The questions which I propose to examine in the following paper, although not themselves dependent on chance, have arisen entirely from games in which it predominates. To determine some method of betting upon a number of successive events, (and the probability of each of which is either equal to, or less than one-half,) by which a profit shall be realised after a considerable number of them have been decided, is a problem which has occupied the attention, and exhausted the efforts, of one set of speculators, as completely as that of the quadrature of the circle has defeated the labours of another. The first and most simple plan, is that of doubling the stake whenever a loss occurs. This is well known, and has been so frequently practised, as to have acquired a peculiar name; it is technically called the martingal; it requires for its success, that the person who employs it have the power of leaving off whenever he please, and that he have the command of an unlimited capital. If the chance of the events happening is one-third instead of one-half, the stake must be tripled.