On the influence of signs in mathematical reasoning / by Charles Babbage.

  • Babbage, Charles, 1791-1871.
Date:
1826
Catalogue details

Licence: Public Domain Mark

Credit: On the influence of signs in mathematical reasoning / by Charles Babbage. Source: Wellcome Collection.

Provider: This material has been provided by The Royal College of Surgeons of England. The original may be consulted at The Royal College of Surgeons of England.

    12/58 (page 10)
    accomplish his object, but the results soon became so complicated that little expectation could be formed of succeeding by that means. In this difficulty Mr. Spence introduced into the equation to be solved an arbitrary quantity a, which is merely employed as a letter by whose powers the resulting series may be arranged ; if the attempt is now made by continually eliminating, a series arises proceeding according to the powers of a, and equations are found for determining their coefficients; finally, the arbitrary quantity a having performed its office is made equal to unity, and the result is the solution of the equation. The success of this plan depends entirely on breaking into a number of separate parts a very complicated expression, each of these portions being separately reducible to known laws. On resolving into their separate parts a vast variety of questions which have occurred, it has been found that the number of individual difficulties is by no means so large as had been originally supposed; many of very different kinds have been found to depend perhaps on the same integral, or on the solution of the same equation. In proportion to the number of questions which are reduced to these new difficulties, they themselves assume importance, and the celebrity which always attaches to those who remove obstacles regarded as insuperable by their predecessors, in- duces many to attempt the solution of these purely abstract ques- tions. Perhaps these ultimate points of reference may not from their nature admit of a comparison with, or reduction to, existing tran- scendents : the labour and ingenuity employed in the attempt are not however thrown away; relations are discovered by which, from a certain number of particular cases numerically given, all others may be readily calculated, approximations are discovered for determining the cases which are required as data, and, finally, they are arranged in tables and accompanied by rules for their employment, by which, as far as results in pure numbers are