the number of ways in which I can select 1, 2, 3, 4, 5, 6, 7, and 8 things, out of 8 things. In general language, if I wish to know in how many ways m things can be selected in combinations out of n things, I must look in the nth line, and take the m+1th number, counting from the left, as the answer. In how many ways, for instance, can a sub-committee of five be chosen out of a committee of nine. The answer is 126, and is the sixth number in the tenth line; it will be found equal to 9.8.7.6.5 which our previous formula (p. 206) would

I. 2 3 4.5'

give.

The full utility of the figurate numbers will be more apparent when we reach the subject of probabilities, but I may give an illustration or two in this place. In how many ways can we arrange four pennies as regards head and tail? The question amounts to asking in how many ways we can select o, 1, 2, 3, or 4 heads out of 4 heads, and the fifth line of the triangle gives us the complete answer, thus-

We can select No head and 4 tails in way.

heads and I tail in 4 ways.

heads and o tail in 1 way.

The total number of different cases is 16, or 2, and when we come to the next chapter, it will be found that these numbers give us the respective probabilities of all throws with four pennies.

I gave in p. 205 a calculation of the number of ways in which eight planets can meet in conjunction; the reader will find all the numbers detailed in the ninth line of the arithmetical triangle. The sum of the whole line is 28 or 256; but we must subtract a unit for the case where no planet appears, and 8 for the 8 cases in which only one

planet appears; so that the total variety of conjunctions is 28-1-8 or 247.

If an organ has twelve stops, we find in the thirteenth line the numbers of combinations which we can draw, O, 1, 2, 3, &c., at a time; the total number of modes of varying the sound is no less than 2121 or 4095 m. If a number be the product of n prime factors, we find in the n+1th line the numbers of divisors, being the product of 1, 2, 3, or more of the prime factors; and the whole number of divisors of the number is the sum of the numbers in the line, subtracting unity, or 2” — I.

One of the most important scientific uses of the arithmetical triangle, consists in the information which it gives concerning the comparative frequency of divergencies from an average. Suppose, for the mere sake of argument, that all persons were naturally of equal stature of five feet, but enjoyed during youth seven independent chances of growing one inch in addition. Of these seven chances, one, two, three, or more, may happen favourably to any individual, but as it does not matter what the chances are, so that the inch is gained, the question really turns upon the number of combinations of o, 1, 2, 3, &c.,. things out of seven. Hence the eighth line of the triangle give us a complete answer to the question, as follows:Out of every 128 people

m Bernouilli, 'De Arte Conjectandi,' trans. by Masères, p. 64.

By taking a proper line of the triangle, an answer may be had under any more natural supposition. This theory of comparative frequency of divergence from an average, was first adequately noticed by M. Quetelet, and has lately been employed in a very interesting and bold manner by Mr. Galton, in his work on 'Hereditary Genius.' We shall afterwards find that the theory of error, to which is made the ultimate appeal in cases of quantitative investigation, is founded upon the comparative numbers of combinations as displayed in the triangle.

Connection between the Arithmetical Triangle and the Logical Abecedarium.

There exists a close connection between the arithmetical triangle described in the last section, and the series of combinations of letters called the Logical Abecedarium. The one is to mathematical science what the other is to logical science. In fact the figurate numbers, or those exhibited in the triangle, are obtained by summing up the logical combinations. Accordingly, just as the total of the numbers in each line of the triangle was twice as great as that for the preceding line (p. 210), so each column of the Abecedarium (p. 109) contained twice as many combinations as the preceding one. The like correspondence would also exist between the sums of all the lines of figures down to any particular line, and of the combinations down to any particular column.

By examining any one column of the Abecedarium, we shall also find that the combinations naturally group themselves according to the figurate numbers. Take the combinations of the letters A, B, C, D; they consist of all the ways in which I can choose four, three, two, one, or none of the four letters, filling up the vacant spaces

The numbers, it will be noticed, are exactly the same as those in the fifth line of the arithmetical triangle, and an exactly similar correspondence would be found to exist in the case of each other column of the Abecedarium.

Numerical abstraction, it has been asserted, consists in overlooking the kind of difference, and retaining only a consciousness of its existence (p. 177). While in logic, then, we have to deal with each combination as a separate kind of thing, in arithmetic we can distinguish only the classes which depend upon more or less positive terms being present, and the numbers of these classes immediately produce the numbers of the arithmetical triangle. It may here be pointed out that there are two modes

in which we can calculate the whole number of combinations of certain things. Either we may take the whole number at once as shown in the Abecedarium, in which case the number will be some power of two, or else we may calculate successively, by aid of permutations, the number of combinations of none, one, two, three, and so on. Hence we arrive at a necessary identity between two series of numbers. In the case of four things we shall have

In a general form of expression we shall have

n n. (n−1) n (n-1) (n-2)

n

2 = 1 +

+

I

+

+ &c.,

3

the terms being continued until they cease to have any value. Thus we have arrived at a proof of simple cases of the Binomial Theorem, of which each column of the Abecedarium is an exemplification. It may be shown that all other mathematical expansions likewise arise out of simple processes of combination, but the more complete consideration of this subject must be deferred.

Possible Variety of Nature and Art.

We cannot adequately understand the difficulties which beset us in certain branches of science, unless we gain a clear idea of the vast number of combinations or permutations which may be possible under certain conditions. Thus only can we learn how hopeless it would be to attempt to treat nature in detail, and exhaust the whole number of events which might arise. It is instructive to consider, in the first place, how immensely great are the numbers of combinations with which we deal in many arts and amusements.