the letters would be absent. But the formation of our vocal organs prevents us from using the far greater part of these conjunctions of letters. At least one vowel must be present in each word; more than two consonants cannot usually be brought together; and to produce words capable of smooth utterance a number of other rules must be observed. To determine exactly how many words might exist in the English language under these circumstances, would be an exceedingly complex problem, the solution of which has never been attempted. The number of existing English words may perhaps be said not to exceed one hundred thousand, and it is only by investigating the combinations presented in the dictionary, that we can learn the Laws of Euphony or calculate the possible number of words. In this example we have an epitome of the work and method of science. The combinations of natural phenomena are limited by a great number of conditions which are in no way brought to our knowledge except so far as they are disclosed in the examination of nature.
It is often a very difficult matter to determine the numbers of permutations or combinations which may exist under various restrictions. Many learned men puzzled themselves in former centuries over what were called Protean verses, or verses admitting many variations in accordance with the Laws of Metre. The most celebrated of these verses was that invented by Bernard Bauhusius, as follows:1—
One author, Ericius Puteanus, filled forty-eight pages of a work in reckoning up its possible transpositions, making them only 1022. Other calculators gave 2196, 3276, 2580 as their results. Wallis assigned 3096, but without much confidence in the accuracy of his result.2 It required the skill of James Bernoulli to decide that the number of transpositions was 3312, under the condition that the sense and metre of the verse shall be perfectly preserved.
In approaching the consideration of the great Inductive problem, it is very necessary that we should acquire correct notions as to the comparative numbers of combinations which may exist under different circumstances. The doctrine of combinations is that part of mathematical science which applies numerical calculation to determine the numbers of combinations under various conditions. It is a part of the science which really lies at the base not only of other sciences, but of other branches of mathematics. The forms of algebraical expressions are determined by the principles of combination, and Hindenburg recognised this fact in his Combinatorial Analysis. The greatest mathematicians have, during the last three centuries, given their best powers to the treatment of this subject; it was the favourite study of Pascal; it early attracted the attention of Leibnitz, who wrote his curious essay, De Arte Combinatoria, at twenty years of age; James Bernoulli, one of the very profoundest mathematicians, devoted no small part of his life to the investigation of the subject, as connected with that of Probability; and in his celebrated work, De Arte Conjectandi, he has so finely described the importance of the doctrine of combinations, that I need offer no excuse for quoting his remarks at full length.
“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged or combined together, it often happens that men, even of the best understandings and greatest circumspection, are guilty of that fault in reasoning which the writers on logic call the insufficient or imperfect enumeration of parts or cases: insomuch that I will venture to assert, that this