by some person acquainted with the usual order of sequence. This conclusion is quite irresistible, and rightly so; for there are but two suppositions which we can make as to the reason of the cards being in that particular order:

1. They have been intentionally arranged by some one who would probably prefer the numerical order.

2. They have fallen into that order by chance, that is, by some series of conditions which, being wholly unknown in nature, cannot be known to lead by preference to the particular order in question.

The latter supposition is by no means absurd, for any one order is as likely as any other when there is no preponderating tendency. But we can readily calculate by the doctrines of permutation the probability that fifty-two objects would fall by chance into any one particular order. Fifty-two objects can be arranged in

52 x 51 x 50 x .... x 4 × 3 × 2 × 1 or 8066 x (10)64 possible orders, the number obtained requiring 68 places of figures for its full expression. Hence it is excessively unlikely, and, in fact, practically impossible, that any one should ever meet with a pack of cards arranged in perfect order by pure accident. If we do meet with a pack so arranged, we inevitably adopt the other supposition, that some person having reasons for preferring that special order, has thus put them together.

We know that of the almost infinite number of possible orders the numerical order is the most remarkable; it is useful as proving the perfect constitution of the pack, and it is the intentional result of certain games. At any rate, the probability that intention should produce that order is incomparably greater than the probability that chance should produce it; and as a certain pack exists in that order, we rightly prefer the supposition which most probably leads to the observed result.

By a similar mode of reasoning we every day arrive, and validly arrive, at conclusions approximating to certainty. Whenever we observe a perfect resemblance between two objects, as, for instance, two printed pages, two engravings, two coins, two foot-prints, we are warranted in asserting that they proceed from the same type, the same plate, the same pair of dies, or the same boot. And why? Because it is almost impossible that with different types, plates, dies, or boots some minute distinction of form should not be discovered. It is barely possible for the hand of the most skilful artist to make two objects. alike, so that mechanical repetition is the only probable explanation of exact similarity. We can often establish with extreme probability that one document is copied from another. Suppose that each document contains 10,000 words, and that the same word is incorrectly spelt in each. There is then a probability of less than in 10,000 that the same mistake should be made in each.

If we meet with a second error occurring in each document, the probability is less than 1 in 10,000 × 9999, that such two coincidences should occur by chance, and the numbers grow with extreme rapidity for more numerous coincidences. We cannot indeed make any precise calculations without taking into account the character of the errors committed, concerning the conditions of which we have no accurate means of estimating probabilities. Nevertheless, abundant evidence may thus be obtained as to the derivation of documents from each other. In the examination of many sets of logarithmic tables, six remarkable errors were found to be present in all but two, and it was proved that tables printed at Paris, Berlin, Florence, Avignon, and even in China, besides thirteen sets printed in England, between the years 1633 and 1822, were derived directly or indirectly from some

common source". With a certain amount of labour, it is possible to establish beyond reasonable doubt the relationship or genealogy of any number of copies of one document, proceeding possibly from parent copies now lost. Tischendorf has thus investigated the relations between the manuscripts of the New Testament now existing, and the same work has been performed by German scholars for several classical writings.

Principle of the Inverse Method.

The inverse application of the rules of probability entirely depends upon a proposition which may be thus stated, nearly in the words of Laplace. If an event can be produced by any one of a certain number of different causes, the probabilities of the existence of these causes as inferred from the event, are proportional to the probabilities of the event as derived from these causes. In other words, the most probable cause of an event which has happened is that which would most probably lead to the event supposing the cause to exist; but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would have happened if the cause existed. Suppose, to fix our ideas clearly, that E is the event, and C, C, C, are the three only conceivable causes. If C1 exist, the probability is p1 that E would follow; if C2 and C, exist, the like probabilities are respectively p2 and Then as Pi is to P2, so is the probability of C, being the actual cause to the probability of C, being it; and, similarly, as p, is to pз, sO is the probability of C, being the actual cause to the By a very simple mathematical

probability of C, being it.

P3.

a Lardner, 'Edinburgh Review,' July 1834, p. 277.

b Mémoires par divers Savans,' tom. vi.; quoted by Todhunter in his History of Theory of Probability,' p. 458.

process we arrive at the conclusion that the actual probability of C, being the cause is

and the similar probabilities of the existence of C, and

The sum of these three fractions amounts to unity, which correctly expresses the certainty that one cause or other must be in operation.

We may thus state the result in general language. If it is certain that one or other of the supposed causes exists, the probability that any one does exist is the probability that if it exists the event happens, divided by the sum of all the similar probabilities. There may seem to be an intricacy in this subject which may prove distasteful to some readers; but this intricacy is essential to the subject in hand. No one can possibly understand the principles of inductive reasoning, unless he will take the trouble to master the meaning of this rule, by which we recede from an event to the probability of each of its possible causes.

This rule or principle of the indirect method is that which common sense leads us to adopt almost instinctively, before we have any comprehension of the principle in its general form. It is easy to see, too, that it is the rule which will, out of a great multitude of cases, lead us most often to the truth, since the most probable cause of an event really means that cause which in the greatest number of cases produces the event. But I have only met with one attempt at a general demonstration of the principle. Poisson imagines each possible cause of an event to be represented by a distinct ballot-box, containing black and white balls, in such ratio that the probability of a white ball being drawn is equal to that of the event

happening. He further supposes that each box, as is possible, contains the same total number of balls, black and white; and then, mixing all the contents of the boxes together, he shows that if a white ball be drawn from the aggregate ballot-box thus formed, the probability that it proceeded from any particular ballot-box is represented by the number of white balls in that particular box, divided by that total number of white balls in all the boxes. This result corresponds to that given by the principle in question.

Thus, if there be three boxes, each containing ten balls in all, and respectively containing seven, four, and three white balls, then on mixing all the balls together we have fourteen white ones; and if we draw a white ball, that is if the event happens, the probability that it came out of the first box is; which is exactly equal to

7

10 ,the 3 10 + 10 + 10

fraction given by the rule of the Inverse Method.

Simple Applications of the Inverse Method.

In many cases of scientific induction we may apply the principle of the inverse method in a simple manner. If only two, or at the most a few hypotheses, may be made as to the origin of certain phenomena, or the connection of one phenomenon with another, we may sometimes easily calculate the respective probabilities of these hypotheses. It was thus that Professors Bunsen and Kirchhoff established, with a probability little short of certainty, that iron exists in the sun. On comparing the spectra of sunlight and of the light proceeding from the incandescent vapour of iron, it became apparent that at least sixty bright lines in the spectrum of iron coincided with dark

c Poisson, 'Recherches sur la Probabilité des Jugements,' Paris, 1837, pp. 82, 83.