Classical mechanics

“ The mechanical formulas of science are admirably adapted to the purpose for which they were designed, that is, the handling of matter, but they are misleading as applied to living beings, and especially to the human mind, which is the farthest removed from the realm of material mechanics. ”

“ Practical Mechanics is, in the most pre-eminent sense, a scientific art; and it may be truly asserted, that almost all the great combinations of modern mechanism, and many of its refinements and nicer improvements, are creations of pure intellect, grounding its exertion upon a moderate number of very elementary propositions in theoretical mechanics and geometry. ”

“ Sequel to the Generalization of the Principles of Mechanics.—Period of Mathematical Deduction.—Analytical Mechanics. WE have now finished the history of the discovery of Mechanical Principles, strictly so called. The three Laws of Motion, generalized in the manner we have described, contain the materials of the whole structure of Mechanics; and in the remaining progress of the science, we are led to no new truth which was not implicitly involved in those previously known. ”

“ The conceptions of force and matter, of action and reaction, of momentum and inertia, with the reasonings in which they are involved, cannot be evaded by any substitution of lines or symbols for the conceptions. Any attempts at such substitution would render the study of Mechanics useless as a preparation of the mind for physical science; and would, indeed, except counteracted by great natural clearness of thought on such subjects, fill the mind with confused and vague notions, quite unavailing for any purposes of sound reasoning. ”

“ It is perfectly natural that the principle of excluded perpetual motion should have been first developed in the simple domain of pure mechanics. Towards the transference of that principle into the domain of general physics the idea contributed much that all physical phenomena are mechanical phenomena. ”

“ Mechanics owes its power to follow out these postulates to its abstract character. Precisely because it regards only those aspects of the actual physical order which are consistent with the fundamental postulate of describability by general formulæ, Mechanics is constrained to be a purely abstract and hypothetical science. For since every actual process involves the appearance of the qualitatively novel, and since all concrete quality is in its essence unique, no actual process can be merely mechanical. ”

“ A mechanical hypothesis is the better, the more extensive the range of phenomena whose laws it serves to deduce from the axioms of mechanics; and the perfection of such a hypothesis would be, if it could, by means of one connected system of suppositions, be made to form a basis for all branches of molecular physics. ”

“ That this bet would have been a perfectly safe one must be quite evident to any person who has the slightest knowledge of practical mechanics, and yet the fundamental idea which is embodied in this and the other examples which we have just given, forms the basis of almost all the attempts which have been made to produce a perpetual motion by purely mechanical means. ”

“ This is due to the fact that a certain quantity, which is known to us most familiarly as mechanical work, on account of its qualitative transformability and quantitative constancy, can be shown to be a constituent of every physical phenomenon, that is, every phenomenon which belongs to the field of mechanics, physics, and chemistry. ”

“ Nor do we maintain that the application and development of principles is a matter of so much importance to the philosophy of science, as the advance towards and to them. Still, there are many circumstances in the latter stages of the progress of the science of Mechanics, which well deserve notice, and make a rapid survey of that part of its history indispensable to our purpose. ”

“ There is a pure Science of Motion, which does not depend upon observed facts, but upon the Idea of motion. It may also be termed Pure Mechanism, in opposition to Mechanics Proper, or Machinery, which involves the mechanical conceptions of force and matter. ”

“ Many of our illustrations will be designedly drawn from very commonplace sources: by this means I would try to impress upon you that mechanics is not a science that exists in books merely, but that it is a study of those principles which are constantly in action about us. Our own bodies, our houses, our vehicles, all the implements and tools which are in daily use—in fact all objects, natural and artificial, contain illustrations of mechanical principles. ”

“ No genuine advance could ever be obtained in Mechanics by applying to the subject the Ideas of Space and Time merely:—no advance in Chemistry, by the use of mere Mechanical Conceptions:—no discovery in Physiology, by referring facts to mere Chemical and Mechanical Principles. Mechanics must involve the Conception of Force;—Chemistry, the Conception of Elementary Composition;—Physiology, the Conception of Vital Powers. Each science must advance by means of its appropriate Conceptions. Each has its own field, which extends as far as its principles can be applied. ”

“ But although physical measurements become inevitable as soon as we wish to obtain a concrete definition, an objective criterion of equal durations, we are able to proceed in our mechanical deductions in a purely mathematical way without further appeal to experiment. The principle of inertia, together with the other fundamental principles of mechanics, enables us, therefore, to place mechanics on a rigorous mathematical basis, and rational mechanics is the result. ”

“ There is no harm done, for example, in directing attention to mechanical actions, such as that of gravity, the lever, &c., even in departments, notably in physics and in physiology, beyond the range of mechanics proper. ”

“ If the beautiful and skilful structures of the middle ages prove that mechanics then existed as a science, mechanics must have existed as a science also among the builders of the Cyclopean walls of Greece and Italy, or of our own Stonehenge; for the masses which are there piled on each other, could not be raised without considerable mechanical skill. ”

“ These teachers will find nothing more important, in teaching elementary mechanics to their scholars, than to inform them that this mechanics has had its day, that a new mechanics where the notions of mass and of time have a wholly different value replaces it ”

“ Mechanics went on inventing "perpetual motions," but, however plausible they looked, somehow or other they wouldn't work. Do away with friction to the greatest possible extent; overcome one obstacle here and another there; increase the time the machine would run indefinitely—still, to a state of equilibrium would it come at last. From these empirical observations mechanics learned, what philosophers might have learned years ago, that experience affords the only sure test of the validity of our notions. ”

“ And in like manner, the Calculus of Operations comes after Algebra, both in order of evolution and in order of abstractness. In Mechanics, the more concrete relations of forces exhibited in the lever, inclined plane, etc., were understood before the more abstract relations expressed in the laws of resolution and composition of forces; and later than the three abstract laws of motion as formulated by Newton came the still more abstract law of inertia. ”

“ Because the evolution of mechanics antedates that of the other branches of physics, mechanics has largely served as a model for the formal organization of the other physical sciences, just as geometry, which has been handed down to us from antiquity in the very elaborate form of Euclid, has largely been used as a model for scientific work in general. ”

“ The greatest advances of science have always consisted in some successful formulation, in clear, abstract, and communicable terms, of what was instinctively known long before, and of thus making it the permanent property of humanity. By Newton's principle of the equality of pressure and counterpressure, whose truth all before him had felt, but which no predecessor had abstractly formulated, mechanics was placed by a single stroke on a higher level. Our statement might also be historically justified by examples from the scientific labors of Stevinus, S. Carnot, Faraday, J. R. Mayer, and others. ”

“ For the equality of the weights which balance each other in scales, wholly depends upon the equality of the arms: we can know that the weights are equal only by proving that the arms are equal. And when by this means we have obtained a system of weights,—a set of equal units of force, then does a science of mechanics become possible. Whence, indeed, it follows, that rational mechanics could not possibly have any other starting-point than the scales. ”

“ In mechanics, the problems of the composition and decomposition of forces have a physical signification, inasmuch as they presuppose the data of experience; suppress these data and there remains only a composition of lines which mean nothing when we call them forces. Therefore mechanics is only a system of geometrical applications. ”

“ But even to those who, having an acquaintance with mathematics, have by its means acquired a knowledge of mechanics, experimental illustrations may still be useful. By actually seeing the truth of results with which you are theoretically familiar, clearer conceptions may be produced, and perhaps new lines of thought opened up. Besides, many of the mechanical principles which lie rather beyond the scope of elementary works on the [Pg 2] subject are very susceptible of being treated experimentally ”

“ Since it is only complete and all-embracing knowledge which can be in the last resort a completely self-contained and self-explaining system, we must expect to find that the concepts employed in the mechanical interpretation of the physical order lead us into contradiction the moment we try to treat them as a complete account of the concrete nature of the whole of Reality. This is shown more particularly in two ways. On the one hand, the application of the categories of Mechanics to the whole of Reality leads inevitably to the indefinite regress. ”

“ And so, too, is it with the projectile: determination of its course by deduction from mechanical laws, primarily ignores all deviations caused by its shape and by the resistance of the air. The science of rational mechanics is a science which consists of such ideal truths, and can come into existence only by thus dealing with ideal cases. It remains impossible so long as attention is restricted to concrete cases presenting all the complications of friction, plasticity and so forth. ”

“ In his Preface, Newton says, “Rational Mechanics must be the science of the Motions which result from any Forces, and of the Forces which are required for any Motions, accurately propounded and demonstrated. For many things induce me to suspect, that all natural phenomena may depend upon some Forces by which the particles of bodies are either drawn towards each other, and cohere, or repel and recede from each other: and these Forces being hitherto unknown, philosophers have pursued their researches in vain. ”

“ Just as the cord in our experiments cannot be straight when any force, however small, is pulling it downwards at the centre, so it is impossible by any exertion of force to straighten the long wire. Some further illustrations of this principle will be given in our next lecture, and with one application of it the present will be concluded.
21. One of the most important practical problems in mechanics is to make a small force overcome a greater. ”

“ The science of mechanics presupposes matter, weight, impenetrability, the possibility of communicating motion by impact, inertia and so forth as ultimate facts, calls them forces of nature, and their necessary and orderly appearance under certain conditions a law of nature. Only after this does its explanation begin, and it consists in indicating truly and with mathematical exactness, how, where and when each force manifests itself, and in referring every phenomenon which presents itself to the operation of one of these forces. ”

“ Mechanics deals with the laws of energy and motion, physics with molecular changes of matter, but neither physics nor mechanics would have gone far, had they stopped to answer the questions as to what motion, energy, matter are in themselves. ”

“ For long before the development of mechanics the conviction of its truth existed and even contributed to that development. Its power of conviction, therefore, must have more universal and deeper roots. We shall revert to this point.
II. MECHANICAL PHYSICS.
It cannot be denied that an unmistakable tendency has prevailed, from Democritus to the present day, to explain all physical events mechanically. Not to mention earlier obscure expressions of that tendency we read in Huygens the following: [51]
"There can be no doubt that light consists of the motion of a certain substance. ”

“ That the dynamical equilibrium of the systems in the stationary stages is governed by the ordinary laws of mechanics, while these laws do not hold for the passing of the systems between the different stationary states. ”

“ Statics and Dynamics.—We must first turn our attention to a technical distinction of Mechanics into two portions, according as the forces about which we reason produce rest, or motion; the former portion is termed Statics, the latter Dynamics. If a stone fall, or a weight put a machine in motion, the problem belongs to Dynamics; but if the stone rest upon the ground, or a weight be merely supported by a machine, without being raised higher, the question is one of Statics.
3. Equilibrium.—In Statics, forces balance each other, or keep each other in equilibrium. ”

“ Notwithstanding, however, the superior ingenuity of modern artists, in mechanics, which these scientific inventions discover, it seems to be a thing absolutely impossible, that any piece of mechanism should be invented, which, possessing perfect mechanical motion, should appear to exert the intelligence of a reasoning agent. ”