is spread over four times as much space, so that a portion of B equal in size to A would only be attracted one fourth as much.

It is plain, then, that as the distance from S increases the attraction decreases, and as the distance decreases the attraction increases, showing an inverse ratio. We also see that while the attraction of one of B's squares is four times less than A's, it is only twice as far from 8; hence, to ascertain the diminution of attraction at B, we must square its distance from S compared with A's distance. C is nine times as large as A and three times as far from 8; the attraction of one of its squares will be one ninth of A's.

Since all bodies attract one another we should naturally suppose that any two bodies on the earth's surface would come together, as two books placed upon a table; but the superior attraction of the earth binds them to the table, and this neutralizes their mutual And. attraction. on table

56. Effect of Gravitation on the Planets. It is by the influence of gravitation that the planets are retained in their orbits. Their motion is the same as if they had been projected into space with an impulse, and then continually drawn from the right lines along which inertia tends to carry them by the attraction of the sun. The planets also attract the sun, but their masses being exceedingly small in comparison with that of the sun, their effects in disturbing its position are very small. The orbits of the planets are ellipses, differing but little from circles.

57. The Force of Gravity is that force of attraction which the earth exerts upon all bodies, tending to draw them towards its centre.

As has been stated, it is only a particular case of Universal Gravitation. It is, therefore, subject to the same law, that is, it varies directly as the mass of the body acted upon, and inversely as the square of its distance from the centre of the earth.

The shape of the earth has been shown by careful measurement to be that of a spheroid, that is, of a sphere slightly flattened at the

poles. The mean radius is a little less than 4,000 miles. On account of the flattening of the earth at the poles, different points are at slightly different distances from the centre, and consequently the force of gravity varies slightly at different places on the surface. For ordinary purposes, however, we may regard the earth as a perfect sphere, and the force of gravity as constant all over its surface.

58. Vertical and Horizontal Lines. - A VERTICAL LINE is a line along which a body falls freely. All vertical lines are directed towards the centre of the earth, but for places near together they may be regarded as parallel.

In Fig. 24, the lines ao and bo are vertical, but if they are not far apart, their convergence is so small that they may be taken as parallel. If, however, their distance apart is considerable, they cannot

be regarded as parallel. A man standing erect has his body in a vertical, and it may happen that two persons on opposite sides of the globe, as at E and E', may both stand erect, and yet their heads be turned in exactly opposite directions, their feet being turned towards each other. Points where this may happen are said to be antipodes.

A HORIZONTAL LINE, or PLANE, at any place is one which is perpendicular to a vertical line at that place. The surface of still water is horizontal, or level. For small areas this surface may be regarded as a plane, but when a large surface is considered, as the ocean, it must be regarded as curved, conforming to the general outline of the earth's surface.

Upon the principle of verticals and horizontals all of our instruments for levelling and making astronomical observations are constructed.

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59. Weight. — The WEIGHT of a body is due to the forced of gravity, acting upon all its particles, but it must not be eig confounded with the force of gravity. Weight is only the effect of gravity when resisted; when gravity is unresisted it produces quite another effect, that is, motion.

At the same place the weights of bodies are proportional to their masses, or the quantities of matter which they contain. We shall see hereafter that the weight of bodies may be determined by means of the balance; the force of gravity is determined by the velocity which it can impart to a body in a certain time, as will be shown more fully hereafter.

60. Centre of Gravity. The CENTRE OF GRAVITY of a body is that point through which the direction of its weight always passes.

We have seen that the weight of a body is the resultant of the action of gravity upon all of its particles. Now, whatever may be the form of a body, or whatever its position, the direction of its weight always passes through a single point. This point is the centre of gravity. Hence, in calculations, the weight of a body may be considered as concentrated in the centre of gravity.

The vertical line which passes through the centre of gravity is called the line of direction.

In the case of solids of regular figure and uniform density, the centre of gravity is at the centre of the figure. Thus the centre of gravity of a sphere, a cube, or a regular octahedron, is in each case at the centre. In a cylinder it is at the centre of the axis; in a parallelopipedon, at the intersection of its diagonals; in a pyramid, on its axis at one fourth of its length from the base.

In plates or sheets of uniform thickness and density, the centre of gravity is at the centre of the surface, or rather at the middle of the short line which joins the centres of the opposite surfaces.

When the surface is of irregular outline the position of the centre of gravity may be found in the following way:

Suspend the body by any part of its edge so that it can move freely, and, by means of a plumb-line, mark on it a vertical line from the point of suspension; again suspend it from some other point of the edge and mark the vertical line; the point where these lines intersect will show the centre of gravity.

By a similar method the position of the centre of gravity in any solid body may be determined; for it will always be found at the intersection of any two lines of direction.

In some cases the centre of gravity is not within the substance of the body itself, as, for example, in a ring, a box, or a cask; yet its position may be determined in precisely the same way.

61. Equilibrium of Heavy Bodies.-The centre of gravity being the point at which the weight is applied, it follows that, if this point is held fast by any support whatever, the effect of the weight is completely counteracted, and the body will be in a state of equilibrium.

If a body has but a single point of support, it can be in equilibrium only when its centre of gravity lies somewhere on a vertical through that point.

If a body has but two points of support, it can be in equilibrium only when its centre of gravity lies in a vertical drawn through some point of the line joining these two points. An example is shown in Fig. 25, which represents a man standing on stilts. To be in equilibrium, his centre of gravity must be exactly over the line joining the feet of his stilts.

If a body has three supports not in a straight line, it will be in equilibrium when the centre of gravity lies on a vertical drawn through any point of the triangle formed by joining these points. An example is shown in Fig. 26, which represents a three-legged table. The centre of gravity being at g, the table will be in equilibrium so long as the vertical through that point pierces the triangle formed by uniting the feet of the table.

62. Different Kinds of Equilibrium. - When bodies. are acted upon by the force of gravity alone, and have one or more points of support, three kinds of equilibrium may exist: Stable, Unstable, and Neutral Equilibrium.

1. Stable Equilibrium. - A body is in stable equilibrium when, on being slightly disturbed from its state of rest, it tends of itself to return to that state.

This will be the case when the centre of gravity is lower in its position of rest than it is in any of the neighboring po