cedent events, disconnected from the general body of physical science.

Events, again, may often exhibit a regularity of succession or preponderance of character, which the simple formula will not take into account. For instance, the majority of the elements recently discovered are metals, so that the probability of the next discovery being that of a metal, is doubtless greater than we calculated (p. 258). At the more distant parts of the planetary system, there are symptoms of disturbance which would prevent our placing much reliance on any inference from the prevailing order of the known planets to those undiscovered ones which may possibly exist at great distances. These and all like complications in no way invalidate the theoretic truth of the formulas, but render their sound application much more difficult.

Erroneous objections have been raised to the theory of probability, on the ground that we ought not to trust to our à priori conceptions of what is likely to happen, but should always endeavour to obtain precise experimental data to guide us. This course, however, is perfectly in accordance with the theory, which is our best and only guide, whatever data we possess. We ought to be always applying the inverse method of probabilities so as to take into account all additional information. When we throw up a coin for the first time, we are probably quite ignorant whether it tends more to fall head or tail upwards, and we must therefore assume the probability of each event as. But if it shows head in the first throw, we now have very slight experimental evidence in favour of a tendency to show head. The chance of two heads is now slightly greater than 1, which it appeared to be at first,2 and as we go on throwing the coin time after time, the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experience. As Laplace remarks, we ought always to have regard to such considerations in common life. Events when closely scrutinized will hardly ever prove to be quite independent, and the slightest pre

1 J. S. Mill, System of Logic, 5th edition, bk. iii. chap. xviii. § 3. 2 Todhunter's History, pp. 472, 598

ponderance one way or the other is some evidence of connection, and in the absence of better evidence should be taken into account.

The grand object of seeking to estimate the probability of future events from past experience, seems to have been entertained by James Bernoulli and De Moivre, at least such was the opinion of Condorcet; and Bernoulli may be said to have solved one case of the problem.1 The English writers Bayes and Price are, however, undoubtedly the first who put forward any distinct rules on the subject.2 Condorcet and several other eminent mathematicians advanced the mathematical theory of the subject; but it was reserved to the immortal Laplace to bring to the subject the full power of his genius, and carry the solution of the problem almost to perfection. It is instructive to observe that a theory which arose from petty games of chance, the rules and the very names of which are forgotten, gradually advanced, until it embraced the most sublime problems of science, and finally undertook to measure the value and certainty of all our inductions.

Fortuitous Coincidences.

We should have studied the theory of probability to very little purpose, if we thought that it would furnish us with an infallible guide. The theory itself points. out the approximate certainty, that we shall sometimes be deceived by extraordinary fortuitous coincidences. There is no run of luck so extreme that it may not happen, and it may happen to us, or in our time, as well as to other persons or in other times. We may be forced by correct calculation to refer such coincidences to a necessary cause, and yet we may be deceived. All that the calculus of probability pretends to give, is the result in the long run, as it is called, and this really means in an infinity of cases. During any finite experience, however long, chances may be against us. Nevertheless the theory is the best guide we can have. If we always think and act according to its well-interpreted indications,

1 Todhunter's History, pp. 378, 379.

2 Philosophical Transactions, [1763], vol. iii. p. 370, and [1764], vol. liv. p. 296. Todhunter, pp. 294-3m

we shall have the best chance of escaping error; and if all persons, throughout all time to come, obey the theory in like manner, they will undoubtedly thereby reap the greatest advantage.

No rule can be given for discriminating between coincidences which are casual and those which are the effects of law. By a fortuitous or casual coincidence, we mean an agreement between events, which nevertheless arise from wholly independent and different causes or conditions, and which will not always so agree. It is a fortuitous coincidence, if a penny thrown up repeatedly in various ways always falls on the same side; but it would not be fortuitous if there were any similarity in the motions of the hand, and the height of the throw, so as to cause or tend to cause a uniform result. Now among the infinitely numerous events, objects, or relations in the universe, it is quite likely that we shall occasionally notice casual coincidences. There are seven intervals in the octave, and there is nothing very improbable in the colours of the spectrum happening to be apparently divisible into the same or similar series of seven intervals. It is hardly yet decided whether this apparent coincidence, with which Newton was much struck, is well founded or not, but the question will probably be decided in the negative.

It is certainly a casual coincidence which the ancients noticed between the seven vowels, the seven strings of the lyre, the seven Pleiades, and the seven chiefs at Thebes.2 The accidents connected with the number seven have misled the human intellect throughout the historical period. Pythagoras imagined a connection between the seven planets and the seven intervals of the monochord. The alchemists were never tired of drawing inferences from the coincidence in numbers of the seven planets and the seven metals, not to speak of the seven days of the week.

A singular circumstance was pointed out concerning the dimensions of the earth, sun, and moon; the sun's diameter was almost exactly 110 times as great as the

1 Newton's Opticks, Bk. I., Part ii. Prop. 3; Nature, vol. i. p 286 2 Aristotle's Metaphysics, xiii, 6. 3.

earth's diameter, while in almost exactly the same ratio the mean distance of the earth was greater than the sun's diameter, and the mean distance of the moon from the earth was greater than the moon's diameter. The agreement was so close that it might have proved more than casual, but its fortuitous character is now sufficiently shown. by the fact, that the coincidence ceases to be remarkable when we adopt the amended dimensions of the planetary system. A considerable number of the elements have atomic weights, which are apparently exact multiples of that of hydrogen. If this be not a law to be ultimately extended to all the elements, as supposed by Prout, it is a most remarkable coincidence. But, as I have observed, we have no means of absolutely discriminating accidental coincidences from those which imply a deep producing cause. A coincidence must either be very strong in itself, or it must be corroborated by some explanation or connection with other laws of nature. Little attention was ever given to the coincidence concerning the dimensions of the sun, earth, and moon, because it was not very strong in itself, and had no apparent connection with the principles of physical astronomy. Prout's Law bears more probability because it would bring the constitution of the elements themselves in close connection with the atomic theory, representing them as built up out of a simpler substance.

In historical and social matters, coincidences are frequently pointed out which are due to chance, although there is always a strong popular tendency to regard them as the work of design, or as having some hidden meaning. If to 1794, the number of the year in which Robespierre fell, we add the sum of its digits, the result is 1815, the year in which Napoleon fell; the repetition of the process gives 1830 the year in which Charles the Tenth abdicated. Again, the French Chamber of Deputies, in 1830, consisted of 402 members, of whom 221 formed the party called "La queue de Robespierre," while the remainder, 181 in number, were named "Les honnêtes gens." If we give to each letter a numerical value corresponding to its place in the alphabet, it will be found that the sum of the values of the letters in each name exactly indicates the number of the party.

A number of such coincidences, often of a very curious character, might be adduced, and the probability against the occurrence of each is enormously great. They must be attributed to chance, because they cannot be shown to have the slightest connection with the general laws of nature; but persons are often found to be greatly influenced by such coincidences, regarding them as evidence of fatality, that is of a system of causation governing human affairs independently of the ordinary laws of nature. Let it be remembered that there are an infinite number of opportunities in life for some strange coincidence to present itself, so that it is quite to be expected that remarkable conjunctions will sometimes happen.

In all matters of judicial evidence, we must bear in mind the probable occurrence from time to time of unaccountable coincidences. The Roman jurists refused for this reason to invalidate a testamentary deed, the witnesses of which had sealed it with the same seal. For witnesses independently using their own seals might be found to possess identical ones by accident.1 It is well known that circumstantial evidence of apparently overwhelming completeness will sometimes lead to a mistaken judgment, and as absolute certainty is never really attainable, every court must act upon probabilities of a high amount, and in a certain small proportion of cases they must almost of necessity condemn the innocent victims. of a remarkable conjuncture of circumstances.2 Popular judgments usually turn upon probabilities of far less amount, as when the palace of Nicomedia, and even the bedchamber of Diocletian, having been on fire twice within fifteen days, the people entirely refused to believe that it could be the result of accident. The Romans believed that there was fatality connected with the name of Sextus.

"Semper sub Sextis perdita Roma fuit."

The utmost precautions will not provide against all contingencies. To avoid errors in important calculations,

1 Possunt autem omnes testes et uno annulo signare testamentum Quid enim si septem annuli una sculptura fuerint, secundum quod Pomponio visum est ?-Justinian, ii. tit. x. 5.

2 See Wills on Circumstantial Evidence, p. 148.