of his idea of a difference machine for computing tables. What is the fundamental idea of the method of differences? Write down the square numbers in the first column, the differences
| Squares | First Differences |
Second Differences |
Third Differences |
| 1 | 3 | 2 | 0 |
| 4 | 5 | 2 | 0 |
| 9 | 7 | 2 | 0 |
| 16 | 9 | 2 | 0 |
| 25 | 11 | 2 | |
| 36 | 13 | ||
| 49 |
between the successive squares in the second column, and the differences of the first difference in the third column; these last are constant, consequently the next differences are all zero. To compute a table of squares, then, it is only necessary to add to a square the preceding first and second differences, thus 49+13 + 2 = 64, etc. In the case of logarithms and other transcendental functions there is no difference which becomes zero, but when a certain number of figures only are required, there is a difference which is zero within a certain range. Hence within that range the same process of calculation may be applied as for a function which has a certain order of differences constant. To calculate tables by a machine only a device for adding is required; to insure accuracy in the printed tables Babbage thought it necessary that the machine which computes the results should also print them.
By 1822 Babbage had constructed a small model having two orders of differences and applicable to computing numbers of from six to eight places. It could compute squares, triangular numbers, values of , and values of any function of which the second difference was constant and not greater than about 1000. He exhibited this model to the Royal Astronomical Society and was subsequently awarded a gold medal on account of it. He also wrote a public letter to Sir Humphrey Davy, then president of the Royal Society, explaining the utility of his invention. Through what had been published the
