Popular Science Monthly/Volume 69/September 1906/The Development of Mechanics

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THE DEVELOPMENT OF MECHANICS
By S. E. SLOCUM, Ph. D.

UNIVERSITY OF CINCINNATI

THE history of mechanics affords a notable instance of what may be called the relativity of science. In the course of its development three distinct sets of mechanical principles have been formulated, each having served in its turn as the foundation of a complete system of mechanics. The first set of principles may be regarded as the first mental image which man formed of the causes underlying the natural motion of material bodies, and, although admirable in many respects, was necessarily somewhat crude and naive. With increased mentality came the formation of a new image, showing a greater maturity of thought than the first and offering a more powerful method of analysis. Finally, in recent times, a third image has been formed, which, although not essentially different in content from the others, exhibits a greater refinement of thought and expression. It is the purpose in what follows to outline briefly these three stages of development, and sketch the chief characteristics of each.

The first scientific development of mechanics arose from investigations concerning the equilibrium and motion of the simple machines in common use, such as the lever, inclined plane and pulley. This order of development was inevitable for the twofold reason that these implements had become familiar by centuries of use, and that they made a direct appeal to the understanding through the grosser and more elementary sensations of weight and pressure. In the second century, b.c., these investigations culminated in Archimedes's famous statement of the principle of the lever, but for seventeen centuries thereafter this statement remained the only instance of correct reasoning on natural phenomena. Apparently human experience did not yet suffice to extend the interpretation of natural law, as witnessed by the Ptolemaic system of astronomy, and Aristotle's division of motions into natural and violent; a classification which served rather to obscure than elucidate the subject.

In the latter part of the fifteenth century a fresh start was made, and the principle of the lever, handed down from Archimedes, was further investigated and generalized by Guido Ubaldi and Leonardo da Vinci. In 1586 these results were extended by Simon Stevin, who, by hanging a string of fourteen balls over a triangular support, established the properties of the inclined plane, and generalized his results by stating the triangle of forces. These pioneers were followed by a host of lesser investigators, and by the middle of the sixteenth century this activity had resulted in the establishment of that branch of mechanics which is now called statics.

The next step was the introduction of the fundamental elements of time and mass in an attempt to investigate the laws of motion. At first little progress was made, as the misconception prevailed that a constant supply of force was necessary to keep a body in motion. Prolonged experiment and investigation, however, gradually resulted in a clearer understanding of these phenomena, and finally led to a correct statement of the first law of motion by the great Italian philosopher Galileo Galilei. Subsequent investigation of the motion of projectiles and falling bodies led Galileo to the two great ideas of inertia and the accelerating action of force, and enabled him to also state the second and third laws of motion. In addition to these great discoveries, Galileo generalized the law of equilibrium by stating the principle of virtual velocities, thus giving the first general solution of all problems in statics.

For the next century the development of mechanics consisted chiefly in an application of the principles of statics to liquids and gases. The only notable advance in mechanical principles during this period was made by Hu} r ghens, who, in connection with his invention of the pendulum clock, investigated the center of oscillation and was thus led to a more general statement of the third law of motion.

The four fundamental ideas of space, time, force and mass were now firmly established, but until the time of Newton found expression only in an inorganic mass of facts and principles. Newton's discovery of gravitation, however, led to such a broad generalization of these ideas as to make possible a systematic treatment of the subject, and mechanics as a science may be said to date from the publication of his famous Principia in 1686. Newton's claim to preeminence, therefore, rests not on the discovery of new mechanical principles, but on the immeasurably greater service of bringing all natural phenomena under the reign of universal law.

Only one element was now lacking to complete the series of independent fundamental statements necessary to constitute the foundation of a complete system of mechanics. There still remained the establishment of a general relation between these fundamental concepts, and after eighty years of experiment and investigation along the lines indicated by Newton, this relation was furnished by d'Alembert in the statement of his famous principle.

This closed the first stage of development. The image was now complete, and henceforth a system of mechanics based on this foundation must be a purely deductive science. The subsequent history of mechanics verifies this statement, for since the time of d'Alembert no essentially new principle has been discovered, and Gauss may be quoted as authority for saying that none ever can be.

The second stage of development was characterized by the elaboration of the system of mechanics formulated by Archimedes, Galileo, Newton and d'Alembert. In the course of this process a new view of the fundamental ideas underlying the subject was attained, which resulted in establishing mechanics upon an entirely different basis. The first step in this direction was made by Euler, and consisted in replacing the geometrical methods of Newton and his predecessors by those of analysis. Euler thus laid the foundation for a system of analytical mechanics which was brought to its perfection by Lagrange in his generalized equations of motion.

This new representation of mechanics was followed by the establishment in the early part of the last century of two great principles; the principle of least action and the principle of the conservation of energy. It is important to note in this connection, however, that each of these principles is deducible from that of d'Alembert, and, consequently, that their establishment did not increase the number of independent fundamental postulates.

The first of these principles dates back to the attempt of Maupertuis to establish on theological grounds a principle of similar nature but of much more limited scope. This attempt, although fruitless in itself, served to direct thought in a new channel, and finally led Gauss to the statement of his 'Principle of Least Constraint.' This in turn led investigators to the idea that all natural phenomena present a maximum or a minimum, and induced Euler and Jacobi to seek expressions whose conditions for a minimum would give the equations of motion. From this it was but a step to the establishment of Hamilton's principle, which consists in the analytical statement that the variations of work and energy vanish for the initial and final configurations. As Hamilton's principle includes both conservative and non-conservative systems, it constitutes a generalization of the principle of least action.

This second principle, like the first, was the product of evolution, as the ideas underlying it had been the subject of investigation from the time of Leibnitz and Descartes. The principle did not assume definite form, however, until the middle of the nineteenth century, when it was stated by several investigators almost simultaneously as the law of the conservation of energy. The names most closely associated with this principle are those of Mayer, Joule and Helmholtz, and it is curious to note that each of these scientists arrived at his results by a different process; Mayer by philosophical reasoning, Joule by experimentation and Helmholtz by mathematical analysis.

The establishment of this law marked the close of the second stage of development. Energy replaced force as a fundamental idea, and a new system of mechanics resulted, founded on the relations between space, time, mass and energy, as embodied in Hamilton's principle.

Although a comparatively short time has elapsed since the establishment of energetics as the basis of mechanics, a third stage of development is already clearly marked. To characterize each stage by a single word, the first may be called constructive, the second deductive and the third, or present stage, critical. To the founders of the first two systems the concepts of force and mass, although more artificial than the intuitive ideas of space and time, were probably no less axiomatic. With the growth of modern scientific criticism, however, came the desire to go back of intuition, if that be possible, and subject the foundations of science to the last analysis. As the result of this tendency the foundations of the first two systems were found open to certain objections, which have been admirably expressed by the late Heinrich Hertz. The chief objection to the first system is in relation to the idea of force, any definition of which seems to involve its author in certain logical difficulties somewhat similar to those encountered in attempting to define a straight line. In the second system, criticism is aimed not at the fundamental concepts, but at the relation between them as expressed in Hamilton's principle, the objections to which are twofold: namely, that it has no simple, natural interpretation, and that it seems to endow matter with the attributes of thought and volition. A further objection is made to both systems on the ground of a certain redundancy in the fundamental ideas, three fundamental concepts being both necessary and sufficient, according to Kirchhoff, for the development of a complete system of mechanics.

In view of these and other objections, Hertz and his followers have outlined an ideal system of mechanics based upon three elements only: namely, space, time and mass. To supplement the deficiency caused by the lack of a fourth element without increasing the number of fundamental concepts, Hertz has introduced the idea of concealed motions acting in connection with those visible to the senses. This idea was originated by Lord Kelvin in his theory of vortex atoms, and was further developed by Maxwell in his attempt to explain electromagnetic action. The first complete treatment of concealed motions, however, was given by Helmholtz, and in the hands of his pupil Hertz it has proved a powerful instrument in establishing mechanics upon a more satisfactory basis.

What the future of mechanics may be it is of course impossible to predict. However, the brief review of its development that has just been given suggests that the foundations have reached bed rock, and that future effort must be directed toward the enlargement of the superstructure and its adaptation to the growing needs of humanity.