Jonas: Seventeenth Century and After… (V)

V

All this is far from obvious. In fact, all appearances are on the side of the opposite, Aristotelian view. In our common experience, bodies do come to rest when the force propelling them ceases to act: the wagon does stop moving when no longer pulled or pushed; and the pulling or pushing, when done by us, is felt to produce the motion from moment to moment. Nor is there anything obvious about a circular motion not being a simple, unitary act. The Galilean revolution has this in common with the Copernican revolution that it replaces the testimony of the senses with an abstraction that directly contradicts but indirectly grounds it. Such an abstraction — and a highly artificial one — is the concept of a quantity of motion as an invisible object, treated like an abiding unit that can be combined with other such units, added to or subtracted from them, much as static, numerable entities can. It leads directly to the cardinal, no less artificial, concept of composite motions whose ultimate elements are simple motions, viz., uniform rectilinear motions. In particular, an accelerated motion (e.g., in free fall) is the cumulative sum of continuous increments of motion produced by the continuous operation of a force (e.g., gravity) which is at any moment a new cause (as to Aristotle   was the force causing the mere continuation of a motion). And likewise, any curvilinear motion is a composite of at least two motions, a tangential and a radial, of which the first represents mere persistence in a state, and the second the continuous incremental input of new force — i.e., again "acceleration" (even if the resultant translatory velocity is uniform). Furthermore, the effect of any "single" input of force, once its work is done, can only be uniform rectilinear motion in which the increment and the antecedent are fused (as after the impact of one billiard ball on another). Or, the motion of a body at any one, infinitesimal moment of its progress is uniform rectilinear motion, and so the total (e.g., parabolic) trajectory can be conceived as the composite of an infinite number of "simple" (tangential) motions with varying values. Its "law" is then the law of the variation, which can be computed from the forces at work — as the forces in turn can be inferred from the geometrical properties of the path.

Finally, in a last step of abstraction, the duality velocity-direction which defines motion, when considered with respect to force and the changes produced by it, reduces to one single datum vis-a-vis the concept of acceleration to which in turn all possible changes of this datum reduce: increase as well as decrease of velocity, as well as any change of direction, can equally be represented as different values of "acceleration." And velocity (absorbing "direction" into its concept) comprises rest among its possible values simply as the zero value: the concept of "inertia" extends as a constant magnitude over all those values. All this is set in a neutral, homogeneous space continuum which knows of no privileged directions, and it concerns bodies with no preference for specific places, differing — with respect to mechanics — only in certain quantitatively specifiable magnitudes.

This novel conceptual scheme — whose novelty and boldness cannot be overstated — was clearly one grand prescription for the mathematical analysis and synthesis of motions (what was first called the "resolutive" and the "compositive" methods). Motions, being resultants, could be resolved into their simple components and, vice versa, constructed from them. Three important developments promoted by the new conceptualization should be noted here.

The first is the geometrizing of nature and consequently the mathematization of physics. Kepler, Galileo, and Descartes   were equally convinced (though for different ostensible reasons) that geometry is the true language of nature and must therefore also be the method of its investigation, which is to decode its sensuous message. This growing conviction was raised by Descartes   to the dignity of a metaphysical principle when he split reality into the two mutually exclusive realms of the res cogitans and the res extensa — the world of mind and the world of matter: the latter is in its essence nothing but "extension"; therefore nothing but determinations of extension, i.e., geometry, are required for a scientific knowledge of the external world. (The defect of this overstatement was that it left no room for the concept of energy, which is an "intensive" rather than a purely "extensive" term: Leibniz   and others set out to remedy this defect of Cartesian extremism.)

Secondly the program of an analysis of motions necessitated a new mathematics, of which Descartes  ’ analytical geometry was only the first step. The reduction of a complex motion to simple motions involves, as we have seen, breaking it down to infinitesimal portions (any curvilinear motion being a composite of an infinite number of tangential motions which, of course, must be conceived as infinitesimal): the answer to the mathematical task thereby posed was the infinitesimal calculus, invented simultaneously (or almost so) by Leibniz   and Newton  .

Thirdly, the conceptual analysis of motions permitted an actual dissociation of its component parts in suitably set up experiments: it thus inspired an entirely new method of discovery and verification, the experimental method. It must be realized that the controlled experiment, in which an artificially simplified nature is set to work so as to display the action of single factors, is toto coelo different from the observation, however attentive, of "natural" nature in its unprocessed complexity, and also from any non-analytical trying-out of its responses to our probing interventions. It essentially differs, in one word, from experience as such. What the experiment aims at — the isolation of factors and their quantification — and is designed to secure by the selective arrangement of conditions, presupposes the theoretical analytic we have described; and it repays theory by its results. Galileo’s inclined plane, which made the vertical component in the motion of the balls clearly distinguishable from the horizontal, is a classical example of such analytic experiment.