dent upon the speed of projection it will commence to return. If, however, the pace be as much as or more than the critical value of seven miles a second, the body will continue its journey, and the attraction of the earth will not be sufficient to recall it.
This point is of so much importance that I am inclined to dwell on it a little longer in hope of removing the paradox which it may seem to involve. For as in the ascent of the body the velocity must ever be lessened by the attractive force pulling it back, it might seem obvious that the initial velocity must be ultimately overcome if only long enough time be granted. But this is not a valid objection. It can be shown that the difficulty is something like that which arises in the well-known case of a geometric series. If we add a half to a quarter, and that to an eighth and that to a sixteenth, and that to a thirty-second, and so on indefinitely, we shall make an infinite number of additions, and yet, though the number of the quantities added together may be infinite, the total that they produce can never be more than finite. This can be more simply seen by a very elementary illustration. If you eat half a cake to-day, and a quarter of it to-morrow, and an eighth the next day, and a sixteenth the day after, and a thirty-second the next, and so on, ever consuming one day exactly half what was left the day before, the cake will never entirely disappear. It would, in fact, last for ever, notwithstanding the fact that an infinitely great number of portions had been abstracted from it. Somewhat similar is the nature of the operation by which the attraction of the earth gradually reduces the ascending velocity of the moving body. All that the earth can effect by attraction is to reduce the pace by the extent of seven
