History topic: Theories of gravitation


One of the most obvious properties of the world is that if you hold a stone in your hand and release it, then the stone falls. Although we now know that the Moon orbits the Earth and the planets orbit the Sun because of the same force that makes the stone fall, this was not suspected by early scholars. Aristotle put forward his ideas on why objects fall to Earth, and also on motion in general, in works written around 330 BC. Aristotle writes in Book II of Physics:-

Some existing things are natural, while others are due to other causes. Those that are natural are ... the simple bodies such as earth, fire, air and water; for we say that these things and things of this sort are natural. All these things evidently differ from those that are not naturally constituted, since each of them has within itself a principle of motion and stability in place ... A nature is a type of principle and cause of motion and stability within these things to which it primarily belongs ... A nature, then, is what we have said; and the things that have a nature are those that have this sort of principle. All things are substances, for a substance is a sort of subject, and a nature is invariably in a subject. The things that are in accordance with nature include both these and whatever belongs to them in their own right, as travelling upward belongs to fire ...

So Aristotle argues that the stone falls because it has a "nature within it" which causes its motion to its natural place which is the centre of the Earth. Natural motion of the heavenly bodies, according to Aristotle, is circular. Other ideas which he put forward were that an object moving at constant speed requires a continuous force acting on it to maintain that speed. He also argued that force can only be applied to an object through contact.

As well as natural motion there is unnatural motion, as for example when a stone is thrown upwards. However such unnatural motions are short lived (since continual application of force is required to maintain motion) and the "natural desire" of the stone to return to the Earth takes over and then natural motion returns the stone to Earth. This theory survived through the 16th century and the following illustration from a book published in 1547 shows how Aristotle's theory was applied to cannonballs. In this illustration the cannonball first travels in a straight line due to its initial unnatural motion, then when this force is spent the cannonball follows the arc of a circle until it begins its natural motion vertically downwards to the Earth.

Aristotle also believed that heavier objects fell more rapidly than lighter ones. This of course is a reasonable assumption, for if you hold a feather in one hand and a brick in the other and let go each at the same time then the brick will certainly hit your toe first. This, of course, is because of resistance of air, but to Aristotle it was simply clear that the heavier object fell more rapidly.

Although these ideas of Aristotle are completely wrong, they are important since they were accepted for nearly two thousand years. Perhaps most significantly for the development of science in Europe, Aristotle's ideas of physics were taken on board by the Christian Church. This had the effect of making it much more difficult for scientists to demonstrate that Aristotle's physics was false.

Although Galileo and Kepler did not put forward theories of gravitation as such, nevertheless they each made significant contributions which set the scene for later developments in gravitational theory. Their contributions which we now discuss were not thought at the time to be in any way connected, for Kepler's contribution concerned the orbits of the planets round the Sun while that of Galileo concerned motion and the acceleration of falling objects. We now know that both contributed to an understanding of gravity, but at the period when they worked no connection between falling objects and planetary motion was suspected.

Kepler's first two laws of planetary motion are: (1) a planet moves round the Sun in an ellipse with the Sun at one focus, and (2) a line joining the planet to the Sun sweeps out equal areas in equal times as the planet describes its orbit. As well as giving a simple mathematical model for planetary motion, these laws were highly significant since they stated for the first time that the motions of the heavenly bodies are not composed of circular motion.

Galileo showed that Aristotle was wrong to claim that for a body to continue to move at a constant speed, a force must be applied. On the contrary, he showed that a body which is not acted on by any force will continue in constant motion. Galileo experimented by letting objects fall toward the Earth, and he discovered that they underwent constant acceleration. Carrying out experiments, Galileo showed that the distance a body travels as it falls is proportional to the square of the time. However, stating the results in this way does little to show how brilliant they were, for they overturned the ideas of gravity as put forward by Aristotle which had been accepted as fact for nearly 2000 years.

It was not just experimental evidence that Galileo used to show that Aristotle's theory of falling objects was incorrect, he showed by logical argument that Aristotle's ideas do not make sense. In Dialogue Concerning the Two Chief Systems of the World (1632) Galileo argues as follows. Suppose we have two stones, the first being lighter than the second. Release the two stones from a height to fall to Earth. Stone 2, being heavier than stone 1, falls more rapidly. If they are joined together, argues Galileo, then the combined object should fall at a speed somewhere between that of the light stone and that of the heavy stone since the light stone by falling more slowly will retard the speed of the heavier. But if we think of the two stones tied together as a single object, then Aristotle says it falls more rapidly than the heavy stone. How do the stones know if they are one object or two? Aristotle's theory is inconsistent. Galileo had produced a stunningly clever "thought experiment" which is always more significant in experimental evidence which could easily have been incorrectly collected.

There are other stunningly innovative ways of thinking by Galileo. Up until this time people had watched a pendulum swinging and seen the bob of the pendulum going back and forward. Galileo, however, saw that the bob went up and down. He essentially saw the movement as composed of vertical and horizontal components with gravity being the force in the vertical direction. Although the time taken for the bob to rise and fall depended on the length of the pendulum, it did not depend on the weight of the bob. Again his argument showed that, ignoring the resistance of the air, the time taken for an object to fall from a given height did not depend on its weight. Rolling balls down inclined planes also allowed Galileo to examine the downwards component of the speed of fall.

We should realise that there are two aspects to questions about gravity. There is the physics of the situation which involves finding the equations which govern bodies acted on by gravity, and for this a proper understanding of the laws of mechanics is required. Then there is the more philosophical question of why the laws hold, what is the mechanism which leads to the observed gravitational effects. Aristotle's theory covered both aspects with a description of how bodies reacted under gravity and also a reason for the effects, namely the "natural tendency" of a stone to produce motion to bring it to its natural place on the Earth. Both aspects are, of course, quite incorrect. Galileo made a staggering advance in understanding mechanics and the way bodies move under gravity, but he did not tackle the reasons for objects to behave as they do. Descartes, on the other hand, produced an explanation of gravity based on his philosophical principles.

Descartes believed in rationalism. Thus he believed that knowledge must be acquired by reasoning rather than by using the senses which can, he argued, be easily deceived. Galileo's thought experiment is a good example of the use of logical reasoning. He did not entirely reject the idea of using observations and conducting experiments but rather used these to guide his intuition which provided him with first principles from which mathematical and logical deduction would then lead to knowledge. He wrote in 1644 in Principles of Philosophy:-

... my consideration of matter in corporeal things involves absolutely nothing apart from divisions, shapes and motions.

In many ways these ideas by Descartes were of fundamental importance in the development of science since he claimed that everything could be analysed mathematically. All one needed to do was to construct a mathematical theory of mechanics and all would be understood. He proposed two forces, one associated with the rest state of a body dependent only on its mass, and one associated with maintaining the motion of a body associated with motus, the product of its speed and mass. (Motus is not momentum since mass times speed has no direction.) To Descartes the only properties that a body possessed were a mass and a speed and his mechanics was based on a conservation of motus. From this he deduced seven rules of motion which described how movements between bodies could be communicated.

Where did this take Descartes in thinking about gravity? Most obviously if a stone is released and falls to earth, then some matter must act on it to force it to do so. However, since we are not aware of such matter, it must be of a new form of matter, call it the ether. Another deduction which followed from Descartes' way of thinking was that a vacuum could not exist. This was in no way deduced from any experiment, just simply a logical deduction that the concept of a vacuum was meaningless since if something existed it could not be nothing. All of these ideas were put forward by Descartes in 1644.

Now once everything was filled with ethereal matter one had to consider what happened when a solid body, such as a planet, moved through the ether. Clearly the ether must be forced away from the front of the object and it will then flow round it in a circular motion. As bodies were in motion, the universe must consist of eddies, vortices and whirlpools. If, Descartes argued, the ether was perpetually in vortex motions caused by bodies passing through it, then the ether in turn would cause motion in bodies situated in it. These vortices were the cause of gravity; they held the planets in their orbits round the Sun. The Earth was surrounded by ethereal material which circulated rapidly round the Earth. This light, rapidly moving material would flow away from the Earth, and heavy, slow moving bodies like a stone would fall towards the centre of the Earth to replace it.

This theory of gravitation was certainly a pleasing one. Gone were the "natural tendency" and "desire" of a stone to move to the Earth as Aristotle had suggested. It had been replaced by a purely mechanical system which explained gravity in terms of purely mechanical terms concerning interactions between matter. It was quickly accepted as a good metaphysical system, but when the details were closely examined they did not quite match with observations. Huygens pointed out in 1669 that if the ether flowed in a circular motion round the Earth's axis then gravitational attraction should be towards the axis rather than towards the centre of the Earth. He tried to give a modification of the vortex motion which was equal in all directions. He failed to explain, however, how the ether could move in all directions without interfering with itself.

In 1675 Malebranche published Recherche de la vérité which revised Descartes' theory of forces, showing that no force could be associated with the rest state of a body, so everything depended on Descartes' motus force. As a result he was only left with four of Descartes' seven rules for motion. He also modified the theory by assuming that apparently hard bodies were permeated by the ether. Leibniz came up with other objections, proposing new ideas in Acta erditorum in 1686. He showed that motus is not conserved in collisions, forcing Malebranche to replace motus by momentum. (This is mass times velocity and has a direction.) Leibniz returned to the idea of two forces, one being a "dead force" which will produce motion in a stone released from rest above the Earth, the second force being a "live force" which he defined to be the mass times the square of the velocity. In some respects Leibniz's first force marked a return to Aristotle's "natural tendency" of a stone to move to the Earth. It was a theory which explained observational evidence of gravity well, but it was strongly rejected by followers of Descartes since it was not based on rational explanations.

One year after Leibniz published his article in Acta erditorum, Newton published the Philosophiae naturalis principia mathematica or Principia as it is always called. In it Newton proposed universal gravitation and the inverse square law of attraction between any two bodies even separated in a vacuum. The attraction was the product of the two masses divided by the square of their distance apart. A stone falling to Earth was subject to precisely the same force which kept the Moon in its orbit about the Earth and the planets in their orbits around the Sun. Newton deduced Kepler's three laws of planetary motion from the inverse square law of attraction. Comets, he showed, were subject to the same gravitational forces and these forces gave a simple explanation of tides. See the article Orbits and gravitation for more information on how Newton discovered his theory of gravitation.

The Principia also gives arguments against Descartes' theory of gravitation. Newton showed that Kepler's first and second law could not both hold under Descartes' theory. He also asked how comets were able to cross vortices strong enough to keep the planets in their orbits without any effect on their own orbits. It was a devastating attack on Descartes' vortex theory of gravitation and put forward a brilliantly simple theory from which so much could be explained. It had one major problem, however. How could two objects attract each other across a vacuum? This was a theory like that of Aristotle's, but now a stone did not have just a "natural tendency" to fall to Earth, far more incredibly it had a "natural tendency" to be attracted to every other object in the universe. Descartes had a rational explanation for why stones fell to Earth and why planets orbited the Sun. Newton's theory, however, was based on some magic power of matter to be attracted to all other matter and this was anything but rational.

How did Newton answer the critics who asked: how is action at a distance possible? He did not try to. Rather he said that he was explaining how things worked but not attempting to explain why they worked that way. For some this was good enough, for others this was such an unsatisfactory state of affairs that they had to reject Newton's theory. Newton himself wrote in a letter to Dr Bentley dated 25 February 1693 (see for example [1], page 302):-

It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and effect 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 through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me a great absurdity, and 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 to the consideration of my reader.

There was no simple solution to the problems that the different theories posed. Many, such as Huygens, accepted the inverse square law but rejected the idea of attraction. Such people looked for other mechanisms which would lead to the inverse square law but produced by a modification of the vortex ideas of Descartes. Malebranche, for example, proposed little vortices within bigger vortices which made the Earth look as if it were producing a "boiling" effect on the ether which produced a radial flow of ether to explain the stone moving towards the centre of the Earth when released. Despite many attempts to modify Descartes' vortex theory, it could never satisfactorily account for the observed world and eventually Newton's theory became universally accepted.

For later developments in the theory of gravitation, the reader should now go to our article General relativity.

Article by: J J O'Connor and E F Robertson


April 2003

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Gravitation.html]