jectile passes over, and the central force acts in right lines directed to a center, the case will be different, and the path of the projectile will be no longer a parabola, but some other curve, such as a circle, or ellipsis. The effect of the central force is to cause the moving body on which it acts, to deviate gradually from the straight line in which it would proceed, if undisturbed by any other force, and to describe an incurvated line, which will always be concave towards the central force; whe ther it approach towards, recede from, or keep at the same distance from the center of force.

Suppose the line AB* to be a tangent to the circle described about the center C, the place of the central force, and the point of contact to be A. If a body were projected by any force in the line AB, it would continue to move in it, if undisturbed by the central force: but as soon as this begins to act, it will gradually cause the projectile to leave the direction AB, and to move in some other line, between the line AB and the line AC; viz. in the diagonal of a parallelogram, whose sides are proportional to the two forces, that acted upon it. If the central force be supposed to act incessantly, while the projectile force acts by a single impulse, there must be a continual deviation from the direction AB, and even from the diagonal of a parallelogram; so that the diagonal will degenerate into a curve which is concave to the central force, and convex towards AB.

The distance of the projectile from the central force may continue invariable, through the whole of its progress, when the central and projectile forces are so balanced, that the one draws it as much towards the center of force, as the other draws it from it. What this balance is, we shall hereafter see, and demonstrate to

* See Plate III. Fig. 2.

you, 'that when the velocity of the projectile is such as it would acquire by falling freely one half of the way to the center, it will, in that case, keep continually at the same distance from the center, and revolve in a circle; of which this is a known property, that in every point a ray drawn from the center is perpendicular to the direction of the projectile, which is a tangent to the circle.

If this angle, instead of being a right angle, be obtuse on the side to which the body moves, then it will recede from the center of force, and continue to fly from it, as long as it remains obtuse; but if it be acute, the moving body will approach the center, as long as the said angle continues to be acute; the effect of the said central force being in the first case to retard the recess of the moving body from the center, and in the other case to accelerate its approach.

If, while the projectile increases its distance from the center of force, the central force retain sufficient strength to make the obtuse angle above mentioned to become less and less obtuse, till it terminate in a right angle at last, that is, when the line from the center of force to the moving body becomes perpendicular to the curve; from that moment the moving body will no longer recede from the center, but in its subsequent motion it will again descend, as the angle now becomes acute; and in its approach to the central force it will describe a curve, in all respects similar to the curve described in its ascent; provided the central force continues to act every moment with the same strength at equal distances from it. Because if we suppose that the motion of the projectile was stopped, when the aforesaid angle became a right angle, and the body thrown back in a contrary direction to that in which it

was moving, and with the same velocity, it must, by the joint operation of these forces, describe the same curve over again, which it described in its ascent, and acquire the same velocity with which it was projected at first. The curve therefore which it describes in its progressive motion, while it is approaching the center of force, being described by the operation of the same forces, must be altogether the same.

Now if the curve, described from the time that the said angle is a right angle, until it become a right angle again, first increasing in its obtuseness to its maximum, and then decreasing, be a complete semi-ellipsis, described in absolute space, then the other part of the curve, which the projectile will describe, in its progress, while the said angle increases, and then decreases, in the degree of its acuteness, must also be a complete semi-ellipsis; and the body will have arrived at the precise point, from whence its motion began. And this being the case in every subsequent revolution, the points where the said angles become right angles, which are called the apsides of the orbit, will remain stationary.

If the portion of the orbit described within the aforesaid limits be less than a semi-ellipsis, the projec tile will have arrived at its greatest distance from the center of force, before a complete revolution round the central force was finished in absolute space. From this point however, where its distance is greatest, called its aphelium, it will begin to descend again towards the center; and in its next revolution, it will come to its greatest distance before it arrives at the same point again in absolute space; and this being the case in every subsequent revolution, the aphelia or apsides will have a retrograde motion, and will be found in all directions,

from the center, after a certain number of revolutions of the projectile.

But if, on the other hand, the portion of the orbit described between the times, when the revolving body is at its greatest and least distances from the center of force, be more than a semi-ellipsis, it will have performed more than a revolution, when it has arrived at its greatest distance from the center, where we suppose its motion began; and this being the case in every subsequent revolution, its apsides will have a progressive motion, in the direction of the moving body, and by this progressive motion, they will perform in due time a complete revolution round the center of force.

Thus we see how a revolving body, which is constantly attracted by a central force in any given point, may by its projectile velocity be kept from falling into the center of force, and describe about it an endless circuit; sometimes approaching towards the center, and again receding from it, within certain limits; while the central force preserves the same force at the same distance. When we come to astronomy, we shall see that this central force resides in the sun, and that it regulates all the motions of the planets and comets that belong to the solar system.

Before Sir Isaac Newton, no man ever advanced any tolerable and consistent hypothesis to account for the motions of the heavenly bodies. He found by considering the nature and effect of that power, which caused a heavy body to descend in a right line directed to the center of the earth, that it was sufficient to retain the moon in her orbit*, and indeed not only the earth and

If AE the space which the moon describes in a minute of time, it will be found to be 33" nearly. And AF will be the versed

moon, but also all the planets and comets in their orbits round the sun as their center; and therefore concluded that matter was endowed with such a force. Hence he called this central force, by which the planets and comets are retained in their orbits, and all their motions governed, gravitation; without pretending to explain its nature or the mode of its operation. He only called it by the name of gravitation, because he found it to be precisely the same with that principle of gravity, which causes a heavy body to descend towards the center of the earth.

But whether this power be mechanical or not, we cannot determine, with sufficient certainty. From accurate experiments and mathematical inductions, he has accurately explained the laws of its operation, and every day's experience confirms the conclusion he has drawn, from what he had observed, concerning this extensive principle of motion; which reaches the whole solar system, and possibly all the systems of the universe. Observation proves that this principle increases in force, as the squares of the distances decrease, from the central

sine of 33", equal to 16 feet in the moon's orbit. She performs her revolution in 27d 7h 43′=39343′. If 39343′ : 360o :: Ì': 33" of a degree. As the radius of the earth is 3985 miles, the radius of the moon's orbit is 60 times as great, viz. 239,100 miles=1,262,448,000 feet. Now AC: AF:: rad. vers. s. 1,33′′ :: 1,262,448,000: 16 feet nearly. Therefore the force which keeps the moon in her orbit, would cause her to descend 16 feet in a minute of time, if her projectile force were destroyed. We have proved before, in considering the descent of heavy bodies by the force of gravity, that a heavy body would fall 16 feet in a minute of time, if it were removed 60 semi-diameters from the center of the earth. Therefore the force of gravity is sufficient to retain the moon in her orbit. See Plate IV. Fig. 1.