BOOK I. TRANSLATIONS.

CHAPTER I. STEPS.

INTRODUCTION.

JUST as Geometry teaches us about the sizes and shapes and distances of bodies, and about the relations which hold good between them, so Dynamic teaches us about the changes which take place in those distances, sizes, and shapes (which changes are called motions), the relations which hold good between different motions, and the circumstances under which motions take place.

Motions are generally very complicated. To fix the ideas, consider the case of a man sitting in one corner of a railway carriage, who gets up and moves to the opposite corner. He has gone from one place to another; he has turned round; he has continually changed in shape, and many of his muscles have changed in size during the process.

To avoid this complication we deal with the simplest motions first, and gradually go on to consider the more complex ones. In the first place we postpone the consideration of changes in size and shape by treating only of those motions in which there are no such changes. A body which does not change its size or shape during the time considered is called a rigid body.

The motion of rigid bodies is of two kinds; change of place, or translation, and change of direction or aspect, which is called rotation. In a motion of pure translation, every straight line in the body remains parallel to its original position; for if it did not, it would turn round,

and there would be a motion of rotation mixed up with the motion of translation. By a straight line in the body we do not mean merely a straight line indicated by the shape or marked upon the surface of the body; thus if a box have a movement of translation, not only will its edges remain parallel to their original positions, but the same will be true of every straight line which we can conceive to be drawn joining any two points of the box.

When a body has a motion of translation it is found that every point of it moves in the same way; so that to describe the motion of the whole body it is sufficient to describe that of one point. When a body is so small that there is no need to take account of the differences in position and motion of its different parts, the body is called a particle. Thus the only motion of a particle that we take account of is the motion of translation of any point in it.

A motion of translation mixed up with a motion of rotation is like that of a corkscrew entering into a cork, and is called a twist.

Bodies which change their size or shape are called elastic bodies. Changes in size or shape are called strains.

The science which teaches how to describe motion accurately, and how to compound different motions together, is called Kinematic (Kivnμa, motion). We may conveniently reckon three branches of it, namely,

(Points or particles (Translations).

Kinematic of Rigid Bodies (Rotations and Twists). (Elastic Bodies (Strains).

It is found that the change of motion of any body depends partly on the position of distant bodies and partly on the strain of contiguous bodies. Considered as so depending, the rate of change of motion is called force; and the law just stated, expressing the circumstances under which motions change, is called the law of force.

The science which teaches how to calculate motions in accordance with the law of force is called Dynamic (Svvaμis, force). It is divided into two parts: Static, which treats of those circumstances under which rest or null motion is possible, and Kinetic, which treats of cir