§5. Triads

 

498. So much for the dyadic clause of the law of time. The triadic clause is that time has no limit, and every portion of time is bounded by two instants which are of it, and between any two instants either way round, instants may be interposed such that taking any possible multitude of objects there is at least one interposed event for every unit of that multitude. This statement needs some explanation of its meaning. First what does it mean to say that time has no limit? This may be understood in a topical or a metrical sense. In a metrical sense it means there is no absolutely first and last of time. That is, while we must adopt a standard of first and last, there is nothing in its own nature the prototype of first and last. For were there any such prototype, that would consist of a pair of objects absolutely first and last. This, however, is more than is intended here. Whether that be true or not is a question concerning rather the events in time than time itself. What is here meant is that time has no instant from which there are more or less than two ways in which time is stretched out, whether they always be in their nature the foregoing and the coming after, or not. If that be so, since every portion of time is bounded by two instants, there must be a connection of time ring-wise. Events may be limited to a portion of this ring; but the time itself must extend round or else there will be a portion of time, say future time and also past time, not bounded by two instants. The justification of this view is that it extends the properties we see belong to time to the whole of time without arbitrary exceptions not warranted by experience. Now, between any two events may be interposed not merely one event but a multitude of events greater than that statement would supply, a multitude of events as great as a multitude of objects describable. This may be really so or not, but this is the instinctive law of which we seem to be directly conscious.

499. By virtue of this, time is a continuum. For since the instants, or possible events, are as many as any collection whatever, and there is no maximum collection, it follows that they are more than any collections whatever. They must, therefore, be individually indistinguishable in their very existence — that is, are distinguishable and the parts distinguishable indefinitely, but yet not composed of individuals absolutely self-identical and distinct from one another — that is, they form a continuum. A continuum cannot be disarranged except to an insignificant extent. An instant cannot be removed. You can no more, by any decree, shorten a legal holiday by transferring its last instant to the work-day that follows that feast, than you can take away intensity from light, and keep the intensity on exhibition while the light is thrown into the ash-barrel. A limited line AB may be cut into two, AC and C'B, and its ends joined, C' to A and C to B. That is to say, all this may be done in the imagination. We have a difficulty in imagining such a thing in regard to time. For in order that the time should flow continuously even in imagination from the end of one day into the beginning of a day that does historically come next, all the events must be prepared so that the states of things of these two instants, including states of gradual change, such as velocity, etc., shall be precisely the same. In the case of a line we do not think of this, although it is equally true, because we are unaccustomed to minutely dealing with the facts about single molecules and atoms upon which the cohesion of matter depends. We, therefore, see no particular difficulty in joining any end of a line to any other line's end continuously. This is as true a view as the other. As far as time itself goes, nothing prevents twenty-four hours being cut out and the day before joining continuously to the day after, were there any power that could affect such a result. In such a case, the two instants brought together would be identified, or made one, which sufficiently shows their want of individual self-identity and repugnance to all others.

500. Intimately connected with the division of metaphysically contingent laws into laws which impose, upon inherences of different attributes in the same subject, forms analogous to forms of thought so that they may evade laws of logic and into those laws which have no reference to thought, there is a division of these latter laws into laws which impose, upon different subjects of precisely the same qualities, forms of relationship analogous to metaphysical forms so that they may evade the laws of metaphysics, that is, laws of space, and into laws which do not concern dyads of inherence but only dyads of reaction.

501. According to the metaphysical law of sufficient reason, alike in all respects two things cannot be. Space evades that law by providing places in which two things or any number, which are precisely alike, except that they are located in different places, themselves precisely alike in themselves, may exist. Thus, space does for different subjects of one predicate precisely what time does for different predicates of the same subject. And as time effects its evasion of the logical law by providing a form analogous to a logical form, so space effects its evasion of the metaphysical law by providing a form analogous to a metaphysical form. Namely, as metaphysics teaches that there is a succession of realities of higher and higher order, each a generalization of the last, and each the limit of a reality of the next higher order, so space presents points, lines, surfaces, and solids, each generated by the motion of a place of next lower dimensionality and the limit of a place of next higher dimensionality.

502. The last division of laws was a broad one. Now a posteriori laws are divided into those which are purely dynamical and those which are more or less intellectual, a division somewhat analogous to that of mental association into association by contiguity and by resemblance. The former are the nomological laws of physics. So far as our present science knows them, they are as follows:

503. First, every particle, or mathematically indivisible portion of matter, when not under a force, moves along a ray, or line belonging to a certain family of lines such that any four of them not all cut by each of an infinite multitude of rays is cut by just two rays.

504. Second, there is a firmament, or surface, severing space into worlds; and its properties are, first, that if (A), (B), (C), (D), (E), (F), are any points in a plane P1) section of it, the rays {AB} and {DE} will meet at a point [{AB} {DE}] which is coradial with [{BC} {EF}] and [{CD} {FA}]; secondly, no material particle ever comes to or leaves the firmament, nor does any plane fixedly connected with a particle ever move into or away from tangency with the firmament; and thirdly, if a body is rigid, that is, has only six degrees of freedom, so that all its radiform filaments are fixed when six of its particles are restricted to lying in fixed planes, or when six of its plane films are restricted to passing through fixed points, then, all its possible displacements are subject to the following conditions:

First, if two particles, A and B, of the rigid solid be situated in points [A[1]] and [B[1]] such that the ray {A[1]B[1]} has two points in the firmament, say [C[1]] and [D[1]], then A and B, however displaced, must lie in a ray that has two points in the firmament, and if any ray through [A[1]] has the two points [C[2]] and

                      [Click here to view] [D[2]] in the firmament, then A remaining fixed in [A[1]], B can be displaced so as to occupy the point [{[{C[1]C[2]}{D[1]D[2]}] B[1]}{C[2]D[2]}] or the point [{[{C[1]D[2]}{C[2]D[1]}]B[1]}{C[2]D[2]}]; but A and B can occupy simultaneously no pair of points which they are not necessarily able to occupy by virtue of this statement.

Second, if two particles, A and B, of the rigid solid are situated at points [A[1]] and [B[1]] such that the ray {A[1]B[1]} has no point in the firmament, [then] in any plane containing [A[1]] and [B[1]] let [C] and [D] be the points of tangency of rays tangent to the firmament and passing through [A[1]]. Then through

[Click here to view] [A[1]] take any ray { r} whatever, then [{[{CB[1]} r]D}{[{DB[1]}

r]C}] will be a point where the particle B may be while A is at [A[1]].

Third, if two particles, A and B, of the rigid solid are situated at points [A[1]] and [B[1]] such that the ray {A[1]B[1]} has one

                                       [Click here to view] point in the firmament [C[1]], [then] in any plane through {A[1]B[1]} take any other point [C[2]] on the firmament, and take any point [E] on the ray, {C[1]C[2]}. Then, if { t} is the ray tangent to the firmament at [C[2]], A and B may be simultaneously at [ t{EA[1]} ] and [ t{EB[1]}].

Every radial filament of a rigid body (supposed to fill all space) has its polar conjugate radial filament. Namely, one of these rays is the intersection of two planes tangent to the firmament, while the other passes through the two points of tangency. Every infinitesimal displacement of a rigid body is as if it were a part of a rigid body filling all space, and having two motions in one of which all the particles in one ray are fixed while all the plane films through its polar conjugate remain in the same plane, while in the other motion the reverse is the case.

505. Third, the effect of force upon a particle is to produce, while that force lasts, a component acceleration of the particle proportional to and in the ray of the force, and the resultant of such component accelerations is the same as if in each infinitesimal time, the different components acted successively, but each for a time equal to the whole of the infinitesimal time.

506. Fourth, the effect of a force between two particles is to give them opposite accelerations along the ray through them, these accelerations being inversely as certain quantities, called the masses of the accelerated particles, which masses are constant throughout all time.

507. Fifth, so far as force acts between pairs of particles regarded as mere occupiers of points, it depends upon the relative positions of the particles.

508. Sixth, it remains at present uncertain how the phenomena of elasticity, etc. are to be accounted for; but it is certain that all force cannot be positional attractions and repulsions. There is therefore some law additional to the last.

509. Seventh, all particles at a greater distance than a decimetre from one another attract one another nearly inversely as the square of the distance, the constant modulus being 6.658 x 10-8 (Boys).1)

510. Eighth, particles closer together are known to attract one another more strongly, and it seems probable, although it is far from proved, that there are at least two kinds of particles attracting one another differently; but here our ignorance begins to be almost complete.

 


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