The objects of instruction in purely scientific mechanics and physics are, first, to produce in the student that improvement of the understanding which results from the cultivation of natural knowledge, and that elevation of mind which flows from the contemplation of the order of the universe; and secondly, if possible, to qualify him to become a scientific discoverer.

The third and intermediate kind of instruction, which connects the first two... relates to the application of scientific principles to practical purposes. It qualifies the student to plan a structure or a machine for a given purpose, without the necessity of copying some existing example, and to adapt his designs to situations to which no existing example affords a parallel. It enables him to compute the theoretical limit of the strength or stability of a structure, or the efficiency of a machine of a particular kind—to ascertain how far an actual structure or machine fails to attain that limit—to discover the cause of such shortcomings—and to devise improvements for obviating such causes; and it enables him to judge how far an established practical rule is founded on reason, how far on mere custom, and how far on error.

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But in practical science, the question is—What are we to do?—a question which involves the necessity for the immediate adoption of some rule of working. In doubtful cases, we cannot allow our machines and our works of improvement to wait for the advancement of science; and if existing data are insufficient to give an exact solution of the question, that approximate solution must be acted upon which the best data attainable show to be the most probable. A prompt and sound judgment in cases of this kind is one of the characteristics of a Practical Man in the right sense of that term.

Science Of Energetics. Although the mechanical hypothesis just mentioned may be useful and interesting as a means of anticipating laws, and connecting the science of thermodynamics with that of ordinary mechanics, still it is to be remembered that the science of thermodynamics is by no means dependent for its certainty on that or any other hypothesis, having been now reduced, to a system of principles, or general facts, expressing strictly the results of experiment as to the relations between heat and motive power. In this point of view the laws of thermodynamics may be regarded as particular cases of more general laws, applicable to all such states of matter as constitute Energy, or the capacity to perform work, which more general laws form the basis of the science of energetics, — a science comprehending, as special branches, the theories of motion, heat, light, electricity, and all other physical phenomena.

A physical theory, like an abstract science, consists of definitions and axioms as first principles, and of propositions, their consequences; but with these differences:—first, That in an abstract science, a definition assigns a name to a class of notions derived originally from observation, but not necessarily corresponding to any existing objects of real phenomena, and an axiom states a mutual relation amongst such notions, or the names denoting them; while in a physical science, a definition states properties common to a class of existing objects, or real phenomena, and a physical axiom states a general law as to the relations of phenomena; and, secondly,—That in an abstract science, the propositions first discovered are the most simple; whilst in a physical theory, the propositions first discovered are in general numerous and complex, being formal laws, the immediate results of observation and experiment, from which the definitions and axioms are subsequently arrived at by a process of reasoning differing from that whereby one proposition is deduced from another in an abstract science, partly in being more complex and difficult, and partly in being to a certain extent tentative, that is to say, involving the trial of conjectural principles, and their acceptance or rejection according as their consequences are found to agree or disagree with the formal laws deduced immediately from observation and experiment.

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The ill-success of the projects of misdirected ingenuity has very naturally the effect of driving those men of practical skill, who, though without scientific knowledge, possess prudence and common sense, to the opposite extreme of caution, and of inducing them to avoid all experiments, and to confine themselves to the careful copying of successful existing structures and machines; a course which, although it avoids risk, would, if generally followed, stop the progress of all improvement. A similar course has sometimes... been adopted by men possessed of scientific as well as practical skill: such men having, in certain cases, from deference to popular prejudice, or from a dread of being reputed us theorists, considered it advisable to adopt the worse and customary design for a work in preference to a better but unusual design.

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Discrepancy between theory and practice, which in sound physical and mechanical science is a delusion, has a real existence in the minds of men; and that fallacy, through rejected by their judgments, continues to exert and influence over their acts.

[O]f that scientifically practical skill which produces the greatest effect with the least possible expenditure of material and work, the instances are comparatively rare. In too many cases we see the strength and the stability which ought to be given by the skilful arrangement of the parts of a structure supplied by means of clumsy massiveness, and of lavish expenditure of material, labour, and money; and the evil is increased by a perversion of the public taste, which causes works to be admired, not in proportion to their fitness for their purposes, or to the skill evinced in attaining that fitness, but in proportion to their size and cost.

The ascertainment and illustration of truth are the objects; and structures and machines are looked upon merely as natural bodies are; namely, as furnishing experimental data for the ascertaining of principles and examples for their application.

Sir John Herschel's "Outlines of Astronomy"—[is] a work in which one of the most profound mathematicians in the world has succeeded admirably in divesting of all mathematical intricacy the explanation of the principles of that natural science which employs higher mathematics most.

This law (regarding the theoretical efficiency of heat engines by Mr. Joule), and the law of the maximum efficiency of heat engines, are particular cases of a general law which regulates all transformation of energy, and is the basis of the Science of Energetics.

In the original discovery of a proposition of practical utility, by deduction from general principles and from experimental data, a complex algebraical investigation is often not merely useful, but indispensable; but in expounding such a proposition as a part of practical science, and applying it to practical purposes, simplicity is of the importance:—and... the more thoroughly a scientific man has studied higher mathematics, the more fully does he become aware of this truth—and... the better qualified does he become to free the exposition and application of principles from mathematical intricacy.