The cycloid is remarkable as being that path, with the exception of the perpendicular, through which a body will move with the greatest velocity; suppose, for example, a body is to descend from any one point to any other, by means of some force acting on it, together with its weight: a person unacquainted with mechanics would say at once, that a straight line is the path it must take to effect this in the shortest possible time, since that is the shortest of all lines that can be drawn between two points. Undoubtedly it is the shortest; notwithstanding which, however, the body would be longer in traversing it, than in moving through a cycloid. If a body were to move through a space of fifty or a hundred yards, by its weight and some other force acting together, the way it must take to do this in the shortest possible time is by moving in a cycloid. It is supposed that birds which build in the rocks possess an instinctive knowledge of this fact, and drop or fly down from height to height in this course. There is certainly a general resemblance between the curved path they describe on such occasions, and the cycloid, but it would be difficult to establish the fact by experiment. Man, however, has founded upon this principle some applications of great value in practical mechanics. In Switzerland, and in several parts of Germany, for example, slides have been constructed along the sides of mountains, by which the timber felled near their summits is conducted with extreme rapidity to the distant valleys.
Note 25, p. [171].--Billiards.
This interesting game is of French origin (billiard, of bile, and from the Latin, pila, a ball). It was hailed as a favourite diversion at the court of Henry III. of France; and was thence communicated to all the courts of modern Europe. To the novice it may appear as a game of accidents and chances, but experience has enabled us to determine the effects of the stroke given to a ball with wonderful precision; and it is quite extraordinary to observe the accuracy with which an accomplished player can effect his object, by measuring with his eye the angle at which he should make the stroke, the position of the ball with respect to the cushion, and the distance of the point of the ball from its centre, at which it should be struck. By such skilful management the ball may be made to take directions which would, at first view, be regarded as contrary to all the known laws of motion, such, for instance, as passing round an object, such as a hat placed on the table, and to strike a ball behind it into a pocket.
Upon this subject the reader should consult a work by M. Mingaud, which has been translated and published by John Thurston, the celebrated billiard-table maker of Catherine Street, Strand. We understand that a still more complete work may be expected from the same source.
Note 26, p. [172].--Collision of bodies.
In investigating the effects produced upon bodies by collision, it is necessary to distinguish between elastic and non-elastic substances, since their motions after impact are governed by very different laws.
If two bodies, void of elasticity, move in one right line, either the same or contrary ways, so that one body may strike directly against another, let the sum of their motions before the stroke, if they move the same way, and the difference of their motions, if contrary ways, be divided into two such parts as are proportional to the quantities of matter in the bodies, and each of those parts will respectively exhibit the motion of each body after the stroke: for example, if the quantities of matter in the bodies be as two to one, and their motions before the stroke as five and four, then the sum of their motions is nine, and the difference is one; and therefore, when they move the same way, the motion of that body, which is as two, will, after the stroke, be six, and the motion of the other, three; but, if they move in contrary directions, the motion of the greater body after the stroke will be two-thirds of one, and of the lesser body one-third of one; for, since the bodies are void of elasticity, they will not separate after the stroke, but move together with one and the same velocity; and, consequently, their motions will be proportional to their quantities of matter; and it follows from the fact of action and reaction being equal, that no motion is either lost or gained by the stroke when the bodies move the same way; because, whatever motion one body imparts to the other, so much must it lose of its own; and, consequently, the sum of their motions before the stroke is neither increased nor diminished by the stroke, but is so divided between the bodies, as that they may move together with one common velocity; that is, it is divided between the bodies in proportion to their quantities of matter: but it is otherwise, where the bodies move in opposite directions, or contrary ways, for then the smaller motion will be destroyed by the stroke, as also an equal quantity of the greater motion, because action and reaction are equal; and the bodies, after the stroke, will move together equally swift, with the difference only of their motions before the stroke; consequently, that difference is, by means of the stroke, divided between them in proportion to their quantities of matter.
The several particular cases, concerning the collision of bodies, may be reduced to four general ones; viz.
1st. It may be, that one body only is in motion at the time of the stroke.
2nd. They may both move one and the same way.