in its action, and not tied to reproduce hard and persistent forms;
it may then produce generalised pictures out of its past experiences
quite automatically. It has no difficulty in reducing images to the
same scale, owing to our constant practice in watching objects as
they approach or recede, and consequently grow or diminish in
apparent size. It readily shifts images to any desired point of the
field of view, owing to our habit of looking at bodies in motion to
the right or left, upward or downward. It selects images that
present the same aspect, either by a simple act of memory or by a
feat of imagination that forces them into the desired position, and
it has little or no difficulty in reversing them from right to left,
as if seen in a looking-glass. In illustration of these generalised
mental images, let us recur to the boat, and suppose the speaker to
continue as follows:--"The boat was a four-oared racing-boat, it was
passing quickly to the left just in front of me, and the men were
bending forward to take a fresh stroke." Now at this point of the
story the listener ought to have a picture well before his eye. It
ought to have the distinctness of a real four-oar going to the left,
at the moment when many of its details still remained unheeded, such
as the dresses of the men and their individual features. It would be
the generic image of a four-oar formed by the combination into a
single picture of a great many sight memories of those boats.
In the highest minds a descriptive word is sufficient to evoke
crowds of shadowy associations, each striving to manifest itself.
When they differ so much from one another as to be unfitted for
combination into a single idea, there will be a conflict, each being
prevented by the rest from obtaining sole possession of the field of
consciousness. There could, therefore, be no definite imagery so
long as the aggregate of all the pictures that the word suggested of
objects presenting similar aspects, reduced to the same size, and
accurately superposed, resulted in a blur; but a picture would
gradually evolve as qualifications were added to the word, and it
would attain to the distinctness and vividness of a generic image
long before the word had been so restricted as to be individualised.
If the intellect be slow, though correct in its operations, the
associations will be few, and the generalised image based on
insufficient data. If the visualising power be faint, the
generalised image will be indistinct.
I cannot discover any closer relation between high visualising power
and the intellectual faculties than between verbal memory and those
same faculties. That it must afford immense help in some professions
stands to reason, but in ordinary social life the possession of a
high visualising power, as of a high verbal memory, may pass quite
unobserved. I have to the last failed in anticipating the character
of the answers that my friends would give to my inquiries, judging
from my previous knowledge of them; though I am bound to say that,
having received their answers, I could usually persuade myself that
they were justified by my recollections of their previous sayings
and conduct generally.
The faculty is undoubtedly useful in a high degree to inventive
mechanicians, and the great majority of those whom I have questioned
have spoken of their powers as very considerable. They invent their
machines as they walk, and see them in height, breadth, and depth as
real objects, and they can also see them in action. In fact, a
periodic action of any kind appears to be easily recalled. But the
powers of other men are considerably less; thus an engineer officer
who has himself great power of visual memory, and who has
superintended the mathematical education of cadets, doubts if one in
ten can visualise an object in three dimensions. I should have
thought the faculty would be common among geometricians, but many of
the highest seem able somehow to get on without much of it. There is
a curious dictum of Napoleon I. quoted in Hume's _Précis of Modern
Tactics_, p. 15, of which I can neither find the original authority
nor do I fully understand the meaning. He is reported to have said
that "there are some who, from some physical or moral peculiarity of
character, form a picture (_tableau_) of everything. No matter what
knowledge, intellect, courage, or good qualities they may have,
these men are unfit to command." It is possible that "tableau"
should be construed rather in the sense of a pictorial composition,
which, like an epigrammatic sentence, may be very complete and
effective, but not altogether true.
There can, however, be no doubt as to the utility of the visualising
faculty when it is duly subordinated to the higher intellectual
operations. A visual image is the most perfect form of mental
representation wherever the shape, position, and relations of
objects in space are concerned. It is of importance in every
handicraft and profession where design is required. The best workmen
are those who visualise the whole of what they propose to do, before
they take a tool in their hands. The village smith and the carpenter
who are employed on odd jobs employ it no less for their work than
the mechanician, the engineer, and the architect. The lady's maid
who arranges a new dress requires it for the same reason as the
decorator employed on a palace, or the agent who lays out great
estates. Strategists, artists of all denominations, physicists who
contrive new experiments, and in short all who do not follow routine,
have need of it. The pleasure its use can afford is immense. I have
many correspondents who say that the delight of recalling beautiful
scenery and great works of art is the highest that they know; they
carry whole picture galleries in their minds. Our bookish and wordy
education tends to repress this valuable gift of nature. A faculty
that is of importance in all technical and artistic occupations,
that gives accuracy to our perceptions, and justness to our
generalisations, is starved by lazy disuse, instead of being
cultivated judiciously in such a way as will on the whole bring the
best return. I believe that a serious study of the best method of
developing and utilising this faculty, without prejudice to the
practice of abstract thought in symbols, is one of the many pressing
desiderata in the yet unformed science of education.
NUMBER-FORMS.
Persons who are imaginative almost invariably think of _numerals_ in
some form of visual imagery. If the idea of _six_ occurs to them,
the word "six" does not sound in their mental ear, but the figure 6
in a written or printed form rises before their mental eye. The
clearness of the images of numerals, and the number of them that can
be mentally viewed at the same time, differs greatly in different
persons. The most common case is to see only two or three figures at
once, and in a position too vague to admit of definition. There are
a few persons in whom the visualising faculty is so low that they
can mentally see neither numerals nor anything else; and again there
are a few in whom it is so high as to give rise to hallucinations.
Those who are able to visualise a numeral with a distinctness
comparable to reality, and to behold it as if it were before their
eyes, and not in some sort of dreamland, will define the direction in
which it seems to lie, and the distance at which it appears to be.
If they were looking at a ship on the horizon at the moment that the
figure 6 happened to present itself to their minds, they could say
whether the image lay to the left or right of the ship, and whether
it was above or below the line of the horizon; they could always
point to a definite spot in space, and say with more or less
precision that that was the direction in which the image of the
figure they were thinking of, first appeared.
Now the strange psychological fact to which I desire to draw
attention, is that among persons who visualise figures clearly there
are many who notice that the image of the same figure invariably
makes its first appearance in the same direction, and at the same
distance. Such a person would always see the figure when it first
appeared to him at (we may suppose) one point of the compass to the
left of the line between his eye and the ship, at the level of the
horizon, and at twenty feet distance. Again, we may suppose that he
would see the figure 7 invariably half a point to the left of the
ship, at an altitude equal to the sun's diameter above the horizon,
and at thirty feet distance; similarly for all the other figures.
Consequently, when he thinks of the series of numerals 1, 2, 3, 4,
etc., they show themselves in a definite pattern that always
occupies an identical position in his field of view with respect to
the direction in which he is looking.
Those who do not see figures with the same objectivity, use
nevertheless the same expressions with reference to their mental
field of view. They can draw what they see in a manner fairly
satisfactory to themselves, but they do not locate it so strictly in
reference to their axis of sight and to the horizontal plane that
passes through it. It is with them as in dreams, the imagery is
before and around, but the eyes during sleep are turned inwards and
upwards.
The pattern or "Form" in which the numerals are seen is by no means
the same in different persons, but assumes the most grotesque
variety of shapes, which run in all sorts of angles, bends, curves,
and zigzags as represented in the various illustrations to this
chapter. The drawings, however, fail in giving the idea of their
apparent size to those who see them; they usually occupy a wider
range than the mental eye can take in at a single glance, and compel
it to wander. Sometimes they are nearly panoramic.
These Forms have for the most part certain characteristics in common.
They are stated in all cases to have been in existence, so far as
the earlier numbers in the Form are concerned, as long back as the
memory extends; they come into view quite independently of the will,
and their shape and position, at all events in the mental field of
view, is nearly invariable. They have other points in common to
which I shall shortly draw attention, but first I will endeavour to
remove all doubt as to the authenticity and trustworthiness of these
statements.
I see no "Form" myself, and first ascertained that such a thing
existed through a letter from Mr. G. Bidder, Q.C., in which he
described his own case as a very curious peculiarity. I was at the
time making inquiries about the strength of the visualising faculty
in different persons, and among the numerous replies that reached me
I soon collected ten or twelve other cases in which the writers
spoke of their seeing numerals in definite forms. Though the
information came from independent sources, the expressions used were
so closely alike that they strongly corroborated one another. Of
course I eagerly followed up the inquiry, and when I had collected
enough material to justify publication, I wrote an account which
appeared in _Nature_ on 15th January 1880, with several illustrations.
This has led to a wide correspondence and to a much-increased store
of information, which enables me to arrive at the following
conclusions. The answers I received whenever I have pushed my
questions, have been straightforward and precise. I have not
unfrequently procured a second sketch of the Form even after more
than two years' interval, and found it to agree closely with the
first one. I have also questioned many of my own friends in general
terms as to whether they visualise numbers in any particular way.
The large majority are unable to do so. But every now and then I
meet with persons who possess the faculty, and I have become
familiar with the quick look of intelligence with which they receive
my question. It is as though some chord had been struck which had
not been struck before, and the verbal answers they give me are
precisely of the same type as those written ones of which I have now
so many. I cannot doubt of the authenticity of independent statements
which closely confirm one another, nor of the general accuracy of
the accompanying sketches, because I find now that my collection is
large enough for classification, that they might be arranged in an
approximately continuous series. I am often told that the
peculiarity is common to the speaker and to some near relative, and
that they had found such to be the case by accident. I have the
strongest evidence of its hereditary character after allowing, and
over-allowing, for all conceivable influences of education and
family tradition.
Last of all, I took advantage of the opportunity afforded by a
meeting of the Anthropological Institute to read a memoir there on
the subject, and to bring with me many gentlemen well known in the
scientific world, who have this habit of seeing numerals in Forms,
and whose diagrams were suspended on the walls. Amongst them are
Mr. G. Bidder, Q.C., the Rev. Mr. G. Henslow, the botanist;
Prof. Schuster, F.R.S., the physicist; Mr. Roget, Mr. Woodd Smith,
and Colonel Yule, C.B., the geographer. These diagrams are given
in Plate I. Figs. 20-24. I wished that some of my foreign
correspondents could also have been present, such as M. Antoine
d'Abbadie, the well-known French traveller and Membre de l'Institut,
and Baron v. Osten Sacken, the Russian diplomatist and entomologist,
for they had given and procured me much information.
I feel sure that I have now said enough to remove doubts as to the
authenticity of my data. Their trustworthiness will, I trust, be
still more apparent as I proceed; it has been abundantly manifest to
myself from the internal evidences in a large mass of correspondence,
to which I can unfortunately do no adequate justice in a brief memoir.
It remains to treat the data in the same way as any other scientific
facts and to extract as much meaning from them as possible.
The peculiarity in question is found, speaking very roughly, in about
1 out of every 30 adult males or 15 females. It consists in the
sudden and automatic appearance of a vivid and invariable "Form" in
the mental field of view, whenever a numeral is thought of, in which
each numeral has its own definite place. This Form may consist of a
mere line of any shape, of a peculiarly arranged row or rows of
figures, or of a shaded space.
I give woodcuts of representative specimens of these Forms, and very
brief descriptions of them extracted from the letters of my
correspondents. Sixty-three other diagrams on a smaller scale will
be found in Plates I., II. and III., and two more which are coloured
are given in Plate IV.
[Illustration: ]
D.A. "From the very first I have seen numerals up to nearly 200,
range themselves always in a particular manner, and in thinking of a
number it always takes its place in the figure. The more attention I
give to the properties of numbers and their interpretations, the
less I am troubled with this clumsy framework for them, but it is
indelible in my mind's eye even when for a long time less
consciously so. The higher numbers are to me quite abstract and
unconnected with a shape. This rough and untidy [8] production is
the best I can do towards representing what I see. There was a
little difficulty in the performance, because it is only by catching
oneself at unawares, so to speak, that one is quite sure that what
one sees is not affected by temporary imagination. But it does not
seem much like, chiefly because the mental picture never seems
_on_ the flat but _in_ a thick, dark gray atmosphere deepening in
certain parts, especially where 1 emerges, and about 20. How I get
from 100 to 120 I hardly know, though if I could require these
figures a few times without thinking of them on purpose, I should
soon notice. About 200 I lose all framework. I do not see the actual
figures very distinctly, but what there is of them is distinguished
from the dark by a thin whitish tracing. It is the place they take
and the shape they make collectively which is invariable. Nothing
more definitely takes its place than a person's age. The person is
usually there so long as his age is in mind."
[Footnote 8: The engraver took much pains to interpret the meaning
of the rather faint but carefully made drawing, by strengthening
some of the shades. The result was very very satisfactory, judging
from the author's own view of it, which is as follows:--"Certainly
if the engraver has been as successful with all the other
representations as with that of my shape and its accompaniments,
your article must be entirely correct."]
T. M. "The representation I carry in my mind of the numerical series
is quite distinct to me, so much so that I cannot think of any
number but I at once see it (as it were) in its peculiar place in
the diagram. My remembrance of dates is also nearly entirely
dependent on a clear mental vision of their _loci_ in the diagram.
This, as nearly as I can draw it, is the following:--"
[Illustration: ]
"It is only approximately correct (if the term 'correct' be at all
applicable). The numbers seem to approach more closely as I ascend
from 10 to 20, 30, 40, etc. The lines embracing a hundred numbers
also seem to approach as I go on to 400, 500, to 1000. Beyond 1000 I
have only the sense of an infinite line in the direction of the arrow,
losing itself in darkness towards the millions. Any special number
of thousands returns in my mind to its position in the parallel
lines from 1 to 1000. The diagram was present in my mind from early
childhood; I remember that I learnt the multiplication table by
reference to it at the age of seven or eight. I need hardly say that
the impression is not that of perfectly straight lines, I have
therefore used no ruler in drawing it."
J.S. "The figures are about a quarter of an inch in length, and in
ordinary type. They are black on a white ground. The numeral 200
generally takes the place of 100 and obliterates it. There is no
light or shade, and the picture is invariable."
[Illustration: ]
etc. etc.
120+---------------
|
|
|110
|
30 40 50 60 70 80 90 |
/
20|
|
|
10|
|
1|
In some cases, the mental eye has to travel along the faintly-marked
and blank paths of a Form, to the place where the numeral that is
wanted is known to reside, and then the figure starts into sight. In
other cases all the numerals, as far as 100 or more, are faintly
seen at once, but the figure that is wanted grows more vivid than its
neighbours; in one of the cases there is, as it were, a chain, and
the particular link rises as if an unseen hand had lifted it. The
Forms are sometimes variously coloured, occasionally very
brilliantly (see Plate IV.). In all of these the definition and
illumination vary much in different parts. Usually the Forms fade
away into indistinctness after 100; sometimes they come to a dead
stop. The higher numbers very rarely fill so large a space in the
Forms as the lower ones, and the diminution of space occupied by
them is so increasingly rapid that I thought it not impossible they
might diminish according to some geometrical law, such as that which
governs sensitivity. I took many careful measurements and averaged
them, but the result did not justify the supposition.
It is beyond dispute that these forms originate at an early age;
they are subsequently often developed in boyhood and youth so as to
include the higher numbers, and, among mathematical students, the
negative values.
Nearly all of my correspondents speak with confidence of their Forms
having been in existence as far back as they recollect. One states
that he knows he possessed it at the age of four; another, that he
learnt his multiplication table by the aid of the elaborate mental
diagram he still uses. Not one in ten is able to suggest any clue as
to their origin. They cannot be due to anything written or printed,
because they do not simulate what is found in ordinary writings or
books.
About one-third of the figures are curved to the left, two-thirds to
the right; they run more often upward than downward. They do not
commonly lie in a single plane. Sometimes a Form has twists as well
as bends, sometimes it is turned upside down, sometimes it plunges
into an abyss of immeasurable depth, or it rises and disappears in
the sky. My correspondents are often in difficulties when trying to
draw them in perspective. One sent me a stereoscopic picture
photographed from a wire that had been bent into the proper shape.
In one case the Form proceeds at first straightforward, then it
makes a backward sweep high above head, and finally recurves into
the pocket, of all places! It is often sloped upwards at a slight
inclination from a little below the level of the eye, just as
objects on a table would appear to a child whose chin was barely
above it.
It may seem strange that children should have such bold conceptions
as of curves sweeping loftily upward or downward to immeasurable
depths, but I think it may be accounted for by their much larger
personal experience of the vertical dimension of space than adults.
They are lifted, tossed and swung, but adults pass their lives very
much on a level, and only judge of heights by inference from the
picture on their retina. Whenever a man first ventures up in a
balloon, or is let, like a gatherer of sea-birds' eggs, over the
face of a precipice, he is conscious of having acquired a much
extended experience of the third dimension of space.
The character of the forms under which historical dates are
visualised contrast strongly with the ordinary Number-Forms. They
are sometimes copied from the numerical ones, but they are more
commonly based both clearly and consciously on the diagrams used in
the schoolroom or on some recollected fancy.
The months of the year are usually perceived as ovals, and they as
often follow one another in a reverse direction to those of the
figures on the clock, as in the same direction. It is a common
peculiarity that the months do not occupy equal spaces, but those
that are most important to the child extend more widely than the rest.
There are many varieties as to the topmost month; it is by no means
always January.
The Forms of the letters of the alphabet, when imaged, as they
sometimes are, in that way, are equally easy to be accounted for,
therefore the ordinary Number-Form is the oldest of all, and
consequently the most interesting. I suppose that it first came into
existence when the child was learning to count, and was used by him
as a natural mnemonic diagram, to which he referred the spoken words
"one," "two," "three," etc. Also, that as soon as he began to read,
the visual symbol figures supplanted their verbal sounds, and
permanently established themselves on the Form. It therefore existed
at an earlier date than that at which the child began to learn to
read; it represents his mental processes at a time of which no other
record remains; it persists in vigorous activity, and offers itself
freely to our examination.
The teachers of many schools and colleges, some in America, have
kindly questioned their pupils for me; the results are given in the
two first columns of Plate I. It appears that the proportion of
young people who see numerals in Forms is greater than that of adults.
But for the most part their Forms are neither well defined nor
complicated. I conclude that when they are too faint to be of
service they are gradually neglected, and become wholly forgotten;
while if they are vivid and useful, they increase in vividness and
definition by the effect of habitual use. Hence, in adults, the two
classes of seers and non-seers are rather sharply defined, the
connecting link of intermediate cases which is observable in
childhood having disappeared.
These Forms are the most remarkable existing instances of what is
called "topical" memory, the essence of which appears to lie in the
establishment of a more exact system of division of labour in the
different parts of the brain, than is usually carried on. Topical
aids to memory are of the greatest service to many persons, and
teachers of mnemonics make large use of them, as by advising a
speaker to mentally associate the corners, etc., of a room with the
chief divisions of the speech he is about to deliver. Those who feel
the advantage of these aids most strongly are the most likely to
cultivate the use of numerical forms. I have read many books on
mnemonics, and cannot doubt their utility to some persons; to myself
the system is of no avail whatever, but simply a stumbling-block,
nevertheless I am well aware that many of my early associations are
fanciful and silly.