Passages from the Life of a Philosopher - Charles Babbage

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1st Dif- ference. 2nd Dif- ference. 3rd Dif- ference. 1 1 3 3 1 2 4 6 4 1 3 10 10 5 1 4 20 15 6 5 35 21 6 56

      He will at once perceive that this Table of the number of cannon balls contained in a triangular pyramid can be carried to any extent by simply adding successive differences, the third of which is constant.

      The next step will naturally be to inquire how any number in this Table can be calculated by itself. A little consideration will lead him to a fair guess; a little industry will enable him to confirm his conjecture.

      〈NUMBER IN EACH PILE.〉

      It will be observed at p. 49 that in order to find {56} independently any number of the Table of the price of butchers’ meat, the following rule was observed:—

      Take the number whose tabular number is required.

      Multiply it by the first difference.

      This product is equal to the required tabular number.

      Again, at p. 53, the rule for finding any triangular number was:—

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 is the number of marbles in the 5th group.

      Now let us make a bold conjecture respecting the Table of cannon balls, and try this rule:—

Take the number whose tabular number is required, say 5
Add 1 to that number 6
Add 1 more to that number 7
Multiply all three numbers together 2)210
Divide by 2 105

      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.

       Table of Contents

      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.

      〈ADDITION.〉

      The process of Addition includes two distinct parts—1st. The first consists of the addition of any one digit to another digit; 2nd. The second consists in carrying the tens to the next digit above.

      Let us take the case of the addition of the two following numbers, in which no carriages occur:—

       6023

       1970

       7993

      It will be observed that, in making this addition, the mind acts by successive steps. The person adding says to himself—

       0 and 3 make three,

       7 and 2 make nine,

       9 and 0 make nine,

       1 and 6 make seven.

      〈CARRIAGE.〉

      In the following addition there are several carriages:—

       2648

       4564

       7212

      The person adding says to himself—

      4 and 8 make 12: put down 2 and carry one.

      1 and 6 are 7 and 4 make 11: put down 1 and carry one.

      1 and 5 are 6 and 6 make 12: put down 2 and carry one.

      1 and 4 are 5 and 2 make 7: put down 7. and carry non

      Now, the length of time required for adding one number to another is mainly dependent upon the number of figures to {59} be added. If we could tell the average time required by the mind to add two figures together, the time required for adding any given number of figures to another equal number would be found by multiplying that average time by the number of digits in either number.

      When we attempt to perform such additions by machinery we might follow exactly the usual process of the human mind. In that case we might take a series of wheels, each having marked on its edges the digits 0, 1, 2, 3, 4, 5, 6, 7, 8,

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