P :: Pp Qq; and therefore Qg= Q P Pp. Hence, sub tracting, it follows that C'q = Cp, or Q: P:: Cp Q : Cq. And therefore C is still the center of gravity of the bodies P, Q, when they are come into the positions p, q; that is, the center of gravity has not been put in motion. Also if the two bodies P, Q, not attracting or repelling each other, move towards each other with uniform velocities which are in the ratio of Q to P, and impinge; the spaces described in any time (as Pp, Qq) will be in the same ratio of Q to P, and, as above, the center of gravity will be at rest. And when the bodies impinge on each other, the velocities of each will either be destroyed, or destroyed and generated in an opposite direction; and in either case, since the mutual pressure is equal on both, the accelerating forces which destroy and generate velocity, will be in the ratio of Q to P, as in Prop. 17. Therefore the velocities destroyed and generated are in the same ratio as the original velocities. Therefore if the whole velocity of one body is destroyed, the whole velocity of the other body also is destroyed, and the bodies are both at rest, and their center of gravity is still at rest after impact. But if the velocities be destroyed, and velocities generated in an opposite direction, these new velocities will also be in the ratio of the original velocities, because the accelerating forces at every instant are so, (Ax. 9); and therefore the spaces described in any time by the new velocities will be in the same ratio; and therefore, as before, it may be shown that C is still the center of gravity of P, Q. Therefore, under all the circumstances stated, the center of gravity remains at rest. Q. E. D. Examples to Propositions 4, 5, 6, 7, 10, 17, 18. By means of these Propositions, we can solve such Examples as the following: When a body falls freely by the action of gravity, the quantity a in Prop. 4 is 32 feet, the unit of time being one second, and v = gt. Also (Prop. 6, Cor. 2) v = 1 gt2. Ex. 1. To find the velocity acquired by a body which falls by gravity for 30 seconds. v = gt = = 32 x 30 = 960 feet per second. 2. To find the space fallen through in the same time, sgt 16 x 302 14400 feet. 3. = = To find in what time a body falls through 1024 feet. 1024 : = 16 × t2, t2 = 64, t = 8 seconds. 4. To find the velocity acquired in the same space, v = gt = 32 × 8 = 256 feet per second. 5. A body is projected directly upwards, with a velocity of 1000 feet a second; how high will it go? By Prop. 7, the height will be that through which a body must fall to acquire the same velocity. 6. A body is projected with a velocity of 32 feet a second in a direction which makes with the horizon half a right angle: to find the time of flight and the range. In this case m=n; therefore, by Prop. 10, 7. A cannon ball is projected with a velocity of 1600 feet a second, in a direction which rises 3 feet in 4 feet horizontal: find the time of flight and the 8. An inelastic body A impinges directly on another inelastic body B at rest, with a velocity of 10 feet a second; A being 3 and B 2 ounces, find the velocity after impact. If a be the velocity of both after impact, the velocity lost by A is 10 x, and the velocity gained by B is x. Hence the momentum lost by A is 3 x (10); and that gained by B is 2 x : and these are equal by Prop. 18; therefore 3 (10x) = 2x, 30 = 3x + 2x, x = 6. 9. The bodies being perfectly elastic, find the motions after impact. In perfectly elastic bodies, the velocity lost by A and the velocity gained by B in the restitution of the figure are equal to the velocity lost by A and gained by B in the compression. Now the velocity lost by A in the compression is 106 or 4; therefore the whole velocity lost by A is 8, and its remaining velocity 2. And the velocity gained by B in the compression is 6, and therefore the whole velocity gained by B is 12, which is B's velocity after impact. 10. A body A (3 ounces) draws B (2 ounces) over a fixed pully find the space described in one second from rest. By Prop. 17, the accelerating force is as is, it is 5 3 2 ; that 3+2 of gravity; and the space in a second is as the force therefore the space described in one second REMARKS ON MATHEMATICAL REASONING, AND ON THE LOGIC OF INDUCTION. SECT. I. On the Grounds of Mathematical 1. THE study of a science, treated according to a rigorous system of mathematical reasoning, is useful, not only on account of the positive knowledge which may be acquired on the subjects which belong to the science, but also on account of the collateral effects and general bearings of such a study, as a discipline of the mind and an illustration of philosophical principles. Considering the study of the mathematical sciences with reference to these latter objects, we may note two ways in which it may promote them ;-by habituating the mind to strict reasoning, and by affording an occasion of contemplating some of the most important mental processes and some of the most distinct forms of truth. Thus mathematical studies may be useful in teaching practical logic and theoretical metaphysics. We shall make a few remarks on each of these topics. |