cation 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.' i

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, and we can then repeat the same process ad infinitum. The fourth line of figures, for instance, contains 1, 3, 3, 1; 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 almost impracticable to continue the table. The longest table of the numbers which I have found is given in Fortia's Traité des Progressions' (p. 80), where they are given up to the fortieth line and the ninth column.

i Bernouilli, 'De Arte Conjectandi,' translated by Francis Maseres, London, 1795, p. 75.

On carefully examining these numbers, we shall find that they are connected with each other by an almost unlimited series of relations, a few of the more simple of which may be noticed.

1. 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.

2. 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

O

These numbers evidently differ each from the previous one by the series of natural numbers. Their employment has been explained, and the first 20,000 of the numbers calculated and printed by E. de Joncourt in a small quarto volume, which was published at the Hague, in 1762.

The fourth column contains the pyramidal numbers, so called because they correspond to the number 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

P

contain the trianguli-pyramidal, the pyramidi-pyramidal numbers, and so on.k

3. 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 thirdcolumn; the third difference of those in the fourth, and so on. The triangle is thus seen to be a complete classification of all numbers according as they have unity for any of their differences.

4. Every number in the table is equal to the sum of the numbers which stand higher in the next column to the left, beginning with the next line above; thus 84 is equal to the sum of 28, 21, 15, 10, 6, 3, I.

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

6. 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 1 or 2-1; the first two lines give 3 or 22-1; the first three lines 7 or 23-1; the first six lines give 63 or 26-1; or, speaking in general language, the sum of the first n lines is 2" — I.

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

7. It follows that the sum of the numbers in any one line is equal to the sum of those in all the preceding lines diminished by a unit. For the sum of the nth line is, as already shewn, 2"-1, and the sum of the first n 1 lines is 2"-1-1, or less by a unit.

-

This enumeration of the properties of the figurate numbers does not approach completeness; a considerable, perhaps 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.' 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 will have at least two places.

The

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. 206), for the number of combinations of m things out of n, for all possible values of m and n. Out 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 or 21, which is the third number in the eighth line. The combinations of three things out of seven are or 35, which appears fourth in the eighth line.

are

7×6

IX 2

7×6×5

I X2 X3

In a similar manner, in the fifth, sixth, seventh, and eighth columns of the eighth line I find it stated in how many ways I can select combinations of 4, 5, 6, and 7 things out of 7. Proceeding to the ninth line, I find in succession