NOTE (A.) REFERRED TO AT PAGE 7.

E

D

IMAGINE the ends of a thread to be fastened at F and S, two points in any plane surface, and the length of the thread to be greater than the distance between the points: And let the point of a pencil be put in the doubling or bight of the thread at E, so that its parts between the pencil and the points to which

A

S

F

G

its ends are fastened may be stretched into straight lines ES, and EF. Suppose now the point of the pencil to be moved along the plane, while it keeps the thread thus extended, -it will trace a curve line AEDPG, which is called an ellipse.

Each of the points F and S to which the ends of the thread are fastened, is called a focus of the ellipse; and C, the middle of the straight line FS, which joins the foci, is called its centre. The distance between either focus and the centre, is called the eccentricity of the ellipse.

The straight line AP which passes through the foci and centre, and is terminated by the ellipse, is called the transverse or greater axis, and the straight line DG, which, passing through the centre, is perpendicular to the greater axis, and is terminated by the ellipse, is called the conjugate or lesser axis.

From the way in which an ellipse is generated, it has evidently this distinguishing property; the sum of straight lines drawn from any point in the curve to the foci, is always a line of the same magnitude, and equal to the transverse axis.

The path of a planet in space being an ellipse, (as has been explained in page 7,) having the sun in S, one of its foci, the point P, the extremity of the greater axis nearest to the sun, is called the perihelion, and A, the other extremity, is called the aphelion. The axis itself is sometimes called the line of the apsides.

Half the greater axis of the elliptic orbit is evidently a mean between the aphelion and perihelion distances, that is, it is as much greater than the one, as it is less than the other; and since a line drawn from either extremity of the lesser axis to either focus is equal to half the greater axis, it follows, that when a planet is at D or G, either extremity of the lesser axis, it is then at its mean distance from the sun. The mean distances of all the planets have been given in the Tabular View of the Solar System fronting page 4.

An imaginary line S e, supposed to join the centres of the sun and any planet, is called the radius vector. This line has the remarkable property of always sweeping over equal areas in equal times, while the planet moves in its orbit round the sun.

NOTE (B.) REFERRED TO AT PAGE 10.

THERE is nothing with which we are more familiar, in their effects at least, than the laws of motion; nor does any phenomenon in nature present itself more frequently to our observation, than the gravitation of matter. It is by these conjointly that the planetary motions are produced; and although this subject has engaged minds of the highest order, and, considered in its full extent, has required the greatest effort of human thought, without, after all, being completely brought under its dominion; yet some of its more simple doctrines are not difficult to be understood; at least they admit of illustrations which may satisfy to a certain extent that incipient thirst for knowledge, which it is most desirable to excite in the minds of youth.

Every body, when left entirely to itself, and without any support, descends towards the Earth's surface; and setting aside the resistance of the air, all bodies, whatever be their shape, and of whatever matter composed, if left perfectly free, descend equally quick with an increasing velocity. This phenomenon, so common as hardly to draw the attention of the vulgar, is yet full of instruction. It is a law of motion established by experience, "that a body perseveres in its state of rest, or of uniform motion in a straight line, till by some external influence it be made to change that state." The change in the state of a body from rest to motion proves, that it is urged towards the Earth by some cause, the nature of which is entirely unknown, otherwise than by the effect it produces: this force gives the body weight, and hence we denominate it gravity.

Farther, we observe that the quickness with which the body descends, which is called its velocity, increases uniformly; that is, it receives equal increments or augmentations in equal times. Hence we conclude that the force has always the same intensity at the same place of the Earth; in this respect it is said to be constant, and in respect of its producing equal changes in the velocity in equal times, it is called an uniformly accelerating force.

The two facts which are constantly under our view, namely, that any body, when put in motion, continues to move uniformly in a straight line, unless it be affected by some external force, and that every particle of matter is urged. downwards in the direction of a perpendicular to the Earth's surface by a force which constitutes what we call its weight, -serve to explain the nature of the motion which is produced when a body is by any means projected obliquely. into the air. In this case, were it not for the force of gravity, the body would have continued its motion uniformly in a straight line with the velocity communicated to it by the propelling power. It is, however, constantly drawn aside, or deflected, from that line by the force of gravity, which acts independently of the propelling force, just as if the body had fallen from a state of rest. Thus there are produced in the body two distinct tendencies to motion, one the uniform rectilineal motion with which it was projected, and the other the uniformly accelerated motion downwards, produced by the action of gravity. These by their union produce that motion in a curve line with which all are familiar, but which it required the penetrating mind of a Galileo fully to explain. He shewed that the path of a body projected obliquely into the air would, if it met with no resistance from that medium, be a parabola, a curve, which does not, like the ellipse, return into itself, but which goes off continually into two branches, thus forming a space open on one side, but unlimited in magnitude.

There are two examples of this kind of motion, with which all youth are familiar; the motion of an arrow shot from a bow, and the motion of a ball projected obliquely into the air by a blow from a mallet.* In the first of these the oblique motion is gradually communicated by a pressure, namely, the elasticity of the bow; in the second, it appears to be communicated in an instant by the blow given to the ball. This, however, is only in appearance; no motion can be communicated in an instant; that produced by a blow might be communicated in any assigned time, however short, by a sufficiently great pressure, as we see exemplified in the case of a ball shot from a cannon.

It was natural to enquire whether terrestrial gravity extended to all distances from the earth; and if so, whether it produced equal effects in the same time at different distances. If it extend indefinitely, then the moon must be within its influence; and hence the important question, May not the moon's motion be produced in the same way as the curvilineal motions we observe at the earth's surface, by the joint action of terrestrial gravity, and a rectilineal motion originally impressed on that body? These important questions engaged the mind of Newton, who resolved them completely. He found by strict and conclusive reasoning, that the force of gravity extends to the moon, but that it decreases as the distance increases, following this law; at a distance of twice the radius of the earth from its centre, the force of gravity is 1-4th only of what we observe at the surface; at the distance of thrice the radius, it is 1-9th; at the distance of 4 radii, it is 1-16th, and so on This law is briefly expressed by saying, that the force of gravity is inversely proportional to the square of the distance. He also proved, that not only does the moon gravitate towards the earth, but also that the earth

As in the game of Golf.