Cor. 1. If a body, which is urged by a constant and uniform force, move through any space, it will move through twice that space in the same time by the velocity acquired. For the sum of all the spaces described by that force, 1 + 2 + 3, &c. to n, was shewn to be n2; and the sum of all the spaces described by the last velocity will be n + n + n, &c. to n terms, the sum of which is n2. But n2 is double of n2; therefore the space described by the last velocity is double the by the accelerating force. space described Cor. 2. If a body is acted on by a constant and uniform force, the space described from the beginning of the motion is as the velocity acquired and the time of moving. Cor. 3. Universally, in all bodies urged by any constant and uniform forces, during any times, the spaces passed over are as the forces and squares of the times di.rectly, and the quantity of matter reciprocally. Cor. 4. The product of the force and square of the time, is as the product of the body and space described. Cor. 5. The product of the force and time is as the product of the quantity of matter and velocity. Cor. 6. The product of the body and the square of its velocity, is as the product of the force and space described. Scholium. 109. If a body be acted on by any two constant and uniformly accelerating forces, f, F, during the respective times tand T, at the end of which are generated the velocity is v, V, and describes the spaces s and S, then, Now, if one of the forces be the force of gravity at the earth's surface, and be represented by unity or 1, and the time T = 1′′; then the corresponding space S has been found by experiment to be 16 feet, and therefore its velocity V2 S = 32 feet. Above, substitute 1 in the place of F, 1" in the place of T, 161, feet in the place of S, and 32 feet in the place of V; and we have, These theorems will be expressed in words afterwards, when we come to treat of the motion of bodies on inclined planes. When any quantity or quantities are given, or are always the same, they must be left out; thus, By Art. 108, Cor. 3, s : S :: ft2: b Art. the body is constant, therefore b case, s: S :: f to F T2 fte: FT2. F T2 B -; but in this or, which is the same, s: S :: Also, such quantities as are proportional to each other must be left out. Thus, if the quantity of matter be proportional to the force, as all bodies are in respect to their gravity,* then the space described is as the square of the velocity. The space is also as the square of the time. Hence the velocity is as the time. We may now show the application of these proportions to falling bodies. * The weights of all bodies in the same place are proportional to the quantities of matter they contain, without any regard to their bulk, figure, or kind. For twice the matter will be twice as heavy, thrice the matter thrice as heavy, and so on. ON GRAVITY. 110. Gravity is that power or force which causes bodies to approach each other. This universal principle, which pervades the whole system of nature, may be enunciated as follows. The mutual tendency of two bodies towards each other increases in the same proportion as their masses are increased, and the square of their distance is decreased; and it decreases in proportion as their masses are decreased, and as the square of their distance is increased. Thus, if any two bodies A and B be placed in free space at a given distance from each other, and if we suppose the mass of A to be double the mass of B, then B will move towards A with double the velocity with which A moves towards B. Also, if at any given distance A tends towards B with a given force, at double that distance A will tend towards B with one-fourth of that force, at treble the distance with one-ninth of that force, and so on. Philosophers have formed theories of various kinds to account for this universal principle. Some have considered that it is produced by particles emanating from a centre or centres. But, as Dr. Paley* very justly observes, we are totally at a loss to comprehend how particles streaming from a centre can possibly draw bodies toward that centre. The impulse, if impulse there be, is all the other way. * We would particularly recommend to our readers the perusal of Paley's Natural Theology, it being a work replete with useful information. We are equally at a loss if we consider that the effect is produced by a conflux of particles flowing towards a centre, and carrying down all bodies along with it; for, if such a fluid exists, it must act very powerfully, and at the same time offer no resistance whatever to bodies moving in it, which is contrary to the known constitution of fluids. We are also utterly unable to conceive how one body can act upon another at a distance, or, in other words, that a body can act where it is not; and it appears no more ridiculous to assert that a body can act when it ceases to exist, than to assert that a body can act where it does not exist. Indeed, every hypothesis relating to the cause of gravity is embarrassed with insuperable difficulties. It seems, as it were, to be among the arcana of the Almighty; and, until a theory can be advanced which can clear up all these difficulties, it is more prudent to conclude, with the illustrious Newton, "that in absence of the secondary cause of gravity, we may attribute it to the final cause of all things, the finger of God, the constant impression of divine power." TERRESTRIAL GRAVITY. 111. Terrestrial Gravity is that force by which bodies are urged towards the centre of the earth, and it is measured by the velocity generated in a second of time. As has been already remarked, experiments shew that a falling body describes 16 feet in the first second, and it has then acquired a velocity of 32 feet, which is therefore the true measure of the force of gravity. 2 There appears to be an inequality of the action of gravity upon different kinds of matter near the surface of the earth. But this arises entirely from the resistance which they meet with in passing through the air; for, in the exhausted receiver of an air-pump, all bodies fall equally; a guinea acquires no greater velocity than a feather; an |