Passages from the Life of a Philosopher

by Charles Babbage

• Number of
  Operation
  Cards used.
• 1 a. Ascertain the number of possible roots by applying Sturm’s theorem to the coefficients.
• 2 b. Find a number greater than the greatest root.
• 3 c. Substitute the powers of ten (commencing with that next greater than the greatest root, and {132} diminishing the powers by unity at each step) for the value of x in the given equation.
• Continue this until the sign of the resulting number changes from positive to negative.
• The index of the last power of ten (call it n), which is positive, expresses the number of digits in that part of the root which consists of whole numbers. Call this index n + 1.
• 4 d. Substitute successively for x in the original equation 0 × 10ⁿ, 1 × 10ⁿ, 2 × 10ⁿ, 3 × 10ⁿ, . . . . 9 × 10ⁿ, until a change of sign occurs in the result. The digit previously substituted will be the first figure of the root sought.
• 5 e. Transform the original equation into another whose roots are less by the number thus found.
• The transformed equation will have a real root, the digit, less than 10ⁿ.
• 6 f. Substitute 1 × 10ⁿ⁻¹, 2 × 10ⁿ⁻¹, 3 × 10ⁿ⁻¹, &c., successively for the root of this equation, until a change of sign occurs in the result, as in process 4.
• This will give the second figure of the root.
• This process of alternately finding a new figure in the root, and then transforming the equation into another (as in process 4 and 5), must be carried on until as many figures as are required, whether whole numbers or decimals, are arrived at.
• 7 g. The root thus found must now be used to reduce the original equation to one dimension lower. {133}
• 8 h. This new equation of one dimension lower must now be treated by sections 3, 4, 5, 6, and 7, until the new root is found.
• 9 i. The repetition of sections 7 and 8 must go on until all the roots have been found.

Now it will be observed that Professor Mosotti was quite ready to admit at once that each of these different processes could be performed by the Analytical Machine through the medium of properly-arranged sets of Jacquard cards.

His real difficulty consisted in teaching the engine to know when to change from one set of cards to another, and back again repeatedly, at intervals not known to the person who gave the orders.

The dimensions of the algebraic equation being known, the number of arithmetical processes necessary for Sturm’s theorem is consequently known. A set of operation cards can therefore be prepared. These must be accompanied by a corresponding set of variable cards, which will represent the columns in the store, on which the several coefficients of the given equation, and the various combinations required amongst them, are to be placed.

The next stage is to find a number greater than the greatest root of the given equation. There are various courses for arriving at such a number. Any one of these being selected, another set of operation and variable cards can be prepared to execute this operation.

Now, as this second process invariably follows the first, the second set of cards may be attached to the first set, and the engine will pass on from the first to the second process, and again from the second to the third process. {134}

But here a difficulty arises: successive powers of ten are to be substituted for x in the equation, until a certain event happens. A set of cards may be provided to make the substitution of the highest power of ten, and similarly for the others; but on the occurrence of a certain event, namely, the change of a sign from + to −, this stage of the calculation is to terminate.

Now at a very early period of the inquiry I had found it necessary to teach the engine to know when any numbers it might be computing passed through zero or infinity.

The passage through zero can be easily ascertained, thus: Let the continually-decreasing number which is being computed be placed upon a column of wheels in connection with a carrying apparatus. After each process this number will be diminished, until at last a number is subtracted from it which is greater than the number expressed on those wheels.

Thus let it be	00000,00000,00000,00423
Subtract	00000,00000,00000,00511
	99999,99999,99999,99912