W. Brown 
355 
those of Dietze*, who employed a metronome and asked the subject to compare 
series of beats of different lengths with one another (without counting). His 
purpose was to determine how many beats could be correctly estimated without 
counting. The result was found to vary with the rapidity of the beats, the most 
favourable interval between the beats being '2 to '3 sec. 
(b) Simple Arithmetical Pi'ocesses. The psychology of addition, subtraction, 
and other simple arithmetical operations has been repeatedly studied by intro- 
spection, either casual or controlled. An example of the former is to be found in 
Binet's Psychologie des Grands Galcidateurs et Joueurs d'Echecs, and of the latter 
in an article in Vol. xvii. of the Am. J. P. (1906)f, 
(c) Higher Mathematical Processes. As regards the psychological analysis of 
higher mathematical processes, very little work has thus far been done. In almost 
all cases the data have been school and college marks or class-lists. The method 
employed has been generally that of correlation, but the coefficients of correlation 
have been worked out between mathematical ability as a whole and the equally 
complex mental capacities involved in the other main subjects of the school or 
college curriculum, e.g. Classics, French, English, etc. No instance of analysis of 
the inner relations of mathematical intelligence is known to the writer. 
III. Collection and Analysis of Data. 
Towards the close of the year 1908, the writer examined in mathematics a 
group of 83 boys belonging to the middle forms of the classical side of an 
English public school. The group consisted of five sets or forms, viz. U. V'^, 
U. V^, L. VI, U. VI and VII ; but all the boys were examined on the same 
three papers — Geometry, Arithmetic, and Algebra — and they had all been working 
along the same lines and in the same environment. The examination papers were 
set by one of the masters, and the writer was expected to return the boys' answers, 
marked, within the week. Although this arrangement meant that the work had 
to be done at high pressure, it was on the whole a fortunate one so far as the 
research was concerned, since it almost certainly rendered the standard of marking 
much more steady than the latter would have been if the marking had extended 
over a longer time. 
The papers were first marked according to ordinary school standards for the 
sake of the school examination, and then according to a differential system of 
* Dietze : "Ueber den Bewusstseinsumfang," Philos. Studien, ii. 1885. 
t C. E. Browne : "The Psychology of the Simple Arithmetical Processes," Am. J. P. Vol. xvii. Jan. 
1906 ; see also Frank D. Mitchell : "Mathematical Prodigies," Am. J. P. Vol. xviii. Jan. 1907. 
Kecently I have myself had excellent opportunities of studying these arithmetical processes intro- 
spectively. The processes seem to go on almost mechanically, but are yet controlled from moment to 
moment by auditory and kinaesthetic imagery. When any distraction occurs, either from without or 
from within, visual imagery is immediately called up, and this is sufficiently stable to allow me to pause 
for a moment and " collect my thoughts." The various forms of imagery present seem to act as controls 
of the thought, but otherwise to be quite inactive, the "driving power" coming from the standing 
purpose to add, subtract, etc. (Of. supra, p. 353, Ach's results.) 
