For, by Cor. 2. Prop. 3. C=

position. Therefore P2=R3.

R 1

=

2PPRR'

by the sup

But as none of the orbits of the solar system are exactly circular, it is unnecessary to pursue this theory any farther, we shall therefore proceed to other propositions, which will be found in fact to agree with what we find in the heavens.

PROPOSITION V.

If one body revolve round another by a force directed to any fixed center, it will describe, by a ray drawn from the center of force, areas proportional to the times.

Let the body be supposed to move in the direction AI* by the projectile force alone; instead of going in this direction it will, by the influence of the central force placed in S, move in the direction AB, and if permitted to move on in that direction, without any farther disturbance from the central force, it would move from B to C, in the same time that it moved from A to B, making BC=AB. But by the influence of the central force, which alone is sufficient to carry it to N, it will neither proceed in the direction BN nor BC, but in the direction BD: and at D it will have such velocity as would carry it over the space DE=BD, in the same time: but by the action of the central force, it is prevented from moving in the direction DE, and bent into the line DF, the diagonal of the parallelogram DKFE; for by the approach of the planet to the center of force, it has now acquired such strength as to carry the planet over the space DK by its influence alone. In the next instant of time, it would move over FG=DF in the direction DF, with the velocity acquired in F. But at the end of the said time, it will be found in the point H,

* See Plate IV. Fig. 2.

having described FH the diagonal of the parallelogram FLHG.

Now I say that all these spaces ASB,BSC,DSE,FSH, described in equal times, are equal: For, because AB= BC, the triangles ASB and BSC are equal, as having the same base and altitude; But BSC=BSD, having the same base BS, and being between the same parallels BS and CD, therefore ABS BSD. For the same reason, BDS=DSE, having the equal bases BD and DE and the same altitude; and DSE=DSF, having the same base DS, and being between the same parallels DS and EF. Therefore, &c. Q.E.D.

=

Now if these bases AB,BD,DF,FH, be supposed indefinitely small, or in their nascent state, they will coincide with the curve described by the revolving body.

Cor. 1. The velocity of the revolving body is everywhere inversely as a perpendicular to the tangent of the orbit in the place of the body, from the center of force; because the velocities are as the bases of the equal triangles ASB,BSD,DSE, &c. described in equal times; but the bases of equal triangles are inversely as their altitudes; that is, inversely as the perpendiculars to the tangents, for the tangents and bases coincide in their

nascent or evanescent state.

Cor. 2. The times of describing the equal areas are directly as the said perpendiculars, because they are inversely as the velocities; for the greater the velocity is, by which any arch is described, the sooner the revolving body passes over it.

Cor. 3. As any area of the orbit is to the time of its description, so is the area of the whole orbit to the periodical time; and therefore the periodical time of a planet's revolution is proportional to the area of the orbit

directly, and inversely to any sector described in a given time.

In our lecture on projectiles we found, that a body would move in the curve of a parabola, if the central force acted with an invariable degree of strength, and the projectile velocity were such as would be acquired by falling through a space equal to one fourth of the parameter of that diameter, which passes through the point of projection; now, one fourth of the parameter is equal to the focal distance. The projectile will therefore move in a parabola, when thrown with a velocity acquired by falling through the focal distance; and this is the necessary adjustment between the central and projectile forces, in the description of a parabola. But we have also seen, that the projected body will move in a circle, whose center is the focus of the parabola, and whose radius is the focal distance, if projected with the velocity acquired in falling through half that distance. Therefore, since the velocities acquired in falling through any given distances are proportional to the square roots of these distances; it follows, that the velocity necessary for the description of a circle is to the velocity necessary for the description of a parabola, as I to 2, the radius of the circle; and the focal distance of the parabola being the same.

These two curves then require this exact adjustment between the central and projectile forces for their description; and the reason is, because the central force is supposed in both these cases to act with the same invariable degree of strength, being always at the same distance from the body revolving in a circle, and being at an infinite distance from it when moving in a parabola, as the center of this curve is at an infinite distance from its vertex.

As the parabola is the utmost limit of an ellipsis on the one side, when the curve flies open, and the center and focus recede to an infinite distance from the vertex; and the right line is the utmost limit of the ellipsis on the other side, when the conjugate axis vanishes; it is evident, that any velocity less than what is necessary for the description of a parabola, will make the revolving body move in an ellipsis, which therefore requires no certain adjustment between the central and projectile forces. If the velocity necessary for the description of a parabola be diminished in any proportion between the ratio of 2:1. the ellipsis described will lie within the parabola, and without the circle, and the center of force will be in the focus nearest to the place of projection, as the body moving without the circle recedes from the center of force: and if it be diminished in a still greater proportion, even down to nothing, the ellipsis described will lie within the circle, until it finally degenerate into the radius of the circle, and the center of force will be in the focus most distant from the point of projection. Lastly, if the parabolic velocity be increased in any degree, the projected body will move in a hyperbola by virtue of a centrifugal force, placed in the focus of the opposite hyperbola; as the focus of the ellipsis, after having receded to an infinite distance, when it flew open into the parabola, comes to a finite distance on the other side; and the curve becomes convex to the center of force. Now, as neither the parabola nor hyperbola are curves that return into themselves, if the projectile move in either of these, it is evident, that it will continually recede from the center of force, and never more return to the place of projection.

But in order that one body may revolve in the curve of an ellipsis round another placed anywhere in the axis

directly, and inversely to any sector described in a given time.

In our lecture on projectiles we found, that a body would move in the curve of a parabola, if the central force acted with an invariable degree of strength, and the projectile velocity were such as would be acquired by falling through a space equal to one fourth of the parameter of that diameter, which passes through the point of projection; now, one fourth of the parameter is equal to the focal distance. The projectile will therefore move in a parabola, when thrown with a velocity acquired by falling through the focal distance; and this is the necessary adjustment between the central and projectile forces, in the description of a parabola. But we have also seen, that the projected body will move in a circle, whose center is the focus of the parabola, and whose radius is the focal distance, if projected with the velocity acquired in falling through half that distance. Therefore, since the velocities acquired in falling through any given distances are proportional to the square roots of these distances; it follows, that the velocity necessary for the description of a circle is to the velocity necessary for the description of a parabola, as I to 2, the radius of the circle; and the focal distance of the parabola being the same.

These two curves then require this exact adjustment between the central and projectile forces for their description; and the reason is, because the central force is supposed in both these cases to act with the same invariable degree of strength, being always at the same distance from the body revolving in a circle, and being at an infinite distance from it when moving in a parabola, as the center of this curve is at an infinite distance from its vertex.