and also in the menstrual motion of the moon round the earth, this is a matter of tolerably simple observation; for by good instruments the apparent diameter of the body's disk, (to which its distance from us must be inversely proportionate,) as also the rate of its motion in the ecliptic, can at all times be measured with considerable accuracy. And now you have seen that if the force, by which any of these bodies is made to deviate from a uniform rectilinear motion, be continually directed to a single point, such a motion round that point must ensue as is actually found in nature.

A. True, with respect to the varying velocity of the motion; but you have not shown me that it ought to be performed in the curve of an ellipse.

B. No, I must send you to other masters for that. But in the meanwhile you will observe, that the fact of the motions not being performed in exact circles evinces clearly that the bodies are not fixed in hollow crystalline spheres or zones, as was once thought; no such clumsy contrivance will now be sufficient to save appearances.

A. They have nothing then to support them but their own energies. But is the direction of an attractive force to one fixed point the only possible supposition which can account for the motions being performed, as we see them, in free space?

Prop. XVI.

B. You shall judge. Recurring to the diagram in Prop. XV., the fact is, that the body moves from B to D instead of C, so that the triangle SBD is described instead of the triangle SBC, and that these two triangles are equal. But, by Prop. XII., these equal triangles must be between the same parallels, that is, CD must be parallel to BS. And again, the line BE (being the direction of the force which caused the body to deviate from BC into BD) must be one side of a parallelogram, of which BD is the diagonal, and CD the opposite side: BS therefore and BE coincide in position; that is, the disturbing force is directed to the point S. And the same thing may be proved in every successive triangle. Thus then you see that the general law, which Kepler found to govern the seeming irregularities of the planetary motions, resolves itself necessarily, (if the first law of motion be only conceded,) into the universal Newtonian principle of attraction to one point.

DIALOGUE IV.

A. YOU seemed to me yesterday to have proved that the moon is attracted to the centre of the earth, and each planet to the centre of the sun.

B. That is unquestionably the fact; but it would be a fact of little importance, if we were not able further to ascertain the degree and measure of this attraction.

A. How is that possible?

B. The attraction of the earth, by which unsupported bodies all round the globe are found constantly to fall in lines perpendicular to the surface, and therefore meeting in the centre, may be measured by the space through which (if the resistance of the air be removed) they all alike fall in a given time; and more accurately by mathematical deduction from the length of a pendulum which performs its vibrations in such a time. But this will only show the force of gravity or attraction here, on the surface of the earth. Whether, as we ascend into the sky, that force is every where the same, or varies with the distance, requires another consideration. And the best way of ascertaining the fact is to weigh the moon.

A. That seems rather more difficult than to travel thither.

B. I assure you it has been done, and by men, like ourselves,

66

Standing on earth, not rapt above the pole." But it is first necessary to determine the distance of that body from the globe which we inhabit.

A. This appears rather more feasible; but I cannot easily divine how it should be attempted.

B. The fixed stars, from whatever part of the earth they may be viewed, always maintain the same positions relatively to each other. These therefore may justly be said to be at an immeasurable distance from us. The sun, when viewed at the same instant from distant parts of the earth, changes its bearings to the fixed stars in a manner scarcely perceptible. But it is quite otherwise with the moon. Observe, as nearly as possible, by the help of a telescope, the moment when exactly one half of her disk is illuminated, so that the side averse from the sun is a straight line. Then measure, by an accurate instrument, the angular distance between the two bodies; and you will not be able to distinguish that distance from a right angle. Now, as there can be no doubt that the moon is a globular body, and borrows all her light from the sun, the angle which she makes with the earth and sun, at the moment I have described, must be a right angle also; for it is the moment when she shows just half of that face to us which she presents entire to the sun. Here then is a triangle, two of whose angles, to all appearance, are right angles.

ance.

A. But that you have shown to be impossible. B. It is impossible in reality, but yet, from the defect of our organs, may be possible in appearThe truth evidently is, that what we take for a right angle is something less; and that the angle which the sun makes with the earth and moon amounts to the difference. Now if this angle be so inconsiderable, the sides of the triangle which include it must be far greater than the base, or the distance of the sun from us than that of the moon.

A. This I will allow. But how does this fact serve our present purpose?

B. I meant to show you that the sun might, scarcely less than a star, be considered as a fixed point, when we would measure the distance of the moon from the earth. And this being allowed, I will introduce you to the simplest method, of those that can be much depended on, for ascertaining that distance. An eclipse of the moon, I think you will admit, is caused by the shadow of the earth.

A. There can be no doubt of it.

B. Watch then the moment, in such an eclipse, when the two extremities of the moon's luminous horns are in the same vertical line; the centre of the earth's shadow will then be at the same altitude above the horizon as the centre of the moon. A. How does that appear?

B. Thus: Let the circle of which M is the Prop.XVII.

A.