274
Babbage's Calculating Engine.
July,
| 50
|
| 110
|
| 194
|
| 302
|
| 434
|
| 590
|
| 770
|
| 974
|
| 1202
|
| 1454
|
| 1730
|
Proceeding with this series in the same way, we obtain the following series of third differences:—
| 60
|
| 84
|
| 108
|
| 132
|
| 156
|
| 180
|
| 204
|
| 228
|
| 252
|
| 276
|
Proceeding in the same way with these, we obtain the following for the series of fourth differences:—
| 24
|
| 24
|
| 24
|
| 24
|
| 24
|
| 24
|
| 24
|
| 24
|
| 24
|
It appears, therefore, that in this case the series of fourth differences consists of a constant repetition of the number 24. Now, a slight consideration of the succession of arithmetical operations by which we have obtained this result, will show, that by reversing the process, we could obtain the table of fourth powers by the mere process of addition. Beginning with the first numbers
in each successive series of differences, and designating the table and the successive differences by the letters T, D1 D2 D3 D4, we have then the following to begin with:—
| T |
D1 |
D2 |
D3 |
D4
|
| 1 |
15 |
50 |
60 |
24
|
