NATURE 
281 

THURSDAY, FEBRUARY 9g, 1871 

THE POWER OF NUMERICAL 
DISCRIMINATION 
fe is well known that the mind is unable through the 
eye to estimate any large number of objects without 
counting them successively. A small number, for in- 
stance three or four, it can certainly comprehend and 
count by an instantaneous and apparently single act of 
mental attention. The limits of this power have been the 
subject of speculation or experiment among psychologists, 
and Sir William Hamilton thus sums up almost the whole 
of what is known about it :— 
“ Supposing that the mind is not limited to the simul- 
taneous consideration of a single object, a question arises, 
How many objects can itembrace atonce? . . . . I 
find this problem stated and differently answered by 
different philosophers, and apparently without a knowledge 
of each other. By Charles Bonnet, the mind is allowed 
to have a distinct notion of six objects at once ; by Abra- 
ham Tucker the number is limited to four ; while Destutt 
Tracy again amplifies it to six. The opinion of the first 
and last of these philosophers appears to me correct. 
You can easily make the experiment for yourselves, but 
you must beware of grouping the objects into classes. If 
you throw a handful of marbles on the floor, you will find it 
difficult to view at once more than six, or seven at most, 
without confusion ; but if you group them into twos, or 
threes, or fives, you can comprehend as many groups as 
you can units, because the mind considers these groups 
only as units ; it views them as wholes, and throws their 
parts out of consideration. You may perform the experi- 
ment also by an act of imagination.” (Lectures, vol. i. 
Pp. 253-4) 
This subject seemed to me worthy of more systematic 
investigation, and it is one of the very few points in psy- 
chology which can, as far as we yet see, be submitted to 
experiment. I have not found it possible to decide con- 
clusively in the manner Hamilton suggests, whether 4 or 
5 or 6 is the limit, nor do imaginative acts of experiment 
seem likely to advance exact knowledge. Probably the 
limit is not really a definite one, and it is almost sure to 
vary somewhat in different individuals. 
I have investigated the power in my own case in the 
following manner. A round paper box 4} inches in 
diameter, lined with white paper, and with the edges cut 
down so as to stand only } inch high, was placed in the 
middle of a black tray. A quantity of uniform black beans 
was then obtained, and a number of them being taken 
up casually were thrown towards the box so that a wholly 
uncertain number fell into it. At the very moment when 
the beans came to rest, their number was estimated with- 
out the least hesitation, and then recorded together with 
the real number obtained by deliberate counting. The 
whole value of the experiment turns upon the rapidity of 
the estimation, for if we can really count five or six by a 
single mental act, we ought to be able to do it unerringly 
at the first momentary glance. 
Excluding a few trials which were consciously bad, and 
some in which the number of beans was more than 15, 
VOL, III. 

I made altogether 1,027 trials, and the following table 
contains the complete results :— 

Actuat Numsers. 





Estimated 
Numbers. = — 
sila 5 6| 7| 8 9) |) x01) zx 51 x2: | 30 mae leee 
3 23 | as 
4 | / 65 | | 
5 } | 102 7 | | 
6 4 120 18 | 
7 I | 20 (1X5 | 30 2 
8 | 25 | 76 | 24 6 I 
9 28 | 76 | 37 | 11 I 
Io x) x8))} -46°| x9) 4 
II 2) 16 | 26 | 17 7 2 
I2 | 2 12 19 It 3 2 
13 | 1. 3h) <6: 3s |an 
14 I ey | eo 
15 | Te) iiakee fl | ae 
Totals.. | 23 | 65 |107 147 (156 |135 122 7 69 | 45 | 26 | rq | xr 

The above table gives the number of trials in which 
each real number was correctly or incorrectly guessed ; 
thus in 120 cases 6 was correctly guessed ; in 7 cases 
it was mistaken for 5, and in 20 for 7. So far as my 
trials went, there was absolute freedom from error in the 
numbers 3 and 4, as might have been expected; but I 
was surprised to find that several times I fell into error as 
regards 5, which was wrongly guessed in 5 per cent. of the 
cases. Abraham Tucker thus appears more correct as to 
my power than the other philosophers. 
But in reality the question is not to be so surely decided 
by the trial of the few first numbers, as by endeavouring 
to obtain some general law pervading the whole series of 
trials. Calculating the average error of estimation in the 
case of each number, without regard to the direction of 
the error, we get the following numbers :— 
Bh Aes On 7 Oy * Oi TO. ih 120 ih eae 
OOP, S827) AA AT, 65 “80 1:73) den Tense 
These numbers vary pretty regularly in an apparently 
linear manner, except that in the case of the numbers 9 
and 12, the result is too smal!. The error is simply 
proportional to the excess of the real number over 44, or 
obeys a law expressed in the formula (z being the real 
number)— 
error = 7 X (# — 4%) 
When we calculate the constant # for each number it 
comes as follows :— 
5 Ga 7, Fie) Avene BUT NG: MEY SS yi Sis 
Shlz) 1220 -TlOMI27 Ook “1D; 125) “008: 127.1288 ton 
These numbers are sufficiently equal to enable us to 
take the average o°116 as a good result, and the formula 
then becomes— 
error = o'116 X (# — 4}) 
| or approximately— 
ln 
error = 
This is a purely empirical law, the meaning or value of 
which I cannot undertake to explain. The most curious 
point is that it seems to confirm my previous conclusion 
that my own power of estimating the number /ve is not per- 
fect. The limit of complete accuracy, if there were one, 
would be neither at 4 nor 5, but half-way between them ; 
but this is a result as puzzling as one of the uninterpretable 
symbols in mathematics, just, for instance, like the fac- 
torial of a fractional number. But I give it for what it 
may be worth. 
Q 
