INERTIA
Newton, Mathematical Principles of Natural Philosophy,
translated by A. Motte and F. Cajori, 1729
Inertia
Kepler is said to have introduced the word "inertia" ("Traegheit"), meaning something like "resistance to motion" (Barbour 1989, 328; Whewell 1847, 191). Later, Newton adopted this word for expressing "a power of resisting"; the word appears in his remark following Definition III, Principia:
The vis insita, or innate force of matter, is a power of resisting, by which every body , as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.
This force is always proportional to the body whose force it is and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inert nature of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia (vis inertiae) or force of inactivity. But a body only exerts this force when another force, impressed upon it, endeavors to change its condition; it is resistance so far as the body, for maintaining its present state, opposes the force impressed; it is impulse so far as the body, by not easily giving way to the impressed force of another, endeavors to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; not are those bodies always truly at rest, which commonly are taken to be so. (Motte and Cajori translation, 1962, 2)
I may add here Julian Barbour's interpretation of the difference between Kepler's notion and Newton's notion: Kepler's idea may be renderd as:
mv = F,
whereas Newton's idea is, as we commonly understand,
ma = F
(Barbour 1989, 329). Moreover, in this context "inertia" means "inertial mass", and this should not be confused with the "inertia" in the law of inertia (Newton's first law):
Law I
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.
Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time. (Motte and Cajori translation, 1962, 13)
As regards the law of inertia, there are too many things to be said, and you should refer, e.g., to Barbour (1989), and read chapters on Galileo, Descartes, Huygens, and Newton. As an example of the use of the law of inertia, see the following construction of parabola (by Galileo).
This is impressive enough. Notice that the series 1, 4, 9, 16, ... generates Galileo's famous series of odd numbers 1, 3, 5, 7, ... (take the difference between consecutive two terms). But more impressive is Huygens's insight on centrifugal force. Almost everyone today knows that a particle, initially fixed on a wheel, uniformly rotating on a horizontal plane, will go straight ahead if released at a certain point (see the following figure). The same law governs the movement of a stone released from a sling (David shooting at Goliath), and the movement of a hammer released by Murofushi, after quick turns of rotation.
But wait! Why does a person inside a whirling cylinder feel a strong pressure against the wall of the cylinder? The force seems to act along the radial direction (hence the name, centrifugal force), not along the tangential direction! Huygens worked on this subject and gave an answer.
This is another manifestation of the law of inertia. But in order to see the centrifugal force, you have to imagine how this inertial motion may look, from the rotating frame fixed on the wheel (a good exercise for grasping the significance of relativity!). This way, you may be able to see what sort of force is necessary for keeping the particle on the wheel (this force is instantaneous and continually changing, and it is related to acceleration, via Newton's second law, which was anticipated by Huygens).
This example is taken from Barbour (1989, 487-489; figures are adapted from his Fig. 9.9 and 9.10), but the original comes from Huygens (his paper on centrifugal force, De Vi Centrifuga written 1659). Huygens contributed to the clarification of the centrifugal force, but our point is "how crucial was the law of inertia" in this clarification. Now, Huygens saw that the paths (seen from the rotating frame; i.e. the initial position B moves to E, F, and M etc., consecutively, so that the trajectory of the particle grows as EK, FL, MN, etc.) of the particle grow as shown by blue lines. And his insight was: the length EK, FL, MN grow as squares, 1, 4, 9, 16, etc., if the time intervals are taken small enough. That is, Huygens found the same pattern as Galileo's series of odd numbers in the phenomena of centrifugal force on the rotating wheel! Notice that the force acts along the radial direction, for the observer fixed on the wheel; and hence seen from him/her, the initial motion of the released particle is also along the radial direction.
As a last example of the use of the law of inertia, let me briefly illustrate Newton's derivation of Kepler's second law (area law; that is, the line connecting the sun and a planet describes equal areas in equal times). For this derivation, Newton uses the so-called "Parallelogram rule" (used by Galileo and many others), the rule for obtaining the resultant motion from two motions acting at the same time (this rule can be eventually derived from Newton's first and second law).
All right. Suppose a particle (or a planet) is moving at B to C; if this motion is inertial, it reaches C in a unit time. See the following Figure. But when the particle is at C, a force acts (instantaneously) and attracts the particle toward A; if this force were acting alone, it would cause the particle to move to D in a unit time. So the particle should be deflected; but how? Here, Newton (around 1679-1680) effectively used the law of inertia and the parallelogram rule (notice CDFE is the parallelogram in question), and derived the following resultant motion.
This satisfies Kepler's area law, and if you make time interval smaller and smaller, you can obtain Kepler's law exactly, for a continuous curve. Moreover, Newton went much further. Newton eventually found that any motion satisfying Kepler's laws, also satisfies the inverse square law (the same form as that of Newton's law of universal gravitation); forces acting between the sun and planets according to this law are hidden under Kepler's laws. For a readable reconstruction of this, the reader is referred to Barbour (1989), 549-556.
See also Newtonian mechanics [Japanese]
References
Barbour, Julian (1989) Absolute or Relative Motion?, vol. 1, Cambridge Univ. Press, 1989. (Discovery of Dynamics, paperback reprint, Oxford Univ. Press, 2001)
Sir Isaac Newton's Mathematical Principles, Motte and Cajori translation, Univ. of California Press, 1962.
Whewell, W. (1847) Philosophy of the Inductive Sciences, 2nd ed., Part I, Frank Cass and Co., 1967.
Last modified March 30, 2003. (c) Soshichi Uchii
