Galileo discovered the empirical
relationships of gravitational acceleration and the phenomena of inertia. Later,
A. Gravitational and Parabolic Motion, Inertia and Inertial Motion
Galileo Galilei (1561 – 1642) “was trained in the medieval Aristotelian tradition.” (Goldberg, p. 22) Aristotle had conjectured that “heavy objects fall down faster than light ones.” (Gamow, 1961, p. 35) In order to test Aristotle’s conjecture, Galileo (in 1589) took a light cannonball and a heavy cannonball to the top of the Leaning Tower of Pisa and simultaneously dropped them. To everyone’s amazement they appeared to hit the ground at the same instant. (Figure 4.1A)
Both of such objects appeared to go faster and faster as they fell. The inquisitive Galileo wanted to know what relationship governs this motion. So in order to dilute or slow down the fall of a ball he rolled it down an inclined plane. (Gamow, 1961, p. 35; Figure 4.1B) After several years of trial and error, in about 1604, Galileo finally discovered the mathematical relationship of gravitational acceleration near the surface of the Earth. The total distance covered by a free-falling terrestrial object “during a certain period of time” is proportional to the square of that time. (Id., p. 36; see Figure 4.2A)
As a by-product of these and similar experiments, Galileo
also discovered (in about 1608) the law that governs the path of a projectile
near the surface of the Earth. (Cohen,
1960, p. 212; Figure 4.1C) The trajectory of a projectile has two
independent components (Gamow, 1961, p. 39), “a vertical component that follows
the law of free fall (just as if there were no horizontal component) and a
horizontal component of forward motion that is uniform (just as if there were
no vertical component).” (Cohen, 1960,
p. 212) The distance of the horizontal
component is proportional to the time elapsed.
Thus, Galileo demonstrated and concluded that terrestrial objects (such as wagons) and projectiles (such as arrows and cannonballs) do tend to continue in motion through space or along the ground even after the force has been withdrawn. He called this theoretical phenomenon of continuous uniform rectilinear motion of matter without applied force, ‘inertia’ or inertial motion.
According to Aristotle’s point of view, a stone released from the top of the mast of a sailing ship will fall vertically down and land close to the stern of the ship. By the end of the 16th century, several people had actually tested Aristotle’s conjectures and found that the stone instead falls to the base of the mast.  (see Figure 5.1) However, no one could explain why this paradox occurs. No one, that is, except Galileo after 1608.
The reason for the paradox was the physical phenomenon of inertia and the resulting inertial motions of moving objects.  Objects on the surface of the Earth which share a common lateral inertial motion, such as the ship, the man on the mast, and the stone, maintain this common lateral inertial motion relative to the Earth, even when the force is withdrawn and they become physically detached from one another, like the falling stone. The detached stone also tends to accelerate downward toward the Earth with a parabolic trajectory due to the forces of gravity.
Why does the falling stone share the common lateral inertial motion of the man on the mast and of the ship? One reason is that the lateral motion is perpendicular to the force of gravity; therefore, there is no opposing force in the lateral direction, so (ignoring the effects of air) inertial motion is sustained. Another reason is that they are all material bodies and have mass (m), similar velocity (v) and thus similar momentum. After the force on the stone is withdrawn (by the man dropping it), the stone continues to move inertially in common with the ship because of its material ‘momentum:’ its mass times its velocity (mv). (Goldberg, p. 52) On the other hand, a photon or a ray of light theoretically does not have any mass, and therefore it cannot exhibit inertia, inertial motion, or lateral material momentum like the stone. Such material concepts of mass, velocity, inertia, inertial motion, and momentum are all irrelevant to non-material light.
Descartes was one of the first philosophers to fully understand the concept of inertia and to extend it to the motions of heavenly bodies. He theorized that a planet that continuously moves uniformly and rectilinearly (substantially in a straight line) does not require a force to maintain such motion. (Goldberg, p. 46; Cohen, 1960, p. 210) This is the classical law of celestial inertial motion (the persistence of motion without apparent applied force), which we observe as the perpetual motions of the planets, the stars, and the galaxies.
B. Newton’s Three Laws of Motion
Early in the Principia,
“Every body continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by forces impressed upon it.”  (Newton, Principia [Motte, Vol. 1, p. 13])
unlike Galileo, who merely described the orbital motions of the planets as
inertial, Newton asserted that such inertial motion must have constant direction of motion
in a straight line as well as a constant magnitude (speed). In other words,
astronomer Johannes Kepler (1570 – 1631), who described the three laws of
planetary motion in 1609 and 1618, also introduced the Latin word ‘inertia’
(meaning ‘laziness’) to physics. (Cohen,
1960, p. 210) Galileo described uniform
inertial motion as where the increments of distance, time and speed “repeat
itself always in the same manner.” (Id.,
pp. 88 – 89; Figure 4.3A)
French philosopher Rene Descartes (1596 – 1650) first clearly described the
phenomenon of inertia as a ‘state’ in his unpublished book, ‘Le Monde,’ (Cohen,
1960, p. 210; Goldberg, p. 53), and French scientist Pierre Gassendi (1592 –
1655) first published a description of the law and he also tested it with
(Cohen, 1960, p. 211; Harrison, pp. 125 – 126)
“The change of motion [acceleration] is proportional to the motive force impressed; and is made in the direction of the right [straight] line in which that force is impressed.” (Newton, Principia [Motte, Vol. 1, p. 13])
This law described the motion of a body that is not in equilibrium, vis.,
where a force is acting on the body and is not counterbalanced by another
beginning of the Principia,
It might be claimed that Galileo anticipated Newton’s second law in his projectile experiments, because they combine two independent forces (one impact, i.e. the propulsion of a cannonball) and (one continuous, vis. gravity) in two different directions, into the combined (uniform velocity and uniform acceleration) motion of a mass (a projectile), which results in a parabolic trajectory in a third direction. (Figure 4.2) But Galileo never took the next step and synthesized these motions, forces, masses and accelerations into a generalized law.
“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.” (Newton, Principia [Motte, Vol. 1, p. 13])
The impact force and acceleration motion of the relatively small mass of a bullet propelled out the barrel of a rifle is equivalent to the force and acceleration motion of the more massive rifle in the opposite direction. For example, when a bazooka is fired, the explosion (force in the barrel) causes the action motion of the projectile out the front of the barrel and the equivalent reaction motion and force of the exhaust out the rear of the barrel. (see Figure 4.3C) Because these opposite forces and motions are equivalent they offset each other, and the person who holds the bazooka does not feel any recoil; in effect, the bazooka remains in equilibrium.
In the third part of the Principia, entitled, “The
System of the World,”
C. Covariance and Invariance
covariance of the interaction between the different variable quantities (F, m,
a) that invariantly results in the same fundamental law of motion (
There is also
another form or set of transformation equations that relate the motions of the same
mechanical experiment in two different inertial frames of reference, in
order to demonstrate that
Toward the end of the 19th century, the mathematical concept of covariance was extended to the concept of ‘invariance’ for certain magnitudes and properties of matter. An ‘invariant magnitude or property’ is one that does not change for any observer regardless of his or the object’s velocity.  (Goldberg, p. 81) The magnitudes and properties of matter which were considered by late 19th century scientists to be invariant included: the mass of an object, its length, its other dimensions, its shape and its color. (see Resnick, 1968, pp. 11, 15)
Very importantly, it should be pointed out at this early juncture that Einstein drastically changed the above mechanical and empirical meaning of algebraic ‘covariance’ for his Special Theory to mean the algebraic ‘invariance’ of physical laws and physical magnitudes with respect to the Lorentz transformations. For Einstein, the term ‘covariance’ meant the transformation (or translation) of any algebraic law, equation, physical phenomenon or magnitude (including Maxwell’s constant transmission velocity of light at c) from one inertial reference frame to another inertial reference frame by means of his radical Lorentz transformation equations or the equivalent. (see Chapters 21 and 27)
When so transformed by Lorentz transformations, the classical laws of physics (including mechanics, electrodynamics and optics) would be distorted and would dramatically change from their classical meaning or magnitudes, but still they would remain invariant “with respect to Lorentz transformations.” (see Einstein, Relativity, p. 48) In other words, the distorted laws of physics and their magnitudes would be the same in each inertial reference frame.
All of these arbitrary and radical mathematical changes to physics were invented for one primary ad hoc purpose: to mathematically and artificially keep Maxwell’s law for the constant transmission velocity of light at c, at the absolute magnitude of c relative to all inertial reference frames, regardless of their linear motions, so that the velocity of light relative to such reference frames could never mathematically be c – v or c + v. (see the Preamble) Rohrlich referred to this artificial mathematical result as “Einstein’s fiat.” (Rohrlich, pp. 55 – 62)
All of the fundamental concepts described in this chapter will be important to a full understanding of the phenomena and theories discussed in the chapters to follow.
D. Failures of Classical Mechanics
important developments of Newtonian mechanics
and gravitation theory were made in the late eighteenth and early nineteenth
century. Such men as Lagrange, Laplace,
and Hamilton, who were both mathematicians, theoretical physicists and
astronomers, provided much more powerful mathematical techniques than were
available to Newton. They made it
possible to predict astronomical events with very great precision. But during this whole period of the
elaboration and extension of
Classical mechanics also failed to recognize the correct relationship between matter, energy and mass. (Chapter 32) All of the above failings, in turn, may have affected the computation and mathematical description of positions, accelerations, momentum, resistance, time, and other values in classical mechanics and celestial mechanics.
Such theoretical failings must of course be adequately remedied by
current physics. Nevertheless,
 Galileo demonstrated the path of an arrow, the path of a fired cannonball, and the path of any other spherical projectile on a wide slightly inclined plane, sometimes called a ‘wedge.’ (Cohen, 1960, p. 112; Figure 4.1C)
 By 1609, Galileo had mathematically confirmed such parabolic motions. (Cohen, 1960, p. 212) Galileo’s mathematical language was geometry. “He compared speed to speed, position to position, time to time…” Algebra did not become popular until the 18th century. (Goldberg, p. 22)
 On Earth such motion gradually slows to a stop because of the resistance of the air and the friction of a surface. On the other hand, in empty space the object will a priori coast at the same velocity and in a straight line (although slightly curved due to the distant forces of gravity), possibly forever.
 But Galileo’s concept of inertial motion did not continue in a straight line forever. Rather, it was limited to straight segments and segments that curved at great distances, because Galileo could not grasp the concept of spatial infinitely. (Cohen, 1960, pp. 117 – 119, 122)
 We must assume that the effects of the air and wind on the stone are negligible.
 On or near the surface of the Earth, terrestrial inertial motion is usually lateral relative to the Earth’s surface, because of the downward force of gravity. However, in the vacuum of space, far from gravitational influences, celestial inertial motion may occur in any direction.
 We shall refer to this subject again in various Chapters.
 However, the concept of celestial inertial motion without applied force is also an impossible idealization, because the gravitational forces of other celestial objects that produce orbital or curved motions are not taken into account.
 The idea that inertial motion can be a ‘state’ of motion or a ‘state’ of rest was asserted by both Galileo and Descartes. (Cohen, 1960, p. 216)
 It also follows that bodies with the same quantity of matter (mass) have the same inertia. (Cohen, 1960, p. 157)
 Although Galileo postulated that uniform motion on a “plane would be perpetual if the plane were of infinite extent” (Cohen, 1960, p. 117), he could not imagine this happening. At great distances Galileo imagined inertial motion to be curvilinear. (Id., pp. 119, 112, 124) In effect, Galileo was really describing uniform momentum.
 Is the motion of celestial bodies ever rigorously uniform straight-line inertial motion? The answer is probably no. A galaxy’s motion through space is about as close as one can get to idealized inertial motion, because its motion appears to be uniform and random and it doesn’t seem to orbit anything. The Sun’s combined galactic motion and slightly orbital motion is also close to straight-line inertial motion, and yet it is ever so slightly curved, because each 200 million years or so the Sun circumnavigates the Milky Way Galaxy. The Earth shares the Sun’s almost uniform straight-line motion at about 225 km/s relative to the core of the galaxy, but because of the Earth’s close proximity to the Sun, the Earth also orbits the Sun at 30 km/s every 365 Earth days. A priori, there can be no perfectly uniform and straight-line inertial motion, because the trajectories of all bodies are affected, more or less, by the gravitational attraction of other objects in the universe.
 Gassendi dropped rocks from the mast of a moving ship, and because of their inertia the rocks landed near the base of the mast, rather than toward the stern. (Harrison, pp. 125 – 126)
turns out that this second law was possibly first conceived and written down by
standard unit of measure for any force is now called a ‘
modern algebraic form of
 Newton asserted that a given force (F) results in a certain acceleration (a) of a body (m), but in order to determine the velocity v of mass m at any instant (during such acceleration), we must also know the duration of time (t) which such force has been applied. Therefore, v = at. (Cohen, 1960, p. 155)
 Again, since ‘mass’ is the quantitative measure of matter’s inertial force of resistance, the magnitude of a body’s mass is often referred to as its ‘inertial mass.’
 In a treatise to follow this one, entitled the Relativity of Gravity, we shall fully discuss Galileo’s, Kepler’s and Newton’s laws of gravity, and Einstein’s attempt to supplant Newton’s law with a radical new ad hoc mathematical theory of gravity (curved spacetime), which he called ‘General Relativity.’
 The concepts of ‘property’ and magnitude ‘invariance’ were absolute concepts, not relative ones. (Goldberg, p. 80)
 For example, it turned out that the term ‘electromagnetic mass’ was actually a misnomer. In reality, electromagnetic mass was just an electromagnetic resistance.
term ‘celestial mechanics’ refers to the application of
 The Latin phrase inter alia means “among other things.”