the relative order or place in the conjunction, we require the number of combinations. Now a selection of 2 out of 8

8.7

is possible in or 28 ways; of 3 out of 8 in

I.2

8.7.6

1.2.3

8.7.6.5 or 56 ways; of 4 out of 8 in or 70 ways; and it 1.2.3.4 may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers of ways are 56, 28, 8, and 1. Thus we have solved the whole question of the variety of conjunctions of eight planets; and adding all the numbers together, we find that 247 is the utmost possible number of modes of meeting.

In general algebraic language, we may say that a group of m things may be chosen out of a total number of n things, in a number of combinations denoted by the formula

n. (n − 1) (n − 2) (n − 3) . . . . (n−m + 1)

I

2

3

4

ቤ

The extreme importance and significance of this formula seems to have been first adequately recognised by Pascal. although its discovery is attributed by him to a friend, M. de Ganières. We shall find it perpetually recurring in questions both of combinations and probability, and throughout the formula of mathematical analysis traces of its influence may be noticed.

The Arithmetical Triangle.

The Arithmetical Triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating. According to Montucla 2" this triangle is in the theory of combinations and changes of order, almost what the table of Pythagoras is in ordinary arithmetic, that is to say, it places at once under the eyes the numbers required in a multitude of cases of this theory." As early as 1544 Stifels had noticed the remarkable properties of these numbers and the mode of their evolution. Briggs, the inventor of the common system of logarithms, was so struck with their importance that he called them the

1 Euvres Complètes de Pascal (1865), vol. iii. p. 302. Montucla states the name as De Gruières, Histoire des Mathématiques, vol. iii. p. 389. 2 Histoire des Mathématiques, vol. iii. p. 378.

Abacus Panchrestus. Pascal, however, was the first who wrote a distinct treatise on these numbers, and gave them the name by which they are still known. But Pascal did not by any means exhaust the subject, and it remained for James Bernoulli to demonstrate fully the importance of the figurate numbers, as they are also called. In his treatise De Arte Conjectandi, he points out their application in the theory of combinations and probabilities, and remarks of the Arithmetical Triangle, "It not only contains the clue to the mysterious doctrine of combinations, but it is also the ground or foundation of most of the important and abstruse discoveries that have been made in the other branches of the mathematics." 1

The numbers of the triangle can be calculated in a very easy manner by successive additions. We commence with unity at the apex; in the next line we place a second unit to the right of this; to obtain the third line of figures we move the previous line one place to the right, and add them to the same figures as they were before removal; we can then repeat the same process ad infinitum. The fourth line of figures, for instance, contains 1, 3, 3, I; moving them one place and adding as directed we obtain :

Carrying out this simple process through ten more steps we obtain the first seventeen lines of the Arithmetical Triangle as printed on the next page. Theoretically speaking the Triangle must be regarded as infinite in extent, but the numbers increase so rapidly that it soon becomes impracticable to continue the table. The longest table of the numbers which I have found is in Fortia's "Traité des Progressions" (p. 80), where they are given up to the fortieth line and the ninth column.

1 Bernoulli, De Arte Conjectandi, translated by Francis Maseres, London, 1795, p. 75.

THE ARITHMETICAL TRIANGLE.

Line.

IO I

36 84 126 126

Examining these numbers, we find that they are connected by an unlimited series of relations, a few of the more simple of which may be noticed. Each vertical column of numbers exactly corresponds with an oblique series descending from left to right, so that the triangle is perfectly symmetrical in its contents. The first column contains only units; the second column contains the natural numbers, 1, 2, 3, &c.; the third column contains a remarkable series of numbers, 1, 3, 6, 10, 15, &c., which have long been called the triangular numbers, because they correspond with the numbers of balls which may be arranged in a triangular form, thus

The fourth column contains the pyramidal numbers, so called because they correspond to the numbers of equal balls which can be piled in regular triangular pyramids. Their differences are the triangular numbers. The numbers of the fifth column have the pyramidal numbers for their differences, but as there is no regular figure of which they express the contents, they have been arbitrarily called the trianguli-triangular numbers. The succeeding columns have, in a similar manner, been said to contain the trianguli-pyramidal, the pyramidi-pyramidal numbers, and so on.1

From the mode of formation of the table, it follows that the differences of the numbers in each column will be found in the preceding column to the left. Hence the second differences, or the differences of differences, will be in the second column to the left of any given column, the third differences in the third column, and so on. Thus we may say that unity which appears in the first column is the first difference of the numbers in the second column; the second difference of those in the third column; the third difference of those in the fourth, and so on. The triangle is seen to be a complete classification of all numbers according as they have unity for any of their differences. Since each line is formed by adding the previous line

1 Wallis's Algebra, Discourse of Combinations, &c., p. 109.

to itself, it is evident that the sum of the numbers in each horizontal line must be double the sum of the numbers in the line next above. Hence we know, without making the additions, that the successive sums must be I, 2, 4, 8, 16, 32, 64, &c., the same as the numbers of combinations in the Logical Alphabet. Speaking generally, the sum of the numbers in the nth line will be 2"-1.

Again, if the whole of the numbers down to any line be added together, we shall obtain a number less by unity than some power of 2; thus, the first line gives I or 21-1; the first two lines give 3 or 22— 1 ; the first three lines 7 or 23— I; the first six lines give 63 or 2o—I; or, speaking in general language, the sum of the first n lines is 2"- I. It follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines increased by a unit. For the sum of the nth line is, as already shown, 2"-1, and the sum of the first n-1 lines is 21, or less by a unit.

This account of the properties of the figurate numbers does not approach completeness; a considerable, probably an unlimited, number of less simple and obvious relations might be traced out. Pascal, after giving many of the properties, exclaims1: "Mais j'en laisse bien plus que je n'en donne; c'est une chose étrange combien il est fertile en propriétés! Chacun peut s'y exercer." The arithmetical triangle may be considered a natural classification of numbers, exhibiting, in the most complete manner, their evolution and relations in a certain point of view. It is obvious that in an unlimited extension of the triangle, each number, with the single exception of the number two, has at least two places.

Out

Though the properties above explained are highly curious, the greatest value of the triangle arises from the fact that it contains a complete statement of the values of the formula (p. 182), for the numbers of combinations of m things out of n, for all possible values of m and n. of seven things one may be chosen in seven ways, and seven occurs in the eighth line of the second column. The combinations of two things chosen out of seven are 1 7×6

1 X 2

or 21, which is the third number in the eighth

1 Euvres Complètes, vol. iii. p. 251.