MS 842 (1908) - A Neglected Argument - Early Drafts

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fact that though every mathematician in those days diligently studied the Mécanique celeste, an important error in the calculation of the moon's mean acceleration remained undetected during all that period.

[various numbers]

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Inductions. A yet fourth theory of induction, that of Laplace, received, by implication, the assent of Mill; and since this theory is taught as correct in all the text-books on the Doctrine of Chances, it behooves me to state, with the utmost brevity, wherein it is demonstrably false and dangerous; and also to show how Mill assented to it.

We have seen that if we devise a method of forming a numerous sample of the class of Ss, that shall be suitable for use in ascertaining, to a certain degree of approximation, what proportion of future experiences

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of Ss would be P, in case existing general conditions should undergo no alteration, then in case there be any definite reason to expect that of the Ss coming to our experience from a certain subclass, say the Ss that are Ts, a markedly different proportion will be P, from the proportion among experienced Ss that are not T, then our suitable method of forming the sample must insure that the proportion of Ss that are Ts shall be so nearly the same in the sample as it will be in our future experience, so long as

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existing general conditions remain unchanged, that no error so large as the intended limit of our approximation need be feared on this account. Moreover, if there be any serious reason to suspect that an experience of an S presenting itself for admission to the sample is so connected with an instance or instances already admitted as to have a special tendency either to be like or to being unlike that instance or body of instances in respect to being or not being P, at least to such a degree as to affect our intended approximation, then our

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suitable method will exclude that instance from the sample. Moreover, if there be any reason to suspect that an instance has in any important degree been brought to our attentive experience by causes connected with its being or not being P, our suitable method will exclude it. Furthermore, supposing that we intend to reach such a degree of approximation that all the figures in the first 2 places of decimals shall exactly be correct, if that a 5 in a certain place of decimals, say the 2nd after the decimal-point, be the largest that will be a matter of no concern in the expression of the probability-limit of the ratio of future Ss that are P to all the Ss, then the suitable rule will provide for the collection of a number of instances equal to 0.45434 x 100P. If, however, we are sure that one tenth