SPECULATIVE DYNAMICS. From The Nation, June 3, 1875. ⁵⁷ “The Mechanism of the Universe and its Primary Effort-exerting Powers. By Augustus Fendler.” Wilmington, Del.: Printed by the “Commercial Printing Company,” 1874. ⁵⁸

Whether when a body moves it is proper to say that it is in motion, or that the motion is in it, is a question often suggested by the language of even the most guarded writers on mathematical dynamics, We follow Professors Thomson and Tait in using “dynamics” in a wide sense, including Statics, in place of “mechanics,” which, though commonly used in this sense, is more properly the theory of machines and mechanical constructions than that of the abstract principles of motion and equilibrium. ⁵⁹ though the strictly mathematical definitions, formulas, theorems, and problems of the science are free from any ambiguity. With what meaning the preposition “in” is used in these expressions is a further and more pertinent question. If with that meaning which the unmathematical language of these writers seems to authorize, then they have really exposed themselves and their readers to the difficulty involved in Zeno’s famous paradox of motion, namely, that since a motion must be either in the place of the moving body or in some other place, and since the moving body does not move in its place, and does not move in any other place, motion is really a contradiction, and therefore, according to logic, an impossibility. The solution of the paradox, for which the science of logic had to establish a distinct principle, recognized that in such expressions the preposition “in” is not properly used in a locative sense, but only in the vaguer sense of appertaining to, or being predicable of, its object. That a body in motion has the attribute of motion (that is, the attribute of having a continuouslychanging distance from some other body, or from some position which is regarded as at rest, or as not having this attribute);

and the other form of the same fact, namely, that the motion in the body is an attribute of the body—are equivalent or entirely accordant expressions of what is signified by the preposition “in.” Zeno’s paradox is logically solved in such terms as these: motion transcends the “sphere” of the locative, or is distinct from both the positive and negative, or the contradictory locative, meanings of “in.” It is neither here nor there as a phenomenon, and yet is not an excluded middle, since the contradiction of this and some other place is a contradiction in relations, both of which are distinct from the nature of motion. Nevertheless, judging by the current language (not mathematical), and the past disputes of mathematicians on the definitions of force and motion—disputes which, after being settled within their own province, have been bequeathed to unmathematical speculators in dynamical philosophy—we should be inclined, at first sight, to allow that such speculators have the warrant of high authority for their attempts at revising the fundamental conceptions of this science. Whether consciously or not, the mathematicians of the seventeenth century and unmathematical sciolists of later times were impelled by this old paradox to a solution of its difficulty by a metaphysical or non-phenomenal conception of the “force of motion,” so called, as something locatively in a moving body, constituting the substantive or sustaining cause of motion; seeing that the phenomenon itself of motion, being a continuous change of distance from a fixed position, could no more properly be in a body than this very distance could be locatively in it.

Newton from the first, and all competent mathematicians of a later time, saw that the mathematical discussion of dynamical problems had no concern with any such metaphysical conception. The supposed cause of the uniformity of motion in a fixed direction which a body has independently of external relations, or vires impresseæ, is not any part of dynamical science. Moreover, the causes of change in the velocities and directions of motion, or these vires impressæ, were conceived in a purely phenomenal or descriptive way, and measured by actually visible and tangible quantities. It was not on account of any speculative inability in Newton to conceive a possible ulterior

cause of gravity that he excluded from mathematical dynamics the search for it, and remained contented with the descriptive quantitative law of its action; but simply because such a research departed in a direction just the opposite of that which led to rigorously-demonstrated explanations of the observed phenomena of nature. If any of these phenomena could have led, “in a mathematical way,” to the law of action in gravitation, Newton’s genius would surely not have failed to deduce it from them. He took gravity with its law for an ultimate fact, simply because it did not follow as a consequence from any other observed laws in the same manner of mathematical deduction in which he had shown that Kepler’s laws follow from it and from the three laws of motion. But even mathematicians, and especially those of Germany, whose men of science are even to this day more given to metaphysics than those of other nations, were for a long time haunted by the metaphysical spectre of a cause called the “force of motion,” and supposed to be needed to keep a body agoing as well as to set it in motion or bring it to rest.

The mathematics of this science, however, deals only with the defined or measured quantitative phenomenal conditions of persistence and change in motion; and the metaphysical mathematicians were so far true to their science as to seek for a measure of this metaphysical cause of motion. A fierce dispute accordingly arose in the seventeenth century, and was continued into the eighteenth, in which the most illustrious men took part, as to whether the “force of motion” should be measured or defined by the velocity directly or by the square of the velocity. But after a bitter contention, prolonged by the rivalries of national honor among European scholars, the question was finally seen to resolve itself into whether the name vis viva, or “force of motion,” ought properly to be given to one or to the other measure. For all mathematical and experimental purposes, these measures were all in all, and were perfectly consistent as measures of different phenomena or relations of motion, if only called by different names. And it was seen that dynamical science could get along perfectly well without any use of the confusing word “force.” But the word continues still to have at least

four distinct meanings in dynamical science—technical meanings related to the use of the word in mathematical reasonings, which are never, however, confounded by mathematicians. All that is really common to them is a vague reference to the production or persistence of states of motion or rest. The real gists of their meanings are in the qualifying terms annexed to them, as in the vis impressa of Newton or the vis mortua of Leibnitz, otherwise called vis acceleratrix, the vis insita or vis inertiæ, the vis viva, and the vis motrix. In place of these names, modern treatises often use, without the substantive word, the terms acceleration (retardation being a minus or algebraically negative acceleration); secondly, mass (the coefficient of velocity, or of its square, in estimating either); thirdly, the momentum; and, fourthly, the energy of motion. But the term energy still has that metaphysical taint of vagueness, even with modern mathematical writers, which so long infected the word “force.” It is still spoken of, both with reference to its actual and potential forms, as if it were something locatively in the moving body, or in a body capable of a defined motion; instead of being only predicably in the permanent internal and the special external conditions, which mathematically determine relative movements and their rates of change. It is not sur prising that an unmathematical speculator in dynamics should be misled by such expressions as the following from the eminent authors, Professors Thomson and Tait, to which many parallel expressions in other authors might be added, namely, “A raised weight, a bent spring, compressed air, etc., are stores of energy which can be made use of at pleasure.” A mathematician, knowing in what terms these antecedent conditions of motion are expressed and measured, understands them to refer only to sensible properties in these “stores,” together with the restraining causes which also have sensible measures, namely, what makes them “stores,” or holds the weight up, or the spring bent, or the air compressed. It is in the being held up, or bent, or compressed—in these antecedent circumstances, as well as in what is locatively in the bodies, that the storing of energy consists; and this energy is also dependent in the case of the raised weight on an equally sensible and measurable outward relation, namely, distance from the ground.

The word “force,” unqualified, but understood to be limited to the meaning and descriptive measure of “accelerative force,” or in a strictly-defined and technical meaning, is still commonly employed in treatises on dynamics. Otherwise it is always qualified, as in the “force of inertia.” All its uses in mathematical language, or the equivalent terms, acceleration, mass, momentum, and energy, refer to precise, unambiguous definitions in the measures of the phenomena of motion, and do not refer to any other substantive or noumenal existence than the universal inductive fact that the phenomena of all actual movements in nature can be clearly, and definitely, or intelligently analyzed into phenomena, and conditions of phenomena, of which these terms denote the measures. In modern dynamics, the mathematical measures of actual phenomena are their real essences, as scientific facts. Even the much-derided Aristotelian doctrine in explanation of the various phenomena of suction—namely, “nature’s abhorrence of a vacuum”—might pass muster in science (though not now as an ultimate principle) if a determination of how much nature’s abhorrence amounts to under defined circumstances were attached to it. The real fault of the principle and its pretended explanations would be paralleled if we should seek to explain the movements of the planets and of falling bodies by “nature’s abhorrence of divorce between bodies”—which is about what the word “attraction” meant to the lively imaginations of Newton’s contemporaries, as with Huygens—without estimating, as the Newtonian law of gravity does, how much this abhorrence amounts to under given external relations. The fact that nature has an abhorrence of a vacuum mathematically dependent on the weight of the liquid forced into it is not impugned by the fact, subsequently discovered, that this weight is balanced by the weight and consequent pressure of the atmosphere, any more than Kepler’s three descriptive and quantitative laws were invalidated by the subsequent deduction of them from the laws of motion and of gravity. Kepler’s laws served, indeed, as the most effective inductive confirmations of these laws and their universality; and Newton’s law of gravity would still hold the honored place it has in science even if it should in future be shown to follow

from independently demonstrated and simpler, more ultimate conditions of changes in motion. Merely speculative explanations of it have no honor at all; for its merits are in its being a precise quantitatively-descriptive law, and on this ground alone it holds its place in mathematical dynamics.

We have said that the word “force,” when used without qualification, has come to mean unambiguously what, for the sake of avoiding ambiguity, Newton called vis impressa; so that in recent treatises the first law of motion is expressed in such terms as these: “A body under the action of no force, or of balanced forces, is either at rest or moves uniformly in a straight line.” Newton’s words were: “Nisi quatenus a viribus impressis.” Now, our author, apparently ignorant of the history of the science, and without any guidance from its mathematics, undertakes to criticise such a statement (Section VI.), simply on the ground that he has chosen, without giving any reasons for it, to give the unqualified word “force” a different meaning in what he is pleased to call an axiom (Axiom VIII., p. 12). He means by it the cause which keeps a body agoing when it moves. Of this cause modern dynamical science knows nothing, except the negative fact stated in the first law of motion, which may be given with even greater clearness without using the word “force” at all—namely, that, independently of properties through which a body is related to other bodies, or independently of such relations, its state of rest or of uniform motion in a fixed direction is unchanged. Behind this fact, except so far as it serves to define the word “force,” or vis impressa, dynamic science does not go; but it goes forward with this and other facts to most fruitful results in mathematical deductions, with which our author does not appear to be at all acquainted. Another fact, the second law of motion, which again may be fully expressed without the use of “force,” is that the change in the component of a velocity in any direction may be measured in terms of a fixed property, namely, mass, and special outward relations, which in general are dependent simply on distances and directions.

Mathematical dynamics knows of no bodies at rest in any absolute sense. All the motions known or considered are relative

motions—namely, continuous changes of distances between bodies, or between these and positions defined by other bodies. It is not known that even the centre or average position of all the masses of the universe is at rest in any absolute sense; so that the absolute motion of no body is known, and the “force” of our author is without any definite measure or utility in mathematical dynamics. The principle of relative motion leaves all measures of motion considered as absolute quite out of the problems of this science, as indeed they are quite beyond our possible knowledge.

One of the principles of mathematical dynamics which staggers the unmathematical sciolist more than any other, and was at first one of the greatest difficulties, even with mathematicians, in the Newtonian theory of gravity—a difficulty repeatedly urged, and brought out from apparently independent meditations by anti-Newtonian heretics—is the doctrine of “action at a distance.” This action, the metaphysicians say, is impossible, and they devote themselves to the invention of media through which force and motion may be communicated, or from which it may be collected (Axiom VII., p. 11), thinking that thereby they are helping out the mathematical genius of Newton by a profounder effort of thought than he was capable of. But with metaphysical action dynamical science has nothing to do. The action at a distance, considered in this science, is simply a change in motion measurably or mathematically dependent on (or a function of) distances from bodies, distances of which nothing is asserted but that they extend indefinitely beyond the masses or the visible and tangible limits of bodies. “A body cannot act where it is not”—“With all my heart,” says Carlyle, “only where is it?” If attractive force is an attribute of bodies (as it is whether or not this force depends on an intangible and invisible medium), then the presence of bodies at a distance from their visible limits must be assumed, so far at least as this attribute is concerned. The color of a body is familiarly known to be distinct from its solid extent, volume, or mass, and is not in the same place; nevertheless, as superficial, is still contiguous with its other sensible qualities. The metaphysical difficulty of believing that the attribute of

attraction may be still more displaced or removed than color, is a difficulty which disappears with its cause, namely, unfamiliarity with the conception. Patient study of mathematical and experimental science has resolved many such difficulties, which are not really logical ones; for whether gravity will ever come in science to be a legitimately derived attribute or property of bodies acting through a medium, or will forever remain, as now, an ultimate phenomenal fact, there is nothing of contradiction or essential opposition to experience in its asserted action at a distance—at a distance, that is, not of course from where it acts, but from the places where other attributes of body are manifested; that is, beyond its visible and tangible limits. Most theories of a gravitative medium have been in fact atomic, and, by the interposition of voids between atoms which is thus made, have really introduced the very action at a distance which the theories were devised to do away with. Indeed, the essential principle of action at a distance is a necessary consequence of the metaphysical axiom (which we are not, however, obliged by positive evidence to accept), that pure continuous matter is incompressible, as in the supposed atoms; and though this action be only on a molecular scale, it is no more possible on this scale than on that larger one of gravitative action which mathematical dynamics is supposed to assume. But, as we have said, no such metaphysical assumption is made in this science.

No student of mathematics, competent to pass an examination in Newton’s “Principia,” not only on its definitions, axioms, and philosophical scholiums, but on its mathematical theorems and problems, could read with any profit, or even with any patience, Mr. Fendler’s speculations. Those parts of the “Principia,” or of more modern treatises, which such thinkers as our author appear to have studied, present themselves to the student who has clearly seen their embodiment in the mathematical deductions and experimental verifications of dynamical science in a wholly different light from that in which such speculative thinkers take them up. The laws or axioms and the definitions of this science are apparently considered by these thinkers as constituting in themselves a complete body of doctrine, capable

of being studied and criticised quite independently of any other mathematics than what they directly involve, whereas they are really integrant parts or elements of a systematic deductive science; and whether or not they are evident at a glance through familiar inductions, or by “intuitions a priori” (as some thinkers will have it), they have their truest proof in the broadest possible tests of experience, through the experimental and observational verifications of their mathematical consequences. Of the nature and force of this kind of proof none but students of mathematical dynamics and experimental physics can be supposed to have any adequate conception. To attempt to criticise the elementary conceptions and first principles of the science in any other way, and especially a priori, or with a simple reference to Vernunft, is really a display of the critic’s incompetency, which is not remedied by a reference of his convictions to ancestral experience, or any other modification of the a priori doctrine, or any treatment of mathematical axioms as philosophical truths. Several modern writers, more distinguished than our author, and especially of late Mr. G. H. Lewes and Mr. H. Spencer, have thus illustrated how a priori too often means no more than ab ignorantia et indolentia. Such writers appear to think that the mathematical deductions of the science are of secondary importance from a philosophical point of view, or are merely illustrative applications of philosophical principles to the processes of nature. But instead of the mathematical body of the science being an appendage to these principles as to an independent body of doctrines, these are themselves chosen and framed, so to speak, or determined in their forms and meanings with reference to the mathematics of a systematic deductive science.