The real number in the 5th pyramid is 35. But the number 105 at which we have arrived is exactly three times as great. If, therefore, instead of dividing by 2 we had divided by 2 and also by 3, we should have arrived at a true result in this instance.
The amended rule is therefore— {57}
| Take the number whose tabular number is required, say | n |
|---|---|
| Add 1 to it | n + 1 |
| Add 1 to this | n + 2 |
| Multiply
these three numbers together | n × (n + 1) × (n + 2) |
| Divide by
1 × 2 × 3. The result is | (n(n + 1)(n + 2))/6 |
This rule will, upon trial, be found to give correctly every tabular number.
By similar reasoning we might arrive at the knowledge of the number of cannon balls in square and rectangular pyramids. But it is presumed that enough has been stated to enable the reader to form some general notion of the method of calculating arithmetical Tables by differences which are constant.
〈ASTRONOMICAL TABLES.〉
It may now be stated that mathematicians have discovered that all the Tables most important for practical purposes, such as those relating to Astronomy and Navigation, can, although they may not possess any constant differences, still be calculated in detached portions by that method.
Hence the importance of having machinery to calculate by differences, which, if well made, cannot err; and which, if carelessly set, presents in the last term it calculates the power of verification of every antecedent term.
Of the Mechanical Arrangements necessary for computing Tables by the Method of Differences.
From the preceding explanation it appears that all Tables may be calculated, to a greater or less extent, by the method of Differences. That method requires, for its successful {58} execution, little beyond mechanical means of performing the arithmetical operation of Addition. Subtraction can, by the aid of a well-known artifice, be converted into Addition.