Meccans

“ To practical Mechanics all the manual arts belong, from which Mechanics took its name. But as artificers do not work with perfect accuracy, it comes to pass that Mechanics is so distinguished from Geometry, that what is perfectly accurate is called Geometrical, what is less so is called Mechanical. But the errors are not in the art, but in the artificers. He that works with less accuracy, is an imperfect Mechanic, and if any could work with perfect accuracy, he would be the most perfect Mechanic of all. ”

“ Size, shape, number, motion, and law are the only and the sufficient principles of explanation. For magnitudes alone are knowable; wherever it is impossible to measure and count, to determine force mathematically, there rigorous, exact science ceases. Nature a system of regularly moved particles of mass; all that takes place mechanical movement, viz., the combination, separation, dislocation, oscillation of bodies and corpuscles; mathematics the organon of natural science! ”

“ 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. ”

“ The greatest generalizations of physical science, such as that forces are proportional to mass and acceleration, are truisms, but, when duly supplemented by specific data, these truisms are the most fruitful sources of useful mechanical knowledge. ”

“ Induction of Force.—The inference of Mechanical Forces from phenomena has been so abundantly practised, that it is perfectly familiar among scientific inquirers. From the time of Newton, it has been the most common aim of mathematicians; and a persuasion has grown up among them, that mechanical forces,—attraction and repulsion,—are the only modes of action of the particles of bodies which we shall ultimately have to consider. ”

“ These subjects are the elements of mechanics; they form the framework which in this and subsequent lectures we shall try to present in a more attractive garb. We shall commence by studying the most remarkable force in nature, a force constantly in action, and one to which all bodies are subject, a force which distance cannot annihilate, and one the properties of which have led to the most sublime discoveries of human intellect. This is the force of gravity. ”

“ Idea of Force.—As the idea of Space is brought out in its full evidence by the study of Geometry, so the idea of Force is called up and developed by the study of the science of Mechanics. It has already been shown, in our scrutiny of the Ideas of the Mechanical Sciences, that Force, the Cause of motion or of equilibrium, involves an independent Fundamental Idea, and is quite incapable of being resolved into any mere modification of our conceptions of space, time, and motion. ”

“ One of these properties is that its particles shall attract one another by a certain force; one of these laws is that this force shall operate by an invariable and fixed rule of direct action, and by an invariable and fixed rule of diminution, according to the distance of the particles from each other; and another law is that a body projected into space, by any force, shall continue to move in a straight line until and unless it is deflected from that line by some other force. ”

“ Therefore Geometry is founded in mechanical practice, and is nothing but that part of universal Mechanics which accurately proposes and demonstrates the art of measuring. But since the manual arts are chiefly conversant in the moving of bodies, it comes to pass that Geometry is commonly referred to their magnitudes, and Mechanics to their motion. In this sense Rational Mechanics will be the science of motions resulting from any forces whatsoever and of the forces required to produce any motions, accurately proposed and demonstrated. ”

“ And if they will please to take any incouragement from so mean and so imperfect endeavours as mine, upon my own experience, I can assure them, without arrogance, That there has not been any inquiry or Problem in Mechanicks, that I have hitherto propounded to my self, but by a certain method (which I may on some other opportunity explain) I have been able presently to examine the possibility of it; and if so, as easily to excogitate divers wayes of performing it: And indeed it is possible to do as much by this method in Mechanicks, as by Algebra can be perform'd in Geometry. ”

“ Longimetry, Altimetry, Levelling, Geodesia, or Surveying, Stereometry, Superficies, and Solids, &c. which see.
Measuring. See Chain.
MECHANICS, a mixed mathematical science, which considers motion and moving powers, their nature and laws, with the effects thereof, in machines, &c. The word is derived from the Greek. That part which considers motion arising from gravity, is sometimes called statics, in contradistinction from that part which considers the mechanical powers and their application, properly called mechanics: it is, in fine, the geometry of motion. ”

“ And behind the weight laid upon the imagination by mass and matter is the perhaps more oppressive weight of custom and convention. “Every body”—so commences Newton’s famous law—“continues in its state of rest or motion in a straight line.” ”

“ It is the structure of the universe that has taught this knowledge to man. That structure is an ever-existing exhibition of every principle upon which every part of mathematical science is founded. The offspring of this science is mechanics; for mechanics is no other than the principles of science applied practically. ”

“ Some mathematicians indeed have maintained that the highest laws of nature with which we are acquainted, the laws of motion and the law of universal gravitation, are not only necessarily true, but are even self-evident and certain à priori, like the truths of geometry. ”

“ As another illustration we may take the case of mechanism and teleology. A system may be defined as "mechanical" when it has a set of determinants that are purely material, such as the positions of certain pieces of matter at certain times. It is an open question whether the world of mind and matter, as we know it, is a mechanical system or not; let us suppose, for the sake of argument, that it is a mechanical system. This supposition—so I contend—throws no light whatever on the question whether the universe is or is not a "teleological" system. ”

“ 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. ”

“ Compounds assumed to be determined by the Mechanical Attraction of the Elements.—When, in consequence of the investigations and discoveries of Newton and his predecessors, the conception of mechanical force had become clear and familiar, so far as the action of external forces upon a body was concerned, it was very natural that the mathematicians who had pursued this train of speculation should attempt to apply the same conception to that mutual action of the internal parts of a body by which they are held together. ”

“ In this lecture I propose to examine some of the forces which exist in our social system, and shall endeavour to estimate them by methods of mathematical procedure and analogical reasoning. We will begin with the old definition of Force as that which puts matter into motion, or which stops, or changes, a motion once commenced. When a mass is in motion, it has a capacity for doing work, which is called Energy ”

“ But what is classification but the perceiving that these objects are not chaotic, and are not foreign, but have a law which is also a law of the human mind? The astronomer discovers that geometry, a pure abstraction of the human mind, is the measure of planetary motion. ”

“ In the phraseology of our mechanics we define matter and force as if they had an independent existence. But we have no conception of inert matter or of disembodied force. All we know of matter is its pressure and its motion. ”

“ Mechanics, like pure mathematics, may proceed from special cases, to problems and methods of extreme generality;—may summon to its aid the curious and refined relations of symmetry, by which general and complex conditions are simplified;—may become more powerful by the discovery of more powerful analytical artifices;—may even have the generality of its principles further expanded, inasmuch as symbols are a more general language than words. ”

“ When I come to speak of the ideas which are involved in our mechanical knowledge, I may, perhaps, be able to bring more clearly into view the necessary truth of general propositions on such subjects. That reaction is equal and opposite to action, is as necessarily true as that two straight lines cannot inclose a space; it is as impossible theoretically to make a perpetual motion by mere mechanism as to make the diagonal of a square commensurable with the side. ”

“ Even the laws of geometry are chiefly of practical importance to us as being a portion of the premises from which the order of the succession of phenomena may be inferred. Inasmuch as the motion of bodies, the action of forces, and the propagation of influences of all sorts, take place in certain lines and over definite spaces, the properties of those lines and spaces are an important part of the laws to which those phenomena are themselves subject. ”

“ After mass and energy there is one physical quantity which plays a very fundamental part in modern physics, known as Action. Action here is a very technical term, and is not to be confused [Pg 153] with Newton’s ‘Action and Reaction.’ In the relativity theory in particular this seems in many respects to be the most fundamental thing of all. ”

“ In cosmical phenomena, everything, in proportion as it is referred to mechanical principles, tends to simplicity,—to permanent uniform forces,—to one common, positive, property. In magnetical and electrical appearances, on the contrary, the application of mechanical principles leads only to a new complexity, which requires a new explanation; and this explanation involves changeable and various forces,—gradations and oppositions of qualities. The doctrine of the universal gravitation of matter is a simple and ultimate truth, in which the mind can acquiesce and repose. ”

“ Büchner. Physics, as I stated to you on another occasion, “makes us acquainted with eight different forces—gravitation, mechanical force, heat, light, electricity, magnetism, affinity, cohesion, which, inseparably united to matter, form and give shape to the world. These [pg 643] forces are, with few exceptions, mutually convertible, so that nothing is lost in the process of conversion” (p. 18) . ”

“ The physical universe has, perhaps, no more general characteristic than this,—its laws are mathematical relations. The law of gravitation, which rules all masses of matter, great or small, heavy or light, at all distances, is a definite numerical law. ”

“ But if a certain clock should be really moved with a weight; we should laugh at a man that would suppose it moved by a spring, and from that principle, suddenly taken up without farther examination, should go about to explain the motion of the index; for certainly the way he ought to have taken should have been, actually to look into the inward parts of the machine, that he might, find the true principle of the proposed motion. ”

“ The Laws of Motion are expressed in terms of Space and Number; the development of the consequences of these laws must, therefore, be performed by means of the reasonings of mathematics; and the science 363 of Mechanics may assume the various aspects which belong to the different modes of dealing with mathematical quantities. Mechanics, like pure mathematics, may be geometrical or may be analytical ”

“ The first four rules are the physics of arithmetic; all beyond is the metaphysics of arithmetic. It flows out of them. Can you comprehend our system of metaphysics until you have clearly and completely mastered our physics? ”

“ Of the physical sciences mechanics was the first to develop in the course of historical evolution. A number of factors contributed to this end—the wide distribution of mechanical phenomena, their significance to human life, and the comparative simplicity of the principles of mechanics, which made it possible to discover them at an early date. Most to be noted is, that of all departments of physics mechanics is the first which lent itself to comprehensive mathematical treatment. ”

“ After working through whole strings of equations, in which the symbols stand for things whose intrinsic nature can never be known to us, he arrives at last at a result which can be interpreted in terms of our own perceptions, and utilized to bring about desired effects in our own lives. What we know about matter, abstract and schematic as it is, is enough, in principle, to tell us the rules according to which it produces perceptions and feelings in ourselves; and it is upon these rules that the practical uses of physics depend. ”

“ For example, there are mechanical laws and equations which admirably explain the facts of motion, yet sentiment is shocked at the attempt to explain Nature on mechanical principles only, and is sustained by common-sense, which sees other facts besides facts of motion, and sees that Nature is not mechanical only. ”