360 An Objective Study of Mathematwal Intelligence 
mainly to the difference in the extent of syllabus covered in the different forms. 
This, at any rate, seems to be the most satisfactory solution of a difficulty which 
the writer himself would be the first to admit. 
The finally corrected values of the correlations are : — 
Alg. Arith. 
•76 +-03 
BD 
•27 + -07 
FG 
•41 ± -06 
Geom. Arith. 
•28 + -07 
CD 
•69 +•04 
BG 
•11 + •08 
Geom. Alg. 
•18 ±-08 
Arith. 
G 
•65 + -04 
DG 
•11 + ^08 
AB 
•42 + -06 
Alg. G 
•45 + -06 
CF 
•20 + ^07 
AC 
•64 + -05 
Geom. 
G 
•39 + ^07 
CI 
•05+^08 
AD 
•31 + ^07 
CG 
•28 +-07 
EH 
•33 +^07 
BC 
-.57 + -05 
IG 
•00 +^08 
FI 
•04+^08 
Several interesting results can be obtained from these values by applying the 
theory of multiple correlation. Thus, limiting our attention for the moment to 
Geometry, Arithmetic, and Algebra, it is possible for us to deduce the degree 
of correlation between any two of them on the assumption that the third ability is 
constant throughout the series. It is given by the formula 
= , ^" ^"^"^ . [See above, p. 359.1 
V(l-r,3^)(l-n/) 
The partial correlation coefficients in the present case are : — 
Geom. Alg. - -05 ± -OS, Geom. Arith. -23 + '07, Alg. Arith. -75 + -03. 
We may perhaps infer from these values that geometrical ability is only related 
to algebraical ability through the mediation of arithmetical ability*. 
Taking now four variates, e.g. A, B, C, and D, it is, in a similar way, possible 
to deduce the degree of correlation between any two, say C and D, on the 
assumption that the other two abilities are constant throughout the series. 
Putting the numbers 1, 2, 3, 4 in the place of A, B, C, D, the partial correlation 
coefficient 
V 1 - i\i - - r.J + 2r,.2rur^ Vl - r,^^ - - r^i + '2.ri2rnr-^ ' 
= •93. 
There is thus a very close relation indeed between memory of propositions in 
geometry, and the power of recognising general relations in a particular case 
in geometry. 
Obviously, the method could be extended indefinitely. 
A glance at the table of corrected coefficients suggests the following additional 
results : — 
(1) the ability to do percentage and proportion sums in arithmetic is more 
closely related to essential geometrical ability than to essential algebraical ability ; 
* This result harmonises with the view that mathematical reasoners fall into two types, the so-called 
"geometrical" (or "intuitional") and the "analytical" (or "logical") types. See H. Poincar6 : ia 
Valeur de la Science, pp. 11 — 15. 
t See Karl Pearson : " Selection, etc." Phil. Trans. Vol. 200 a, p. 31. 
