cular energy, is analogous to that of the magnets of the dynamo; being unable to produce energy in the smallest quantity, but able to direct its transformation in one way rather than in another.* This, however, appears a very unsatisfactory analogy. The static force of the magnets belongs to the same order of being with the current of electricity, being related to it somewhat as pressure to motion; while Will is not a physical force, but is of another order of being from matter and its forces.

Another possible reply is, that the Will may determine the time and manner of the transformation of energy, somewhat in the same way that, in mechanism, a very small force is able to guide the action of a very great one. For instance, the steam-engines which propel a large ship, though they work up to several thousand horse-power, can be started or stopped by the will of the engineer moving a lever with the exertion of an amount of muscular force almost infinitesimally smaller than that of the engines which he controls. And it would be possible indefinitely to diminish the muscular power needed, until the gentlest finger-touch on an electric button was sufficient to control the most powerful engines. In such arrangements there is no relation whatever between the magnitudes of the controlling and the controlled forces ;the magnitude of the controlled force may be indefinitely increased, while that of the controlling force remains unchanged. Is not this a significant symbol of the control of Will over the muscular forces?

But in reply to this, it is urged that the analogy altogether fails unless it were possible for the will of the engineer to control the engines without the exertion of muscular power at all; and however this may be diminished by refinement of mechanism, it can never be reduced absolutely to nothing.

Sir John Herschel saw the difficulty, and appears to have concluded that the Will can and does produce energy, though. in quantity so minute as to be incapable of experimental proof. This is cutting the knot rather than untying it.

*We think this suggestion, though quite differently expressed, is fundamentally identical with one made on the same subject in an article on Atomic Theories in the North British Review, March, 1868, by the late Prof. Fleeming Jenkin, and now published in the collected edition of his papers. + We quote from Sir John Herschel's Familiar Lectures on Scientific Subjects (Strahan, 1866), page 468.

"The actual force necessary to be originated to give rise to the utmost imaginable exertion of animal power in any case, may be no greater than is required to remove a single material molecule from its place through a space inconceivably minute, no more in comparison with the dynamical force disengaged, directly or indirectly, by the act, than the pull of a hair trigger

A French writer of our time,-Professor Armand Sabatier, of Montpellier,-has proposed to cut the knot in another way, by denying the absolute uniformity of the order of Nature. He admits, of course, that all motions on the largest scale, that is to say, those of the celestial bodies, and indeed of all masses which are visible to the unassisted eye, are absolutely determined; but he maintains that this is not true of those molecular motions which modern science has proved to exist everywhere; and, as he truly remarks, it is not in the greatest but in the minutest motions that the nature of matter is in any degree revealed to us. Light consists of undulations in an ethereal substance, moving, so long as the light is not polarised, in every plane at right angles to the direction of the ray; and the heat of bodies consists of vibrations of their molecules, moving, no doubt, in every direction at once. These motions are, in M. Sabatier's opinion, in some degree undetermined, and not subject to any rigid law of uniformity; and he finds traces of the same indeterminism in some motions which are on a sufficiently large scale to be visible under the microscope. One instance of this which he mentions is that of the "Brownian" motions of minute particles suspended in water or other liquids.† These movements are of very small amplitude; but incessant, of quite sensible rapidity, and in every direction at once. They are well seen in ink when it is sufficiently thick to make them visible, and it is these motions which prevent ink from losing its properties as such by the subsidence of the black particles.

We cannot think there is any truth in Sabatier's hypothesis as regards inorganic nature. To say that the minutest motions are undetermined, is to say that below a certain limit of magnitude the laws of motion cease to be absolutely true.

in comparison with the force of the mine which it explodes. But without the power to make some material disposition, to originate some movement, or to change, at least temporarily, the amount of dynamical force appropriate to some one or more material molecules, the mechanical results of human or animal volition are inconceivable. It matters not that we are ignorant of the mode in which this is performed. It suffices to bring the origination of dynamical power, to however small an extent, within the domain of acknowledged personality.'

* In a series of articles entitled Evolution et Liberté, in the Revue Chrétienne, of April, May, September, and October, 1885.

+ So named after the eminent botanist, Robert Brown, who first called attention to their importance. Professor Jevons (Quarterly Journal of Science, April, 1878), offers what appears to be a satisfactory explanation of these motions as being due to minute disturbances of electric equilibrium. They are precisely analogous to the motions of pith balls in a well-known electric experiment.

Now, the laws of motion are perfectly simple; though not mathematical in the nature of their evidence,-for they are proved only by experiment, and have not that self-evidencing character which belongs to mathematical truth,-yet they are mathematical in form; though the proof that they are absolutely true is never perfectly complete, yet every increase in the accuracy and perfection of astronomical knowledge brings us nearer to such absolute proof; and it seems extremely improbable that they should be subject to any limit whatever. The Brownian motions, the motions of the molecules of gases, the undulatory motion which constitutes light,-all these, however minute, are motions, and we cannot doubt that they are rigidly subject to the laws of motion. It is uncertain how far chemical actions can be resolved into the motions of atoms, accompanied in many cases by transformations of energy, as in the case of heat produced by combustion; but the law of the absolute invariability of chemical properties and actions,—the proof of which, it is true, can never be complete, though every increase of chemical knowledge. strengthens it,-makes it probable, with a probability approaching indefinitely near to certainty, that the laws of chemical action admit of no more limitation or exception than the laws of motion. We consequently hold with scientific men generally, that all inorganic actions, on whatever scale of magnitude, whether planetary or atomic, are determined by the laws of motion with a certainty which, though not mathematical in its nature, is equal to mathematical certainty.

An

But is absolute, determinism true in mathematics? attempt has been made by Professor Boussinesq, of Lille, to show that this is not the fact ;*-that absolute determinism, though generally true in mathematics, is not always so, and therefore is not necessarily always true in mechanics. He chiefly makes use in his argument of what are called singular solutions. We must here state when and how a singular solution arises, for the term is by no means self-explaining.

A set of curves are drawn which we shall call C, C', C", &c. They are not in general mathematically similar, but constitute a family, varying continuously from curve to curve according to a definite law. They are indefinite in number and indefinitely near to each other, and are so drawn that C intersects with C', C' with C", and so on.

A curve S, which is generally of a totally distinct kind from the curves C, is drawn through these intersections; and the

* See Paul Janet's article in the Contemporary Review, June, 1878

curves C, at the points where they intersect each other, are tangential to S; that is to say, they touch it without intersecting it; so that the relation of S to the curves C is somewhat like that of a circle to its tangents. S is called the envelope of the curves C, and it is "singular," that is to say unique, and not one of a family like the curves C.

The following diagram will give an idea of the relation of the curves C to S.

Every line, straight or curved, may be described as produced by the motion of a point P,-this is actually the case when it is drawn by a pencil,—and consequently the equation which describes the direction of a curve at any place may also be read as describing the direction of the motion of P at that place. Equations usually speak a perfectly unambiguous language, but in singular solutions an exception arises; the equation which describes the direction of motion at that point of any C where it touches S will be equally satisfied by P either continuing to move along its C, or at that point leaving the C and moving along S. So that the equation which describes the direction of the motion of P at any point of S does not absolutely determine its path, but leaves undetermined which of two paths it is to take; those paths being along curves of unlike kinds.

1

Where there is thus mechanical indetermination, there is, or may be, room for voluntary determination to enter. An agency like the Will, which is not properly a force inasmuch as it cannot exert energy, may nevertheless determine the motion of a point along one of these two curves rather than the other. It is no objection to this that the indetermination shown in a singular solution cannot be realised under experimental conditions. It is impossible to do this, just as it is impossible to make a cone stand on its apex. But it does not seem by any means impossible that it may be realised among

molecular or atomic actions; and the actions in those dim. recesses of the brain where alone the Will acts on matter can be only an atomic scale.

This argument appears to us of much importance; as showing that absolute determinism is not a mathematical truth. But we do not think it throws any light on the actual modus operandi of Freedom. The processes of life are not mechanical, and its laws are not resultants from the physical and chemical properties of the substances of which the organism is composed. Even if all physiological processes could be referred to chemical laws, this would not be true of the morphological processes which build up tissues and organs; and though it might conceivably be true that the law of Habit, in virtue of which every action tends to become easier with repetition and to repeat itself, was a merely physical law like that whereby "streams their channels deeper wear ;"* yet the law of heredity, whereby habits and tendencies of all kinds, both active and formative, tend to be reproduced in the offspring, cannot be merely physical and mechanical. In all life, even the merely organic life of vegetables, there is something as, absolutely inscrutable as the ultimate properties of matter; and it seems to us probable, though not capable of demonstration, that Sabatier's theory of a certain limited indeterminism, though untrue of inorganic matter, is true of organic nature. As he reminds us, we do not there find, either in form or in function, the rigid mathematical uniformity of inorganic nature. Variation, though generally very slight, is universal; no two trees in a forest, no two leaves on a tree, are exactly alike; the same is true probably of the physiological processes of all organisms, and certainly of the muscular actions of animals; and even if Darwin's theory of the origin of all organic forms by natural selection among spontaneous variations is unsatisfactory and insufficient, he has at least made it obvious that it is this fact of variability,-so utterly unlike any property of any part of inorganic nature,-which makes the evolution of organic forms possible. It is asserted, no doubt, by those with whom absolute determinism is an article of scientific faith, that organic variations are absolutely determined, partly by differences and changes in the environment of the organism, and partly by the laws of its development. This may be true. It is at present, and may ever remain, impossible to prove either absolute determinism or a certain limited indeterminism in the organic world. Sabatier only insists that his opinion

* "Time but the impression stronger makes,

As streams their channels deeper wear."--Burns.