1()8 PART ILSOME IMPORTANT METHODOLOGICAL TERMS. 



It has been objected that complete induction is the equivalent 

 of simple enumeration, but psychologically this does not appear 

 to be so. The value of the complete enumeration, exactly as 

 in incomplete induction, resides in the comprehensive summing 

 up. 1 -f- 1 is commonly written 2, or 1 -f- 1 == 2 ; but really 

 1 -f- 1 is not equal to 2, it is not even equal to 1 -f- 1 ; it is 

 1 -f- I. 1 The 2 holds together the two 1's, and represents a new 

 fact. That is to say, since the one matter of consequence is 

 to arrive at a general statement, it follows that it is immaterial 

 whether we reach it by inference or by enumeration. 2 Indeed, 

 the word All indicates a process of thought not involved in the 

 enumeration, say, of twenty objects, for, abstractly speaking, 

 twenty objects may be but a portion of "all" objects, whereas 

 by the term All we judge that the class consists of no fewer 

 and no more than twenty objects. "All the books in this lib- 

 rary are English books" (Jevons), expresses a qualitatively 

 different statement from "435 books in this library are English 

 books". A study of the mathematical notions of children be- 

 tween the ages of three and five would help to fix these delicate 

 methodological and psychological distinctions. 



Sonne mit ihren Umlaufszeiten." (C. S. Cornelius, Uber die Bedeutung des 

 Causalprinzips in der Naturwissenschaft, Halle, 1867, p. 7.) 



"One would naturally suppose that the colours and lines of mother-of-pearl 

 were due to the chemical or physical character of the substance itself. Sir 

 David Brewster, however, happened to take an impression of a piece of 

 mother-of-pearl in beeswax and resin, and was surprised to see the colours 

 reproduced upon its surface. He then took a number of other impressions 

 in balsam, gum-arabic, lead, etc., and found the iridescent colours repeated 

 in every case. In this way he proved that the colours were caused by the 

 form of the substance, and not by its chemical qualities or physical com- 

 position." (J. E. Creighton, An Introduction to Logic, pp. 202-203.) 



"To determine its position [the position of the cirriped] he studied the 

 structure of as many genera as possible. Dr. J. E. Gray, who had already 

 collected a large amount of material for a monograph of the group, turned 

 it over to Darwin." (Frank Cramer, op. cit., pp. 49-50.) In order to ascer- 

 tain whether the primrose and the cowslip were different forms of the same 

 species, "he transplanted cowslips from the fields inlo a shrubbery, and 

 then into highly manured land; the next year they were protected from in- 

 sects, artificially fertilised", and seed grown, which was sown in a hot-bed. 

 The young plants were set out, some in very rich soil, some in stiff, poor 

 clay, some in old peat, and others in pots in the greenhouse 765 in all." 

 (Ibid., pp. 82-83.) 



"The presence of the fat-soluble factor . . . has also been found present 

 in many oils and fats derived from the animal kingdom, as for example, cod- 

 liver oil, shark-liver oil, beef fat, the fats of kidneys, heart muscle and liver 

 tissues, herring oil, cod oil, salmon oil, and whale oil." (Report on . . . 

 Vita mines, p. 21.) 



" Sir Charles Lyell was preparing a third edition of his Principles, and, as 

 was his habit, visited every site in Europe where any discovery of note 

 had been made." (A. Keith, op. cit., p. 48.) 



1 On the philosophical aspect of this problem, see Leon Brunschvicg, 

 Les etapes de la philosophic mathe~matique, Paris, 1912, ch. 21. 



"The problem of all inferential processes is naturally this, from given 

 data or premisses to develop as much new truth as possible." (Lotze, Logic, 

 vol. 1, pp. 133-134.) It is new truths in which mankind is interested. 



