# Gravity

Gravity or gravitation is a natural phenomenon by which all things with energy are brought toward (or gravitate toward) one another, including stars, planets, galaxies and even light and sub-atomic particles. Gravity is most accurately described by Albert Einstein's general theory of relativity, which describes gravity not as a force but as a consequence of the curvature of spacetime caused by the uneven distribution of mass/energy; and resulting in gravitational time dilation, where time lapses more slowly in lower (stronger) gravitational potential. However, for most applications, gravity is well approximated by Newton's law of universal gravitation, which postulates that gravity causes a force where two bodies of mass are directly drawn (or 'attracted') to each other according to a mathematical relationship, where the attractive force is proportional to the product of their masses and inversely proportional to the square of the distance between them.

## Quotes
### A–C

* There are some oddities in the perspective with which we see the world. The fact that we live at the bottom of a deep gravity well, on the surface of a gas covered planet going around a nuclear fireball 90 million miles away and think this to be normal is obviously some indication of how skewed our perspective tends to be, but we have done various things over intellectual history to slowly correct some of our misapprehensions.
** Douglas Adams, Speech at Digital Biota 2, Cambridge, UK, (1998).
* Anaxagoras... speaks absurdly concerning the permanency of the infinite: for he says that the infinite itself supports itself; and this because it is in itself: for nothing else contains it. As if where any thing is it is naturally there. But this is not true: for a thing may be situated in a certain place by force, and not where it is naturally adapted to be. If, therefore, the whole is by no means moved; for that which is established in itself, and is in itself, is necessarily immoveable; yet it should be said why it is not naturally adapted to be moved: for it is not just that he who thus speaks should be dismissed; since there may be any thing else which is not moved; but nothing hinders it from being naturally adapted to be moved: for earth also is not borne along; nor if it were infinite could it be locally moved, in consequence of being restrained by the middle. It would not, however, remain in the middle, because there is not any other place into which it could be moved, but because it is not natural to it so to be moved. Though, indeed, it might be said, that the earth supports itself. If, therefore, this is not the cause of the permanency of the earth if it were infinite, but its gravity is the cause; and that which is heavy abides in the middle, and the earth is in the middle: in like manner also, the infinite will abide in itself, through some other cause, and not because it is infinite, and will itself support itself. At the same time likewise, it is manifest, that it is necessary every part of it should abide: for as the infinite abides supported in itself; so likewise whatever part of it is assumed, will abide in itself: for the places of the whole and the part are of the same species; as of the whole earth and a clod, the place is downward; and of the whole of fire, and a spark, the place is upward. So that if the place of the infinite is in itself, there will be the same place also of a part of the infinite. It will abide therefore, in itself. And, in short, it is evident that it is impossible to say that there is an infinite body, and at the same time that there is a certain place for bodies, if every sensible body either possesses gravity or levity. And if it is heavy, it has a natural tendency to the middle; but if light, it naturally tends upward: for it is necessary that an infinite body should be such. But it is impossible that the whole should be passive in either way, or the half in both ways: for how will you divide? Or how will one part of the infinite be above, and another below? Or how will it have extremes or a middle? Further still; every sensible body is in place; but the species and differences of place are upward and downward, before and behind, to the right hand and to the left: and these things not only thus subsist with relation to us, and by position, but have a definite subsistence in the universe itself. But it is impossible that these things should be in the infinite: and in short it is impossible that there should be an infinite place.
** Aristotle, Physics (ca. 330 BCE) as quoted in Works, Vol. 1, The Physics or Physical Auscultation of Aristotle (1806) Tr. Thomas Taylor.
* Moreover, force, gravity, and words of that kind, are often, and not unwisely, used in the concrete; in such a way that they know the movement of the body, the difficulty of resistance, etc. But when they are used by philosophers to signify certain natures, precise and abstract from all these, which are neither subject to the senses, nor can be understood by any power of the mind, nor can they be shaped by the imagination, they in turn give rise to errors and confusion.
** George Berkeley, De Motu (1721) in The works of George Berkeley Volume 1 of 4. Alexander Campbell Fraser. Ed. De Motu: Paragraph 6, p. 503
* GRAVITATION, n. The tendency of all bodies to approach one another with a strength proportion to the quantity of matter they contain -- the quantity of matter they contain being ascertained by the strength of their tendency to approach one another. This is a lovely and edifying illustration of how science, having made A the proof of B, makes B the proof of A.
** Ambrose Bierce, The Cynic's Dictionary (1906); republished as The Devil's Dictionary (1911).
* Our two greatest problems are gravity and paper work. We can lick gravity, but sometimes the paperwork is overwhelming.
** Wernher von Braun as quoted in the Chicago Sun Times (10 Jul 1958)
* In order that a pendulum may continue to make the same number of oscillations in a given time, it must be shortened as it is carried towards the equator; and the variation of its length in different latitudes affords an accurate measurement of the force of gravity. But the force of gravity has known relation to the figure of the earth, which, therefore, may be determined, by observing the length of the seconds pendulum at different points on its surface.
** Sir David Brewster in: The Edinburgh encyclopaedia, conducted by D. Brewster, Edinburgh encyclopaedia, 1830, p. 355.
* Cotes's Preface [to the 2nd edition of Principia] is of historical importance... It is interpreted as advocating the theory of "action at a distance", and the theory that gravity is an innate property of matter. Phrases in Newton's Principia appear to carry a similar implication. ...In these expressions, the "bodies" or the "corpuscles" are represented as active, as "attracting." They are not passive like a chip of wood carried about by a eddy in a pool, or like a planet passively swept through space by a Cartesian vortex. It was easy, therefore, to jump to an inference that in the Newtonian theory, gravity was an innate, inherent property of matter. ...such an interpretation was made by writers on the European continent, for example by Huygens, Lalande, [Jean Baptiste] Bordas-Demoulin, and others. Thus, after the publication of the Principia in 1687, Huygens... abandoned the explanation of the planetary motion by Descartes' theory of vortices, and published his adherence to Newton's celestial mechanics. But Huygens did not accept the view that gravitation was an innate property of matter, a view which he attributed to Newtonian philosophy. On this point Huygens rejected what interpreted to be the tenet of Newton, and continued his adhesion to the tenet of Descartes. While the reader of the first edition of Principia had some justification in attributing to Newton the view that gravity was an innate property of matter, they were nevertheless mistaken. In the first edition Newton had made no explicit declaration on this point. ...Newton was no more a believer in gravity as an innate property of bodies than was Descartes. But the readers of the first edition of Principia had no means of knowing this.
** Florian Cajori, Explanatory Appendix, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World (1934) Isaac Newton, Florian Cajori, ed. R. T. Crawford, revision of Andrew Motte's translation.
* Newtonian action at a distance is spoken of as "immediate action." Newton, on the other hand, postulates an agent and gives it time to act. To be sure, in his calculations of gravitational attractions, he assumes, as a necessary approximation (having no experimental data on the speed of propagation of gravitational action), that the action is instantaneous, but not so in his talks on gravity. In a letter to Boyle he considers the cause of gravitation between two approaching bodies. They "make the ether between them begin to rarify"; and again, in his hypothesis on light, he says, "So may the gravitating attraction of the earth be caused by the continual condensation of some other such like ethereal spirit... in such a way... as to cause it [this spirit] from above to descend with great celerity for a supply; in which descent it may bear down with it the bodies it pervades, with force proportional to the superficies of all their parts it acts upon."
** Florian Cajori, Explanatory Appendix, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World (1934) ibid.
* Laplace made the assumption that the transmission of gravity is not instantaneous, and he found that in order to produce the known effects in the secular acceleration of the moon, gravity must travel seven million times faster than the speed of light. ...Laplace's calculation has been found to be incomplete, and his velocity of gravity illusory.
** Florian Cajori, Explanatory Appendix, Sir Isaac Newton's Mathematical Principles of Natural Philosophy and His System of the World (1934) ibid. Reference: Laplace, Mécanique céleste, Livre x, close of Ch. VII.
* For the better part of my last semester at Garden City High, I constructed a physical pendulum and used it to make a “precision” measurement of gravity. The years of experience building things taught me skills that were directly applicable to the construction of the pendulum. Twenty-five years later, I was to develop a refined version of this measurement using laser-cooled atoms in an atomic fountain interferometer.
** Steven Chu in: Steven Chu – Biographical, Nobleprize.org.

### D–F

* The essence of Riemann's discoveries consists in having shown that there exist a vast number of possible types of spaces, all of them perfectly self-consistent. When, therefore, it comes to deciding which one of these possible spaces real space will turn out to be, we cannot prejudge... Experiment and observation alone can yield us a clue. To a first approximation, experiment and observation prove space to be Euclidean, and this accounts for our natural belief... merely by force of habit. But experiment is necessarily innacurate, and we cannot foretell whether our opinions will not have to be modified when our experiments are conducted with greater accuracy. Riemann's views thus place the problem of space on an empirical basis excluding all a priori assertions on the subject. ...the relativity theory is very intimately connected with this empirical philosophy; for... Einstein is compelled to appeal to a varying non-Euclideanism of four-dimensional space-time in order to account with extreme simplicity for gravitation. ...had the extension of the universe been restricted on a priori grounds... to three dimensional Euclidean space, Einstein's theory would have been rejected on first principles. ...as soon as we recognise that the fundamental continuum of the universe and its geometry cannot be posited a priori... a vast number of possibilities are thrown open. Among these the four-dimensional space-time of relativity, with its varying degrees of non-Euclideanism, finds a ready place.
** A. D'Abro, Forward, The Evolution of Scientific Thought from Newton to Einstein (1927).
* With the new views advocated by Riemann... the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure of geometry for space from place to place. ... Riemann's exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann's works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclidean space, thus inverting in a certain sense Riemann's ideas. But in Clifford's day this belief was mathematically untenable. Furthermore, the physical exploration of space seemed to yield unvarying Euclideanism. ...it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann's ideas to victory. (In all fairness to Einstein... he does not appear to have been influenced directly by Riemann.) Nor were Clifford's hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion, and electricity. ... Einstein had been led to recognize that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was fundamental was... one of Space-Time... possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann's ideas to space-time instead of to space... He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity... We had called their effects gravitational effects... never suspecting that they were the result of those very local variations in the geometry for which our search had been in vain....the theory of relativity is the theory of the space-time metrical field.
** A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927).
* Let us revert to the metrical field, as defining the space-time structure. Although Riemann had attributed the existence of the structure, or metrical field, of space to the binding forces of matter, there is not the slightest indication in Einstein's special theory that any such view is going to be developed later on; in fact, it does not appear that Einstein was influenced in the slightest degree by Riemann's ideas. ...in the special theory, the problem of determining whence the structure, or field, arises, what it is, what causes it, is not even discussed in a tentative manner. Space-time, with its flat structure, is assumed to be given or posited by the Creator. But in the general theory the entire situation changes when Einstein accounts for gravitation, hence for a varying lay of the metrical field, in terms of a varying non-Euclidean structure of space-time around matter. We are then compelled to recognise not only that the metrical field regulates the behaviour of material bodies and clocks, as was also the case in the special theory, but, furthermore, that a reciprocal action takes place and that matter and energy in turn must affect the lay of the metrical field. But we are still a long way from Riemann's view that the field is not alone affected but brought into existence by matter; and it is only when we consider the cosmological part of Einstein's theory that this idea of Riemann's may possibly be vindicated. And here we come to a parting of the ways with de Sitter and Eddington on one side, Einstein and Thirring on the other, and Weyl somewhere in between the two extremes.
** A. D'Abro, The Evolution of Scientific Thought from Newton to Einstein (1927).
* In the year 1900 Max Planck wrote... $E=hv$, where $E$ is the energy of a light wave, $v$ is its frequency, and $h$ is... Plank's constant. It said that energy and frequency are the same thing measured in different units. Plank's constant gives you a rate of exchange for for converting frequency into energy... But in the year 1900 this made no physical sense. Even Plank himself did not understand it. ...Now Hawking has written down an equation which looks rather like Plank's equation... $S=kA$, where $S$ is the entropy of a black hole, $A$ is the area of its surface, and $k$ is... Hawking's constant. Entropy means roughly the same thing as the heat capacity of an object. ...Hawking's equation says that entropy is really the same thing as area. The exchange rate... is given by Hawking's constant... But what does it really mean to say that entropy and area are the same thing? We are as far away from understanding that now as Planck was of understanding quantum mechanics in 1900. ...[T]his equation will emerge as a central feature of the still unborn theory which will tie together gravitation and quantum mechanics and thermodynamics.
** Freeman Dyson, Infinite in All Directions: Gifford Lectures given at Aberdeen, Scotland April-November 1985 (1988) pp. 21-22 (paperback, 1989).
* Light-waves in passing a massive body such as the sun are deflected through a small angle. This is additional evidence that the Newtonian picture of gravitation as a tug is inadequate. You cannot deflect waves by tugging at them, and clearly another representation of the agency which deflects them must be found.
** Arthur Eddington, The Nature of the Physical World (1928) p. 122–.
* The Newtonian scheme says that the planet tends to move in a straight line, but the sun's gravity pulls it away. Einstein says that the planet tends to take the shortest route and does take it.
** Arthur Eddington, The Nature of the Physical World (1928) p. 133.
* Einstein's law of gravitation controls a geometrical quantity curvature in contrast to Newton's law which controls a mechanical quantity of force.
** Arthur Eddington, The Nature of the Physical World (1928) p. 133.
* Enter superstring theory. The concept that particles are really tiny strings dates from the 1960s, but it took on wings in 1974, when John Schwarz... and Joel Scherk... came to terms with what had been an ugly blemish in their calculations. String theory kept predicting the existence of a particle with zero mass and a spin of two. Schwarz and Scherk realized that this unwelcome particle was nothing other than the graviton, the quantum carrier of gravitational force (Although there is no quantum theory of gravity yet, it is possible to specify some of the characteristics of the quantum particle thought to convey it.) This was liberating: The calculations were saying not only that string theory might be the way to a fully unified account of all particles and forces but that one could not write a string theory without incorporating gravity. Ed Witten... recalled that this news constituted "the greatest intellectual thrill of my life.
** Timothy Ferris, The Whole Shebang: A State of the Universe Report (1998) p. 221.
* We postulate: It shall be impossible, by any experiment whatsoever performed inside such a box, to detect a difference between an acceleration relative to the nebulae and gravity. That is, an accelerating box in some gravitational field is indistinguishable from a stationary box in some different gravitational field. How much like Einstein this sounds, how reminiscent of his postulate of special relativity! We know the principle of equivalence works for springs, (as we knew special relativity worked for electrodynamics), and we extend it by fiat to all experiments whatsoever. We are used to such procedures by now, but how originally brilliant it was in 1911—what a brilliant, marvelous man Einstein was!
** Richard Feynman (1962-63) Feynman Lectures on Gravitation (1995), Lecture 7.
* One other test of gravity... is the question of whether the pull is exactly proportional to the mass... and changes in velocity are inversely proportional to the mass... That means that two objects of different mass will change their velocity in the same manner in a gravitational field. ...That is Galileo's old experiment from the Leaning Tower of Pisa. ...How accurate is it? It was measured in an experiment by ...Eötvös in 1909 and ...by Dicke, and is known to one part in 10,000,000,000. ...[S]uppose you wanted to know whether the pull is exactly proportional to the inertia. The earth is going around the sun, so the things are thrown out by inertia. But they are attracted by the sun to the extent that they have mass... So if they are attracted to the sun in a different proportion from that thrown out by inertia, one will be pulled towards the sun, and the other away from it, and so, hanging them on opposite ends of a rod on another Cavendish quartz fiber, the thing will twist towards the sun. It does not twist at this accuracy, so we know that the sun's attraction to two objects is exactly proportional to its coefficient of inertia; in other words, its mass.
** Richard Feynman The Character of Physical Law (1965) pp. 29-30, video 36:36.
* Poetry had a much more serious beginning than is usually imagin'd, and... the Muses have of late days mightily deviated from their original Gravity.
** Bernard Le Bovier de Fontenelle, The History of Oracles, and the Cheats of the Pagan Priests (1688)

### G

* Edward Witten is fond of declaring that string theory had already made a dramatic and experimentally confirmed prediction: "String theory had the remarkable property of predicting gravity." What Witten means by this is that both Newton and Einstein developed theories of gravity because their observations of the world clearly showed them that gravity exists, and that, therefore, it required an accurate and consistent explanation. On the contrary, a physicist studying string theory—even if he or she was completely unaware of general relativity—would be inexorably led to it by the string framework.
** Brian Greene, The Elegant Universe (2003).
* Much as Kaluza found that a universe with five spacetime dimensions provided a framework for unifying electromagnetism and gravity, and much as string theorists found that a universe with ten spacetime dimensions provided a framework for unifying quantum mechanics and general relativity, Witten found that a universe with eleven spacetime dimensions provided a framework for unifying all string theories.
** Brian Greene, The Fabric of the Cosmos (2003).
* He used to say the left-hand side of his equation is beautiful and the right-hand side is ugly. Much of what he was doing in the latter part of his career was trying to move the right-hand side to the left... and understand matter as a geometrical structure. To build matter itself from geometry—that in a sense is what string theory does. ...especially in a theory like the heterotic string which is inherently a theory of gravity in which the particles of matter as well as the other forces of nature emerge in the same way that gravity emerges from geometry. Einstein would have been pleased with this, at least with the goal, if not the realization. ...He would have liked the fact that there is an underlying geometrical principle — which, unfortunately, we don’t really yet understand.
** "David Gross" interview, Superstrings: A Theory of Everything? (1992) ed. P. C. W. Davies, Julian Brown.
* It turns out that the energy of a gravitational field—any gravitational field—is negative. During inflation, as the universe gets bigger and bigger and more and more matter is created, the total energy of matter goes upward by an enormous amount. Meanwhile, however, the energy of gravity becomes more and more negative. The negative gravitational energy cancels the energy in matter, so the total energy of the system remains whatever it was when inflation started—presumably something very small. ...This capability for producing matter in the universe is one crucial difference between the inflationary model and the previous model.
** Alan Guth, "A Universe in Your Backyard," in Third Culture: Beyond the Scientific Revolution (1996) ed. John Brockman, p. 279.
* The miracle of physics that I'm talking about here is something that was actually known since the time of Einstein's general relativity; that gravity is not always attractive. Gravity can act repulsively. Einstein introduced this in 1916... in the form of the cosmological constant, and the original motivation of modifying the equations of general relativity to allow this was because Einstein thought that the universe was static, and he realized that ordinary gravity would cause the universe to collapse if it was static. ...The fact that general relativity can support this gravitational repulsion, still being consistent with all the principles that general relativity incorporates, is the important thing which Einstein himself did discover..
** Alan Guth, The Early Universe (2012) Lecture 1: Inflationary Cosmology: Is Our Universe Part of a Multiverse? Part I, MITOpenCourseware (OCW) course 8.286, Massachusetts Institute of Technology.
* Inflation takes advantage of this possibility... to let gravity be the repulsive force that drove the universe into the period of expansion that we call the Big Bang. In fact, when one combines general relativity with conventional ideas, now, in particle physics there really is a pretty clear indication, I should say, not quite a prediction... that at very high energy densities one expects to find states of matter which literally turn gravity on its head and cause gravity to become repulsive.
** Alan Guth, The Early Universe (2012) Lecture 1 ibid.
* What it takes to produce a gravitational repulsion is a negative pressure. According to general relativity, it turns out... both pressures and energy densities can produce gravitational fields, unlike Newtonian physics, where it's only mass densities that produce gravitational fields.
** Alan Guth, The Early Universe (2012) Lecture 1 ibid.
* A positive pressure produces an attractive gravitational field... Positive pressures are just sort of normal pressures and attractive gravity is normal gravity, so normal pressures produce normal gravity, but it is possible to have negative pressures, and negative pressures produce repulsive gravity, and that's the secret of what makes inflation possible.
** Alan Guth, The Early Universe (2012) Lecture 1 ibid.
* The gravitational repulsion created by this small patch of repulsive gravity material would be, then, the driving force of the Big Bang and it would cause the region to undergo exponential expansion... there is a certain doubling time, and if you wait the same amount of time it doubles again, and if you wait the same amount of time it doubles again... and it's because these doublings build up so dramatically, it doesn't take very much time to build the whole universe. In about 100 doublings this tiny patch of 10$^{-28}$ cm can become large enough, not to be the universe, but to be a small marble-sized region which will then ultimately become the observed universe, as it continues to coast outward after inflation ends.
** Alan Guth, The Early Universe (2012) Lecture 1 ibid.

### H–K

* Einstein is the only figure in the physical sciences with a stature that can be compared with Newton. Newton is reported to have said "If I have seen further than other men, it is because I stood on the shoulders of giants." This remark is even more true of Einstein who stood on the shoulders of Newton. Both Newton and Einstein put forward a theory of mechanics and a theory of gravity but Einstein was able to base General Relativity on the mathematical theory of curved spaces that had been constructed by Riemann while Newton had to develop his own mathematical machinery. It is therefore appropriate to acclaim Newton as the greatest figure in mathematical physics and the Principia is his greatest achievement.
** Stephen Hawking, "Newton's Principia" in 300 Years of Gravitation. (1987) by Hawking and W. Israel, p. 4.
* Bodies like the earth are not made to move on curved orbits by a force called gravity; instead, they follow the nearest thing to a straight path in a curved space, which is called a geodesic. A geodesic is the shortest (or longest) path between two nearby points.
** Stephen Hawking, A Brief History of Time (1988).
* The universe would have expanded in a smooth way from a single point. As it expanded, it would have borrowed energy from the gravitational field, to create matter. As any economist could have predicted, the result of all that borrowing, was inflation. The universe expanded and borrowed at an ever-increasing rate. Fortunately, the debt of gravitational energy will not have to be repaid until the end of the universe.
** Stephen Hawking, The Beginning of Time (1996).
* 'I should here have described some Clocks and Time-keepers of great use, nay absolute necessity in these and many other Astronomical observations, but that I reserve them for some attempts that are hereafter to follow, about the various wayes I have tryed, not without good success of improving Clocks and Watches and adapting them for various uses, as for accurating Astronomy, completing the Tables of the fixt stars to Seconds, discovery of Longitude, regulating Navigation and Geography, detecting the properties and effects of motions for promoting secret and swift conveyance and correspondence, and many other considerable scrutinies of nature: And shall only for the present hint that I have in some of my foregoing observations discovered some new Motions even in the Earth it self, which perhaps were not dreamt of before, which I shall hereafter more at large describe, when further tryalls have more fully confirmed and compleated these beginnings. At which time also I shall explaine a Systeme of the World, differing in many particulars from any yet known, answering in all things to the common Rules of Mechanicall Motions: This depends upon three Suppositions. First, that all Cœlestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers, whereby they attract not only their own parts, and keep them, from flying from them, as we may observe the Earth to do, but that they do also attract all the other Cœlestial Bodies that are within the sphere of their activity; and consequently that not only the Sun and the Moon have an influence upon the body and motion of the Earth, and the Earth upon them, but that Mercury also, Venus, Mars, Saturne, and Jupiter by their attractive powers, have a considerable influence upon its motion as in the same manner the corresponding attractive power of the Earth hath a considerable influence upon every one of their motions also. The second supposition is this, That all bodys whatsoever that are put into direct and simple motion, will so continue to move forward in a streight line, till they are by some other effectual powers deflected and bent into a Motion describing a Circle, Ellipsis, or some other more compounded Curve Line. The third supposition is, That these attractive powers are so much the more powerful in operating, by how much nearer the body wrought upon is to their own Centers. Now what these several degrees are I have not yet experimentally verified;'—But these degrees and proportions of the power of attraction in the celestiall bodys and motions, were communicated to Mr. Newton by R. Hooke in the yeare 1678, by letters, as will plainely appear both by the coppys of the said letters, and the letters of Mr. Newton in answer to them, which are both in the custody of the said R. H., both which also were read before the Royall Society at their publique meeting, as appears by the Journall book of the said Society.—'but it is a notion which if fully prosecuted as it ought to be, will mightily assist the astronomer to reduce all the Cœlestiall motions to a certaine rule, which I doubt will never be done true without it. He that understands the natures of the Circular Pendulum and Circular Motion, will easily understand the whole ground of this Principle, and will know where to find direction in nature for the true stating thereof. This I only hint at present to such as have ability and opportunity of prosecuting this Inquiry, and are not wanting of Industry for observing and calculating, wishing heartily such may be found, having my self many other things in hand which I would first compleat, and therefore cannot so well attend it. But this I durst promise the Undertaker, that he will find all the great Motions of the World to be influenced by this Principle, and that the true understanding thereof will be the true perfection of Astronomy.'
** Robert Hooke, Attempt to Prove the Motion of the Earth (1674) pp. 27-28, read to the Royal Society 1671, written 1670. Inserted comment (in italics) by John Aubrey, Brief Lives. Note: this statement precedes the publication of Newton's Principia by 16 to 17 years.
* Having enumerated some of the most remarkable Proprieties of Gravity, we come in the next place to consider what may be the Cause thereof. And first, I believe I shall not need to say much against the Opinion of Intelligent Matter, which supposes every part of Matter to act understandingly, for that being supposed, all Philosophy is vain, and there needs no farther Inquiry into Nature. And secondly, I have as little to say to its Cousin-german Opinion, viz. the Regimen of an Hylarchick Spirit. And 3ly, The Epicurean Atoms seem to me to give as little of Explanation almost as either of the former. And 4ly, For the Peripatetick Doctrine of tendency to the Center of the Universe, besides that the Foundation is false, the Earth being proved not to be in the Center, 'tis not yet understood what the tendency is. 5ly, The Cartesian Doctrine and that of Mr. Hobbs are both insufficient, because they do not give any reason why Bodies should descend towards the Center under or near the Poles. 6ly, Nor will the Magnetism of Gilbert or Kepler serve; for, as I shall afterwards shew, that is a Propriety distinct from Gravity, and of quite another nature.
** Robert Hooke, "A Discourse on the Nature of Comets" [& of Gravity and Gravitation] (1682, soon after Michaelmas) The Posthumous Works of Robert Hooke (1705).
* I could... largely explain, and plainly evince, that the Motions of several Bodies at a distance, are caused by the internal motion of the founding Body; and that this Power of moving is every way propagated by the ambient Medium, which excites in solid Bodies at a distance, a similar Motion. I could farther also prove, that every one of these distinct internal Motions of Bodies, as that of Light, and that of Sound, have distinct and differing Mediums, by which those Motions are communicated from the affecting to the affected Body: And so I conceive also that the Medium of Gravity may be distinct and differing both from that of Light, and from that of Sound. I conceive then, that the Gravity of the Earth may be caused by some internal Motion of the internal or central Parts of the Earth; which internal and central Motion may be caused, generated and maintained by the Motion of the external and all the intermediate parts of its Body: So that the whole Globe of the Earth may contribute to this Motion, as it will happen to a Globe of Glass or solid Mettal, to any part of which no internal Motion can be communicated, without at the same time affecting the whole with the same Motion. And I shall most plainly and evidently prove, when I come to the Explication of Magnetism, that this is undeniably performed and effected by this means.
** Robert Hooke, "A Discourse on the Nature of Comets" [& of Gravity and Gravitation] (1682) ibid.
* If light takes the path with the least time between two points, and light beams bend under the influence of gravity, then the shortest distance between two points is a curved line. Einstein was shocked by this conclusion: If light could be observed traveling in a curved line, it would mean that space itself is curved.
** Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (1995).
* Einstein independently discovered Riemann's original program, to give a purely geometric explanation to the concept of "force." …To Riemann, the bending and warping of space causes the appearance of a force. Thus forces do not really exist; what is actually happening is that space itself is being bent out of shape. The problem with Riemann's approach... was that he had no idea specifically how gravity or electricity and magnetism caused the warping of space. ...Here Einstein succeeded where Riemann failed.
** Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (1995).

### L–M

* Many other powers in nature, have an analogy to gravity, but extend to less distances, and observe laws somewhat different. It has been found very difficult to account for them mechanically. For this purpose, some have imagined certain effluvia to proceed from bodies, or atmospheres environing them; others have invented vortices; but all their attempts have hitherto proved unsatisfactory. That such powers take place in nature, and contribute to produce its chief phænomena, is most evident; but their causes are very obscure, and hardly accessible by us. In all the cases when bodies seem to act upon each other at a distance, and tend towards one another without any apparent cause impelling them, this force has been commonly called attraction and this term is frequently used by Sir Isaac Newton. But he gives repeated cautions that he pretends not, by the use of this term, to define the nature of the power, or the manner in which it acts. Nor does he ever affirm, or insinuate, that a body can act upon another at a distance, but by the intervention of other bodies. It is of the utmost importance in philosophy to establish a few general powers in nature, upon unquestionable evidence, to determine their laws, and trace their consequences, however obscure the causes of those powers may be; and this he has done with great success.
** Colin MacLaurin, An Account of Sir Isaac Newton's Philosophical Discoveries: in Four Books (1748 or 1750) see 1775 edition.
* However commodious the term attraction may be, to avoid an useless and tedious circumlocution, yet because it was used by the school-men to cover their ignorance, the adversaries of Sir Isaac Newton's philosophy have taken an unjust handle from his use of this term, after all his precautions, to depreciate and even ridicule his doctrines; by which they only convince us that they neither understand them, nor have impartially and duly considered them. Mr. Leibnitz made use of this same term, in the same sense with Sir Isaac Newton, before he set up in opposition to him; and it is often to be met with in the writings of the most accurate philosophers, who have used it without always guarding against the abuse of it, as he has done. A term of art has been often employed by crafty men, with too much success, to raise a dislike against their opponents, and mislead the unwary, and to disgust them from enquiring into the truth; but such disingenuity is unworthy of philosophers. No writer hath appeared against Sir Isaac Newton, of late, by whom this argument, tho' altogether groundless, is not insisted on at great length; and sometimes adorned with the embellishments of wit and humour; but if the reader will take the trouble to compare their descriptions with Sir Isaac Newton's own account, he will easily perceive how little it was minded by them; and that the sum of all their art and skill amounts to this only, that they were able to expose a creature of their own imagination. Possibly some unskilful men may have fancied that bodies might attract each other by some charm or unknown virtue, without being impelled or acted upon by other bodies, or by any other powers of whatever kind; and some may have imagined that a mutual tendency may be essential to matter, tho' this is directly contrary to the inertia of body described above; but surely Sir Isaac Newton has given no ground for charging him with either of these opinions: he has plainly signified that he thought that those powers arose from the impulses of a subtile ætherial medium that is diffused over the universe, and penetrates the pores of grosser bodies. It appears from his letters to Mr. Boyle that this was his opinion early; and if he did not publish it sooner, it proceeded from hence only, that he found he was not able from experiment and observation to give a satisfactory account of this medium, and the manner of its operation, in producing the chief phænomena of nature.
** Colin MacLaurin, An Account of Sir Isaac Newton's Philosophical Discoveries: in Four Books (1748 or 1750).
* Accordingly, we find in his Optical Queries and in his letters to Boyle, that Newton had very early made the attempt to account for gravitation by means of the pressure of a medium, and that the reason he did not publish those investigations "proceeded from hence only, that he found he was not able, from experiment and observation, to give a satisfactory account for this medium, and the manner of its operation in producing the chief phenomena of Nature." The doctrine of direct action at a distance cannot claim for its author the discoverer [Newton] of universal gravitation. It was first asserted by Roger Cotes, in his preface to the Principia... According to Cotes, it is by experience that we learn that all bodies gravitate. We do not learn in any other way that they are extended, movable, or solid. Gravitation, therefore, has as much right to be considered an essential property of matter as extension, mobility, or impenetrability. And when the Newtonian philosophy gained ground in Europe, it was the opinion of Cotes, rather than than that of Newton that became most prevalent, till at last Boscovich propounded his theory, that matter is a congeries of mathematical points, each endowed with the power of attracting or repelling the others according to fixed laws. In his world, matter is unextended, and contact is impossible. He did not forget, however, to endow his mathematical points with inertia.
** James Clerk Maxwell, On Action at a Distance, Proceedings of the Royal Institution of Great Britain, Vol. 7 (1873–1875) pp. 48–49. Note: includes quote from Colin MacLaurin's An Account of Sir Isaac Newton's Philosophical Discoveries: in Four Books (1748 or 1750).
* Could it be, nevertheless, that Einstein's theory is wrong? Might it be necessary to modify it—to find a new theory of gravity that can explain both the stronger gravity and the apparent antigravity being observed today—rather than simply throwing in invisible things to make the standard model work?
** John Moffat, Introduction, Reinventing Gravity (2008) p. xi.
* To physicists such as myself, the huge amount of invisible dark matter needed to make Einstein's theory fit the astrophysical data is reason enough for exploring modified gravity theories.
** John Moffat, Reinventing Gravity (2008) Ch. 9, Other Alternative Gravity Theories, p. 143.
* A large part of the relativity community is in denial—refusing even to contemplate the idea that black holes may not exist in nature, or seriously consider the idea that any kind of new matter such as the new putative dark energy can play a fundamental role in gravity theory.
** John Moffat, Reinventing Gravity (2008) Ch. 14, Do Black Holes Exist In Nature? p. 204.
* Giving up Einstein's theory of gravity is simply unacceptable to many in the community. It may take a new generation of physicists to view the evidence with unclouded eyes.
** John Moffat, Reinventing Gravity (2008) Epilogue, p. 221.
* It offends reason to believe that a well-established natural law can admit of exceptions. A natural law must hold everywhere and always, or be invalid. I cannot believe, for example, that the universal law of gravitation, which governs the physical world, is ever suspended in any instance or at any point of the universe. Now I consider economic laws comparable to natural laws, and I have just as much faith in the principle of the division of labor as I have in the universal law of gravitation. I believe that while these principles can be disturbed, they admit of no exceptions.
** Gustave de Molinari, §II of The Production of Security, tr. J. Huston McCulloch (Auburn, AL: Ludwig von Mises Institute, 2009; orig. 1849), p. 25.

### N

* Were I to assume an hypothesis, it should be this... First, it is to be supposed therein, that there is an æthereal medium much of the same constitution with air, but far rarer, subtler, and more strongly elastic. Of the existence of this medium the motion of a pendulum in a glass exhausted of air almost as quickly as in the open air, is no inconsiderable argument. But it is not to be supposed that this medium is one uniform matter, but compounded, partly of the main phlegmatic body of æther, partly of other various æthereal spirits, much after the manner, that air is compounded of the phlegmatic body of air intermixed with various vapours and exhalations: for the electric and magnetic effluvia, and gravitating principle, seem to argue such variety. Perhaps the whole frame of nature may be nothing but various contextures of some certain æthereal spirits, or vapours, condensed as it were by precipitation, much after the manner, that vapours are condensed into water, or exhalations into grosser substances, though not so easily condensible; and after condensation wrought into various forms; at first by the immediate hand of the Creator; and ever since by the power of nature; which, by virtue of the command, increase and multiply, became a complete imitator of the copies set her by the protoplast. Thus perhaps may all things be originated from æther.
** Isaac Newton (1675) "An Hypothesis explaining the Properties of Light, discoursed of in my several Papers." History of Royal Society of London, Vol. iii. (1757) ed., Thomas Birch pp. 249–250.
* So may the gravitating attraction of the earth be caused by the continual condensation of some other such like ethereal spirit, not of the main body of phlegmatic ether, but of something very thinly and subtilely diffused through it, perhaps of an unctuous or gummy, tenacious, and springy nature, and bearing much of the same relation to ether, which the vital aereal spirit, requisite for the conservation of flame and vital motions, does to air. For, if such an ethereal spirit may be condensed in fermenting or burning bodies, or otherwise coagulated in the pores of the earth and water, into some kind of humid active matter, for the continual uses of nature, adhering to the sides of those pores, after the manner that vapours condense on the sides of a vessel; the vast body of the earth, which may be every where to the very centre in perpetual working, may continually condense so much of this spirit, as to cause it from above to descend with great celerity for a supply; in which descent it may bear down with it the bodies it pervades, with force proportional to the superficies of all their parts it acts upon; nature making a circulation by the slow ascent of as much matter out of the bowels of the earth in an aereal form, which for a time constitutes the atmosphere; but being continually buoyed up by a new air, exhalations and vapours rising underneath, at length, (some part of the vapours which return in rain excepted,) vanishes again into the ethereal spaces, and there perhaps in time relents, and is attenuated into its first principle: for nature is a perpetual worker, generating fluids out of solids, and solids out of fluids, fixed things out of volatile, and volatile out of fixed, subtile out of gross, and gross out of subtile; some things to ascend, and make the upper terrestrial juices, rivers, and atmosphere; and by consequence others to descend for a requital to the former.
** Isaac Newton, (1675) as quoted by Stephen Peter Rigaud, Appendix No. XX, Extract from Newton's Hypothesis on Light, Historical Essay on the First Publication of Sir Isaac Newton's Principia (1838) Note: from Thomas Birch, History of Royal Society of London, Vol. iii. (1757) pp. 249-251.
* I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis, and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy.
** Isaac Newton, Letter to Robert Hooke (15 February 1676) [5 February 1676 (O.S.)]
* The truth is, my notions about things of this kind are so indigested, that I am not well satisfied myself in them; and what I am not satisfied in, I can scarce esteem to fit to be communicated to others; especially in natural philosophy, where there is no end of fancying. But because I am indebted to you... I could not forbear to take the opportunity of conveying this to you... I shall set down one conjecture more... it is about the cause of gravity. For this end I will suppose aether to consist of parts differing from one another in subtilty by indefinite degrees; that in the pores of bodies there is less of the grosser aether, in proportion to the finer, than in open spaces; and consequently, that in the great body of the earth there is much less of the grosser aether, in proportion to the finer, than in the regions of the air; and that yet the grosser aether in the air affects the upper regions of the earth, and the finer aether in the earth the lower regions of the air, in such a manner, that from the top of the air to the surface of the earth, and again from the surface of the earth to the centre thereof, the aether is insensibly finer and finer. Imagine now any body suspended in the air, or lying on the earth, and the aether being by the hypothesis grosser in the pores, which are in the upper parts of the body, than in those which are in its lower parts, and that grosser aether being less apt to be lodged in those pores than the finer aether below, it will endeavour to get out and give way to the finer aether below, which cannot be, without the bodies descending to make room above for it to go out into.
** Isaac Newton, Letter to Robert Boyle (1679) Isaaci Newtoni Opera (Horsley's ed.) Vol. 4 (1782) pp. 385–394.
* 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 mathematical principles of philosophy; for all the difficulty of philosophy seems to consist in this — from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena...
** Isaac Newton, Preface, Philosophiae Naturalis Principia Mathematica (1687).
* 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 centers of other like systems, these being form'd 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.
** Isaac Newton, Scholium Generale (1713; 1726) The Mathematical Principles of Natural Philosophy (1729) Tr. Andrew Motte (1846).
* In the beginning of the year 1665 I found the method of approximating Series and the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of tangents of Gregory and Slusius, and in November had the direct method of Fluxions, and the next year in January had the Theory of Colours, and in May following I had entrance into the inverse method of Fluxions. And the same year I began to think of gravity extending to the orb of the Moon, and having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere, from Kepler's Rule of the periodical times of the Planets being in a sesquialterate proportion of their distances from the centers of their orbs I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve: and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the earth, and found them answer pretty nearly. All this was in the two plague years of 1665 and 1666, for in those days I was in the prime of my age for invention, and minded Mathematicks and Philosophy more than at any time since. What Mr Hugens has published since about centrifugal forces I suppose he had before me. At length in the winter between the years 1676 and 1677 I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet must revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis and with a radius drawn to that center describe areas proportional to the times. And in the winter between the years 1683 and 1684 this Proposition with the Demonstration was entered in the Register book of the R. Society. And this is the first instance upon record of any Proposition in the higher Geometry found out by the method in dispute. In the year 1689 Mr Leibnitz, endeavouring to rival me, published a Demonstration of the same Proposition upon another supposition, but his Demonstration proved erroneous for want of skill in the method.
** Issac Newton, (ca. 1716) A Catalogue of the Portsmouth Collection of Books and Papers Written by Or Belonging to Sir Isaac Newton (1888) Preface
** Also partially quoted in Sir Sidney Lee (ed.), The Dictionary of National Biography Vol.40 (1894).
* You sometimes speak of Gravity as essential and inherent to Matter. Pray do not ascribe that Notion to me; for the Cause of Gravity is what I do not pretend to know, and therefore would take more Time to consider it.
** Isaac Newton, Letter II, Four letters from Isaac Newton to doctor Bentley, containing some Arguments in Proof of a Deity (1756).
* It is inconceivable that inanimate brute Matter, should, without the mediation of something else, which is not material, operate on and affect other Matter without mutual Contact, as it must be, if Gravitation in the sense of Epicurus, be essential and inherent in it. And this is one Reason why I desired you would not ascribe innate Gravity to me. That Gravity should be innate, inherent and essential to Matter so that one Body may act upon another at a Distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed, from one to another, is to me so great an Absurdity, that I believe that no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain Laws; but whether this Agent be material or immaterial, I have left for the Consideration of my Readers.
** Isaac Newton, Letter III, Four letters from Isaac Newton to doctor Bentley, containing some Arguments in Proof of a Deity (1756).

### O–T

* Weyl considered an aspect about general relativity... the nonpreservation of direction in a curved space. ...[He] decided to consider the possibility that length was also not preserved. ...To effect this change mathematically, Weyl had to make a slight modification in the structure of general relativity. He assumed that in addition to the usual metric (set of numbers or variables) that described the gravitational field, there was another one related to length. ...amazingly when the result was analyzed Maxwell's equations mysteriously appeared. It almost seemed as if a bit of magic had occurred and scientists quickly became interested in the miracle. ...but with detailed analysis the theory was shown to be flawed. Einstein was the first to put his finger on the flaw. ...Weyl soon acknowledged the flaw and laid his theory to rest. It may have been a failure (actually it was not an entire failure; a similar idea is used today in modern field theory), but it did accomplish something important: it got people interested in the possibility that the electromagnetic and gravitational field could be unified. Einstein soon began working on an alternative theory, as did others.
** Barry Parker, Einstein's Dream: The Search for a Unified Theory of the Universe (1986).
* How... can we understand the connexion between Force and Matter? Matter is known to us only through its manifestations of Force: our ultimate test of Matter is the ability to resist: abstract its resistance and there remains nothing but empty extension. Yet, on the other hand, resistance is equally unthinkable apart from Matter—apart from something extended. Not only... are centres of force devoid of extension unimaginable; but, as an inevitable corollary, we cannot imagine either extended or unextended centres of force to attract and repel other such centres at a distance, without the intermediation of some kind of matter. ...the hypothesis of Newton, equally with that of Boscovich, is open to the charge that it supposes one thing to act upon another through a space which is absolutely empty—a supposition which cannot be represented in thought. This charge is indeed met by the introduction of a hypothetical fluid existing between the atoms or centres. But the problem is not thus solved: it is simply shifted, and re-appears when the constitution of this fluid is inquired into.
** Herbert Spencer, First Principles (1864) p. 61.
* But if justice be a natural principle, then it is necessarily an immutable one; and can no more be changed—by any power inferior to that which established it—than can the law of gravitation, the laws of light, the principles of mathematics, or any other natural law or principle whatever; and all attempts or assumptions, on the part of any man or body of men—whether calling themselves governments, or by any other name—to set up their own commands, wills, pleasure, or discretion, in the place of justice, as a rule of conduct for any human being, are as much an absurdity, an usurpation, and a tyranny, as would be their attempts to set up their own commands, wills, pleasure, or discretion in the place of any and all the physical, mental, and moral laws of the universe.
** Lysander Spooner, §I of "The Science of Justice (Continued)," Chap. II of Natural Law; or The Science of Justice (1882), pp. 11–12.
* According to Newton's law of gravity, every object in the universe attracts every other object... with a gravitational force... $F={\frac {mMG}{R^{2}}}$... almost as famous as $E=mc^{2}$... On the left side is the force, $F$, between two masses... On the right side, the bigger mass is $M$ and the smaller mass is $m$. ...The last symbol... $G$, is a numerical constant called Newton's constant. ...Ironically, Newton never knew the value of his own constant. ...$G$ was too small to measure until the end of the eighteenth century. ...Cavendish found that the force between a pair of one-kilogram masses separated by one meter is approximately 6.6 x 10$^{-11}$ newtons. (The Newton is... about one-fifth of a pound.) ...Newton had one lucky break... the special mathematical properties of the inverse square law. ...[B]y the miracle of mathematics, you can pretend that the entire mass is located at a single point. This... allowed Newton to calculate the escape velocity... $Escape\;velocity={\sqrt {2MG/R}}$ ... the bigger the mass [$M$] and the smaller the radius $R$, the larger the escape velocity.
** Leonard Susskind, The Black Hole War: My Battle with Stephen Hawking to make the World Safe for Quantum Mechanics (2008)
* I would like to emphasize at the opening of this symposium that the often quoted ratio M/L is in fact the ratio V$^2$r/L of the directly observable quantities V, r, and L. This ratio V$^2$r/L can only be interpreted as an indicator of mass to light ratio if we assume that Newton's law of gravitational attraction is correct on the scale of galaxies. Since Keplerian behavior is essentially never seen in extra-galactic systems, I might be so bold as to suggest that the validity of Newton's law should be seriously questioned. I hope that the observers who have definite evidence that Keplerian behavior has been observed in any system will emphasize that evidence at this meeting.
** Joel E. Tohline: "Internal kinematics and dynamics of galaxies". Proceedings of the IAU Symposium. vol. 100. D. Reidel. 1983. p. 10.
* Men of splendid talents are generally too quick, too volatile, too adventurous, and too unstable to be much relied on; whereas men of common abilities, in a regular, plodding routine of business, act with more regularity and greater certainty. Men of the best intellectual abilities are apt to strike off suddenly, like the tangent of a circle, and cannot be brought into their orbits by attraction or gravity — they often act with such eccentricity as to be lost in the vortex of their own reveries. Brilliant talents in general are like the ignes fatui; they excite wonder, but often mislead. They are not, however, without their use; like the fire from the flint, once produced, it may be converted, by solid, thinking men, to very salutary and noble purposes.
** John Trusler, Dictionary of Burning Words of Brilliant Writers: A Cyclopædia of Quotations from the Literature of All Ages (1895) p. 575.

### U–Z

* A new theory by the author has been added, which draws the physical inferences consequent on the extension of the foundations of geometry beyond Reimann... and represents an attempt to derive from world-geometry not only gravitational but also electromagnetic phenomena. Even if this theory is still only in its infant stage, I feel convinced that it contains no less truth than Einstein's Theory of Gravitation—whether this amount of truth is unlimited or, what is more probable, is bounded by the Quantum Theory.
** Hermann Weyl, Preface to 3rd Edition (1919), Space—Time—Matter (1952) Tr. Henry L. Brose (orig. title: Raum, Zeit, Materie (1918).
* It is quite easy to include a weight for empty space in the equations of gravity. Einstein did so in 1917, introducing what came to be known as the cosmological constant into his equations. His motivation was to construct a static model of the universe. To achieve this, he had to introduce a negative mass density for empty space, which just canceled the average positive density due to matter. With zero total density, gravitational forces can be in static equilibrium. Hubble's subsequent discovery of the expansion of the universe, of course, made Einstein's static model universe obsolete. ...The fact is that to this day we do not understand in a deep way why the vacuum doesn't weigh, or (to say the same thing in another way) why the cosmological constant vanishes, or (to say it in yet another way) why Einstein's greatest blunder was a mistake.
** Frank Wilczek & Betsy Devine, Longing for the Harmonies: Themes and Variations from Modern Physics (1987).
* The dark matter really does look like matter. It does not look like a modification of gravity.
* Frank Wilczek, Why Is Gravity So Elusive? Frank Wilczek, Erik Verlinde, Laura Mersini-Houghton. YouTube (4 December 2017). 26:01 of 40:44
* String theory is extremely attractive because gravity is forced upon us. All known consistent string theories include gravity, so while gravity is impossible in quantum field theory as we have known it, it is obligatory in string theory.
** Edward Witten, as quoted by Michio Kaku, Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the 10th Dimension (1995).
* Even though it is, properly speaking, a postprediction, in the sense that the experiment was made before the theory, the fact that gravity is a consequence of string theory, to me, is one of the greatest theoretical insights ever.
** Edward Witten, as quoted by John Horgan, The End of Science: Facing the Limits of Knowledge in the Twilight of the Scientific Age (1996).
* If supersymmetry plays the role in physics that we suspect it does, then it is very likely to be discovered by the next generation of particle accelerators, either at Fermilab... or at CERN... Discovery of supersymmetry would be one of the real milestones in physics, made even more exciting by its close links to still more ambitious theoretical ideas. Indeed, supersymmetry is one of the basic requirements of "string theory," which is the framework in which theoretical physicists have had some success in unifying gravity with the rest of the elementary particle forces. Discovery of supersymmetry would would certainly give string theory an enormous boost.
** Edward Witten, Foreward (June 30, 1999) Supersymmetry: Unveiling the Ultimate Laws of Nature (2000) by Gordon Kane.
* It was found [in the 1970s], unexpectedly and without anyone really having a concept for it, that the rules of perturbation theory can be changed in a way that makes relativistic quantum gravity inevitable rather than impossible. The change is made by replacing point particles by strings. Then Feynman graphs are replaced by Riemann surfaces, which are smooth - unlike the graphs, which have singularities at interaction vertices. The Riemann surfaces can degenerate to graphs in many different ways. In field theory, the interactions occur at the vertices of a Feynman graph. By contrast, in string theory, the interaction is encoded globally, in the topology of a Riemann surface, any small piece of which is like any other. This is reminiscent of how non-linearities are encoded globally in twistor theory.
** Edward Witten, "The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G. W. Gibbons, E. P. S. Shellard & S. J. Rankin.
* Replacing particles by strings is a naive-sounding step, from which many other things follow. In fact, replacing Feynman graphs by Riemann surfaces has numerous consequences: 1. It eliminates the infinities from the theory. ...2. It greatly reduces the number of possible theories. ...3. It gives the first hint that string theory will change our notions of spacetime. Just as in QCD, so also in gravity, many of the interesting questions cannot be answered in perturbation theory. In string theory, to understand the nature of the Big Bang, or the quantum fate of a black hole, or the nature of the vacuum state that determines the properties of the elementary particles, requires information beyond perturbation theory... Perturbation theory is not everything. It is just the way the [string] theory was discovered.
** Edward Witten, "The Past and Future of String Theory" in The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics (2003) ed. G. W. Gibbons, E. P. S. Shellard & S. J. Rankin.
* It is observed by Bacon, in his essay on the opinions of Parmenides, that the most ancient philosophers, Empedocles, Anaxagoras, Anaximenes, Heraclitus, and Democritus, submitted their minds to things as they found them; but that Plato made the world subject to ideas, and Aristotle made even ideas, as well as all other things, subservient to words; the minds of men beginning to be occupied, in those times, with idle discussions and verbal disputations, and the correct investigation of nature being wholly neglected. Plato entertained, however, some correct notions respecting the distinction of denser from rarer matter by its greater inertia; and it would be extremely unjust to deny a very high degree of merit to Aristotle's experimental researches, in various parts of natural philosophy, and in particular to the vast collection of real information contained in his works on natural history. Aristotle attributed absolute levity to fire, and gravity to the earth, considering air and water as of an intermediate nature. By gravity the ancients appear in general to have understood a tendency towards the centre of the earth, which they considered as identical with that of the universe; and as long as they entertained this opinion, it was almost impossible that they should suspect the operation of a mutual attraction in all matter, as a cause of gravitation. The first traces of this more correct opinion respecting it are found in the works of Plutarch.
** Thomas Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (1845) Vol. 1, Lecture LX. On the History of Terrestrial Physics, p. 582.

### The Origins of Modern Science (1949)

    Herbert Butterfield
* William Gilbert published a famous book on the magnet in 1600 and laid himself open to the gibes of Sir Francis Bacon for being one of those people so taken by their pet subject of research that they could only see the whole universe transposed into terms of it. Having made a spherical magnet called a terrella, and having found that it revolved when placed in a magnetic field, he decided that the whole earth was a magnet, that gravity was a form of magnetic attraction, and that the principles of the magnet accounted for the workings of the Copernican system as a whole. Kepler and Galileo were both influenced by this view, and with Kepler, it became an integral part of his system, a basis for the doctrine of almost universal gravitation.
* Descartes was liable to be misled by too easy an acceptance of data that had been handed down by scholastic writers. ...two grand Aristotelian principles helped to condition the form of the universe as he reconstructed it—first, the view that a vacuum is impossible, and secondly, the view that objects could only influence one another if they actually touched—there could be no such thing as attraction, no such thing as action at a distance. ...Descartes insisted that every fraction of space should be fully occupied all the time by continuous matter... infinitely divisible. The particles were... packed so tightly that one of them could not move without communicating the commotion to the rest. The matter formed whirlpools in the skies, and it was because the planets were caught each in its own whirlpool that they were carried around... all similarly caught in a larger whirlpool, which had the sun as its centre... Gravity itself was the result of these whirlpools of invisible matter which had the effect of sucking things down towards their centre. ...In the time of Newton the system of Descartes and the theory of vortices or whirlpools proved to be vulnerable to both mathematical and experimental attack.
* In the middle of the 1660s, Borelli, Newton, Huygens and Hooke were wrestling with various parts of the same planetary problems, some of them treading on one another's heels in the study of the nature of light... in England, experiments with the pendulum clock had started independently, and Christopher Wren, William Croone, William Balle and Laurence Rooke appear to have unaugurated the enquiry into laws of motion, Robert Hooke performing most of the experiments. The 1670s must represent one of the greatest decades in the scientific revolution, if not the climax... and in both London and Paris... achievements... were of a remarkable nature. So far as the gravitational theory... our attention ought to be directed not merely to Newton... but to the combined operations of the English group. The Royal Society... following Baconian principles, sought to collect... the data necessary for the establishment of the Copernican hypothesis... ideally... "freely communicating their methods and pooling their gains." Here the names... in the forefront are... Isaac Newton, Robert Hooke, Edmond Halley and Christopher Wren.
* Hooke... followed Bacon in his attempt to demonstrate that the effects of gravity on a body must diminish as the body was sunk into the bowels of the earth. He sought to discover how far the effects were altered at great heights or in the region of the equator; and he threw light on the problem by observations and experiments on the pendulum. From the globular shapes of the heavenly bodies and the stable conformations of the ridges on the moon he deduced that the moon and the planets had gravity; and by 1666 he saw the motion of a comet (for example) as incurvated by the pull of the sun... and suggested that the motion of the planets might be explicable on the kind of principles that account for the motion of a pendulum. In 1674 he was suggesting that by this route one could arrive at a mechanical system of the planets which would be "the true perfection of astronomy." He pointed out that... account must be taken of the force which all heavenly bodies must be presumed to be exerting on one another. ...By 1678 he had formulated the idea of gravitation as a universal principle; and by 1679 he, too, had discovered that the diminution of the force of gravity is proportional to the square of the distance. ...But ...Hooke did not produce the mathematical demonstrations of his system.

## See also

* Non-Euclidean geometry
* Spacetime
* Theory of relativity
* Gravitational wave

## External links

* Encyclopedia of Mathematics
** Gravitation
** Theory of Gravitation