# Truth and Error or the Science of Intellection/Chapter 11
## CHAPTER XI DYNAMICS

A citizen of a township must obey the laws of the township. The same person is also a citizen of the county subject to the laws of the county, a citizen of the state subject to the laws of the state, a citizen of the United States subject to the laws of the United States, and finally he is a citizen of the world, subject to international law. Thus a man belongs to a hierarchy of governmental incorporations in which he may demand rights and must perform duties of allegiance.
In the same manner every atom of matter in the lowest body exists in a hierarchy of bodies. An atom of hydrogen exists in the molecule of water. The same atom exists also in the sea, the earth-moon body, the solar system, and the galaxy. Now this atom of hydrogen partakes in the specific or special mode of motion of every body in this hierarchy. We may consider the motion of the atom of hydrogen in the atom itself, if it is a compound body as some chemists suppose; then we may consider it in the molecule, then in the tide, then in the earth in rotation, then in the earth-moon body on an axis within the earth, then in the earth in revolution in the solar system, and then in the galaxy with the solar system, and if there be a system of galaxies we may consider it in such body.
This atom has components of path in an atom, in a molecule, in the tide, in the earth, in the earth-moon body, in the solar system body, in the galaxy body, and finally in another system which includes the galaxy, if there be such a system. If we consider the path of an atom in any one of the incorporations in the hierarchy, we can describe it in terms of dimensions of space, as space is limited by the periphery of that particular body; but when we attempt to describe its motion in two different members of the hierarchy, we are compelled to enlarge our conception of space, for the path of a particle in the atom is modified by its path in the molecule. Then if we consider the path of the atom in the tide we must still further modify our concept of it; then if we consider also the path of the atom in the terrestrial motion about the axis of the earth, we must again modify our concept of it; then if we consider also its path in the earth-moon body, the solar system body, and the galaxy body, we have at last a concept of the path of the atom in a hierarchy of bodies. If we desire, therefore, to conceive of the path followed by the atom of hydrogen directed by all its incorporations combined, we must imagine it determined by all bodies of the hierarchy, and thus to be spiral or vortical. I shall hereafter call this path a hierarchal path.
Descartes conceived this path to be vortical, and taught that the ether in moving in a vortical path carried with it the celestial bodies, and thus explained their revolution. I believe that he properly conceived the nature of the path which a particle describes in a hierarchy of bodies, but of the cause of this path he was in error when he considered that the whole body of ether describes the same path in a vortex.
We may describe the motion of a particle in any one of its incorporations, neglecting it in the other members of the hierarchy, and such a description is legitimate if it be understood as motion in the one incorporation; or we may describe the motion of a particle in two incorporations, but in order to do so it is necessary to use the terms of the space of the higher incorporation. This plan must be continued through all the incorporations if we try to describe all of the deflections of path which are experienced by the atom. If we consider the path of a particle of matter in every one of the hierarchy of bodies severally, we get as many systems of motion as there are bodies, and they seem, when thus narrowly and imperfectly considered, to be incongruous; but when we consider all of these paths concomitantly as hierarchal motion in terms of the space of the highest body, they are made congruous.
Every particle in the universe is in motion, which motion is probably constant in rate of speed. Motion is not only speed, but also path. While the speed in the ultimate particle is constant, the path is variable in direction. This is the proposition I am trying to maintain.
Of ponderable matter, as it is found in terrestrial and celestial systems, all particles are making a grand excursion of the universe. There is no star that does not proceed on this journey, nor is there any body of matter in the earth which does not proceed with the earth in its journey. Ethereal matter does not seem to proceed in this manner from position to position throughout the universe, but the motion of each particle seems to be confined to an environment of other particles, and vibrates back and forth or around and around within its narrow environment. A particle of ponderable matter never returns to the position which it occupies at any one instant of time, so far as we can determine by reasoning. Every position occupied by a particle is instantaneously evacuated, and another particle, either of ponderable matter or of ethereal matter, takes its place.
As there is a hierarchy of bodies, and as there is a hierarchy of paths for every particle of ponderable matter, so there is a hierarchy of freedoms of motion. Take three rods, fasten them together by their central points so that they extend in coördinate directions. The three rods will constitute a body of rods, and although the three are incorporated, that is, fixed to one another, the body has three degrees of freedom. Fix the ends of these rods to a stone quarry, and the three-rod body becomes a component part of the earth body, but still has three degrees of freedom. Then the same three-rod body has three degrees of freedom in the earth-moon body, the solar system body, and the galaxy body. Now we are compelled to believe, by reasoning based on facts observed in modern time, that the molecular bodies and the atomic bodies of the three-rod body have every one three degrees of freedom. This reasoning in molecular science is no less cogent than that in astronomical science, for chemistry gives the same freedom to atoms and molecules that astronomy gives to stars and systems.
We are compelled to conceive of the rigidity of the solid state as the homologue of the astronomical state, and as we know that the rigidity of the astronomical state is a mode of established motion, so we conclude that the rigidity of the solid state is a mode of established motion. Thus the concept is made that man stands between two realms of bodies, the vast or astronomical and the minute or molecular, and that which is observed in astronomy is repeated in chemistry. The astronomic world is the correlative of the molecular world. If there is no gap in this reasoning every particle of matter has a constant rate of speed which is subdivided among the paths of the hierarchy of incorporations to which it belongs. To this form we are compelled to reduce the concept of the persistence of motion or the correlation of forces; for if speed is constant in the atom the forces of the universe are correlative, or, to use a better term, are reciprocal. This conclusion that speed is constant in the particle is necessitated, and hence is valid if we accept the fundamental doctrine of modern chemistry that bodies are composed of discrete particles.
Motion can be diverted in any body of the hierarchy without increasing the speed of the particle. Nature never seems to add to or to subtract from the speed of the particle, although the motion of a molar body may seem to be derived from another body so long as we consider only the molar motion. But when we consider the motion of the particles of the body in their higher and lower incorporations, we find that the apparent added motion is deflection. This is illustrated in the earth-moon body when it rotates about its axis, and thus deflects the motions both of the earth and the moon in their common paths around the sun. So, if a body suspended above the earth falls to the earth, its path with the earth in its course is deflected, and the path of the earth in its course is also deflected. In a falling body we observe not only the deflection of terrestrial motion, but the falling body itself is composed of molecules and atoms which are in motion, and the earth also is composed of molecules and atoms in motion, and these paths are also deflected by the falling of the body. The deflection of their terrestrial motion is but the reciprocal of their deflection in molecular motion. When a body, say of water, loses heat it gains the strength of structure, which is a force, and hence a mode of motion which it exhibits as ice. The body does not transmit its speed of particle to another body, but only induces a corresponding change in that other body from solid strength or rigidity to heat motion by deflecting molecular paths. Thus motion as speed cannot be dissipated. When water is evaporated the particles of vapor which are produced still have the same amount of motion as speed, and when water and carbonic acid are built into wood, their motion remains as the solid strength of the wood in another mode of molecular path. Here we see that rigidity or solid strength is a mode of motion as path. Thus it is that motion as speed is persistent in the particle, but as path it is variable.
Every particle in the wooden ball rolling on the floor has astronomical path, molecular path, and molar path. Consider one of these particles moving with the three kinds of motion as three constituents of path, and we realize that its speed is very great, and that the path which it traverses is greatly composite; that is, composed of deflected parts, in a hierarchy of bodies. If such a particle had its composite path straightened into a right-line path it would quickly pass out of the sphere of the solar system from whatever point within the system it might start, and in whatever direction the right-line path extended. But the molecule remains within the solar system because its stellar path is composite, and it remains within the ball because its molar path is composite, and it remains within the molecule because its molecular path is composite.
When the ball was started molar path was developed, and when it stopped that molar path was ended. We must not suppose that molar motion as speed came out of nothing and vanished into nothing. We resort to preëxisting molecular motion to explain it. We say that the molar motion was derived from the molecular motion of the hand that set the ball rolling, and that it was transformed into molecular motion in the wall which destroyed the molar motion. In making this explanation we assume that motion as speed went out of the hand into the ball, and then out of the ball into the wall. Is this true? Was the speed of the molecular motion in the hand diminished and the speed of the molecular motion in the wall increased? Did motion as speed go out of the hand into the ball? There was a change in the motion of the hand, and a change in the motion of the ball. In what did this change consist? We know that in part at least it consisted in a change of paths. The molecular paths in the hand must have had their directions changed, and the molecular paths in the ball must have had their directions changed. Is this change of direction all, or is there a transference of speed so that one loses while the other gains? The whole problem is narrowed to this issue: That which we call acceleration pertains wholly to deflection, or in very small part to speed, as loss of speed by one and gain by another.
There is still another set of relations to be considered. A body is composed of particles; in order that they should remain within the sphere of the body their paths must be composite, and in order that their paths may be composite there must be a sufficient number of internal collisions to deflect them and retain them within that sphere. If the body itself is moved the paths of the several particles in the average must thus be rendered less composite; that is, the number of collisions must be diminished. The motion of the body as such, therefore, is accomplished by diminishing the deflections within the body, and thus straightening their paths. The translatory motion of a body is a straightening of the paths of the particles of which the body is composed.
Imagine a man walking in a circle of ten feet radius. The sphere of his motion is within the circumference. He may soon walk a mile and never be more than twenty feet away from any given point in the circumference; change his direction so that his path is straightened, and he may soon be a mile away. A body of men walking in a circle remain together as a body within the circumference of the circle as it moves with the earth; change their paths to a cycloid directed to a distant point, and the body of men will move away in that direction; change their paths to parallel right lines, and as a body they may soon be a mile away and still in a circle. A division of an army may be maneuvering in a field as divisions, brigades, regiments, battalions, companies, and platoons, and yet remain in the same field enclosed by a wall; without walking the individual men with any greater speed you may march them to another twenty miles away, and they will lie down to rest at night with no less fatigue than if they had been maneuvering in the enclosed field.
In the same manner the molecules of the wooden ball are in motion within the theater of the ball, so that they do not pass beyond its boundaries; yet impose upon each molecule a change of direction in such a manner that they all move a little more in one course, and a translation of the ball is affected by a change of direction in the motion of its constituent molecules, and the ball still remains as an incorporated body. It is thus possible to explain the molar motion of the ball as a change in direction of the motion of its molecular parts, without assuming an increase of speed in the parts, but only a development of speed in the body by the deflection of its particles. By such an assumption the molar motion perceived by vision would be legitimately derived from the molecular motion known by higher reason, and appear as a change of direction in the molecular motions of the ball. No motion as speed would be created or destroyed, while the apparent molar motion would be explained by a change of direction in molecular motions, very minute as compared with the composite paths of the several molecules and atoms.
When we consider the total motions of the atoms of the ball shot from a cannon’s mouth, an inconceivably small change of direction in the motion of every atom, as compared with the complexity of its path, would fully account for the flight of the ball as projected by dynamite.
Now we know of deflection and that it arises from collision, and we know of no other change in motion. Acceleration as increase of speed cannot, in the nature of the case, be demonstrated, for it may always be explained as deflection, and can never be explained without deflection. If acceleration is explained as deflection, it is explained by referring it to a known cause, and adequately explained.
It is illegitimate to assume an unknown and unknowable cause when a known cause is sufficient for the explanation. We may, therefore, affirm that the acceleration of a body is the deflection of its particles.
At the Brooklyn meeting of the American Association for the Advancement of Science in 1894, I read a paper on this subject, in which I tried to demonstrate that motion is constant in the particle. In the foregoing statement I have put this demonstration in another form. I now propose to give it in a new form by the method of reductio ad absurdum.
Newton taught that inertia is resistance to change of state, either as rest or direction of motion, and Newton also referred to the ambiguity of the term rest without pointing out the nature of this ambiguity. We have seen from the foregoing discussion that rest is absence of molar motion, and that molar motion is created by deflecting molecular motion. Hence the acceleration of a body is reduced to the deflection of its particles, as we have already seen. Following Newton, it is taught in the text-books of physics that inertia is resistance to deflection and acceleration; therefore, reduced to the simplest terms, inertia is resistance to deflection.
PROPOSITION
When two bodies collide their particle paths are deflected, but their particle speeds are unchanged.
First, assume that one body, A, has the mode of motion called rest, and that after the collision it has molar motion; then its molecular motions are deflected. Then assume that their speeds are accelerated; then the particle motions of B also must be deflected and accelerated, if action and reaction are equal in deflection and speed. Therefore, motion as force is created, which is absurd. But Newton’s law says that action and reaction are equal and in opposite directions; therefore, action and reaction result only in particle deflection.
Second, assume that A is at rest, and that at collision B is brought to rest, and thus that B has the speed of its particles diminished; then motion as force is annihilated, which is absurd, but action and reaction being equal as deflection no speed is lost to either.
Third, assume that the particles of A are deflected and their speed accelerated, and that the increase of particle speed in A is derived from the particle speed of B; then action and reaction as speed are not equal, but while both are equally deflected A has more speed, B less, and the more equals the less, with opposite signs. Then A after collision, having more speed than B after collision, has more inertia, which is absurd; therefore, when bodies collide their particle paths are deflected, but their particle speeds are unchanged.
Let this argument be stated in brief:
First, the tendency of modern investigation is to explain all forces as derived from modes of motion. Great progress has been made in this direction, and the theory is widely accepted.
Second, all understood forces are collisions.
Third, if all forces are collisions the motions from which they result obey the third law of motion, that action and reaction are equal and in opposite directions. By this law it is seen that no motion as speed can be lost or gained by any particle of matter.
Fourth, by collision paths can be changed, but motion as speed cannot be transmitted by one particle to another.
Fifth, in starting or stopping molar motion there is an apparent creation and annihilation of motion, but this appearance is known to be an illusion. It is known to be in part deflection, and can all be thus explained; and if the third law of motion is valid it is thus explained.
It must clearly be understood that the above argument does not deny that molar motion as speed can be created or destroyed; it simply affirms that molar motion cannot be created from nothing, and that it cannot be annihilated, but that it comes from molecular motion and returns to molecular motions. Every particle of which we have knowledge is a constituent of many bodies in a hierarchy of bodies, and what is here affirmed is that the acceleration of a body in speed is deflection of its particles, that the particles themselves are not accelerated in speed, and further that embodiment itself is always a result of deflection in the particle embodied. A molar body may have its molar motion increased or diminished in speed by deflecting its molecular motions. If the speed of a molar body be changed, the direction of its molecular particles must necessarily be changed. This proposition is self-evident. The third law of motion is equally simple. The law here demonstrated affirms that acceleration in one embodiment is deflection in another, and it makes valid Newton’s law, which would be an absurdity were the law here demonstrated untrue; and if untrue, the persistence of motion is an absurdity, and with it the persistence of energy falls to the ground.
When the concept of persistence of speed in the particle is once gained, there follows from it a series of corollaries which are demonstrations of axioms of scientific experience, but which otherwise have no demonstration. The following are examples:
PROPOSITION
Gravity, as inversely proportional to the square of the distance, is persistent in the mass.
Assuming that force is motion and gravity force, then if the particle can lose any of its speed it can lose gravity, which is absurd; and if in the collision of a body speed is transferred from its particles to the particles of another body, then the other body must weigh more, which also is absurd; therefore, gravity, as inversely proportional to the square of the distance, is persistent in the mass.
Speed is not a property which can run away by leaping from one particle to another and from one body to another; it is not an occult something—a mystery, a nothing. It is the speed of a particle.
We have seen that when particles in motion have incident paths they collide and their paths are deflected; hence, all motion is directed motion. Collision or impulse is the first mode of force in which action and reaction are exhibited. Then we note how right-line paths are divided into components by collision, becoming deflected paths; then how by systematic collisions they may be developed into revolution. Then we consider that particles may be incorporated in a body with their several particles revolving around a common center, and this revolution of the particles is rotation of the body. Thus by incorporation the motions of particles may be correlated by rotation and revolution, as exhibited in celestial bodies.
In the case of two stellar orbs revolving about a common center, as the earth and the moon, it is plain that gravity causes the deflection of both bodies inversely proportional to their masses. Here acceleration is chiefly deflection, being positive at perigee and negative at apogee. So, in the revolution of the sun and the earth about a common axis, acceleration is chiefly deflection, being positive at perihelion and negative at aphelion. Thus we have a well-known astronomical example of acceleration, and find it deflection and increase or decrease of bodily speed, and now we must refer this acceleration of speed in the body to deflection in the particles of which it is composed.
It is taught in astronomy that in the revolution of a planet the area of the radius vector is equal for equal times. This doctrine is made simple and plain when the nature of acceleration is understood.
In an ethereal medium of particles moving with a persistent speed, two bodies will mutually intercept collisions with the ethereal medium inversely proportional to the square of their distance apart, which is an explanation of the law of gravity, and is the theory of La Sage in terms of motion.
On page 642, Vol. iv, article 22, of Bowditch’s translation of La Place’s Méchanique Céleste it is stated:
“If gravitation be produced by the impulse of a fluid directed towards the center of the attracting body, the preceding analysis, relative to the impulse of the solar light, will give the secular equation depending on the successive transmission of the attractive force.”
After proving this proposition and obtaining the secular equation of the attracting body from the successive transmission of gravity, the cause of the moon is discussed, and La Place decides that:
“We must suppose that the gravitating fluid has a velocity which is at least a hundred millions of times greater than that of light; or at least we must suppose, in its action on the moon, that it has at least that velocity to counteract her gravity towards the earth. Therefore, mathematicians may suppose, as they have heretofore done, that the velocity of the gravitating fluid is infinite.”
The theory of La Sage is stated in terms of a fluid transmitted from one body to another. We now know that waves, not fluids, are transmitted in the case of heat and light, and in a like manner gravity as deflection must be considered as wave action or vibration in some form. With these principles the instantaneous action of gravity is simple and self-evident, for speed is not transmitted, but only deflection is caused.
Every particle has constant motion as speed which cannot be increased or diminished, and the absurdity of perpetual motion should be called the absurdity of perpetual collision between two bodies without other deflection. The particles collide because of impinging paths; they are deflected and their paths are turned apart, and they cannot be made to collide again until other external collisions bring their paths together. If the particle A is deflected after one collision, to be once more deflected, another collision is necessary. It is thus that the absurdity of perpetual collision can be simply demonstrated.
After such an analysis the doctrine of virtual velocities is self-evident; and there are many other consequences of this law which, properly understood, would make many propositions of physics self-evident.
Motion as speed is constant in the particle. The particle, of whatever order it may be in the members of the hierarchy, is accelerated by deflecting its particles. The principles or laws of dynamics are all corollaries of this fundamental law; hence dynamics may be taught as a deductive science. Thus we have the mathematics of number, the mathematics of space, and the mathematics of motion, all fundamentally deductive sciences.