Fourthly, That Gravity does equally affect all Bodies, without regard either to their Matter, Bulk, or Figure; so that the Impediment of the Medium being removed, the most compact and most loose, the greatest and smallest Bodies would descend the same Spaces in equal Times; the Truth thereof will appear from the Experiment I before-cited. In these two last Particulars, is shewn the great difference between Gravity and Magnetism, the one affecting only Iron, and that towards its Poles, the other all Bodies alike in every part. As a Corollary, from hence it will follow, that there is no such thing as positive Levity, those things that appear light, being only comparatively so; and whereas several things rise and swim in Fluids, 'tis because, Bulk for Bulk, they are not so heavy as those Fluids; nor is there any Reason why Cork, for Instance, should be said to be light, because it swims on Water, any more than Iron, because it swims on Mercury.
Fifthly, That this Power increases as you descend, and decreases as you ascend from the Center, and that in the Proportion of the Squares of the Distances therefrom reciprocally, so as at a double Distance to have but a quarter of the Force; this Property is the Principle on which Mr. Newton has made out all the Phænomena of the Cœlestial Motions, so easily and naturally, that its Truth is past Dispute. Besides that, it is highly rational, that the attractive or gravitating Power should exert it self more vigorously in a small Sphere, and weaker in a greater, in proportion as it is contracted or expanded; and if so, seeing that the Surfaces of Spheres are as the Squares of their Radii, this Power, at several Distances, will be as the Squares of those Distances reciprocally; and then its whole Action upon each Spherical Surface, be it great or small, will be always equal. And this is evidently the Rule of Gravitation towards the Centers of the Sun, Jupiter, Saturn and the Earth, and thence is reasonably inferred, to be the general Principle observed by Nature, in all the rest of the Cœlestial Bodies.
These are the principal Affections of Gravity, from which the Rules of the Fall of Bodies, and the Motion of Projects are Mathematically deducible. Mr. Isaac Newton has shew'd how to define the Spaces of the Descent of a Body, let fall from any given height, down to the Center, supposing the Gravitation to increase, as in the fifth Property; but considering the smallness of heighth, to which any Project can be made ascend, and over how little an Arch of the Globe it can be cast by any of our Engines, we may well enough suppose the Gravity equal throughout, and the Descents of Projects in parallel Lines, which in Truth are towards the Center, the difference being so small as by no means to be discovered in Practice. The Opposition of the Air, 'tis true, is considerable against all light Bodies moving through it, as likewise against small ones (of which more hereafter) but in great and ponderous Shot, this Impediment is found by Experience but very small, and may safely be neglected.
Propositions concerning the Descent of Heavy Bodies, and the Motion of Projects.
Prop. I. The Velocities of Falling Bodies, are proportionate to the Times from the beginning of their Falls.
This follows, for that the Action of Gravity being continual, in every Space of Time, the falling Body receives a new Impulse, equal to what it had before, in the same Space of Time, received from the same Power: For Instance, in the first Second of Time, the falling Body has acquired a Velocity, which in that time would carry it to a certain Distance, suppose 32 Foot, and were there no new Force, would descend at that rate with an equable Motion: But in the next Second of Time, the same Power of Gravity continually acting thereon, superadds a new Velocity equal to the former; so that at the end of two Seconds, the Velocity is double to what it was at the end of the first, and after the same manner may it be proved to be triple, at the end of the third Second, and so on. Wherefore the Velocities of falling Bodies, are proportionate to the Time of their Falls, Q. E. D.
Plate 4. pag. 310.
Prop. II. The Spaces described by the Fall of a Body, are as the Squares of the Times, from the beginning of the Fall.
Demonstration. Let AB (Fig. 9. Tab. 4.) represent the Time of the Fall of a Body, BC perpendicular to AB, the Velocity acquired at the end of the Fall, and draw the Line AC; then divide the Line AB representing the Time, into as many equal Parts as you please, as b, b, b, b, &c. and through these Points draw the Lines bc, bc, bc, bc, &c. parallel to BC, 'tis manifest that the several Lines, bc, represent the several Velocities of the falling Body, in such Parts of the Time as Ab is of AB, by the former Proposition. It is evident likewise, that the Area ABC is the Sum of all the Lines bc being taken, according to the Method of Indivisibles, infinitely many; so that the Area ABC represents the Sum of all the Velocities, between none and BC supposed infinitely many; which Sum is as the Space descended in the Time represented by AB. And by the same Reason the Areas Abc, will represent the Spaces descended in the Times Ab; so then the Spaces descended in the Times AB, Ab, are as the Areas of the Triangles ABC, Abc, which by the 20th of the 6 of Euclid, are as the Squares of their Homologous Sides AB, Ab, that is to say, of the Times: Wherefore the Descents of falling Bodies, are as the Squares of the Times of their Fall, Q. E. D.