354 An Objective Study of 3lathematlcal Intelligence 
ally a psychology as well as a logic ; but this potentiality has not as yet been 
actualised by the psychologists, the reason being, as is shown by the cases 
previously mentioned, that introspection alone is insufficient to give us the 
requisite data*. 
Apart from the experiments mentioned earlier in this paper, most of the 
experimental work on the psychology of reasoning has taken the form of a study 
of " tests " of elementary physical and mental processes accepted problematically as 
symptoms of degrees of intelligence. The history of this movement is already 
recorded with sufficient fulness in Meumann's Experimentelle Pddagogik, Bd. I. 
(Leipzig, 1907). Hitherto the results obtained have been either conflicting or 
negative. In particular, the hypothesis, put forward by one investigator, that 
general intelligence and general power of sensory discrimination are identical, has 
not been confirmed by recent work. 
Mention should, however, be made of the fact that the problem of reasoning 
has been attacked more directly in a recent research by Willis L. Gard, entitled 
"A Preliminary Study of the Psychology of Reasoning," communicated to the 
American Journal of Psychology^ by Prof. Sanford. The .experiments took the 
form of getting the subject to solve simple mathematical puzzles, e.g. to fill up 
gaps in partially worked out division and multiplication suras. It seems somewhat 
doubtful whether puzzles are ever a good test of intelligence, since they really 
involve a good deal of specialisation of interest and ability. 
II. Historical. 
Turning now to the subject-proper of this paper, viz. the study of mathematical 
intelligence, we find that earlier experimental work is limited to the analysis of 
the mental processes involved in the simplest forms of arithmetical operation, — 
counting, adding, subtracting, etc. 
(a) Counting. As regards counting, experiments have been worked to decide 
the rival claims of the intuitional method (of Pestalozzi) and the counting method 
in the teaching of number. These two methods are based on the spatial and 
temporal groupings of units, respectively, and the experiments therefore fall 
naturally into two classes. The first includes the experiments of CattellJ, Lay§, 
Warrenj], MessengerlT and Helene Nanu**, who all worked with the tachistoscope, 
exposing to the subjects' view lines or dots, varying in number and arrangement, 
for very short lengths of time. Experiments belonging to the second class are 
* See William Brown : "Educational Psychology in the Secondary School," Journal of Philosophy, 
Psychology, and Scientific Methoth, Vol. vii. No. 1, Jan. 6, 1910. 
t Vol. XVIII. Oct. 1907. 
J Cattell; " Ueber die Zeit des Erkenuens u.s.w." Philos. Studien, in. 1886. 
§ Lay : Fiihrer dnrch den ersten Rechenunterricht, 1 Aiifl., 1898. 
II Warren : Princetown Contributions to Psychology, ii. 1898, Vol. 3. 
H Messenger: " The Perception of Number," Psychol. Review, Monograph Supplement, v. 1903. 
** Helene A. Nanu : " Zur Psychologie der Zahlauffassung," Wiirzburger Dissertation, 1904. 
