One day Mosotti, who had been unavoidably absent from the previous meeting, when a question of great importance had been discussed, again joined the party. Well aware of the acuteness and rapidity of my friend's intellect, I asked my other friends to allow me five minutes to convey to Professor Mosotti the substance of the preceding sitting. After putting a few questions to Mosotti himself, he placed before me distinctly his greatest difficulty.
He remarked that he was now quite ready to admit the power of mechanism over numerical, and even over algebraical relations, to any extent. But he added that he had no conception how the machine could perform the act of judgment sometimes required during an analytical inquiry, when two or more different courses presented themselves, especially as the proper course to be adopted could not be known in many cases until all the previous portion had been gone through.
I then inquired whether the solution of a numerical equation of any degree by the usual, but very tedious proceeding of approximation would be a type of the difficulty to be explained. He at once admitted that it would be a very eminent one.
For the sake of perspicuity and brevity I shall confine my present explanation to possible roots.
I then mentioned the successive stages:—
Number of Operation | ||
1
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a. | Ascertain the number of possible roots by applying Sturm's theorem to the coefficients.
|
2
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b. | Find a number greater than the greatest root.
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3
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c. | Substitute the powers of ten (commencing with that next greater than the greatest root, and 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. |
