by experiments. and hydrodynamics became a part of mechanics. Another science... was astronomy. Improvements... led to... more accurate determinations of the motions of the planets... When the phenomena of electricity and magnetism were discovered, the... forces were compared to the gravitational forces... Finally, in the nineteenth century, even the theory of heat could be reduced..."

"The first difficulty arose in the discussion of the electromagnetic field in... Faraday and Maxwell. In Newtonian mechanics the gravitational force had been considered as given... In the work of Faraday and Maxwell... the field of force... became the object of the investigation... they tried to set up equations of motion for the fields, not primarily for the bodies... This change led back to a point of view...held... before Newton. An action could... be transferred... only when the two bodies touched... Newton had introduce a very new and strange hypothesis by assuming a force that acted over a long distance. Now in the theory of fields... action is transferred from one point to a neighboring point... in terms of differential equations. ...the description of the electromagnetic fields... by Maxwell's equations seemed a satisfactory solution of the problem of force. ...The axioms and definitions of Newton had referred to bodies and their motion; but with Maxwell the fields... seemed to have acquired the same degree of reality as the bodies in Newton's theory. This view... was not easily accepted; and to avoid such a change in the concept of reality... many physicists believed that Maxwell's equations actually referred to the deformations of an elastic medium... the ether... the medium was so light and thin that it could penetrate into other matter and could not be seen or felt. ...[H]owever ...it could not explain the complete absence of any longitudinal light waves."

"I shall try to sum up the main obstacles which arrested the progress of science for such an immeasurable time. The first was the splitting of the world into two spheres, and the mental split which resulted from it. The second was the geocentric dogma, the blind eye turned on the promising line of thought which had started with the Pythagoreans and stopped abruptly with Aristarchus of Samos. The third was the dogma of uniform motion in perfect circles. The fourth was the divorcement of science from mathematics. The fifth was the inability to realize that a body at rest tended to stay at rest, a body in motion tended to stay in motion. The main achievement of the first part of the scientific revolution was the removal of these five cardinal obstacles. This was done chiefly by three men: Copernicus, Kepler and Galileo. After that, the road was open to the Newtonian synthesis; from there on the journey led with rapidly gaining speed to the atomic age."

"The history of the development of mechanics is quite indispensable to a full comprehension of the science in its present condition. It also affords a simple and instructive example of the processes by which natural science generally is developed."

"These independent objects of Newtonian physics might move, touch each other, collide, or even, by a certain stretch of the imagination, act at a distance: but nothing could penetrate them except in the limited way that light penetrated translucent substances. This world of separate bodies, unaffected by the accidents of history or geographic location, underwent a profound change with the elaboration of the new concepts of matter and energy that went forward from Faraday and von Mayer through Clerk-Maxwell and Willard Gibbs and Ernest Mach to Planck and Einstein. The discovery that solids, liquids, and gases were phases of all forms of matter modified the very conceptions of substance, while the identification of electricity, light, and heat as aspects of a protean energy, and the final break-up of "solid" matter into particles of this same ultimate energy lessened the gap, not merely between various aspects of the physical world, but between the mechanical and the organic. Both matter in the raw and the more organized and internally self-sustaining organisms could be described as systems of energy in more or less stable, more or less complex, states of equilibrium."

"Galileo's comprehension of the concept of acceleration, which he defined as a change of velocity either in magnitude or direction... was an abstract idea that no one seems to have thought much about before. And in using it to test the still accepted Aristotelian precept that a moving object requires a force to maintain it, Galileo easily demonstrated that it is not motion but rather acceleration which cannot occur without an external force. Deliberately rejecting common sense as a prejudiced witness, he let nature herself speak in the form of a "hard, smooth and very round ball" rolling down a "very straight" ideal groove lined with polished parchment, and then rolling up another groove, clocking each roll "hundreds or times"... he showed that, while downward motion (helped by gravity force) makes speed increase and upward motion (hindered by gravity force) makes speed decrease, there is always a "boundary case" in between... where speed remains constant (without any appreciable force)—and that, by reducing friction, this boundary case can be made to approach a horizontal level where gravity has no effect. Similarly testing... he also drafted a law of falling bodies: "that the distances traversed, during equal intervals of time... stand to one another in the same ratio as the odd numbers beginning with unity." And his beautiful analysis of a cannonball's trajectory into horizontal and vertical components... was one day to be of enormous help to Isaac Newton in solving the riddle of gravity."

"Newton proposed that the particles of the air (we would call them molecules), were motionless in space and were held apart by repulsive forces between them... He assumed that the repulsive force was inversely proportional to the distance between the particles...He showed that, on the basis of this assumption, a collection of static particles in a box would behave exactly as Boyle had found. His model led directly to Boyle's law. Probably the greatest scientist ever, Newton managed to get the right answer from a model that was wrong in every possible way."

"The founders of modern science - for instance, Galileo, Kepler, and Newton - were mostly pious men who did not doubt God’s purposes. Nevertheless they took the revolutionary step of consciously and deliberately expelling the idea of purpose as controlling nature from their new science of nature. They did this on the ground that inquiry into purposes is useless for what science aims at: namely, the prediction and control of events. To predict an eclipse, what you have to know is not its purpose but its causes. Hence science from the seventeenth century onwards became exclusively an inquiry into causes. The conception of purpose in the world was ignored and frowned on. This, though silent and almost unnoticed, was the greatest revolution in human history, far outweighing in importance any of the political revolutions whose thunder has reverberated through the world."

"Newton did not show the cause of the apple falling, but he shewed a similitude between the apple and the stars. By doing so he turned old facts into new knowledge; and was well content if he could bring diverse phenomenon under "two or three Principles of Motion" even "though the Causes of these Principles were not yet discovered.""

"Newtonian mechanics does not apply to all situations. If the speeds of the interacting bodies are very large —an appreciable fraction of the speed of light —we must replace Newtonian mechanics with Einstein’s special theory of relativity, which holds at any speed, including those near the speed of light. If the interacting bodies are on the scale of atomic structure (for example, they might be electrons in an atom), we must replace Newtonian mechanics with quantum mechanics. Physicists now view Newtonian mechanics as a special case of these two more comprehensive theories. Still, it is a very important special case because it applies to the motion of objects ranging in size from the very small (almost on the scale of atomic structure) to astronomical (galaxies and clusters of galaxies)."

"The foundational achievement of classical mechanics is to establish that the first point is faulty. It is fruitful, in that framework, to allow a broader concept of the character of physical reality. To know the state of a system of particles, one must know not only their positions, but also their velocities and their masses. Armed with that information, classical mechanics predicts the system’s future evolution completely. Classical mechanics, given its broader concept of physical reality, is the very model of Einstein Sanity."

"This book tells the story of the researches that are traditionally lumped together under the label "radiation theory" and revolving, loosely speaking around the familiar heat-and-light exchange (hot bodies emit light and radiate; the absorption of light, especially sunlight, is warming)."

"These days it is common knowledge that short waves are more powerful than long ones, as the very short ones, known as x-rays, damage living tissues. It took half-a-century to learn this fact: it was one of the great discoveries of young Albert Einstein of 1905. When he announced it leading researchers found it most incredible..."

"The development of quantum mechanics in the beginning of the twentieth century was a unique intellectual adventure, which obliged scientists and philosophers to change radically the concepts they used to describe the world. After these heroic efforts, it became possible to understand the stability of matter, the mechanical and thermal properties of materials, the interaction of radiation and matter, and many other properties of the microscopic world that had been impossible to understand with classical physics. A few decades later, that conceptual revolution enabled a technological revolution, at the root of our information-based society. It is indeed with the quantum mechanical understanding of the structure and properties of matter that physicists and engineers were able to invent and develop the transistor and the laser—two key technologies that permit the high-bandwidth circulation of information, as well as many other scientific and commercial applications."

"The realization of the importance of entanglement and the clarification of the quantum description of single objects have been at the root of a second quantum revolution, and the John Bell was its prophet."

"The blackbody oven embodied an... instance of radiation interacting with matter. ...Planck first... derived an empirical equation to fit the data. ...His more ambitious aim now was to find a theoretical entropy-energy connection applicable to the blackbody problem. ...Ludwig Boltzmann interpreted the second law of thermodynamics as a "probability law." If the relative probability or disorder for the state of the system was W, he concluded, then the entropy S of the system in that state was proportional to the logarithm of W,S ∝ lnW ...Plank applied this to the blackbody problem by writingS = k lnW (1)for the total entropy of the vibrating molecules... "resonators"—in the blackbody oven's walls... k is now called Boltzmann's constant. ...Boltzmann's theory taught the lesson that conceivably—but against astronomically unfavorable odds—any macroscopic process can reverse... contradicting the second law of thermodynamics. Boltzmann's conclusions seemed fantastic to Planck, but by 1900 he was becoming increasingly desparate, even reckless... The counting procedure Planck used to calculate the disorder W... was borrowed from... Boltzmann's theoretical techniques. He considered... that the total energy of the resonators was made up of small indivisible "elements," each one of magnitude ε. It was then possible to evaluate W as a count of the number of ways a certain number of energy elements could be distributed to a certain number of resonators... His argument would not succeed unless he assumed that the energy ε of the elements was proportional to the frequency with which the resonators vibrated, ε ∝ v, or ε = hv, with h the proportionality constant."

"The second quantum revolution... was the term coined by... Alain Aspect to describe the changes in physics, the beginnings of which date back to the 1960s. ...he brought together two different threads. The first one embraced the emergence of the awareness of the importance of... entanglement. ...It started a conceptual revolution, including the perspective of building quantum computers... The second thread derives from physicists' ability to isolate, control, and observe single quantum systems such as electrons, neutrons and atoms. Finally these threads merged into a new field of research entitled quantum information. In Aspect's formulation... he posited two quantum revolutions taking place in the twentieth century. The first one, in the first half of the century, created the scientific theory that describes the behavior of atoms, radiation, and their interactions. The second one occurred in the second half and is still evolving... intellectual aspects... arose from the renewal of research on the foundations of quantum physics."

"Intuitively, Planck knew that his work was as important as Newton's in paving a new physics, as he privately confided to his son in 1901. Because of his own conservative beliefs, Planck felt stymied by the revolutionary nature of his own ideas... In truth, Planck was only underestimating the importance of his work, if anything. In retrospect... it started a revolution in science—the quantum revolution—compared to which the Copernican revolution pales."

"The development of the quantum ideas themselves occurred in discrete jumps, quantum leaps of the creative insights of a few people."

"The start of the revolution that produced the old quantum theory is moved from the end of 1900 to 1906... The preceding crisis...resulted from the difficulties in reconciling Planck's derivation with the tenets of classical physics. Planck's change in vocabulary—from "resonator" to "oscillator" and from "element" to "quantum"—is the central symptom of incommensurability. It signals the changed meaning of the quantity hv from a mental subdivision of the energy continuum to a physically separable atom of energy. That my critics continue to apply the term "energy quantum" to pre-1906 papers and lectures in which Planck consistently used "energy element" reveals something of the difficulty of reversing the gestalt switch that took place during that year and those which followed. ...Boltzmann's probabilistic derivation of the entropy of a gas ...illustrates the problem to which the concept of paradigm was a response. The derivation was not reduced to rules but instead served as a model to be applied by means of analogy. As a result, when its application was transferred from gases to radiation, Planck and Lorentz could invent different analogies with which to effect change."

"Quantum mechanics is revolutionary because it overturned scientific concepts that seemed to be so obvious and so well confirmed by experience that they were beyond reasonable question, but it is an incomplete because we still do not know precisely where quantum mechanics will lead us—nor even why it must be true!"

"In the brief period between 1900 and 1935 there occurred one of the most astonishing outbursts of scientific creativity in all of history. ...no other historical era has crammed so much scientific creativity, so many discoveries of new ideas and techniques, into so few years. Although a few outstanding individuals dominate... they were assisted in their work by an army of talented scientists and technicians."

"In the years 1925 and 1926 modern quantum theory came into being fully fledged. These anni mirabiles remain an episode of great significance in the folk memory of the theoretical physics community... Werner Heisenberg had been struggling to understand the details of atomic spectra. ...Heisenberg's discovery came to be known as matrix mechanics. ...In 1925 matrices were... mathematically exotic to the average theoretical physicist... Prince Louis de Broglie... made the bold suggestion that if undulating light also showed particle-like properties... one should expect particles such as electrons to manifest wavelike properties... by generalizing the Planck formula. The latter had made the particlelike property of energy proportional to the wavelike property of frequency. De Broglie suggested that... particlelike... momentum... should analogously be related to... wavelength, with Planck's universal constant again... These equivalences provided... for translating from particles to waves, and vice versa. In 1924, de Broglie laid out these ideas in his doctoral thesis. ...To attain a full dynamical theory, a further generalization... allowed the incorporation of interactions... This is the problem that Schrödinger succeeded in solving. Early in 1926 he published the famous equation... led to its discovery by exploiting an analogy drawn from optics."

"The quantum revolution describes our deepest insight, so far, into the physical structure of nature. It is comparable only with the Copernican revolution, switching from a finally oriented anthropomorphic description of physical phenomena to one using general laws with initial or boundary conditions, connected with the names Kepler, Galileo, and Newton, or with the change from tangible mass points as basic structures to Faraday's and Maxwell's field concepts and, shortly before quantum theory, with the relativization of space and time by the lonely genius Einstein."

"Quantum mechanics, created early this century in response to certain experimental facts which were inexplicable according to previously held ideas (...'classical physics'), caused three great revolutions. In the first place it opened up a completely new set range of phenomena to which the methods of physics could be applied. ...The second revolution was the apparent breakdown of determinism, which had always been an unquestioned ingredient and an inescapable prediction of classical physics. ...The outcome of any particular experiment is not, even in principle, predictable, but is chosen at random from a set of possibilities; all that can be predicted is the probability of particular results when the experiment is repeated many times. ...even if we had complete knowledge of the initial state... The third revolution ...challenged the basic belief, implicit in all science and indeed in almost the whole of human thinking, that there exists an objective reality ...that does not depend for its existence on its being observed."

"Above all, the ominous clouds of those phenomena that we are with varying success seeking to explain by means of the quantum of action, are throwing their shadows over the sphere of physical knowledge, threatening no one knows what new revolution."

"The spectrum of the sun was first observed, in 1666, by Newton, who allowed light coming from a small round opening in a shutter to pass through a glass prism. This spectrum was most impure; and a pure spectrum was not obtained until, in 1802, Wollaston repeated Newton's experiment, replacing the round opening by a slit parallel to the edge of the prism. He observed several dark lines crossing the spectrum, which limited, as he thought, the different spectral colors."

"It is manifest that everything in the world, whether it be substance or accident, produces rays in its own manner like a star... Everything that has actual existence in the world of the elements emits rays in every direction, which fill the whole world."

"Mohammedan science made a great advance on that of the Greeks... [in] optics, and this was very largely a by-product of medicine. Optics comes much more in the doctor's province, especially in tropical and sub-tropical countries where there is a prevalence of eye diseases ...This goes right back to the ns, who were doing ...cataract operations, between 2000 and 2500 BC. Therefore... they introduced into science something ...which the Greeks did not have ...the lens. The Greeks ...knew that the mirror could focus the rays—that was due to Archimedes—but they did not have any lenses."

"Phenomena were accounted for by taking into consideration the frictional resistances that would interfere with rapid vibrations of the electrons. When these frictional resistances were weak, oscillatory disturbances, such as rays of light, could be propogated through the , which was then termed transparent (glass). When these frictional forces were considerable, the light ray was unable to set the electrons into vibration; its energy was consumed in the attempt, and as a result it could not proceed; the dielectric was then opaque (ebonite, sulphur)."

"Kepler's theory of vision introduced the concept of optical images that is the basis of modern ."

"The wave theory furnishes the simplest possible explanation of interference phenomena. On the other hand it has considerable difficulty in explaining the rectilinear propagation of light. In this respect the analogy between sound and light seems to break down, for sound does not travel in straight lines. ...This analogy between sound and light presents still further contradictions when polarization phenomena are under consideration. It was these contradictions which prevented for a long time the general recognition of the wave theory in spite of the simple explanation which it offers of interference. The difficulties were not removed until a too close analogy between sound and light was given up."

"[T]he explanation of the rectilinear propagation of light from the standpoint of the wave theory presents difficulties. To overcome these difficulties Huygens made the supposition that every point P which is reached by a light-wave may be conceived as the source of elementary light-waves, but that these elementary waves produce an appreciable effect only upon the surface of their envelope. ...Fresnel replaced Huygens' arbitrary assumption that only the envelope of the elementary waves produces appreciable light effects by the principle that the elementary waves in their criss-crossing influence one another in accordance with the principle of interference. Light ought then to appear not only upon the enveloping surface, but everywhere where the elementary waves reinforce one another; on the other hand, there should be darkness wherever they destroy one another. ...[I]t is possible to deduce from this Fresnel-Huygens principle not only the laws of diffraction, but also those of straight-line propagation, reflection, and refraction."

"The result of my work has been the most extraordinary, the most unforeseen, and the happiest, that ever was; for, after having performed all the equations, multiplications, antitheses, and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for the s which Monsieur Descartes has established."

"Newton's proof of the law of refraction is based on an erroneous notion that light travels faster in glass than in air, the same error that Descartes had made. This error stems from the fact that both of them thought that light was corpuscular in nature."

"To complete the theory of reflexion and on the undulatory hypothesis, it will be necessary to show what becomes of those oblique portions of the secondary waves, diverging in all directions from every point of the reflecting or refracting surfaces... which do not conspire to form the principal wave. But to understand this, we must enter on the doctrine of the interference of the rays of light,—a doctrine we owe almost entirely to the ingenuity of Dr. Young, though some of its features may be pretty distinctly traced in the writings of Hooke, (the most ingenious man, perhaps, of his age,) and though Newton himself occasionally indulged in speculations bearing a certain relation to it. But the unpursued speculations of Newton, and the appercus of Hooke, however distinct, must not be put in competition, and, indeed, ought scarcely to be mentioned with the elegant, simple, and comprehensive theory of Young,—a theory which, if not founded in nature, is certainly one of the happiest fictions that the genius of man has yet invented to group together natural phenomena, as well as the most fortunate in the support it has unexpectedly received from whole classes of new phenomena, which at their first discovery seemed in irreconcileable opposition to it. It is, in fact, in all its applications and details one succession of felicities insomuch that we may almost be induced to say, if it be not true, it deserves to be so. The limits of this Essay, we fear, will hardly allow us to do it justice."

"We must never forget that it is principles, not phenomena,—the interpretation, not the mere knowledge of facts,—which are the objects of enquiry to the natural philosopher. As truth is single, and consistent with itself, a principle may be as completely and as plainly elucidated by the most familiar and simple fact, as by the most imposing and uncommon phenomenon. The colours which glitter on a soap-bubble are the immediate consequence of a principle the most important from the variety of phenomena it explains, and the most beautiful, from its simplicity and compendious neatness, in the whole science of optics. If the nature of periodical colours can be made intelligible by the contemplation of such a trivial object, from that moment it becomes a noble instrument in the eye of correct judgment; and to blow a large, regular, and durable soap-bubble may become the serious and praise-worthy endeavour of a sage, while children stand round and scoff, or children of a larger growth hold up their hands in astonishment at such waste of time and trouble. To the natural philosopher there is no natural object unimportant or trifling. From the least of nature's works he may learn the greatest lessons."

"Perhaps the most striking example of the services which have been rendered to Science by the contemplation of various models, many or all of which have ultimately been found to be inadequate for complete representation, is to be found in the history of Optics. The various forms of the corpuscular theory, and of the wave theory, of Light were all attempts to represent the phenomena by models, the value of which had to be estimated by developing their Mathematical consequences, and comparing these consequences with the results of experiments. The adynamical theory of Fresnel, the elastic solid theory of the ether developed by Navier, Cauchy, Poisson, and Green, the labile ether theory developed by Cauchy and Kelvin, and the rotational ether theory of MacCullagh were all efforts of the kind... indicated; they were all successful in some greater or less degree in the representation of the phenomena, and they all stimulated Physicists to further efforts to obtain more minute knowledge of those phenomena. Even such an inadequate theory as that of Fresnel led to the very interesting observation by Humphry Lloyd of the phenomenon of conical refraction in crystals, as the result of the prediction by Rowan Hamilton that the phenomenon was a necessary consequence of the Mathematical fact that Fresnel's wave surface in a biaxal crystal possesses four conical points."

"There can be no doubt that light consists of the motion of a certain substance. For if we examine its production, we find that here on earth it is principally fire and flame which engender it, both of which contain beyond doubt bodies which are in rapid movement, since they dissolve and destroy many other bodies more solid than they: while if we regard its effects, we see that when light is accumulated, say by concave mirrors, it has the property of combustion just as fire has, that is to say, it disunites the parts of bodies, which is assuredly a proof of motion, at least in the true philosophy, in which the causes of all natural effects are conceived as mechanical causes. Which in my judgment must be accomplished or all hope of ever understanding physics is renounced."

"Velocity of transverse undulations in our hypothetical medium, calculated from the electromagnetic experiments of 'MM'. Kohlrausch and Weber, agrees so exactly with the velocity of light calculated from the optical experiments of M. Fizeau, that we can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena."

"That light is not itself a substance may be proved from the phenomenon of interference. A beam of light from a single source is divided by certain optical methods into two parts, and these, after travelling by different paths, are made to reunite and fall upon a screen. If either half of the beam is stopped, the other falls on the screen and illuminates it, but if both are allowed to pass, the screen in certain places becomes dark, and thus shows that the two portions of light have destroyed each other. Now, we cannot suppose that two bodies when put together can annihilate each other; therefore light cannot be a substance. ... What we have proved is that one portion of light can be the exact opposite of another portion... Such quantities are the measures, not of substances, but always of processes taking place in a substance. We therefore conclude that light is... a process going on in a substance... so that when the two portions [of light] are combined no process goes on at all. ...the light is extinguished when the difference of the length of the paths is an odd multiple of... a half wave-length. ...we see on the screen a set of fringes consisting of dark lines at equal intervals, with bright bands of graduated intensity between them. ...if the two rays are polarized ...when the two planes of polarization are parallel the phenomena of interference appear as above ...As the plane turns ...light bands become less distinct ...at right angles ...illumination of the screen becomes uniform, and no trace of interference can be discovered. ...The process may, however, be an electromagnetic one ...the electric displacement and the magnetic disturbance are perpendicular to each other, either ...supposed to be in the plane of polarization."

"Helmholtz's invention of the ophthalmoscope in 1851 marks an epoch in ."

"Thomas Young... attained equal eminence by his discoveries in connection with the undulatory theory of light, in which he was the first to assert the principle of interference, and that of transverse vibrations... The remarkable fact that Young, of whom Helmholtz says that he came a generation too soon, remained scientifically unrecognised and popularly almost unknown to his countrymen, has been explained by his unfortunate manner of expression and the peculiar channels through which his labours were announced to the world. ...[S]everal great names contributed, by the authority they commanded, to oppose Young's claims to originality and renown. Lord Brougham, shielded by the powerful anonymity of the ',' and ostentatiously parading the authority of Newton, submitted the views of Young to a ruthless and unfair criticism, the popular influence of which Young probably never overcame. The great authority on optics, Brewster, who has enriched that science by such a number of experiments and observations of the first importance, never really adopted the theories of Young and Fresnel."

"[W]ith regard to light, that it consists of vibrations was almost proved by the phenomena of , while those of polarisation showed the excursions of the particles to be perpendicular to the line of propogation; but the phenomena of dispersion, etc., require additional hypotheses which may be very complicated. Thus, the further progress of molecular speculation appears quite uncertain. If hypotheses are to be tried haphazard, or simply because they will suit certain phenomena, it will occupy the mathematical physicists of the world say half a century on the average to bring each theory to the test, and since the number of possible theories may go up into the trillion, only one of which can be true, we have little prospect of making further solid additions to the subject in our time."

"Contemporary with Vitellio and Peccam was... Roger Bacon, a man of almost universal genius, and who wrote on almost every branch of science. He frequently quotes Alhazen on the subject of optics, and seems to have carefully studied his writings, as well as those of other Arabians, which were the fountains of natural knowledge in those days, and which had been introduced into Europe by means of the Moors in Spain. Notwithstanding the pains this great man took with the subject of opticks, it does not appear that, with respect to theory, he made any considerable advance upon what Alhazen had done before him."

"Descartes subscribed to the doctrine of instantaneous propagation, but with him something new emerged: for his was the first uncompromisingly mechanical theory that asserted the instantaneous propagation of light in a material medium... Indeed, mechanical analogies had been used to explain optical phenomena long before Descartes, but the Cartesian theory was the first clearly to assert that light itself was nothing but a mechanical property of the luminous object and of the transmitting medium. It is for this reason that we may regard Descartes' theory of light as legitimate starting point of modern physical optics."

"One of the most distinguished and prolific mathematicians in the medieval tradition of Arabic Islamic science, al-Hasan ibn al-Haytham (Latinized as Alhacen or Alhazen) became known in Europe in the thirteenth century as the author of the monumental book on optics—the mathematical theory of vision. In his Kitâb al-Manâ zir (De aspectibus), the eleventh-century scholar offered a new solution to the problem of vision, combining experimental investigations of the behavior of light with inventive geometrical proofs and constant forays into the psychology of visual perception — all systematically tied together to form a coherent alternative to the Euclidean and Ptolemaic theories of "visual rays" issuing from the eye."

"[I]t was Ibn al-Haytham's early embrace of empiricism and trust in mathematical proof that underlay the revolutionary project of his mature magnum opus, the Optics, the book that pointed the science of vision in the direction later pursued in seventeenth-century Europe. It was wholly composed of systematically arranged experiments and geometrical proofs, all expressed in clear, consistent vocabulary and orderly exposition. The Latin translation influenced medieval European scientists and philosophers such as Roger Bacon and Witelo. But the book came into its own later, when it attracted the attention of mathematicians like Kepler, Descartes, and Huygens, thanks in part to 's edition published in Basel in 1572."

"Of those who ascribe perception to something other than similarity, Alcmaeon states... the difference between men and animals. For man, he says, differs from other creatures "inasmuch as he alone has the power to understand. Other creatures perceive by sense but do not understand"; since to think and to perceive by sense are different processes and not, as Empedocles held, identical. not Empedocles held He next speaks of the senses severally. ...Eyes see through the water round about. And the eye obviously has fire within, for when one is struck flashes out. Vision is due to the gleaming,—that is to say, the transparent character of that which [in the eye] reflects the object; and sight is the more perfect, the greater the purity of this substance."

"Anaxagoras holds that sense perception comes to pass by means of opposites, for the like is unaffected by the like. He then essays to review each separately. Accordingly he maintains that seeing is due to the reflection in the pupil, but that nothing is reflected in what is of like hue, but only in what is of a different hue. Now with most this contrast of hue occurs by day, but with some by night, and this is why the latter are keen of vision by night. But, in general, night the rather is of the eye's own hue. Furthermore, there is reflection by day, he holds, because the light is a contributing cause of reflection, and because the stronger of two colours is regularly reflected better in the weaker."

"[W]hen, in 1815, a young French military engineer, named Augustin Jean Fresnel, returning from the Napoleonic wars, became interested in the phenomena of light, and made some experiments concerning diffraction which seemed to him to controvert the accepted notions of the materiality of light, he was quite unaware that his experiments had been anticipated... He communicated his experiments and results to the French Institute, supposing them to be absolutely novel. That body referred them to a committee, of which... the dominating member was Dominique Francois Arago... [who] at once recognized the merit of Fresnel's work, and soon became a convert to the theory. He told Fresnel that Young had anticipated him as regards the general theory, but that much remained to be done, and he offered to associate himself with Fresnel in prosecuting the investigation. Fresnel was not a little dashed to learn that his original ideas had been worked out by another while he was a lad, but he... went ahead with unabated zeal. ... [A] bitter feud ensued, in which Arago was opposed by the "Jupiter Olympus of the Academy," Laplace, by the only less famous Poisson, and by the younger but hardly less able Biot. So bitterly raged the feud that a life-long friendship between Arago and Biot was ruptured forever. The opposition managed to delay the publication of Fresnel's papers, but Arago continued to fight with his customary enthusiasm and pertinacity, and at last, in 1823, the Academy yielded, and voted Fresnel into its ranks, thus implicitly admitting the value of his work."

"It was in May 1801 that I discovered, by reflecting on the beautiful experiments of Newton, a law which appears to me to account for a greater variety of interesting phenomena than any other optical principle that has yet been made known. I shall endeavour to explain this law by a comparison. Suppose a number of equal waves of water to move upon the surface of a stagnant lake, with a certain constant velocity, and to enter a narrow channel leading out of the lake. Suppose then another similar cause to have excited another equal series of waves, which arrive at the same channel, with the same velocity, and at the same time with the first. Neither series of waves will destroy the other, but their effects will be combined: if they enter the channel in such a manner that the elevations of one series coincide with those of the other, they must together produce a series of greater joint elevations; but if the elevations of one series are so situated as to correspond to the depressions of the other, they must exactly fill up those depressions, and the surface of the water must remain smooth; at least I can discover no alternative, either from theory or from experiment. Now, I maintain that similar effects take place whenever two portions of light are thus mixed; and this I call the general law of the interference of light. I have shown that this law agrees, most accurately, with the measures recorded in Newton's Optics, relative to the colours of transparent substances, observed under circumstances which had never before been subjected to calculation, and with a great diversity of other experiments never before explained. This, I assert, is a most powerful argument in favour of the theory which I had before revived: there was nothing that could have led to it in any author with whom I am acquainted, except some imperfect hints in those inexhaustible but neglected mines of nascent inventions, the works of the great Dr. Robert Hooke, which had never occurred to me at the time that I discovered the law; and except the Newtonian explanation of the combinations of tides in the Port of Batsha."

"The Angles of Refexion and Refraction, lie in one and the same Plane with the Angle of Incidence."

"The Angle of Reflexion is equal to the Angle of Incidence."

"If the reflected or refracted Ray be returned directly back to the Point of Incidence, it shall be refracted into the Line before described as the incident Ray."

"Refraction out of a rarer Medium into a denser, is made toward the Perpendicular; that is, so that the Angle of Refraction be less than the Angle of Incidence."

"The Sine of Incidence is either accurately or very nearly in a given Ratio to the Sine of the Refraction."

"Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action (caeteris paribus [all else being equal]) strongest at the least distance?"

"Do not the Rays which differ in Refrangibility differ also in Flexibility; and are they not by their different inflexions separated from one another, so as after separation to make the Colours in the three Fringes... ? And after what manner are they inflected to make those Fringes?"

"Are not the Rays of Light in passing by the edges and sides of Bodies, bent several times backwards and forwards, with a motion like that of an Eel? And do not the three Fringes of colour'd Light... arise from three such bendings?"

"Do not the Rays of Light which fall upon Bodies, and are reflected or refracted, begin to bend before they arrive at the Bodies; and are they not reflected, refracted, and inflected, by one and the same Principle, acting variously in various Circumstances?"

"Do not Bodies and Light act mutually upon one another; that is to say, Bodies upon Light in emitting, reflecting, refracting and inflecting it, and Light upon Bodies for heating them, and putting their parts into a vibrating motion wherein heat consists?"

"Do not several sorts of Rays make Vibrations of several bignesses, which according to their bigness excite Sensations of several Colours, much after the manner that the Vibrations of the Air, according to their several bignesses excite Sensations of several Sounds? And particularly do not the most refrangible Rays excite the shortest Vibrations for making a Sensation of deep violet, the least refrangible the largest form making a Sensation of deep red, and several intermediate sorts of Rays, Vibrations of several intermediate bignesses to make Sensations of several intemediate Colours?"

"Is not the Heat of the warm Room convey'd through the Vacuum by the Vibrations of a much subtiler Medium than Air, which after the Air was drawn out remained in the Vacuum? And is not this Medium the same with that Medium by which Light is refracted and reflected and by whose Vibrations Light communicates Heat to Bodies, and is put into Fits of easy Reflexion and easy Transmission? ...And do not hot Bodies communicate their Heat to contiguous cold ones, by the Vibrations of this Medium propagated from them into the cold ones? And is not this Medium exceedingly more rare and subtile than the Air, and exceedingly more elastick and active? And doth it not readily pervade all Bodies? And is it not (by its elastick force) expanded through all the Heavens?"

"Doth not this Æthereal Medium in passing out of Water, Glass, Crystal, and other compact and dense Bodies into empty Spaces, grow denser and denser by degrees, and by that means refract the Rays of Light not in a point, but by bending them gradually in curve Lines? And doth not the gradual condensation of this Medium extend to some distance from the Bodies, and thereby cause the Inflexions of the Rays of Light, which pass by the edges of dense Bodies, at some distance from the Bodies?"

"Is not this Medium [æther] much rarer within the dense Bodies of the Sun, Stars, Planets, and Comets, than in the empty celestial Spaces between them? And in passing from them to great distances, doth it not grow denser and denser perpetually, and thereby cause the gravity of those great Bodies towards one another, and of their parts towards the Bodies; every Body endeavouring to go from the denser parts of the Medium towards the rarer? ...And though this Increase of density may at great distances be exceeding slow, yet if the elastick force of this Medium be exceeding great, it may suffice to impel Bodies from the denser parts of the Medium towards the rarer, with all that power which we call Gravity. And that the elastic force of this Medium is exceeding great, may be gather'd from the swiftness of its Vibrations."

"As Attraction is stronger in small Magnets than in great ones in proportion to their Bulk, and Gravity is greater in the Surfaces of small Planets than in those of great ones in proportion to their bulk, and small Bodies are agitated much more by electric attraction than great ones; so the smallness of the Rays of Light may contribute very much to the power of the Agent by which they are refracted."

"And so if any one would suppose that Æther (like our Air) may contain Particles which endeavour to recede from one another (for I do not know what this Æther is) and that its Particles are exceedingly smaller than those of Air, or even than those of Light: The exceeding smallness of its Particles may contribute to the greatness of the force by which those Particles may recede from one another, and thereby make that Medium exceedingly more rare and elastick than Air, and by consequence exceedingly less able to resist the motions of projectiles, and exceedingly more able to press upon gross Bodies, by endeavouring to expand it self."

"Are not all hypotheses erroneous, in which light is supposed to consist of a Pression or Motion, propagated through a fluid medium? ...If Light consisted only in Pression propagated without actual Motion, it would not be able to agitate and heat the Bodies which refract and reflect it. If it consisted in Motion propogated to all distances in an instant, it would require an infinite force every moment, in every shining Particle, to generate that Motion. And if it consisted in Pression or Motion, propogated either in an instant or in time, it would bend into the Shadow. For Pression or Motion cannot be propogated in a Fluid in right Lines, beyond an Obstacle which stops part of the Motion, but will bend and spread every way into the quiescent Medium which lies beyond the Obstacle. Gravity tends downwards, but the Pressure of Water arising from Gravity tends every way with equal Force, and is propogated as readily, with as much force sideways as downwards, and through crooked passages as through straight ones. The Waves on the Surface of stagnating Water, passing by the sides of a broad Obstacle which stops part of them, bend afterwards and dilate themselves gradually into the quiet Water behind the Obstacle. The Waves, Pulses or Vibrations of the Air, wherein Sounds consist, bend manifestly, though not so much as Waves of Water. For a Bell or a Cannon may be heard beyond a Hill which intercepts the sight of the sounding Body, and Sounds are propogated as readily through crooked Pipes as through straight ones. But light is never known to follow crooked Passages nor to bend into the Shadow. For the fix'd Stars by the Interposition of any Planets cease to be seen. And so do parts of the Sun by Interposition of the Moon, Mercury or Venus. The Rays which pass very near to the edges of any Body, are bent a little by the action of the Body, as we shew'd above; but this bending is not towards but from the Shadow, and is perform'd only in the passage of the Ray by the Body, and at a very small distance from it."

"Whoever has considered what a number of properties and effects of light are exactly similar to the properties and effects of bodies of a sensible bulk, will find it difficult to conceive that light is any thing else but very small and distinct particles of matter: which being incessantly thrown out from shining substances, and every way dispersed by reflection from all others, do impress upon our organs of seeing that peculiar motion, which is requisite to excite in our minds the sensation of light. But for the present purpose it is sufficient to observe that light consists of parts, both successive in the same lines and contemporary in several lines: because in the same place, you may stop that which comes one moment, and let pass that which comes presently after; and at the same time, you may stop it in one place, and let it pass in another. For that part of the light which is stopt cannot be the same with that which is let pass."

"The least light or part of light, which may be stopt alone, without the rest of the light, or propagated alone, or do or suffer any thing alone, which the rest of the light doth not or suffereth not, is called a Ray of light. That rays of light are straight, is evident enough from the shadows of bodies; or from the appearance of light passing through little holes into a dark room full of dust or smoke; or because bodies cannot be seen through the bore of a bended pipe; or because they cease to be seen by the interposition of other bodies, as the fixt stars by the interposition of the moon and planets; and the parts of the sun by the interposition of the Moon, Mercury or Venus. Rays of light may therefore be represented by straight lines, not Mathematical but Physical, which are described by the motion of the parts or particles of light: and the point which a ray possesses in falling upon any surface may be considered as a Physical Point."

"When a ray of light falls obliquely upon a smooth polished surface, it is turned out of its way either by reflection or refraction in the following manner. Imagine the paper upon which this figure is drawn to be perpendicular to the surface of stagnating water, and to cut it in the line RS, and that a ray of light, coming in the air along the line AC, falls upon RS at the point C. Then supposing the line PCQ... to be perpendicular to the surface of the water, if the ray be reflected, or turned back at C into the air again, it will describe a straight line CB, inclined to the perpendicular PC at an angle PCB exactly equal to the angle PCA."

"But if the ray that came along AC goes into the water at C, it will not proceed straight forward, but being refracted or bent at C, it will describe another straight line CE inclined to the perpendicular CQ at a lesser angle ECQ than the angle ACP; and the line CE will always be so situated, that when any circle, described about the center C, cuts the line CA in A and CE in E, the perpendiculars AD and EF, drawn from A and E to the line PQ shall always bear the same proportion to each other; whatever be the magnitude of the angle ACP. In water the line EF is always three quarters of AD. ... [T]he angle ACP [is] the Angle of Incidence, BCP the Angle of Reflection, ECQ the Angle of Refraction; the line AD the Sine of Incidence, that is, of the angle of incidence; and EF the Sine of Refraction, that is, of the angle of refraction."

"The foregoing properties of Reflection and Refraction being discovered and established by repeated experiments upon light and bodies of all sorts both fluid and solid, without any exception yet known; and being the principal foundation of the whole science of Opticks, are called the Laws of Reflection and Refraction; and are expressed by Sir Isaac Newton..."

"[T]he optical philosophers of antiquity had satisfied themselves that vision is performed in straight lines;— ...they had fixed their attention upon those straight lines, or rays, as the proper object of the science;—they had ascertained that rays reflected from a bright surface make the angle of reflection equal to the angle of incidence;—and they had drawn several consequences from these principles. We may add... the art of perspective, which is merely a corollary from the doctrine of rectilinear visual rays... The ancients practised this art, as we see in the pictures which remain to us; and we learn from Vitruvius, that they also wrote upon it. , who had been instructed by Eschylus... was the first author on this subject, and Anaxagoras, who was a pupil of Agatharchus, also wrote an Actinographia, or doctrine of drawing by rays... The moderns re-invented the art in the flourishing times of their painting... about the end of the fifteenth century; and... we have treatises on it."

"Alhazen... asserted (lib. vii.), that "refraction takes place towards the perpendicular;" and reference is made to experiment for the proof. On the same ground he states that the quantities of refraction differ according to the magnitudes of the angles which the (primœ lineœ) directions of incidence make with the perpendicular to the surface; and moreover (which shows accuracy as well as distinctness,) that the angles of refraction do not follow the proportion of the angles of incidence."

"In Roger Bacon's works we find a tolerably distinct explanation of the effect of a convex glass; and in the work of Vitellio... the effect of refraction at the two surfaces of a glass globe is clearly traced. ...Vitellio had obtained experimentally a number of measures of the refraction out of air into water and into glass. Out of these facts no rule had yet been collected, when, in 1604 Kepler published his "Supplement to Vitellio." ...Kepler attempted to reduce to law the astronomical observations of Tycho,—devising an almost endless variety of possible formulæ, tracing their consequences with undaunted industry, and relating with a vivacious garrulity, his disappointments and his hopes,— ...he proceeded in the same manner with regard to Vitellio's Tables of Observed Refractions. He tried a variety of constructions by triangles, conic sections, &c., without being able to satisfy himself, and he at last is obliged to content himself with an approximate rule, which makes the refraction partly proportional to the angle of incidence, and partly to the secant of that angle. In this way he satisfies the observed refractions within a difference of less than half a degree each way. When we consider how simple the law of refraction is, (that the ratio of the sines of the angles of incidence and refraction is constant for the same medium,) it appears strange that a person attempting to discover it, and drawing triangles for the purpose, should fail; but this lot of missing what afterwards seems to have been obvious, is a common one in the pursuit of truth."

"The person who did discover the Law of the Sines, was Willebrord Snell, about 1621; but the law was first published by Descartes who had seen Snell's papers. Descartes does not acknowledge this... and after his manner, instead of establishing its reality by reference to experiment, he pretends to prove à priori that it must be true, comparing, for this purpose, the particles of light, to balls striking a substance which accelerates them."

"Descartes... showed considerable skill in tracing the consequences of the principle when once adopted. In particular we must consider him as the genuine author of the explanation of the rainbow. It is true, that Fleischer and Kepler had previously ascribed this phenomenon to the rays of sunlight which, falling on drops of rain, are refracted into each drop, reflected at its inner surface, and refracted out again. Antonio de Dominis had found that a glass globe of water, when placed in a particular position with respect to the eye, exhibited bright colours; and had hence explained the circular form of the bow, which, indeed, Aristotle had done before. But none of these writers had shown why there was a narrow bright circle of a certain definite diameter; for the drops which send rays to the eye after two refractions and a reflection, occupy a much wider space in the heavens. Descartes assigned the reason for this in the most satisfactory manner, by showing that the rays which, after two refractions and a reflection, come to the eye at an angle of about forty-one degrees with their original direction, are far more dense than those in any other position. He showed, in the same manner, that the existence and position of the secondary bow resulted from the same laws."

"[I]n 1672 Newton gave the true explanation of the facts; namely, that light consists of rays of different colours and different refrangibility. ...[T]he impression which this discovery made, both upon Newton and upon his contemporaries, shows how remote it was from the then accepted opinions. There appears to have been a general persuasion that the coloration was produced, not by any peculiarity in the law of refraction itself, but by some collateral circumstance,—some dispersion or variation of density of the light, in addition to the refraction. Newton's discovery consisted in teaching distinctly that the law of refraction was to be applied, not to the beam of light in general, but to the colours in particular."

"When Newton produced a bright spot on the wall of his chamber, by admitting the sun's light through a small hole in his window-shutter, and making it pass through a prism, he expected the image to be round; which, of course, it would have been, if the colours had been produced by an equal dispersion in all directions, but to his surprise he saw the image, or spectrum, five times as long as broad. He found that no consideration of the different thickness of the glass, the possible unevenness of its surface, or the different angles of rays proceeding from the two sides of the sun, could be the cause of this shape .He found, also, that the rays did not go from the prism to the image in curves; he was then convinced that the different colours were refracted separately, and at different angles; and he confirmed this opinion by transmitting and refracting the rays of each colour separately. ...Newton's opinions were not long in obtaining general acceptance; but they met with enough of cavil and misapprehension to annoy extremely the discoverer... impatient alike of stupidity and of contentiousness."

"[Anthony] Lucas of Liege repeated Newton's experiments, and obtained Newton's result, except that he never could obtain a spectrum whose length was more than three and a half times its breadth. Newton... persisted in asserting that the image would be five times as long as broad... We now know that the dispersion, and consequently the length, of the spectrum, is very different for different kinds of glass, and it is very probable that the Dutch prism was really less dispersive than the English one. The erroneous assumption which Newton made in this instance, he held by to the last; and was thus prevented from making [another] discovery."

"Newton was attacked by... Hooke and Huyghens. These philosophers, however, did not object so much to the laws of refraction of different colours, as to some expressions used by Newton, which, they conceived conveyed false notions respecting the composition and nature of light. Newton had asserted that all the different colours are of distinct kinds, and that, by their composition they form white light. ...Hooke maintained that all natural colours are produced by various combinations of two primary ones, red and violet; and Huyghens held a similar doctrine, taking, however, yellow and blue for his basis. ...These writers also had both of them adopted an opinion that light consisted in vibrations; and objected to Newton... involving the hypothesis that light was a body. Newton appears to have had a horror of the word hypothesis, and protests against its being supposed that his "theory" rests on such a foundation."

"The doctrine of the unequal refrangibility of different rays is clearly exemplified in the effects of lenses, which produce images more or less bordered with colour, in consequence of this property. The improvement of telescopes was, in Newton's time, the great practical motive for aiming at the improvement of theoretical optics. Newton's theory showed why they were imperfect... The false opinion... that the dispersion must be the same when the refraction is the same led him to believe that the imperfection was insurmountable, and made him turn his attention to the construction of reflecting instead of refracting telescopes. But the rectification of Newton's error was a further confirmation of the general truth of his principles in other respects; and since that time, the soundness of the Newtonian law of refraction has hardly been questioned among physical philosophers."

"Göthe not only adopted and strenuously maintained the opinion that the Newtonian theory was false, but he framed a system of his own to explain the phenomena of colour. ...Göthe's views are, in fact, little different from those of Aristotle and Antonio de Dominis though more completely and systematically developed. ...it is not difficult to point out the peculiarities in Göthe's intellectual character which led to his singularly unphilosophical views on this subject. ...[H]e appears, like many persons in whom the poetical imagination is very active, to have been destitute of the talent and the habit of geometrical thought. In all probability, he never apprehended clearly and steadily those relations on which the Newtonian doctrine depends. ...[P]robably ...he had conceived the "composition" of colours in some way altogether different from that which Newton understands by composition. What Göthe expected to see, we cannot clearly collect; but we know... his intention of experimenting with a prism arose from his speculations on the rules of colouring in pictures; and we can easily see that any notion of the composition of colours which such researches would suggest, would require to be laid aside before he could understand Newton's theory of the composition of light."

"Sir David Brewster... contests Newton's opinion, that the coloured rays into which light is separated by refraction are altogether simple and homogeneous, and incapable of being further analysed or modified. For he finds that by passing such rays through coloured media, (as blue glass for instance,) they are not only absorbed and transmitted in very various degrees, but that some of them have their colour altered; which cannot be conceived otherwise than as a further analysis of them, one component colour being absorbed and the other transmitted. ...The whole subject of the colours, of objects both opake and transparent, is still in obscurity. Newton's conjectures concerning the causes of the colours of natural bodies, appear to help us little; and his opinions on that subject are to be separated altogether from the important step which he made in optical science, by the establishment of the true doctrine of refractive dispersion."

"The discovery that the laws of refractive dispersion of different substances were such as to allow of combinations which neutralized the dispersion without neutralising the refraction, is one which has hitherto been of more value to art than to science."

"Euler observed, that a combination of lenses which does not colour the image must be possible, since we have an example of such a combination in the human eye; and he investigated mathematically the conditions requi site for such a result. Klingenstierna... also showed that Newton's rule could not be universally true."

", in 1757, repeated Newton's experiment, and obtained an opposite result. He found that when an object was seen through two prisms, one of glass and one of water, of such angles that it did not appear displaced by refraction, it was coloured. Hence it followed that, without being coloured, the rays might be made to undergo refraction; and that thus, substituting lenses for prisms, a combination might be formed, which should produce an image without colouring it, and make the construction of an achromatic telescope possible."

"Euler at first hesitated to confide in Dollond's experiments; but he was assured of their correctness by Clairaut, who had throughout paid great attention to the subject; and those two great mathematicians, as well as D'Alembert, proceeded to investigate mathematical formulæ which might be useful in the application of the discovery. The remainder of the deductions, which were founded upon the laws of dispersion of various refractive substances, belongs rather to the history of art than of science. Dollond used at first, for his achromatic object-glass, a lens of crown-glass, and one of flint-glass; afterwards, two lenses of the former substance, including between them one of the latter. He also adjusted the curvatures of his lenses in such a way as to correct imperfections arising from the spherical form of the glasses, as well as the fault of colour. Afterwards Blair, and more recently Mr. Barlow, have used fluid media along with glass lenses, in order to produce improved object-glasses; and various mathematicians, as Sir J. Herschel and Professor Airy among ourselves, have simplified and extended the investigation of the formulæ which determine the best combinations of lenses in the object-glasses and eye-glasses of telescopes, both with reference to spherical and chromatic aberrations."

"The phenomena of ocular spectra and complementary colours... forms a curious chapter in the history of those illusions which take their origin in the eye... [A]fter looking fixedly at a bright light or a striking colour for a few moments the eye preserves an impression of the object for a certain time. A very light window looked at intently for several seconds will leave the impression of its cross bars on the retina for several minutes, the colour of the image changing at every movement of the eye. The same effect may be observed when looking at the setting sun, or a flaring gas light. If the light at which we look is coloured we shall see the complementary colour in the impression left on the retina. Sir David Brewster was one of the first to notice and experiment upon these very interesting facts."

"[T]he complementary of any colour is that which is necessary to make white light. ...The impression left by the setting sun is of this character. At first, while the eye is open, the image is black, then brownish red, with a light blue border; but if the eye be shut suddenly, it becomes green, with a red border, the brilliancy of colour being apparently in proportion to the strength of the impression. These spectra may be perceived for a long time, if the eye is gently rubbed with the finger now and then. Some eyes are more impressionable in this respect than others, and Beyle gives an instance of an individual who saw the spectrum of the sun for years, whenever he looked at a bright object. A modern instance of this occurred lately to an amateur astronomer who was looking at an eclipse of the sun. He unfortunately used a glass that was not sufficiently smoked and the image... remained on his retina for months after. This... afforded an instance of the necessity of attention in order to see any object, for after the first few days he only became sensible of [it] when his attention was called to it by some accidental circumstance. These facts were so inexplicable to Locke that he consulted Newton on the subject, and was surprised to learn that the great philosopher himself had suffered for several months from a sun-spectrum in the eye."

"We can say with every confidence that "the Providence which shapes our ends," knows our wants better than we do ourselves, and bestows on us the things we ought to have asked for instead of those we have asked for. We shall find a very simple proof of this in the history of the discovery of the velocity of light."

"A short time after the invention of the telescope and the consequent discovery of Jupiter's satellites, Römer... was engaged in a series of observations... to determine the time which one of these bodies took to revolve round its planet. The method employed by Römer was to observe the successive s of the satellite and to notice the interval that elapsed between each of them. But it at last happened that the interval between the two occultations, which was about forty five hours, became prolonged by periods of 8, 13, and 16 minutes, during that half of the year when the earth was receding from the planet, while it became proportionally cut short during [earth's approach]. Römer was struck by a happy idea he suspected instantly that... an interval of time sufficiently long [was required] to allow the light that had left the satellite immediately after its disappearance to reach the eye of the observer. ...[T]he farther off the earth was from the satellite the longer was the interval of time between its disappearance and that of the arrival of the last portions of its light upon the earth ...It was thus that Römer explained the difference between the calculated and observed time of the occultation and he saw that he was on the threshold of a great discovery. ...he saw that light propagated itself through space with a certain velocity and that the fact... just mentioned furnished the precise means of measuring it. Thus the occultation of the satellite was retarded one second for every 185,000 miles that the earth is distant from Jupiter; the reason being that a ray of light takes a second to travel this distance... because the velocity of light is... 185,000 miles per second."

"There are many persons... whom we should astonish, and possibly enrage, by asserting... that we could cause darkness by means of light, that silence could be produced by sound, or cold by heat. ...[B]y throwing a second stone into the water we form another series of undulations which are mutually destroyed when they encounter each other. It is the same with the peculiar fluid which, existing throughout space, is thrown in a state of undulation by incandescent bodies; by opposing one set of waves to another we obtain rest as a result. This fact was first observed by Grimaldi in 1665 and Dr. Thomas Young was the first to offer an explanation. Fresnel used it with great success at the beginning of the century to demonstrate the truth of the undulatory theory, by showing that it could not be explained by any other."

"All experiments in which the eye of the investigator is provided with good optical instruments are distinguished, as is well known, by a high degree of precision; and some of the most important discoveries could not have been made without these instruments. Up to the present time, in experiments on diffraction there has been no instrument, except a magnifying-glass, which could be used with profit; and this may perhaps be one of the reasons why in this field of physical optics we are so backward, and why we know so little of the laws of this modification of light. ...[I]t is most to be desired that these laws should be exactly known; and this is specially so because a knowledge of them makes the nature of light itself better known at the same time."

"If sunlight is admitted into a darkened room through a small opening and falls upon a dark screen some distance away, which has a narrow aperture, and if the light which passes through this slit is allowed to fall upon a white surface or a piece of ground-glass placed a short distance behind the screen, one sees... that the illuminated portion of the white surface is larger than the narrow slit in the screen, and that it has colored edges—in short, that the light through the slit is inflected or diffracted. The narrower the openings, so much the greater is the inflection. The shadow of every body which is placed in a beam of sunlight entering a darkened room through a small opening is bounded by fringes of color which are, moreover, for any given distance of the surface on which the shadow is received, of the same size for bodies of all kinds of matter. The shadow of a narrow object, such as a hair, has, in addition to the outer fringes, others within the shadow, which change with the thickness of the hair, but in other respects are similar to the outer ones. Since the colored fringes are very small, and since most of the light is lost through absorption at the surface on which the shadow is cast, no great accuracy could be expected with the methods which have been used up to this time to observe diffraction phenomena; and this is all the more true because by these methods it is impossible to measure the angles of inflection of the light which alone can make us acquainted with the laws of diffraction. Up to the present, these angles from which the path of the diffracted light can be learned have been calculated from the dimensions of the colored bands and their distance from the diffracting body; but assumptions have been made which... do not agree with the truth, and which, therefore, give false results."

"In order to receive in the eye all the light diffracted through a narrow opening, and to see the phenomena strongly magnified; still more in order to directly measure the inflection of the light, I placed in front of the objective of a -telescope a screen in which there was a narrow vertical opening which could be made wider or narrower by means of a screw. By means of a heliostat I threw sunlight into a darkened room through a narrow slit so that it fell upon this screen, through whose opening the light was therefore diffracted. I could then observe through the telescope the phenomena produced by the diffraction, magnified, and yet seen with sufficient brightness; and at the same time I could measure the angles of inflection of the light by means of the theodolite."

"Descartes's theory of light rapidly displaced the conceptions which had held sway in the Middle Ages. The validity of his explanation of was, however, called in question by his fellow-countryman Pierre de Fermat... and a controversy ensued which was kept up by the Cartesians long after the death of their master. Fermat eventually introduced a new fundamental law, from which he proposed to deduce the paths of rays of light. This was the celebrated Principle of Least Time, enunciated in the form, "Nature always acts by the shortest course." From it the law of reflection can readily be derived, since the path described by light between a point on the incident ray and a point on the reflected ray is the shortest possible consistent with the condition of meeting the reflecting surfaces. In order to obtain the law of refraction, Fermat assumed that "the resistance of the media is different," and applied his "method of maxima and minima" to find the paths which would be described in the least time from a point of one medium to a point of the other. In 1661 he arrived at the solution. "The result of my work," he writes, "has been the most extraordinary, the most unforeseen and the happiest, that ever was; for, after having performed all the equations, multiplications, antitheses and other operations of my method, and having finally finished the problem, I have found that my principle gives exactly and precisely the same proportion for the refractions which Monsieur Descartes has established." His surprise was all the greater, as he had supposed light to move more slowly in dense than in rare media, whereas Descartes had... been obliged to make the contrary supposition."

"Although Fermat's result was correct, and, indeed, of high permanent interest, the principles from which it was derived were metaphysical rather than physical in character, and consequently were of little use for the purpose of framing a mechanical explanation of light. Descartes' theory therefore held the field until the publication in 1667§ of the Micrographia of Robert Hooke..."

"Hooke, who was both an observer and a theorist, made two experimental discoveries which concern [us]... but in both of these, as it appeared, he had been anticipated. The first was the observation of the iridescent colours which are seen when light falls on a thin layer of air between two glass plates or lenses, or on a thin film of any transparent substance. These are generally known as the "colours of thin plates," or ""; they had been previously observed by Boyle. Hooke's second experimental discovery, made after... Micrographia, was that light in air is not propagated exactly in straight lines, but that there is some illumination within the geometrical shadow of an opaque body. This observation had been published in 1665 in a posthumous work of Francesco Maria Grimaldi... who had given to the phenomenon the name ."

"Hooke's theoretical investigations on light were of great importance, representing as they do the transition from the Cartesian system to the fully developed theory of undulations. He begins by attacking Descartes' proposition, that light is a tendency to motion rather than an actual motion. "There is," he observes, "no luminous Body but has the parts of it in motion more or less"; and this motion is "exceeding quick." Moreover, since some bodies (e.g. the diamond when rubbed or heated in the dark) shine for a considerable time without being wasted away, it follows that whatever is in motion is not permanently lost to the body, and therefore that the motion must be of a to-and-fro or vibratory character. The amplitude of the vibrations must be exceedingly small, since some luminous bodies (e.g. the diamond again) are very hard, and so cannot yield or bend to any sensible extent."

"(Ptolemy) left in his Optics, the earliest surviving table of angles of refraction from air to water. ...This table, quoted and requoted until modern times, has been admired... A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order... As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience."

"Descartes maintained his confidence in the instantaneity of light. ...Yet in his derivation of the law of refraction, Descartes reasoned that light traveled faster in a dense medium than in one less dense. He seems to have had no qualms about comparing infinite magnitudes!"

"Fermat had recourse to the principle of the economy of nature. Heron and Olympiodorus had pointed out in antiquity that, in reflection, light followed the shortest possible path, thus accounting for the equality of angles. During the medieval period Alhazen and Grosseteste had suggested that in refraction some such principle was also operating, but they could not discover the law. Fermat, however, not only knew (through Descartes) the law of refraction, but he also invented a procedure—equivalent to the differential calculus—for maximizing and minimizing a function of a single variable. … Fermat applied his method … and discovered, to his delight, that the result led to precisely the law which Descartes had enunciated. But although the law is the same, it will be noted that the hypothesis contradicts that of Descartes. Fermat assumed that the speed of light in water to be less than that in air; Descartes' explanation implied the opposite."

"Regardless of the prophetic value of Dirac’s description [on interference] his was probably the first discussion... including a coherent beam of light. In other words, Dirac wrote the first chapter in laser optics."

"Feynman uses Dirac's notation to describe the quantum mechanics of stimulated emission... he applies that physics to... dye molecules... In this regard, Feynman could have predicted the existence of the tunable laser."

"The intimate relation between interference and has its origin in the interference equation itself."

"Multiple-prism arrays were first introduced by Newton (1704) in his book . In that visionary volume Newton reported on arrays of nearly isosceles prisms in additive and compensating configurations to control the propagation path and the dispersion of light. Further, he also illustrated slight beam expansion in a single isosceles prism."

"If we were to question a man of average education, or even one... [in] the cultured class, as to what he conceives to be the nature of Mathematical Science, and as to what he thinks are the aims... we should probably receive a somewhat vague... impression that Mathematics is concerned with calculations... involving a copious use of symbols and diagrams entirely unintelligible to the uninitiated. ...of no interest to anyone except a few individuals who have an unaccountable taste for such things, and happen to be endowed with a peculiar transcendental faculty... unnecessary for other people, and which he... is quite happy without... [O]ur friend would probably admit that... subjects, such as Arithmetic and Mensuration... are extremely useful... and that anything beyond them is of little or no concern to the world in general."

"In the scientific world... decidedly vague and narrow conceptions of the functions of Mathematical thinking are current. In such circles, the notion is extremely common that the sole function of Mathematics is to provide the means of carrying out... calculations... and thus that Mathematics plays in them a comparatively humble part analogous to that of a mechanical tool."

"Mathematical thinking...has played a most important part in the formation of the concepts with which the Physical Sciences work... [I]t has reduced the originally vague conceptions which arise in connection with physical observation to precise forms in which they can be exhibited as measurable quantities."

"Mathematical thinking, in a more or less explicit form, pervades every department of human activity. The grocer... The Engineer... The Philosopher, in his reflections on spatial and temporal relations, on number and quantity, on matter and motion, is in a region of thought in which the boundary between his own domain and that of the Mathematician is almost non-existent. The Epistemologist has always to take Mathematical knowledge as a kind of touchstone on which to test his theories of the nature of knowledge. The dominant views in various departments of philosophical thinking have been modified in important points by the results of recent Mathematical research, and will... in the future, be further modified..."

"Mathematical thought is... the most all-pervading and the most highly specialized department of mental activity."

"The more closely men scrutinized natural phenomena, at first for practical reasons, and later from intellectual curiosity, the more things and processes they found to have aspects which are measurable, and the more they were able to employ their developing Mathematical processes and concepts for the precise characterization of various aspects of the world of phenomena."

"But... the natural development of Mathematical thought, starting as it did in connection with the more obvious aspects of sensuous experience, under the pressure of physical needs, brings it to a region reaching far beyond that in which the primitive intuitions of time, space, and matter formed the exclusive subject matter of the Science."

"[T]he Engineer, like the Physicist, has constantly to make use of Mathematical methods; but as his ultimate aim is to harness the forces of nature and use them to obtain practical results, rather than to bring their relations under general laws and concepts as the theoretical Physicist does, he is perhaps less directly concerned than the Physicist with the part which Mathematical Science has played in the formation of the concepts... He uses applied Mathematics, and applied Physics, and is apt to take both of them more or less as ready-made products, although he cannot do so beyond a certain point without grave danger to his efficiency as a scientific engineer."

"In former times the Mathematician and the Physicist were usually one and the same man. ...it was in the nineteenth century that the increasing complexity of both Sciences produced that separation ...which has become continually more marked, and has reached its extreme... in our own time. ...The chief drawback is that each specialist, from lack of interest in, and knowledge of, the progress of the other great department, is apt to miss that large source of inspiration in his own study which is supplied by the other one."

"I remember... at a Board meeting at Cambridge, the subject of Bessel's functions came into the discussion... to include them in an examination syllabus. Their utility in connection with Applied Mathematics having been referred to, a very great Pure Mathematician who was present ejaculated—"Yes, Bessel's functions are very beautiful functions, in spite of their having practical applications." It would have been interesting to have heard what this great man would have said if he had known that Professor Perry would one day propose the desecration of these beautiful functions by recommending them as suitable playthings for young boys."

"Speaking... from personal experience, one of the effects of prolonged study of some of the more abstract branches of Mathematics, as for example the Theory of Functions, is that one begins to take the greatest interest in, and to be most attracted by... aspects of the subject which are most remote from the interests of the Physicist. One gets into an attitude... in which the kind of well-behaved functions, without abnormal singularities... appear to have a somewhat bourgeois aspect, in their comparatively uninteresting respectability. In the mind of one who makes a minute and prolonged study of the peculiarities which Fourier's series may present, a[n]... effect of that study is that the ordinary Fourier's series, which converge everywhere... normally, begin to acquire a certain tameness... which deprives them of interest. The failure of convergence of Taylor's series becomes to some Mathematical students a matter of greater interest than that presented by the series in the ordinary cases in which... they are fitted for purposes of application."

"Mathematical thinking has played a very important part in the formation of the fundamental concepts of the Physicist; very often this part has been a dominant one. Many of these concepts could only have received a precise meaning and... taken definite forms as the result of the work of Mathematicians... the result of a long train of previous Mathematical thinking. For example, the conception of Energy, and the exact meaning of the... law of the Conservation of Energy, emerged as results of the development of the abstract side of molar mechanics, which determined the mode in which the of moving bodies and as work are defined as measurable quantities. Only by the transference and extension of these notions to the molecular domain did the conception involved in the modern doctrine become possible. The doctrine... had been established before Joule and Mayer commenced their work, and was a necessary presupposition of their further development. Joule was able to determine the only owing to the fact that mechanical work was already regarded as a measurable quantity, measured in a manner which had been fixed in the course of the development of the older Mathematical Mechanics. The notion of Potential, fundamental in Electrical Science, and which every Physicist, and every Electrical Engineer, constantly employs, was first developed as a Mathematical conception during the eighteenth century in connection with the theory of the attractions of gravitating bodies. It was transferred to the electrical domain by George Green and others, together with a good deal of detailed mathematics connected with it which had previously been applied to the function."

"The ultimate aim of the Physicist, even [the]... experimental[ist], is much higher than that of attaining to a merely empirical knowledge of facts. His real object is to classify facts in such a way as to refer them to general laws which are of a more or less abstract character, and which involve concepts of schematic representations that require... the aid of the Mathematician."

"The man of true Physical instincts, endowed with the great faculty of scientific imagination, possessed for example by Lord Kelvin in a very remarkable degree, is for ever imagining models which shall enable him... to represent and depict the course of actual physical processes. The possibility and consistency of such models require Mathematical Analysis for their investigation. The Mathematician may also, by tracing the necessary consequences of the postulation of a model of a particular type, formulate crucial tests in accordance with which further experiments will decide whether a... model can be retained at least provisionally, or whether it must be rejected as inadequate... and must give place to some other model..."

"Perhaps the most striking example of the services which have been rendered to Science by the contemplation of various models, many or all of which have ultimately been found to be inadequate for complete representation, is to be found in the history of Optics. The various forms of the corpuscular theory, and of the wave theory, of Light were all attempts to represent the phenomena by models, the value of which had to be estimated by developing their Mathematical consequences, and comparing these consequences with the results of experiments. The adynamical theory of Fresnel, the elastic solid theory of the ether developed by Navier, Cauchy, Poisson, and Green, the labile ether theory developed by Cauchy and Kelvin, and the rotational ether theory of MacCullagh were all efforts of the kind... indicated; they were all successful in some greater or less degree in the representation of the phenomena, and they all stimulated Physicists to further efforts to obtain more minute knowledge of those phenomena. Even such an inadequate theory as that of Fresnel led to the very interesting observation by Humphry Lloyd of the phenomenon of conical refraction in crystals, as the result of the prediction by Rowan Hamilton that the phenomenon was a necessary consequence of the Mathematical fact that Fresnel's wave surface in a biaxal crystal possesses four conical points."

"Although the theoretical Physicist has for his real aim the formulation of abstract schemes for the description and correlation of the physical phenomena which he observes, with the Mathematician processes of abstraction must go very much further than with the Physicist."

"A strong tendency of Mathematics in its later developments is to split up notions, originally undivided, into components, and to proceed to deal with these components in isolation, and often in separate branches of study."

"[D]uring the last half-century, number and measurable quantity have been separated... the idea of number alone has been recognized as the foundation upon which Mathematical Analysis rests, and the theory of extensive magnitude is now regarded as a separate department in which the methods of Analysis are applicable, but as no longer forming part of the foundation upon which Analysis itself rests. For the purposes of analysing the implications of the methods employed, and of pushing those methods to the highest possible degree of development, this kind of separation is indispensable, and has led to the very abstract form..."

"For the Physicist on the other hand, it is essential that abstraction should not go nearly so far... Too much abstraction... would entail the penalty that he would lose his way in a field which is barren for his purposes, and would lead to a loss of contact with the phenomenal world. To do what the Mathematician does, and must do... would be fatal to the Physicist, to whom above all things a large degree of concreteness in his conceptions is indispensable."

"Not only did Mathematical Science take its origin in the necessities and interests... in the physical world, but at every stage of its development the problems of Physics have been the source of the ideas which have directed the Mathematician, and from which new paths of investigation have been suggested..."

"But every great problem... from the physical side... has given rise to a train of ideas... and has started a host of questions... [which] have led him in most cases far beyond the original domain..."

"The most abstract branches of modern Mathematics, the theory of functions, real and complex, the theories of groups and of s—all arose originally from physical beginnings, but have reached out into vast developments... remote from the physical region. At any moment one... of these developments may become urgently necessary for the purposes of Physics, and may thus be in a position to pay back some of the debt they owe to the parent from whose side they have wandered so far."

"The question is often asked... why Mathematicians cannot restrict themselves more to those aspects of their Science which bring them in contact with Physics, and which are concerned with what often receives the question-begging and ambiguous name of reality; a word that has an indefinite number of shades of meaning, varying with every difference in philosophical view, but which in this connection is generally associated with... the physical world. Why... do modern Mathematicians... wander away from the source... from which its ever-renewed inspiration has been received, in order to lose themselves in a transcendentalism which, in its aloofness from physical investigation, condemns them to an endless and barren immersion in abstractions of their own creation? ...cut ...off from the roots of the Science?"

"To stop short at a point dictated by considerations of applicability to Physics is impossible to those to whom clear and thoroughly defined conceptions are a desideratum, the lack of which in any department of their study leaves them no rest. To attempt to confine the activities of Mathematicians by imposing... a restriction of the nature... above... would be to attempt to strangle the Science as a progressive development."

"Mathematics can in the long run be developed to the highest degree of perfection, not only from the point of view of specialists within its own domain, but also as constituting an essential component of the intellectual life and stock of ideas of the world, only on the condition that it is allowed full freedom of self-expression."

"The utilitarian notion... has the fatal limitation that it attempts to assign limits to what is, or may in the future become, useful, in accordance with a more or less arbitrarily restricted standard of what constitutes utility."

"When the exigencies of Physics suggest to the Mathematician some... special problem for solution, he is impelled to search for some generalization, some law, under which a whole class of analogous processes or problems can be subsumed. ...The Physicist also ...is really occupied in attempting to exhibit ...some general law under which a whole class of phenomena can be subsumed. A narrow utilitarianism would be as fatal to the growth of Physics as to that of Mathematics."

"The Mathematical Physicist plays a part of supreme importance as an intermediary and interpreter between the Pure Mathematician and the experimental Physicist. ...he must follow... progress both in Mathematics... and in experimental Physics. ...[I]n spite of some brilliant exceptions, the Mathematical Physicist does not... take as prominent a part as was formerly the case, especially during the nineteenth century, the age of Maxwell, Kelvin, Stokes, Helmholtz. ...In earlier times, when ...molar mechanics, and especially , occupied the centre of the interests... the passage from the observation of concrete phenomena to their abstract Mathematical representation was comparatively easy. The observational work was simpler and less technical... highly equipped physical laboratories had not yet come into existence... The Mathematical Physicists and Astronomers of the eighteenth century were largely engaged in working out the detailed implications of the law of gravitation, and had commenced, largely under the influence of the idea of , to work out problems such as... the vibrations of strings and other bodies."

"In the... [early] nineteenth century the centre of physical interest passed on to such subjects as Hydrodynamics, the Conduction of Heat, and Elasticity, in which an abstract representation of a body as a continuous plenum... made the problems readily accessible to continuous Mathematical Analysis. ...[M]uch attention was given to... Electricity and Magnetism, and much of the Mathematical Analysis which had been devised for...dealing with problems of gravitational attractions, vibrations, etc., was found, with further development, to be applicable to the new problems... Much of the work, such as that of Ampère... was still carried out under the influence of the idea of , first brought into prominence in connection with the Newtonian law of gravitation, but the idea of the continuous medium gradually became the dominating notion."

"The period in which Physical Mathematics was applied with such great success to continuous media probably reached its culmination in Maxwell's equations of Electrodynamics which are now usually regarded as representing the average effects exhibited when actual discreteness is smoothed out."

"In our own time the centre of physical interest has transferred itself, in connection with Electromagnetism, to the molecular and sub-molecular domain, in which discrete objects become the subject of scrutiny."

"The boundaries between Physics and Chemistry have been broken down. In this region of investigation... Physicists... have been rewarded by the discovery, during the last two decades, of a crowd of remarkable facts, probably destined to have the most far-reaching influence upon our conceptions of the material world."

"It has been said that the Theory of Numbers is a subject which has never been soiled by any practical application. Who can be absolutely sure that even so apparently transcendental a branch of thought as this will always remain undefiled by the contaminating touch of physical application?"

"In the history of Science it is possible to find many cases in which the tendency of Mathematics to express itself in the most abstract forms has proved to be of ultimate service in the physical order of ideas. Perhaps the most striking example is to be found in the development of abstract Dynamics. The greatest treatise which the world has seen, on this subject, is Lagrange's Mécanique Analytique, published in 1788. ...conceived in the purely abstract Mathematical spirit ...Lagrange's idea of reducing the investigation of the motion of a dynamical system to a form dependent upon a single function of the of the system was further developed by Hamilton and Jacobi into forms in which the equations of motion of a system represent the conditions for a stationary value of an integral of a single function. The extension by Routh and Helmholtz to the case in which "ignored co-ordinates" are taken into account, was a long step in the direction of the desirable unification which would be obtained if the notion of potential energy were removed by means of its interpretation as dependent upon the kinetic energy of concealed motions included in the dynamical system. The whole scheme of abstract Dynamics thus developed upon the basis of Lagrange's work has been of immense value in theoretical Physics, and particularly in statistical Mechanics... But the most striking use of Lagrange's conception of generalized co-ordinates was made by Clerk Maxwell, who in this order of ideas, and inspired on the physical side by... Faraday, conceived and developed his dynamical theory of the Electromagnetic field, and obtained his celebrated equations. The form of Maxwell's equations enabled him to perceive that oscillations could be propagated in the electromagnetic field with the velocity of light, and suggested to him the Electromagnetic theory of light. Heinrich Herz, under the direct inspiration of Maxwell's ideas, demonstrated the possibility of setting up electromagnetic waves differing from those of light only in respect of their enormously greater length. We thus see that Lagrange's work... was an essential link in a chain of investigation of which one result... gladdens the heart of the practical man, viz. wireless telegraphy."

"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."

"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:"

"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."

"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."

"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 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 ACP \cdot l = \iint uydxdywhere 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 becomesr \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^3dxThis 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."

"Three Lemmas which present no difficulty are given and demonstrated [by James Bernoulli]: 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...]"

"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."

"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."

"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: 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. 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 ..."

"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 formtension ∝ (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 obtainds = \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 equationEk^2 ad^3 y + Pa (dx)^2 dy + Qx(dx)^2 dy = 0is 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 expressionP = \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."

"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. <!--(77-78.)p.45"

"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."

"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. 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."

"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."

"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."

"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..."

"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."

"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."

"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... ."

"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."

"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."

"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."

"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. ). ..."

"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 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."

"We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe."

"If the general picture, however, of a Big Bang followed by an expanding Universe is correct, what happened before that? Was the Universe devoid of all matter and then the matter suddenly somehow created? How did that happen? In many cultures, the customary answer is that a God or Gods created the Universe out of nothing. But if we wish to pursue this question courageously, we must of course ask the next question: where did God come from? If we decide that this is an unanswerable question, why not save a step and conclude that the origin of the Universe is an unanswerable question? Or, if we say that God always existed, why not save a step, and conclude that the Universe always existed? That there's no need for a creation, it was always here. These are not easy questions. Cosmology brings us face to face with the deepest mysteries, questions that were once treated only in religion and myth."

"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."

"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."

"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..."

"[W]e stress... the wide range of validity exhibited by s in theoretical physics. ...[I]t has ...been demonstrated how they can be employed to derive equations of optics, dynamics of particles and rigid bodies, and electromagnetism. In addition, physicists have succeeded in formulating the laws of elasticity and hydrodynamics as variational principles, and even Einstein's law of gravitation was included in this category by Hilbert, who found a scaler function... for which \partial\int\mathfrak{h}\,dx_0\,dx_1\,dx_2\,dx_3=0 is equivalent to Einstein's law. This function has been called the "curvature," an identification which induced Whittaker to describe Hilbert's principle in the laconic words, "gravitation simply represents a continual effort of the universe to straighten itself out.""

"The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)in all regions free from matter and electromagnetic field, where G_{\mu \nu} is the contracted Riemann-Christoffel tensor associated with the fundamental tensor g_{\mu \nu}, and \lambda is the ."

"An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy."

"In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world."

"The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world..."

"O. Heckmann has pointed out that the non-static solutions of the field equations of the general theory of relativity with constant density do not necessarily imply a positive curvature of three-dimensional space, but that this curvature may also be negative or zero. There is no direct observational evidence for the curvature, the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the "cosmological constant" &lambda; was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of &lambda;."

"The determination of the coefficient of expansion h depends on the measured red-shifts, which do not introduce any appreciable uncertainty, and the distances of the extra-galactic nebulae, which are still very uncertain. The density depends on the assumed masses of these nebulae and on the scale of distance, and involves, moreover, the assumption that all the material mass in the universe is concentrated in the nebulae. It does not seem probable that this latter assumption will introduce any appreciable factor of uncertainty."

"Although... the density... corresponding to the assumption of zero curvature and to the coefficient of expansion... may perhaps be on the high side, it... is of the correct order of magnitude, and we must conclude that... it is possible to represent the facts without assuming a curvature of three-dimensional space. The curvature is, however, essentially determinable, and an increase in the precision of the data derived from observations will enable us in the future to fix its sign and to determine its value."

"Why should not the space be there already, and the material system expand into it..? ...[I]f the speed of recession continues to increase outwards, it will ere long approach the speed of light, so that something must break down. The result is that the system becomes a ... such a system cannot expand without the space also expanding. ...[E]xpansion of space has often been given too much prominence ...and readers have been led to think that it is more directly concerned in the explanation of the motions of the nebule than is... the case. ...If we adopt open space we encounter certain difficulties (not necessarily insuperable) which closed space entirely avoids; and we do not want... speculation as to the solution of difficulties which need never arise. If we wish to be noncommittal, we shall naturally work in terms of a closed universe of finite radius R, since we can at any time revert to an infinite universe by making R infinite."

"The immediate results of introducing the cosmical term into the law of gravitation was the appearance... of two universes—the Einstein universe and the de Sitter universe. Both were closed spherical universes; so that a traveller going on and on in the same direction would at last find himself back at the starting-point... Both claimed to be static universes... thus they provided a permanent framework within which the small-scale systems—galaxies and stars—could change and evolve. ...[H]owever ...in de Sitter's universe there would be an apparent recession of remote objects ...At that time only three radial velocities were known, and these ...lamely supported de Sitter ...2 to 1. ...But in 1922 ...V. M. Sipher furnished me ...measures of 40 spiral nebulæ for ...my book Mathematical Theory of Relativity. ...[T]he majority had become 36 to 4 ..."

"The situation has been summed up in the statement that Einstein’s universe contains matter but no motion and de Sitter’s contains motion but no matter. ...[T]he actual universe containing both matter and motion does not correspond exactly to either... Which is the better choice for a first approximation? Shall we put a little motion into Einstein’s world of inert matter, or... a little matter into de Sitter’s ?"

"The choice between Einstein’s and de Sitter’s models... [W]e are not now restricted to these... extremes; we have... the whole chain of intermediate solutions between motionless matter and matterless motion... [W]e can pick... the right proportion of matter and motion to correspond with what we observe. ...[E]arlier... it was the preconceived idea that a static solution was a necessity... an unchanging background of space. ...[T]his ...should strictly have barred... de Sitter’s solution, but ...it was the precursor of the other non-static solutions..."

"[I]nvestigation of non-static solutions was carried out by A. Friedmann in 1922. His solutions were rediscovered in 1927 by Abbé G. Lemaître, who brilliantly developed the astronomical theory... and... remained unknown until 1930... In the meantime the solutions had been discovered... by H. P. Robertson, and through him... interest was... realised. The astronomical application, stimulated by Hubble and Humason’s observational work on the spiral nebule, was also being rediscovered, but it had not been carried so far as in Lemaître’s paper."

"The intermediate solutions of Friedmann and Lemaitre are "expanding universes." Both the material system and the closed space, in which it exists, are expanding. At one end we have Einstein’s universe with no motion and therefore in equilibrium. Then... we have model universes showing more and more rapid expansion until we reach de Sitter’s... The rate of expansion increases all the way along the series and the density diminishes; de Sitter’s universe is the limit when the average density of celestial matter approaches zero. The series of expanding universes then stops... but because there is nothing left to expand."

"[T]he most satisfying theory would be one which made the beginning not too unæsthetically abrupt. This... can only be satisfied by an Einstein universe with all... major forces balanced. Accordingly, the primordial state of things... is an even distribution of s and electrons, extremely diffuse and filling all (spherical) space, remaining nearly balanced for an exceedingly long time until its inherent instability prevails. ...[T]he density of this distribution can be calculated ...[at] about one proton and electron per litre. ...[S]mall irregular tendencies accumulate, and evolution gets under way. ...[T]he formation of condensations ultimately ...become the galaxies; this ...started off an expansion, which ...automatically increased in speed until ...now manifested ...in the recession of the spiral nebulae. As the matter drew closer... in the condensations... evolutionary processes followed—evolution of stars... of... more complex elements... of planets and life."

"Within the galaxy the average world-curvature is... thousands of times greater than Lamaître's average for the universe... his formulæ are inapplicable. The result... only the intergalactic distances expand. The galaxies... are unaffected... —s, stars, human observers and their apparatus, atoms—are entirely free from expansion. Although the cosmical repulsion or expansive tendency is present in all of these... it is checked by much larger forces... [T]he demarcation between permanent and dispersing systems is... abrupt. It corresponds to the distinction between periodic and aperiodic phenomena."

"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."

"All change is relative. The universe is expanding relatively to our common material standards; our material standards are shrinking relatively to the size of the universe. The theory of the "expanding universe" might also be called the theory of the "shrinking atom". ...[T]ake the... universe as our standard of constancy... he sees us shrinking... only the intergalactic spaces remain the same. The earth spirals round the sun in an ever‑decreasing orbit. ...Our years will ...decrease in geometrical progression in the cosmic scale of time. ... Owing to the property of geometrical progressions an infinite number of our years will add up to a finite cosmic time; so that what we should call the end of eternity is an ordinary finite date in the cosmic calendar. But on that date the universe has expanded to infinity in our reckoning, and we have shrunk to nothing in the reckoning of the cosmic being. ...When the last act opens the curtain rises on midget actors rushing through their parts at frantic speed. Smaller and smaller. Faster and faster. One last microscopic blurr of intense agitation. And then nothing."

"If the astronomers are right, it is a straightforward conclusion from the observational measurements that the system of galaxies is expanding—or, since the system of the galaxies is all we know—that the universe is expanding. There is no subtlety or metaphysics about it ...But are we sure of the observational facts? Scientific men are rather fond of saying pontifically that one ought to be quite sure of one's observational facts before embarking on theory. Fortunately those who give this advice do not practice what they preach. Observation and theory get on best when they are mixed together, both helping one another in the pursuit of truth. It is a good rule not to put overmuch confidence in a theory until it has been confirmed by observation. I hope I shall not shock the experimental physicists too much if I add that it is also a good rule not to put overmuch confidence in the observational results that are put forward until they have been confirmed by theory. So in starting to theorise about the expanding universe I am not taking it for granted that the observational evidence which we have been considering is entirely certain."

"It is scarcely true... that we observe these velocities of recession. We observe a shift of the spectrum to the red; and although such... is usually due to recession... it is not inconceivable that it should arise from another cause."

"[I]t was theory that first suggested a systematic recession of the spiral nebulae and so led to a search for this effect. The theoretical possibility was first discovered by W. de Sitter in 1917. Only three radial velocities were known at that time, and they... lamely supported his theory by... 2 to 1. Since then... support is far more unanimous... mainly due to V. M. Slipher... and M. L. Humason... The linear law of proportionality between speed and distance was found by E. H. Hubble. Meanwhile the theory has also developed, and... taken the form... associated with... A. Friedman and G. Lemaître."

"The theory of relativity predicts... a... force... we call the cosmical repulsion... directly proportional to the distance... It is so weak... we can leave it out of... motions of the planets... or any motion within... our... galaxy. ...[S]ince it increases... to the distance we... if we go far enough, find it significant."

"I have said the repulsion is proportional to the distance... Distance from what? From anywhere you like. ...Cosmical repulsion is a dispersing force tending to make a system expand uniformly—not diverging from any centre in particular, but such that all internal distances increase at the same rate. That corresponds precisely to the kind of expansion we observe in the system of the galaxies."

"I have said that relativity theory predicts a force of cosmical repulsion. ...[R]elativity theory does not talk of anything so crude as force; it describes... curvature of space-time. But for practical purposes... nearly equivalent to the Newtonian force of gravitation... [T]he actual relativity effect is represented with sufficient accuracy by a force of cosmical repulsion... up to the greatest distances... we... observe."

"The galaxies exert on one another their ordinary gravitational attraction approximately according to Newton's law. This makes them tend to cling together. So we... have a contest of two forces, Newtonian attraction... and cosmical repulsion... If our theory is right cosmical repulsion must have got the upper hand... Having got the advantage, cosmical repulsion will keep it; because, as the nebulae become further apart, their mutual attraction will become weaker..."

"is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid."

"[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the ' of the surface into the third dimension—whence it derives its name..."

"[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties."

"Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}... [C]onsider... a "small circle" of radius r (measured on the surface!)... its perimeter L and area A... are clearly less than the corresponding measures 2\pi r and \pi r^2... in the Euclidean plane. ...for sufficiently small r (i.e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)"

"In the sum \sigma of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180&deg;]; it can... be shown that this "spherical excess" is given by 2)\sigma - \pi = K \deltawhere \delta is the area of the spherical triangle and the angles are measured in s (in which 180&deg; = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!"

"[T]he space constant K... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space."

"These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point."

"The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...), V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)."

"In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space."

"We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K."

"What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles..."

"Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature K... the kind of structure exhibited... in the neighborhood of a ".""

"What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry."

"In what respect... does the general theory of relativity differ...? The answer is: in its universality; the force of gravitation in the geometrical structure acts equally on all matter. There is here a close analogy between the gravitational mass M...(Sun) and the inertial mass m... (Earth) on the one hand, and the heat conduction k of the field (plate)... and the coefficient of expansion c... on the other. ...The success of the general relativity theory... is attributable to the fact that the gravitational and inertial masses of any body are... rigorously proportional for all matter."

"The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish."

"Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" d... and the distance r which appears as an element of the geometry."

"All the light which is radiated... will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure... in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows...4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + ...), or r = d (1 + \frac{K d^2}{6} + ...)."

"[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature... we must express N, or equivalently V, to which it is assumed proportional, in terms of d. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);...plotting N against... d and comparing... with the formula (5), it should be possible operationally to determine the "curvature" K."

"This... is an outrageously over-simplified account of the assumptions and procedures..."

"The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing..."

"Hubble was inclined, from about 1936, to reject the Doppler-effect interpretation of the red shifts and to regard the nebulae as stationary; but theoretical cosmologists, notably McVittie... and Heckmann... severely criticized Hubble’s method...and disputed his conclusions. Although these criticisms... came to be generally accepted, it still seemed that the available data were open to rival interpretations, depending on the method of analysis..."

"At last, in 1949, the... ... was ready... Humason... succeeded in photographing the spectra of two remote galaxies in the . These exhibited red-shifts which, on the Doppler interpretation, indicated... one-fifth of the velocity of light. [I]n 1956, with... photoelectric equipment attached... [W. A.] Baum obtained a red-shift... recessional velocity of about two-fifths of the velocity of light."

"[I]n... 1952, Baade... announced that Hubble’s entire distance scale was in error... According to Baade, the distances formerly assigned to all extragalactic objects must be multiplied by a factor of about two. Later it was generally accepted that this... was probably nearer three. ...[I]t followed that the sizes of all such objects had been underestimated. ...Therefore ...this nebula must be... twice as far away... [T]he average absolute magnitude at maximum brightness of novae... in the Milky Way attain on the average... 7.4, whereas those... in the Andromeda... 5.7... [T]he apparent anomaly could be removed by placing... Andromeda... rather more than twice as far as previously. ...[[w:Distance measure|[E]xtragalactic distances]] had ...been underestimated because of an error in converting... relative distances of s into an absolute scale. ...Baade's revision ...applied only to extragalactic objects... [and] had momentous consequences concerning the size and , for the scale of both was correspondingly increased."

"An important new survey of the law relating red-shifts and magnitudes published in 1956 by Humason, Mayall and Sandage suggested... that the expansion of the universe may have been faster in the past... so that its age may be somewhat less than that estimated on the hypothesis of uniform expansion. But... caution, for a recent review (1958) by Sandage of Hubble's criteria for constructing the extragalactic distance-scale has revealed that, not only must his Cepheid criterion be corrected but also... the brightest star criterion..."

"As for Hubble’s brightest star criterion, Sandage... has shown that objects in the of galaxies which Hubble believed to be highly luminous stars are... regions of glowing of intrinsic luminosity... two magnitudes brighter... If Sandage’s result is accepted, then the distances of all galaxies beyond those in which Cepheids can be detected... must be augmented by a factor... between 5 and 10... with the result that the rate of increase of velocity with distance will be reduced to between 5O and 100 kilometres per second per megaparsec. Consequently, taking 80 as a rough average... the , if it has expanded uniformly, will have to be increased to about 13-5 thousand million years. If... it was expanding more rapidly in the past... this... might be reduced to about 9 thousand million years."

"In preparing this version in English of Fourier's celebrated treatise on Heat, the translator has followed faithfully the French original. He has, however, appended brief foot-notes, in which will be found references to other writings of Fourier and modern authors on the subject, distinguished by the initials [Alexander Freeman] A. F."

"The notes marked R.L.E. are... from... memoranda on the margin of a copy of... Robert Leslie Ellis."

"Primary causes are unknown to us; but are subject to simple and constant laws, which may be discovered by observation, the study of them being the object of natural philosophy."

"Heat, like gravity, penetrates every substance of the universe, its rays occupy all parts of space. The object of our work is to set forth the mathematical laws which this element obeys. The theory of heat will hereafter form one of the most important branches of general physics."

"Archimedes... explained the mathematical principles of the equilibrium of solids and fluids. ... Galileo, the originator of dynamical theories, discovered the laws of motion of heavy bodies. Within this new science Newton comprised the whole system of the universe."

"The successors of these philosophers have extended these theories, and given them an admirable perfection: they have taught us that the most diverse phenomena are subject to a small number of fundamental laws..."

"[T]he same principles regulate all the movements of the stars, their form, the inequalities of their courses, the equilibrium and the oscillations of the seas, the harmonic vibrations of air and sonorous bodies, the transmission of light, capillary actions, the undulations of fluids, in fine the most complex effects of all the natural forces, and thus has the thought of Newton been confirmed: quod tam paucis tam multa praestet geometria gloriatur [from so little to so much stands the glory of Geometry.]"

"[M]echanical theories... do not apply to the effects of heat... a special order of phenomena, which cannot be explained by the principles of motion and equilibrium."

"We have... instruments adapted to measure many of these effects... but... not the mathematical demonstration of the laws..."

"I have deduced these laws... in the course of several years with the most exact instruments..."

"To found the theory, it was... necessary to distinguish and define... the elementary properties which determine the action of heat... a very small number of general and simple facts; whereby every... problem... is brought back to... mathematical analysis."

"[T]o determine... movements of heat, it is sufficient to submit each substance to three fundamental observations. ...[B]odies ...do not possess in the same degree the power to contain heat, to receive or transmit it across their surfaces, nor to conduct it through the interior of their masses. These are the three... qualities... our theory... distinguishes and shews how to measure."

"No diurnal variation can be detected at the depth, of about three metres [ten feet]; and the annual variations cease to be appreciable at a depth much less than sixty metres."

"Radiant heat which escapes from the surface of all bodies, and traverses elastic media, or spaces void of air, has special laws... The mathematical theory... I... formed gives an exact measure of them. It consists... in a new which... serves to determine... effects... direct or reflected."

"The principles of the theory are derived, as are those of rational mechanics, from a very small number of primary facts..."

"The differential equations of... heat [propagation] express the most general conditions, and reduce... physical questions to... pure analysis... not less rigorously established than... equations of equilibrium and motion. ...[W]e have always preferred demonstrations analogous to... the theorems... of statics and dynamics. These equations... receive a different form, when they express the distribution of luminous heat in transparent bodies, or the movements in the interior of fluids occasioned by changes of temperature and density. ...[I]n... natural problems which... most concerns us... the limits of temperature differ so little that we may omit... variations of... coefficients."

"The same theorems which have made known... the equations of... [heat] movement.., apply... to... problems of general analysis and dynamics whose solution has... long... been desired."

"[T]he same expression whose abstract properties geometers had considered, and which... belongs to general analysis, represents... the motion of light in the atmosphere... determines the laws of diffusion of heat in solid matter, and enters into... the theory of probability."

"The analytical equations... which Descartes was the first to introduce into the study of curves and surfaces, are not restricted to... figures, and... rational mechanics; they extend to all general phenomena. There cannot be a language more universal and more simple, more free from errors and... obscurities, [i.e.,] more worthy to express the invariable relations of natural things."

"[[Mathematical analysis|[M]athematical analysis]] is as extensive as nature... it defines all perceptible relations, measures times, spaces, forces, temperatures; this... science is formed slowly, but it preserves every principle... acquired; it grows and strengthens... incessantly in the midst of... variations and errors of... mind. Its chief attribute is clearness; it has no marks to express confused notions. It brings together phenomena the most diverse, and discovers the hidden analogies which unite them."

"If matter escapes us, as that of air and light, by its extreme tenuity, if bodies are placed far... in the immensity of space, if man wishes to know the... heavens at successive epochs... if the actions of gravity and of heat are exerted in the interior of the earth at depths... inaccessible, mathematical analysis can yet lay hold of the laws of these phenomena. It makes them present and measurable, and seems... a faculty of the... mind destined to supplement the shortness of life and... imperfection of... senses... more remarkable, it follows the same course in the study of all phenomena; it interprets... by the same language, as if to attest the unity and simplicity of the... the universe, and to make... evident that... order which presides over all natural causes."

"The problems of the theory of heat present... simple and constant dispositions which spring from the general laws of nature; and if the order... in these phenomena could be grasped... it would produce... impression comparable to... musical sound."

"In this work we have demonstrated all the principles of the theory of heat, and solved all the fundamental problems... [W]e wished to shew the actual origin of the theory and its gradual progress."

"The subjects of these memoirs will be, the theory of radiant heat, the problem of the terrestrial temperatures, that of the temperature of dwellings, the comparison of theoretic results with... experiments, lastly the demonstrations of the differential equations of the movement of heat in fluids."

"The new theories explained in our work are united for ever to the mathematical sciences, and rest like them on invariable foundations; all the elements... they... possess they will preserve, and... acquire greater extent. Instruments will be perfected and experiments multiplied. The analysis which we have formed will be deduced from more general, ...[i.e,] more simple and more fertile methods... For all substances... determinations will be made of all... qualities relating to heat, and of the variations of the coefficients which express them. At different stations on the earth observations will be made, of the temperatures of the ground at... depths, of the intensity of the solar heat and its effects... in the atmosphere, in the ocean and in lakes; and the constant temperature of the heavens proper to the planetary regions will become known. The theory... will direct... these measures, and assign their precision. No considerable progress can... be made... not founded on experiments... for mathematical analysis can deduce from general and simple phenomena the expression of the laws of nature; but... application of these laws... demands... exact observations."

"The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory... is to demonstrate these laws; it reduces... researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment."

"[T]he action of heat is always present, it penetrates all bodies and spaces, it influences the processes of the arts, and occurs in all the phenomena of the universe."

"When heat is unequally distributed among the different parts of a solid mass, it tends to attain equilibrium, and passes slowly from the parts which are more heated to those which are less; and... it is dissipated at the surface, and lost in the medium or in the void."

"The tendency [of heat] to uniform distribution and the spontaneous emission which acts at the surface of bodies, change continually the temperature at their different points."

"The problem of the propagation of heat consists in determining what is the temperature at each point of a body at a given instant, supposing that the initial temperatures are known."

"If we expose to... continued... uniform... source of , the same part of a metallic ring, whose diameter is large, the molecules nearest... the source will be first heated, and, after a... time, every point of the solid will have... nearly the highest temperature... it can attain... not the same at different points... [and] less and less... [the] more distant from that [source] point..."

"When the temperatures have become permanent, the source... supplies, at each instant, a quantity of heat which... compensates for that... dissipated at all the points of the external surface of the ring."

"If now the source be suppressed, heat will continue to be propagated in the [ring's] interior... but that... lost in the... void, will no longer be compensated... by the... source, so that all... temperatures will... diminish... until... equal to the temperatures of the surrounding medium."

"Whilst the temperatures are permanent and the source remains [continued and uniform], if at every point of the mean circumference of the ring an ordinate be raised perpendicular to the plane of the ring, whose length is... the fixed temperature at that point, the curved line which passes through the ends of these ordinates will represent the... state of the temperatures..."

"[T]he thickness of the ring is supposed... sufficiently small for the temperature to be... equal at all points of the same section perpendicular to the mean circumference."

"When the [heat] source is removed, the line which bounds the ordinates... at the different points will change its form continually."

"The problem consists in expressing, by one equation, the variable form of this curve, and in thus including in a single formula all the successive [temperature] states of the solid."

"Let z be the constant temperature at point m [on] the mean circumference [of the ring], x the distance of this point from the [heat] source [point o], that is to say the length of the arc of the mean circumference, included between the point m and the point o... z is the highest temperature which the point m can attain by virtue of the constant action of the source, and this permanent temperature z is function f(x) of the distance x. The first part of the problem consists in determining the function f(x) which represents the permanent [temperature] state of the solid."

"Consider next the variable state... as soon as the [heat] source has been removed; denote by t the time... passed since the... source [removal], and by v the... temperature at... m after the time t. v will be a... function F(x, t) of the distance x and the time t; the object... is to discover this function F(x, t), of which we only know as yet that the initial value... f(x) = F(x, o)."

"If we place a solid homogeneous... sphere or cube, in a medium... [of] constant temperature... for a... long time, it will acquire at all its points... [the] temperature... of the fluid. Suppose the mass to be withdrawn... to transfer... to a cooler medium, heat will begin to be dissipated at its surface; the temperatures at different points of the mass will not be... the same, and if we suppose it divided into an infinity of layers by surfaces parallel to its external surface, each of those layers will transmit, at each instant, a certain quantity of heat to the layer which surrounds it. If... each molecule carries a separate thermometer... the state of the solid will from time to time be represented by the variable system of... these thermometric heights. It is required to express the successive states by analytical formulae, so that we may know at any... instant the temperatures... and compare the quantities of heat which flow during the same instant, between two adjacent layers, or into the surrounding medium."

"If the mass is spherical, and we denote by x the distance... from the centre... t the time... cooling, and by v the variable temperature of the point m... all points... at the same distance x... have the same temperature v. This quantity v is a certain function F(x, t) of the radius x and... time t... such that it becomes constant whatever... value of x, when... [t=0]; for... the temperature at all points is the same at... emersion. The problem consists in determining... [F(x, t)]."

"[D]uring... cooling... heat escapes, at each instant, through the external surface, and passes into the medium... [and] this quantity is not constant; it is greatest at the beginning of... cooling. If... we consider the variable state of the internal spherical surface... [at] radius... x... there must be at each instant a... quantity of heat which traverses that surface, and passes through that part... more distant from the centre. This continuous flow of heat is variable like that through the external surface, and both are quantities comparable with each other; their ratios are numbers whose varying values are functions of the distance x, and of the time t... elapsed. It is required to determine these functions."

"[T]he effects of the propagation of heat depend in... every solid substance, on three elementary qualities... its capacity for heat, its own conducibility, and the exterior conducibility."

"[I]f two bodies of the same volume and of different nature have equal temperatures, and if the same quantity of heat be added to them, the increments of temperature are not the same; the ratio of these increments is the, ratio of their capacities for heat."

"The proper or interior conducibility of a body expresses the facility with which heat is propagated in passing from one internal molecule to another."

"The external or relative conducibility of a solid body depends on the facility with which heat penetrates the surface, and passes from this body into a given medium, or... from the medium into the solid. The last property is modified by the... polished state of the surface... also according to the medium in which... immersed; but the interior conducibility can change only with the nature of the solid."

"These three elementary qualities are represented... by constant[s], and the theory... indicates experiments suitable for measuring their values. As soon as... determined... problems relating to the propagation of heat depend only on numerical analysis."

"[T]here is no mathematical theory which has a closer relation... with public economy, since it serves to give clearness and perfection to the practice of the numerous arts... founded on... heat."

"To complete our theory it was necessary to examine the laws which radiant heat follows, on leaving the surface of a body. ...[T]he intensities of the different rays, which escape in all directions from any point in the surface of a heated body, depend on the angles which their directions make with the surface at the same point. We have proved that the intensity of a ray diminishes as the ray makes a smaller angle with the element of surface, and that it is proportional to the sine of that angle."

"[A] very extensive class of phenomena exists, not produced by mechanical forces, but resulting simply from the presence and accumulation of heat. This part of natural philosophy cannot be connected with dynamical theories, it has principles peculiar to itself..."

"In whatever manner the heat was at first distributed, the system of temperatures altering more and more, tends to coincide... with a definite state which depends only on the form of the solid. In the ultimate state the temperatures of all the points are lowered in the same time, but preserve amongst each other the same s: in order to express this property the analytical formulae contain terms composed of exponentials and of quantities analogous to ."

"Several problems of mechanics present analogous results, such as the isochronism of oscillations, the multiple of sonorous bodies. ...As to those results which depend on changes of temperature... mathematical analysis has outrun observation, it has supplemented our senses, and has made us in a manner witnesses of regular and harmonic vibrations in the interior of bodies."

"These considerations present a singular example of the relations which exist between the abstract science of numbers and natural causes."

"[T]he functions obtained by successive differentiations, which are employed in the development of infinite series and in the solution of numerical equations, correspond also to physical properties. The first of these functions, or the properly so called, expresses in geometry the inclination of the tangent of a curved line, and in dynamics the velocity of a moving body when the motion varies; in the theory of heat it measures the quantity of heat which flows at each point of a body across a given surface. Mathematical analysis has therefore necessary relations with sensible phenomena; its object is not created by human intelligence; it is a pre-existent element of the universal order, and is not in any way contingent or fortuitous; it is imprinted throughout all nature."

"The theory of heat will always attract the attention of mathematicians, by the rigorous exactness of its elements and the analytical difficulties... and above all by the extent and usefulness of its applications; for all its consequences concern... general physics, the operations of the arts, domestic uses and civil economy."

"Of the nature of heat uncertain hypotheses only could be formed, but the knowledge of the mathematical laws to which its effects are subject is independent of all hypothesis; it requires only an attentive examination of the chief facts which common observations have indicated, and which have been confirmed by... experiments."

"The action of heat tends to expand all bodies, solid, liquid or gaseous; this is the property which gives evidence of its presence."

"When all the parts of a solid homogeneous body... are equally heated, and preserve without any change the same quantity of heat, they have also and retain the same density."

"The temperature of a body equally heated in every part, and which keeps its heat, is that which the indicates when it is and remains in perfect contact with the body in question. Perfect contact is when the thermometer is completely immersed in a fluid mass, and, in general, when there is no point of the external surface of the instrument which is not touched by one of the points of the solid or liquid mass whose temperature is to be measured."

"[D]ifferent bodies placed in the same region, all of whose parts are and remain equally heated, acquire also a common and permanent temperature."

"Of... the action of heat, that which seems simplest and most conformable to observation, consists in comparing this action to that of light. Molecules separated from one another reciprocally communicate, across empty space, their rays of heat, just as shining bodies transmit their light."

"All bodies have the property of emitting heat through their surface; the hotter they are the more they emit; the intensity of the emitted rays changes very considerably with the state of the surface."

"Every surface which receives rays of heat from surround ing bodies reflects part and admits the rest : the heat which is not reflected, but introduced through the surface, accumulates within the solid; and so long as it exceeds the quantity dissipated by irradiation, the temperature rises."

"[M]olecules which compose... bodies are separated by spaces void of air, and have the property of receiving, accumulating and emitting heat. Each of them sends out rays on all sides, and at the same time receives other rays from the molecules which surround it."

"The effects of heat can by no means be compared with those of an elastic fluid whose molecules are at rest. It would be useless to attempt to deduce from this hypothesis the laws of [heat] propagation... The free state of heat is the same as that of light; the active state... is then entirely different from that of gaseous substances. Heat acts in the same manner in a vacuum, in elastic fluids, and in liquid or solid masses, it is propagated only by way of radiation, but its sensible effects differ according to the nature of bodies."

"Heat is the origin of all elasticity; it is the repulsive force which preserves the form of solid masses, and the volume of liquids. In solid masses, neighbouring molecules would yield to their mutual attraction, if its effect were not destroyed by the heat which separates them. This elastic force is greater according as the temperature is higher; which is the reason... bodies dilate or contract when their temperature is raised or lowered."

"The equilibrium... in the interior of a solid mass, between the repulsive force of heat and the molecular attraction, is stable; [i.e.,] it re-establishes itself when disturbed... If the molecules are arranged at [equilibrium] distances.., and if an external force begins to increase this distance without any change of temperature, the effect of attraction begins by surpassing that of heat, and brings back the molecules to their original position, after a multitude of oscillations... A similar effect is exerted in the opposite sense when a mechanical cause diminishes the primitive distance of the molecules; such is the origin of the vibrations of sonorous or flexible bodies, and of all the effects of their elasticity."

"[T]he mode of action of heat always consists, like... light, in... reciprocal communication of rays... but it is not necessary to consider the phenomena under this aspect... to establish the theory of heat. ...T[]he laws of equilibrium and propagation of radiant heat, in solid or liquid masses, can be rigorously demonstrated, independently of any physical explanation, as the necessary consequences of common observations."

"[T]he quantity of heat which one of the molecules receives from the other is proportional to the difference of temperature of the two molecules... it is null, if the temperatures are equal..."

"Denoting by v and v^\prime the temperatures of two equal molecules m and n...p their extremely small distance [apart], and... dt, the infinitely small... instant, the quantity of heat which m receives from n during this instant will be... (v^\prime - v) \theta (p) \cdot dt. We denote by \theta (p) a certain function of the distance p which, in solid bodies and in liquids, becomes [zero] nothing when p has a sensible magnitude. The function is the same for every point of the same given substance... [but] varies with the nature of the substance."

"The quantity of heat which bodies lose through their surface is subject to the same principle. If we denote by \sigma the area, finite or infinitely small, of the surface, all of whose points have the temperature v, and if a represents the temperature of the... air, the coefficient h being the... external conducibility, we shall have \sigma h (v - a) dt as the expression for the quantity of heat which this surface \sigma transmits to the air during... instant dt. ...h may... be considered as having a constant value, proper to each state of the surface, but independent of the temperature."

"[C]onsider... the uniform movement of heat in the simplest case, which is... an infinite... solid body formed of some homogeneous substance... enclosed between two parallel and infinite planes; the lower plane A is maintained... at a constant temperature a... the upper plane B is... maintained... at... fixed temperature b, ...less than... a; the problem is to determine... the result... if... continued for an infinite time. ...In the final and fixed state... the permanent temperature... is... the same at all points of the same section parallel to the base... [D]enoting by z the height of an intermediate section... from the plane A... e the whole height or distance AB, and... v the temperature of the section whose height is z, we must have v = a + \frac{b - a}{e} z. ...[I]f the temperatures were at first established in accordance with this law, and... the... surfaces A and B... always kept at... temperatures a and b, no change would happen."

"By what precedes we see... Heat penetrates the mass gradually across the lower plane: the temperatures of the intermediate sections are raised, but can never exceed nor even quite attain a certain limit... this limit or final temperature is different for different intermediate layers, and decreases in arithmetic progression from the fixed temperature of the lower plane to the fixed temperature of the upper plane. ... [D]uring each division of time, across a section parallel to the base, or a... portion of that section, a certain quantity of heat flows, which is constant... the same for all the intermediate sections; it is equal to that which proceeds from the source, and to that which is lost... at the upper surface..."

"[T]o compare... the intensities of the constant flows of heat... propagated uniformly in the two solids, that is... the quantities of heat which, during unit of time, "cross unit of surface of each of these bodies. The ratio of these intensities is that of the two quotients \frac{a - b}{e} and \frac{a^\prime - b^\prime}{e^\prime}. ...[D]enoting the first flow by F and the second by F^\prime we... have \frac{F}{F^\prime} = \frac{a - b}{e}\div\frac{a^\prime - b^\prime}{e^\prime}."

"Suppose... in the second solid, the permanent temperature a^\prime ...is that of boiling water, 1... b^\prime is that of melting ice, 0... distance e^\prime is the unit of measure... [Then \frac{a^\prime - b^\prime}{e^\prime} = \frac{1-0}{1} = 1.] [D]enote by K the constant flow of heat which, during unit of time... would cross unit of surface in this [second] solid, if it were formed of a given substance; K expressing a certain number of units of heat, that~is to say a certain... [multiple] of the heat necessary to convert a kilogramme of ice into water... [T]o determine the constant flow F, in a solid... of the same substance, the \frac{F}{K} = \frac{a - b}{e} \div 1 or F = K \frac{a - b}{e}. ... F denotes the quantity of heat which, during the unit of time, passes across a unit of area of the surface taken on a section parallel to the base."

"Thus the thermometric state of a solid enclosed between two parallel infinite plane sides whose perpendicular distance is e, and which are maintained at fixed temperatures a and b, is represented by the two equations:v = a + \frac{b - a}{e} z, and F = K \frac{a - b}{e} or F = -K\frac{dv}{dz}The first... expresses the law according to which the temperatures decrease from the lower side to the opposite side, the second indicates the quantity of heat which, during a given time, crosses a definite part of a section parallel to the base."

"We have taken... coefficient K... to be the measure of the specific conducibility of each substance; this... has... different values for different bodies. It represents... the quantity of heat which, in a homogeneous solid formed of a given substance and enclosed between two infinite parallel planes, flows, during one minute, across a surface of one square metre taken on a section parallel to the extreme planes, supposing that these two planes are maintained, one at the temperature of boiling water, the other at the temperature of melting ice, and that all the intermediate planes have acquired and retain a permanent temperature."

"The chief elements of the method we have followed are these: 1st. We consider... the general condition given by the partial differential equation, and all the special conditions which determine the problem... and we... form the analytical expression which satisfies all... these conditions."

"2nd. We first perceive that this expression contains an infinite number of terms, into which unknown constants enter, or that it is equal to an which includes one or more arbitrary functions. In the first instance, [i.e.], when the general term is affected by the symbol \textstyle \sum , we derive from the special conditions a definite , whose roots give the values of an infinite number of constants. The second instance... when the general term becomes... infinitely small... the sum of the series is... changed into a definite integral."

"3rd. We can prove by the fundamental theorems of algebra, or even by the physical nature of the problem, that the transcendental equation has all its roots real, in number infinite."

"4th. In elementary problems, the general term takes the form of a sine or cosine; the roots of the definite equation are either whole numbers, or real or irrational quantities, each... included between two definite limits. In more complex problems, the general term takes the form of a function given implicitly by means of a differential equation integrable or not. However it may be, the roots of the definite equation exist, they are real, infinite in number. This distinction of the parts of which the integral must be composed, is very important, since it shews... the form of the solution, and the necessary relation between the coefficients."

"5th. It remains only to determine the constants which depend on the initial state; which is done by elimination of the unknowns from an infinite number of equations of the first degree. We multiply the equation which relates to the initial state by a differential factor, and integrate it between defined limits, which are most commonly those of the solid in which the movement is effected. There are problems in which we have determined the coefficients by successive integrations, as may be seen in... the temperature of dwellings. In this case we consider the exponential integrals, which belong to the initial state of the infinite solid... The most remarkable of the problems... and the most suitable for shewing... our analysis, is... the movement of heat in a cylindrical body. In other researches, the determination of the coefficients would require processes of investigation... we do not... know. But... without determining the values of the coefficients, we can always acquire an exact knowledge of the problem, and of the natural course of the phenomenon... the chief consideration is that of simple movements."

"6th. When the expression sought contains a definite integral, the unknown functions... under the... integration are determined, either by the theorems... we have given... or by a more complex process... in the Second Part. These theorems can be extended to any number of variables. They belong in some respects to an inverse method of definite integration; since they serve to determine under the symbols \textstyle \int and \textstyle \sum unknown functions... such that the result of integration is a given function. The same principles are applicable to... problems of geometry... general physics, or... analysis, whether the equations contain finite or infinitely small differences, or... both. The solutions... obtained by this method are complete, and consist of general integrals. ...[O]bjections... are devoid of... foundation..."

"7th. ...[E]ach of these solutions gives the equation proper to the phenomenon, since it represents it distinctly throughout the... extent of its course, and... determine[s] with facility all its results numerically. The functions... obtained by these solutions are then composed of a multitude of terms... finite or infinitely small: but the form of these expressions is... [not] arbitrary; it is determined by the physical character of the phenomenon. For this reason, when the value of the function is expressed by a series into which exponentials relative to the time enter, it is of necessity... since the natural effect whose laws we seek, is... decomposed into distinct parts, corresponding to the... terms of the series. The parts express so many simple movements compatible with the special conditions; for each one of these movements, all the temperatures decrease, preserving their primitive ratios. In this composition we ought not to see a result of analysis due to the linear form of the differential equations, but an actual effect which becomes sensible in experiments. It appears also in dynamical problems in which we consider the causes which destroy motion; but it belongs necessarily to all problems of the theory of heat, and determines the nature of the method which we have followed for the solution..."

"8th. The mathematical theory of heat includes : first, the exact definition of all the elements of the analysis; next, the differential equations; lastly, the integrals appropriate to the fundamental problems. The equations can be... [obtained] in several ways; the same integrals can also be obtained, or other problems solved, by introducing certain changes in the course of the investigation. ...[T]hese researches do not constitute a method different from our own; but confirm and multiply its results."

"9th. ...[The objection] that the transcendental equations which determine the exponents having imaginary roots... would... [of necessity] employ the terms which proceed from them, and... would indicate a periodic character in part of the phenomenon... has no foundation, since the equations in question have.... all their roots real, and no part of the phenomenon can be periodic."

"10th. It has been alleged that... to solve... problems of this kind, it is necessary to resort in all cases to a... form of the integral... denoted as general... but this distinction has no foundation... the use of a single integral... in most cases... complicating... unnecessarily."

"11th. It has been supposed that the method which consists in expressing the integral by a succession of exponential terms, and in determining their coefficients by means of the initial state, does not solve the problem of a prism which loses heat unequally at its two ends; or that, at least, it would be very difficult to verify in this manner the solution derivable from the integral ( \gamma ) by long calculations. We shall perceive, by a new examination, that our method applies directly to this problem, and that a single integration even is sufficient."

"12th. We have developed in series of sines of multiple arcs functions which appear to contain only even powers of the variable, \cos x for example. We have expressed by convergent series or by definite integrals separate parts of different functions, or functions discontinuous between certain limits, for example that which measures the ordinate of a triangle. Our proofs leave no doubt of the exact truth of these equations."

"13th. We find in the works of many geometers results and processes of calculation analogous to those... we... employed. These are particular cases of a general method, which... it became necessary to establish in order to ascertain... the mathematical laws of the distribution of heat. This theory required an analysis... one principal element of which is the... expression of separate functions [f(x)], or of parts of functions... f(x) which has values existing when... x is included between given limits, and whose value is always nothing, if the variable is not included between those limits. This function measures the ordinate of a line which includes a finite arc of arbitrary form and coincides with the axis of abscissae in all the rest of its course. This motion is not opposed to the general principles of analysis; we might even find... first traces... in the writings of Daniel Bernouilli...Cauchy...Lagrange and Euler. It had always been regarded as manifestly impossible to express in a series of sines of multiple arcs, or at least in a trigonometric , a function which has no existing values unless the values of the variable are included between certain limits, all the other values of the function being nul. But this point of analysis is fully cleared up, and it remains incontestable that separate functions, or parts of functions, are exactly expressed by trigonometric convergent series, or by definite integrals. We have insisted on this... since we are not concerned... with an abstract and isolated problem, but with a primary consideration intimately connected with the most useful and extensive considerations. Nothing has appeared to us more suitable than geometrical constructions to demonstrate the truth of these new results, and to render intelligible the forms which analysis employs far their expression."

"14th. The principles which have served to establish for us the analytical theory of heat, apply directly to the investigation of the movement of waves in s, a part of which has been agitated. They aid also the investigation of the s of elastic laminae, of stretched flexible surfaces, of plane elastic surfaces of very great dimensions, and apply in general to problems which depend upon the theory of elasticity. The property of the solutions which we derive from these principles is to render the numerical applications easy, and to offer distinct and intelligible results, which really determine the object of the problem, without making that knowledge depend upon integrations or eliminations which cannot be effected. We regard as superfluous every transformation of the results of analysis which does not satisfy this primary condition."

"In this groundbreaking study, arguing that previous theories of mechanics... did not explain the laws of heat, Fourier set out to study the mathematical laws governing heat diffusion and proposed that an infinite mathematical series may be used to study the conduction of heat in solids. Known... as the 'Fourier Series', this work paved the way for modern mathematical physics. ...This book will be especially helpful for mathematicians... interested in trigometric series and their applications."

"Between 1807 and 1811... Fourier... developed a mathematical theory of heat conduction... independent of the caloric hypothesis, but... was not published until 1822... as Théorie analytique de la chaleur... Fourier set the study of the theory of heat in the tradition of rational mechanics, basing it on differential equations... The heat transmitted between... molecules was proportional to the difference in their temperature and a function of the distance between them... [and] varied with the nature of the... substance. ...Fourier did not rely upon... speculation about the nature of heat. ...[W]hat was important was not what heat was, but what it did, in a given experimental setting."

"[O]ne can hardly imagine someone with a broader background than Fourier, more uniquely situated to simultaneously tackle problems of pure thought as well as in the physical world around him, perhaps in the same stroke of the pen. In the introduction of The Analytical Theory of Heat, he made no secret about the fact that he intended to do just that, with mathematics as his language and tool."

"Newton’s glory was to fulfil, in his Principia of 1687, Galileo's hope of geometrizing gravitation. The ingredients of his solution were Descartes' first law of motion (the principle of inertia), Galileo's rules of and composition of velocities, and Kepler's rules of planetary orbiting. Newton showed that Kepler's rules following form Galileo's, the principle of inertia, and the assumption that a planet falls towards the Sun along the line joining their centres. Since Kepler's rules allowed the substitution of an area for a time, Newton could reduce the problem of the magnitude of gravitational acceleration to a problem in geometry."

"Only around the end of the nineteenth century did scientists come across a few observations that did not fit well with Newton's laws, and these led to the net revolution in physics - the theory of relativity and quantum mechanics."

"Lex I. Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressi cogitur statum illum mutare. Lex II. Mutationem motus proportionalem esse vi motrici impressa; & fieri secundum lineam rectam qua vis illa imprimatur. Lex III. Actioni contrariam semper & aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales & in partes contrarias dirigi. (English translation: Law I. Every body perseveres in its state of rest or of uniform motion in a straight line, except in so far as it is compelled to change that state by the forces impressed upon it. Law II. The change of motion is proportional to the motive force impressed upon it; and takes place along the straight line on which that force is impressed. Law III. To an action there is always an opposite and equal reaction: or the actions of two bodies on each other are always equal and directed in opposite directions.)"