Principles of explanation
3.1. Number and space in the harmony of the spheres
3.2. Explanation of change in Aristotelian cosmology
3.3. Galileo on motion
3.4. Cartesian physics
3.5. Early concepts of force
3.6. Newton’s dynamics
3.7. Absolute and relative space, time, and motion
Number and space
in the harmony of the spheres
chapter discusses some fundamental and irreducible principles of explanation operative in the physical sciences. Before the Copernican era, especially quantity and space were crucial principles of explanation, both in ancient and medieval
philosophy (3.1-3.2). The Copernican revolution transformed these into quantitative and spatial relations, and developed two new relational principles, pure kinetic motion, and physical interaction.
Together with matter and action by contact, local motion became the focus of Galileo’s and Descartes’ mechanical philosophy, propagating
the explanation of motion by motion (3.3-3.4). Kepler and Newton explained change of motion by a force, introducing experimental philosophy as a correction to mechanism (3.5-3.6). Newton discussed the problem of absolute or relative time, space, and motion
(3.7). In order to make clear the novelty of these principles, some earlier attempts at explanation will be reviewed first.
Pythagoras and the rational numbers
In the sixth century BC, Pythagoras and his school tried to reduce cosmic relations to pure numbers and their ratios. Rational means
both reasonable and proportional. As proportions of natural numbers, rational numbers are intelligible, and with their help the universe may be understood.
For instance, the Pythagoreans
discovered that the musical tones sounding harmoniously together, stand to each other as integral numbers. These tones were not related to frequencies or wavelengths, as they are considered now, but to the linear dimensions of some musical instruments. Two
strings of unequal length producing different tones differ by an octave if the lengths of the string are as 1:2, by a fifth if the proportion is 2:3, whereas a fourth corresponds to the ratio 3:4. It inspired Copernicans like Stevin, Galileo, Kepler, Mersenne,
and Huygens to study harmonics (2.6). Galileo’s
father, Vincenzio Galilei, came into conflict with Aristotelian philosophers about the theory of musical consonants, invented by the Pythagoreans. He argued that this theory was no longer in accord with new musical practices. Both Simon Stevin and Christiaan
Huygens tried to found a new theory, leading up to Huygens’ division of the octave in 31 tones. Marin Mersenne discovered that any tone produced by a musical instrument is accompanied by a number of harmonics.
The Pythagoreans attached much significance to numbers like 10, being the sum of 1, 2, 3, and 4. These four numbers were supposed to determine a point, a line, a plane, and a body, respectively. Accordingly, some Pythagoreans assumed
ten celestial bodies or celestial spheres. A significant result of Greek astronomy, ascribed to Pythagoras, was the identification of the morning star and the evening star as the same planet, Venus (though the terms are also applied to other bright celestial
bodies). In order to restore the number of ten planets, some Pythagoreans postulated a counter-earth, moving on the other side of the central fire, and therefore not observable. The central fire is never seen because the earth is only inhabitable at the side
turned away from the fire.
Perhaps influenced by Babylonian mythology, the emphasis shifted from the number 10 to the no less holy number of 7. The motions of the now seven planets stand to each other as the tones of the octave,
displaying the harmony of the spheres. Kepler employed in his Harmonice mundi a musical notation to describe the elliptic motion of the planets, even if he recognized only six planets.
The turn from arithmetic to geometry
Although influenced by the Pythagorean tradition, Plato made an important shift. Whereas the Pythagoreans stressed numerical proportions, Plato turned to spatial relations. Plato based a theory
of matter on the Pythagorean discovery of the existence of exactly five regular polyhedrons (2.6). The four elements, earth, water, air, and fire, introduced by Empedocles, corresponded respectively with the cube, the icosahedron, the octahedron, and the tetrahedron.
The dodecahedron with its twelve pentagonal faces characterizes the fifth element, quintessence or ether, from which the heavenly bodies are made. The argument for this correspondence is quite vague.
The shift from the numerical to the spatial principle of explanation was caused by a crisis in the Pythagorean brotherhood, which led to its disbandment. This crisis occurred shortly after the discovery
of the famous Pythagorean theorem. It implies that in a square with side 1, the square of the diagonal equals 2. The Pythagoreans proved that the length of this diagonal cannot be expressed by a rational number. The length of the diagonal is thus not rational,
it is irrational, unreasonable, and unintelligible.
The starting point of the Pythagorean school, to explain everything with the help of rational proportions, was shipwrecked on a simple
spatial problem. The Pythagoreans could not overcome the irreducibility of spatial relations to purely numerical ones. As a result, Alexandrian mathematicians inspired by Plato, such as Euclid, Ptolemy, and Archimedes, directed themselves to the development
of geometry, as a principle of explanation besides the numerical one. Apollonius of Perga studied the sections of a plane with a double cone, discovering and naming the ellipse, the parabola, and the hyperbola. For the Copernicans, the Pythagorean-Platonic
tradition became a source of inspiration to build a mathematical physics. However, first they had to transform numerology into calculation, numbers into quantities.
3.2. Explanation of change
in Aristotelian cosmology
In ancient and medieval philosophy motion was never a principle of explanation. It was always considered a result of an explanation. Like any kind of change, local motion had to be explained, and could not be used as explanans.
In Copernicanism, motion became a new principle of explanation.
In order to understand the revolutionary character of this move, we have first to review Aristotle’s theory of change,
to which it was opposed. The Aristotelian scheme of explanation was one of the answers given to a problem first formulated by Parmenides of Elea, circa 500 BC. Parmenides identified being with intelligibility. On logical grounds, Parmenides proved change to be unintelligible, and thus non-existing.
is cannot have come into being. If it did, it came either from what is or what is not. But it did not come from what is, since if it is existent it did not come to be but already is; nor from what is not, for the non-existent cannot generate anything.’
By means of some well-known paradoxes, Parmenides’ disciple, Zeno of Elea, tried to prove that motion is an illusion. He argued that Achilles, a legendary athlete, would never be able to overtake a tortoise, for if he would have covered the distance which at first separated him from the animal, it would have moved on. Concerning these
paradoxes, several views are conceivable.
First, it may be admitted that the arguments are correct, and that motion is illusory. This view was shared by Parmenides, Zeno, and Plato.
The observable world to which motion belongs is deceptive. It is merely an appearance, a shadow of the real, unchangeable world of ideas.
Second, one may accept Zeno’s arguments, but maintain that motion is real. This means the recognition that motion cannot be explained starting
from the categories Zeno applied. In his arguments we find only numbers and spatial distances as elements. Zeno succeeded in proving that these principles are not sufficient to explain motion. Shortly before Zeno, the Pythagoreans had proved that spatial relations
cannot be explained by relations between integral numbers (3.1). Zeno’s paradoxes can be interpreted as demonstrating that motion cannot be explained by numerical and spatial relations. This way to consider Zeno’s paradoxes means to accept motion as an unexplained principle of explanation. It became the route of the Copernicans (3.3).
The third view is that Zeno’s arguments are incorrect, and that motion is not illusory. This approach led Aristotle to his realistic theory of change.
For the Greek philosophers, the ideal state of the universe was a static equilibrium in which only
natural motions would occur, to be explained by numerical and spatial relations. Aristotle was far more realistic than his predecessors. He valued observations higher than Plato did, and accepted change as a problem to be solved.
According to Aristotle’s Physics, the explanation of change must start from two unchangeable principles, eternal form and eternal matter, both universal. The forms are very similar to Plato’s
ideas. The main difference concerns their relation with observable things, which Plato considered unreliable copies of the ideas. These should primarily be understood in a mathematical sense. The ideal triangle, a subject matter of geometry, is only approximately
realized in triangular bodies. Aristotle was more inspired by his biological studies, and his forms refer to species of animals or plants. Unlike mathematical ideals, these can only be studied by careful observation.
Any individual is a combination of form and matter, is formed matter, is substance. Forms and matter are eternal and unchangeable. Only substances can change. This view prevented Aristotle and his followers from appreciating that also relations and even motion itself might be variable.
Strictly unformed matter, materia prima (first matter), is characterized as absence of form and therefore does no more exist in any concrete
sense than pure form. But matter can also
be conceived in a less absolute sense. When a sculptor creates a sculpture, its design is changed as little as the marble, the material used. Only the concrete piece of marble changes, because the sculptor adds a new form to its matter. This form existed beforehand
in the imagination of the artist, and the perfect form does not change. Changing a substance is the process (kinesis) by which it attains a new form.
Potentiality and actuality
In order to explain such a process, it is not sufficient to have insight into the form and matter concerned.
The matter involved must have the potential to attain the appropriate form. It is possible to make a sculpture out of a piece of marble, but not of sand or water. Marble has the potential to become a statue, which water has not. This is even more clear in living nature. A chicken-egg has the potential to
become a chicken, and never becomes a duck or a horse.
Besides form, matter and potential, a fourth principle of explanation is needed, the efficient cause, the actualization
of the potential. The chicken-egg must be hatched, the marble must be worked on. Only if these four causes are found is the explanation complete, according to Aristotle.
Instead of potential
and actual, Aristotle also speaks of the final cause or destiny, and the efficient cause. In this case, the process from potentiality to actuality is treated separately. It is not always possible to distinguish the four causes. Sometimes the final cause or end appears identical with the form to be achieved.
physics changed the four Aristotelian causes dramatically. The formal cause was replaced by natural law (8.6). The material cause was resolved in the schemes of matter and motion or matter and force. End, purpose, or destiny disappeared as a principle of explanation
in physics, though it remained relevant in the study of animal behaviour and of human activity. Only the efficient cause survived, as the cause of motion, or of change of motion.
Four kinds of change
Aristotle distinguished four kinds of change in individuals, in decreasing order of importance: change of essence or of nature,
generation and corruption, coming into being and passing away, for instance the birth of a caterpillar, or the death of a butterfly; qualitative change of properties, like the change of a caterpillar into a butterfly; quantitative change
of degree, increase or decrease, for instance the growth of a caterpillar; and finally change of position, for example the local motion of our caterpillar on a tree. Although the least far-reaching, no change is possible without local motion.
Every change has two termini, a beginning (its matter) and an end (its form). For the first kind of motion, generation and corruption, these are contradictories, x and non-x. The other three have contraries
like hot and cold as termini. Therefore, Aristotle rejected the infinity of the cosmos. If the cosmos were infinite, the elements could move infinitely far away, without an end. Uniform circular motion is the only kind of motion having neither beginning nor
end. It lacks contraries, it lacks potential
existence. Celestial bodies, moving uniformly in perfect circles, are completely actual, hence unalterable, eternal, incorruptible, ungenerable, unengendered, and impassive. These cosmic properties are found as a logical consequence of Aristotle’s
theory of change. Even in this case, local motion is the basis of the actuality of the bodies concerned.
For a natural change no external cause is needed. We still distinguish between
natural and unnatural death – the latter needs explanation. The alteration of an acorn into an oak is a natural process, and does not require an external cause. But the change of an oak into a pile of boards is an unnatural process, in need of an external
cause. The free fall of a heavy body can be prohibited, if it is sustained. Similarly, natural processes like the growth of a plant can be prohibited, for instance, because of lack of water. But as soon as such external impediments are removed, the process
will occur according to its nature, without any external cause.
Four terrestrial elements
Besides the four causes and four types of change, Aristotle distinguished four terrestrial elements. Plato related the elements to the regular polyhedrons, but this could not serve Aristotle’s
theory of change.
Generation and corruption always involves a mixing of elements. The celestial bodies are made of a single element, ether, because they cannot be generated or corrupted.
Aristotle related the terrestrial elements to termini of change. These are pairs of contrary properties, like warm and cold, dry and moist, up and down. Earth is dry and cold, water moist and cold, air hot and moist, and fire hot and dry. Earth and water are
heavy, and by their nature move downward. Fire and air are light, moving upwards. The upward and downward motions are opposite, hence point to imperfection, and to the existence of at least two elements, heavy earth and light fire. The contrary qualities of heavy and light were never related to density. Only neo-Platonic scholars like Benedetti and Galileo studied density as a quantitative property (1.4).
Empedocles’ four elements, if severed from the distinction between gravity and levity, are consistent both with Aristotelian and Copernican views. Up till the 19th century they remained the basis of medical and psychological
theories. Galileo connected the elements with the senses. In chemistry, the four elements were abandoned in the 18th century.
The theory of the elements enabled Aristotle to criticize
the older atomic views. As a reply to Parmenides’ denial of variability, the atomists accepted only local motion as possible change. In various ways, from the time of Galileo most Copernicans were atomists. They considered atomism not in the first place
as a theory of the structure of matter, but as an ontological foundation of their world view, in which local motion does not play a subordinate, but rather a leading part.
Aristotle on natural motion
Before Galileo’s Discorsi, the most important treatise of motion was
Aristotle’s Physics. It
can only be understood in its relation to his cosmology, and the theory of the elements, displayed in Aristotle’s On the heavens. Just like Plato, Aristotle put the earth in the centre of the universe.
If the cosmos were in equilibrium, it would consist of a set of perfect concentric spheres. At the centre is the sphere of the heavy element earth, surrounded by the less heavy element water, the light element air, and lightest of
all, fire. In or near the sphere of fire one observes lightning, comets, meteors, and aurora borealis (northern lights). It occupies the periphery of the sublunary spheres rather than their centre, as some Pythagoreans assumed. The sphere of fire is contained
in the lunar sphere, the lower boundary of the heavenly space in which the celestial bodies move around.
According to Aristotle, the lunar sphere constitutes a sharp division between
the celestial and terrestrial realms. The celestial space is ordered, the sublunary space is disordered. In the heavens, everything is perfect: the spherical shape of the celestial bodies; their unchangeability and incorruptibility; their circular uniform
motion. On the other side, the sublunary sphere is imperfect. The separation of earth, water, air, and fire into concentric spheres is disturbed, and the four elements are mixed. Here one finds not only natural motion, but also unnatural, violent motions.
Natural motion in the sublunary spheres is vertical and rectilinear. It is directed toward the centre for heavy bodies, and toward the periphery for light bodies. Even this natural motion is caused
by an unnatural, artificial initial state, deviating from the ideal state of equilibrium. Sublunary natural motion of a body means motion to its natural place, and can only occur if it is not in its natural place to begin with. This shows that also for Aristotle
spatial position has a high priority as a principle of explanation. Natural motion is explained by spatial arguments.
The natural motion of the heavens, too, is not without cause. It
is caused by the divine unmoved mover, who, at the uppermost periphery of the cosmos, is nothing but pure thought, thinking about itself – thought returning to itself. The rotation of the heavens is caused by the love of this god, by striving to become
god-like, perfect. It is a final cause, not an efficient one. Circular motion is returning in itself. Ultimately, all motion is caused by the prime mover as an end, and the centre of the cosmos is unmoving.
Aristotle’s cosmos is an organized whole, in which everything has its natural place. It is, however, not a living organism. Aristotle did not adhere to astrology, which was not influential in Athens of his time. However, when
during ages to come astrology and alchemy came to the forefront, their leading idea of the microcosmos-macrocosmos correspondence easily fitted into Aristotle’s cosmology.
Violent motion and the impetus theory
Aristotle distinguished natural motion from violent, artificial motion,
motion influenced by a force. Natural
motion is motion according to the nature of a thing, whether it is heavy or light. A heavy body falls downward, because it is heavy. This is an internal cause. For a natural motion no external cause is needed, as it is for any unnatural motion. The relation
between natural and violent motion is rest. Natural motion is the actualization of some potential, and naturally ends when the body has achieved its end, its natural place. Violent motion is contrary to the nature of a thing. Therefore, by force of its nature,
everything resists violent motion. For Aristotle, rest is ontologically different from motion, both natural and violent, and is superior to both. Clearly, natural motion is explained by spatial circumstances, violent motion by a force. Motion is not a principle
of explanation itself.
During the Middle Ages, Aristotle’s theory of local motion caused much discussion. In particular, it is by no means clear what kind of force causes the motion
of an arrow. Apparently, it is the force of the bended bow. But this force ceases to act as soon as the arrow has left the bow, whereas the motion does not cease. Aristotle rejected any kind of action except action by contact.
In order to solve this problem, the 14th-century scholars Jean Buridan and others at Paris developed the impetus theory, assuming that motion
can be caused either by an internal or by an external motor. The external motor is the force, the internal motor is called the impetus. The bow does not only supply an external force to the arrow, but also an internal impetus. During the motion, the impetus
decreases until it is exhausted, and the motion ceases. In the theories of Aristotle and his disciples it was unimaginable that a body would partake in a natural and a violent motion simultaneously. An arrow shot obliquely would move rectilinearly until its
impetus is exhausted. Only then it would begin to fall.
The observed curved path of motion is contradicted by the theory, and hence deceptive.
Impetus was supposed to be proportional to the quantity of matter in the body and to its
motion. In the 17th century this was transformed into mass and velocity, but these magnitudes were not defined yet in the 14th century. Nevertheless, the impetus can be recognized as a predecessor of the modern concept of linear momentum. However, impetus
was considered as the cause of the motion, whereas the later momentum is merely a measure of motion, quantity of motion. The transformation of impetus into momentum has been a laborious process, and is a fruit of the Copernican revolution.
Although the medieval impetus theorists assumed that a falling body is increasing its impetus, and they also studied uniformly accelerated motion, before the 16th century the two were never related. Buridan arrived at the important insight that the speed of a falling body at any instant depends on the path traversed since the start of the motion, rather than on the distance to the body’s natural place.
During the Middle Ages, projectile motion was never considered a proof against the validity of Aristotle’s views. Even the impetus theorists tried to solve the problem within the context of Aristotle’s physics. Only in
the 17th century the emergence of a new theory of motion became possible after the distinction between celestial and terrestrial physics was destroyed, in particular after Galileo’ attack on Aristotle’s cosmology (3.3).
The beauty of Aristotle’s theory of change
theory of change is one of the most beautiful intellectual achievements ever made. It is not only completely logical, coherent and consistent; it is also in harmony with common sense. Apart from being falsified by classical physics, it has only three flaws:
the uncertain status of light, being both terrestrial and celestial; projectile motion; and, perhaps most important, the lack of insight that distinctions like hot and cold are not contraries, but allow of gradual transitions. The first could be tentatively
solved by abolishing the distinction between celestial and terrestrial physics; the second required a new view on motion; and the third led to the introduction of measurement in physics.
Aristotelian philosophy, local motion was only one kind of change, and change was a process from potentiality to actuality. The Copernican revolution implied the slow and gradual transition from local motion as a process to inertial motion as a state.
The first steps were taken by Copernicus himself, who contended that the earth’s daily rotation is not in need of any explanation besides the spherical shape of the earth. The natural motion of a sphere is rotation. The final steps were made by Huygens and Leibniz, who rigorously stated the relativity of motion, and by Newton, who associated the idea of inertia with the idea of mass. Meanwhile, Galileo, Beeckman, Descartes, Mersenne,
and Huygens founded mechanical philosophy.
3.3. Galileo on motion
Any new theory of motion could only have a chance after Aristotle’s cosmology was abolished. Meanwhile, Copernicus’ system could only be considered an interesting mathematical exercise. Kepler’s attempt to replace the Aristotelian
system by a Platonic one (2.6) was abortive, because Plato shared Aristotle’s distinction between celestial and terrestrial physics. The absolute split between the perfect heavens and the imperfect earth constitutes the heart of Aristotelian cosmology
– so much so that Galileo found it necessary to devote the entire first day of his Dialogue to the demolition of this distinction.
Unlike Brahe and Kepler, Galileo Galilei
was not a professional astronomer, although as a professor of mathematics at Padua, he had to teach Ptolemaic astronomy. As an able instrument maker he earned more money than the university paid him. Learning about the recent invention of the spyglass, he
built a telescope, and directed it to the heavens. He observed the moon, the sun, the planets, and the fixed stars, and discovered Jupiter’s four moons. In 1610 he published his discoveries in Sidereus nuncius (The message of the stars),on which his attack on Aristotle’s cosmology
was based. After this publication, which drew a lot of attention, Galileo discovered the phases of Venus and studied the motion of sunspots.
Galileo’s main argument was to show, on the basis of observations, that the celestial bodies are not perfect, not incorruptible, and not
unalterable. He discussed the mountains on the moon, in order to show that the moon is not perfectly spherical. He argued that the earth, like the moon and Venus, reflects the light of the sun, by pointing to the so-called secondary light of the moon. This occurs shortly before or after new moon, when alongside a small sickle
the dark part of the moon is perceptible.
Galileo explained why this phenomenon is not observable at first or last quarters. He showed that the sunspots are continuously generated and corrupted. He pointed to Tycho’s observations showing that the new generable and corruptible stars (novae) are celestial objects, not sublunary ones.
On the other hand, he argued that circular motion pertains to terrestrial as well as celestial objects. In his discussion of magnetism, Galileo
let Sagredo state that a magnet exerts both circular and linear motions, and should therefore, according to Aristotle, be composed of celestial and terrestrial matter.
Galileo interpreted Copernican cosmology in a realistic way. Colliding with Aristotle’s no less realistic cosmology, he came into conflict
with the Catholic Church (8.3). Despite its prohibition in 1633, Galileo’s Dialogue terminated Aristotle’s cosmology. After him, Descartes, Huygens, Newton, and other Copernicans did no longer bother to refute it.
Galileo’s Dialogue made an end to the separation of terrestrial and celestial physics. In both realms the same kind of explanations could now be applied. The relevance of this achievement can hardly be
Motion as a principle of explanation
Whereas the Pythagoreans were confronted with the irreducibility of the spatial principle of explanation, Zeno stumbled on the irreducibility of the kinetic mode (3.1). With hindsight, we could say that the Copernicans accepted Zeno’s
challenge head-on, by making motion
a new principle of explanation. In ancient and medieval philosophy motion was never a principle of explanation. It was always pursued as the result of an explanation. Like any kind of change, local motion was an explanandum and could
not be used as explanans.
In Copernicanism, the study of motion plays a central part. The characteristic trait of the Copernican system is not its heliocentrism, but the assumption
that the earth is moving. Motion was introduced as a principle of explanation. Copernicus argued that the apparent retrograde motion of the planets is caused by the real motion of the earth (2.4).
The principal objection against Copernicanism concerned the doctrine of the moving earth. In order to avoid it, Tycho Brahe proposed his compromise
system (2.5). As an astronomer, he recognized the advantages of Copernicus’ theory. In his earth-centred system, with five planets circling around the sun, the sun has a much more prominent position than in Ptolemy’s. Tycho considered the physical
and theological objections against the earth’s motion insurmountable.
In contrast, Galileo Galilei felt inspired by the difficulties engendered by the earth’s
motion. Between 1609,
when he made his provocative astronomical discoveries, and 1633, when he was convicted by the papal Inquisition, Galileo was the main agitator in favour of Copernicanism (8.3). After this episode, he published his ideas on the theory of motion in Discorsi
(1638), avoiding mentioning Copernican views. Most work on this theory was done at Padua, where he was a university professor from 1592 to 1610.
Galileo’s claim to be the first to have discovered the sunspots is unjustified, but the quality of
his observations and his reasoning were unsurpassed during his lifetime. He observed that some sunspots do not change their dimensions during several days, except that their apparent width decreases if the sunspots move from the centre towards the circumference of the solar disc. Based on a careful measurement,
he demonstrated that this phenomenon could only be explained if the sunspots are situated on the surface of the sun, which is spherical and rotates in about thirty days about its axis. Thus he explained the apparent change of the sunspots by the motion of
the sun. In a similar way he explained why the apparent speed of the sunspots changes during the sun’s motion. According to Galileo, in the heliocentric system the sun has only one motion, rotation around its own axis in thirty days. Assuming the earth
at rest, one has to describe two more motions to the sun: the daily and annual motion around the earth. The daily motion of the sun would imply that the axis of the sun’s own rotation changes continually, which is dynamically hard to believe.
Galileo’s mechanical philosophy
Galileo understood motion to
be sui generis, not to be explained, but to be used as an explanation. Explanation of motion by motion is short for the adoption of one or two principles of motion in order to explain other kinds of motion. It is not necessary to explain
the primary or natural motions themselves. The principle of inertia could be formulated as: a body on which no external force is acting, moves because it moves. The principle of relativity, first explored by Galileo, implies that inertial motion is
a relation. Besides inertial (circular) motion, Galileo considered the uniformly accelerated motion of free fall as a principle of explanation.
In Il saggiatore (The assayer,
1623) Galileo presented the program of the rising mechanical philosophy, to reduce all physical phenomena to matter, quantity, shape, and motion:
‘... whenever I conceive any material
or corporeal substance, I immediately feel the need to think of it as bounded, and as having this or that shape; as being large or small in relation to other things, and in some specific place at any given time; as being in motion or at rest; as touching or
not touching some other body; and as being one in number, or few, or many. From these conditions I cannot separate such a substance by any stretch of my imagination.’
This became the nucleus of mechanical philosophy. For instance, Galileo explained heat as motion of corpuscles:
‘I do not believe that in addition to shape, number, motion, penetration, and touch there is any other quality in fire corresponding to “heat”.’
In suit of Benedetti, he explained sound as motion caused by the periodic motion of a string,
‘… the waves which are produced by the vibrations of a sonorous body, which spread through the air, bringing to the tympanum of the ear a stimulus which the mind translates into sound.’
As a consequence, Galileo distinguished objective from subjective
properties, or primary from secondary qualities, as he called them:
‘To excite in us tastes, odors, and sounds I believe that nothing is required in external bodies except shapes,
numbers, and slow or rapid movements. I think that if ears, tongues and noses were removed, shapes and numbers and motions would remain, but not odors or tastes or sounds.’
Galileo considered motion as a state of a system. A body, moving or at rest, is physically completely unaffected by which of these two states it is in, and being in one or the other in no way changes it. Therefore, rest and motion are not contraries, as was taught by Aristotle. Both are states of motion. Although this was only gradually understood, it implies that the attribution of the state of rest or motion to a given body is only possible in relation to another one.
In 1612 Galileo formulated the first clear expression of the principle of inertia, arguing from a discussion of balls moving on a plane. If the plane is tilted, a downward motion accelerates, whereas an upward motion decelerates.
‘And therefore, all external impediments removed, a heavy body on a spherical surface concentric with the earth will be indifferent to rest and to movements toward any part of the horizon.
And it will maintain itself in that state in which it has once been placed; that is, if placed in a state of rest, it will conserve that; and if placed in movement toward the west (for example), it will maintain itself in that movement.’
Discorsi recognized two fundamental or natural motions: uniform circular motion (at constant speed), and the uniform accelerated motion
of free fall (at constant acceleration). Both occur without external cause, and are idealized states. The third day of Discorsi, entitled ‘Change of position – De motu locali’, introduced these carefully by the axiomatic method,
starting with the words:
‘My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning
which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated.’
Galileo recognized first uniform circular motion as primary, not in need of explanation. It concerns planets turning around the sun, and the earth
rotating about its axis, as well as a terrestrial body moving without friction on a horizontal plane, for instance, a ball on a smooth surface, without air resistance. Galileo described the horizontal motion sometimes as being approximately rectilinear, but
in principle it was circular.
Galileo’s second natural motion is free fall in a vacuum. In this case, acceleration instead of velocity is a constant. Again, no external cause is needed. Gravity, the source of the motion, is an intrinsic property of the falling body. But it is not its cause, for the motion of fall is independent of the weight of the body, as Galileo found shortly before 1610.
two kinds of natural motion are not contrary to each other. First, one kind can change into the other. A ball uniformly accelerated on an inclined plane may continue its motion at constant speed on a horizontal plane. Secondly, they have a common source, for
gravity is the source of all motion. When discussing the inertial motion of a ball on a horizontal plane, Galileo lets the motion start from an inclined plane. The uniform motion of the planets he explains with the help of a Platonic myth about the fall of the planets from a common point toward their present orbits. Gravity is even considered a measure of motion. Thirdly, two natural motions can be composed; a body is able to perform two motions simultaneously. This applies to a ball rolling down an inclined plane, or to the earth combining its daily and annual motions. It also applies to the combination of a horizontal uniform motion and a vertical accelerated motion. This enabled Galileo to explain the path of a cannon ball,
the time-honoured problem of projectile motion. He emphasized that his theory is able to explain why the cannon reaches farthest if the ball is fired at an angle of 45o. This fact was empirically known for quite a long time, but never explained.
Galileo on circular motions
Galileo connected both natural motions
to circular motions in the following way.
On a horizontal plane he considered a number of objects starting simultaneously from the same point, moving at the same speed in various directions. At any instant, their positions constitute a circle of which the common starting point is the centre. Next
he considered in one vertical plane a number of objects moving simultaneously on planes with various angles of inclination. If they start simultaneously from the same point, at any subsequent moment their positions lie on a circle having the horizontal through
the common starting point for a tangent. Galileo concluded:
‘The two kinds of motion occurring in nature give rise therefore to two infinite series of circles, at once resembling
and differing from each other … (this constitutes) … a mystery related to the creation of the universe.’
Galileo’s universe was composed of circles. Uniform circular motion is primary, not in need of explanation. It concerns planets turning around the sun, and the earth rotating about its axis, as well as a terrestrial body moving without friction on a horizontal
plane, for instance, a ball on a smooth surface, without air resistance. Galileo describes the horizontal motion sometimes such as to be approximately rectilinear, but in principle it is circular. The beauty of the Copernican system was expressed in uniform circular motion not confined to the heavens.
In the 14th century, the impetus theory was applied to circular motion, and some scholars came close to the idea of inertia (3.2). Jean Buridan,
for instance, observed that a heavy millstone, having much impetus if turned around, is inclined to persist in its motion. For the same reason he suggested that the celestial spheres are not in need of a cause to continue their motion. Galileo agreed with
this opinion. In an ordered universe only finite and terminate motions, that is, uniform circular motions, do not disorder the parts of the universe. This view prohibited Galileo from accepting Kepler’s discovery of non-uniform, elliptic motion. If comets were celestial bodies, they would move in oval trajectories, putting an end to the supremacy of circular
motion. For this reason Galileo assumed comets to be atmospheric phenomena.
Galileo’s concept of inertia concerned celestial bodies, moving uniformly in circles around the sun, or turning around their axis, as well as terrestrial
objects moving frictionless on a horizontal plane.
This horizontal motion, too, was circular, for the earth is spherical. In Galileo’s world, only circular motion could be uniform.
Before Galileo, Giovanni Battista Benedetti had argued that in a vacuum all bodies having the same density would fall with the same speed. Initially Galileo adhered to Benedetti’s view that this speed is determined by the density of the falling body. He supposed the speed of fall to be proportional to the difference between the densities of the body
and the medium in which it falls. By this law he tried to account for the upward force exerted by the medium, and to disprove the distinction between light and heavy bodies, essential in Aristotle’s physics.
Only gradually, Galileo realized that a falling body is subject to three forces: the force of friction, dependent on the medium; the upward force of buoyancy, dependent on the density of the falling body relative to that of the medium;
and the downward force of gravity. Because he wanted to base the law of fall on experience, he could not start with fall in a vacuum, because that is not empirically available. Besides, its existence was denied by philosophers. Instead he estimated the contributions of friction and buoyancy, and discussed a situation in which both could be neglected. Galileo showed that buoyancy depends on
the relative specific weight of the falling body and the medium. Therefore, in a hypothetical void (which specific weight is zero), buoyancy is zero. The same applies to friction. This means that in a void, all bodies would fall with the same speed, independent
of their density. Acceleration instead
of velocity would be a constant. Again,
no external cause is needed. Gravity, the source of the motion, is an intrinsic property of the falling body. But it is not its cause, for the motion of fall is independent of the weight of the body, as Galileo found shortly before 1610. According to Aristotle a heavier body falls faster than a lighter one. Galileo refuted this
by a simple thought experiment. Consider two falling bodies of unequal weight connected by a string. On the one hand, the lighter one should delay the heavier one, but because the combination is even heavier, it should fall faster. After the invention of the air-pump, Boyle’s experiments (1669) confirmed that in an exhausted receiver a feather ‘descended like a dead weight’.
The axiom, that free fall implies a constant acceleration equal for all bodies, enabled Galileo to explain a large variety of phenomena:
‘… in the principle (of accelerated motion) laid down in this treatise (Galileo) has established a new science dealing with a very old subject … he deduces from a single principle
the proofs of so many theorems … the door is now opened, for the first time, to a new method fraught with numerous and wonderful results which in future years will command the attention of other minds …’
His decision to exclude physical elements from his theory of motion was prompted by his wish to make the most of motion as a principle of explanation.
In Discorsi Galileo presented his law for
the motion of falling bodies as an axiom, but he found it in an experiment with balls rolling down an inclined plane, slowing down that motion considerably. Galileo performed his experiments before 1606, but he published them only in 1638. Marin Mersenne in 1647 and others observed that Galileo could impossibly have found his law with the stated accuracy, if he measured time with a kind of water clock as described in Discorsi, or any other clock then available. However, he could have determined the equality
of successive time intervals with the help of musical beats, with the stated accuracy. This did not allow Galileo of calculating the acceleration of the ball at various angles of inclination, or to extrapolate this to find the acceleration of free fall. This may have been fortunate, for he would have found
a value significantly different from the value found later by Christiaan Huygens from his experiments on pendulums. Only in the 18th century, physicists discovered that a rolling ball divides its energy between the kinetic energy of its centre of mass and the rotation energy around this point, such that the acceleration of a rolling
ball is less than that of a gliding body along the same plane. Still, Galileo concluded that the acceleration of free fall follows the same pattern as in the case of balls rolling down an inclined plane.
Galileo’s experiments led him to the insight that the velocity at any instant is determined by the time elapsed since the start of the motion, not by its distance to the end point. This refuted Aristotle’s view of fall as a motion toward
the body’s natural place. Initially,
Galileo made a mistake by connecting the velocity with the path covered since the start of the movement. Shortly before 1610, he found that the increase of velocity is proportional to the time passed since the start. This became his definitive law of fall. For all bodies moving in a vacuum, whether vertically or along an inclined plane, the speed increases proportionally to time, the proportionality constant depending on the angle
With this law, Galileo established that the distances, covered in equal times by a ball on an inclined plane, are in the same proportion as the odd numbers. This means that the path covered since the start is proportional to the square of the time passed.
As a result, after Galileo time became
the most important parameter in mechanics. It symbolizes the shift from space (the covered path) to motion (the kinetic time) as the new principle of explanation.
that pendulums of equal length are isochronous, their periods being independent of the amplitude or the mass of the bob. He found the period to be proportional to the square root of the length of the pendulum. The isochrony of pendulums introduced a mechanical standard of time. After Galileo the problem of the measurement of time became
urgent, for navigation as well as for mechanics and astronomy. It also created the problem of the nature of time itself (3.7).
The earth’s motion and the tides
In his Dialogue, Galileo took pains to make the double motion of the earth acceptable. He does so in
two ways. First, he disproves the advanced arguments against the moving earth. Why don’t we feel a constant eastern wind? Because the atmosphere turns around with the earth. Why does a falling stone arrive at the foot of a tower, and not slightly to the west? Because the stone partakes in
the motion of the earth. Here, Galileo applies the relativity of motion. Thus the unobservability of the earth’s motion is explained by the motion of everything moving together with the earth. Galileo’s arguments refute the objections against the earth’s motion, but do not prove it.
Therefore, Galileo next looked for positive evidence, for phenomena only explainable by the motion of the earth. He found them in the retrograde motion of the planets (Copernicus’ argument), the observed motion of the sunspots,
and eventually in the not yet observed stellar parallax. Galileo thought that also the tides are such a phenomenon. He was so much convinced of this that he wanted to call his book Dialogue on the tides. This was refused by the papal censor, who agreed
to an impartial discussion of the structure of the cosmos, but would not allow the suggestion that the earthly motion could be proved. Galileo discussed his tidal theory in the fourth day of Dialogue.
He rejected Kepler’s idea that the tides are caused by the moon. According to Galileo the double motion of the earth, daily and annual, causes accelerations and decelerations, leading to a periodic motion of water in its basin. He stressed that the details of the tidal motion depend
on the shape of the basin, by which he tried to meet the objection that his theory implies a period of one day, whereas the actual period is about twice a day.
In 1687, Newton proved Kepler to be right, because the gravitational pull by the sun and the moon determines the tides. Nevertheless, Galileo, too, was not far off the track, in particular with respect to the relevance of the shape of the basin. Within the context of what he knew of mechanics, his theory of the tides is a marvellous achievement.
It shows an awareness of the relevance of acceleration, of inertia, and of resonance. Also in Newton’s explanation, it is essential that the earth moves around the common centre of mass of the earth-moon system, and this motion being circular is accelerated.
Hence, the aim of Galileo to prove the earth’s motion by the tidal theory is also achieved if it is replaced by Newton’s theory.
3.4. Cartesian physics
Between circa 1620 and 1650, René Descartes or Cartesius was a leading philosopher, regardless whether this word is taken in the 17th-century meaning
of scientist, or in its modern meaning.
In particular the twenty years he lived in the Dutch Republic (1629-1649) were very fruitful. His first published book, Discours de la méthode, dates from 1637, but its three appendices, La dioptrique, Les méteors,
and La géometrie, were written some time before. La géometrie contributed significantly to analytical geometry, which Descartes considered the paradigm of each science. The certainty provided by geometry is warranted by its
method, and in order to arrive at the same level of reliability, each science should proceed by the same method. Principia philosophiae (1644) and its extended translation, Les principes de la philosophie (1647), present Descartes’
theories of motion.
Descartes advanced Galileo’s program of explaining motion by motion in several respects, such as the law of inertia, the law of conservation of motion, the
problem of collision, the mechanical properties of matter, and the properties of light. Descartes was convinced that with his method he could solve all problems of his time. As a program to replace the Aristotelian scheme of explanation by a mechanistic one,
Cartesian physics exerted a large influence. Though in many respects a cul-de-sac, Cartesian physics constitutes an essential part of the Copernican revolution.
After Galileo, Descartes
became the main founder of mechanical philosophy,
attempting to reduce macroscopic phenomena to microscopic ones, to be explained by matter, quantity, shape, and motion. The transfer of motion only occurred by impact between mutually impenetrable material particles. Phenomena that could not be reduced in
this mechanical way were excluded from physics.
Descartes divided reality into res extensa, the objective physical world, essentially extension identified with matter; and res
cogitans, the subjective mental world, which essence is thought, the human mind. He assumed the two worlds to interact via the pineal gland (near the centre of the human brain, between the two hemispheres), the ‘principal seat of the soul’, the source of ‘clear and distinct ideas’.
Descartes identified space with matter. All matter is space, and all space is material, presumably with a varying density. A vacuum
is unintelligible. Matter is infinitely divisible in a mathematical sense. For this reason, Descartes is usually considered not to have been an atomist. However, on physical grounds he assumed the existence of particles, with a minimum dimension. He even distinguished three kinds of corpuscles into which space-filling matter was differentiated. Normal bodies are composed of coarse matter. Interplanetary space is filled with fine matter, and the pores in both are filled by the finest material, composing the sun and the stars, and responsible for the transmission
of light. Particles could only differ because of their spatial shape and magnitude.
Aristotle defined a substance as any individual combination of form and matter (3.2). Essential and
accidental properties determine a material body. Essential properties were contrary: a substance was dry or moist, cold or warm, heavy or light. Properties like colour and taste were accidental. Like Galileo, Descartes did not distinguish essential and accidental,
but primary and secondary properties.
This distinction follows from the Platonist proposition that the real world is not necessarily the world as we perceive it. Primary qualities belong to objects as they really are. Secondary qualities such as heat or colour have no independent existence apart
from the senses. The primary properties of matter are related to extension or motion – volume, quantity of motion, hardness, impenetrability, etc. Other qualities are secondary, and should be reduced to primary, mechanical properties. An example is Descartes’ reduction of magnetism to the motion of cork-screw particles fitting holes in magnetic materials
like iron (4.1).
If the Aristotelians talked about primary or manifest properties, they referred to sensory experience. Under Plato’s influence, Galileo, Descartes, and other mechanists
considered sensory experience to be secondary, in need of explanation on mechanical principles.
Galileo connected the principle of inertia with uniform circular motion (3.3), but his disciples took linear inertial motion for granted. Before Galileo published anything on motion, Isaac Beeckman (who discussed his views with Descartes) distinguished two kinds of inertial motion: uniform rectilinear motion, and (like Copernicus) uniform rotation of
a heavy body around its axis.
Descartes accepted the infinity of the universe, and therefore did not share Aristotle’s and Galileo’s caution against rectilinear inertial motion. He posited the law, that a body on
which no force is acting moves rectilinearly at constant speed as long as it does not collide with other bodies. However, this situation is imaginary, because Descartes believed that space is filled with matter, and therefore actual motion can only occur in a vortex or whirlpool.
The introduction of the principle of inertia generated a problem unknown in Aristotelian physics – the problem of how motion can change. For Aristotle, celestial motion never changes, it being eternally circular. Sublunary natural
motion simply ceases as soon as the body has arrived at its natural place. Violent motion ceases when the driving force no longer acts. Local motion does not change, but is a kind of change, change of position.
After the establishment of the principle of inertia the question arose how motion can be started, halted, or changed in direction. Clearly this can only be done by some external force, for if no external force is acting on a body,
it continues its motion. Galileo never posited this problem in his published work, but Descartes did. It is the main problem of his physics. According to the mechanist program of explaining motion by motion, any movement can be changed only by another moving body. The only conceivable possibility
for this is a collision between the two bodies.
The problem of collision forms the heart of Cartesian physics. Descartes introduced quantity of motion as a measure of motion,
operationally defined as the product of volume (quantity of matter) and speed. This definition differs from the later definition of linear momentum in two ways. First, Descartes took volume to be quantity of matter, because he identified matter with extension.
Newton would amend this by taking mass, operationally defined as the product of volume and density, as quantity of matter (1.4). Next, Descartes considered velocity to be a scalar magnitude, like speed. Later, Huygens corrected this, by assuming velocity to
be a vector, as we now call it, having direction as well as magnitude. In its corrected form, this is now called the law of conservation of momentum, the product of mass and velocity. During a collision (whether elastic or not) one body may transfer
some quantity of motion to one or more other bodies, but only such that the vector sum of all momentums is conserved. For Descartes, collision was the only conceivable way to change the motion of a body. The concept of force could only be applied as a derivative of action by contact.
Descartes considered his law of conservation of motion to be
clear and distinct, therefore evidently true, not subject to empirical scrutiny. He assumed quantity of motion to be indestructible, because it is natural. At the creation, God supplied the cosmos with a quantity of motion, never to change afterwards.
Descartes elaborated these ideas into seven laws
of impact. With only one exception,
these laws are contradicted by the results of experiments with colliding objects. Admitting this, Descartes observed that his laws concern circumstances which cannot be realized in concrete reality. They concerned collisions between bodies in a vacuum, and
a vacuum is impossible.
‘The proofs of all this are so certain, that even if our experience would show us the contrary, we are obliged to give credence to our mind rather than
to our senses.’
In his theory of impact, Descartes treated rest and motion as contraries. Besides the concept of quantity of motion, he applied quantity of rest, inertia. In a collision, if the body at rest is larger
than the moving one, the quantity of rest dominates the quantity of motion. The effects of spatial extension and motion are, respectively, quantity of rest (inertia) and quantity of motion (momentum).
Descartes needed the distinction
between rest and motion to explain the existence of bodies moving as a whole. The parts of the body move together with the whole, but are at rest with respect to each other. If Descartes would not have introduced the idea of rest, the idea of universal motion
would have excluded the existence of extended bodies.
Without admitting it in plain words, Descartes assumed some kind of absolute space, a space as seen by God. Elsewhere, Descartes
contended that motion can only be relative.
This dilemma arose from his identification of space and matter. If matter is the same as space, local motion as change of position is strictly speaking impossible. The only possibility to create motion in a plenum arises when spatial parts exchange their positions.
Hence, real motion occurs in vortices, circular motion returning into itself. Real vortex motion in a plenum is relative motion, and the non-existing idealized rectilinear motion in a void is absolute.
Descartes applied his view of vortex motion in a plenum to the celestial bodies. The sun turns around its axis, as was discovered by Galileo and others (3.3), and it drags along the surrounding matter, and hence the planets. According to Descartes,
a rotating planet creates its own vortex, dragging around satellites like the moon.
In Christiaan Huygens’ work, mechanical philosophy reached its acme. He rejected Newton’s views on force, accepting only matter, quantity, shape, and motion as principles of explanation, but he also took distance from Descartes’ philosophy of clear and distinct ideas. He
valued observations and experiments as sources of knowledge much higher than Descartes did. Galileo, Huygens, and Newton denied that rest and motion are contraries. They treated rest as a state of motion, with zero velocity. In a beginning movement, the object
starting from rest passes through all degrees of speed, until arriving at its final speed. This is the foundation of the principle of relativity, whether Galilean or Einsteinian. If every movement has a relative character, there cannot be a fundamental distinction between rest and motion. When Huygens applied
the principle of relativity to Descartes’ laws of impact, all but one turned out to be false. Huygens corrected these, and together with Wallis and Wren, he solved the problem of elastic and inelastic collision.
Descartes had assumed that the vortex motion of matter around the rotating earth caused a centripetal motion of all bodies having density less than the whirling matter. Huygens thought he was able to demonstrate this effect in an experiment
(1668). He put pieces of sealing wax into a pail of water. If the pail was kept rotating, the pieces of wax moved to the periphery. But if the rotation were interrupted, all pieces moved to the centre, ‘… in one piece, which presented me the effect
of gravity.’ One objection against
Descartes’ theory was that the density of the imperceptible whirling matter would have to be larger than the density of all bodies falling to the earth. Moreover it is difficult to understand why gravity is directed to the centre, rather than to the
axis of the earth’s rotation. In an ingenious way, Huygens sought to meet these objections. He published his theory, developed in 1667, only in 1690, three years after the much more successful theory of Newton, which Huygens admiringly but critically
Explaining gravity from mechanical motion, Descartes was the first to relate celestial motion with the motion of a falling body. Galileo never connected gravity with the natural motion of the planets. Kepler compared gravity with magnetism, and magnetism with planetary motion, but never gravity with planetary motion. Kepler rejected the Aristotelian
view of gravity as a natural tendency of heavy bodies toward the centre of the universe. He sustained Copernicus’ view that gravity is a mutual corporeal affection between cognate bodies tending to unite them, but this did not inspire him to connect gravity with planetary motion.
3.5. Early concepts of force
Johannes Kepler always considered himself a Copernican, because he adhered to the idea of the moving earth. Yet he broke away from Copernicus’ fundamental idea of
uniform, circular motion (2.6). Kepler’s first two laws, published in Astronomia nova (1609), indicate exactly the failure of Platonic and Copernican models. The planetary orbits are not circular but elliptic, and the planetary motion is not
uniform, but varies according to the area law. Hence it should not be amazing that many Copernicans after Kepler rejected his results, and held to the uniform circular motion. For the mechanists like Galileo, Descartes, and Huygens, Kepler’s views did not fit their program of explaining motion by motion. Moreover, Kepler’s calculations were very difficult to understand and only comprehensible by professional astronomers.
It took quite a long time before other astronomers confirmed Kepler’s discoveries.
Kepler realized that planetary motion deviating from uniform circular motion needs a non-kinematical
explanation, which he sought in a kind of force.
Aristotle knew forces only as causes of violent motion. Since Archimedes’ works were rediscovered, force became an important concept in the study of equilibrium situations. In both cases, forces were restricted to the sub-lunar realm. Kepler was the first to use the concept of a force to planetary motion. Initially, in his Mysterium cosmographicum (1597), Kepler assumed an animistic view, supposing each planet’s motion to be conducted
by a soul. But in a footnote added in 1621, he wrote that everywhere the word soul should be replaced by force, the moving soul of the planets, the cause of planetary motion.
Kepler conjectured that the sun exerts an influence on the planets, pushing them around in their orbital motion. Like Aristotle, Kepler supposed
the force keeping a body in violent motion to be proportional to its speed. Because a planet’s velocity is largest if it is closest to the sun, Kepler concluded this force to be inversely proportional to the distance from the sun. He supposed it to be
tangential, directed along the planetary orbit. It was by no means attractive, i.e., directed towards the sun. Kepler suggested that the rotation of the sun causes the revolution of the planets. Kepler estimated the period of the sun’s revolution to be about three days, and he was disappointed to learn from Galileo’s investigation of the sunspots that the actual period is thirty days.
Like Galileo, Kepler admired William Gilbert’s book De magnete (On the magnet, 1600). Gilbert was a halfway Copernican, accepting the diurnal rotation of the earth, but not committing himself to its annual motion. Contrary to Peregrinus, who in 1269 ascribed the properties of the compass needle to the
rotation of the heavens, Gilbert
ascribed these to the rotating earth, which he considered a huge magnet (4.1). He believed magnetism to be the driving force for the diurnal rotation of the earth.
this view, and he assumed that the force exerted by the sun on the planets is also magnetic, as well as the influence of the moon on the tides. Galileo and Descartes rejected both ideas, because they wanted to explain motion by motion. Galileo executed this program with respect to the tides (3.3), whereas Descartes
gave a mechanical explanation of magnetism (4.1).
Manifest and occult principles
The Copernicans rejected the Aristotelian distinction between celestial and terrestrial physics in two ways. The first was assuming that the same laws apply to both realms. This line was
followed by Galileo, Descartes, and finally by Newton. The second way was to assume that the same force acts universally at the earth and in the heavens. This is Kepler’s line, who considered magnetism in this capacity, and again Newton’s,
who demonstrated gravity to be the force determining planetary motion as well as the motion of falling bodies. ‘The force which retains the moon in its orbit is that very force which we commonly call gravity.’
The second line was first pursued by astrologers, starting from the assumption of a parallel development of celestial and terrestrial events.
Kepler was the last Copernican to be sympathetic to astrology. After him, astrology became definitely occult (dark). The Aristotelians called qualities occult if these could not be reduced to the manifest qualities, directly observable with the senses, such
as hot and cold, moist and dry, hard and soft (3.2).
Also gravity and levity were manifest properties. Their key example of an occult property was magnetism.
Mechanists considered properties occult if these could not be reduced to the
clear and evident principles of mechanics, to matter, quantity, shape, and motion. Hence they were proud of Descartes’ achievement, the reduction of magnetism to the motion of cork-screw particles, and the explanation of gravity by vortex motion (3.4).
They objected to Newton’s theory of gravity, with its inherent principles of attraction and action at a distance, which they considered occult. Not being a mechanist philosopher, Newton rejected this view. For him, gravity was a manifest property,
universally shared by all bodies, no less than their extension, hardness, impenetrability, mobility, and inertia.
The novelty introduced by Kepler and Newton concerns force as a dynamic principle, as a cause of change of motion. Aristotle too connected
force with violent motion, but this motion was not variable.
Influenced by Archimedes, who studied the problem of the lever (and perhaps by the medieval scholar Jordanus Nemorarius),
Tartaglia, Benedetti, Stevin, Galileo, Torricelli, and Huygens developed the static principle of force. The most important example of force was weight, the only kind of force considered by Archimedes, and problems concerning the centre of gravity were very prominent during the 17th century. The static concept of force
and pressure was also applied in hydrostatics and aerostatics (9.5).
3.6. Newton’s dynamics
Isaac Newton’s ideas on mechanics, gravity, and planetary motion were shaped between circa 1665 and 1685, during his years at Cambridge, quite long after the lifetimes of Kepler, Galileo, and
Descartes, whose works he used and criticized.
He arrived at the insight that besides matter and motion, force as a new principle of explanation was required, independent of motion. He became the most important representative of experimental philosophy.
Newton took pains to demonstrate that he was a true disciple of Copernicus, though not in the mechanist line of Galileo, Descartes, and Huygens, but in Kepler’s line. From Kepler he inherited the laws of planetary motion (which
he fitted into his theory of gravitation), Kepler’s respect for observations, and the idea of force as a dynamic principle. Later on, we shall discuss the theory of gravity (9.4). The present section deals with the dynamical concept of force in the context of mechanics.
the introduction to Principia (merely 28 pages in the English edition), Isaac Newton presented a summary of mechanics he was about to use. It contains operational definitions, axioms or laws of motion, several theorems, and a philosophical exposition
of his ideas on space and time, commenting:
‘Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments.’
He introduced a new operational definition of mass (1.4), and discussed the metrics of time, space, and inertial motion (3.7). He wrought a synthesis of 17th-century physics (7.1), but this did not include Cartesian mechanical philosophy,
which he severely criticized.
Innate force and impressed force
Mechanists like Descartes and Huygens could only conceive of force as an effect of motion and action by contact. Huygens considered a centrifugal acceleration as caused by a uniform circular
motion. Whereas anyone before them considered uniform circular motion to be natural, not in need of a driving force, Hooke and Newton realized that a centripetal force is required to maintain it. Their novelty was to introduce an entirely new view
of force as a dynamic principle, as a cause of change of motion. Newton’s concept of force is not as univocal as it is now. He distinguished several kinds of force (Latin: vis).
Vis inertiae, the force of inertia (not to be confused with the inertial
forces to be mentioned below), is also called vis insita or innate force. It is for any body proportional to its quantity of matter, now called its mass. Conform his predecessors, Newton initially considered vis inertiae the force keeping a body in
its inertial state. In Principia it became the force that resists any change of an inertial state of rest or uniform motion. If besides the innate force no other force is acting on the body, the latter moves uniformly and rectilinearly. We call this
Newton’s first law, but he got it from Descartes (3.4), perhaps via Huygens’ Horologium oscillatorium (1673).
What we now call a force is Newton’s vis impressa, an external force acting on a body. Its accelerating effect on the body’s
motion is inversely proportional to the body’s mass, to the inertial force. Newton considered the case of an external force acting during a short time. The effect of an impulse (the product of an impressed force and its duration) is a change
of the quantity of motion (linear momentum), which Newton operationally defined as the product of mass and velocity. This is Newton’s force law, the second law of motion. It is nowadays better known as the product of mass and acceleration. This refers
to a continually acting force, and was not explicitly given by Newton, but he used it all the same.
Like Huygens, Newton considered velocity to have direction as well as magnitude. The
same applies to impressed force, for which he discussed the law of composition. Impressed forces acting on the same body can balance each other. An unbalanced force acting on a body changes its speed, direction, or both.
suggested that the force law had been accepted by all his contemporaries, even by Galileo. Maybe he referred to the static concept of force (3.5). But Newton’s dynamic interpretation of force as the cause of changing motion was original, if he did not take it from Robert Hooke, who made him aware that
uniform circular motion requires a centripetal force. The principle of inertia connects the static and dynamic views of force. The static view is applied in equilibrium situations. In the dynamic view, if all forces acting on a body balance each other, the
body is at rest or moves uniformly along a straight line.
One may wonder why Newton introduced the law of inertia as an axiom, because at first sight it can be derived from the force
law. If the net force on a body is zero, its acceleration is zero; hence its velocity is constant. However, according to both common sense and Aristotelian physics, violent motion ceases if the force ceases to act. Hence, without an unbalanced external force, a body would remain at rest (apart from natural motion). The second law alone does not refute this view. Only in combination with the first law the common sense view was
The force law states that a body moving under the influence of an external unbalanced force accelerates, but it does not specify with respect to what the acceleration is determined.
The answer is that the acceleration is measured with respect to an inertial system. Apparently, this only shifts the problem, because now the question arises, with respect to what one can speak of an inertial system. The answer to this question is given by
the first law: an inertial system is a body moving without the influence of an external force. Newton defined inertial motion with respect to absolute space, but he admitted that this absolute motion cannot be measured (3.7). This means that (as a matter of
principle) the first law confirms the existence of inertial systems.
Like Huygens, Newton considers velocity to have direction as well as magnitude. The same applies to force, for which
he discusses the law of composition.
Hence, an impressive force is present if speed, direction or both change.
Centrifugal or centripetal
The distinction between force as caused by motion, and force as cause of motion is manifest with respect to circular motion. Only since the introduction
of linear inertia, excluding circular inertial motion, it has become clear that circular motion is accelerated.
The first to investigate this problem was Christiaan Huygens. He derived
the correct formula for the acceleration. Faithful to the program of explaining motion by motion, he introduced the centrifugal force as a result of circular motion. Hooke and Newton, on the other hand, supposing that acceleration needs a force, introduced the centripetal force as a cause of circular motion. It is a real, physical force like a magnetic or gravitational force.
Since then, in Newtonian mechanics, centrifugal force is considered an apparent
force, an inertial force (not to be confused with Newton’s vis inertia, mentioned above). An inertial force like the centrifugal force or the Coriolis force only occurs in a non-inertial reference system, a rotating reference system,
for example. An inertial force does not satisfy Newton’s third law. It would be wrong to consider the centrifugal force the reaction to the centripetal force, because these forces act on the same body, whereas the law of action and reaction
refers to the forces between two bodies. The centrifugal force balances the centripetal force only in the rotating reference system, in which the body is supposed to be at rest. In an inertial reference system, the centripetal force causes the body to accelerate,
whereas the centrifugal force does not exist.
Action and reaction
Newton also ascribed the third law to others, but that is no more than false modesty. The famous law of action and reaction was brand-new:
‘To every action there is always opposed an
equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.’
Like Descartes, Newton assumed that the motion of a body can only be changed by another body. However, this change is not caused by the motion
of the second body, but by the force acting between them, either by contact or at a distance. Because this force has a reciprocal nature, the motion of the second body is changed as well. The law of conservation of linear momentum was for Descartes
an axiom, but it now became a theorem (a corollary) to be derived from Newton’s laws. For Newton, force takes precedence over motion.
The third law may be considered the constitutional law of experimental philosophy. As we shall
see, it allows of measuring impressed forces. It distinguishes Newtonian dynamics from Cartesian or Leibnizian mechanics. Unlike inertia (the innate force), the impressed force is not a property of bodies, but a relation between them.
The dualism of force and matter
three axioms or laws of motion lie at the foundation of the Newtonian dualism of force and matter. Throughout his life, Newton maintained an ambivalent position with respect to this dualism, because he was under the spell that matter could only be passive. To accept that matter could be active
would make it independent of God. Therefore Descartes identified matter with extension, and he assumed that God had created an invariant quantity of motion. By introducing force as a new principle of explanation, Newton made matter more active than the mechanists
would allow. By vis insita, the force of inertia, each body resists change of motion. Circular motion requires a vis centripetalis as a cause, instead of Huygens’ vis centrifugalis as an effect. Matter became interactive
as a source of vis impressa, subject to the law of action and reaction, first of all the mutual force of gravity, later also as the source of electricity and magnetism. Matter turned out to have specific properties, having mass and chemical,
electric, or magnetic properties, contrary to the mechanist view that matter can only have spatial extension and shape. In order to maintain God’s sovereignty over matter, Newton emphasized that any kind of force is subject to laws. Between 1700 and
1850, the matter-force dualism became the inspiration for the development of electricity (electric charge and Coulomb force), magnetism (magnetic force and pole-strength), and thermal physics (temperature difference as a force, and heat considered as matter).
Newton’s third law made interaction a new principle of explanation. It interprets a force as a relation between two interacting bodies, on a par with the earlier discussed relations of quantity of matter, spatial distance, and
relative motion. Though an actual force may partly depend on mass or spatial distance, as is the case with gravitational force, or on relative motion, as is the case with friction, a force is conceptually different from quantitative, spatial, or kinetic relations.
In contrast to impressed force as a relation between bodies, Descartes’ quantity of motion, Huygens’ linear momentum, and Leibniz’ vis viva were supposed to be variable
properties of a moving body, transferable to other bodies. In the 18th century, disciples of Descartes and Leibniz quarrelled about the priority of momentum and vis viva. The Newtonian scholar Jean d’Alembert demonstrated these concepts to be
equally useful, momentum being the time-integral of the Newtonian force acting on a body, and vis viva being its space-integral. This means that force is the cause of change of momentum, and vis viva is the ability to perform work. But this compromise proposal was evidently unacceptable for both parties, because it would imply the recognition
of the priority of the Newtonian force.
By the three laws of motion, force is attributed a higher status than momentum. In the second law, impressed force and momentum occur at the same level, and the first law mentions neither. But in the third law only impressed forces
occur. Moreover, the law of conservation of linear momentum and Kepler’s second law Newton could derive from his third law. Euler did the same for the law of conservation of angular momentum in general. In 1847, Helmholtz showed this to apply to the law of conservation of energy as well.
This derivation depends on the rather severe condition that all forces can be reduced to so-called central forces, working between point-masses. This condition was acceptable for most Newtonians, because it fitted into their atomistic views. But it
was far less acceptable for the Cartesians, who rejected atomism in favour of the identification of matter and space, making bodies extensive in principle.
The third law maintained its
priority over the conservation laws until the end of the 19th century. Only in the field theories of Maxwell and Einstein, implying a return to Cartesian views of matter and space, it turned out that the above mentioned condition is too narrow. Since the 20th
century, physicists prefer to reverse the situation. Newton’s third law is shown to be a consequence of the law of conservation of momentum, under certain conditions. Hence, in modern physics the conservation laws have a higher status than Newton’s
laws of motion, and the concepts of energy and momentum are more important than the concept of force.
This does not mean that conservation laws were absent in Newtonian mechanics and
physics. Related to the material side of the matter-force dualism, several conservation laws were developed after Newton’s death – the laws of conservation of matter in chemistry, of electric charge, of magnetic pole-strength, and of heat.
3.7. Absolute and relative space,
time, and motion
It is often said that the shift from the geocentric to the heliocentric world view implies that mankind no longer held the centre of the universe, and had to be content
with a more modest position.
This is typical hindsight. It was by no means the view of Copernicus, Kepler, and Galileo, who knew the background of ancient and medieval cosmology better than our present-day world viewers. In this cosmology, the central position of the earth was by no means
considered important. The earth, including its inhabitants, was considered imperfect, occupying a very low position in the cosmological hierarchy. With the advent of the heliocentric world view, man was not ‘bereft from his central place’, but
was ‘placed into the heavens’, the earth becoming a planet at the same level as the perfect celestial bodies.
‘As for the earth, we seek rather to ennoble and perfect it when we strive to make it like the celestial bodies, and, as it were, place
it in the heaven.
The Copernican view of the cosmos was of great influence on the concepts of space and time. In Aristotelian physics space is finite, bounded by the starry sphere, but time is infinite. Aristotle recognized neither beginning nor end of the cosmos and this embarrassed his medieval disciples. The Christian world
view requires a beginning, the Creation, as well as an end, the return of Jesus Christ to the earth.
In the Aristotelian view of the cosmos determined by the form-matter motif, the earth
stood still at the centre of the universe, which as a whole was not very much larger than the earth. Of course, Aristotle knew that the dimensions of the earth are much smaller than those of the sphere of the stars, but the latter was considered to be small
enough to take the argument of stellar parallax seriously (2.4). When Copernicus introduced the annual motion of the earth, he had to enlarge the minimum dimension of the starry sphere such that the distance between the earth and the sun becomes negligible
compared to it. This turned out to be a step towards the idea of an infinite universe. Copernicus, Kepler, and Galileo, however, still considered the cosmos to be spatially finite. Descartes, on the other hand, identified physical space with mathematical, Euclidean space, and therefore took it to be infinite.
physics the place of an object is its immediate environment.
The natural place of the element earth is water, surrounding the earth. The natural place of the sphere of fire is above the sphere of air and below the lunar sphere. The place of Saturn is the sphere to which it is attached, above Jupiter’s sphere and
below the starry sphere. Descartes agreed that the place of a body is its environment. On the other hand, he realized that the position of a body can be determined with respect to a coordinate system, and is not in need of material surroundings. He vacillated between the views that motion is
relative and that it is absolute.
Inspired by his view on inertia, Newton devoted one quarter of his summary of mechanics to a scholium on space, time, and motion. He did not intend to give definitions of these concepts, ‘as being known to all.’ His first aim was to make a distinction between absolute and relative time. In this context the term relative appears to
differ from the now usual one, implying that the unit and the zero point of time are arbitrary. By relative time Newton meant time as actually measured by some clock.
true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable)
measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.’
Some clocks may be more accurate than others, but in principle no measuring instrument is absolutely accurate. By absolute time Newton meant
a universal standard or metric of time, independent of measuring instruments. No one before Newton posed the problem of distinguishing the standard of time from the way it is measured. It could only be raised in the context of experimental philosophy. After Newton, the establishment of a reliable metric for any measurable quantity became standard practice in the physical sciences (1.4).
Aristotle defined time as the measure of change, but his physics was never developed into a quantitative theory of change, and this conceptual definition did not become operational. Galileo discovered the isochrony of the pendulum.
Its period of oscillation depends only on the length of the pendulum, and is independent of the amplitude (as long as it is small compared to the pendulum’s length) and of the mass of the bob. Experimentally, this can be checked by comparing several
pendulums, oscillating simultaneously. Pendulums provided the means to synchronize clocks.
In 1659 Huygens derived the pendulum law making use of the principle of inertia, but apparently
he did not see the inherent problem of time. Like Aristotle and Galileo, he just assumed the daily motion of the fixed stars (or the diurnal motion of the earth) to be uniform, and thus a natural measure of time. But Newton’s theory of universal gravitation
applied to the solar system showed that the diurnal motion of the earth may very well be irregular. It is a relative measure of time in Newton’s sense.
The problem of
absolute time, space, and motion is most pregnant expressed in Newton’s first law, the law of inertia:
‘Every body continues in its state of rest, or of uniform motion in
a right line, unless it is compelled to change that state by forces impressed upon it.’
Uniform motion means that equal distances are traversed in equal times. This means that the absolute standard of time is operationally defined
by the law of inertia itself. The accuracy of any actual clock should be judged by the way it confirms this law. The law of inertia is a genuine axiom, because there is no experimental way to test it.
However, Newton did not follow this path. The only way he saw to solve the problem was to postulate an absolute clock, together with an absolute space. Newton admitted that the velocity of an inertial moving body can never be determined with respect
to this absolute space, but he maintained that non-uniform motion with respect to absolute space can be determined experimentally. He hung a pail of water on a rope, and made it turn. Initially, the water remained at rest and its surface horizontal. Next, the water began rotating, and its surface became concave. If ultimately the rotation of the
pail was arrested abruptly, the water continued its rotation, maintaining a concave surface. Newton concluded that the shape of the surface was determined by the absolute rotation of the fluid, independent of the state of motion of its immediate surroundings.
Observation of the shape of the surface allowed him to determine whether the fluid was rotating or not. In a similar way, Jean Foucault’s pendulum experiment (1851) demonstrated the earth’s rotation without reference to some extraterrestrial reference
system, such as the fixed stars. Both Newton and Foucault supplied physical arguments to sustain their views on space as independent of matter. Descartes’ mechanical philosophy identified matter with space. In his mechanics and theory of gravity, Newton
had to distinguish matter from space and time. In the 18th and 19th centuries Newton’s views on space and time became standard.
Gottfried Leibniz and Samuel Clarke (the latter acting on behalf of Newton) discussed these views in 1715-1716, each writing five letters. Leibniz held that space as the order of
simultaneity or co-existence, and time as the order of succession, only serve to determine relations between material particles. Denouncing absolute space and time, he said that only relative space and time are relevant. But it is clear that relative now means
something different from Newton’s intention. Apparently Leibniz did not understand the relevance of the principle of inertia for the problem of the metrics of space and time.
The debate focussed on theological questions. For Newton and virtually all his predecessors and contemporaries, considerations of space and time
were related to God’s eternity and omnipresence.
This changed significantly after Newton’s death, when scientists took distance from theology. This does not mean that later physicists were not faithful Christians. For instance, Michael Faraday was a pious and active member of the strongly religious Sandemanians, but he separated his faith firmly from his scientific
work. Natural theology remained influential during the 18th and 19th centuries, but its focus shifted to biology and geology, and after Newton it had no significant influence on the contents of classical physics.
Leibniz’ rejection of absolute space and
time was repeated by Ernst Mach in the 19th century, who in turn influenced Albert Einstein, although later Einstein took distance from Mach’s opinions. Mach denied the conclusion drawn from Newton’s pail experiment. He said that the same effect should be expected if it were possible to rotate the starry universe instead of the pail with water. The rotating mass of the stars would have the effect of making the surface of the fluid
concave. This means that the inertia of any body would be caused by the total mass of the universe. It has not been possible to find a mathematical theory (not even the general theory of relativity) or any experiment giving the effect predicted by Mach. Mach’s principle, stating that rotational motion is just as relative as linear uniform motion, is therefore unsubstantiated. Whereas inertial motion is sui generis, independent of physical causes, accelerated
motion with respect to an inertial system always needs a physical explanation.
Newton treated the metric of time independent of the metric of space. Einstein showed these metrics to
be related. Both Newtonian and relativistic mechanics use the law of uniform time to introduce inertial systems. An inertial system is a spatial and temporal reference system in which the law of inertia is valid. It can be used to measure accelerated motions
as well. Starting with one inertial system, all others can be constructed by using either the Galileo group or the Lorentz group, both reflecting the relativity of motion and expressing the symmetry of space and uniform time. Both start from the axiom that kinetic time is uniform. In the classical Galileo group, the unit of time is the same in all reference systems. In the relativistic Lorentz group, the unit of speed (the speed of light)
is a universal constant. Late 19th-century measurements decided in favour of the latter. In special relativity, the Lorentz group of all inertial systems serves as an absolute standard for temporal and spatial measurements.
Time as measured by a clock is called uniform if the clock correctly shows that a subject on which no net force is acting moves uniformly. This appears to be circular reasoning. On the one side, the uniformity of motion means equal distances in equal times. On the other hand, the equality of temporal intervals is determined by a clock subject to the norm
that it represents uniform motion correctly.
This circularity is unavoidable, meaning that the uniformity of kinetic time is an axiom that cannot be proved, an expression of a fundamental law. Uniformity is a law for kinetic time, not an intrinsic property of time. There is nothing
like a stream of time, flowing independently of the rest of reality. Time only exists in relations between events, as Leibniz maintained, although he did not understand the metrical character of time. The uniformity of kinetic time expressed by the
law of inertia asserts the existence of motions being uniform with respect to each other. If applied by human beings constructing clocks, the law of inertia becomes a standard. A clock does not function properly if it represents
a uniform motion as non-uniform. But that is not all.
Whereas the law of inertia allows of projecting kinetic time on a linear scale, time can also be projected on a circular scale, as displayed on a traditional clock, for instance. The possibility of establishing the
equality of temporal intervals is actualized in uniform circular motion, in oscillations, waves, and other periodic processes, on an astronomical scale as in pulsars, or at a sub-atomic scale, as in nuclear magnetic resonance. Besides the kinetic
aspect of uniformity, the time measured by clocks has a periodic character as well. Whereas inertial motion is purely kinetic, the explanation of any periodic phenomenon requires some physical cause besides the principle of inertia. Mechanical clocks depend on the regularity of a pendulum or a balance,
based on the force of gravity or of a spring. Huygens and Newton proved that a system moving with a force directed to a centre and proportional to the distance from that centre is periodic. This is the case in a pendulum or a spring. Electronic clocks apply
the periodicity of oscillations in a quartz crystal.
Periodicity has always been used for the measurement of time. The days, months, and years refer to periodic motions of celestial
bodies moving under the influence of gravity. The modern definition of the second depends on atomic oscillations. The periodic character of clocks allows of digitalizing kinetic time, each cycle being a unit, whereas the cycles are countable. The uniformity of time as a universal law for kinetic relations and the periodicity of
all kinds of periodic processes determined by physical interactions reinforce each other. Without the uniformity of inertial motion, periodicity cannot be understood, and vice versa.
the end of the 19th century, Ernst Mach and Henri Poincaré suggested that the uniformity of time is merely a convention. One has no intuition of the equality of successive time intervals. This philosophical idea would have the rather absurd consequence, that the periodicity of oscillations, waves, and other natural rhythms would also be based on a convention. More relevant is to observe that physicists are able to explain many kinds of periodic motions and processes based on laws presupposing the uniformity of kinetic time as a fundamental axiom.
Motion and interaction as mutually independent principles of explanation
In Newton’s work impressed force is the most important concept besides matter. This may be called the strongest rupture with the mechanists, who wanted to explain motion by motion. For Galileo and Descartes, matter was characterized by quantity,
extension, shape, and motion. Motion
could only be caused by motion. Newton
emphasized that perceptibility and tangibility were characteristic of matter as well. The property of matter to be able to act upon things cannot be grounded on extension alone. Newton introduced a new principle of explanation, now called interaction. Besides
quantitative, spatial, and kinetic relations, interactions turn out to be indispensable for the explanation of natural phenomena.
Galileo and Descartes showed motion to be a principle
of explanation independent of the quantitative and spatial principles. This led them to the law of inertia, now called Newton’s first law. Descartes assumed that all natural phenomena should be explained by motion as well as matter, conceived to be identical
with space. Newton relativized this kinetic principle, by demonstrating the need of another irreducible principle of explanation, the physical principle of interaction. However, Newton only made a start. For, as a Copernican inspired by the idea that the earth moves, his real interest was in the explanation of all kinds of motion, including accelerated motion. The full exploration
of the physical principle of explanation did not occur during the Copernican era, but in the succeeding centuries.
Although this Copernican commitment partly justifies Dijksterhuis’
view that Newton fulfilled the ‘mechanization of the world picture’, the distinction between Cartesianism and Newtonianism is important enough to shed some doubt on this view. It is not improbable that Dijksterhuis considered the Copernican era
too much from the viewpoint of late 19th-century mechanism, which included a revival of Cartesianism.
 The idea of fundamental principles of explanation or modal aspects of being and human experience is due to Herman Dooyeweerd and Dirk Vollenhoven. Because explanation and experience always occur in relations, I prefer
to call these aspects relation frames, Stafleu 2002, 2011, 2015.
 Cohen 1980b; Levenson 1994. So many Copernicans were interested in music, that Drake 1970, 53 states that modern physical science is ‘the offspring of music wedded to mathematics.’
 Initially the Pythagorean
universe was geocentric. Sometimes the sphere of the fixed stars was included, sometimes the central fire, sometimes both. In the latter case, the counter-earth was superfluous. See Dreyer 1906, chapter 2; Heath 1913, chapter 6, 12; Guthrie 1962-1981, I, 282-301.
Kepler 1619; Koyré 1961, 326-343.
 Plato, Timaeus, 1179-1186.
 Popper 1963, chapter 2.
 Guthrie 1962-1981, II, 1-79.
 Guthrie 1962-1981, II, 28. This is not literally Parmenides’ text, but a paraphrase.
 Salmon (ed.) 1970, 5-16, 45-58. The main source of Zeno’s paradoxes is Aristotle, Physics, VI, 2, 9; VIII, 8. See Clavelin 1968, 34-48; Guthrie 1962-1981, II, 2, 91-96.
 See Plato’s parable
of the cave, Plato, Republic VII, 747-750.
 Mathematically, local motion implies the continuous succession of temporal instants. This appears to imply a paradox, too. Like the rational and real numbers, points on a continuous line are ordered, yet no point has
a unique successor. Since there are infinitely many other points between A and B, one cannot say that a point A is directly succeeded by a point B. Yet, a uniformly or accelerating moving thing passes the points of its path
successively. Therefore, continuous succession of temporal moments cannot be reduced to quantitative and/or spatial relations.
 Aristotle, Physics, I, 7.
 Aristotle, Physics, V, 2.
 Aristotle, Physics, I 8, 9.
 Aristotle, Physics, III, 1. By the distinction between eternal form and matter, and changeable substance with its potential and actual properties, Aristotle avoided Parmenides’ problem.
Metaphysics, I, 3, V, 2; , Physics, II, 3, 7.
 Aristotle, Metaphysics, XII, 2; Physics, III, 1. In Physics, V, 1 and elsewhere Aristotle only mentions three kinds of motion, because the first is not treated in Physics, but in On
generation and corruption.
 Aristotle, Physics, VIII, 7, 9.
 Aristotle, Physics, VIII, 8, 9.
 Aristotle, On the heavens, I, 2, 3.
 The pseudo-Aristotelian Questions of mechanics, extensively annotated by Galileo, only became available in Europe after 1525; see Clavelin 1968, chapter 1.
 Aristotle, Physics, IV, 8.
 Aristotle on projectile motion: Physics, VIII, 10; see Koyré 1939, 51. Galileo was the first scholar to recognize that the trajectory of a projectile is curved right from the start. For his critique of Aristotle’s
theory, see Galileo 1632, 151.
 Clavelin 1968, 96-97. The latter was Aristotle’s opinion, see Aristotle, On the heavens, I, 8.
 Copernicus 1543, 38-40 (I, 4). This shows that Copernicus accepted spatial causes besides the kinetic one explaining retrograde motion.
 Galileo 1610. Sidereus nuncius can also be translated as the starry messenger, Drake (ed.) 1957, 19.
 Galileo 1613. The publication of this book caused Galileo’s first encounter with clerical censors, who ‘… adamantly refused a layman the right to meddle with Scripture’, see Shea 1986,
Galileo 1632, 71-78.
 Galileo 1610, 42-45; 1632, 67-69, 91-99.
 Probably, Galileo did not know that the same explanation had already been given by Leonardo da Vinci and by Maestlin, see Clavelin 1968, 199-203.
 Galileo 1613, 98; 1632, 54, 58.
 Galileo 1632, 51. The new stars named after Brahe (1597) and Kepler (1604) actually were supernovae, the last occuring in the Milky Way and visible with the naked eye.
 I am not aware of any
Copernican who actually discussed Zeno’s paradoxes. However, Spinoza’s views resemble those of Parmenides, and his views on motion those of Zeno.
 In a logical sense an explanation has a causal character, but this cause is not necessarily a physical one.
 Drake 1978, 42. It is a matter of dispute whether Galileo’s study of motion was inspired by Copernicanism. Although he openly adhered to it only after 1609, it appears that he accepted the Copernican system after
about 1590, see Drake 1975; Dijksterhuis 1950, 372 (IV: 85); Clavelin 1968, 177.
 Galileo 1613 consists of three letters to the Augsburg merchant Mark Welser, in which Galileo criticized Christoph Scheiner’s anonymous interpretation of the sunspots.
 Galileo 1613, 107-109; 1632, 345-356.
See Drake’s note on page 486 to Galileo 1632, 354.
 Galileo 1623, 277-278.
 Galileo 1638, 98-99. See Drake 1970, chapter 2.
 Galileo 1623,276-277. Taste, odour, sound, touch are connected, respectively, with water (fluids), fire, air, earth. Vision (‘the sense eminent above all others’) is related to light, implicitly referring
to the ether. See Plato, Timaeus, 1186-1192.
 Koyré 1939, 130-131; Descartes 1647, 77-78. In this respect, Copernicanism was preceded by the 14th-century Paris scholars, see Dijksterhuis 1950, 193-194 (II: 108).
 Galileo 1632, 116: motion does
Galileo 1632, 21: rest is an infinite degree of slowness.
 Galileo 1613, 113-114; 1632, 145-148.
 Galileo 1632, 19; 1638, 215; 1613, 113.
 Galileo 1638, 161; Koyré 1939, 181.
 Galileo 1632, 145-148.
 Galileo 1632, 20-21; 1638, 261.
 Galileo 1638, 264-269.
 Galileo 1632, 175. A composite motion is as natural as a simple motion, see Galileo 1632, 235.
 Galileo 1638, 276. Tartaglia mentioned this fact in 1531, see Drake 1970, 26.
 Galileo 1638, 192-193; Koyré 1961, 119. However, Drake 1970, chapter 12-13 argues against the view that Galileo was obsessed by uniform circular motion, that he adhered to circular inertial motion, and that he
seriously believed the planets to move in uniform circular motion around the sun.
 Galileo 1638, 193-194.
 Galileo 1632, 259 even considered the motion of animals to be primarily circular.
 Galileo 1632, 19; 1638, 215; 1613, 113.
 Galileo 1632, 118-119.
 Galileo 1632, 28; 1638, 215, 244, 251.
 Galileo 1613, 113-114; 1632, 31-32, 147; see Clavelin 1968, 372-374.
 Koyré 1939, 26-27; Galileo 1638, 62.
 Galileo 1638, 72-84.
 Galileo 1638, 161; Koyré 1939, 181.
 Galileo 1638, 62-64.
 Galileo 1638, 242-243.
 Duhem 1908, 110-114. Duhem described Galileo’s conscious decision to abstain from a physical explanation of free fall as positivism or instrumentalism. To consider Galileo an instrumentalist is highly
improbable, however, considering his incessant struggle against an instrumentalist interpretation of Copernicanism. If he had been an instrumentalist, he would never have had a conflict with the Inquisition. Galileo 1613, 97: ‘… the true constitution
of the universe … exists; it is unique, true, real, and could not possibly be otherwise …’; see Galileo 1615, 166; Kolakowski 1966, 28-29; Dijksterhuis 1950, 372-374 (IV: 84-88).
 Galileo 1638, 178-179.
Dijksterhuis 1950, 400-402 (IV: 130-132).
 Settle1961; Drake 1978, 88-90.
 Huygens experimented with a conical pendulum and (like Riccioli in Almagestum Novum (1651) before him) determined the length of a pendulum having a period of one second. His measurement of the fall in one second
of 15.096 Paris feet could be trusted to about +0.01 feet, Harper 2002, 182. Huygens’ value corresponds to 9.807 m/sec2 for the acceleration of gravity at Paris, the modern value being 9.8087
m/sec2, Harper 2002, 197.
 Aristotle, On the heavens, I, 8.
 Galileo 1638, 167; Koyré 1939, 65 ff; Hanson 1958, 37 ff, 89; Finocchiaro 1973, 86 ff. The same mistake was made by Albert of Saxony, see Clavelin 1968, 99, and later by Descartes. Beeckman, misunderstanding Descartes’
proof, arrived in 1618 at the correct law of fall, twenty years before Galileo published his discovery.
 Galileo 1638, 74; Drake 1970, 39-40.
 Galileo 1632, 221-222; 1638, 153, 175.
 Galileo 1638, 96-97.
 Galileo 1632, 125-133 mentions all arguments against terrestrial motion, and refutes them on p. 133-218. See Finocchiaro 1980, 208; Copernicus 1543, 42-46 (I, 7, 8).
 Galileo’s use of the principle
of relativity to refute arguments against the earth’s motion differs from applying it in a mathematical theory, as Huygens did, several decades later.
 Galileo 1632, 416-465; Finocchiaro 1973, 16-18.
 Kepler 1609, 26-27 (Preface); see Koyré 1961, 194; Galileo 1632, 462.
 Galileo’s theory of tidal motion probably dates from about 1596, see Drake 1978, 36-44. Galileo developed this theory further in 1616. It circulated as a manuscript, Discorsi sopra il flusso e reflusso del
mare. In 1619, Francis Bacon rejected Galileo’s theory, because of his (Bacon’s) observations, published in 1616. Galileo rejected the observations as far as contradicting his theory. Nevertheless, Shea’s (1972, 186) characterization
of Galileo’s theory as a skeleton in the cupboard is wide of the mark.
 Newton 1687, 435-440.
 Dijksterhuis 1950, 444-460 (IV: 194-220); Scott 1952; Gaukroger 1995; 2006, 289-322; Clarke 2006.
 Descartes 1647, 48, 53-54.
 Descartes 1649, 351-355, 359-362.
 Descartes 1647, 53, 65-73; Kant 1781-1787, A 20-21, B 5-6, 11-12, 36.
 Descartes 1647, 74, 82.
 Van der Hoeven 1961,109-120.
 Descartes 1647, 159; 1664, 24-25.
 Burtt 1924, 63-71, 106-111, 115-121.
 Descartes 1647, 53, 65-73.
 Koyré 1939, 237. Westfall 1980, 11 states that by 1661 only two significant figures had embraced the principle of inertia, Descartes and Huygens.
 Van Berkel 1983. Although Beeckman never published his work, his correspondence with Descartes and others was quite influential.
 Descartes 1647, 85; 1664, 38. See Scott 1952, on Descartes’ physics.
 Galileo 1638, 269-272 discusses impact, announcing a separate treatise, the so-called fifth or sixth day, published posthumously (1718).
 Descartes 1647, 86-88; Koyré 1965, 77-78; Van der Hoeven 1961, 120-139; Deason 1986, 168: ‘The mechanical world view rested on a single, fundamental assumption: matter is passive. It possesses
no active, internal forces.’
 Descartes 1637, 21.
 Descartes 1647, 89-94.
 Descartes 1647, 93. See Hübner 1976, 304: ‘… Descartes’ rules of impact describe fundamental processes within nature as God sees them.’
 Koyré 1965, 77; Harman 1982a,
Descartes 1647, 76-79.
 Dijksterhuis 1950, 503 (IV, 282).
 Galileo 1632, 21-28; 1638, 162-166.
 Huygens 1690, 132-133; Dijksterhuis 1950, 507-509 (IV: 288-290).
 Descartes 1647, 210-214.
 Kepler 1609, 25-26 (Introduction); Koyré 1961, 194.
 Descartes admitted that the planetary orbits are not perfectly circular, but he did not accept Kepler’s laws, see Descartes 1647, 117. Descartes was aware of the need to distinguish motion from ‘action,
or inclination to move’, but he reduced action to motion, Descartes 1637, 88.
 On the development of the concept of force, see Jammer 1957, chapter 5-7; Cohen 1974.
 Kepler 1609, 26 (Introduction); Koyré 1961, 194; see Kepler 1609, chapter 32-39 on force; Koyré 1961, 185-224.
 Kepler 1609, 34 (Introduction), 228 (chapter 34); Galileo 1632, 345.
 Galileo 1613, 106; 1615, 212-213.
 Gilbert 1600; Kepler 1609, 229 (chapter 34), 329, 331 (chapter 57); Galileo 1632, 399-414.
 Kepler 1609, chapter 34, 57; Koyré 1961, 208.
 Heilbron 1979, 19-43.
 Newton 1687, 398-400. However, as a force, gravity is not a property of a body apart from other bodies by which it is attracted.
 Drake 1970, chapter 1.
 The standard work on Newton is Westfall 1980. See also Dijksterhuis 1950, 509-539; Alexander 1956; Cohen 1971; 1980; 1985; McMullin 1978; Hall 1992; Cohen, Smith (eds.) 2002.
 Nevertheless, Newton read little
or nothing of Kepler’s works, see Cohen 1980, 189.
 Cohen 1973, 322-327; Harman 1982a,13-17; Dijksterhuis 1950, 512-515 (IV: 295-297). In fact, Newton’s views on the activity of matter, strongly influenced by his alchemical research, changed continuously throughout
his life, see McMullin 1978. Elkana 1974, 16: ‘… while great discoveries are being made, the very concepts which the discoverers think in, or work with, are in a state of flux; only after they had completed their discovery and formulated the results
mathematically, does the new concept emerge and become fixed; this fixing of the concept carries with it the results of the last discovery out of which it emerged.’
 For a discussion of Newton’s laws of motion, see Hanson 1965; Ellis 1965; Nagel 1961, 174-202; Cohen 1980, 171-193; Gaukroger 2010, chapter 2. Attempts to reduce the principle of inertia to the principle of sufficient
reason may be called question-begging, see Gaukroger 2010, 308.
 Newton 1687, 14-17.
 Cohen 2002, 68-70. Mach (1883) and Hertz (1894), denying causality, reduced Newtonian force to matter and motion, by defining force (F) as the product of mass (m) and acceleration (a): F=ma,
reducing Newton’s second law to an operational definition.
 Common sense assumes that each body experiences a frictional force, dependent on speed, in a direction opposite to the velocity. Accordingly, if the total force on a body is zero, the body would be at rest.
A unique reference system would exist in which all bodies on which no forces act would be at rest. This would agree with Aristotle’s mechanics, but it contradicts both the classical principle of relativity and the modern one.
 Newton 1687, 14-17.
Huygens’ De vi centrifuga was written in 1659 and published posthumously in 1703, see Van Helden 1980, 150. An excerpt was published as an appendix to his Horologium oscillatorium (1673), which Newton studied.
 Newton 1687, 2-6.
In 1674 Hooke observed that circular motion requires an unbalanced force, see Westfall 1980, 382-383. Westfall 1980, 416: ‘The new insight into circular motion, however, constituted the very heart of the orbital dynamics on which he was at work. Only
the principle of inertia would allow the unique leap of the imagination which made the new dynamics possible, the recognition of the dynamic identity of uniform circular motion and uniformly accelerated motion in a straight line. Heretofore in the history
of mechanics, these two motions had been treated as irreducible opposites. What Newton perceived were the possibilities that opened up if he treated the two as dynamically the same.’
 McMullin 1978, 2, 29-56.
 Jammer 1957, chapter 9. In contrast to Newton’s dualism of matter and force, we find the Cartesians’ dualism of motion and matter (identified with space), and Leibniz’ dualism of motion and force (dynamism).
Iltis 1970; Szabo 1977, 47-85; Jammer 1957, 165-166; Papineau 1977.
 e.g., Kuhn1957, 3; cf. Burtt 1924, 18-20.
 Koyré 1961, 114-115; Lovejoy 1936, 101-108.
 On Descartes’ and Newton’s concepts of time and space, see Koyré 1957; 1965, 79-95; Jammer 1954; Burtt 1924, Ch. 4, 7.
 Galileo 1632, 319-320 observes that there is no proof that the universe is finite. Aristotle’s assumption that the universe is finite and has a centre depends on his view that the starry sphere moves.
Aristotle, Physics, IV, 2, 4.
 Also Galileo was aware of the principle of a Cartesian coordinate system, see Galileo 1632, 12-14.
 Landes 1983. During the Middle Ages, the establishment of temporal moments (like noon or midnight, or the date of Eastern) was more important than the measurement of temporal intervals, which was only relevant for astronomers.
Mechanical clocks came into use since the 13th century, with a gradually increasing accuracy.
 Kant 1781-1787, A 19 ff, B 33 ff recognized its relevance.
 Newton 1687, 10-11.
 Grant 1981, 254-255: ‘Newton’s absolute, infinite, three-dimensional, homogeneous, indivisible, immutable, void space, which offered no resistance to the bodies that moved and rested in it, became the accepted
space of Newtonian physics and cosmology for some two centuries.’
 Alexander (ed.) 1956; Grant 1981, 247-255. Grant 1981, 250: ‘It was less a genuine dialogue than two monologues in tandem ...’
 Cohen, Smith (eds.) 2002, 5: ‘Abandoning Newtonian space and time in the manner Leibniz called for would entail abandoning the law of inertia as formulated in the seventeenth century, a law at the heart of Leibniz’s
 Newton 1687, 545-546 (General scholium, 1713); Jammer 1954; Grant 1981, 240-247.
 Grant 1981, 255: ‘… scientists gradually lost interest in the theological implications of a space that already possessed properties derived from the deity. The properties remained with the space. Only God
departed.’ Ibid. 264: ‘It was better to conceive God as a being capable of operating wherever He wished by His will alone rather than by His literal and actual presence. Better that God be in some sense transcendent rather than omnipresent, and
therefore better that He be removed from space altogether. With God’s departure, physical scientists finally had an infinite, three-dimensional, void frame within which they could study the motion of bodies without the need to do theology as well.’
Mach 1883, 279-286; see Grünbaum 1963, chapter 14; Disalle 2002.
 Mach 1883, 286-290.
 Pais 1982, 288: ‘… to this day Mach’s principle has not brought physics decisively farther.’
 In 1831 Évariste Galois introduced a group as a mathematical structure describing symmetries. In physics, groups were first applied in relativity theory, and since 1925 in atomic, molecular, and solid state physics.
One of the first text books on quantum physics (Weyl 1928) dealt with the theory of groups.
 Margenau 1950, 139.
 Maxwell 1877, 29; Cassirer 1921, 364. The uniformity of time is sometimes derived from a ceteris paribus argument. If one repeats a process at different moments under exactly equal circumstances, there is no
reason to suppose that the process would proceed differently, and its duration should be the same.
 Periodicity is not only a kinetic property, but a spatial one as well, as in crystals. In a periodic wave, the spatial periodicity is expressed in the wavelength, the temporal one in the period, both repeating themselves
 In the 20th century, a second became defined as the duration of 9,192,631,770 periods of the radiation arising from the transition between two hyperfine levels of the atom caesium 133. This number gives an impression of the accuracy
in measuring the frequency of electromagnetic microwaves.
 Mach 1883, 217: ‘Die Frage, ob eine Bewegung an sich gleichförmig sei, hat gar keinen Sinn. Ebensowenig können wir von einer “absoluten Zeit” (unabhängig von jeder Veränderung)
sprechen.’ (‘The question of whether a motion is uniform in itself has no meaning at all. No more can we speak of an “absolute time”, independent of any change.’) Poincaré 1905, chapter 2; Reichenbach 1956,
116-119; Grünbaum 1968, 19, 70; Carnap 1966, chapter 8 poses that the choice of the metric of time rests on simplicity: the formulation of natural laws is simplest if one sticks to this convention.
 Einstein observed that the equality of covered
distances provides a problem as well, because spatial relations are subject to the order of simultaneity, dependent on the state of motion of the clocks used for measuring uniform motion.
 According to Reichenbach 1956, 117 it is an ‘empirical fact’ that different definitions give rise to the same ‘measure of the flow of time’: natural, mechanical, electronic or atomic clocks,
the laws of mechanics, and the fact that the speed of light is the same for all observers.
 Galileo 1623, 277-278; Koyré 1939, 179.
 Dijksterhuis 1950, 503.
 Dijksterhuis 1950, 515.