first differences. It is evident in the present instance that that column represents the natural numbers. But we already know that the first difference of the natural numbers is constant and equal to unity. It appears, therefore, that a Table of these numbers, representing the group of marbles, might be constructed to any extent by mere addition—using the number 1 as the first number of the Table, the number 1 as the first Difference, and also the number 1 as the second Difference, which last always remains constant.
Now as we could find the value of any given number of pounds of meat directly, without going through all the previous part of the Table, so by a somewhat different rule we can find at once the value of any group whose number is given.
Thus, if we require the number of marbles in the fifth group, proceed thus:—
| Take the number of the group | 5 |
| Add 1 to this number, it becomes | 6 |
| —— | |
| Multiply these numbers together | 2)30 |
| —— | |
| Divide the product by 2 | 15 |
This gives 15, the number of marbles in the 5th group.
If the reader will take the trouble to calculate with his pencil the five groups given above, he will soon perceive the general truth of this rule.
We have now arrived at the fact that this Table—like that of the price of butchers' meat—can be calculated by two different methods. By the first, each number of the Table is calculated independently: by the second, the truth of each number depends upon the truth of all the previous numbers.
Perhaps my young friend may now ask me, What is the use of such Tables? Until he has advanced further in his
