The term Natural Philosophy was used by Newton, and is still used in British Universities, to denote the investigation of laws in the material world, and the deduction of results not directly observed. Observation, classification, and description of phenomena necessarily precede Natural Philosophy in every department of natural science. The earlier stage is, in some branches, commonly called Natural History; and it might with equal propriety be so called in all others. Our object is twofold : to give a tolerably complete account of what is now known of Natural Philosophy, in language adapted to the non-mathematical reader ; and to furnish, to those who have the privilege which high mathematical acquirements confer, a connected outline of the analytical processes by which the greater part of that knowledge has been extended into regions as yet unexplored by experiment. We commence with a chapter on Motion, a subject totally independent of the existence of Matter and Force. In this we are naturally led to the consideration of the curvature and tortuosity of curves, the curvature of surfaces, distortions or strains, and various other purely geometrical subjects. The Laws of Motion, the Law of Gravitation and of Electric and Magnetic Attractions, Hookes Law, and other fundamental principles derived directly from experiment, lead by mathematical processes to interesting and useful results, for the full testing of which our most delicate experimental methods are as yet totally insufficient. A large part of the present volume is devoted to these deductions; which, though not immediately proved by experiment, are as certainly true as the elementary
laws from which mathematical analysis has evolved them. The analytical processes which we have employed are, as a rule, such as lead most directly to the results aimed at, and are therefore in great part unsuited to the general reader.
We adopt the suggestion of Ampere, and use the term Kinematics for the purely geometrical science of motion in the abstract. Keeping in view the proprieties of language, and following the example of the most logical writers, we employ the term Dynamics in its true sense as the science which treats of the action of force, whether it maintains relative rest, or produces acceleration of relative motion. The two corresponding divisions of Dynamics are thus conveniently entitled Statics and Kinetics.
One object which we have constantly kept in view is the grand principle of the Conservation of Energy. According to
modern experimental results, especially those of Joule, Energy is as real and as indestructible as Matter. It is satisfactory to find that Newton anticipated, so far as the state of experimental science in his time permitted him, this magnificent modern generalization.
We desire it to be remarked that in much of our work, where we may appear to have rashly and needlessly interfered with methods and systems of proof in the present day generally accepted, we take the position of Restorers, and not of Innovators. In our introductory chapter on Kinematics, the consideration of Harmonic Motion naturally leads us to Fourier s Theorem, one of the most important of all analytical results as regards usefulness in physical science. In the Appendices to that chapter we have introduced an extension of Greens Theorem, and a
treatise on the remarkable functions known as Laplace's Goefficients. There can be but one opinion as to the beauty and utility of this analysis of Laplace ; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.In the simplified and symmetrical form in which we give it, it will be found quite within the reach of readers moderately familiar with modern mathematical methods. In the second chapter we give Newton's Laws of Motion in his own words, and with some of his own comments—every attempt that has yet been made to supersede them having ended in utter failure. Perhaps nothing so simple, and at the same time so comprehensive, has ever been given as the foundation of a system in any of the sciences. The dynamical use of the Generalized Coordinates of Lagrange, and the Varying
Action of Hamilton, with kindred matter, complete the chapter.
The third chapter,"Experience," treats briefly of Observation and Experiment as the basis of Natural Philosophy. The fourth chapter deals with the fundamental Units, and the chief Instruments used for the measurement of Time, Space, and Force. Thus closes the First Division of the work, which is strictly preliminary, and to which we have limited the present issue. This new edition has been thoroughly revised, and very considerably extended. The more important additions are to be found in the Appendices to the first chapter, especially that devoted to Laplace's Coefficients ; also at the end of the second chapter, where a very full investigation of the " cycloidal motion' of systems is now given; and in Appendix B', which describes a number of continuous calculating machines invented and constructed since the publication of our first edition. Agreat improvement has been made in the treatment of Lagrange's Generalized Equations of Motion. We believe that the mathematical reader will especially profit by a perusal of the large type portion of this volume ; as he will thus be forced to think out for himself what he has
been too often accustomed to reach by a mere mechanical application of analysis. Nothing can be more fatal to progress than a too confident reliance on mathematical symbols ; for the student is only too apt to take the easier course, and consider the formula and not the fact as the physical reality. In issuing this new edition, of a work which has been for several years out of print, we recognise with legitimate satisfaction the very great improvement which has recently taken place in the more elementary works on Dynamics published in this country, and which we cannot but attribute, in great part, to our having effectually recalled to its deserved position Newton's system of elementary definitions, and Laws of Motion. We are much indebted to Mr Burnside and Prof. Chrystal for the pains they have taken in reading proofs and verifying formulas ; and we confidently hope that few erratums of serious
consequence will now be found in the work.
W. THOMSON.
P. G. TAIT.