The calculating engine

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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:—

Adding each number to the number on its left, and repeating 24, we get the following as