Elementary statics - Goodwin

The following treatise is not a mere reprint, but may be regarded as a new and modified edition of my Elementary Treatise on Statics. The structure of the book has been considerably changed, and I have given up, in deference to the opinion of several friends, the conversational form which I adopted in my former work for purposes of explanation. Doubtless there were disadvantages attaching to the method which I then employed but I am bound to say, that in abandoning it I have felt that there were also advantages for the loss of which I am not sure that I have been able to compensate.

I have adhered to the division of the subject into experimental and demonstrative Mechanics; I believe that this mode of treatment is very advantageous, and that even those students who are intending to proceed to the higher branches of Mechanics and to higher modes of treatment will find considerable advantage in regarding the elementary principles as in a certain sense capable of experimental demonstration, and in knowing mechanical truths not only as the results of mathematical reasoning, but as truths familiarly exhibited to the eye. The excellent apparatus devised by Professor Willis, and referred to in the note of p. 11, makes the experimental treatment of the subject possible and easy for every Lecture-Boom; and I think that the duty of dealing with the science of Mechanics in this way, of course in subordination to a genuine mathematical treatment, cannot be pressed too strongly upon all teachers and lecturers.

1. In the following elementary treatise it will be assumed that the reader is acquainted with Plane Geometry as found in Euclid's Elements, with Algebra, and with Plane Trigonometry. It would be possible to gain some knowledge of Mechanics without having first studied the latter two branches of pure Mathematics, but it is so very much to the advantage of the student to postpone his entrance upon mixed Mathematics until he has become familiar with the elements of the subjects just mentioned, that I shall assume him to have adopted this order of study.

2. Having spoken of pure and mixed Mathematics, let me explain what is meant by the terms, and point out the distinction between these two great branches of science.

Pure Mathematics are concerned with the conceptions of space and number, or space and quantity. Geometry is the science of space. Algebra that of number or quantity. Trigonometry, as usually treated, combines both space and   quantity. All the higher branches of pure Mathematics involve the same conceptions, and may in fact be classed either as Geometrical or Algebraical, or as partly Geometrical and partly Algebraical.

Mixed Mathematics is the term used to include all those sciences, in which mathematical reasoning is employed upon conceptions other than those of space and number. For example: Mechanics, or the science of Force; Optics; Hydrostatics; Astronomy.

3. The superior difficulty of this latter class of sciences will be perceived at once. Geometry is made to depend upon definitions and axioms, the meaning and the truth of which are easily seen. There is no difficulty in understanding what is meant by a triangle, a circle, a parallelogram; the mind easily grants that the whole is greater than its part, that if equals be added to equals the wholes will be equal, that two straight lines cannot include a space, and so on. The definitions and axioms of number are more simple still. But when we wish to reduce the science of force to mathematical calculation, it is not so easy to devise definitions and axioms upon which we can reason: what is force? how is its effect to be measured? what principles can we lay down, as the ground upon which our reasoning can be conducted?

4. It will be the purpose of the following pages to answer these questions; that is, to shew how mechanical problems, or problems involving the conception of force, can be reduced to mathematical reasoning. Let me introduce the subject by observing that the science of mechanics divides itself into two great branches. If I throw a ball into the air, it is a mechanical problem to find the path which the ball will describe, the time which will elapse before it strikes the ground, the manner in which it will rebound after striking, the path which it will describe after the rebound, and so on. This would be called a dynamical problem, and belongs to the more difficult branch of the subject. There are much easier problems: for example; I take a steelyard and suspend a weight from its shorter arm, and another weight from its longer arm, and it is a mechanical problem to find the proportion of the weights when the arms are given, or the proportion of the arms when the weights are given. This would be called a statical problem. The fundamental distinction is, that in the former case we have motion, in the latter we have none.

5. In the present treatise we shall be concerned only with the simpler branch of Mechanics, which is called Statics: that is, we shall be concerned with problems, in which the forces are so balanced one against another that they do not produce any motion.

It will be observed, however, that although in those problems with which we shall be engaged there be no motion, there is a tendency to motion. A book upon a table does not move, because it is supported by the table; but it has a tendency to move, and would move, if the table were taken away. If I push against a heavy block of stone, I shall probably not move it, because the friction will be so great as to prevent it from moving; but it has a tendency to move, and would move, if I were strong enough to move it. Hence we define force by reference to motion, or tendency to motion; and we say that,

Any cause which produces or tends to produce motion in a body is called force,

This is the formal definition of force, and applies equally to Statics and Dynamics. Now let me say a few words upon the meaning of the term body, which occurs in the definition.