On the Philosophy of Discovery, Chapters Historical and Critical

|(k)| It is recognized that what Newton—besides the foundation of the analytical treatment, the development of which, by the bye, has of itself rendered superfluous, or indeed rejected much which belonged to Newton's essential Principles and glory—has added to the Keplerian Laws is the Principle of Perturbations,—a Principle whose importance we may here accept thus far (hier in sofern anzuführen ist); namely, so far as it rests upon the Proposition that the so-called attraction is an operation of all the individual parts of bodies, as being material.

|(l)| It lies in this, that matter in general assigns a center for itself (sich das centrum setzt), and the figure of the body is an element in the determination of its place; that collective bodies of the system recognize a reference to their Sun (sich ihre Sonne setzen), but also the individual bodies themselves, according to the relative position with regard to each other into which they come by their general motion, form a momentary relation of their gravity (schwere) towards each other, and are related to each other not only in abstract spatial relations, but at the same time assign to themselves a joint center, which however is again resolved [into the general center] in the universal system.

As to what concerns the features of the path, to show how the fundamental determinations of Free Motion are connected with the Conception, cannot here be undertaken in a satisfactory and detailed manner, and must therefore be left to its fate. The proof from reason of the quantitative determinations of free motion can only rest upon the determinations of Conceptions of space and time, the elements whose relation (intrinsic not extrinsic) motion is.

|(m)| That, in the first place, the motion in general is a motion returning into itself, is founded on the determination of particularity and individuality of the bodies in general (§ 269), so that partly they have a center in themselves, and partly at the same time their center in another.

|(n)| These are the determinations of Conceptions which form the basis of the false representatives of Centripetal Force and Centrifugal Force, as if each of these were self-existing, extraneous to the other, and independent of it; and as if they only came in contact in their operations and consequently externally. They are, as has already been mentioned, the Lines which must be drawn for the mathematical determinations, transformed into physical realities.

Further, this motion is uniformly accelerated, (and—as returning into itself—in turn uniformly retarded). In motion as free, Time and Space enter as different things which are to make themselves effective in the determination of the motion (§ 266, note).

|(o)| In the so-called Explanation of the uniformly accelerated and retarded motion, by means of the alternate decrease and increase of the magnitude of the Centripetal Force and Centrifugal Force, the confusion which the assumption of such independent Forces produces is at its greatest height.

|(p)| According to this explanation, in the motion of a Planet from the Aphelion to the Perihelion, the centrifugal is less than the centripetal force, and on the contrary, in the Perihelion itself, the centrifugal force is supposed to become greater than the centripetal. For the motion from the Perihelion to the Aphelion, this representation makes the forces pass into the opposite relation in the same manner. It is apparent that such a sudden conversion of the preponderance which a force has obtained over another, into an inferiority to the other, cannot be anything taken out of the nature of Forces.

|(q)| On the contrary it must be concluded, that a preponderance which one Force has obtained over another must not only be preserved, but must go onwards to the complete annihilation of the other Force, and the motion must either, by the Preponderance of the Centripetal Force, proceed till it ends in rest, that is, in the Collision of the Planet with the Central Body, or till by the Preponderance of the Centrifugal Force it ends in a straight line. But now, if in place of the suddenness of the conversion, we suppose a gradual increase of the Force in question, then, since rather the other Force ought to be assumed as increasing, we lose the opposition which is assumed for the sake of the explanation; and if the increase of the one is assumed to be different from that of the other, (which is the case in some representations,) then there is found at the mean distance between the apsides a point in which the Forces are in equilibrio. And the transition of the Forces out of Equilibrium is a thing just as little without any sufficient reason as the aforesaid suddenness of inversion. And in the whole of this kind of explanation, we see that the mode of remedying a bad mode of dealing with a subject leads to newer and greater confusion.—A similar confusion makes its appearance in the explanation of the phænomenon that the pendulum oscillates more slowly at the equator. This phænomenon is ascribed to the Centrifugal Force, which it is asserted must then be greater; but it is easy to see that we may just as well ascribe it to the augmented gravity, inasmuch as that holds the pendulum more strongly to the perpendicular line of rest.

§ 240

|(r)| And now first, as to what concerns the Form of the Path, the Circle only can be conceived as the path of an absolutely uniform motion. Conceivable, as people express it, no doubt it is, that an increasing and diminishing motion should take place in a circle. But this conceivableness or possibility means only an abstract capability of being represented, which leaves out of sight that Determinate Thing on which the question turns.

The Circle is the line returning into itself in which all the radii are equal, that is, it is completely determined by means of the radius. There is only one Determination, and that is the whole Determination.

But in free motion, in which the Determinations according to space and according to time come into view with Differences, in a qualitative relation to each other, this Relation appears in the spatial aspect as a Difference thereof in itself, which therefore requires two Determinations. Hereby the Form of the path returning into itself is essentially an Ellipse.

|(s)| The abstract Determinations which produces the circle appears also in this way, that the arc or angle which is included by two Radii is independent of them, a magnitude with regard to them completely empirical. But since in the motion as determined by the Conception, the distance from the center, and the arc which is run over in a certain time, must be comprehended in one determinateness, [and] make out a whole, this is the sector, a space-determination of two dimensions: in this way, the arc is essentially a Function of the Radius Vector; and the former (the arc) being unequal, brings with it the inequality of the Radii.

|(t)| That the determination with regard to the space by means of the time appears as a Determination of two Dimensions,—as a Superficies-Determination,—agrees with what was said before (§ 266) respecting Falling Bodies, with regard to the exposition of the same Determinateness, at one while as Time in the root, at another while as Space in the Square. Here, however, the Quadratic character of the space is, by the returning of the Line of motion into itself, limited to a Sector. These are, as may be seen, the general principles on which the Keplerian Law, that in equal times equal sectors are cut off, rests.

This Law becomes, as is clear, only the relation of the arc to the Radius Vector, and the Time enters there as the abstract Unity, in which the different Sectors are compared, because as Unity it is the Determining Element. But the further relation is that of the Time, not as Unity, but as a Quantity in general,—as the time of Revolution—to the magnitude of the Path, or, what is the same thing, the distance from the center.

|(u)| As Root and Square, we saw that Time and Space had a relation to each other, in the case of Falling Bodies, the case of half-free motion—because that [motion] is determined on one side by the conception, on the other by external [conditions]. But in the case of absolute motion—the domain of free masses—the determination attains its Totality. The Time as the Root is a mere empirical magnitude; but as a component (moment) of the developed Totality, it is a Totality in itself,—it produces itself, and therein has a reference to itself; as the Dimensionless Element in itself, it only comes to a formal identity with itself, the Square; Space, on the other hand, as the positive Distribution (aussereinander) [comes] to the Dimension of the Conception, the Cube.

|(v)| Their Realization preserves their original difference. This is the third Keplerian Law, the relation of the Cubes of the Distances to the Squares of the Times;—a Law which is so great on this account, that it represents so simply and immediately Reason as belonging to the thing: while on the contrary the Newtonian Formula, by means of which the Law is changed into a Law for the Force of Gravity, shows the Distortion, Perversion and Inversion of Reflexion which stops half-way.

Additions to new Edition. § 269

The center has no sense without the circumference, nor the circumference without the center. This makes all physical hypotheses vanish which sometimes proceed from the center, sometimes from the particular bodies, and sometimes assign this, sometimes that, as the original [cause of motion] … It is silly (läppisch) to suppose that the centrifugal force, as a tendency to fly off in a Tangent, has been produced by a lateral projection, a projectile force, an impulse which they have retained ever since they set out on their journey (von Haus aus). Such casualty of the motion produced by external causes belongs to inert matter; as when a stone fastened to a thread which is thrown transversely tries to fly from the thread. We are not to talk in this way of Forces.

|(w)| If we will speak of Force, there is one Force, whose elements do not draw bodies to different sides as if they were two Forces. The motion of the heavenly bodies is not a being pulled this way or that, such as is thus imagined; it is free motion: they go along, as the ancients said, as blessed Gods (sie gehen als selige Götter einher). The celestial corporeity is not such a one as has the principle of rest or motion external to itself.

|(x)| Because stone is inert, and all the earth consists of stones, and the other heavenly bodies are of the same nature,—is a conclusion which makes the properties of the whole the same as those of the part. Impulse, Pressure, Resistance, Friction, Pulling, and the like, are valid only for an existence of matter other than the celestial. Doubtless that which is common to the two is matter, as a good thought and a bad thought are both thoughts; but the bad one is not therefore good, because it is a thought.

Appendix KDEMONSTRATION THAT ALL MATTER IS HEAVY(Cam. Phil. Soc. Feb. 22, 1841.)

The discussion of the nature of the grounds and proofs of the most general propositions which the physical sciences include, belongs rather to Metaphysics than to that course of experimental and mathematical investigation by which the sciences are formed. But such discussions seem by no means unfitted to occupy the attention of the cultivators of physical science. The ideal, as well as the experimental side of our knowledge must be carefully studied and scrutinized, in order that its true import may be seen; and this province of human speculation has been perhaps of late unjustly depreciated and neglected by men of science. Yet it can be prosecuted in the most advantageous manner by them only: for no one can speculate securely and rightly respecting the nature and proofs of the truths of science without a steady possession of some large and solid portions of such truths. A man must be a mathematician, a mechanical philosopher, a natural historian, in order that he may philosophize well concerning mathematics, and mechanics, and natural history; and the mere metaphysician who without such preparation and fitness sets himself to determine the grounds of mathematical or mechanical truths, or the principles of classification, will be liable to be led into error at every step. He must speculate by means of general terms, which he will not be able to use as instruments of discovering and conveying philosophical truth, because he cannot, in his own mind, habitually and familiarly, embody their import in special examples.

Acting upon such views, I have already laid before the Philosophical Society of Cambridge essays on such subjects as I here refer to; especially a memoir "On the Nature of the Truth of the Laws of Motion," which was printed by the Society in its Transactions. This memoir appears to have excited in other places, notice of such a kind as to show that the minds of many speculative persons are ready for and inclined towards the discussion of such questions. I am therefore the more willing to bring under consideration another subject of a kind closely related to the one just mentioned.

The general questions which all such discussions suggest, are (in the existing phase of English philosophy) whether certain proposed scientific truths, (as the laws of motion,) be necessary truths; and if they are necessary, (which I have attempted to show that in a certain sense they are,) on what ground their necessity rests. These questions may be discussed in a general form, as I have elsewhere attempted to show. But it may be instructive also to follow the general arguments into the form which they assume in special cases; and to exhibit, in a distinct shape, the incongruities into which the opposite false doctrine leads us, when applied to particular examples. This accordingly is what I propose to do in the present memoir, with regard to the proposition stated at the head of this paper, namely, that all matter is heavy.

At first sight it may appear a doctrine altogether untenable to assert that this proposition is a necessary truth: for, it may be urged, we have no difficulty in conceiving matter which is not heavy; so that matter without weight is a conception not inconsistent with itself; which it must be if the reverse were a necessary truth. It may be added, that the possibility of conceiving matter without weight was shown in the controversy which ended in the downfall of the phlogiston theory of chemical composition; for some of the reasoners on this subject asserted phlogiston to be a body with positive levity instead of gravity, which hypothesis, however false, shows that such a supposition is possible. Again, it may be said that weight and inertia are two separate properties of matter: that mathematicians measure the quantity of matter by the inertia, and that we learn by experiment only that the weight is proportional to the inertia; Newton's experiments with pendulums of different materials having been made with this very object.

I proceed to reply to these arguments. And first, as to the possibility of conceiving matter without weight, and the argument thence deduced, that the universal gravity of matter is not a necessary truth, I remark, that it is indeed just, to say that we cannot even distinctly conceive the contrary of a necessary truth to be true; but that this impossibility can be asserted only of those perfectly distinct conceptions which result from a complete development of the fundamental idea and its consequences. Till we reach this stage of development, the obscurity and indistinctness may prevent our perceiving absolute contradictions, though they exist. We have abundant store of examples of this, even in geometry and arithmetic; where the truths are universally allowed to be necessary, and where the relations which are impossible, are also inconceivable, that is, not conceivable distinctly. Such relations, though not distinctly conceivable, still often appear conceivable and possible, owing to the indistinctness of our ideas. Who, at the first outset of his geometrical studies, sees any impossibility in supposing the side and the diagonal of a square to have a common measure? Yet they can be rigorously proved to be incommensurable, and therefore the attempt distinctly to conceive a common measure of them must fail. The attempts at the geometrical duplication of the cube, and the supposed solutions, (as that of Hobbes,) have involved absolute contradictions; yet this has not prevented their being long and obstinately entertained by men, even of minds acute and clear in other respects. And the same might be shewn to be the case in arithmetic. It is plain, therefore, that we cannot, from the supposed possibility of conceiving matter without weight, infer that the contrary may not be a necessary truth.

Our power of judging, from the compatibility or incompatibility of our conceptions, whether certain propositions respecting the relations of ideas are true or not, must depend entirely, as I have said, upon the degree of development which such ideas have undergone in our minds. Some of the relations of our conceptions on any subject are evident upon the first steady contemplation of the fundamental idea by a sound mind: these are the axioms of the subject. Other propositions may be deduced from the axioms by strict logical reasoning. These propositions are no less necessary than the axioms, though to common minds their evidence is very different. Yet as we become familiar with the steps by which these ulterior truths are deduced from the axioms, their truth also becomes evident, and the contrary becomes inconceivable. When a person has familiarized himself with the first twenty-six propositions of Euclid, and not till then, it becomes evident to him, that parallelograms on the same base and between the same parallels are equal; and he cannot even conceive the contrary. When he has a little further cultivated his geometrical powers, the equality of the square on the hypothenuse of a right-angled triangle to the squares on the sides, becomes also evident; the steps by which it is demonstrated being so familiar to the mind as to be apprehended without a conscious act. And thus, the contrary of a necessary truth cannot be distinctly conceived; but the incapacity of forming such a conception is a condition which depends upon cultivation, being intimately connected with the power of rapidly and clearly perceiving the connection of the necessary truth under consideration with the elementary principles on which it depends. And thus, again, it may be that there is an absolute impossibility of conceiving matter without weight; but then, this impossibility may not be apparent, till we have traced our fundamental conceptions of matter into some of their consequences.

The question then occurs, whether we can, by any steps of reasoning, point out an inconsistency in the conception of matter without weight. This I conceive we may do, and this I shall attempt to show.

The general mode of stating the argument is this:—the quantity of matter is measured by those sensible properties of matter which undergo quantitative addition, subtraction and division, as the matter is added, subtracted and divided. The quantity of matter cannot be known in any other way. But this mode of measuring the quantity of matter, in order to be true at all, must be universally true. If it were only partially true, the limits within which it is to be applied would be arbitrary; and therefore the whole procedure would be arbitrary, and, as a method of obtaining philosophical truth, altogether futile.

We may unfold this argument further. Let the contrary be supposed, of that which we assert to be true: namely, let it be supposed that while all other kinds of matter are heavy (and of course heavy in proportion to the quantity of matter), there is one kind of matter which is absolutely destitute of weight; as, for instance, phlogiston, or any other element. Then where this weightless element (as we may term it) is mixed with weighty elements, we shall have a compound, in which the weight is no longer proportional to the quantity of matter. If, for example, 2 measures of heavy matter unite with one measure of phlogiston, the weight is as 2, and the quantity of matter as 3. In all such cases, therefore, the weight ceases to be the measure of the quantity of matter. And as the proportion of the weighty and the weightless matter may vary in innumerable degrees in such compounds, the weight affords no criterion at all of the quantity of matter in them. And the smallest admixture of the weightless element is sufficient to prevent the weight from being taken as the measure of the quantity of matter.

But on this hypothesis, how are we to distinguish such compounds from bodies consisting purely of heavy matter? How are we to satisfy ourselves that there is not, in every body, some admixture, small or great, of the weightless element? If we call this element phlogiston, how shall we know that the bodies with which we have to do are, any of them, absolutely free from phlogiston?

We cannot refer to the weight for any such assurance; for by supposition the presence and absence of phlogiston makes no difference in the weight. Nor can any other properties secure us at least from a very small admixture; for to assert that a mixture of 1 in 100 or 1 in 10 of phlogiston would always manifest itself in the properties of the body, must be an arbitrary procedure, till we have proved this assertion by experiment: and we cannot do this till we have learnt some mode of measuring the quantities of matter in bodies and parts of bodies; which is exactly what we question the possibility of, in the present hypothesis.

Thus, if we assume the existence of an element, phlogiston, devoid of weight, we cannot be sure that every body does not contain some portion of this element; while we see that if there be an admixture of such an element, the weight is no longer any criterion of the quantity of matter. And thus we have proved, that if there be any kind of matter which is not heavy, the weight can no longer avail us, in any case or to any extent, as a measure of the quantity of matter.

I may remark, that the same conclusion is easily extended to the case in which phlogiston is supposed to have absolute levity; for in that case, a certain mixture of phlogiston and of heavy matter would have no weight, and might be substituted for phlogiston in the preceding reasoning.

I may remark, also, that the same conclusion would follow by the same reasoning, if any kind of matter, instead of being void of weight, were heavy, indeed, but not so heavy, in proportion to its quantity of matter, as other kinds.

On all these hypotheses there would be no possibility of measuring quantity of matter by weight at all, in any case, or to any extent.

But it may be urged, that we have not yet reduced the hypothesis of matter without weight to a contradiction; for that mathematicians measure quantity of matter, not by weight, but by the other property, of which we have spoken, inertia.

To this I reply, that, practically speaking, quantity of matter is always measured by weight, both by mechanicians and chemists: and as we have proved that this procedure is utterly insecure in all cases, on the hypothesis of weightless matter, the practice rests upon a conviction that the hypothesis is false. And yet the practice is universal. Every experimenter measures quantity of matter by the balance. No one has ever thought of measuring quantity of matter by its inertia practically: no one has constructed a measure of quantity of matter in which the matter produces its indications of quantity by its motion. When we have to take into account the inertia of a body, we inquire what its weight is, and assume this as the measure of the inertia; but we never take the contrary course, and ascertain the inertia first in order to determine by that means the weight.