Now as each seven-fold or less-than-seven-fold statement is complementary to a nine-fold or more-than-nine-fold statement, it follows that the complete number of types will be 159 × 2 + 78 = 396.
It appears then that the types of statement concerning four classes are only about 26 times as numerous as those concerning three classes, fifteen in number, although the number of possible combinations is 256 times as great.
Professor Clifford informs me that the knowledge of the possible groupings of subdivisions of classes which he obtained by this inquiry has been of service to him in some applications of hyper-elliptic functions to which he has subsequently been led. Professor Cayley has since expressed his opinion that this line of investigation should be followed out, owing to the bearing of the theory of compound combinations upon the higher geometry.2 It seems likely that many unexpected points of connection will in time be disclosed between the sciences of logic and mathematics.
Distinction between Perfect and Imperfect Induction.
We cannot proceed with advantage before noticing the extreme difference which exists between cases of perfect and those of imperfect induction. We call an induction perfect when all the objects or events which can possibly come under the class treated have been examined. But in the majority of cases it is impossible to collect together, or in any way to investigate, the properties of all portions of a substance or of all the individuals of a race. The number of objects would often be practically infinite, and the greater part of them might be beyond our reach, in the interior of the earth, or in the most distant parts of the Universe. In all such cases induction is imperfect, and is affected by more or less uncertainty. As some writers have fallen into much error concerning the functions and