He might also want to know how many marbles the thirtieth or any other distant group might contain. Perhaps he might go to papa to obtain this information; but I much fear papa would snub him, and would tell him that it was nonsense—that it was useless—that nobody knew the number, and so forth. If the boy is told by papa, that he is not able to answer the question, then I recommend him to pay careful attention to whatever that father may at any time say, for he has overcome two of the greatest obstacles to the acquisition {52} of knowledge—inasmuch as he possesses the consciousness that he does not know—and he has the moral courage to avow it.[13]
[13] The most remarkable instance I ever met with of the distinctness with which any individual perceived the exact boundary of his own knowledge, was that of the late Dr. Wollaston.
If papa fail to inform him, let him go to mamma, who will not fail to find means to satisfy her darling’s curiosity. In the meantime the author of this sketch will endeavour to lead his young friend to make use of his own common sense for the purpose of becoming better acquainted with the triangular figures he has formed with his marbles.
〈SECOND DIFFERENCE CONSTANT.〉
In the case of the Table of the price of butchers’ meat, it was obvious that it could be formed by adding the same constant difference continually to the first term. Now suppose we place the numbers of our groups of marbles in a column, as we did our prices of various weights of meat. Instead of adding a certain difference, as we did in the former case, let us subtract the figures representing each group of marbles from the figures of the succeeding group in the Table. The process will stand thus:—
| Number of the Group. | Table. | 1st Difference. | 2nd Difference. |
|---|---|---|---|
| Number of Marbles in each Group. | Difference between the number of Marbles in each Group and that in the next. | ||
| 1 | 1 | 1 | 1 |
| 2 | 3 | 2 | 1 |
| 3 | 6 | 3 | 1 |
| 4 | 10 | 4 | 1 |
| 5 | 15 | 5 | 1 |
| 6 | 21 | 6 | |
| 7 | 28 | 7 |
It is usual to call the third column thus formed the column of {53} 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. | |