as his friend Roberval had also done. He considered the infinitely small quantities introduced in his method of drawing tangents, and of resolving maxima and minima, as derived from finite differences; and, as Laplace remarks, he has extended his method to a case, when the variable quantity is irrational. He was, therefore, very near to the method of fluxions; with the principle of it, he was perfectly acquainted;-and so at the same time were both Roberval and Wallis, though men much inferior to Fermat. The truth is, that the disco

very of the new calculus was so gradually approximated, that more than one had come quite near it, and were perfectly acquainted with its principles, before any of the writings of Newton or Leibnitz were known. That which must give, in such a case, the right of being considered as the true inventor, is the extension of the principle to its full range; connecting with it a new calculus, and new analytical operations; the invention of a new algorithm with corresponding symbols. These last form the public acts, by which the invention becomes known to the world at large, the judge by which the matter must be finally decided. Great, therefore, as is the merit of Fermat, which no body can be more willing than ourselves to acknowledge; and near as he was to the greatest invention of modern times, we cannot admit that his property in it is to be put on a footing with that of

Newton or of Leibnitz;—we should fear, that, in doing so, we were violating one of the most sacred and august monuments that posterity ever raised in honour of the dead.

It has been already stated, that, when all the different ways in which an event can fall out, are equally possible and independent of one another, the fraction which expresses the probability, that the event may have certain conditions, is one which has for its numerator all the cases favourable to such conditions, and for its denominator all the cases possible. But when the event that happens affects that which is to follow, the question becomes sometimes of considerable difficulty. Laplace mentions one case, simple indeed, but important in its application. Suppose a fact to be transmitted through twenty persons ;-the first communicating it to the second, the second to the third, &c.; and let the probability of each testimony be expressed by (that is, suppose that of ten reports made by each witness, nine only are true,) then at every time the story passes from one witness to another, the evidence is reduced to of what it was before. -Thus, after it has passed the whole twenty, the evidence will be found to be less than 1.

1

9

"The diminution of evidence by this sort of transmission may," says Laplace, "be compared to the extinction of light by the interposition of several pieces of glass; a small number of pieces

will be sufficient to render an object entirely invisible which a single piece allowed to be seen very distinctly. Historians do not seem," he adds, "to have paid sufficient attention to this degradation of the probability of facts when seen across a great number of successive generations."

It does not appear, however, that the diminu. tion of evidence here supposed is a necessary consequence of transmission from one age to another.

It

may hold in some instances; but in those that most commonly occur, no sensible diminution of evidence seems to be produced by the lapse of time. Take any ancient event that is well attested, such, for example, as the retreat of the ten thousand, and we are persuaded it will be generally admitted, that the certainty of that event having taken place is as great at this moment as it was on the return of the Greek army, or immediately after Xenophon had published his narrative. The calculation of chances may indeed be brought to depose in favour of it; for the probability will be found to be very small, that any considerable interpolation or change in the supposed narrative of Xenophon could have taken place without some historical document remaining to inform us of such a change. The combination of the chances necessary to produce and to conceal such an interpolation is in the highest degree improbable; and the authority of Xenophon remains, on that account,

the same at this moment that it was originally. The ignorance of a transcriber, or the presumption of a commentator, may vitiate and alter a passage; but there is a virtue in sense and consistency by which they restore themselves. The greatest danger that an ancient author runs is when a critic like Bentley is turned loose upon his text. Yet there is no fear but that, in the arguments by which he would recommend his alterations, he will leave a sufficient security against their being received.

There is an error on the subject of chance, and of cases that are equally possible, against which it is necessary to guard.

Some writers argue as if regular events were less possible than irregular, and that in the game, for example, of Cross and Pile, a combination in which Cross would happen twenty times in succession, is less easy for nature to produce than one in which Cross and Pile are mixed together without regularity. This however is not true; for it is to suppose that the events which have already taken place, affect those that are to follow; and this, in what relates to chance, cannot be admitted. The regular combinations happen more rarely than the irregular, only because they are less numerous. If we look for a particular cause as acting in the cases where symmetry occurs, it is not because we suppose the symmetrical arrangement to be less possi ble than other; but it is improbable that chance

any

has produced it, because the symmetrical arrangements are few and the asymmetrical may be without number. We see on a table, for instance, letters so disposed as to make the word Constantinople; and we immediately conclude that this arrangement is not the effect of chance: not that it is less possible for chance to produce it, than any other given arrangement of the same fourteen letters-for if it were not a word in any language, we would never suspect the existence of design-but because the word being in use amongst us, it is incomparably more probable that this arrangement of the letters is the work of design, than of chance.

"Events may be so extraordinary that they can hardly be established by testimony. We would not give credit to a man who would affirm that he saw an hundred dice thrown in the air, and that they all fell on the same faces. If we had ourselves been spectators of such an event, we would not believe our own eyes, till we had scrupulously examined all the circumstances, and assured ourselves that there was no trick nor deception. After such an examination, we would not hesitate to admit it, notwithstanding its great improbability; and no one would have recourse to an inversion of the laws of vision in order to account for it. This shows that the probability of the continuance of the laws of nature is superior, in our estimation, to every other evidence, and to that of historical facts the best es