Physics books

"I have found the theory that the atoms of the radio-active bodies are undergoing spontaneous disintegration extremely serviceable not only in correlating the known phenomena but also in suggesting new lines of research."

"The rapid advance of our knowledge of radio-activity has been dependent on the information already gained by research into the electric properties of gases."

"The close of the old and the beginning of the new century have been marked by a very rapid increase of our knowledge of that most important but comparatively little known subject-the connection between electricity and matter."

"The study of the radio-active substances and of the discharge of electricity through gases has supplied very strong experimental evidence in support of the fundamental ideas of the existing atomic theory. It has also indicated that the atom itself is not the smallest unit of matter but is a complicated structure made up of a number of smaller bodies."

"A great impetus to the study of this subject was initially given by the experiments of Lenard on the cathode rays and by Rontgen's discovery of the X rays."

"An examination of the conductivity imparted to a gas by the X rays led to a clear view of the mechanism of the transport of electricity through gases by means of charged ions."

"This ionization theory of gases has been shown to afford a satisfactory explanation not only of the passage of electricity through flames and vapors, but also of the complicated phenomena observed when a discharge of electricity passes through a vacuum tube."

"Study of the cathode rays showed that they consisted of a stream of material particles, projected with great velocity, and possessing an apparent mass small compared with that of the hydrogen atom. The connection between the cathode [rays] and Röntgen rays [i.e., X rays] and the nature of the latter were also elucidated. Much of this admirable experimental work on the nature of the electric discharge has been done by Professor J.J. Thomson and his students in the Cavendish Laboratory Cambridge."

"An examination of natural substance, in order to see if they gave out dark radiations similar to X rays, led to the discovery of the radio active bodies which possess the property of spontaneously emitting radiations, invisible to the eye, but readily detected by their action on photographic plates, and their power of discharging electrified bodies."

"In order to explain the phenomena of radio-activity, a theory has been put forward which regards the atoms of the radio-active elements as suffering spontaneous disintegration, and giving rise to a series of radio-active substances which differ in chemical properties from the parent elements. The radiations accompany the breaking-up of the atoms, and afford a comparative measure of the rate at which the disintegration takes place. This theory is found to account in a satisfactory way for all the known facts of radio-activity, and welds a mass of disconnected facts into one homogeneous whole."

"On this view, the continuous emission of energy from the active bodies is derived from the internal energy inherent in the atom, and does not in any way contradict the law of the conservation of energy. At the same time, however, it indicates that an enormous store of latent energy is resident in the radio-atoms themselves. This store of energy has previously not been observed, on account of the impossibility of breaking up into simpler forms the atoms of the elements by the action of the chemical or physical forces at our command."

"On this theory we are witnessing in the radio-active bodies a veritable transformation of matter. This process of disintegration was investigated, not by direct chemical methods, but by means of the property possessed by the radio-active bodies of giving out specific types of radiation."

"Except in the case of a very active element like radium, the process of disintegration takes place so slowly, that hundreds if not thousands of years would be required before the amount transformed would come within the range of detection of the balance or the spectroscope. In radium, however, the process of disintegration takes place at such a rate that it should be possible within a limited space of time to obtain definite chemical evidence on this question."

"The recent discovery that helium can be obtained from radium adds strong confirmation to the theory; for helium was indicated as a probable disintegration product of the radio-active elements before this experimental evidence was forthcoming. If the production of helium by radium is completely substantiated, the further study of radio-active bodies promises to open up new and important fields of chemical inquiry."

"In this book the experimental facts of radio-activity and the connection between them are interpreted on the disintegration [of atoms] theory. Many of the phenomena observed can be investigated in a quantitative manner... the agreement of any theory with the facts, which it attempts to explain, must ultimately depend upon the results of accurate measurement."

"The value of any working theory depends upon the number of experimental facts it serves to correlate, and upon its power of suggesting new lines of work. In these respects the disintegration theory, whether or not it may ultimately be proved to be correct, has already been justified by its results."

"The term "radio-active" is now generally applied to a class of substances, such as uranium, thorium, radium, and their compounds, which possess the property of spontaneously emitting radiations capable of passing through plates of metal and other substances opaque to ordinary light."

"A strongly radio-active body like radium is able to cause marked phosphorescence and fluorescence on some substances placed near it."

"The most remarkable property of the radio-active bodies is their power of spontaneously and continuously radiating energy at a constant rate, without, as far as is known, the action upon them of any external exciting cause. The phenomena at first sight appear to be in direct contradiction to the law of conservation of energy, since no obvious change with time occurs in the radiating material. The phenomena appear still more remarkable when it is considered that the radio active bodies must have been steadily radiating energy since the time of their formation in the earth's crust."

"Immediately after Röntgen's discovery of the production of X rays, several scientists were led to examine if any natural bodies possessed the property of giving out radiations which could penetrate metals and other substances opaque to light. As the production of X rays seemed to be in some way connected with cathode rays, which cause strong fluorescent and phosphorescent effects on various bodies, the substances first examined were those that were phosphorescent when exposed to light. ...there seems to be little doubt that the effects are due to short ultra-violet light waves, capable of passing through certain substances opaque to ordinary light. These effects, while interesting in themselves, are of quite a distinct character from those shown by the radio-active bodies which will now be considered."

"The first important discovery in the subject of radio-activity was made in February, 1896, by M. Henri Becquerel, who found that a uranium salt [potassium uranyl sulfate], the double sulphate of uranium and potassium, emitted some rays which gave an impression on a photographic plate enveloped in black paper. These rays were also able to pass through thin plates of metals and other substances opaque to light. ...It was at first natural to suppose that the emission of these rays was in some way connected with the power of phosphorescence, but later observations showed that there was no connection whatever between them. ...The amount of action on the photographic plate does not depend on the particular compound of uranium employed, but only on the amount of uranium present in the compound. The non-phosphorescent are equally active with the phosphorescent compounds. The amount of radiation given out is unaltered if the active body is kept continuously in darkness. The rays are given out by solutions, and by crystals which have been deposited from solutions in the dark and never exposed to light. This shows that the radiation cannot be due in any way to the gradual emission of energy stored up in the crystal in consequence of exposure to a source of light."

"These radiations from uranium are persistent, and as far as observations have yet gone, are unchanged either in intensity or character, with lapse of time. Observations to test the constancy of the radiations for long periods of time have been made by Becquerel. ...No perceptible weakening of the radiation has been observed over a period of four years. Mme Curie has made determinations of the activity of uranium over a space of five years... but found no appreciable variation during that period."

"Since the uranium is thus continuously radiating energy from itself, without any known source of excitation, the question arises whether any known agent is able to affect the rate of its emission. No alteration was observed when the body was exposed to ultra-violet light or to ultra-red light or to X rays. ..The intensity of the uranium radiation is not affected by a variation of temperature between 200 ºC and the temperature of liquid air."

"The radiations from uranium are analogous, as regards their photographic and electrical actions, to [[w:X-ray|Röntgen rays], but compared with the rays from an ordinary X ray tube, these actions are extremely feeble. While with Röntgen rays a strong impression is produced on a photographic plate in a few minutes or even seconds, several days' exposure to the uranium rays is required to produce a well-marked action, even though the uranium compound, enveloped in black paper, is placed close to the plate. The discharging action, while very easily measurable by suitable methods, is also small compared with that produced by X rays from an ordinary tube."

"The absence of reflection, refraction, or polarization in the penetrating rays from uranium necessarily follows in the light of our present knowledge of the rays. It is now known that the uranium rays, mainly responsible for the photographic action, are deviable by a magnetic field, and are similar in all respects to cathode rays, i.e. the rays are composed of small particles projected at great velocities. The absence of the ordinary properties of transverse light waves is thus to be expected."

"The rays from uranium are complex in character, and in addition to the penetrating deviable rays, there is also given off a radiation very readily absorbed by passing through thin layers of metal foil, or by traversing a few centimeters of air. The photographic action due to these rays is very feeble in comparison with that of the penetrating rays, although the discharge of electrified bodies is mainly caused by them. Besides these two types of rays, some rays are emitted which are of an extremely penetrating character and are non-deviable by a magnetic field. These rays are difficult to detect photographically, but can be readily examined by the electric method."

"The question naturally arose whether the property of spontaneously giving out penetrating radiations was confined to uranium and its compounds, or whether it was exhibited to any appreciable extent by other substances. By the electrical method, with an electrometer of ordinary sensitiveness, any body which possesses an activity of the order of 1/100 of that of uranium can be detected. With an electroscope of special construction, such as has been designed by C.T.R. Wilson for his experiments on the natural ionization of air, a substance of activity 1/10000 and probably 1/100000 of that of uranium can be detected."

"Mme Curie made a detailed examination by the electrical method of the great majority of known substances, including the very rare elements, to see if they possessed any activity. In cases when it was possible, several compounds of the elements were examined. With the exception of thorium and phosphorus, none of the other substances possessed an activity even of the order of 1/100 of uranium."

"The ionization of the gas by phosphorus does not, however, seem to be due to a penetrating radiation like that found in the case of uranium, but rather to a chemical action taking place at its surface. The compounds of phosphorus do not show any activity, and in this respect differ from uranium and the other active bodies."

"Le Bon has also observed that quinine sulphate, if heated and then allowed to cool, possesses for a short time the property of discharging both positively and negatively electrified bodies. It is necessary, however, to draw a sharp line of distinction between phenomena of this kind and those exhibited by the naturally radio-active bodies. While both, under special conditions, possess the property of ionizing the gas, the laws controlling the phenomena are quite distinct in the two cases. ...the activity of the naturally radio active bodies is spontaneous and permanent. It is exhibited by all compounds and is not as far as is yet known altered by change in the chemical or physical conditions."

"The discharging and photographic action alone cannot be taken as a criterion as to whether a substance is radio-active or not. It is necessary in addition to examine the radiations, and to test whether the actions take place through appreciable thicknesses of all kinds of matter opaque to ordinary light. For example, a body giving out short waves of ultra-violet light can be made to behave in many respects like a radio-active body. As Lenard has shown, short waves of ultra-violet light will ionize the gas in their path, and will be rapidly absorbed in the gas. They will produce strong photographic action, and may pass through some substances opaque to ordinary light. The similarity to a radio-active body is thus fairly complete as regards these properties. On the other hand, the emission of these light waves, unlike that of the radiations from an active body, will depend largely on the molecular state of the compound, or on temperature and other physical conditions."

"But the great point of distinction lies in the nature of the radiations from the bodies in question. In one case the radiations behave as transverse waves, obeying the usual laws of light waves, while in the case of a naturally active body, they consist for the most part of a continuous flight of material particles projected from the substance with great velocity. Before any substance can be called "radio-active" in the sense in which the term is used to describe the properties of the natural radio-active elements, it is thus necessary to make a close examination of its radiations; for it is inadvisable to extend the use of the term "radio-active" to substances which do not possess the characteristic radiating properties of the radio-active elements which we have described, and the active products which can be obtained from them. Some of the pseudo active-bodies will however be considered later."

"Thorium compounds gives out a material emanation made up of very small particles themselves radio-active. The emanation behaves like a radio-active gas; it diffuses rapidly through porous substances like paper, and is carried away by a current of air. ...In addition to giving out an emanation, thorium behaves like uranium in emitting three types of radiation, each of which is similar in properties to the corresponding radiation from uranium."

"It seemed probable that the large activity of some of these minerals, compared with uranium and thorium, was due to the presence of small quantities of some very active substance, which was different from the known bodies thorium and uranium. This supposition was completely verified by the work of M. and Mme Curie, who were able to separate from pitchblende by purely chemical methods two active bodies, one of which in the pure state is over a million times more active than the metal uranium. This important discovery was due entirely to the property of radio-activity possessed by the new bodies. The only guide in their separation was the activity of the products obtained. ...The activity of the specimens thus served as a basis of rough qualitative and quantitative analysis, analogous in some respects to the indication of the spectroscope."

"The chief difficulty lay in the fact that pitchblende is a very complex mineral and contains in varying quantities nearly all the known metals. ...The analysis of pitchblende by chemical methods, using the procedure sketched above, led to the discovery of two very active bodies, polonium and radium. The name polonium was given to the first substance discovered by Mme Curie in honor of the country of her birth. The name radium was a very happy inspiration of the discoverers, for this substance in the pure state possesses the property of radio-activity to an astonishing degree."

"Radium is extracted from pitchblende by the same process necessary to separate barium, to which it is very closely allied in chemical properties. ...The chloride of radium is less soluble than that of barium and can be separated from it by the method of fractional crystallization."

"Both polonium and radium exist in infinitesimal quantities in pitchblende. In order to obtain a few decigrammes of very active radium, it is necessary to use several tons of pitchblende, or the residues obtained from the treatment of uranium minerals. ...M. and Mme Curie were indebted for their first working material to the Austrian government, who generously presented them with a ton of the treated residue of uranium materials, from the State manufactory."

"Mme Curie has utilized the coloration of the crystals of radiferous barium as a means of controlling the final process of purification. The crystals of salts of radium and barium deposited from acid solutions are indistinguishable. The crystals of radiferous barium are at first colorless, but in the course of a few hours, become yellow, passing to orange and sometimes to a beautiful rose color. The rapidity of this coloration depends on the amount of barium present. Pure radium crystals do not color, or at any rate not as rapidly as those containing barium. The coloration is a maximum for a definite proportion of radium, and this fact can be utilized as a means of testing the amount of barium present. When the crystals are dissolved in water the coloration disappears."

"Giesel has observed that pure radium bromide gives a beautiful carmine color to the Bunsen flame. If barium is present in any quantity only the green color due to barium is observed and a spectroscopic examination shows only the barium lines. This carmine coloration of the Bunsen flame is a good indication of the purity of the radium."

"Since the preliminary announcement of the discovery of radium, Giese has devoted a great deal of attention to the separation of radium, polonium, and other active bodies from pitchblende. ...Using the method of fractional crystallization of the bromide instead of the chloride he has been able to prepare considerable quantities of pure radium. By this means the labor of final purification of radium has been much reduced. He states that six or eight crystallizations with the bromide are sufficient to almost completely free the radium from the barium."

"Mme Curie has made successive determinations of the atomic weight of the new element with specimens of steadily increasing purity. ...In these experiments about 0.l gram of pure radium chloride has been obtained by successive fractionations. The difficulty involved in preparing a quantity of pure radium chloride large enough to test the atomic weight may be gauged from the fact that only a few centigrams of fairly pure radium, or a few decigrams of less concentrated material, are obtained from treatment of about 2 tons of the mineral from which it is derived. ...Runge and Precht have examined the spectrum of radium in a magnetic field, and have shown the existence of series analogous to those observed for calcium, barium, and strontium. These series are connected with the atomic weights of the elements in question, and Runge and Precht have calculated by these means that the atomic weight of radium should be 258--a number considerably greater than the number 225 obtained by Mme Curie by means of chemical analysis. Marshall Watts, on the other hand, using another relation between the lines of the spectrum, deduced the value obtained by Mme Curie. Considering that the number found by Mme Curie agrees with that required by the periodic system, it is advisable in the present state of our knowledge to accept the experimental number rather than the one deduced by Runge and Precht from spectroscopic evidence."

"There is no doubt that radium is a new element possessing remarkable physical properties. The detection and separation of this substance, existing in such minute proportions in pitchblende, has been due entirely to the characteristic property we are considering, and is the first notable triumph of the study of radio-activity."

"On account of its enormous activity the radiations from radium are very intense: a screen of zinc sulphide, brought near a few centigrams of radium bromide, is lighted up quite brightly in a dark room, while brilliant fluorescence is produced on a screen of platino-barium cyanide. An electroscope brought near is almost instantly discharged, while a photographic plate is immediately affected. At a distance of one meter, a day's exposure to the radium rays would produce a strong impression. The radiations from radium are analogous to those of uranium, and consist of the three types of rays: easily absorbed, penetrating, and very penetrating. Radium also gives rise to an emanation similar to that of thorium, but with a very much slower rate of decay. The radium emanation retains its activity for several weeks, while that of thorium lasts only a few minutes. The emanation obtained from a few centigrams of radium illuminates a screen of zinc sulphide with great brilliancy. The very penetrating rays of radium are able to light up an X ray screen in a dark room, after passage through several centimeters of lead and several inches of iron."

"All the salts of radium are naturally phosphorescent. The phosphorescence of impure radium preparations is in some cases very marked. All the radium salts possess the property of causing rapid colorations of the glass vessel which contains them. For feebly active material the color is usually violet, for more active material a yellowish-brown and finally black."

"Polonium was the first of the active substances obtained from pitchblende. It has been investigated in detail by its discoverer Mme Curie. ...This active substance... is so closely allied in chemical properties to bismuth that it has so far been found impossible to effect a complete separation. Partial separation of polonium can be made by successive fractionations."

"The polonium prepared by Mme Curie differs from the other radio-active bodies in several particulars. In the first place the radiations include only very easily absorbable rays. The two penetrating types of radiation given out by uranium, thorium, and radium are absent. In the second place the activity does not remain constant, but diminishes continuously with the time. Mme Curie found that the polonium lost half its original activity in the course of eleven months."

"The decay of the activity of polonium with time has led to the view that polonium is not a new active substance, but merely active bismuth, i.e. bismuth which in some way had been made active by admixture with radio-active bodies. The activity of any product is not necessarily a proof that a radio-element is present, for it has been shown that many inactive elements become active by association with active matter. The activity of these substances, when removed from the active element, is however only transient, and decays gradually with the time. This activity is not due to the presence of the radio-element itself. For example, barium separated from radium is strongly active, although the spectroscopic examination shows no trace of the radium lines."

"The discussion of the nature of polonium was renewed by the discovery of Marckwald that a substance similar to polonium, of which the activity did not decay with time, could be separated from pitchblende. ...The radiations from Marckwald's substance are similar to of polonium, for no penetrating rays are present. The radiations are very intense. They have a marked photographic action, and cause many substances, like zinc oxide and the diamond, to phosphoresce brightly. The strong luminosity of the under these rays can be utilized to distinguish the diamond imitations, for glass is only slightly phosphorescent in comparison. ...Marckwald ...states that his preparations have preserved their activity unchanged while the polonium of the Curies undoubtedly loses its activity in the course of a few years."

"Debierne has obtained from pitchblende a very active substance which he named actinium. This active substance is precipitated with the iron group, and appears to be very closely allied in chemical properties to thorium, though it is many thousand times more active. It is very difficult to separate from thorium and the rare earths. ...Debierne has obtained a substance comparable in activity with radium. The separation, which is difficult and laborious, has so far not been carried far enough to bring out any new lines in the spectrum. Actinium gives out easily absorbed and penetrating deviable rays like those of radium, and also a radio-active emanation, which is more allied to the emanation of thorium than to that of radium. The emanation has a distinctive rate of decay; it loses its activity in the course of a few seconds, while the thorium emanation loses half its activity in one minute. The distinctive character of the radiations and emanations, together with the permanence of the activity; make it very probable that actinium will prove to be a new element of very great activity."

"Giesel also has obtained from pitchblende a radio-active substance which in many respects is similar to the actinium of Debierne. The active substance belongs to the group of cerium earths, and is precipitated with them. The method of preparation of this material is the same as that employed for the separation of the rare earths. This substance is similar in radio-active behavior to thorium, but intensely active in comparison. From the method of separation, thorium itself cannot be present except in minute quantity. ...If a piece of paper is placed in a small closed vessel containing the active material, in a short time the paper itself becomes powerfully active. This is especially the case if it is moistened with water."

"In the words of the celebrated English mathematician, Edward Wright, I doubt not that our united efforts "will find the heartiest approval among all intelligent men and children of magnetic science.""

"Not only does Gilbert frequently make use of what he terms "words new and unheard-of," besides attaching to many others a signification far different from that generally recognized at this day, but, what is worse, he retains to a great extent the terminology of the mediaeval scholastic philosophers."

"It is known that in the philosophy of the schoolmen (as in that of Aristotle) form—forma—means that which added to matter—materia—constitutes the true nature of the thing. Matter per se is indifferent, indefinite; form gives it definiteness. The earth is informed with verticity—that is its prime distinction. When any portion of the earth loses verticity it loses its forma—is deformate. To restore to it verticity, is to reformate it, or to informate it. Portions of the earth that are deformate are, as it were, effete, excrementitious, waste matter."

"England's great poet, John Dryden, tells us: "It is almost impossible to translate verbally and well at the same time; for the Latin (a most severe and compendious language) often expresses that in one word which either the barbarity or the narrowness of modern tongues cannot supply in more. ...But since every language is so full of its own proprieties that what is beautiful in one is often barbarous, nay, sometimes nonsense, in another, it would be unreasonable to limit a translator to the narrow compass of his author's words; it is enough if he choose out some expression which does not vitiate the sense.""

"To give here such an analysis as Gilbert's admirable work merits would be impracticable but the short review of it made by Dr. John Robison deserves... reproduction as follows..."

"It is curious to mark the almost perfect sameness of Dr Gilbert's sentiments and language with those of Lord Bacon. They both charge, in a peremptory manner, all those who pretend to inform others, to give over their dialectic labours, which are nothing but ringing changes on a few trite truths, and many unfounded conjectures, and immediately to betake themselves to experiment.""

"He has pursued this method on the subject of magnetism, with wonderful ardour, and with equal genius and success; for Dr. Gilbert was possessed both of great ingenuity, and a mind fitted for general views of things. The work contains a prodigious number and variety of observations and experiments, collected with sagacity from the writings of others, and instituted by himself with considerable expense and labour."

"It would indeed be a miracle if all Dr. Gilbert's general inferences were just, or all his experiments accurate. It was untrodden ground. But on the whole, this performance contains more real information than any writing of the age in which he lived, and is scarcely exceeded by any that has appeared since. We may hold it with justice as the first-fruits of the Baconian or experimental philosophy."

"This work of Dr Gilbert's relates chiefly to the loadstone, and what we call magnets; that is, pieces of steel which have acquired properties similar to those of the loadstone. But he extends the term magnetism and the epithet magnetic, to all bodies which are affected by loadstones and magnets, in a manner similar to that in which they affect each other. In the course of his investigations, indeed, he finds that these bodies are only such as contain iron in some state or other; and in proving this limitation he mentions a great variety of phenomena which have a considerable resemblance to those which he allows to be magnetical, namely, those which he called electrical, because they were produced in the same way that amber is made to attract and repel light bodies. He marks, with care, the distinctions between these and the characteristic phenomena of magnets. He seems to have known, that all bodies may be made electrical, while ferruginous substances alone can be made magnetical."

"It is not saying too much of this work of Dr. Gilbert's to affirm, that it contains almost everything that we know about magnetism. His unwearied diligence in searching every writing on the subject, and in getting information from navigators, and his incessant occupation in experiments, have left very few facts unknown to him."

"We ascribe it to the general indolence of mankind, who do not take the trouble of consulting originals, where things are mixed with others which they do not want, or treated in a way, and with a painful minuteness, which are no longer in fashion. We earnestly recommend it [De Magnete] to the perusal of the curious reader. He will (besides the philosophy) find more facts in it than in the two large folios of Scarella."

"The manner in which "this great man arrived to discover so much of magnetical philosophy" and "all the knowledge he got on this subject," we are told by Sir Kenelm Digby, "was by forming a little load-stone into the shape of the earth. By which means he compassed a wonderful designe, which was, to make the whole globe of the earth maniable; for he found the properties of the whole earth, in that little body; which he therefore called a terrella, or little earth; and which he could manage and try experiences upon, at his will. And in like manner, any man that hath an aim to advance much in natural sciences, must endeavour to draw the matter he enquireth, of into some small modell, or into some kinde of manageable method; which he may turn and wind as he pleaseth. And then let him be sure, if he hath a competent understanding, that he will not misse his mark.""

"Amongst the many other ingenious contrivances frequently alluded to in his book, Gilbert mentions the versorium, an iron needle moving freely upon a point, with which he was enabled to measure excited electricity. He is besides the inventor of "two most ingenious and necessarie Instruments for Sea men to find out thereby the latitude of any place upon sea or land, in the darkest night, that is without the helpe of Sunne, Moone or Starre." These instruments are described in Thomas Blunderville's quarto work entitled "The Theoriques of the seven Planets, shewing their diverse motions... printed at London 1602.""

"In the present volume will be found photo lithographic reproductions of three... title-pages. ...The 1628 is the most elaborate of all known Gilbert title-pages. As described by Prof. Sir Wm. Thomson (Lord Kelvin), it is "in the form of a monument, ornamented with commemorative illustrations of Gilbert's theory and experiments, and a fantastic indication of the earliest European mariner's compass, a floated loadstone, but floating in a bowl on the sea and left behind by the ship sailing away from it! In the upper left-hand corner is to be seen Gilbert's terrella and orbis virtutis. The terrella is a little globe of loadstone, which he made to illustrate his idea that the earth is a great globular magnet. ...The meaning of the little bars bordering the terrella is explained in Gilbert's book... where he alludes to the application of bits of fine iron wire as long as a barley-corn, etc., etc."

"The orbis virtutis is simply Gilbert's expression for what Faraday called the field of force, that is to say, the space round a magnet, in which magnetic force is sensibly exerted on another magnet, as, for instance, a small needle, properly placed for the test."

"Gilbert's word virtue expresses even more clearly than Faraday's word force the idea urged so finely by Faraday, and proved so validly by his magneto-optic experiment, that "there is a real physical action of a magnet through all the space round it tho' no other magnet be there to experience force and show its effects.""

"The only known writing of Gilbert in English is in the form of a letter dated 14th February (?1602) which appears at the end of William Barlowe's "Magneticall Advertisements or divers observations concerning the loadstone," quarto London 1616, and reads as follows: To the Worshipfull my good friend, Mr. William Barlowe at Easton by Winchester. "...you have shewed mee more—and brought more light than any man hath done. ...Johannes Franciscus Sagredus... a great Magneticall man... writeth that hee hath conferred with divers learned men of Venice and with the Readers of Padua and reporteth wonderfull liking of my booke, you shall have a coppy of the letter: Sir, I propose to adjoyne an appendix of six or eight sheets of paper to my booke after a while, I am in hand with it of some new inventions, and I would have some of your experiments, in your name and invention put into it, if you please, that you may be knowen for an augmenter of that art. So for this time in haste I take my leave the xiiyth of February. Your very loving friend, W. GILBERT." His intention to print the short appendix was never carried into effect."

"In his epistle to Dr. Walter Charleton (physician in ordinary to King Charles I), the celebrated English poet, John Dryden, predicts that "Gilbert shall live till loadstones cease to draw, Or British fleets the boundless ocean awe.""

"To the most learned Mr. William Gilbert, the distinguished London physician and father of the magnetic philosophy: a laudatory address concerning these books on magnetism, by Edward Wright."

"In truth, in my opinion, there is no subject-matter of higher importance or of greater utility to the human race upon which you could have brought your philosophical talents to bear. For by the God-given favor of this stone has it come about that the things which for so many centuries lay hid—such vast continents of the globe, so infinite a number of countries, islands, nations, and peoples—have been, almost within our own memory, easily discovered and oft explored, and that the whole circle of the globe has been circumnavigated more than once by our own Drake and Cavendish."

"Sailors of old were often beset, as we learn from the histories, by an incredible anxiety and by great peril, for, when storms raged and the sight of sun and stars was cut off, they knew not whither they were sailing, neither could they by any means or by any device find out."

"Since the magnetic pointer does not always regard the same northern spot in every locality, but usually varies therefrom, either to the east or to the west, tho' it nevertheless hath and holds ever the same variation in the same place, wherever that may be; it has come about that by means of this variation (as it is called) closely observed and noted in certain maritime regions, together with an observation of the latitude, the same places can afterward be found by navigators when they approach and come near to the same variation. ...Thanks to this magnetic indication, that ancient geographical problem, how to discover the longitude, would seem to be on the way to a solution."

"The iron pointer suspended freely and with the utmost precision in equilibrium on its axis, and then touched and excited with a loadstone, dips down to a fixed and definite point below the horizon (e.g. in the latitude of London it dips nearly 72 degrees) and there stands. But because of the wonderful agreement and congruency manifested in nearly all and singular magnetic experiments, equally in the earth itself and in a terrella, (i.e. a spherical loadstone), it seems (to say the least) highly probable and more than probable that the same pointer... will at the equator stand in equilibrium on the plane of the horizon."

"It is highly probable that in proceeding a very short distance from south to north (or vice versa) there will be a pretty sensible change in the dip; and thus the dip being carefully noted once and the latitude observed, the same place and the same latitude may thereafter be very readily found by means of a dip instrument even in the darkest night and in the thickest weather."

"If these books of yours on the Loadstone contained nought save this one method of finding the latitude from the magnetic dip, now first published by you, even so our British mariners as well as the French, the Dutch, the Danes, whenever they have to enter the British sea or the strait of Gibraltar from the Atlantic Ocean, will justly hold them worth no small sum of gold."

"Hardly twenty years after the English artificer, Robert Norman, had in 1576, devised the inclinatorium, which enabled him to determine the dip or inclination of the magnetic needle, Gilbert boasted that, by means of this instrument, he could ascertain a ship's place in dark starless nights. Gilbert commends the method as applicable aere caliginoso [dark atmosphere]; and Edward Knight, the English mathematician, in the introduction which he added to his master's great work, describes this proposal as "worth much gold." Having fallen into the same error with Gilbert of presuming that the isoclinal lines coincided with the geographical parallel circles, and that the magnetic and geographical equators were identical, he did not perceive that the proposed method had only a local and very limited application."

"That discovery of yours, that the entire globe is magnetical, albeit to many it will seem to the last degree paradoxical, nevertheless is buttressed and confirmed by so many and so apposite experiments... that no room is left for doubt or contradiction."

"I come therefore to the cause of magnetic variation—a problem that till now has perplexed the minds of the learned; but no one ever set forth a cause more probable than the one proposed now for the first time in these your books on the Loadstone. The fact that the magnetic needle points due north in the middle of the ocean and in the heart of continents—or at least in the heart of their more massive and more elevated parts—while near the coasts there is, afloat and ashore, an inclination of the needle toward those more massive parts, just as happens in a terrella that is made to resemble the earth globe in its greater elevation at some parts and shows that it is weak or decayed or otherwise imperfect elsewhere: all this makes exceedingly probable the theory that the variation is nothing but a deviation of the magnetic needle to those more powerful and more elevated regions of the globe."

"All those who have imagined or accepted certain "respective points" as well as they who speak of magnetic mountains or rocks or poles, will begin to waver as soon as they read these your books on the Loadstone and will of their own accord come over to your opinion."

"As for... the circular motion of the earth and the terrestrial poles, though many will deem it the merest theorizing, still I do not see why it should not meet with indulgence even among those who do not acknowledge the earth's motion to be spherical, seeing that even they cannot readily extricate themselves from the many difficulties that result from a diurnal motion of the whole heavens."

"Which theory is the more probable, that the equinoctial circle of the earth may make a rotatary movement of one quarter of an English mile, (60 miles being equal to one degree on the earth's equator in one second of time... or that the equator of the primum mobile in the same time, with inexpressible celerity, makes 5,000 miles and that in the twinkling of an eye it makes about 50 English miles, surpassing the velocity of a flash of lightning, if they are in the right who most strenuously deny the earth's motion?"

"It does not seem to have been the intention of Moses or the prophets to promulgate nice mathematical or physical distinctions: they rather adapt themselves to the understanding of the common people and to the current fashion of speech, as nurses do in dealing with babes."

"While we devoutly acknowledge and adore the inscrutable wisdom of the triune Godhead, having with all diligence investigated and discerned the wondrous work of his hands in the magnetic movements, do hold it to be entirely probable, on the ground of experiments and philosophical reasons not few, that the earth while it rests on its centre as its basis and foundation, hath a spherical motion nevertheless."

"Your work, I say, that has been kept back for so many years, your New Physiology of the Loadstone and of the Great Magnet (i.e. the Earth)—a philosophy never to be sufficiently admired; let it go forth into the light of publicity; for believe me... these your books on the Loadstone (De Magnete) will do more to perpetuate your memory than would the monument of any Magnate (Magnatis cujusvis) erected over your grave."

"Every day, in our experiments, novel, unheard-of properties came to light: and our Philosophy became so widened, as a result of diligent research, that we have attempted to set forth, according to magnetic principles, the inner constitution of the globe and its genuine substance, and in true demonstrations and in experiments that appeal plainly to the senses, as though we were pointing with the finger, to exhibit to mankind Earth, mother of all."

"Even as geometry rises from certain slight and readily understood foundations to the highest and most difficult demonstrations, whereby the ingenious mind ascends above the aether: so does our magnetic doctrine and science in due order first show forth certain facts of less rare occurrence; from these proceed facts of a more extraordinary kind; at length, in a sort of series, are revealed things most secret and privy in the earth, and the causes are recognized of things that, in the ignorance of those of old or through the heedlessness of the moderns, were unnoticed or disregarded."

"Why should I, in so vast an ocean of books whereby the minds of the studious are bemuddled and vexed; of books of the more stupid sort whereby the common herd and fellows without a spark of talent are made intoxicated, crazy, puffed up; are led to write numerous books and to profess themselves philosophers, physicians, mathematicians, and astrologers, the while ignoring and contemning men of learning: why, I say, should I add aught further to this confused world of writings, or why should I submit this noble and (as comprising many things before unheard) of this new and inadmissible philosophy to the judgment of men who have taken oath to follow the opinions of others, to the most senseless corrupters of the arts, to lettered clowns, grammatists, sophists, spouters, and the wrong-headed rabble to be denounced, torn to tatters, and heaped with contumely."

"Armed loadstone. One that is furnished with an iron helmet or cap."

"Cuspis (point). The end of a magnetized versorium."

"Crotch. Name sometimes given to the end not touched and excited, although in some instruments both ends are commonly so designated, according as they are most convenient for excitation by the loadstone."

"Declinatorium. A bar or needle movable vertically on its axis and that is excited with a loadstone; used in the dip instrument."

"Electrics. Bodies that attract in the same way as amber."

"Excited magnetic body. One such as iron or steel that acquires magnetism from a loadstone or natural magnet."

"Magnetic coition. This phrase is used rather than attraction because magnetic movements do not result from attraction of one body alone but from the coming together of two bodies harmoniously (not the drawing of one by the other)... the coition is always vigorous, even though heavy substances make opposition."

"Magnetized versorium. An iron bar or needle resting on a point (electroscope) and put in motion—excited—by the loadstone or natural magnet."

"Ostensio. Physical demonstration (opposed to theory)."

"Paralleletically. In the direction of a parallel of latitude."

"Sphere of influence. The entire space over which the force of a loadstone extends."

"Sphere of coition. The entire space over which the smallest magnetic body moves toward a loadstone."

"Verticity. Polar strength—activity (or what in Gilbert's day was understood as energy); not gyrating, vertiginous, but turning power: nor is it polar revolution, but a directing virtue, an innate turning vigor (virtus convertens)."

"Jules Henry Poincaré... who anticipated many aspects of relativity theory, once put it this way... Suppose... everything in the universe became a thousand times larger than before. ...would you be able to tell that anything had changed? Is there an experiment you could perform that would prove you had altered in size? No, said Poincaré... "Larger" means larger in relation to something else. ...Size, then, is relative. There is no absolute way to measure an object."

"G. J. Whitrow points out in his book The Structure and Evolution of the Universe, a very simple explanation [for the Michelson–Morley experiment] would immediately occurred to everyone: the earth doesn't move!"

"The strangest explanation [for the Michelson–Morley experiment] was put forth by an Irish physicist, George Francis Fitzgerald. Perhaps, he said, the ether wind puts pressure on a moving object, causing it to shrink a bit in the direction of motion. To determine the length of a moving object, its length at rest must be multiplied by the following simple formula, in which \scriptstyle v^2 is the velocity of the object multiplied by itself, \scriptstyle c^2 is the velocity of light multiplied by itself: \scriptstyle \sqrt{1-\frac{v^2}{c^2}}."

"The speed of light in an unobtainable limit; when this is reached the formula becomes \scriptstyle \sqrt{1-\frac{c^2}{c^2}} which reduces to 0. ...In other words, if an object could obtain the speed of light, it would have no length at all in the direction of its motion!"

"FitzGerald's theory was put into elegant mathematical form by the Dutch physicist Hendrick Antoon Lorentz, who had independently thought of the same explanation. ...The theory came to be known as the Lorentz-FitzGerald contraction theory."

"Lorentz made an important addition to his original theory. He introduced changes in time. Clocks, he said, would be slowed down by the ether wind, and in just such a way as to make the velocity of light always measure 299,800 meters per second."

"There were many other experiments that had created a highly unsatisfactory state of affairs with respect to theory about electromagnetic phenomenoa. If the Michelson-Morley test had never been made, the special theory would still have been formulated."

"Einstein, following the steps of Ernst Mach, took a bolder view. The reason Michelson and Morley were unable to detect an ether wind, Einstein said, is simple: There is no ether wind. He did not say that there was no ether; only... [that the ether] is of no value in measuring uniform motion."

"Classical physics—the physics of Isaac Newton—made clear that if you are on a uniformly [non-accelerating] moving object, say a train car that is closed on all sides so you cannot see the scenery as you go by, there is no mechanical experiment by which you can prove that you are moving. ...If you toss a ball straight up in the air, it comes straight down again. This is exactly what would happen if you standing still."

"The special theory of relativity carries the classical relativity of Newton forward another step. It says that in addition to being unable to detect the train's motion by a mechanical experiment, it is also impossible to detect its motion by an optical experiment."

"Imagine two spaceships, A and B. There is nothing in the cosmos except these two ships. They move toward each other at uniform speed. ...To speak of an absolute motion of either ship is to say something that has no meaning. There is only one reality: a relative motion that brings the ships together at uniform speed."

"It is not possible to measure uniform motion in any absolute way."

"In the special theory of relativity, the speed of light becomes... a new absolute. ...Regardless of the motion of its source, light always moves through space with the same constant speed."

"There is no absolute time throughout the universe by which absolute simultaneity can be measured. Absolute simultaneity of distant events is a meaningless concept."

"If an astronaut traveled as fast as light his clock would stop completely."

"If two spaceships are in relative motion, an observer on each ship will measure the other ship as contracted slightly in the direction of its motion. ...The theory does not say that each ship is shorter than the other; it says that astronauts on each ship measure the other ship as shorter."

"Two ships are passing each other with uniform speed close to that of light. As they pass, a beam of light on the other ship is sent from the ceiling to the floor. There it strikes a mirror and is reflected back to the ceiling again. You will see the path of this light as a V [shape]. If you had sufficiently accurate instruments (of course no such instrument exists), you could clock the time it takes this light beam to traverse the V-shaped path. By dividing the length of the path by the time, you obtain the speed of light. ...an astronaut inside the other ship is doing the same thing [measuring his light beam's speed]. From his point of view... the light simply goes down and up along the same line, obviously a shorter distance than along the V that you observed. ...he also obtains the speed of light. ...But his light path is shorter. ...There is only one possible explanation: his clock is slower. Of course, the situation is perfectly symmetrical. If you send a beam down and up inside your ship, he will see its path as V-shaped. He will deduce that your clock is slower."

"All three variables—length, time, mass—are covered by the same Lorentz contraction [ \scriptstyle \sqrt{1-\frac{v^2}{c^2}} ]... Length and the rate of clocks vary in the same direct proportion, so the formula is the same for each. Mass... varies in the inverse proportion... \scriptstyle \frac{1} {\sqrt{1-\frac{v^2}{c^2}}}."

"If the ships could attain a relative speed equal to that of light, observers on each ship would think the other ship had shrunk to zero in length, acquired an infinite mass, and that time on the other ship had slowed to a full stop! If inertial mass did not vary in this way, then the steady application of force, such as the force supplied by rocket motors, could keep increasing a ship's velocity until it passed the speed of light. ...When the ship has contracted to one-tenth its rest length, its relativistic mass has become ten times as great. ...ten times as much force is required to produce the same increase in speed."

"The speed of light can never be reached. If it were reached, the outside observer would find that the ship had shrunk to zero length, had acquired an infinite mass, and was exerting an infinite force with its rocket motors. Astronauts inside the ship would observe no changes in themselves, but they would find the cosmos hurtling backward with the speed of light, cosmic time at a standstill, every star flattened to a disk and infinitely massive."

"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."

"Judicious Reader, There was published some years since in Rome a salutiferous edict which... imposed a seaonable silence upon the Pythagorean opinion of the mobility of the earth. ...that decree was not the production of sober scrutiny but of ill-informed passion... consultors altogether ignorant of astronomical observations ought not to clip the wings of speculative wits with rash prohibitions."

"I thought fit... to appear openly upon the theatre of the World as a witness of the naked truth."

"It is my resolution... to give foreign nations to see of this matter... And, collecting all the speculations of mine that concern the Copernican system, to let them know that... there proceed from this climate not only doctrines for the health of the soul but also ingenious discoveries for the delight of the mind."

"I have personated the Copernican... proceeding upon the hypothesis purely mathematical; striving by every artifice to represent it superior not to that of the immobility of the Earth absolutely but as it is defended by some who, claiming to profess the Peripatetic doctrine, retain of it no more than the name, and are content, foregoing the old ways, to adore shadows, not philosophizing with their own intelligence but with the sole remembrance of a few principles badly understood."

"I will endeavour to show that all experiments that can be made upon the Earth are insufficient means to conclude for its mobility but are indifferently applicable to the Earth, movable or immovable..."

"We will examine the celestial phenomena that make for the Copernican hypothesis, as if it were to prove absolutely victorious, adding by the way certain new observations which yet serve only for astronomical facility, not for natural necessity."

"I will propose an ingenious fancy. ...the unknown problem of the tides might receive some light, admitting of the Earth's motion. ...I have thought good to lay down those probabilities that would render it credible, admitting that the Earth did move."

"I hope that by these considerations the world will come to know that, if other nations have navigated more than we, we have not studied less than they; and that our returning to assert the Earth's stability and to take the contrary only for a mathematical fantasy, proceeds not from lack of acquaintance with others' ideas thereof but... from those reasons that piety, religion, the knowledge of the Divine Omnipotence, and a consciousness of the incapacity of man's understanding dictate to us."

"I chanced, many years ago, as I lived in the stupendous city of Venice, to converse frequently with the Signor Giovan Francesco Sagredo, a man of noble extraction and most acute intellect. There came thither from Florence, at the same time, Signor Filippo Salviati, whose least glory was the eminence of his blood and magnificence of his estate, a sublime intellect that knew no more exquisite pleasure than elevated speculations. In the company of these two I often discoursed of these matters before a certain Peripatetic philosopher, who seemed to have no greater obstacle in understanding the truth than the fame he had acquired by Aristotelian interpretations. Now, seeing that inexorable fate has deprived Venice and Florence of those two great lights in the early summer of their years, I did resolve... to perpetuate their lives to their honor in these leaves... Nor shall the honest Peripatetic want his place, to whom, for his excessive affection towards the commentaries of Simplicius, I thought fit, without mentioning his own name, to leave that of the author he so much respected. Let these two great souls, ever venerable to my heart, please to accept this public monument of my never dying love; and let the remembrance of their eloquence assist me in delivering to posterity the considerations that I have promised."

"There had casually taken place... several discourses at times between these gentlemen which had rather inflamed than satisfied in their minds the thirst they had for learning; whereupon they took the wise resolution to meet together for certain days in which, all other business set aside, they might betake themselves with more ordered speculation to contemplate the wonders of God in heaven and on earth."

"We have in our age new accidents and observations, and such, that I question not in the least, but if Aristotle were now alive, they would make him change his opinion; which may be easily collected from the very manner of his discoursing: For when he writeth that he e­steemeth the Heavens inalterable, &c. because no new thing was seen to be begot therein, or any old to be dissolved, he seems im­plicitely to hint unto us, that when he should see any such accident, he would hold the contrary; and confront, as indeed it is meet, sensible experiments to natural reason: for had he not made any reckoning of the senses, he would not then from the not seeing of any sensible mutation, have argued immutability."

"I do believe for certain, that he [Aristotle] first procured by help of the senses, such experiments and obser­vations as he could, to assure him as much as it was possible, of the conclusion, and that he afterwards sought out the means how to demonstrate it: For this, the usual course in demonstrative Sciences, and the reason thereof is, because when the conclusion is true, by help of resolutive Method, one may hit upon some pro­position before demonstrated, or come to some principle known per se; but if the conclusion be false, a man may proceed in in­finitum, and never meet with any truth already known; but ve­ry oft he shall meet with some impossibility or manifest absurdi­ty."

"Nor need you question but that Pythagoras a long time be­fore he found the demonstration for which he offered the Hecatomb, had been certain, that the square of the side subtending the right angle in a rectangle triangle, was equal to the square of the other two sides: and the certainty of the conclusion condu­ced not a little to the investigating of the demonstration, un­derstanding me alwayes to mean in demonstrative Sciences."

"What ever was the method of Aristotle, and whether his arguing à priori preceded sense à posteriori, or the contrary; it sufficeth that the same Aristotle preferreth (as hath been oft said) sensible ex­periments before all discourses..."

"If this of which we dispute, were some point of Law, or other part of the Studies called Humanity, wherein there is neither truth nor falshood, if we will give sufficient credit to the acutenesse of the wit, readinesse of answers, and the general practice of Writers, then he who most aboundeth in these, makes his reason more probable and plausible; but in Natural Sciences, the conclusions of which are true and necessary, and wherewith the judgment of men hath nothing to do, one is to be more cautious how he goeth about to maintain anything that is false; for a man but of an ordinary wit, if it be his good for­ tune to be of the right side, may lay a thousand Demosthenes and a thousand Aristotles at his feet. Therefore reject those hopes and conceits, wherewith you flatter yourself, that there can be any men so much more learned, read, and versed in Authors, than we, that in despite of nature, they should be able to make that become true, which is false."

"If I was demanded what my first apprehension, and pure natural reason dictated to me concerning the production of things like or unlike there above, I would alwayes reply, that they are most different, and to us altogether unimaginable, for so me thinks the riches of Nature, and the omnipotence of our Creator and Governour, do require."

"I ever accounted extraordinary madnesse that of those, who would make humane comprehension the measure of what na­ture hath a power or knowledge to effect; whereas on the con­trary there is not any the least effect in Nature, which can be fully understood by the most speculative wits in the world. This their so vain presumption of knowing all, can take beginning from no­thing, unlesse from their never having known anything; for if one hath but once onely experienced the perfect knowledg of one onely thing, and but truly tasted what it is to know, he shall perceive that of infinite other conclusions, he understands not so much as one."

"The having a perfect knowledg of nothing, maketh some believe they understand all things."

"Your discourse is very concluding; in confirmation of which we have the example of those who understand, or have known some thing, which the more knowing they are, the more they know, and freely confesse that they know little; nay, the wisest man in all Greece, and for such pronounced by the Oracle, openly professed to know that he knew nothing."

"It must be granted therefore, either that Socrates or that the Oracle itself was a lyar, that declaring him to be most wise, and he confessing that he knew himself to be most ig­norant."

"Neither one nor the other doth follow, for that both the assertions may be true. The Oracle adjudged Socrates the wi­sest of all men, whose knowledg is limited; Socrates acknowledgeth that he knew nothing in relation to absolute wisdome, which is infinite; and because of infinite, much is the same part as is little, and as is nothing (for to arrive... to the infinite number, it is all one to accumulate thousands, tens, or ciphers,) therefore Socrates well perceived his wisdom to be nothing, in comparison of the infinite knowledg which he wanted. But yet, because there is some knowledg found amongst men, and this not equally shared to all, Socrates might have a greater share thereof than others, and therefore verified the answer of the Oracle."

"I think I very well understand this particular amongst men, Simplicius, there is a power of operating, but not equally dispensed to all; and it is without question that the power of an Emperor is far greater than that of a private person; but, both this and that are nothing in comparison of the Divine Omnipotence. Amongst men, there are some that better understand Agriculture than many others; but the knowledg of planting a Vine in a trench, what hath it to do with the knowledg of ma­king it to sprout forth, to attract nourishment, to select this good part from that other, for to make thereof leaves, another to make sprouts, another to make grapes, another to make raisins, ano­ther to make the huskes of them, which are the works of most wise Nature? This is one only particular act of the innumerable, which Nature doth, and in it alone is discovered an infinite wisdom, so that Divine Wisdom may be concluded to be infinitely infinite."

"Do we not say that the judicious discovering of a most lovely Statua in a piece of Marble, hath sublimated the wit of Buonarruotti far above the vulgar wits of other men? And yet this work is onely the imitation of a meer aptitude and disposition of exteriour and superficial mem­bers of an immoveable man; but what is it in comparison of a man made by nature, composed of as many exteriour and inte­riour members, of so many muscles, tendons, nerves, bones, which serve to so many and sundry motions? but what shall we say of the senses, and of the powers of the soul, and lastly, of the understanding? May we not say, and that with reason, that the structure of a Statue falls far short of the formation of a living man, yea more of a contemptible worm?"

"I must have recourse to a Philosophical distinction, and say that the understanding is to be taken two ways, that is intensivè, or extensivè; and that extensive, that is, as to the multitude of intel­ligibles, which are infinite, the understanding of man is as nothing, though he should understand a thousand propositions; for that a thousand, in respect of infinity is but as a cypher: but taking the understanding intensive, (in as much as that term imports) intensively, that is, perfectly some propositions, I say, that humane wis­dom understandeth some propositions so perfectly, and is as abso­lutely certain thereof, as Nature herself; and such are the pure Mathematical sciences, to wit, Geometry and Arithmetick: in which Divine Wisdom knows infinite more propositions, because it knows them all; but I believe that the knowledge of those few compre­hended by humane understanding, equalleth the divine, as to the certainty objectivè, for that it arriveth to comprehend the neces­sity thereof, than which there can be no greater certainty."

"As to the truth, of which Mathematical demonstrations give us the knowledge, it is the same, which the divine wisdom knoweth; but this I must grant you, that the manner whereby God knoweth the infinite propo­sitions, of which we understand some few... his is done at one single thought or intuition; and whereas we... beginning from one of the most simple, and taking that for its definition, do proceed with argumentation to another..."

"What other, is that proposition, that the square of the side subtending the right angle in any triangle, is equal to the squares of the other two, which include it, but onely the Paralellograms being upon common bases, and between parallels equal amongst themselves?"

"These inferences, which our intellect apprehendeth with time and a gradual motion, the Divine Wisdom, like light, penetrateth in an instant, which is the same as to say, hath them alwayes pre­sent..."

"Our understanding, both as to the manner and the multitude of the things comprehended by us, is infinitely surpast by the Divine Wisdom; but yet I do not so vilifie it, as to repute it absolutely nothing; yea rather, when I consider how many and how great misteries men have understood, discovered, and contrived, I very plainly know and understand the mind of man to be one of the works, yea one of the most ex­cellent works of God."

"If I behold a statue of some excellent master, I say with my self: "When wilt thou know how to chizzle away the refuse of a piece of Marble, and discover so lovely a figure as lyeth hid therein? When wilt thou mix and spread so many colors upon a Cloth, or Wall, and represent therewith all visible objects, like a Michael Angelo, a Raphaello, or a Tizvano? If I behold what invention men have had in comparting Musical intervals, in establishing Precepts and Rules for the management thereof with admirable delight to the ear, when shall I cease my astonishment? What shall I say of such and so various instruments of that Art? The reading of excellent Poets, with what admiration doth it swell anyone who attentively considereth the invention of concepts and their explanation? What shall we say of Architecture? What of Navigation? But, above all other stupendous inventions, what sublimity of mind was that in him, that imagined to himself to find out a way to communicate his most secret thoughts to any other person, though very far distant from him either in time or place, speaking with those that are in the Indies, speaking to those who are not yet born, nor shall be this thousand, or ten thousand years? And with how much facility? but by the various collection of twenty-four little letters upon a paper? Let this be the Seal of all the admirable inventions of man and the close of our Discourse for this day. For the warmer hours being past, I suppose that Salviatus hath a desire to go and take the cool air in his Gondelo..."

"The sum of yesterdayes conferences were an examination of the Principles of Ptolomy and Copernicus, and which of their opinions is the more probable and rational; that, which affirmeth the sub­stance of the Cœlestial bodies to be ingenerable, incorruptible, un­alterable, impassible, and in a word, exempt from all kind of change, save that of local, and therefore to be a fifth essence, quite different from this of our Elementary bodies, which are generable, corrup­tible, alterable, &c. or else the other, which taking away such deformity from the parts of the World, holdeth the Earth to en­joy the same perfections as the other integral bodies of the universe; and esteemeth it a moveable and erratick Globe, no lesse than the Moon, Jupiter, Venus, or any other Planet."

"Do you question whether Aristotle, had he but seen the novelties discovered in Hea­ven, would not have changed his opinion, amended his Books, and embraced the more sensible Doctrine; rejecting those silly Gulls, which too scrupulously, go about to defend what ever he hath said; not considering, that if Aristotle were such a one as they fancy him to themselves, he would be a man of an untracta­ble wit, an obstinate mind, a barbarous soul, a stubborn will, that accounting all men else but as silly sheep, would have his Oracles preferred before the Senses, Experience, and Nature herself?"

"Because it is more easie for a man to sculk under anothers shield than to shew himself openly, they tremble, and are affraid to stir one step from him; and rather than they will admit some alterations in the Heaven of Aristotle, they will impertinently de­ny those they behold in the Heaven of Nature."

"We need a Guid in unknown and uncouth wayes, but in champion places, and open plains, the blind only stand in need of a Leader; and for such, it is better that they stay at home. But he that hath eyes in his head, and in his mind, him should a man choose for his Guid. Yet mistake me not, thinking that I speak this, for that I am against hearing of Aristotle; for on the contrary, I commend the reading, and diligently studying of him; and onely blame the servile giving ones self up a slave unto him, so, as blindly to subscribe to what ever he delivers, and without search of any farther reason thereof, to receive the same for an inviolable decree. Which is an abuse, that carrieth with it ano­ther great inconvenience, to wit, that others will no longer take pains to understand the validity of his Demonstrations. And what is more shameful, than in the middest of publique disputes, whilest one person is treating of demonstrable conclusions, to hear another interpose with a passage of Aristotle, and not sel­dome writ to quite another purpose, and with that to stop the mouth of his opponent? But if you will continue to study in this manner, I would have you lay aside the name of Philosophers; and call your selves either Historians or Doctors of Memory, for it is not sit, that those who never philosophate, should usurp the honourable title of Philosophers."

"Therefore Simplicius, come either with arguments and demonstrations of your own, or of Aristotle, and bring us no more Texts and na­ked authorities, for our disputes are about the Sensible World, and not one of Paper."

"Whatsoever motion may be ascribed to the Earth, it is necessary that it be to us, (as inhabitants upon it, and conse­quently partakers of the same) altogether imperceptible, and as if it were not at all, so long as we have regard onely to terrestrial things..."

"If we consider onely the immense magnitude of the Starry Sphere, compared to the smalness of the Terrestrial Globe, contained therein so many mil­lions of times; and moreover weigh the velocity of the motion which must in a day and night make an entire revolution thereof, I cannot perswade my self, that there is any man who believes it more reasonable and credible, that the Cœlestial Sphere turneth round, and the Terrestrial Globe stands still."

"He which should hold it more ra­tional to make the whole Universe move, and thereby to salve the Earths mobility, is more unreasonable than he that being got to the top of your Turret, should desire, to the end onely that he might behold the City, and the Fields about it, that the whole Country might turn round, that so he might not be put to the trouble to stir his head."

"Motion is so far Motion, and as Motion operateth, by how far it hath relation to things which want Motion: but in those things which all equally partake thereof it hath nothing to do, and is as if it never were. And thus the Merchandises with which a ship is laden, so far move, by how far leaving London, they pass by France, Spain, Italy, and sail to Aleppo, which London, France, Spain &c. stand still, not moving with the ship: but as to the Chests, Bales and other Parcels, wherewith the ship is stow'd and and laden, and in respect of the ship it self, the Motion from Lon­don to Syria is as much as nothing; and nothing altereth the re­lation which is between them: and this, because it is common to all, and is participated by all alike: and of the Cargo which is in the ship, if a Bale were romag'd from a Chest but one inch onely, this alone would be in that Cargo, a greater Motion in respect of the Chest, than the whole Voyage of above three thousand miles, made by them as they were stived together."

"I hold it [the concept or relative motion] to be much more antient: and suspect that Aristotle in receiving it from some good School, did not fully under­stand it, and that therefore, having delivered it with some altera­tion, it hath been an occasion of confusion amongst those, who would defend whatever he saith. And when he writ, that what­ soever moveth, doth move upon something immoveable, I suppose that he equivocated, and meant, that whatever moveth, moveth in respect to something immoveable; which proposition admitteth no doubt, and the other many."

"We having divided the Universe into two parts, one of which is necessarily moveable, and the other immoveable; for the obtaining of whatsoever may depend upon, or be required from such a motion, it may as well be done by making the Earth alone, as by making all the rest of the World to move: for that the operation of such a motion consists in nothing else, save in the relation or habitude which is between the Cœlestial Bodies, and the Earth, the which relation [with an exchange in the two terms] is all that is changed. Now if for the obtaining of the same effect ad unguem [to a nail's thickness], it be all one whe­ther the Earth alone moveth, the rest of the Universe standing still; or that, the Earth onely standing still, the whole Universe moveth with one and the same motion; who would believe, that Nature (which by common consent, doth not that by many things, which may be done by few) hath chosen to make an innumerable number of most vast bodies move, and that with an unconceivable velocity, to perform that, which might be done by the moderate motion of one alone about its own Centre? ...which self same effect falls out exactly in the same manner, if, without troubling so great a part of the Universe."

"Nature never doth that by many things, which may be done by a few."

"If you will ascribe this Great Motion to Heaven, you must of necessity make it contrary to the particular motion of all the Orbs of the Planets, each of which without controversie hath its peculiar motion from the West towards the East, and this but very easie and moderate: and then you make them to be hurried to the contrary part, i. e. from East to West, by this most furious diurnal motion: whereas, on the contrary, making the Earth to move in it self, the contrariety of motions is taken away, and the onely motion from West to East is accom­modated to all appearances, and exactly satisfieth every Phœno­menon."

"The Ptolomaique Hypothesis... most unreasonably confoundeth the order, which we assuredly see to be amongst those Cœlestial Bodies, the circumgyration of which is not questionable, but most certain. And that Order is, that according as an Orb is greater, it finisheth its revolution in a longer time, and the lesser, in shorter. ...but if you would have the Earth immoveable, it is necessary, that when you have past from the short period of the Moon, to the others successively bigger, until you come to that of Mars in two years, and from thence to that of the bigger Sphere of Jupiter in twelve years, and from this to the other yet bigger of Saturn, whose period is of thirty years, it is necessary, I say, that you passe to another Sphere incomparably greater still than that, and make this to ac­complish an entire revolution in twenty four hours. ...But the motion of the Earth being granted, the order of the pe­riods will be exactly observed, and from the very slow Sphere of Saturn, we come to the fixed Stars, which are wholly immovea­ble."

"If the Starry Sphere be supposed moveable [there is an] immense disparity between the motions of those stars themselves; of which some would come to move most swiftly in most vast cir­cles, others most slowly in circles very small, according as those or these should be found nearer, or more remote from the Poles. ... And not onely the magnitudes of the circles, and conse­quently the velocity of the motions of these Stars, shall be most different from the circles and motions of those others, but... the self-same Stars shall successively vary its circles and velocities: For that those, which two thousand years since were in the Equinoctial, and consequently did with their motion describe very vast cir­cles, being in our dayes many degrees distant from thence, must of necessity become more slow of motion, and be reduced to move in lesser circles, and it is not altogether impossible but that a time may come, in which some of them which in aforetime had continually moved, shall be reduced by uniting with the Pole, to a state of rest, and then after some time of cessation, shall return to their motion again."

"No thought can comprehend what ought to be the solidity of that immense Sphere, whose depth so stedfastly holdeth fast such a multitude of Stars... Or else, supposing the Heavens to be fluid, as we are with more reason to believe, so as that every Star wandereth to and fro in it, by wayes of its own, what rules shall regulate their motions, and to what pur­pose, so, as that being beheld from the Earth, they appear as if they were made by one onely Sphere? ...nor can I see how the Earth, a pendent body, and equilibrated upon its centre, exposed indif­ferently to either motion or rest, and environed with a liquid am­bient, should not yield also as the rest, and be carried about."

"It sufficeth not to know that it [the nature of a body falling downwards] is streight, but its requi­site to know whether it be uniform, or irregular; that is, whe­ther it maintain alwayes one and the same velocity, or else goeth retarding or accelerating. ...Neither doth this suffice, but its requisite to know ac­cording to what proportion such accelleration is made; a Pro­blem, that I believe was never hitherto understood by any Phi­losopher or Mathematician; although Philosophers, and particu­larly the Peripateticks, have writ great and entire Volumes, touching motion."

"We speak of the peice of Ordinance mounted per­pendicular to the Horizon, that is, of a shot towards our vertical point, and to conclude, of the return of the ball by the same line unto the same peice, though that in the long time which it is se­parated from the peice, the earth hath transported it many miles towards the East; now it seemeth, that the ball ought to fall a like distance from the peice towards the West; the which doth not happen: therefore the peice without having been moved did stay expecting the same. The answer is the same with that of the stone falling from the Tower; and all the fallacy, and equivocati­on consisteth in supposing still for true, that which is in question; for the Opponent hath it still fixed in his conceit that the ball departs from its rest, being discharged by the fire from the piece; and the departing from the state of rest, cannot be, unlesse the immobility of the Terrestrial Globe be presupposed, which is the conclusion of that was in dispute; Therefore, I reply, that those who make the Earth moveable, answer, that the piece, and the ball that is in it, partake of the same motion with the Earth; nay that they have this together with her from nature; and that therefore the ball departs in no other manner from its quiescence, but conjoyned with its motion about the cen­tre, the which by its projection upwards, is neither taken away, nor hindered; and in this manner following, the universal motion of the Earth towards the East, it alwayes keepeth perpendicular over the said piece, as well in its rise as in its return. And the same you see to ensue, in making the experiment in a ship with a bullet shot upwards perpendicularly with a Crosse-bow, which returneth to the same place whether the ship doth move, or stand still."

"I would not do so much wrong to Plato, but yet I may truly say with Aristotle, that he too much lost himself in, and too much doted upon that, his Geometry: for that in conclusion these Mathematical subtilties, Salviatus, are true in abstract, but applied to sensible and Physical matter, they hold not good. For the Mathematicians will very well demonstrate for example, that Sphæra tangit planum in puncto [the sphere touches the plane at the point]; a position like to that in dispute, but when one cometh to the matter, things succeed quite another way. And so I may say of these angles of contact, and these proportions; which all evaporate into Air, when they are applied to things material and sensible."

"I would be loth to leave you in that other which you hold, namely, that a material Sphere doth not touch a plain in one sole point: and I could wish some few hours conversation with some persons conversant in Geometry, might make you a little more intelligent amongst those who know nothing thereof."

"The truth sometimes gaines strength by con­tradiction."

"Things are ex­actly the same in abstract as in con­crete."

"Contact in a sin­gle point is not pe­culiar to the per­fect Spheres onely, but belongeth to all curved figures."

"It is more diffi­cult to find Figures that touch with a part of their sur­face, than in one sole point."

"If any figure can be given to a Solid, the Spherical is the easi­est of all others, as it is likewise the most simple, and holdeth the same place amongst solid figures, as the Circle holdeth amongst the superficial. The description of which Circle, as being more ea­sie than all the rest, hath alone been judged by Mathematicians worthy to be put amongst the postulata belonging to the descri­ption of all other figures."

"The formation of the Sphere is so very easie, that if in a plain plate of hard metal you take an empty or hollow circle, within which any Solid goeth casually re­volving that was before but grosly rounded, it shall, without any other artifice be reduced to a Spherical figure, as perfect as is pos­sible for it to be; provided, that that same Solid be not lesse than the Sphere that would passe thorow that Circle. And that which is yet more worthy of our consideration is, that within the self-same incavity one may form Spheres of several magnitudes."

"The circular Fi­gure only is placed amongst the postu­lata of Mathema­ticians."

"The Sphericall Figure is easier to be made than any other. ...Sphericall Fi­gures of sundry magnitudes may be made with one onely instrument."

"Above all things it must be considered, that the motion of descending grave bodies is not uniform, but departing from rest they go continually accelerating.. But this general notion is of no avail, if it be not known according to what proportion this increase of velocity is made; a conclusion that hath been until our times unknown to all Philoso­phers; and was first found out & demonstrated by the Academick [Galileo], our common friend, who in some of his writings not yet publish­ed... he proveth, how that the acceleration of the right mo­tion of grave bodies, is made according to the numbers uneven beginning ab unitate [from unity], that is, any number of equal times being assigned, if in the first time the moveable departing from rest shall have passed such a certain space, as for example, an ell, in the se­cond time it shall have passed three ells, in the third five, in the fourth seven, and so progressively, according to the following odd numbers; which in short is the same, as if... the [sums of] spaces passed are to each other, as the squares of their times."

"You, Simplicius, as I believe, have gone by boat many times to Padoua, and if you will confess the truth, you never felt in your self the participation of that motion, unless when the boat running a-ground, or encoun­tring some obstacle, did stop, and that you with the other Passen­gers being taken on a sudden, were with danger over-set. It would be necessary that the Terrestrial Globe should meet with some rub that might arrest it, for I assure you, that then you would discern the impulse residing in you, when it should toss you up towards the Stars."

"It may be collected how easily one may be deceived by the bare appearance, or, if you will, representation of the sense. And the accident is, the Moons seeming to follow those that walk the streets in the night, with a pace equal to theirs, whilst they see it go gli­ding along the Roofs of houses, upon which it sheweth just like a cat, that really running along the ridges of houses, leaveth them behind. An appearance that, did not reason interpose, would but too manifestly delude the sight."

"I have twice or thrice observed in the discourses of this Authour, that to prove that a thing is so, or so, he still alledgeth, that in that manner it is conformable with our understanding; or that otherwise we should never be able to conceive of it; or that the Criterium of Philosophy would be overthrown. As if that na­ture had first made mens brains, and then disposed all things in conformity to the capacity of their intellects. But I incline rather to think that Nature first made the things themselves, as she best liked, and afterwards framed the reason of men capable of con­ceiving (though not without great pains) some part of her se­crets."

"Nature first made things as she pleased, and after­wards capacitated mens understand­ings for conceiving of them."

"Although I might very rationally put it in dispute, whe­ther there be any such centre in nature, or no; being that neither you nor any one else hath ever proved, whether the World be fi­nite and figurate, or else infinite and interminate; yet nevertheless granting you, for the present, that it is finite, and of a terminate Spherical Figure, and that thereupon it hath its centre; it will be requisite to see how credible it is that the Earth, and not rather some other body, doth possesse the said centre."

"If I deny his [Aristotle's] assumption, to wit, that the Universe is moveable, all his demonstrations come to nothing, for he onely proveth the Universe to be finite and terminate, for [by assuming] that it is moveable."

"I do not ask the Peripateticks... they, as observant and humble vassals of Aristotle, would deny all the ex­periments and all the observations in the World, nay, would also refuse to see them, that they might not be forced to acknowledg them, and would say that the World stands as Aristotle writeth, and not as nature will have it, for depriving them of the shield of his Authority, with what do you think they would appear in the field?"

"Now if it were true that the centre of the World is the same about which... the Planets, move, it is most certain that it is not the Earth, but the Sun rather that is fixed in the centre of the World. So that as to this first simple and general apprehension, the middle place belongeth to the Sun, and the Earth is as far remote from the centre, as it is from that same Sun."

"The seeing all the Planets one while neerer and ano­ther while farther off from the Earth with so great differences, that for example, Venus when it is at the farthest, is six times more remote from us, than when it is neerest, and Mars riseth almost eight times as high at one time as at another."

"It is argued in the three superiour planets Mars, Jupi­ter, and Saturn, in that we find them alwayes neerest to the Earth when they are in opposition to the Sun, and farthest off when they are towards the conjunction, and this approximatian and recession importeth thus much that Mars neer at hand, ap­peareth very neer 60 times greater than when it is remote. As to Venus in the next place, and to Mercury, we are certain that they revolve about the Sun, in that they never move far from him, and in that we see them one while above and another while below it, as the mutations of figure in Venus necessarily argueth."

"The annual mo­tion of the Earth mixing with the motions of the o­ther Planets pro­duce extravagant appearances."

"As to the operation of the diurnal motion upon the Celestial bodies, it neither was, nor can be other, than to make the Universe seem to run precipitately the contrary way; but this annual motion intermixing with the particular motions of all the planets, produceth very many ex­travagancies, which have disarmed and non-plust all the greatest Scholars in the World."

"The centre of the Celestial conversions of the five planets Saturn, Jupiter, Mars, Venus and Mercury, is the Sun; and shall be likewise the centre of the motion of the Earth, if we do but succeed in our attempt of placing it in Hea­ven."

"My admiration, Sagredus, is very different from yours, you wonder that so few are followers of the Pythagorean Opinion; and I am amazed how there could be any yet left till now that do em­brace and follow it: Nor can I sufficiently admire the eminencie of those mens wits that have received and held it to be true, and with the sprightlinesse of their judgements offered such violence to their own sences, as that they have been able to prefer that which their reason dictated to them, to that which sensible experiments re­presented most manifestly on the contrary. That the reasons against the Diurnal virtiginous revolution of the Earth by you already ex­amined, do carry great probability with them, we have already seen; as also that the Ptolomaicks, and Aristotelicks, with all their Sectators did receive them for true, is indeed a very great argument of their efficacie; but those experiments which apertly contradict the annual motion, are of yet so much more manifestly repugnant, that (I say it again) I cannot find any bounds for my admiration, how that reason was able in Aristarchus and Copernicus, to com­mìt such a rape upon their Sences, as in despight thereof, to make her self mistress of their credulity."

"Although Astronomy in the courses of many ages hath made a great progress in discovering the constitution and motions of the Celestial bodies, yet is it not hitherto arrived at that height, but that very many things remain undecided, and haply many others also undiscovered."

"To say... that the motion of the Earth meeting with the motion of the Lunar Orb, the concurrence of them occasioneth the Ebbing and Flowing [of the seas], is an absolute vanity, not onely be­cause it is not exprest, nor seen how it should so happen, but the falsity is obvious, for that the Revolution of the Earth is not con­trary to the motion of the Moon, but is towards the same way. So that all that hath been hitherto said, and imagined by others, is, in my judgment, altogether invalid. But amongst all the famous men that have philosophated upon this admirable effect of Nature, I more wonder at Kepler than any of the rest, who being of a free and piercing wit, and having the motion ascri­bed to the Earth, before him, hath for all that given his ear and assent to the Moons predominancy over the Water, and to oc­cult properties, and such like trifles."

"We have now, from these four dayes Dis­course, great attestations, in favour of the Copernican Systeme, amongst which these three taken: the first, from the Stations and Retrogradations of the Planets, and from their approaches and recessions from the Earth; the second, from the Suns revolving in it self, and from what is observed in its spots; the third, from the Ebbing and Flowing of the Sea do shew very rational and concluding."

"If, at more leasure go­ing over the things again that have been alledged you meet with any doubts, or scruples not well resolved, you will excuse my oversight, as well for the novelty of the Notion, as for the weaknesse of my wit, as also for the grandure of the Subject, as also finally, because I do not, nor have pretended to that as­sent from others, which I my self do not give to this conceit, which I could very easily grant to be a Chymæra, and a meer paradox..."

"And you Sagredus, although in the Discourses past you have many times, with great applause, declared, that you were pleased with some of my conjectures, yet do I believe, that that was in part more occasioned by the novelty than by the cer­tainty of them, but much more by your courtesie, which did think and desire, by its assent, to procure me that content which we naturally use to take in the approbation and applause of our own matters... your civility hath obliged me to you..."

"So am I also pleased with the ingenuity of Simplicius. Nay, his constancy in maintaining the Doctrine of his Master, with so much strength & undauntedness, hath made me much to love him. ...I ask pardon, if I have sometimes moved him with my too bold and resolute speaking: and let him be assured that I have not done the same out of any inducement of sinister affection, but onely to give him occasion to set before us more lofty fancies that might make me the more knowing."

"As for the past Discourses, and particulatly in this last, of the reason of the Ebbing and Flowing of the Sea, I do not, to speak the truth, very well apprehend the same, but by that slight Idea, what e­ver it be, that I have formed thereof to my self, I confesse that your conceit seemeth to me far more ingenuous than any of all those that I ever heard besides, but yet neverthelesse I esteem it not true and concluding: but keeping alwayes before the eyes of my mind a solid Doctrine... I know that both you being asked, Whether God, by his infinite Power and Wisdome might confer upon the Element of Water the reciprocal motion which we observe in the same in any other way, than by making the containing Vessel to move; I know, I say, that you will answer, that he might, and knew how to have done the same many wayes, and those unimaginable to our shallow understanding: upon which I forthwith conclude, that this being granted, it would be an extravagant boldnesse for any one to goe about to limit and confine the Divine Power and Wisdome to some one particular conjecture of his own."

"This of yours is admirable, and truly Angelical Do­ctrine, to which very exactly that other accords, in like manner divine, which whilst it giveth us leave to dispute, touching the constitution of the World, addeth withall (perhaps to the end, that the exercise of the minds of men might neither be discou­raged, nor made bold) that we cannot find out the works made by his hands. Let therefore the Disquisition permitted and or­dain'd us by God, assist us in the knowing, and so much more admiring his greatnesse, by how much lesse we finde our selves too dull to penetrate the profound Abysses of his infinite Wis­dome."

"Above all, I shall very impatiently wait to hear the Elements of the new Science of our Academick about the natural and violent local Motions. And in the mean time, we may, according to our custome, spend an hour in taking the Air in the Gondola that waiteth for us. FINIS."

"The name of physical science... is often applied in a more or less restricted manner to those branches of science in which the phenomena considered are of the simplest and most abstract kind, excluding the consideration of the more complex phenomena, such as those observed in living beings."

"The first part of physical science relates to the relative position and motion of bodies."

"In all scientific procedure we begin by marking out a certain region or subject as the field of our investigations. To this we must confine our attention, leaving the rest of the universe out of account till we have completed the investigation in which we are engaged."

"A knowledge of the configuration of the system at a given instant implies a knowledge of the position of every point of the system with respect to every other point at that instant."

"The model or diagram is supposed to resemble the material system only in form, not necessarily in any other respect."

"A body so small that, for the purposes of our investigation, the distances between its different parts may be neglected is called a material particle. ...But we cannot treat them as material particles when we investigate their rotation. Even an atom when, we consider it as capable of rotation, must be regarded as consisting of many material particles. The diagram of a material particle is of course a mathematical point, which has no configuration."

"As indicating an operation \overline{AB} is called a Vector, and the operation is completely defined by the direction and distance of the transference. ...All vectors ...are regarded as equal which are parallel (and drawn towards the same parts) and of the same magnitude."

"It appears then that the distance between one thing and another does not depend on any material thing between them, as Descartes seems to assert when he says that if that which is in a hollow vessel were taken out of it without anything entering to fill its place, the sides of the vessel, having nothing between them, would be in contact. This assertion is grounded on the dogma of Descartes, that the extension in length, breadth, and depth which constitute space is the sole essential property of matter. "The nature of matter," he tells us, "or of body considered generally, does not consist in a thing being hard, or heavy, or colored, but only in its being extended in length, breadth, and depth." By thus confounding the properties of matter with those of space he arrives at the logical conclusion, that if the matter within a vessel could be entirely removed the space within the vessel would no longer exist. In fact he assumes that all space must be always full of matter."

"The primary property of matter was indeed distinctly announced by Descartes in what he calls the "First Law of Nature": "That every individual thing, so far as in it lies, perseveres in the same state, whether of motion or of rest.""

"Descartes... never attained to a full understanding of his own words (quantum in se est), and so fell back on his original confusion of matter with space—space being, according to him, the only form of substance, and all existing things but affections of space. This error... forms one of the ultimate foundations of the system of Spinoza."

"I would advise those who study any system of metaphysics to examine carefully that part of it which deals with physical ideas."

"We shall find it more conducive to scientific progress to recognise, with Newton, the ideas of time and space as distinct, at least in thought, from that of the material system whose relations these ideas serve to co-ordinate."

"The idea of Time in its most primitive form is probably the recognition of an order of sequence in our states of consciousness."

"Absolute, true, and mathematical Time is conceived by Newton as flowing at a constant rate, unaffected by the speed or slowness of the motions of material things. It is also called duration."

"Relative, apparent, and common time is duration as estimated by the motion of bodies, as by days months and years."

"There is nothing to distinguish one portion of time from another except by the different events which occur in them, so there is nothing to distinguish one part of space from another except its relation to the place of material bodies. We cannot describe the time of an event except by reference to some other event, or the place of a body except by reference to some other body. All our knowledge both of time and place is essentially relative."

"When a man has acquired the habit of putting words together, without troubling himself to form the thoughts which ought to correspond to them, it is easy for him to frame an antithesis between this relative knowledge and a so-called absolute knowlege, and to point out our ignorance of the absolute position of a point as an instance of the limitation of our faculties. Any one, however, who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge."

"There is a maxim which is often quoted, that "The same causes will always produce the same effects." To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more than once, so that the causes and effects cannot be the same in all respects. What is really meant is that if the causes differ only as regards the absolute time or the absolute place at which the event occurs, so likewise will the effects."

"The following statement... appears... and more capable of application to particular cases: "The difference between one event and another does not depend on the mere difference of the times or the places at which they occur, but only on differences in the nature configuration or motion of the bodies concerned." It follows from this, that if an event has occurred at a given time and place it is possible for an event exactly similar to occur at any other time and place."

"There is another maxim... which asserts "That like causes produce like effects." This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. ...there are ...cases in which a small initial variation may produce a very great change in the final state of the system..."

"The motion of a material particle which has continuous existence in time and space is the type and exemplar of every form of continuity."

"If the motion of a particle is such that in equal intervals of time, however short, the displacements of the particles are equal and in the same direction, the particle is said to move with constant velocity. ...the path of the body will be a straight line, and the length of any part of the path will be proportional to the time of describing it."

"When the velocity of a particle is not constant, its value at any given instant is measured by the distance which would be described in unit of time by a body having the same velocity as that which the particle has at that instant."

"The phrase absolute velocity has as little meaning as absolute position. It is better, therefore, not to distinguish any point in the diagram of velocity as the origin, but to regard the diagram as expressing the relations of all the velocities without defining the absolute value of any one of them."

"The phrase "at rest" means in ordinary language "having no velocity with respect to that on which the body stands," as, for instance, the surface of the earth or the deck of a ship. It cannot be made to mean more than this."

"It is... unscientific to distinguish between rest and motion, as between two different states of a body in itself, since it is impossible to speak of a body being at rest or in motion except with reference, expressed or implied, to some other body."

"The word Acceleration is here used to denote any change in the velocity, whether that change be an increase, a diminution, or a change of direction."

"The process of constructing the diagram of total accelerations, by a comparison of the initial and final diagrams of velocities, is the same as that by which the diagram of displacements was constructed by a comparison of the initial and final diagrams of configuration."

"If the rate of acceleration is constant, it is measured by the total acceleration in a unit of time. If the rate of acceleration is variable, its value at a given instant is measured by the total acceleration in unit of time of a point whose acceleration is constant and equal to that of the particle at the given instant."

"The method of deducing the rate of acceleration from a knowledge of the total acceleration in any given time is precisely analogous to that by which the velocity at any instant is deduced from a knowledge of the displacement in any given time."

"In future... when we use the word acceleration without qualification, we mean... the rate of acceleration."

"In the diagram of configuration we use the capital letters A, B, C, &c., to indicate the relative position of the bodies of the system; in the diagram of velocities we use the small letters, a, b, c, to indicate the relative velocities of these bodies; and in the diagram of accelerations we use the Greek letters α, β, γ, to indicate their relative accelerations."

"Acceleration, like position and velocity, is a relative term and cannot be interpreted absolutely."

"If we take into account the whole phenomenon of the action between... two portions of matter, we call it Stress. This stress, according to the mode in which it acts, may be described as Attraction, Repulsion, Tension, Pressure, Shearing stress, Torsion, &c."

"External or "impressed" force considered with reference to its effect—namely, the alteration of the motions of bodies—is completely defined and described in Newton's three laws of motion. The first law tells us under what conditions there is no external force. The second shows us how to measure the force when it exists. The third compares the two aspects of the action between two bodies as it affects the one body or the other."

"It may thus be shown that the denial of Newton's [first] law is in contradiction to the only system of consistent doctrine about space and time which the human mind has been able to form."

"If a carriage in a railway train moves with constant velocity in a straight line, the external forces which act on it—such as the traction of the carriage in front of it pulling it forwards, the drag of that behind it, the friction of the rails, the resistance of the air acting backwards, the weight of the carriage acting downwards, and the pressure of the rails acting upwards—must exactly balance each other."

"We know that a thread of caoutchoucz` when stretched beyond a certain length exerts a tension which increases the more the thread is elongated. On account of this property the thread is said to be elastic. When the same thread is drawn out to the same length it will, if its properties remain constant, exert the same tension. Now let one end of the thread be fastened to a body, M, not acted on by any other force than the tension of the thread, and let the other end be held in the hand and pulled in a constant direction with a force just sufficient to elongate the thread to a given length; the force acting on the body will then be of a given intensity, F. The body will acquire velocity, and at the end of a unit of time this velocity will have a certain value, V. If the same string be fastened to another body, N, and pulled as in the former case, so that the elongation is the same as before, the force acting on the body will be the same, and if the velocity communicated to N in a unit of time is also the same, namely, V, then we say of the two bodies M and N that they consist of equal quantities of matter, or, in modern language, they are equal in mass."

"If equal quantities of the substance produce equal effects of any kind, we may employ these effects as measures of the quantity of the substance. For instance, if we are dealing with sulphuric acid of uniform strength, we may estimate the quantity of a given portion of it in several different ways We may weigh it, we may pour it into a graduated vessel, and so measure its volume, or we may ascertain how much of a standard solution of potash it will neutralize."

"In abstract dynamics, however, matter is considered under no other aspect than as that which can have its motion changed by the application of force. Hence any two bodies are of equal mass if equal forces applied to these bodies produce, in equal times, equal changes of velocity. This is the only definition of equal masses which can be admitted in dynamics, and it is applicable to all material bodies whatever they may be."

"It is an observed fact that bodies of equal mass, placed in the same position relative to the earth, are attracted equally towards the earth whatever they are made of; but this is not a doctrine of abstract dynamics founded on axiomatic principles, but a fact discovered by observation, and verified by the careful experiments of Newton on the times of oscillation of hollow wooden balls suspended by strings of the same length, and containing gold, silver, lead, glass, sand, common salt, wood, water, and wheat. ...measuring the length of a pendulum which swings seconds."

"The weight of a gramme—that is to say, the force which causes it to fall—may be asserted by letting it fall freely. At the end of one second its velocity will be about 981 centimeters per second if the experiment be in Britain."

"A pound or a gramme is greater in high latitudes than near the equator, and therefore a measurement of force in gravitation measure is of no scientific value unless it is stated in what part of the world the measurement was made."

"Let a unit of force act for unit of time upon unit of mass. The velocity of the mass will be changed, and the total acceleration will be unity in the direction of the force. The magnitude and direction of this total acceleration will be the same whether the body is originally at rest or in motion. For the expression "at rest" has no scientific meaning, and the expression "in motion," if it refers to relative motion, may mean anything, and if it refers to absolute motion can only refer to some medium fixed in space."

"To discover the existence of a medium, and to determine our velocity with respect to it by observation on the motion of bodies, is a legitimate scientific inquiry, but supposing all this done we should have discovered, not an error in the laws of motion, but a new fact in science."

"The effect of a given force on a body does not depend on the motion of that body. Neither is it affected by the simultaneous action of other forces on the body."

"We arrive at the following form of the [Newton's second] law. When any number of forces act on a body, the acceleration due to each force is the same in direction and magnitude as if the others had not been in action."

"When a force, constant in direction and magnitude, acts on a body, the total acceleration is proportional to the interval of time during which the force acts. For if the force produces a certain total acceleration in a given interval of time, it will produce an equal total acceleration in the next, because the effect of the force does not depend upon the velocity which the body has when the force acts on it."

"The total acceleration in a given time is proportional to the force. For if several equal forces act in the same direction on the same body in the same direction, each produces its effect independently of the others."

"The total effect of a force in communicating velocity to a body is therefore proportional to the force and to the time during which it acts conjointly."

"The product of the time of action of a force into its intensity if it is constant, or its mean intensity if it is variable ,is called the Impulse of the force."

"There are certain cases in which a force acts for so short a time that it is difficult to estimate either its intensity or the time during which it acts. But it is comparatively easy to measure the effect of the force in altering the motion of the body on which it acts..."

"The word impulse was originally used to denote the effect of a force of short duration, such as that of a hammer striking a nail. There is no essential difference, however, between this case and any other case of the action of force."

"If a number of equal forces act in the same direction, each on a unit of mass, the different masses will all move in the same manner, and may be joined together into one body without altering the phenomenon. The velocity of the whole body is equal to that produced by one of the forces acting on a unit of mass. Hence the force required to produce a given change of velocity in a given time is proportional to the number of units of mass of which the body consists."

"The momentum of any body is... measured in terms of the momentum of unit of mass moving with unit of velocity which is taken as the unit of momentum."

"The direction of the momentum is the same as that of the velocity, and as the velocity can only be estimated with respect to some point of reference, so the particular value of the momentum depends on the point of reference which we assume. The momentum of the moon, for example, will be very different according as we take the earth or the sun for the point of reference."

"Statement of the Second Law of Motion in Terms of Impulse and Momentum—The change of momentum of a body is numerically equal to the impulse which produces it and is in the same direction."

"If any number of forces act simultaneously on a body, each force produces an acceleration proportional to its own magnitude. Hence if in the diagram of accelerations we draw from any origin a line representing in direction and magnitude the acceleration due to one of the forces, and from the end of this line another representing the acceleration due to another force, and so on, drawing lines for each of the forces taken in any order, then the line drawn from the origin to the extremity of the last of the lines will represent the acceleration due to the combined action of all the forces. Since in this diagram lines which represent the accelerations are in the same proportion as the forces to which these accelerations are due, we may consider the lines as representing these forces themselves. The diagram, thus understood, may be called a Diagram of Forces, and the line from the origin to the extremity of the series represents the Resultant Force."

"An important case is that in which the set of lines representing the forces terminate at the origin so as to form a closed figure. In this case there is no resultant force, and no acceleration. The effects of the forces are exactly balanced and the case is one of equilibrium. The discussion of cases of equilibrium forms the subject of the science of statics. It is manifest that since the system of forces is exactly balanced, and is equivalent to no force at all, the forces will also be balanced if they act in the same way on any other material system, whatever be the mass of that system. This is the reason why the consideration of mass does not enter into statical investigations."

"We have... used the word stress to denote the mutual action between two portions of matter. This word was borrowed from common language, and invested with a precise scientific meaning by the late Professor Rankine to whom we are indebted for several other valuable scientific terms."

"As soon as we have formed for ourselves the idea of a stress, such as the Tension of a rope or the Pressure between two bodies, and have recognized its double aspect as it affects the two portions of matter between which it acts, the third law of motion is seen to be equivalent to the statement that all force is of the nature of stress, that stress exists only between two portions of matter, and that its effects on these portions of matter (measured by the momentum generated in a given time) are equal and opposite. The stress is measured numerically by the force exerted on either of the two portions of matter."

"If... we neglect the weight of the string, each portion of the string is in equilibrium under the action of the tensions at its extremities, so that the tensions at any two transverse interfaces of the string must be the same. For this reason we often speak of the tension of the string as a whole, without specifying any particular section of it, and also the tension between the two bodies, without considering the nature of the string through which the tension is exerted."

"The fact that a magnet draws iron towards it was noticed by the ancients, but no attention was paid to the force with which the iron attracts the magnet. Newton, however, by placing the magnet in one vessel and the iron in another, and floating both vessels in water so as to touch each other, showed experimentally that as neither vessel was able to propel the other along with itself through the water, the attraction of the iron on the magnet must be equal and opposite to that of the magnet on the iron, both being equal to the pressure between the two vessels."

"Newton goes on to point out the consequence of denying the truth of this [third] law. For instance, if the attraction of any part of the earth, say a mountain, upon the remainder of the earth were greater or less than that of the remainder of the earth upon the mountain, there would be a residual force acting upon the system of the earth and the mountain as a whole, which would cause it to move off, with an ever increasing velocity, through infinite space."

"To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus."

"We cannot... regard Newton's statement as an appeal to experience and observation, but rather as a deduction of the third law of motion from the first."

"Whenever the subject admitted of it he had recourse to diagrams, though his fellow students might solve the question more easily by a train of analysis. Many illustrations of this... might be taken from his writings, but in truth it was only one phase of his mental attitude towards scientific questions, which led him to proceed from one distinct idea to another instead of trusting to symbols and equations."

"In January, 1854, Maxwell's undergraduate career closed. He was second wrangler, but shared with Dr Routh, who was senior wrangler, the honor of the first Smith's prize. In due course he was elected fellow of Trinity and place on the staff of College Lecturers."

"During his undergraduateship he had... found time for the study of Electricity. This had already borne fruit and now resulted in the first important memoirs on that subject,—the memoir on Faraday's Lines of Force."

"The labour of drilling classes composed chiefly of young and untrained lads, in the elements of mechanics and physics, was not the work for which Maxwell was specifically fitted."

"The most serious demands upon his powers and upon his time were made by his investigations on the Stability of Saturn's Rings. This was the subject chosen by the Examiners for the Adams Prize Essay to be ajudged in 1857, and was advertised in the following terms:— "The problem may be treated on the supposition that the system of Rings is exactly or very approximately concentric with Saturn and symmetrically disposed about the plane of his equator and different hypotheses may be made concerning the physical constitution of the Rings. It may be supposed (1) that they are rigid; (2) that they are fluid and in part aeriform; (3) that they consist of masses of matter not materially coherent. The question will be considered to be answered by ascertaining on these hypotheses severally whether the conditions of mechanical stability are satisfied by the mutual attractions and motions of the Planets and the Rings." It is desirable that an attempt should also be made to determine on which of the above hypotheses the appearances both of the bright rings and the recently discover dark ring may be most satisfactorily explained; and to indicate any causes to which a change in form such as is supposed from a comparison of modern with the earlier observations to have taken place, may be attributed.""

"The aim of an experiment of illustration is to throw light upon some scientific idea so that the student may be able to grasp it. ...the phenomenon which we wish to observe or to exhibit is brought into prominence, instead of being obscured and entangled among other phenomena, as it would when it occurs in the ordinary course of nature. ...The educational value of such experiments is often inversely proportional to the complexity of the apparatus."

"The student who uses home made apparatus, which is always going wrong, often learns more than one who has the use of carefully adjusted instruments, to which he is apt to trust and which he dares not take to pieces."

"Science appears to us with a very different aspect after we have found out that it is not in lecture rooms only, and by means of the electric light projected on a screen, that we may witness physical phenomena, but that we may find illustrations of the highest doctrines of science in games and gymnastics, in travelling by land and by water, in storms of the air and of the sea, and wherever there is matter in motion."

"This habit of recognising principles amid the endless variety of their action can never degrade our sense of the sublimity of nature or mar our enjoyment of its beauty. On the contrary, it tends to rescue our scientific ideas from that vague condition in which we too often leave them, buried among the other products of a lazy credulity, and to raise them into their proper position among the doctrines in which our faith is so assured, that we are ready at all times to act on them."

"In experimental researches... the ultimate object is to measure something which we have already seen—to obtain a numerical estimate of some magnitude. ...we must find out which of its features are capable of measurement, and what measurements are required in order to make a complete specification of the phenomenon. We must then make these measurements and deduce from them the result which we require to find."

"The opinion seems to have got abroad, that in a few years all the great physical constants will have been approximately estimated, and that the only occupation which will then be left to men of science will be to carry on these measurements to another place of decimals. ...But the history of science shews that even during that phase of her progress in which she devotes herself to improving the accuracy of the numerical measurement of quantities with which she has long been familiar, she is preparing the materials for the subjugation of new regions, which would have remained unknown if she had been contented with the rough methods of her early pioneers."

"The labour of careful measurement has been rewarded by the discovery of new fields of research, and by the development of new scientific ideas... the history of the science of terrestrial magnetism affords us a sufficient example..."

"That celebrated traveller Humboldt was profoundly impressed with the scientific value of a combined effort to be made by the observers of all nations, to obtain accurate measurements of the magnetism of the earth; and we owe it mainly to his enthusiasm for science... that not only private men of science, but the governments of most of the civilised nations... were induced to take part in the enterprise. But the actual working out of the scheme, and the arrangements by which the labours of the observers were so directed as to obtain the best results, we owe to the great mathematician Gauss, working along with Weber, the future founder of the science of electro-magnetic measurement in the magnetic observatory of Göttingen, and aided by the skill of the instrument-maker Leyser. ...Numbers of scientific men joined the Magnetic Union, learned the use of the new instruments and the new methods of reducing the observations; and in every city of Europe... sitting, each in his cold wooden shed, with his eye fixed at the telescope, his ear attentive to the clock, and his pencil recording in his note-book the instantaneous position of the suspended magnet."

"The increase in the accuracy and completeness of magnetic observations which was obtained by the new method opened up fields of research. [There are] disturbances to which the magnetism of our planet is found to be subject. Some of these disturbances are periodic following the regular courses of the sun and moon. Others are sudden, and are called magnetic storms, but, like the storms of the atmosphere, they have their known seasons of frequency. The last and the most mysterious of these magnetic changes is that secular variation by which the whole character of the earth, as a great magnet, is being slowly modified, while the magnetic poles creep on from century to century along their winding track in the polar regions. We have thus learned that the interior of the earth is subject to the influences of the heavenly bodies..."

"We must not suppose that the inner history of our planet is ended."

"The new methods of measuring forces were successfully applied by Weber to the numerical determination of all the phenomena of electricity, and very soon afterwards the electric telegraph, by conferring a commercial value on exact numerical measurements, contributed largely to the advancement, as well as to the diffusion of scientific knowledge."

"It is to Gauss, to the Magnetic Union, and to magnetic observers in general, that we owe our deliverance from that absurd method of estimating forces by a variable standard which prevailed so long even among men of science. It was Gauss who first based the practical measurement of magnetic force (and therefore of every other force) on those long established principles, which, though they are embodied in every dynamical equation, have been so generally set aside, that these very equations, though correctly given... are usually explained... by assuming, in addition to the variable standard of force, a variable, and therefore illegal, standard of mass."

"There is no more powerful method for introducing knowledge into the mind than that of presenting it in as many different ways as we can. When the ideas, after entering through different gateways, effect a junction in the citadel of the mind, the position they occupy becomes impregnable."

"The knowledge of physical science obtained by the combined use of mathematical analysis and experimental research will be of a more solid, available, and enduring kind than that possessed by the mere mathematician or the mere experimenter."

"It is not till we attempt to bring the theoretical part of our training into contact with the practical that we begin to experience the full effect of what Faraday has called "mental inertia"—not only the difficulty of recognising, among the concrete objects before us, the abstract relation which we have learned from books, but the distracting pain of wrenching the mind away from the symbols to the objects, and from the objects back to the symbols. This however is the price we have to pay for new ideas."

"When we have overcome these difficulties, and successfully bridged over the gulph between the abstract and the concrete... we have acquired the rudiment of a permanent mental endowment. When, by a repetition of efforts of this kind, we have more fully developed the scientific faculty, the exercise of this faculty in detecting scientific principles in nature, and in directing practice by theory, is no longer irksome, but becomes an unfailing source of enjoyment, to which we return so often, that at last even our careless thoughts begin to run in a scientific channel."

"Now in the case of study, a great part of our fatigue often arises, not from those mental efforts by which we obtain the mastery of the subject, but from those which are spent in recalling our wandering thoughts; and these efforts of attention would be much less fatiguing if the disturbing force of mental distraction could be removed."

"A man whose soul is in his work always makes more progress than one whose aim is something not immediately connected with his occupation. In the latter case the very motive of which he makes use to stimulate his flagging powers becomes the means of distracting his mind from the work before him."

"There may be some mathematicians who pursue their studies entirely for their own sake. Most men, however, think that the chief use of mathematics is found in the interpretation of nature. Now a man who studies a piece of mathematics in order to understand some natural phenomenon which he has seen, or to calculate the best arrangement of some experiment which he means to make, is likely to meet with far less distraction of mind than if his sole aim had been to sharpen his mind for the successful practice of the Law, or to obtain a high place in the Mathematical Tripos."

"It must be one of our most constant aims to maintain a living connexion between our work and the other liberal studies of Cambridge, whether literary, philological, historical or philosophical."

"There is a narrow professional spirit which may grow up among men of science, just as it does among men who practise any other special business. But surely a University is the very place where we should be able to overcome this tendency of men to become, as it were, granulated into small worlds, which are all the more worldly for their very smallness. We lose the advantage of having men of varied pursuits collected into one body, if we do not endeavour to imbibe some of the spirit even of those whose special branch of learning is different from our own."

"It is not so long ago since any man who devoted himself to geometry, or to any science requiring continued application, was looked upon as necessarily a misanthrope, who must have abandoned all human interests, and betaken himself to abstractions so far removed from the world of life and action that he has become insensible alike to the attractions of pleasure and to the claims of duty. In the present day, men of science are not looked upon with the same awe or with the same suspicion. They are supposed to be in league with the material spirit of the age, and to form a kind of advanced Radical party among men of learning."

"It is true that the history of science is very different from the science of history. We are not studying or attempting to study the working of those blind forces which, we are told, are operating on crowds of obscure people, shaking principalities and powers, and compelling reasonable men to bring events to pass in an order laid down by philosophers. The men whose names are found in the history of science are not mere hypothetical constituents of a crowd, to be reasoned upon only in masses. We recognise them as men like ourselves, and their actions and thoughts, being more free from the influence of passion, and recorded more accurately than those of other men, are all the better materials for the study of the calmer parts of human nature."

"The history of science is not restricted to the enumeration of successful investigations. It has to tell of unsuccessful inquiries, and to explain why some of the ablest men have failed to find the key of knowledge, and how the reputation of others has only given a firmer footing to the errors into which they fell."

"The history of the development, whether normal or abnormal, of ideas is of all subjects that in which we, as thinking men, take the deepest interest. But when the action of the mind passes out of the intellectual stage, in which truth and error are the alternatives, into the more violently emotional states of anger and passion, malice and envy, fury and madness; the student of science, though he is obliged to recognise the powerful influence which these wild forces have exercised on mankind, is perhaps in some measure disqualified from pursuing the study of this part of human nature."

"We cannot enter into full sympathy with these lower phases of our nature without losing some of that antipathy to them which is our surest safeguard against a reversion to a meaner type, and we gladly return to the company of those illustrious men who by aspiring to noble ends, whether intellectual or practical, have risen above the region of storms into a clearer atmosphere, where there is no misrepresentation of opinion, nor ambiguity of expression, but where one mind comes into closest contact with another at the point where both approach nearest to the truth."

"Two theories of the constitution of bodies have struggled for victory with various fortunes since the earliest ages of speculation: one is the theory of a universal plenum, the other is that of atoms and void. The theory of the plenum is associated with the doctrine of mathematical continuity, and its mathematical methods are those of the Differential Calculus, which is the appropriate expression of the relations of continuous quantity. The theory of atoms and void leads us to attach more importance to the doctrines of integral numbers and definite proportions; but, in applying dynamical principles to the motion of immense numbers of atoms, the limitation of our faculties forces us to abandon the attempt to express the exact history of each atom, and to be content with estimating the average condition of a group of atoms large enough to be visible. This method... which I may call the statistical method, and which in the present state of our knowledge is the only available method of studying the properties of real bodies, involves an abandonment of strict dynamical principles, and an adoption of the mathematical methods belonging to the theory of probability. ...If the actual history of Science had been different, and if the scientific doctrines most familiar to us had been those which must be expressed in this way, it is possible that we might have considered the existence of a certain kind of contingency a self evident truth, and treated the doctrine of philosophical necessity as a mere sophism."

"The properties of bodies were investigated by several distinguished French mathematicians on the hypothesis that they are systems of molecules in equilibrium. The somewhat unsatisfactory nature of the results... produced... a reaction in favour of the opposite method of treating bodies as if they were... continuous. This method, in the hands of Green, Stokes, and others, has led to results the value of which does not at all depend on what theory we adopt as to the ultimate constitution of bodies."

"There are innumerable other molecules, whose constants are not approximately, but absolutely identical with those of the first molecule, and this whether they are found on the earth, in the sun, or in the fixed stars. By what process of evolution the philosophers of the future will attempt to account for this identity in the properties of such a multitude of bodies, each of them unchangeable in magnitude, and some of them separated from others by distances which Astronomy attempts in vain to measure, I cannot conjecture. My mind is limited in its power of speculation, and I am forced to believe that these molecules must have been made as they are from the beginning of their existence. ...we cannot ascribe either their existence or the identity of their properties to the operation of any of those causes which we call natural. Is it true then that our scientific speculations have really penetrated beneath the visible appearance of things which seem to be subject to generation and corruption and reached the entrance of that world of order and perfection which continues this day as it was created perfect in number and measure and weight?"

"We may be mistaken. No one has as yet seen or handled an individual molecule, and our molecular hypothesis may, in its turn, be supplanted by some new theory of the constitution of matter; but the idea of the existence of unnumbered individual things, all alike and all unchangeable, is one which cannot enter the human mind and remain without fruit. But what if these molecules, indestructible as they are, turn out to be not substances themselves, but mere affections of some other substance?"

"According to Sir W. Thomson's theory of Vortex Atoms, the substance of which the molecule consists is a uniformly dense plenum, the properties of which are those of a perfect fluid, the molecule itself being nothing but a certain motion impressed on a portion of this fluid, and this motion is shewn, by a theorem due to Helmholtz, to be as indestructible as we believe a portion of matter to be. If a theory of this kind is true, or even if it is conceivable, our idea of matter may have been introduced into our minds through our experience of those systems of vortices which we call bodies, but which are not substances, but motions of a substance; and yet the idea which we have thus acquired of matter, as a substance possessing inertia, may be truly applicable to that fluid of which the vortices are the motion, but of whose existence, apart from the vortical motion of some of its parts, our experience gives us no evidence whatever."

"It has been asserted that metaphysical speculation is a thing of the past and that physical science has extirpated it. The discussion of the categories of existence, however, does not appear to be in danger of coming to an end in our time, and the exercise of speculation continues as fascinating to every fresh mind as it was in the days of Thales."

"As stated repeatedly in this book, John von Neumann's Mathematical Foundations of Quantum Mechanics was an extraordinarily influential work. It is important to recall that the language most commonly used to describe and discuss the measurement process in terms of a collapse or projection of the wave function essentially originates with this classic work. It was von Neumann who so clearly distinguished (in the mathematical sense) between the continuous time-symmetric quantum mechanical equations of motion and the discontinuous, time-asymmetric measurement process. Although much of his contribution to the development of the theory was made broadly within the boundaries of the Copenhagen view, he stepped beyond those boundaries in his interpretation of quantum measurement."

"Thus the formal proof of von Neumann does not justify his informal conclusion: 'It is therefore not, as is often assumed, a question of reinterpretation of quantum mechanics - the present system of quantum mechanics would have to be objectively false in order that another description of the elementary process than the statistical one be possible.' It was not the objective measurable predictions of quantum mechanics which ruled out hidden variables. It was the arbitrary assumption of a particular (and impossible) relation between the results of incompatible measurements either of which might be made on a given occasion but only one of which can in fact be made."

"The physicists didn't want to be bothered with the idea that maybe quantum theory is only provisional. A horn of plenty had been spilled before them, and every physicist could find something to apply quantum mechanics to. They were pleased to think that this great mathematician had shown it was so. Yet the Von Neumann proof, if you actually come to grips with it, falls apart in your hands! There is nothing to it. It's not just flawed, it's silly. If you look at the assumptions made, it does not hold up for a moment. It's the work of a mathematician, and he makes assumptions that have a mathematical symmetry to them. When you translate them into terms of physical disposition, they're nonsense. You may quote me on that: The proof of Von Neumann is not merely false but foolish!"

"There are (at least) two possible ways to formulate precisely (i.e. mathematically) elementary QM. The eldest one, historically speaking, is due to von Neumann in essence, and is formulated using the language of Hilbert spaces and the spectral theory of unbounded operators. A more recent and mature formulation was developed by several authors in the attempt to solve quantum field theory problems in mathematical physics. … The newer formulation can be considered an extension of the former one, in a very precise sense that we shall not go into here, also by virtue of the novel physical context it introduces and by the possibility of treating physical systems with infinitely many degrees of freedom, i.e. quantum fields. In particular, this second formulation makes precise sense of the demand for locality and covariance of relativistic quantum field theories, and allows to extend quantum field theories to a curved spacetime."

"The gist and kernel of mechanical ideas has in almost every case grown up in the investigation of very simple and special cases of mechanical processes; and the analysis of the history of the discussions concerning these cases must ever remain the method at once the most effective and the most natural for laying this gist and kernel bare. Indeed, it is not too much to say that it is the only way in which a real comprehension of the general upshot of mechanics is to be attained."

"I have not in every case been able to avoid the use of the abbreviated and precise terminology of mathematics. To do so would have been to sacrifice matter to form; for the language of everyday life has not yet grown to be sufficiently accurate for the purposes of so exact a science as mechanics."

"The elucidations which I here offer are, in part, substantially contained in my treatise, Die Geschichte und die Wurzel des Satzcs von der Erhaltung der Arbeit (...1872). At a later date nearly the same views were expressed by Kirchhoff (Vorlesungen über mathematische Physik: Mechanik, Leipsic, 1874) and by Helmholtz (Die Thatsachen in der Wahrnehmung, Berlin, 1879), and have since become commonplace enough. Still the matter, as I conceive it, does not seem to have been exhausted, and I cannot deem my exposition to be at all superfluous."

"Regard for the true endeavor of philosophy, that of guiding into one common stream the many rills of knowledge, will not be found wanting in my work, although it takes a determined stand against the encroachments of metaphysical methods."

"The questions here dealt with have occupied me since my earliest youth, when my interest for them was powerfully stimulated by the beautiful introductions of Lagrange to... his Analytic Mechanics, as well as by the lucid and lively tract of Jolly, Principien der Mechanik (...1852). If Duehring's estimable work, Kritische Geschichte der Principien der Mechanik (...1873), did not particularly influence me, it was that at the time of its appearance, my ideas had been... published. ...[T]he reader will... find many points of agreement between Dühring's criticisms and those here expressed."

"The new apparatus for the illustration of the subject, here figured and described, were designed entirely by me and constructed by Mr. F. Hajek... In less immediate connection with the text stand the fac-simile reproductions of old originals in my possession. The quaint and naive traits of the great inquirers, which find in them their expression, have always exerted upon me a refreshing influence in my studies, and I have desired that my readers should share this pleasure with me."

"That branch of physics which is at once the oldest and the simplest and which is therefore as introductory to other departments of this science, is concerned with the motions and equilibrium masses. It bears the name of mechanics."

"The history of the development of mechanics, is... 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."

"An instinctive, irreflective knowledge of the processes of nature will doubtless always precede the scientific, conscious apprehension, or investigation, of phenomena. The former is the outcome of the relation in which the processes of nature stand to the satisfaction of our wants."

"The acquisition of the most elementary truth does not devolve upon the individual alone: it is pre-effected in the development of the race."

"[I]t is necessary to make a distinction between mechanical experience and mechanical science... If we carefully examine the ancient Egyptian and Assyrian monuments, we shall find there pictorial representations of many kinds of implements and mechanical contrivances... long before theory was dreamed of, implements, machines, mechanical experiences, and mechanical knowledge were abundant."

"In the infinite variety of nature many ordinary events occur; while others appear uncommon, perplexing, astonishing, or even contradictory to the ordinary run of things. As long as this is the case we do not possess a well-settled and unitary conception of nature. Thence is imposed the task of... seeking out... those elements that are the same, and... ever present. By this means, on the one hand, the most economical and briefest description and communication are rendered possible; and on the other, when once a person has acquired the skill of recognising these permanent elements throughout the greatest range and variety of phenomena... this ability leads to a comprehensive, compact, consistent, and facile conception of the facts. When... we are everywhere able to detect the same few simple elements, combining in the ordinary manner, then they appear to us as things... familiar... in the phenomena, we feel at home with them, they no longer perplex us, they are explained. It is a process of adaptation of thoughts to facts with which we are here concerned."

"We now propose to enter more minutely into subject of our inquiries, and at the same time, without making the history of mechanics the chief topic discussion, to consider its historical development so far as this is requisite to an understanding of the present state of mechanical science... Apart from the consideration that we cannot afford to neglect the great incentives that it is in our power to derive from the foremost intellects of all epochs, incentives which taken as a whole are more fruitful than the greatest men of the present day are able to offer, there is no grander, no more intellectually elevating spectacle than that of the utterances of the fundamental investigators in their gigantic power. Possessed as yet of no methods, for these were created by their labors, and are only rendered comprehensible to us by their performances, they grapple with and subjugate the object of their inquiry, and imprint upon it the forms of conceptual thought. They that know the entire course of the development of science, will, as a matter of course, judge more freely and more correctly of the significance of any present scientific movement than they, who limited in their views, to the age in which their own lives have been spent, contemplate merely the momentary trend that the course of intellectual events takes at the present moment."

"[E]very enlightening progress made in science is accompanied with a certain feeling of disillusionment. We discover that that which appeared wonderful to us is no more wonderful than other things which we know instinctively and regard as self-evident; nay, that the contrary would be much more wonderful; that everywhere the same fact expresses itself. Our puzzle turns out then to be a puzzle no more; it vanishes into nothingness, and takes its place among the shadows of history."

"The principle of the parallelogram of forces, at which Stevinus arrived and employed, (yet without expressly formulating it,) consists... of following truth. If a body A (Fig. 26) is acted by two forces whose directions coincide with the AB and AC, and whose magnitudes are proportional the lengths AB and AC, these two forces produce the same effect as a single force, which acts in the direction of the diagonal AD of the parallelogram ABCD and is proportional to that diagonal."

"Newton... [with respect to "Absolute, true, and mathematical time... duration" and to "Relative, apparent, and commone time"] stood under the influence of the view of mediæval philosophy, as though he had grown unfaithful to his resolve to investigate only actual facts."

"Time... appears to be some particular and independent thing, on the progress of which the position of the pendulum depends, while the things that we resort to for comparison and choose at random appear to play a wholly collateral part."

"[A]ll things in the world are connected with one another and depend on one another, and...we... and all our thoughts are... a part of nature."

"It is utterly beyond our power to measure the changes of things by time. Quite the contrary, time is an abstraction, at which we arrive by means of the changes of things..."

"[W]e are not restricted to any one definite measure, all being interconnected."

"We arrive at the idea of time,—to express it briefly and popularly,—by the connection of that which is contained in... our memory with that which is contained in... our sense-perception. When we say that time flows on in a definite direction or sense, we mean that physical events generally (and therefore also physiological events) take place only in a definite sense. Differences of temperature, electrical differences, differences of level generally, if left to themselves, all grow less and not greater. ...In all this there is... a peculiar and profound connection of things. To demand at... present... a full elucidation of this matter, is to anticipate, in the manner of speculative philosophy, the results of all future special investigation, that is a perfect physical science."

"Newton has... acted contrary to his expressed intention only to investigate actual facts. No one is competent to predicate things about absolute space and absolute motion; they are pure things of thought, pure mental constructs, that cannot be produced in experience. All our principles of mechanics are... experimental knowledge concerning the relative positions and motions of bodies. ...No one is warranted in extending these principles beyond the boundaries of experience. ...[S]uch an extension is meaningless, as no one possesses the requisite knowledge to make use of it."

"It might be... that... motion is determined by a medium... In such a case we should have to substitute this medium for Newton's absolute space. ...[I]t is easily demonstrable that the atmosphere is not this motion-determinative medium. We should, therefore, have to picture to ourselves some other medium, filling, say, all space, with respect to... which... we have at present no adequate knowledge. In itself such a state of things would not belong to the impossibilities. It is known, from recent hydrodynamical investigations, that a rigid body experiences resistance in a frictionless fluid only when its velocity changes. ...[T]his result is derived theoretically from the notion of inertia; but it might... also be regarded as the primitive fact from which we... start. ...[W]e might... hope to learn more in the future concerning this hypothetical medium; and from the point of view of science it would be in every respect a more valuable acquisition than the forlorn idea of absolute space."

"Let us... examine the point on which Newton, apparently with sound reasons, rests his distinction of absolute and relative motion. If the earth is affected with an absolute rotation about its axis, centrifugal forces are set up in the earth: it assumes an oblate form, the acceleration of gravity is diminished at the equator, the plane of Foucault's pendulum rotates, and so on. All these phenomena disappear if the earth is at rest and the other heavenly bodies are affected with absolute motion round it, such that the same relative rotation is produced. This is, indeed, the case, if we start ab initio from the idea of absolute space. But if we take our stand on the basis of facts, we shall find we have knowledge only of relative spaces and motions. Relatively, not considering the unknown and neglected medium of space, the motions of the universe are the same whether we adopt the Ptolemaic or the Copernican mode of view. Both views are, indeed, equally correct; only the latter is more simple and more practical. The universe is not twice given, with an earth at rest and an earth in motion; but only once, with its relative motions, alone determinable. It is, accordingly, not permitted us to say how things would be if the earth did not rotate. We may interpret the one case that is given us, in different ways. If, however, we so interpret it that we come into conflict with experience, our interpretation is... wrong. The principles of mechanics can, indeed, be so conceived, that even for relative rotations s arise."

"The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their... experimentally-given comportment with respect to the celestial bodies, we should render ourselves culpable of a falsity. When... we say, that a body preserves unchanged its direction and velocity in space, our assertion is... an abbreviated reference to the entire universe. ...[S]uch ...is permitted the original author Newton, Descartes or Galileo] of the principle, because he knows... no difficulties stand in the way of carrying out its implied directions. But no remedy lies in his power, if difficulties of the kind mentioned present themselves; if, for example, the requisite, relatively fixed bodies are wanting."

"When we reflect that the time-factor that enters into the is nothing more than... the measure of the distances (or angles of rotation) of the bodies of the universe, we see that even in the simplest case... apparently... the mutual action of only two masses, the neglecting of the rest of the world is impossible."

"Nature does not begin with elements, as we are obliged to begin with them. It is certainly fortunate... that we can, from time to time, turn aside our eyes from the over powering unity of the All, and allow them to rest on individual details. But we should not omit, ultimately, to complete and correct our views by a thorough consideration of the things which for the time being we left out of account."

"[W]ithin the short span of a human life and with man's limited powers of memory, any stock of knowledge worthy of the name is unattainable except by the greatest mental economy. Science... may be regarded as a minimal problem... the completest possible presentment of facts with the least possible expenditure of thought."

"The function of science... is to replace experience. Thus, on the one hand, science must remain in the province of experience, but, on the other, must hasten beyond it, constantly expecting confirmation, constantly expecting the reverse. Where neither confirmation nor refutation is possible, science is not concerned. Science acts and only acts in the domain of uncompleted experience."

"[T]he age in which the chief development of the science of mechanics took place, was an age of predominantly theological cast. Theological questions were excited by everything, and modified everything. No wonder, then, that mechanics is colored thereby."

"[N]otions of the constancy of the quantity of matter... of motion, of the indestructibility of work or energy... which completely dominate modern physics, all arose under the influence of theological ideas."

"During the entire sixteenth and seventeenth centuries, down to the close of the eighteenth, the prevailing inclination of inquirers was, to find in all physical laws some particular disposition of the Creator."

"Towards the close of the eighteenth century a remarkable change... apparently an abrupt departure from the current trend... but... the logical outcome of the development... After an attempt... to found mechanics on Euler’s principle of least action, Lagrange, in a subsequent treatment... declared his intention of... disregarding theological and metaphysical speculations, as... precarious and foreign to science. He erected a new mechanical system on... different foundations, and no one... will dispute its excellencies. All subsequent scientists of eminence accepted Lagrange’s view, and the present attitude of physics to theology was thus substantially determined."

"Newton never, despite his profound religiosity, mingled theology with the questions of science. ...The same may be said of Galileo and Huygens."

"In fact, science can accomplish nothing by the consideration of individual facts; from time to time it must cast its glance at the world as a whole."

"But now, after a century has elapsed, after our judgment has grown more sober, the world-conception of the encyclopaedists appears to us as a mechanical mythology in contrast to the animistic of the old religions."

"Physical science does not pretend to be a complete view of the world; it simply claims that it is working toward such a complete view in the future. The highest philosophy of the scientific investigator is... toleration of an incomplete conception of the world and the preference for it rather than an apparently perfect, but inadequate conception."

"I am going to talk about what Leibniz and Ernst Mach said about time. ...you can put the Leibnizian/Machian idea into a theory of dynamics, into the way the universe works. ...And the bottom line is this: There is a great likelihood that time does not exist at all, that it is a redundant concept. ...Now for the famous reactions that any of you who have studied the philosophy of science will surely have encountered: The Leibniz-Clarke Correspondence and Ernst Mach's famous book on mechanics. Let me give you some quotations. Leibniz said: "I hold space to be something merely relative." ...His claim is ontological: only relative things exist. ...Only relative changes are real. ...Mach said very similar things many years later."

"We must not be surprised... that... all physicists of the last century saw in classical mechanics a firm and final foundation for all of physics, yes, indeed, for all natural science, and that they never grew tired in their attempts to base Maxwell's theory of electromagnetism, which, in the meantime, was slowly beginning to win out, upon mechanics as well. Even Maxwell and H. Hertz, who in retrospect appear as those who demolished the faith in mechanics as the final basis of all physical thinking, in their conscious thinking adhered throughout to mechanics as the secure basis of physics. It was Ernst Mach who, in his History of Mechanics, shook this dogmatic faith; this book exercised a profound influence upon me in this regard while I was a student. I see Mach's greatness in his incorruptible skepticism and independence; in my younger years, however, Mach's epistemological position also influenced me very greatly, a position which today appears to be essentially untenable."

"My attention was drawn to Ernst Mach's Science of Mechanics by my friend Besso while a student, around the year 1897. The book exerted a deep and persisting impression upon me... owing to its physical orientation toward fundamental concepts and fundamental laws."

"In the year 1883... Ernst Mach, published a book Mechanics and Its Evolution, which turned out to be in many respects a landmark in our understanding of the laws of motion. Mach presented a critical analysis of Newtonian mechanics and directed the attention of scientists to the fact that the "constancy of mass," if the operational definition \frac{m_2}{m_1} = \frac{a_2}{a_1} [compare two masses by pushing them with the same force and comparing accelerations] is used, is an experimental fact and by no means a "philosophical truth" which can be derived from "intelligible principles." There was a possibility that experiments would show a change of mass caused by external circumstances. ...[T]oward the end of the nineteenth century, J. J. Thomson derived from Maxwell's electromagnetic field theory the result that mass points behave like particles with electric charges. In the twentieth century, the motion of fast electrically charged particles was systematically investigated... in the . If electrostatic forces are acting in the direction of the actual velocity, high speed particles (i.e., with speeds comparable to the speed of light) obtain accelerations which are noticeably smaller than the accelerations of slow particles in the same electrostatic field."

"[T]he attempts to eiliminate organismic and theological elements from the law of inertia have proved to be of great consequence in the field of "science proper." They have stimulated research for new physical laws. ...Newton described it [absolute space (and time)] as the "sensorium of God"... mechanics were based on the "accidental fact" that the system of fixed stars was... at rest in "absolute space." However, if we examine the two premises... we can draw the simple conclusion: The law of inertia is valid with respect to the fixed stars. ..."Inertia" is then an interaction between material bodies just as much as gravitation is. This point was stressed in the nineteenth century by Ernst Mach, first in a short paper in 1872, and then more elaborately in his book... Mechanics and its Evolution in 1883..."

"As the correspondence at the Einstein Archives at Princeton reveals, one of the young scientists deeply caught up in Mach's point of view was Michelange (Michele) Besso—Einstein's oldest and closest friend, fellow student, and colleague at the Patent Office in Bern, the only person to whom Einstein gave credit for help (manche wertvolle Anregung) when he published his basis paper on relativity in 1905. It was Besso who introduced Einstein to Mach's work [Science of Mechanics]. ...Besso remained a loyal Machist to the end."

"In his Science of Mechanics... Ernst Mach suspended... the classic formula of the law of inertia because he believed that... [this] had to be rethought. His famous studies of Newton's terminology (in particular... absolute space and absolute time)... led Mach to propose a new formulation in which gravitation appears as a function of the entire mass distribution of the universe. In this critical reconstruction... Mach believed that it had to be shown... that even the most firmly established concepts were merely auxiliary constructs... [which] give a provisional account of relations... between our experiences. And... we tend to forget that we were the ones who introduced our concepts... Mach demands that we... keep dissolving the most firmly established concepts and formulæ... so that we do not perceive them as something independent of ourselves and allow them to become obstacles to our knowledge."

"The aim of this book is to exhibit the scientific connexion of the various steps by which our knowledge of the phenomena of heat has been extended."

"The first of these steps is the invention of the thermometer... The second step is the measurement of quantities of heat, or . The whole science of heat is founded on Thermometry and Calorimetry, and when these operations are understood we may proceed to the third step, which is the investigation of those relations between the thermal and the mechanical properties of substances which form the subject of Thermodynamics."

"The whole of [thermodynamics] depends on the consideration of the Intrinsic Energy of a system of bodies, as depending on the temperature and physical state, as well as the form, motion, and relative position of these bodies. Of this energy, however, only a part is available for the purpose of producing mechanical work, and though the energy itself is indestructible, the available part is liable to diminution by the action of certain natural processes, such as conduction and radiation of heat, friction, and viscosity. These processes, by which energy is rendered unavailable as a source of work, are classed together under the name of the Dissipation of Energy..."

"The last chapter is devoted to the explanation of various phenomena by means of the hypothesis that bodies consist of molecules, the motion of which constitutes the heat of those bodies."

"[I]t has been found necessary to omit everything which is not an essential part of the intellectual process by which the doctrines of heat have been developed, or which does not materially assist the student in forming his own judgment on these doctrines."

"A full account of the most important experiments on the effects of heat will be found in [Robert Vickers] Dixon's 'Treatise on Heat' (Hodges & Smith, 1849)."

"Professor 's treatise contains all that is necessary to be known in order to make experiments on heat. The student may be also referred to [Augustin Privat-]Deschanel's ' Natural Philosophy,' Part II., translated by Professor [J. D.] Everett, who has added a chapter on Thermodynamics; to Professor Rankine's work on the Steam Engine, in which he will find the first systematic treatise on thermodynamics; to Professor Tait's 'Thermodynamics,' which contains an historical sketch of the subject, as well as the mathematical investigations; and to Professor Tyndall's work on 'Heat as a Mode of Motion,' in which the doctrines of the science are forcibly impressed on the mind by well-chosen illustrative experiments."

"Words... which express the same things as the words of primitive language, but express them in a way susceptible of accurate numerical statement, are called scientific terms, because they contribute to the growth science."

"[W]e prefer to form our estimate of the state of bodies from their observed action on some apparatus whose conditions are more simple and less variable than those of our own senses."

"Any substance in which an increase of temperature, however small, produces an increase of volume may be used to indicate changes of temperature."

"[T]he thermo-electric properties of metals, and the variation of their electric resistance with temperature, are also employed in researches on heat."

"Heat... is something which may be transferred from one body to another, so as to diminish the quantity of heat in the first and increase that in the second by the same amount."

"When heat leaves a body, there is either a fall of temperature or a change of state. If no heat enters or leaves a body, and if no changes of state or mechanical actions take place in the body, the temperature of the body will remain constant."

"Heat... may pass out of one body into another just as water may be poured from one vessel into another, and it may be retained in a body for any time, just as water may be kept in a vessel. We have therefore a right to speak of heat as of a measurable quantity, and to treat it mathematically like other measurable quantities so long as it continues to exist as heat."

"[W]e have no right to treat heat as a substance, for it may be transformed into something which is not heat, and [which] is certainly not a substance at all, namely, mechanical work."

"[T]hough we admit heat to the title of a measurable quantity, we must not give it rank as a substance, but must hold our minds in suspense till we have further evidence..."

"[E]vidence is furnished by experiments on friction, in which mechanical work, instead of being transmitted... heat is generated... in an exact proportion to the amount of work lost. We have, therefore, reason to believe that heat is of the same nature as mechanical work, that is, it is one of the forms of Energy."

"In the eighteenth century... the word Caloric was introduced to signify heat as a measurable quantity. ...but the form of the word accommodated itself to the tendency of the chemists... to seek for new 'imponderable substances,' so that the word caloric came to connote... heat as an indestructible imponderable fluid, insinuating itself into the pores of bodies, dilating and dissolving them, and ultimately vaporising them, combining with bodies in definite quantities, and so becoming latent, and reappearing when, these bodies alter their condition."

"[T]he word caloric... soon came to imply the... existence of something material, though probably of a more subtle nature than the then newly discovered gases."

"Caloric resembled... gases in being invisible and in its property of becoming fixed in solid bodies. It differed from them because its weight could not be detected by the finest balances, but there was no doubt in the minds of many eminent men that caloric was a fluid pervading all bodies, probably the cause of all repulsion, and possibly even of the extension of bodies in space."

"[T]he theory of heat as a form of energy is called Thermodynamics."

"The instrument by which quantities of heat are measured is called a ... sufficiently distinct from that of the word Thermometer. The method of measuring heat may be called Calorimetry."

"A certain quantity of heat, with which all other quantities are compared, is called a Thermal Unit. This is the quantity of heat required to produce a particular effect, such as to melt a pound of ice, or to raise a pound of water from one defined temperature to another defined temperature. A particular thermal unit has been called... a ."

"We have... obtained two of the fundamental ideas... the idea of temperature, or the property of a body... with reference to its power of heating other bodies; and the idea of heat as a measurable quantity, which may be transferred from hotter bodies to colder ones."

"[T]he Diffusion of Heat ...invariably transfers heat from a hotter body to a colder one, so as to cool the hotter body and warm the colder... This process would go on till all bodies were brought to the same temperature if it were not for certain other processes... for instance, when combustion or any other chemical process takes place, or when any change occurs in the form, structure, or physical state of the body."

"Three processes of diffusion of heat are commonly recognised—Conduction, , and ."

"Conduction is the flow of heat through an unequally heated body from places of higher to places of lower temperature."

"The term convection is applied to those processes by which the diffusion of heat is rendered more rapid by the motion of the hot substance from one place to another..."

"In , the hotter body loses heat, and the colder body receives heat by means of a process occurring in some intervening medium which does not itself become thereby hot."

"[T]he amount of heat transferred from the hotter to the colder body is invariably greater than the amount, if any, transferred from the colder to the hotter."

"When heat is passing through a body by conduction, the temperature of the body must be greater in the parts from which the heat comes than in those to which it tends, and the quantity of heat which passes through any thin layer of the substance depends on the difference of the temperatures of the opposite sides of the layer. ...[W]e may define conduction as the passage of heat through a body depending on inequality of temperature in adjacent parts of the body. ...[T]he parts of the body through which the heat comes... must be hotter... and the parts higher up the stream of heat still hotter."

"[T]ake a thermometer with a large bulb... and dash a little hot water over the bulb. The fluid will fall in the tube before it begins to rise, showing that the bulb began to expand before the fluid expanded."

"If we make use of a thermometer... if the sun's rays fall on it... while the air immediately surrounding the bulb is at a temperature below freezing. The heat... to which the thermometer... responds, is not conveyed... through the air, for the air is cold... The mode in which the heat reaches the body... without warming the air through which it passes, is called radiation."

"Substances which admit of taking place through them are called Diathermanous. Those which do not allow heat to pass through them without becoming themselves hot are called Athermanous. ...If the body is not perfectly diathermanous it stops more or less of the radiation, and becomes heated itself, instead of transmitting the whole radiation to bodies beyond it."

"The distinguishing characteristic of radiant heat is, that it travels in rays like light, whence the name radiant. These rays have all the physical properties of rays of light, and are capable of reflexion, , interference, and polarisation."

"[I]t is only when the radiation is stopped that it has any effect in heating what it falls on."

"By means of any regular concave piece of metal... pressed when hot against a clear sheet of , first on one side and then on the other, it is easy to make a lens of ice which may be used... as a ; but this experiment... is far inferior in interest to one invented by Professor Tyndall, in which the heat, instead of being concentrated by ice, is concentrated in ice."

"[A]ll hot bodies emit radiation."

"When [a] body is hot enough, its radiations become visible, and the body is said to be red hot. When it is still hotter it sends forth... rays of every colour, and it is... white hot. When a body is too cold to shine visibly, it still shines with invisible heating rays... it does not appear that any body can be so cold as not to send forth radiations."

"[O]ur eyes are sensitive only to particular kinds of rays... coming from some very hot body... directly or after reflexion or scattering at the surface of other bodies."

"between bodies differs from heat as we have defined it—1st, in not making the body hot through which it passes; 2nd, in being of many different kinds."

"[I]n a at rest the at any point must be equal in all directions."

"There are two great classes of fluids. ...the first class, such as water... will partly fill the vessel... Fluids having this property are called s. If... the fluid which we put into the closed vessel be one of the second class, then... it will expand and fill the vessel... Fluids having this property are called es. ...a gas cannot, like a liquid, be kept in an openmouthed vessel."

"The distinction... between a gas and a liquid is that, however large the space may be into which a portion of gas is introduced, the gas will expand and exert pressure on every part of its boundary, whereas a liquid will not expand more than a very small fraction of its bulk, even when the pressure is reduced to zero; and some liquids can even sustain a hydrostatic tension, or negative pressure, without their parts being separated."

"A great many solid bodies are constantly in a state of evaporation or transformation into the gaseous state at their free surface. , , and carbonate of ammonia... if not kept in stoppered bottles, gradually disappear by evaporation, and the vapour... may be recognised by its smell and by its chemical action. ... is continually passing into a state of vapour... and in a dry climate during a long frost large pieces... [gradually] disappear. ...[[w:Iron|[I]ron]] and copper have each a well-known smell. This... may arise from chemical action at the surface, which sets free or some other gas combined with a very small quantity of the metal."

"The original thermometer invented by Galileo was an air thermometer. It consisted of a glass bulb with a long neck. The air in the bulb was heated and then the neck was plunged into a coloured liquid. As the air in the bulb cooled, the liquid rose in the neck, and the higher the liquid the lower the temperature of the air in the bulb. By putting the bulb into the mouth of a patient and noting the point to which the liquid was driven down in the tube, a physician might estimate whether the ailment was of the nature of a fever or not. Such a thermometer has several obvious merits. It is easily constructed, and gives larger indications for the same change of temperature than a thermometer containing any liquid as the thermometric substance. Besides this, the air requires less heat to warm it than an equal bulk of any liquid, so that the air thermometer is very rapid in its indications. The great inconvenience of the instrument as a means of measuring temperature is, that the height of the liquid in the tube depends on the pressure of the atmosphere as well as on the temperature of the air in the bulb. The air thermometer cannot therefore of itself tell us anything about temperature. We must consult the barometer at the same time, in order to correct the reading of the air thermometer. Hence the air thermometer, to be of any scientific value, must be used along with the barometer, and its readings are of no use till after a process of calculation has been gone through. This puts it at a great disadvantage compared with the mercurial thermometer... But if the researches on which we are engaged are of so important a nature that we are willing to undergo the labour of double observations and numerous calculations, than the advantages of the air thermometer may again preponderate."

"The present treatise is the outcome of a suggestion made to me some years ago by Mr R. R. Webb that I should assist him in the preparation of a work on Elasticity. He has unfortunately found himself unable to proceed... and I have therefore been obliged to take upon myself the whole of the... responsibility. I wish to acknowledge... the debt that I owe to him as a teacher of the subject, as well as... for many valuable suggestions..."

"The division of the subject adopted is that... by Clebsch in his classical treatise, where a clear distinction is drawn between exact solutions for bodies all whose dimensions are finite and approximate solutions for bodies some of whose dimensions can be regarded as infinitesimal. The present volume contains the general mathematical theory of the elastic properties of the first class of bodies, and I propose to treat the second class in another volume."

"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."

"In the analysis of strain I have thought it best to follow Thomson and Tait's Natural Philosophy, beginning with the geometrical or rather algebraical theory of finite homogeneous strain, and passing to the physically most important case of infinitesimal strain."

"The discussion of the stress-strain relations rests upon as an axiom generally verified in experience, and on Sir W. Thomson['s] thermodynamical investigation of the existence of the energy-function."

"The theory of elastic crystals adopted is that which has been elaborated by the researches of F. E. Neumann and W. Voigt."

"The conditions of rupture or rather of safety of materials are as yet so little under stood that it seemed best to give a statement of the various theories that have been advanced without definitely adopting any of them."

"In most of the problems considered in the text Saint-Venant's "greatest strain" theory has been provisionally adopted. In connexion with this theory I have endeavoured to give precision to the term ""."

"Among general theorems I have included an account of the deduction of the theory from Boscovich's point-atom hypothesis. This is rendered necessary partly by the controversy that has raged round the number of independent elastic constants, and partly by the fact that there exists no single investigation of the deduction in question which could now be accepted by mathematicians."

"[With regard to] Saint-Venant's theory of the equilibrium of beams... In spite of the work of Prof. Pearson it seems not yet to be understood by English mathematicians that the cross-sections of a bent beam do not remain plane. The old-fashioned notion of a bending moment proportional to the curvature resulting from the extensions and contractions of the fibres is still current. Against the venerable bending moment the modern theory has nothing to say, but it is quite time that it should be generally known that it is not the whole stress, and that the strain does not consist simply of extensions and contractions of the fibres. In explaining the theory I have followed Clebsch's mode of treatment, generalising it so as to cover some of the classes of aeolotropic bodies treated by Saint-Venant."

"[W] are occupied with the principal analytical problems presented by elastic theory. The theory leads in every special case to a system of partial differential equations, and the solution of these subject to conditions given at certain bounding surfaces is required. The general problem is that of solving the general equations with arbitrary conditions at any given boundaries. In discussing this problem I have made extensive use of the researches of Prof. Betti of Pisa, whose investigations are the most general that have yet been given..."

"The case of a solid bounded by an infinite plane and otherwise unlimited is investigated on the lines laid down by Signor Valentino Cerruti, whose analysis is founded on Prof. Betti's general method, and some of the most important particular cases are worked out synthetically by M. Boussinesq's method of potentials. In this connexion I have introduced the last-mentioned writer's theory of "local perturbations", a theory which gives the key to Saint-Venant's "principle of the elastic equivalence of statically equipollent systems of load"."

"The student without previous acquaintance with the subject is advised in all cases to provide the required proofs. It is hoped that he will not then fail to understand the subject for lack of examples, nor waste his time in mere problem grinding."

"The Mathematical Theory of Elasticity is occupied with an attempt to reduce to calculation the state of strain, or relative displacement, within a solid body which is subject to the action of an equilibrating system of forces, or is in a state of slight internal relative motion, and with endeavours to obtain results which shall be practically important in applications to architecture, engineering, and all other useful arts in which the material of construction is solid."

"Alike in the experimental knowledge obtained, and in the analytical methods and results, nothing that has once been discovered ever loses its value or has to be discarded; but the physical principles come to be reduced to fewer and more general ones, so that the theory is brought more into accord with that of other branches of physics, the same general dynamical principles being ultimately requisite and sufficient to serve as a basis for them all."

"[A]lthough, in the case of Elasticity, we find frequent retrogressions on the part of the experimentalist, and errors on the part of the mathematician, chiefly in adopting hypotheses not clearly established or... discredited, in pushing to extremes methods merely approximate, in hasty generalizations, and in misunderstandings of physical principles, yet we observe a continuous progress in all the respects mentioned when we survey the history of the science from the initial enquiries of Galileo to the conclusive investigations of Saint-Venant and Lord Kelvin."

"The first mathematician to consider the nature of the resistance of solids to rupture was Galileo. Although he treated solids as inelastic, not being in possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators."

"[Galileo] endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break it arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem, and, in particular, the determination of this axis is known as Galileo's problem."

"[T]he two great landmarks are the discovery of in 1660, and the formulation of the general equations by Navier in 1821."

"provided the necessary experimental foundation for the theory. When the general equations had been obtained, all questions of the small strain of elastic bodies were reduced to a matter of mathematical calculation."

"Hooke and Mariotte occupied themselves with the experimental discovery of what we now term stress-strain relations. Hooke gave in 1678 the famous law of proportionality of stress and strain which bears his name, in the words "Ut tensio sic vis; that is, the Power of any spring is in the same proportion with the Tension thereof." By "spring" Hooke means... any "springy body," and by "tension" what we should now call "extension," or, more generally, "strain." This law he discovered in 1660, but did not publish until 1676, and then only under the form of an anagram, ceiiinosssttuu. This law forms the basis of the mathematical theory of Elasticity."

"Hooke does not appear to have made any application of [his law] to the consideration of Galileo's problem. This application was made by Mariotte, who in 1680 enunciated the same law independently. He remarked that the resistance of a beam to flexure arises from the extension and contraction of its parts, some of its longitudinal filaments being extended, and others contracted. He assumed that half are extended, and half contracted. His theory led him to assign the position of the axis, required in the solution of Galileo's problem, at one-half the height of the section above the base."

"In the interval between the discovery of Hooke's law and that of the general differential equations of Elasticity by Navier, the attention of those mathematicians who occupied themselves with our science was chiefly directed to the solution and extension of Galileo's problem, and the related theories of the vibrations of bars and plates, and the stability of columns."

"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."

"As soon as the notion of a flexural couple proportional to the curvature was established it could be noted that the work done in bending a rod is proportional to the square of the curvature."

"Daniel Bernoulli suggested to Euler that the differential equation of the elastica could be found by making the integral of the square of the curvature taken along the rod a minimum... Euler, acting on this... was able to obtain the differential equation of the curve..."

"Euler pointed out... that the rod, if of sufficient length and vertical when unstrained, may be bent by a weight attached to its upper end... [and was led] to assign the least length of a column in order that it may bend under its own or an applied weight. Lagrange followed and used his theory to determine the strongest form of column. These two... found [the] length which a column must attain to be bent by its own or an applied weight, and... that for shorter lengths it will be simply compressed, while for greater lengths it will be bent. These researches are the earliest in... elastic stability."

"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... Hooke's Law."

"Coulomb was also the first to consider the resistance [although considered as nonelastic] of thin fibres to torsion... to which Saint-Venant refers under the name I'ancienne thiorie... Coulomb was [also] first to [consider] strain we now call shear, though he considered it in connexion with rupture only... when the shear [permanent set, not an elastic strain] of the material is greater than a certain limit."

"Except Coulomb's, the most important work of the period for the general mathematical theory is the physical discussion of elasticity by Thomas Young. ...[Young,] besides defining his modulus of elasticity, was the first to consider shear as an elastic strain. He called it "detrusion," and noticed that the elastic resistance of a body to shear, [as opposed to] its resistance to extension or contraction, are in general different; but he did not introduce a distinct modulus of rigidity to express resistance to shear. He defined "the modulus of elasticity of a substance" as "a column of the same substance capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length." What we now call "Young's modulus" is the weight of this column per unit of area of its base. This introduction of a definite physical concept, associated with the coefficient of elasticity which descends, as it were from a clear sky, on the reader of mathematical memoirs, marks an epoch in the history of the science."

"During the first period in the history of our science (1638—1820) while these various investigations of special problems were being made, there was a cause at work which was to lead to wide generalizations. This cause was physical speculation concerning the constitution of bodies. In the eighteenth century the Newtonian conception of material bodies, as made up of small parts which act upon each other by means of central forces, displaced the Cartesian conception of a plenum pervaded by "vortices." Newton regarded his "molecules" as possessed of finite sizes and definite shapes, but his successors gradually simplified them into material points. The most definite speculation of this kind is that of Boscovich, for whom the material points were nothing but persistent centres of force. To this order of ideas belong Laplace's theory of capillarity and Poisson's first investigation of the equilibrium of an "elastic surface," but for a long time no attempt seems to have been made to obtain general equations of motion and equilibrium of elastic solid bodies."

"At the end of the year 1820 the fruit of all the ingenuity expended on elastic problems might be summed up as—an inadequate theory of flexure, an erroneous theory of torsion, an unproved theory of the vibrations of bars and plates, and the definition of Young's modulus. But such an estimate would give a very wrong impression of the value of the older researches. The recognition of the distinction between shear and extension was a preliminary to a general theory of strain; the recognition of forces across the elements of a section of a beam, producing a resultant, was a step towards a theory of stress; the use of differential equations for the deflexion of a bent beam and the vibrations of bare and plates, was a foreshadowing of the employment of differential equations of displacement; the Newtonian conception of the constitution of bodies, combined with Hooke's Law, offered means for the formation of such equations; and the generalization of the principle of in the Mécanique Analytique threw open a broad path to discovery in this as in every other branch of mathematical physics."

"Physical Science had emerged from its incipient stages with definite methods of hypothesis and induction and of observation and deduction, with the clear aim to discover the laws by which phenomena are connected with each other, and with a fund of analytical processes of investigation. This was the hour for the production of general theories, and the men were not wanting."

"In... 1821... Fresnel announced his conclusion that the observed facts in regard to the interference of polarised light could be explained only by the hypothesis of transverse vibrations. He showed how a medium consisting of "molecules " connected by central forces might be expected to execute such vibrations and to transmit waves of the required type. Before the time of Young and Fresnel such examples of transverse waves as were known—waves on water, transverse vibrations of strings, bars, membranes and plates—were in no case examples of waves transmitted through a medium; and neither the supporters nor the opponents of the undulatory theory of light appear to have conceived of light waves otherwise than as "longitudinal " waves of condensation and rarefaction, of the type rendered familiar by the transmission of sound."

"The theory of elasticity, and, in particular, the problem of the transmission of waves through an elastic medium now attracted the attention of... Cauchy and Poisson—the former a discriminating supporter, the latter a sceptical critic of Fresnel's ideas. In the future the developments of the theory of elasticity were to be closely associated with the question of the propagation of light, and these developments arose in great part from the labours of these two savants."

"By the Autumn of 1822 Cauchy had discovered most of the elements of the pure theory of elasticity. ...[H]e had generalized the notion of hydrostatic pressure, and he had shown that the stress is expressible by means of six component stresses, and also by means of three purely normal tractions across a certain triad of planes which cut each other at right angles—the "principal planes of stress.""

"[Cauchy] had shown also how the differential coefficients of the three components of displacement can be used to estimate the extension of every linear element of the material, and had expressed the state of strain near a point in terms of six components of strain, and also in terms of the extensions of a certain triad of lines which are at right angles to each other—the "principal axes of strain.""

"[Cauchy] had determined the equations of motion (or equilibrium) by which the stress-components we connected with the forces that are distributed through the volume and with the kinetic reactions. By means of relations between stress-components and strain-components, he had eliminated the stress-components from the equations of motion and equilibrium, and had arrived at equations in terms of the displacements."

"Cauchy obtained his stress-strain relations for isotropic materials by means of two assumptions, viz. : (1) that the relations in question are linear, (2) that the principal planes of stress are normal to the principal axes of strain."

"[Cauchy's] equations... are those which are now admitted for isotropic solid bodies. The methods used in these investigations are quite different from... Navier's... no use is made of the hypothesis of material points and central forces. ...Navier's equations contain a single constant to express the elastic behaviour of a body, while Cauchy's contain two such constants."

"At a later date Cauchy extended his theory to the case of crystalline bodies, and he then made use of the hypothesis of material points between which there are forces of attraction or repulsion."

"Clausius criticized the restrictive conditions which Cauchy imposed upon the arrangement of his material points, but he argued that these conditions are not necessary for the deduction of Cauchy's equations."

"Poisson's] April, 1828... memoir is very remarkable... like Cauchy, [he] first obtains the equations of equilibrium in terms of stress-components, and then estimates the traction across any plane resulting from the "intermolecular" forces. The expressions... involve summations with respect to all the "molecules," situated within the region of "molecular" activity of a given one. Poisson... assumes... summations with respect to angular space about the given "molecule," but not... with respect to distance... The equations of equilibrium and motion of isotropic elastic solids... thus obtained are identical with Navier's."

"Poisson assumed that the irregular action of the nearer molecules may be neglected, in comparison with the action of the remoter ones, which is regular. This assumption is the text upon which Stokes afterwards founded his criticism of Poisson. As we have seen, Cauchy arrived at Poisson's results by the aid of a different assumption. Clausius held that both Poisson's and Cauchy's methods could be presented in unexceptionable forms."

"The revolution which Green effected in the elements of the theory is comparable in importance with that produced by Navier's discovery of the general equations. Starting from what is now called the Principle of the Conservation of Energy he propounded a new method of obtaining these equations."

"[Green] stated his principle and method in the following words:— "In whatever way the elements of any material system may act upon each other, if all the internal forces exerted be multiplied by the elements of their respective directions, the total sum for any assigned portion of the mass will always be the exact differential of some function. But this function being known, we can immediately apply the general method given in the Mécanique Analytique, and which appears to be more especially applicable to problems that relate to the motions of systems composed of an immense number of particles mutually acting upon each other. One of the advantages of this method, of great importance, is that we are necessarily led by the mere process of the calculation, and with little care on our part, to all the equations and conditions which are requisite and sufficient for the complete solution of any problem to which it may be applied.""

"The function here spoken of, with its sign changed, is the potential energy of the strained elastic body per unit of volume, expressed in terms of the components of strain; and the differential coefficients of the function, with respect to the components of strain, are the components of stress. Green supposed the function to be capable of being expanded in powers and products of the components of strain. He therefore arranged it as a sum of homogeneous functions of these quantities of the first, second and higher degrees. Of these terms, the first must be absent, as the potential energy must be a true minimum when the body is unstrained; and, as the strains are all small, the second term alone will be of importance. From this principle Green deduced the equations of Elasticity, containing in the general case 21 constants. In the case of isotropy there are two constants, and the equations are the same as those of Cauchy's first memoir."

"Lord Kelvin has based the argument for the existence of Green's strain-energy-function on the First and Second Laws of Thermodynamics. From these laws he deduced the result that, when a solid body is strained without alteration of temperature, the components of stress are the differential coefficients of a function of the components of strain with respect to these components severally. The same result can be proved to hold when the strain is effected so quickly that no heat is gained or lost by any part of the body."

"Poisson's theory leads to the conclusions that the resistance of a body to compression by pressure uniform all round it is two-thirds of the of the material, and that the resistance to shearing is two-fifths of the Young's modulus. He noted a result equivalent to the first of these, and the second is virtually contained in his theory of the torsional vibrations of a bar."

"The observation that resistance to compression and resistance to shearing are the two fundamental kinds of elastic resistance in isotropic bodies was made by Stokes, and he introduced definitely the two principal moduluses of elasticity... the "modulus of compression" and the "rigidity," as they are now called."

"From Hooke's Law and from considerations of symmetry [Stokes] concluded that pressure equal in all directions round a point is attended by a proportional compression without shear, and that shearing stress is attended by a corresponding proportional shearing strain."

"As an experimental basis for Hooke's Law [Stokes] cited the fact that bodies admit of being thrown into states of isochronous vibration."

"By a method analogous to that of Cauchy's first memoir, but resting on the above-stated experimental basis, [Stokes] deduced the equations with two constants which had been given by Cauchy and Green. Having regard to the varying degrees in which different classes of bodies—liquids, soft solids, hard solids—resist compression and distortion, he refused to accept the conclusion from Poisson's theory that the modulus of compression has to the rigidity the ratio 5 : 3. He pointed out that, if the ratio of these moduluses could be regarded as infinite, the ratio of the velocities of "longitudinal " and " transverse " waves would also be infinite, and then, as Green had already shown, the application of the theory to optics would be facilitated."

"The hypothesis of material points and central forces does not now hold the field. ...Of much greater importance have been the development of the atomic theory in Chemistry and of statistical molecular theories in Physics, the growth of the doctrine of energy, the discovery of electric radiation. It is now recognized that a theory of atoms must be part of a theory of the æther, and that the confidence which was once felt in the hypothesis of central forces between material points was premature. To determine the laws of the elasticity of solid bodies without knowing the nature of the æthereal medium or the nature of the atoms, we can only invoke the known laws of energy as was done by Green and Lord Kelvin; and we may place the theory on a firm basis if we appeal to experiment to support the statement that, within a certain range of strain, the strain-energy-function is a quadratic function of the components of strain, instead of relying, as Green did, upon an expansion of the function in series."

"The problem of curved plates or shells was first attacked from the point of view of the general equations of Elasticity by H. Aron. He expressed the geometry of the middle-surface by means of two parameters after the manner of Gauss, and he adapted to the problem the method which Clebsch had used for plates. He arrived at an expression for the potential energy of the strained shell which is of the same form as that obtained by Kirchhoff for plates, but the quantities that define the curvature of the middle-surface were replaced by the differences of their values in the strained and unstrained states."

"E. Mathieu adapted to the problem [of curved plates or shells ] the method which Poisson had used for plates. He observed that the modes of vibration possible to a shell do not fall into classes characterized respectively by normal and tangential displacements, and he adopted equations of motion that could be deduced from Aron's formula for the by retaining the terms that depend on the stretching of the middle-surface only."

"Lord Rayleigh... concluded from physical reasoning that the middle-surface of a vibrating shell remains unstretched, and determined the character of the displacement of a point of the middle-surface in accordance with this condition. The direct application of the Kirchhoff-Gehring method led to a formula for the potential energy of the same form as Aron's and to equations of motion and boundary conditions which were difficult to reconcile with Lord Rayleigh's theory. Later investigations have shown that the extensional strain which was thus proved to be a necessary concomitant of the vibrations may be practically confined to a narrow region near the edge of the shell, but that, in this region, it may be so adjusted as to secure the satisfaction of the boundary conditions while the greater part of the shell vibrates according to Lord Rayleigh's type."

"Whenever very thin rods or plates are employed in constructions it becomes necessary to consider the possibility of , and thus there arises the general problem of elastic stability. [T]he first investigations... of this kind were made by Euler and Lagrange. ...In all [isolated problems] two modes of equilibrium with the same type of external forces are possible, and the ordinary proof of the determinacy of the solution of the equations of Elasticity is defective."

"A general theory of elastic stability has been proposed by G. H. Bryan. He arrived at the result that the theorem of determinacy cannot fail except in cases where large relative displacements can be accompanied by very small strains, as in thin rods and plates, and in cases where displacements differing but slightly from such as are possible in a rigid body can take place, as when a sphere is compressed within a circular ring of slightly smaller diameter. In all cases where two modes of equilibrium are possible the criterion for determining the mode that will be adopted is given by the condition that the energy must be a minimum."

"The history of the mathematical theory of Elasticity shows clearly that the development of the theory has not been guided exclusively by considerations of its utility for technical Mechanics. Most of the men by whose researches it has been founded and shaped have been more interested in Natural Philosophy than in material progress, in trying to understand the world than in trying to make it more comfortable."

"[D]iscussions... concerning the number and meaning of the elastic constants have thrown light on most recondite questions concerning the nature of molecules and the mode of their interaction."

"Even in the more technical problems, such as the transmission of force and the resistance of bars and plates, attention has been directed, for the most part, rather to theoretical than to practical aspects of the questions. To get insight into what goes on in impact, to bring the theory of the behaviour of thin bars and plates into accord with the general equations—these and such-like aims have been more attractive... than endeavours to devise means for effecting economies in engineering constructions or to ascertain the conditions in which structures become unsafe."

"The... fact that most great advances in Natural Philosophy have been made by men who had a first-hand acquaintance with practical needs and experimental methods has often been emphasized; and, although the names of Green, Poisson, Cauchy show that the rule is not without important exceptions, yet it is exemplified well in the history of our science."

"Whenever, owing to any cause, changes take place in the relative positions of the parts of a body the body is said to be "strained." A very simple example of a strained body is a stretched bar."

"Let l_0 be the length before stretching, and l the length when stretched. Then (l - l_0)/l_0is a number (generally a very small fraction) which is called the extension..."

"Let e denote the extension of the bar, so that its length is increased in the ratio 1 + e : 1 ...[V]olume is increased by stretching the bar, but not in the ratio 1 + e : 1. When the bar is stretched longitudinally it contracts laterally... If the linear lateral contraction is e^\prime, the sectional area is diminished in the ratio (1 - e^\prime)^2 : 1, and the volume in question is increased in the ratio (1 + e) (1 - e^\prime)^2 : 1. In... a bar under tension e^\prime is a certain multiple of e, say \sigma e... [with] \sigma... about \frac{1}{3} or \frac{1}{4} for very many materials. If e is very small and e^2 is neglected, the areal contraction is 2\sigma e, and the cubical dilatation is (1 - 2\sigma)e."

"[M]easure the coordinate z along the length of the [vertical] bar. Any particle of the bar which has the coordinates x, y, z when the weight is not attached will move after the attachment of the weight into a new position. Let the particle which was at the origin move through a distance z_0, then the particle which was at (x, y, z) moves to the point of which the coordinates arex(1 - \sigma e), \qquad y(1 - \sigma e), \qquad z_0 + (z - z_0)(1 + e)."

"Love was the first investigator to present a successful approximation shell theory based on classical elasticity. To simplify the strain-displacement relationships and, consequently, the constitutive relations, Love [in this Treatise] introduced the following assumptions, known as the first approximations and commonly termed the Kirchoff-Love hypotheses..."

"[Love's] early books on elasticity, theoretical mechanics, and calculus have been used by many generations of students, whilst the much enlarged edition of his Treatise on Elasticity is a monumental work."

"History of science"

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

"Engineering"

"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."

"When I recall the days of twenty years ago, when the conception of the physical quantum of 'action' was first beginning to disentangle itself from the surrounding mass of available experimental facts, and when I look back upon the long and tortuous road which finally led to its disclosure, this development strikes me at times as a new illustration of Goethe's saying, that 'man errs, so long as he is striving'."

"The pursuit of a goal, the brightness of which is undimmed by initial failure, is an indispensable condition, though by no means a guarantee, of final success. In my own case, such a goal has been for many years the solution of the question of the distribution of energy in the normal spectrum of radiant heat."

"The discovery by Gustav Kirchhoff that the quality of the heat radiation produced in an enclosure surrounded by any emitting or absorbing bodies whatsoever, all at the same temperature, is entirely independent of the nature of such bodies, established the existence of a universal function, which depends only upon the and the , and is entirely independent of the particular properties of the substance. ...[T]his remarkable function promised a deeper insight into the relation between energy and temperature, which is the principal problem of thermodynamics and therefore also of the entire field of ."

"A most suitable body... seemed H. Hertz's rectilinear oscillator (dipole) whose laws of emission for a given frequency he had just... developed. If a number of such oscillators be distributed in an enclosure surrounded by reflecting walls, there would take place, in analogy with sources and resonators in... sound, an exchange of energy by means of the emission and reception of electro-magnetic waves, and... corresponding to Kirchoff's law should establish itself in the vacuum-enclosure."

"I expected, in a [naive] way... that the laws of classical electrodynamics would suffice, if one adhered sufficiently to generalities and avoided too special hypotheses..."

"I... first developed in... general terms... the laws of the emission and absorption of a linear ... by a... circuitous route which might have been avoided had I used the electron theory which had just been put forward by H. A. Lorentz."

"The outcome of this long series of investigations... was the establishment of a general relation between the energy of a resonator of a definite free frequency and the energy radiation of the corresponding spectral region in the surrounding field in equilibrium with it."

"The remarkable result was obtained that this relation is independent of the nature of the resonator, and in particular of its coefficient of damping—a result which... introduced the simplification that the energy of the radiation could be replaced by the energy of the resonator, so that a simple system of one degree of freedom could be substituted for a complicated system having many degrees of freedom."

"But this result constituted only a preparatory advance towards the attack on the main problem... [M]y original hope [was] that the radiation emitted by the resonator would differ in some characteristic way from the absorbed radiation... The resonator reacted only to those rays which were emitted by itself, and exhibited no trace of resonance to neighbouring spectral regions."

"[M]y suggestion that the resonator might be able to exert a one-sided, i. e. irreversible, action on the energy of the surrounding radiation field called forth the emphatic protest of Ludwig Boltzmann... showing that according to... classical dynamics... the processes I was considering could take place in... the opposite sense. Thus a spherical wave emitted from a resonator when reversed shrinks... continually decreasing... on to the resonator, is absorbed by it, and so permits the resonator to send out again into space the energy formerly absorbed in the direction from which it came."

"[I]t became more and more evident that... an essential link was missing which should lead to... comprehension... The only way out... was to attack the problem from the opposite side, from... thermodynamics, a domain in which I felt more at home."

"[M]y previous studies on the second law of thermodynamics served me here... in that my first impulse was to bring not the but the of the resonator into relation with its energy, more accurately not the entropy itself but its with respect to the energy... [T]his differential coefficient... has a direct physical significance for the irreversibility of the exchange of energy between the resonator and the radiation."

"But as I was... too much devoted to pure phenomenology to inquire more closely into the relation between entropy and probability, I felt compelled to limit myself to the available experimental results. Now, at that time... 1899, interest was centred on the law of the distribution of energy... proposed by W. Wien... On calculating the relation following from this law between the entropy and energy of a resonator the remarkable result is obtained that the reciprocal value of the above differential coeffcient... R, is proportional to the energy. This extremely simple relation can be regarded as an adequate expression of Wien's law..."

"I believed... that the basis of the law of the distribution of energy could be expressed by the theorem that the value of R is proportional to the energy. But in view of the results of new measurements this conception soon proved untenable."

"[T]wo simple limits were established by direct observation for the function R: for small energies proportionality to the energy, for large energies proportionality to the square of the energy. Nothing... seemed simpler than to put in the general case R equal to the sum of a term proportional to the first power and another proportional to the square of the energy... and thus was found a new radiation formula which... has withstood experimental examination fairly satisfactorily."

"But even if this radiation formula should prove to be absolutely accurate it would after all be only an formula found by happy guesswork..."

"I was... occupied with the task of giving it a real physical meaning, and this... led me, along Boltzmann's line... to the consideration of the relation between and probability... after some weeks of the most intense work of my life clearness began to dawn... and an unexpected view revealed itself..."

", according to Boltzmann, is a measure of a physical probability, and the meaning of the second law of thermodynamics is that the more probable a state is, the more frequently will it occur in nature."

"[W]hat one measures are only the differences of entropy, and never entropy itself, and consequently one cannot speak... of the absolute entropy of a state. But nevertheless the introduction of an appropriately defined absolute magnitude of entropy is... recommended... by its help certain general laws can be formulated with great simplicity."

"[T]he case is... the same as with energy. Energy... cannot itself be measured; only its differences can."

"[T]he concept used by our predecessors was not energy but work, and even Ernst Mach, who devoted much attention to the law of but... avoided all speculations exceeding the limits of observation, always abstained from speaking of energy..."

"[I]n the early days of one was content to deal with heats of reaction, that is to say again with differences of energy, until emphasized that... calculations could be... shortened if energies instead of calorimetric numbers were used."

"The additive constant which... remained undetermined for energy was later finally fixed by the relativistic law of the proportionality between energy and inertia."

"As in the case of energy, it is now possible to define an absolute value of entropy, and thus of physical probability, by fixing the additive constant so that together with the energy (or better still, the temperature) the entropy also should vanish."

"Such... led to a comparatively simple method of calculating the physical probability of a given distribution of energy in a system of resonators, which yielded precisely the same expression for entropy as that corresponding to the radiation law; and it gave me particular satisfaction, in compensation for the many disappointments... to learn from Ludwig Boltzmann of his interest and... acquiescence in my... reasoning."

"To work out these probability considerations the knowledge of two universal constants is required, each of... independent meaning, so... evaluation of these... from the radiation law could serve as an a posteriori test whether the... process is merely a mathematical artifice or has a true physical meaning."

"The first constant... is connected with the definition of temperature. If temperature were defined as the mean of a molecule in a , which is a minute energy indeed, this constant would have the value ⅔. But in the conventional scale of temperature the constant ...[instead] assumes an extremely small value... intimately connected with the energy of a single molecule... [I]ts accurate determination would lead to the calculation of the mass of a molecule and... associated magnitudes. This constant is frequently termed Boltzmann's constant, although to the best of my knowledge Boltzmann... never introduced it (...he, as appears from... his statements, never believed it would be possible to determine this constant accurately)..."

"Nothing can better illustrate the rapid progress of experimental physics within the last twenty years than the fact that... [by] a host of methods ...the mass of a single molecule can be measured with almost the same accuracy as that of a planet."

"While at the time when I carried out this calculation on the basis of the radiation law an exact test of the value... was... impossible... it was not long before E. Rutherford and H. Geiger succeeded, by... a direct count of the α-particles, in determining the value of the electrical as 4.65 10-10, the agreement... with my value 4.69 10-10... a decisive confirmation of my theory. ...[M]ethods ...by E. Eegener, R. A. Millikan, and others... have led to a but slightly higher value."

"Much less simple... was the interpretation of the second universal constant of the radiation law... the product of energy and time (...a first calculation to 6.55 10-27 erg. sec.) I called the elementary quantum of action."

"While this constant was absolutely indispensable to... a correct expression for —for only with its aid could be determined the magnitude of the 'elementary region' or 'range' of probability, necessary for the statistical treatment of the problem—it obstinately withstood all attempts at fitting it... into the frame of the classical theory. So long as it could be regarded as infinitely small, that is to say for large values of energy or long periods of time, all went well; but in the general case a difficulty arose... which became the more pronounced the weaker and... more rapid the oscillations."

"The failure placed one before the dilemma: either the quantum of action was only a fictitious magnitude, and... the entire deduction from the radiation law was illusory and a mere juggling with formulae, or there is at the bottom of this method of deriving the radiation law some true physical concept."

"If the latter were the case, the would have to play a fundamental role in physics, heralding the advent of a new state of things, destined, perhaps, to transform completely our physical concepts which since the introduction of the infinitesimal calculus by Leibniz and Newton have been founded upon the assumption of the continuity of all causal chains of events."

"Experience has decided for the second alternative. But that the decision should come so soon... was due not to the examination of the law of distribution of the energy of heat radiation, still less to my special deduction of this law, but to the steady progress of the work of those investigators who have applied the concept of the quantum of action to their researches."

"The first advance... was made by A. Einstein, who... pointed out that the... quanta of energy associated with the quantum of action seemed capable of explaining... a series of remarkable properties of light action discovered experimentally, such as Stokes's rule, the emission of electrons, and the ionization of gases, and on the other hand, by the identification of the expression for the energy of a system of resonators with the energy of a solid body, derived a formula for the specific heat of solid bodies which on the whole represented it correctly as a function of temperature, more especially exhibiting its decrease with falling temperature."

"With regard to specific heat of solid bodies, Einstein's view, which rests on the assumption of a single free period of the atoms, was extended by M. Born and Th. von Karman to the case which corresponds better to reality, viz. that of several free periods; while P. Debye, by a bold simplification of the assumptions as to the nature of the free periods, succeeded in developing a comparatively simple formula for the specific heat of solid bodies which excellently represents its values, especially those for low temperatures obtained by W. Nernst and his pupils, and... compatible with the elastic and optical properties of such bodies."

"But the influence of the quanta asserts itself also in the case of the specific heat of gases. At the very outset it was pointed out by W. Nernst that to the energy quantum of vibration must correspond an energy quantum of rotation, and it was therefore... expected that the rotational energy of gas molecules would also vanish at low temperatures. ...That 'quantized' rotations of gas molecules... do actually occur in nature can no longer be doubted... although a[n] exhaustive explanation of... rotation spectra is still outstanding."

"The inverse of the process of producing light quanta by the impact of electrons is the emission of electrons on exposure to light-rays, or s, and here... energy quanta following from the action quantum and the vibration period play a characteristic role, as was early recognized from the striking fact that the velocity of the emitted electrons depends not upon the intensity but only on the colour of the impinging light. [T]the relations to the light quantum, pointed out by Einstein, have proved successful in every direction... shown especially by R. A. Millikan, by measurements of the velocities of emission of electrons, while the importance of the light quantum in inducing photo-chemical reactions was disclosed by E. Warburg."

"[T]he results... hitherto quoted... taken in their totality, form an overwhelming proof of the existence of the quantum of action, the quantum hypothesis received its strongest support from the theory of the structure of atoms (Quantum Theory of Spectra) proposed and developed by Niels Bohr... the long-sought key to the gates of the wonderland of spectroscopy which since the discovery of spectrum analysis... stubbornly refused to yield. And... once clear, a stream of new knowledge poured in a sudden flood, not only over this... field but into the adjacent territories of physics and chemistry."

"Its first brilliant success was the derivation of Balmer's formula for the spectrum series of and , together with the reduction of the universal constant of Rydberg to known magnitudes; and even the small differences of the Rydberg constant for these two gases appeared as a necessary consequence of the slight wobbling of the massive atomic nucleus (accompanying the motion of electrons around it). As a sequel came the investigation of other series in the visual and especially the X-ray spectrum aided by Ritz's resourceful combination principle, which only now was recognized in its fundamental significance."

"But whoever may have still felt inclined... in the face of this almost overwhelming agreement... to believe it... a coincidence, must... give up... doubt when A. Sommerfeld deduced, by a logical extension of the laws of the distribution of quanta in systems with several degrees of freedom, and by a consideration of the variability of inert mass required by the principle of relativity, that magic formula before which the spectra of both and revealed the mystery of their ... by the most delicate measurements...of F. Paschen..."

"P. Epstein achieved a complete explanation of the of the electrical splitting of spectral lines, P. Debye obtained a simple interpretation of the K-series of the X-ray spectrum investigated by , and then... a long series of further researches... illuminated... the dark secret of atomic structure."

"[T]he quantum of action, which in every one of the many and most diverse processes has always the same value, namely 6.52 10-27 erg. sec., deserves to be... incorporated into the system of the universal s."

"[A]t just the same time as the idea of general relativity arose... nature revealed, precisely... where ...least ...expected, an absolute and strictly unalterable unit, by means of which the amount of action contained in a space-time element can be expressed by a perfectly definite number, and thus is deprived of its former relative character."

"[T]he mere introduction of the quantum of action does not yet mean that a true Quantum Theory has been established. Nay, the path which research has yet to cover... is perhaps not less long than that from the discovery of the velocity of light by Olaf Römer to the foundation of Maxwell's theory of light."

"The difficulties which the introduction of the quantum of action into the well-established classical theory has encountered from the outset... have gradually increased rather than diminished; and although research... has... passed over some of them, the remaining gaps in the theory are the more distressing..."

"[W]hat in Bohr's theory served as the basis of the laws of action consists of certain hypotheses which a generation ago would doubtless have been flatly rejected by every physicist. That with the atom certain quantized orbits [i.e. picked out on the quantum principle] should play a special role could well be granted; somewhat less easy to accept is the further assumption that the electrons moving on these curvilinear orbits, and therefore accelerated, radiate no energy. But that the sharply defined frequency of an emitted light quantum should be different from the frequency of the emitting electron would be regarded... in the classical school as monstrous and almost inconceivable. But numbers decide... the tables have been turned."

"While originally it was a question of fitting in with as little strain as possible a new and strange element into an existing system... generally regarded as settled, the intruder... having won an assured position, now has assumed the offensive; and... is about to blow up the old system... The only question... is, at what point and to what extent this will happen."

"[O]ut of the classical theory the great principles of thermodynamics will not only maintain intact their central position in the quantum theory, but will perhaps even extend their influence."

"The significant part played in the origin of the classical thermodynamics by mental experiments is now taken over in the quantum theory by P. Ehrenfest's hypothesis of the adiabatic invariance; and just as the principle introduced by R. Clausius, that any two states of a material system are mutually interconvertible on suitable treatment by reversible processes, formed the basis for the measurement of , just so do the new ideas of Bohr show a way into the midst of the wonderland he has discovered."

"[O]ne... question... will... lead to an extensive elucidation of the entire problem. What happens to the energy of a light-quantum after its emission? Does it pass outwards in all directions, according to Huygens's wave theory, continually increasing in volume and tending towards infinite dilution? Or does it, as in Newton's emanation theory, fly like a projectile in one direction only? In the former case the quantum would never again be in a position to concentrate its energy at a spot strongly enough to detach an electron from its atom; while in the latter case it would be necessary to sacrifice the chief triumph of Maxwell's theory — the continuity between the static and the dynamic fields — and with it the classical theory of the interference phenomena which accounted for all their details, both alternatives leading to consequences very disagreeable..."

"[S]cience will some day master the dilemma, and what may now appear to us unsatisfactory will appear from a higher standpoint as endowed with a particular harmony and simplicity. But until... [then] the problem of the quantum of action will not cease to stimulate research, and the greater the difficulties encountered in its solution the greater will be its significance for the broadening and deepening of all our physical knowledge."

"The theory of relativity is intimately connected with the theory of space and time. I shall therefore begin with a brief investigation of the origin of our ideas of space and time, although in doing so I know that I introduce a controversial subject. The object of all science, whether natural science or psychology, is to co-ordinate our experiences and to bring them into a logical system. How are our customary ideas of space and time related to the character of our experience?"

"Before the development of the theory of relativity it was known the principle of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, that of the energy tensor, which is important of the further development of the theory of relativity."

"A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is: (Inertial mass) \cdot (Acceleration) = (Intensity of the gravitational field) \cdot (Gravitational mass). It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body."

"A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this a real straight line. The natural, that is, the simplest, generalization of the straight line which is meaningful in the system of concepts of the general (Riemannian) theory of invariants is that of the straightest, or geodesic, line."

"Some try to explain Hubble's shift of spectral lines by means other than the Doppler effect. There is, however, no support for such a conception in the known physical facts."

"It is the essential achievement of the general theory of relativity that it has freed physics from the necessity of introducing the "inertial system" (or inertial systems). This concept is unsatisfactory for the following reason: without any deeper foundation it singles out certain co-ordinate systems among all conceivable ones. It is then assumed that the laws of physics hold only for such inertial systems (e.g. the law of inertia and the law of the constancy of the velocity of light). Thereby, space as such is assigned a role in the system of physics that distinguishes it from all other elements of physical description. It plays a determining role in all process, without in its turn being influenced by them. Though such a theory is logically possible, it is on the other hand rather unsatisfactory. Newton had been fully aware of this deficiency, but he had also clearly understood that no other path was open to physics in his time. Among the later physicians it was above all Ernst Mach who focussed attention on this point."

"One can give good reasons why reality cannot at all be represented by a continuous field. From the quantum phenomena it appears to follow with certainty that a finite system of finite energy can be completely described by a finite set of numbers (quantum numbers). This does not seem to be in accordance with a continuum theory, and must lead to an attempt to find a purely algebraic theory for the description of reality. But nobody knows how to obtain the basis of such a theory."

"In May 1921 Albert Einstein delivered a series of lectures at Princeton University on the broad topic of relativity. The lectures form a unified survey of the basic concepts of relativity. Beginning with the pre-relativity physics of Newton (or perhaps, more correctly, the "three dimensional" relativity of Newton) Einstein lays the foundation for "four dimensional" relativity primarily from the postulational standpoint. The development of special relativity is followed by the formulation of the general theory leading up to the Schwarzschild line element and the cosmological problem."

"The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration: and practical. To practical mechanics all the manual arts belong, from which mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that mechanics is so distinguished from geometry, that what is perfectly accurate is called geometrical, what is less so, is called mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy is an imperfect mechanic; and if any could work with perfect accuracy, he would be the most perfect mechanic of all; for the description if right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn; for it requires that the learner should first be taught to describe these accurately, before he enters upon geometry; then it shows how by these operations problems may be solved."

"Our design not respecting arts, but philosophy, and our subject not manual but natural powers, we consider chiefly those things which relate to gravity, levity, elastic force, the resistance of fluids, and the like forces, whether attractive or impulsive; and therefore we offer this work as the mathematical principles if philosophy; for all the difficulty of philosophy seems to consist in this—from the phænomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phænomena."

"I wish we could derive the rest of the phænomena of nature by the same kind of reasoning from mechanical principles; for I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy."

"[R]ational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated....For I am induced by many reasons to suspect that they may all depend upon certain forces by which the particles of bodies, by some causes hitherto unknown, are either mutually impelled towards each other, and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy."

"In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them."

"An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertiæ only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force."

"I do not define time, space, place and motion, as being well known to all. Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common."

"Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year."

"Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is vulgarly taken for immovable space; such is the dimension of a subterraneous, an æreal, or celestial space, determined by its position in respect of the earth. Absolute and relative space, are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be perpetually mutable."

"Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their superfices, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same thing with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the place of the whole is the same thing with the sum of the places of the parts, and for that reason, it is internal, and in the whole body."

"It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate: for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions."

"Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a ship under sail, the relative place of a body is that part of the ship which the body possesses; or that part of its cavity which the body fills, and which therefore moves together with the ship: and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved."

"Absolute time, in astronomy, is distinguished from relative, by the equation or correction of the vulgar time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality for their more accurate deducing of the celestial motions."

"It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the true, or equable, progress of absolute time is liable to no change."

"The duration or perseverance of the existence of things remains the same, whether the motions are swift or slow, or none at all: and therefore it ought to be distinguished from what are only sensible measures thereof; and out of which we collect it, by means of the astronomical equation. The necessity of which equation, for determining the times of a phenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter."

"As the order of the parts of time is immutable, so also is the order of the parts of space."

"All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be moveable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute motions.But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; and then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of others may be referred.But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions whether any of these do keep the same position to that remote body; it follows that absolute rest cannot be determined from the position of bodies in our regions."

"It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forward, arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, those that are relatively at rest within them, will partake of their motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, beside their translation from near the surrounding ones, partake likewise of their true motions; and though that translation were not made they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel; but, if the shell moves, the kernel will also move, as being part of the whole, without any removal from near the shell."

"[I]f a place is moved, whatever is placed therein moves along with it; and therefore a body, which is moved from a place in motion, partakes also of the motion of its place. Upon which account, all motions, from places in motion, are no other than parts of entire and absolute motions; and every entire motion is composed of the motion of the body out of its first place, and the motion of this place out of its place; and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than by immovable places; and for that reason I did before refer those absolute motions to immovable places, but relative ones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute immovable space."

"The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body."

"The effects which distinguish absolute from relative motion are, the forces of receding from the axe of . For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned about that the cord is strongly twisted, then filled with water, and held at rest together with the water; after, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for some time in this motion; the surface of the water will at first be plain, as before the vessel began to move: but the vessel, by gradually communicating its motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be measured by this endeavour. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavour to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavour to recede from the axis; and this endeavour showed the real circular motion of the water perpetually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect; but relative motions, in one and the same body, are innumerable, according to the various relations it bears to external bodies, and like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavour to recede from the axis of their motions.Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place and motion, their measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. Upon which account, they do strain the sacred writings, who there interpret those words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities themselves with their relations and vulgar measures."

"It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate: for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindermost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the translation of the globes among the bodies, we should find the determination of their motions. But how we are to collect the true motions from their causes, effects, and apparent differences; and, vice versa, how from the motions, either true or apparent, we may come to the knowledge of their causes and effects, shall be explained more at large in the following tract. For to this end it was that I composed it."

"Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon."

"The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."

"To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts."

"We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances."

"[I]t is not to be conceived that mere mechanical causes could give birth to so many regular motions, since the comets range over all parts of the heavens in very eccentric orbits; for by that kind of motion they pass easily through the orbs of the planets, and with great rapidity; and in their aphelions, where they move the slowest, and are detained the longest, they recede to the greatest distances from each other, and thence suffer the least disturbance from their mutual attractions."

"This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another."

"This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite; he is not duration or space, but he endures and is present. He endures for ever, and is every where present; and by existing always and every where, he constitutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no where. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always and every where. Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, nor touched; nor ought he to be worshipped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final causes: we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy."

"Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phænomena, and I frame no hypotheses; for whatever is not deduced from the phænomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy."

"Isaac Newton's Principia Mathematica is indisputably one of the important and influential books ever written, yet it is scarcely read. Latin is not the problem, for translations in English and French have done little to expand its readership. Those who have attempted to read the Principia quickly recognize the problem: The geometrical style of mathematics is almost opaque to the modern reader."

"The first edition of Isaac Newton's famous Principia mathematica (1687) contains only one reference to the Scriptures and one mention of God and . Thus, there is superficial evidence to suggest that this pivotal work of physics is a mostly secular book that is not fundamentally associated with theology and natural theology. The fact that the General Scholium – with its overt theological and natural theological themes – was only added to the Principia a quarter-century later with the second edition of 1713 may also suggest that this theology came as an afterthought and is therefore not integral to the conceptual structure of the Principia. Moreover, the relative paucity of theology in the first edition, combined with the evidence of the appended General Scholium of 1713, could be used as evidence of a ‘theological turn’ in Newton's thought after 1687. This article uses evidence from Newton's private manuscripts to argue that there is more theology in all three editions of the Principia than a simple reading of the published text would imply."

"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."

"To explain all nature is too difficult a task for any one man or even for any one age. 'Tis much better to do a little with certainty, & leave the rest for others that come after you, than to explain all things by conjecture without making sure of any thing."

"My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which I shall premise the following Definitions and Axioms."

"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?"

"Is not Fire a Body heated so hot as to emit Light copiously? For what else is a red hot Iron than Fire? And what else is a burning Coal than red hot Wood?"

"Do not great Bodies conserve their heat the longest, their parts heating one another, and may not great dense and fix'd Bodies, when heated beyond a certain degree, emit Light so copiously, as by the Emission and Re-action of its Light, and the Reflexions and Refractions of its Rays within its Pores to grow still hotter, till it comes to a certain period of heat, such as is that of the Sun?"

"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 intermediate 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 stow, 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 Pression or Motion propagated through a fluid medium?"

"[T]o make way for the regular and lasting Motions of the Planets and Comets, it's necessary to empty the Heavens of all Matter, except perhaps some very thin Vapours, Steams or Effluvia, arising from the Atmospheres of the Earth, Planets and Comets, and from such an exceedingly rare Æthereal Medium...A dense Fluid can be of no use for explaining the Phænomena of Nature, the Motions of the Planets and Comets being better explain'd without it. It serves only to disturb and retard the Motions of those great Bodies, and make the frame of Nature languish: And in the Pores of Bodies, it serves only to stop the vibrating Motions of their Parts, wherein their Heat and Activity consists. And as it is of no use, and hinders the Operations of Nature, and makes her languish, so there is no evidence for its Existence, and therefore it ought to be rejected. And if it be rejected, the Hypotheses that Light consists in Pression or Motion propagated through such a Medium, are rejected with it."

"And for rejecting such a Medium, we have the authority of those the oldest and most celebrated philosophers of ancient Greece and Phoenicia, who made a Vacuum and Atoms and the Gravity of Atoms the first Principles of their Philosophy, tacitly attributing Gravity to some other Cause than dense Matter. Later Philosophers banish the Consideration of such a Cause out of natural Philosophy, feigning Hypotheses for explaining all things mechanically, and referring other Causes to Metaphysicks: Whereas the main Business of natural Philosophy is to argue from Phænomena without feigning Hypotheses, and to deduce Causes from Effects, till we come to the very first Cause, which certainly is not mechanical; and not only to unfold the Mechanism of the World, but chiefly to resolve these such such like Questions."

"What is there in places empty of matter? and Whence is it that the sun and planets gravitate toward one another without dense matter between them? Whence is it that Nature doth nothing in vain? and Whence arises all that order and beauty which we see in the world? To what end are comets, and whence is it that Planets move all one and the same way in Oorbs concentrick, while comets move all manner of ways in Orbs very excentrick; and what hinders the fix'd Stars from falling upon one another?"

"[D]oes it not appear from Phænomena that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite Space, as it were in his Sensory, sees the things themselves intimately, and throughly perceives them, and comprehends them wholly by their immediate presence to himself."

"[U]nless they be exceeding rare, a great Objection arises from the regular and very lasting Motions of the Planets and Comets in all manner of Courses through the Heavens."

"How came the Bodies of Animals to be contrived with so much Art, and for what ends were their several Parts? Was the Eye contrived without Skill in Opticks, and the Ear without Knowledge of Sounds? How do the Motions of the Body follow from the Will, and whence is the Instinct in Animals? Is not the Sensory of Animals that place to which the sensitive Substance is present, and into which the sensible Species of Things are carried through the Nerves and Brain, that there they may be perceived by their immediate presence to that Substance? And these things being rightly dispatch'd, does it not appear from Phænomena that there is a Being incorporeal, living, intelligent, omnipresent, who in infinite Space, as it were in his Sensory, sees the things themselves intimately, and throughly perceives them, and comprehends them wholly by their immediate presence to himself: Of which things the Images only carried through the Organs of Sense into our little Sensoriums, are there seen and beheld by that which in us perceives and thinks. And though every true Step made in this Philosophy brings us not immediately to the Knowledge of the first Cause, yet it brings us nearer to it, and on that account is to be highly valued."

"[T]he main Business of natural Philosophy is to argue from Phænomena without feigning Hypotheses, and to deduce Causes from Effects, till we come to the very first Cause, which certainly is not mechanical; and not only to unfold the Mechanism of the World, but chiefly to resolve these and such like Questions."

"Are not the rays of Light very small Bodies emitted from shining Substances? ... Pellucid Substances act upon Rays of Light at a distance in refracting, reflecting, and inflecting them, and the Rays mutually agitate the Parts of those Substances at a distance for heating them; and this Action and Re-action at a distance very much resembles an attractive Force between Bodies."

"Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the Particles of Light which enter into their Composition? ... The changing of Bodies into Light, and Light into Bodies, is very conformable to the Course of Nature, which seems delighted with Transmutations."

"Have not the small Particles of Bodies certain Powers, Virtues, or Forces, by which they act at a distance, not only upon the Rays of Light for reflecting, refracting, and inflecting them, but also upon one another for producing a great Part of the Phenomena of Nature?"

"How these Attractions may be perform'd, I do not here consider. What I call Attraction may be perform'd by impulse, or by some other means unknown to me. I use that Word here to signify only in general any Force by which Bodies tend towards one another, whatsoever be the Cause. For we must learn from the Phaenomena of Nature what Bodies attract one another, and what are the Laws and Properties of the Attraction, before we enquire the Cause by which the Attraction is perform'd. The Attractions of Gravity, Magnetism and Electricity, react to very sensible distances, and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape Observation; and perhaps electrical Attraction may react to such small distances, even without being excited by Friction."

"[I]t seems probable to me that God, in the Beginning, formed Matter in solid, massy, hard, impenetrable, moveable Particles, of such Sizes and Figures, and with such other Properties, and in such Proportion to Space, as most conduc'd to the end for which he form'd them; and that these primitive Particles, being Solids, are incomparably harder than any porous Bodies compounded of them; even so very hard as never to wear or break in pieces; no ordinary Power being able to divide what God himself made one in the first Creation. While the Particles continue entire, they may compose Bodies of one and the same Nature and Texture in all Ages: But should they wear away, or break in pieces, the Nature of Things depending on them, would be changed."

"As in Mathematicks, so in Natural Philosophy, the Investigation of difficult Things by the Method of Analysis, ought ever to precede the Method of Composition."

"By this way of Analysis we may proceed from Compounds to Ingredients, and from Motions to the Forces producing them; and in general, from Effects to their Causes, and from particular Causes to more general ones, till the Argument end in the most general. This is the Method of Analysis: and the Synthesis consists in assuming the Causes discover'd, and establish'd as Principles, and by them explaining the Phænomena proceeding from them, and proving the Explanations."

"And since Space is divisible in infinitum, and Matter is not necessarily in all places, it may be also allow'd that God is able to create Particles of Matter of several Sizes and Figures, and in several Proportions to Space, and perhaps of different Densities and Forces, and thereby to vary the Laws of Nature, and make Worlds of several sorts in several Parts of the Universe."

"[T]o derive two or three general Principles of Motion from Phænomena, and afterwards to tell us how the Properties and Actions of all corporeal Things follow from those manifest Principles, would be a very great step in Philosophy."

"[T]he Instinct of Brutes and Insects, can be the effect of nothing else than the Wisdom and Skill of a powerful ever-living Agent."

"[W]e are not to consider the World as the Body of God, or the several Parts thereof, as the Parts of God. He is an uniform Being, void of Organs, Members or Parts, and they are his Creatures subordinate to him, and subservient to his Will."

"There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the Business of experimental Philosophy to find them out."

"Now the smallest 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 Chymistry, and the Colours of natural Bodies depend, and which by cohering compose Bodies of a sensible Magnitude."

"[A]ll material Things seem to have been composed of the hard and solid Particles above-mention'd, variously associated in the first Creation by the Counsel of an intelligent Agent. For it became him who created them to set them in order. And if he did so, it's unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature."