- This article covers the physics of gravitation. See also gravity (disambiguation).
Gravitation is the tendency of masses to move toward each other. Weight is a force caused by gravity equal to the product of the acceleration of gravity and the mass of the object. Exactly why two masses separated in space have a gravitational attraction to one another remains largely unknown, despite much research and various theories.
The first mathematical formulation of the theory of gravitation was made by Sir Isaac Newton and proved astonishingly accurate. He postulated the force of "universal gravitational attraction".
Newton's theory has now been replaced by Albert Einstein's theory of general relativity but for most purposes dealing with weak gravitational fields (for example, sending rockets to the Moon or around the solar system) Newton's formulae are sufficiently accurate. For this reason, Newton's law is often used and will be presented first.
Newton's law of universal gravitation
Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel
and pointing straight down to the center of the Earth
Newton's law of universal gravitation states the following:
Every object in the Universe attracts every other object with a force directed along the line of centers for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects.
Considering only the magnitude of the force, and momentarily putting aside its direction, the law can be stated symbolically as follows.
- F is the magnitude of the gravitational force between two objects
G is the gravitational constant, that is approximately : G = 6.67 × 10−11 N m2 kg-2
- m1 is the mass of first object
- m2 is the mass of second object
- r is the distance between the objects
Strictly speaking, this law applies only to point-like objects. If the objects have spatial extent, the force has to be calculated by integrating the force (in vector form, see below) over the extents of the two bodies. It can be shown that for an object with a spherically-symmetric distribution of mass, the integral gives the same gravitational attraction on masses outside it as if the object were a point mass.1
This law of universal gravitation was originally formulated by Isaac Newton in his work, the Principia Mathematica (1687). The history of gravitation as a physical concept is considered in more detail below.
Gravity on a macroscopic scale
Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. In this formulation, quantities in bold represent vectors.
- F12 is the force on object 1 due to object 2
- G is the gravitational constant
- m1 and m2 are the masses of the objects 1 and 2
- r21 = | r1 − r2 | is the distance between objects 1 and 2
is the unit vector from object 2 to 1
It can be seen that the vector form of the equation is the same as the scalar form, except for the vector value of F and the unit vector. Also, it can be seen that F12 = − F21.
Gravitational acceleration is given by the same formula except for one of the factors m:
Einstein's theory of gravitation
Newton's formulation of gravitation is quite accurate for most practical purposes. There are a few problems with it though:
- Newton had shown (given the classical assumptions of the nature of space and time) that there is conservation of angular momentum only if gravitational force is transmitted instantaneously. Einstein's theory of special relativity implies that an assumption of instantaneous propagation of gravitational force is untenable. Einstein's theory of special relativity was successfully built on the backbone of the experimentally supported assumption that there exists some velocity at which signals can be transmitted, the speed of light in vacuum.
- There is no prospect of identifying the mediator of gravity. Newton himself felt the inexplicable action at a distance to be unsatisfactory.
- It does not explain the precession of the perihelion of the orbit of the planet Mercury. This precession is small; the unexplained portion is on the order of one angular second per century (an arc-second per century).
- It predicts that light is deflected by gravity. However, this predicted deflection is only half as much as observations of this deflection, which were made after General Relativity was developed in 1915.
- The observed fact that gravitational and inertial masses are the same (or at least proportional) for all bodies is unexplained within Newton's system. See equivalence principle.
Einstein developed a new theory called general relativity which includes a theory of gravitation, published in 1915. The gravitational aspect of this theory says that the presence of matter "warps" spacetime. Objects in free fall in the universe take geodesics in spacetime. A geodesic is the counterpart of a straight line in Euclidean geometry.
How spacetime curvature simulates gravitational force
The curvature of spacetime considered as a whole implies a rather complex picture that is usually treated with the tools of differential geometry and that requires the use of tensor calculus. It is possible though to understand - at least approximately - the mechanism of gravitation without tensors when the total curvature of spacetime is split into two components:
The above components of the curvature of spacetime, and only these, are responsible for the gravitation according to Einstein's theory.
The effect of the first component, the curvature of space, is negligible in all cases when the velocities of objects are much smaller than the speed of light and when the ratios of masses divided by the distances separating their centers of mass are much smaller than a specific constant, namely the ratio of speed of light squared to Newtonian gravitational constant: . So for the majority of cases in the universe, and certainly for almost all cases in our solar system except precession of perihelion of Mercury and deflection of light rays in the vicinity of the Sun, we may treat the space as flat, as ordinary Euclidean space. It leaves us only with the gravitational time dilation as a possible reason for the illusion of "gravitational force" acting at the distance. Assuming that the ratio of masses to distances between them are smaller than the constant above, the time dilation is tiny, but it is enough to cause "Newtonian gravitational attractive force" as we know it.
The reason for this illusion is this: any mass in the universe modifies the rate of time in its vicinity this way that time runs slower closer to the mass and the change of time rate is controlled by an equation having exactly the same form as the equation that Newton discovered as his "Law of Universal Gravitation". The difference between them is in essence not in form since the Newtonian potential is replaced by the Einsteinian time rate dτ / dt, where τ is the time at a point at vicinity of the mass (the proper time of objects at this point in space, the time that is measured by the clocks in this point) and t is the time at observer at infinity, with the right side of the equation staying the same as in Newtonian equation (with accuracy to irrelevant constants). Because of the same form of both equations, the path of the object that takes an extremum of proper time while traveling, and by this taking a geodesic in spacetime, is the same (with accuracy to the negligible in this case curvature of space) as the Newtonian orbit of this object around the mass. So it looks as if the path of the object were bent by some "force of attraction" between the object and the mass. Since bending of the object's path is clearly visible and the time dilation extremely difficult to notice, a (fictitious) "gravitational force" has been assumed rather than a (real, presently measured with precise enough and formerly unavailable clocks) time dilation as the reason for bending the paths of objects moving in vicinity of masses.
So without any force involved into keeping the traveling object in line the object follows the Newtonian orbit in space just by following a geodesic in spacetime. This is Einstein's explanation why without any "gravitational forces" all the objects follow Newtonian orbits and at the same time why the Newtonian gravitation is the approximation of the Einsteinian gravitation.
In this way the Newton's "Law of Universal Gravitation" that looked to people who tried to interpret it as an equation describing a hypothetical "force of gravitational attraction" acting at a distance (except to Newton himself who didn't believe that "action at a distance" is possible) turned out to be really an equation describing spacetime geodesics in Euclidean space. We may say that Newton discovered the geodesic motion in spacetime and Einstein, by applying Riemannian geometry to it, extended it to the curved spacetime, disclosed the hidden Newtonian physics, and made its math accurate.
How energy is conserved if no forces act at a distance
It often puzzles students of Einstein's gravitation that without any force acting at distance the kinetic energy of a free falling objects changes. The puzzling question is "where is this kinetic energy coming from, when the object is moving down; or going to, when the object is moving up"? The old "gravitational field" of the "attractive force" that was considered to be a repository of this "gravitational energy" in Newton's gravity isn't any good any more since now, if "attractive force" is zero, so is the "gravitational field". We need to identify another repository for this energy.
As we know the total energy of an object is E=mc˛, where m is the so-called "relativistic mass", and c is the speed of light. When an object falls "down" its kinetic energy goes up. Energy has mass and so m goes up. However c2 drops down by the same amount since the falling object gets into space where time is running slower (recall time dilation) and so the speed of light, as observed by the same distant observer who is seeing the increasing kinetic energy, is slower as well (that's why the speed of light is not constant in a gravitational field). If both m and c2 change in opposite directions by the same amount, the product (the total energy of the object) stays the same for a free falling object. That's how the conservation of energy works in Einstein's gravitation.
There is one important result of Einstein's gravitation: to keep the change of c2 the same as change of m there must be a relative increase in amount of space (space curvature) equal to the relative time dilation. Otherwise, if amount of space didn't increase, the speed of light would drop as fast as the rate of time (since speed is proportional to time). But in Einstein's equation for energy c is squared so the internal energy would drop faster than required for total energy being constant. It might be said therefore that the nature has to curve space by the same amount as time gets dilated because of nature's inability to create energy from nothing or to destroy energy to nothing.
How gravitational force is generated with no action at distance
Once we know how internal energy is changing along the path of the body in space we may calculate a gravitational force that will push this body at certain direction when the body is somehow restrained against its free fall. It is a situation when the body sits e.g. on the surface of the earth.
Since a "force" is a derivative of energy along the path then differentiating the internal energy of a body along that path (x) we get
, which is an expression for change in speed of light in "gravitational field" of acceleration , we get the familiar Newtonian expression for gravitational force as .
It is of course not an "attractive force" but a force of inertia with which the body pushes at whatever restrains its inertial movement toward the place of its lower internal energy.
That's why it is said that in Einstein's theory the gravitational force is not a mysterious "force of gravitational attraction" acting at the distance any more but simply an inertial force that the body acts with on a restraint that is restraining the natural free movement of the body. Once the restraint is removed the body "falls", converting its internal (potential) energy into kinetic energy while keeping its total energy intact, as described in the previous section.
How Einstein's gravitation differs from Newton's
Einsteinian gravitation is not just a small modification of Newtonian gravity. Even in the limit in which general relativity can be well approximated by Newton's equations, the gravitational potential of the Newtonian theory only knows about the time dilation portion of the Einsteinian gravitational field. The space curvature is not found in the Newtonian framework at all. In all cases when the space curvature becomes relevant - like in close enough proximity to big enough masses, like stars or in the context of large enough velocities - the curvature of space can't be neglected and the predictions of Newtonian and Einsteinian theories start to differ markedly. Every time such a difference was measured, the Einsteinian theory was much closer to the actual observations - essentially, its predictions were always exact.
In particular the Einsteinian gravitation explained why Mercury's precession differs from Newtonian prediction: since Mercury is the closest planet to the Sun it moves faster than any other planet, and also it is in more curved space than all other planets. This is reflected in the behavior of Mercury and the Einsteinian calculations predict this behavior within observational error.
The other Einsteinian prediction is bending light rays in vicinity of the Sun. Since the Newtonian deflection of the ray corresponds only to the time dilation, and since it happens for the reasons explained in the previous section that the relative curvature of space must be the same as the relative time dilation, the total deflection is twice as big as its Newtonian prediction. The Einsteinian prediction being twice as big as Newtonian is again within the observational error.
Yet despite such an "elegant" simplification of physics (and simpler in physics is more elegant) as Einsteinian elimination of action at a distance, only the observational differences between theories count in science since it is very easy to be mislead by "elegance of logic". As Einstein said "the elegance should concern a tailor rather than a physicist". He also said that "things should be made as simple as possible but not any simpler".
E.g. before 1998 a group of prominent gravity physicists maintained that to make Einstein's field equation even simpler requires to remove Einstein's cosmological constant from it. They advertised this constant as an "Einstein's biggest blunder" (apparently a term coined by Einstein himself). Lack of this constant in Einstein's field equation predicted a decelerating expansion of space, which in turn was strongly advocated by almost all gravity physicists at that time. It was called standard model of cosmology. Proving that the expansion is decelerating due to "tremendous gravitational attraction of all masses of the universe" (in Einsteinian theory where there is no "gravitational attraction" at all) was supposed to be the first proof ever that cosmology is science after all, since finally it would be able to predict something. A team of enthusiastic young astronomers has been appointed to confirm this prediction. In 1998 the results came in. It turned out that the prediction is false: the space of our universe looks as if it were expanding at accelerating rate.
That the universe should look as if it were expanding at an accelerating rate, is also predicted by Einstein's theory. The theory predicts the Hubble constant of the apparent expansion as Ho = c / R, where c is speed of light and R is so called "Einstein's radius of the universe". The acceleration is predicted as . It's however predicted only on the basis of strict conservation of energy, which presently isn't favored by cosmologists. So the further explanations of the subject can't be presented here as contradicting wikipedia's policy of not publishing original research. Curious readers must look elsewhere for the details, or even better, derive the results themselves. It is well within the abilities of high school students interested in physics, once they understand the underlying mathematics.
Units of measurement and variations in gravity
Gravitational phenomena are measured in various units, depending on the purpose. The gravitational constant is measured in newtons times metre squared per kilogram squared. Gravitational acceleration, and acceleration in general, is measured in metres per second squared or in non-SI units such as galileos, gees, or feet per second squared.
The acceleration due to gravity at the Earth's surface is approximately 9.81 m/s2, depending on the location. A standard value of the Earth's gravitational acceleration has been adopted, called g. When the typical range of interesting values is from zero to tens of metres per second squared, as in aircraft, acceleration is often stated in multiples of g. When used as a measurement unit, the standard acceleration is often called "gee", as g can be mistaken for g, the gram symbol. For other purposes, measurements in millimetres or micrometres per second squared (mm/s² or µm/s²) or in multiples of milligalileos (1 mGal = 1/1000 Gal) are typical, as in geophysics. A related unit is the eotvos, which is the unit of the gravitational gradient. Mountains and other geological features cause subtle variations in the Earth's gravitational field; the magnitude of the variation per unit distance is measured in eotvoses.
Typical variations with time are 2 µm/s² (0.2 mGal) during a day, due to the tides, i.e. the gravity due to the Moon and the Sun.
Gravity, and the acceleration of objects near the Earth
The acceleration due to the apparent "force of gravity" that "attracts" objects to the surface of the Earth is not quite the same as the acceleration that is measured for a free-falling body at the surface of the Earth (in a frame at rest on the surface). This is because of the rotation of the Earth, which leads (except at the poles) to a centrifugal force which slightly lessens the acceleration observed. See Coriolis effect.
Comparison with electromagnetic force
The gravitational interaction of protons is approximately a factor 1036 weaker than the electromagnetic repulsion. This factor is independent of distance, because both interactions are inversely proportional to the square of the distance. Therefore on an atomic scale mutual gravity is negligible. However, the main interaction between common objects and the Earth and between celestial bodies is gravity, because gravity is electrically neutral: even if in both bodies there were a surplus or deficit of only one electron for every 1018 protons and neutrons this would already be enough to cancel gravity (or in the case of a surplus in one and a deficit in the other: double the interaction).
In terms of Planck units: the charge of a proton is 0.085, while the mass is only . From that point of view, the gravitational force is not small as such, but because masses are small.
The relative weakness of gravity can be demonstrated with a small magnet picking up pieces of iron. The small magnet is able to overwhelm the gravitational interaction of the entire Earth.
Gravity is small unless at least one of the two bodies is large or one body is very dense and the other is close by, but the small gravitational interaction exerted by bodies of ordinary size can fairly easily be detected through experiments such as the Cavendish torsion bar experiment.
Gravity and quantum mechanics
It is strongly believed that three of the four fundamental forces (the strong nuclear force, the weak nuclear force, and the electromagnetic force) are manifestations of a single, more fundamental force. Combining gravity with these forces of quantum mechanics to create a theory of quantum gravity is currently an important topic of research amongst physicists. General relativity is essentially a geometric theory of gravity. Quantum mechanics relies on interactions between particles, but general relativity requires no particles in its explanation of gravity. Scientists have theorized about the graviton (a particle that transmits the force gravity) for years, but have been frustrated in their attempts to find a consistent quantum theory for it. Many believe that string theory holds a great deal of promise to unify general relativity and quantum mechanics, but this promise has yet to be realized. It never can be for obvious reasons if Einstein's theory is true, due to the non-existence of "gravitational attraction" (explained in the above section "Einstein's Theory of Gravity")
Experimental tests of theories
Today General Relativity is accepted as the standard description of gravitational phenomena. (Alternative theories of gravitation exist but are more complicated than General Relativity.) General Relativity is consistent with all currently available measurements of large-scale phenomena. For weak gravitational fields and bodies moving at slow speeds at small distances, Einstein's General Relativity gives almost exactly the same predictions as Newton's law of gravitation.
Crucial experiments that justified the adoption of General Relativity over Newtonian gravity were the classical tests: the gravitational redshift, the deflection of light rays by the Sun, and the precession of the orbit of Mercury.
General relativity also explains the equivalence of gravitational and inertial mass, which has to be assumed in Newtonian theory.
More recent experimental confirmations of General Relativity were the (indirect) deduction of gravitational waves being emitted from orbiting binary stars, the existence of neutron stars and black holes, gravitational lensing, and the convergence of measurements in observational cosmology to an approximately flat model of the observable Universe, with a matter density parameter of approximately 30% of the critical density and a cosmological constant of approximately 70% of the critical density.
Even to this day, scientists try to challenge General Relativity with more and more precise direct experiments. The goal of these tests is to shed light on the yet unknown relationship between Gravity and Quantum Mechanics. Space probes are used to either make very sensitive measurements over large distances, or to bring the instruments into an environment that is much more controlled than it could be on Earth. For example, in 2004 a dedicated satellite for gravity experiments, called Gravity Probe B, was launched. Also, land-based experiments like LIGO are gearing up to possibly detect gravitational waves directly.
Speed of gravity: Einstein's theory of relativity predicts that the speed of gravity (defined as the speed at which changes in location of a mass are propagated to other masses) should be consistent with the speed of light. In 2002, the Fomalont-Kopeikin experiment produced measurements of the speed of gravity which matched this prediction. However, this experiment has not yet been widely peer-reviewed, and is facing criticism from those who claim that Fomalont-Kopeikin did nothing more than measure the speed of light in a convoluted manner.
The Pioneer anomaly is an empirical observation that the positions of the Pioneer 10 and Pioneer 11 space probes differ very slightly from what would be expected according to known gravitational effects. The possibility of new physics has not been ruled out, as some experiments, such as those involving the Jeff Cook Effect, bring to light a variety of new possibilities, despite very thorough investigation in search of a more prosaic explanation.
Although the law of universal gravitation was first clearly and rigorously formulated by Isaac Newton, the phenomenon was observed and recorded by others. Even Ptolemy had a vague conception of a force tending toward the center of the Earth which not only kept bodies upon its surface, but in some way upheld the order of the universe. Johannes Kepler inferred that the planets move in their orbits under some influence or force exerted by the Sun; but the laws of motion were not then sufficiently developed, nor were Kepler's ideas of force sufficiently clear, to make a precise statement of the nature of the force. Christiaan Huygens and Robert Hooke, contemporaries of Newton, saw that Kepler's third law implied a force which varied inversely as the square of the distance. Newton's conceptual advance was to understand that the same force that causes a thrown rock to fall back to the Earth keeps the planets in orbit around the Sun, and the Moon in orbit around the Earth.
Newton was not alone in making significant contributions to the understanding of gravity. Before Newton, Galileo Galilei corrected a common misconception, started by Aristotle, that objects with different mass fall at different rates. To Aristotle, it simply made sense that objects of different mass would fall at different rates, and that was enough for him. Galileo, however, actually tried dropping objects of different mass at the same time. Aside from differences due to friction from the air, Galileo observed that all masses accelerate the same. Using Newton's equation, F = ma, it is plain to us why:
The above equation says that mass m1 will accelerate at acceleration a1 under the force of gravity, but divide both sides of the equation by m1 and:
Nowhere in the above equation does the mass of the falling body appear. When dealing with objects near the surface of a planet, the change in r divided by the initial r is so small that the acceleration due to gravity appears to be perfectly constant. The acceleration due to gravity on Earth is usually called g, and its value is about 9.8 m/s2 (or 32 ft/s2). Galileo didn't have Newton's equations, though, so his insight into gravity's proportionality to mass was invaluable, and possibly even affected Newton's formulation on how gravity works.
However, across a large body, variations in r can create a significant tidal force.
It's important to understand that while Newton was able to formulate his law of gravity in his monumental work, he was not comfortable with it because he was deeply uncomfortable with the notion of "action at a distance" which his equations implied. He never, in his words, "assigned the cause of this power." In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.
He lamented the fact that 'philosophers have hitherto attempted the search of nature in vain' for the source of the gravitational force, as he was convinced 'by many reasons' that there were 'causes hitherto unknown' that were fundamental to all the 'phenomena of nature.' These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer is yet to be found. While it is true that Einstein's hypotheses are successful in explaining the effects of gravitational forces more precisely than Newton's in certain cases, he too never assigned the cause of this power, in his theories. It is said that in Einstein's equations, 'matter tells space how to curve, and space tells matter how to move,' but this new idea, completely foreign to the world of Newton, does not enable Einstein to assign the 'cause of this power' to curve space any more than the Law of Universal Gravitation enabled Newton to assign its cause. In Newton's own words:
- I wish we could derive the rest of the phenomena 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.
If science is eventually able to discover the cause of the gravitational force, Newton's wish could eventually be fulfilled as well.
It should be noted that here, the word "cause" is not being used in the same sense as "cause and effect" or "the defendant caused the victim to die". Rather, when Newton uses the word "cause," he (apparently) is referring to an "explanation". In other words, a phrase like "Newtonian gravity is the cause of planetary motion" means simply that Newtonian gravity explains the motion of the planets. See Causality and Causality (physics).
A self-gravitating system is a system of masses kept together by mutual gravity. An example is a binary star.
Special applications of gravity
A height difference can provide a useful pressure in a liquid, as in the case of an intravenous drip and a water tower.
A weight hanging from a cable over a pulley provides a constant tension in the cable, also in the part on the other side of the pulley.
Comparative gravities of different planets and Earth's Moon
The standard acceleration due to gravity at the Earth's surface is, by convention, equal to 9.80665 metres per second squared. (The local acceleration of gravity varies slightly over the surface of the Earth; see gee for details.) This quantity is known variously as gn, ge (sometimes this is the normal equatorial value on Earth, 9.78033 m/s²), g0, gee, or simply g (which is also used for the variable local value). The following is a list of the gravitational accelerations (in multiples of g) at the surfaces of each of the planets in the solar system and the Earth's Moon :
Note: The "surface" is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune) in the above table. It is usually specified as the location where the pressure is equal to a certain value (normally 75 kPa?). For the sun, the "surface" is taken to mean the photosphere.
Within the earth, the gravitational field peaks at the core-mantle boundary, where it has a value of 10.7 m/s².
For spherical bodies surface gravity in m/s2 is 2.8 × 10−10 times the radius in m times the average density in kg/m3.
When flying from earth to mars, climbing against the field of the earth at the start is 100 000 times heavier than climbing against the force of the sun for the rest of the flight.
Mathematical equations for a falling body
These equations describe the motion of a falling body under acceleration g near the surface of the Earth. Here, the acceleration of gravity is a constant, g, because in the equations above, r21 would be a constant vector, pointing straight down. In this case, Newton's law of gravitation simplifies to the law
- F = mg
The following equations ignore air resistance and the rotation of the Earth, but are usually accurate enough for heights not exceeding the tallest man-made structures. They fail to describe the Coriolis force, for example. They are extremely accurate on the surface of the Moon, where the atmosphere is almost nil. Astronaut David Scott demonstrated this with a hammer and a feather. Galileo was the first to demonstrate and then formulate these equations. He used a ramp to study rolling balls, effectively slowing down the acceleration enough so that he could measure the time as the ball rolled down a known distance down the ramp. He used a water clock to measure the time; by using an "extremely accurate balance" to measure the amount of water, he could measure the time elapsed. 2
- For Earth, in Metric units: in English units:
For other planets, multiply by the ratio of the gravitational accelerations shown above.
|Distance d traveled by a falling object
under the influence of gravity for a time t:
|Elapsed time t of a falling object
under the influence of gravity for distance d:
|Average velocity va of a falling object
under constant acceleration g for any given time:
|Average velocity va of a falling object
under constant acceleration g traveling distance d:
|Instantaneous velocity vi of a falling object
under constant acceleration g for any given time:
|Instantaneous velocity vi of a falling object
under constant acceleration g, traveling distance d:
Note: Distance traveled, d, and time taken, t, must be in the same system of units as acceleration g. See dimensional analysis. To convert metres per second to kilometres per hour (km/h) multiply by 3.6, and to convert feet per second to miles per hour (mph) multiply by 0.68 (or, precisely, 15/22).
Note 1: Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. Preceded by A Guide to Newton's Principia, by I.Bernard Cohen. University of California Press 1999 ISBN 0-520-08816-6 ISBN 0-520-08817-4
Note 2: See the works of Stillman Drake, for a comprehensive study of Galileo and his times, the Scientific Revolution.