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

  • Charles Babbage
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.

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    and the profit or loss by that bet will be ' ( _ i)“ - v(_ I)*-} ( _ ly a new undetermined letter being again introduced. By these means his wth stake will be represented by M — — ij( — 1— l)'- ... V ( — l)*; and thus all the n cases become reduced into one expression. The influence of this indeterminateness in contributing to the success of the solution of the following problem is very apparent. Supposing a player to regulate his stakes in the manner just described, and that his chances of winning and losing are equal, what will he have profited after p + g events, p of which have been favorable, and g have been unfavorable? At a first view it might perhaps be considered that the data are insufficient for the solution, and that the order in which the events take place ought to be given: the result however of the investigation will prove that the profit or loss is entirely indepen- dent of that consideration. It has already been shewn that his profit on each event will be as follows: u (-ly^v {(-1)“} (-l)^ W(-1)'-V {(-!)“+ (-1)^1 (-1)% u(-iy-v {(_!)“ + (~i)» + (-lyi (- in U {(- i)“+ (-1)^+ (-1)’”} (—1)”. On examining the composition of that collection of terms which multiply V it will appear that the sum of all the above quantities is equal to -vx the sum of all the products two by two which can) be formed out of the quantities ( —in...( — C