strange and peculiar ways? the reply is, that different persons have
natural fancies for different lines and curves. Their handwriting
shows this, for handwriting is by no means solely dependent on the
balance of the muscles of the hand, causing such and such strokes to
be made with greater facility than others. Handwriting is greatly
modified by the fashion of the time. It is in reality a compromise
between what the writer most likes to produce, and what he can
produce with the greatest ease to himself. I am sure, too, that I
can trace a connection between the general look of the handwritings
of my various correspondents and the lines of their Forms. If a
spider were to visualise numerals, we might expect he would do so in
some web-shaped fashion, and a bee in hexagons. The definite
domestic architecture of all animals as seen in their nests and
holes shows the universal tendency of each species to pursue their
work according to certain definite lines and shapes, which are to
them instinctive and in no way, we may presume, logical. The same is
seen in the groups and formations of flocks of gregarious animals
and in the flights of gregarious birds, among which the wedge-shaped
phalanx of wild ducks and the huge globe of soaring storks are as
remarkable as any.
I used to be much amused during past travels in watching the
different lines of search that were pursued by different persons in
looking for objects lost on the ground, when the encampment was
being broken up. Different persons had decided idiosyncracies, so
much so that if their travelling line of sight could have scored a
mark on the ground, I think the system of each person would have
been as characteristic as his Number-Form.
Children learn their figures to some extent by those on the clock. I
cannot, however, trace the influence of the clock on the Forms in
more than a few cases. In two of them the clock-face actually appears,
in others it has evidently had a strong influence, and in the rest
its influence is indicated, but nothing more. I suppose that the
complex Roman numerals in the clock do not fit in sufficiently well
with the simpler ideas based upon the Arabic ones.
The other traces of the origin of the Forms that appear here and
there, are dominoes, cards, counters, an abacus, the fingers,
counting by coins, feet and inches (a yellow carpenter's rule
appears in one case with 56 in large figures upon it), the country
surrounding the child's home, with its hills and dales, objects in
the garden (one scientific man sees the old garden walk and the
numeral 7 at a tub sunk in the ground where his father filled his
watering-pot). Some associations seem connected with the objects
spoken of in the doggerel verses by which children are often taught
their numbers.
But the paramount influence proceeds from the names of the numerals.
Our nomenclature is perfectly barbarous, and that of other civilised
nations is not better than ours, and frequently worse, as the French
"quatre-vingt dix-huit," or "four score, ten and eight," instead of
ninety-eight. We speak of ten, eleven, twelve, thirteen, etc., in
defiance of the beautiful system of decimal notation in which we
write those numbers. What we see is one-naught, one-one, one-two, etc.,
and we should pronounce on that principle, with this proviso, that
the word for the "one" having to show both the place and the value,
should have a sound suggestive of "one" but not identical with it.
Let us suppose it to be the letter _o_ pronounced short as in
"on," then instead of ten, eleven, twelve, thirteen, etc., we might
say _on-naught, on-one, on-two, on-three_, etc.
The conflict between the two systems creates a perplexity, to which
conclusive testimony is borne by these numerical forms. In most of
them there is a marked hitch at the 12, and this repeats itself at
the 120. The run of the lines between 1 and 20 is rarely analogous
to that between 20 and 100, where it usually first becomes regular.
The 'teens frequently occupy a larger space than their due. It is not
easy to define in words the variety of traces of the difficulty and
annoyance caused by our unscientific nomenclature, that are
portrayed vividly, and, so to speak, painfully in these pictures.
They are indelible scars that testify to the effort and ingenuity
with which a sort of compromise was struggled for and has finally
been effected between the verbal and decimal systems. I am sure that
this difficulty is more serious and abiding than has been suspected,
not only from the persistency of these twists, which would have long
since been smoothed away if they did not continue to subserve some
useful purpose, but also from experiments on my own mind. I find I
can deal mentally with simple sums with much less strain if I
audibly conceive the figures as on-naught, on-one, etc., and I can
both dictate and write from dictation with much less trouble when
that system or some similar one is adopted. I have little doubt that
our nomenclature is a serious though unsuspected hindrance to the
ready adoption by the public of a decimal system of weights and
measures. Three quarters of the Forms bear a duodecimal impress.
I will now give brief explanations of the Number-Forms drawn in
Plates I., II., and III., and in the two front figures in Plate IV.
DESCRIPTION OF PLATE I.
Fig. 1 is by Mr. Walter Larden, science-master of Cheltenham College,
who sent me a very interesting and elaborate account of his own case,
which by itself would make a memoir; and he has collected other
information for me. The Number-Forms of one of his colleagues and of
that gentleman's sister are given in Figs. 53, 54, Plate III. I
extract the following from Mr. Larden's letter--it is all for which
I can find space:--
[Illustration: PLATE I. _Examples of Number-Forms_.]
"All numbers are to me as images of figures in general; I see them
in ordinary Arabic type (except in some special cases), and they
have definite positions in space (as shown in the Fig.). Beyond 100
I am conscious of coming down a dotted line to the position of 1
again, and of going over the same cycle exactly as before, _e.g._
with 120 in the place of 20, and so on up to 140 or 150. With higher
numbers the imagery is less definite; thus, for 1140, I can only say
that there are no new positions, I do not see the entire number in
the place of 40; but if I think of it as 11 hundred and 40, I see 40
in its place, 11 in its place, and 100 in its place; the picture is
not single though the ideas combine. I seem to stand near 1. I have
to turn somewhat to see from 30-40, and more and more to see from
40-100; 100 lies high up to my right and behind me. I see no shading
nor colour in the figures."
Figs. 2 to 6 are from returns collected for me by the Rev. A.D. Hill,
science-master of Winchester College, who sent me replies from 135
boys of an average age of 14-15. He says, speaking of their replies
to my numerous questions on visualising generally, that they
"represent fairly those who could answer anything; the boys
certainly seemed interested in the subject; the others, who had no
such faculty either attempting and failing, or not finding any
response in their minds, took no interest in the inquiry." A very
remarkable case of hereditary colour association was sent to me by
Mr. Hill, to which I shall refer later. The only five good cases of
Number-Forms among the 135 boys are those shown in the Figs. I need
only describe Fig. 2. The boy says:--"Numbers, except the first
twenty, appear in waves; the two crossing-lines, 60-70, 140-150,
never appear at the _same time_. The first twelve are the image of a
clock, and 13-20 a continuation of them."
Figs. 7, 8, are sent me by Mr. Henry F. Osborn of Princeton in the
United States, who has given cordial assistance in obtaining
information as regards visualising generally. These two are the only
Forms included in sixty returns that he sent, 34 of which were from
Princeton College, and the remaining 26 from Vassar (female) College.
Figs. 9-19 and Fig. 28 are from returns communicated by Mr. W.H.
Poole, science-master of Charterhouse College, which are very
valuable to me as regards visualising power generally. He read my
questions before a meeting of about 60 boys, who all consented to
reply, and he had several subsequent volunteers. All the answers
were short, straightforward, and often amusing. Subsequently the
inquiry extended, and I have 168 returns from him in all, containing
12 good Number-Forms, shown in Figs. 9-19, and in Fig. 28. The
first Fig. is that of Mr. Poole himself; he says, "The line only
represents position; it does not exist in my mind. After 100, I
return to my old starting-place, _e.g._ 140 occupies the same
position as 40."
The gross statistical result from the schoolboys is as follows:
--Total returns, 337: viz. Winchester 135, Princeton 34, Charterhouse
168; the number of these that contained well-defined Number-Forms
are 5, 1, and 12 respectively, or total 18--that is, one in twenty.
It may justly be said that the masters should not be counted,
because it was owing to the accident of their seeing the Number-Forms
themselves that they became interested in the inquiry; if this
objection be allowed, the proportion would become 16 in 337, or one
in twenty-one. Again, some boys who had no visualising faculty at
all could make no sense out of the questions, and wholly refrained
from answering; this would again diminish the proportion. The
shyness in some would help in a statistical return to neutralise the
tendency to exaggeration in others, but I do not think there is much
room for correction on either head. Neither do I think it requisite
to make much allowance for inaccurate answers, as the tone of the
replies is simple and straightforward. Those from Princeton, where
the students are older and had been specially warned, are remarkable
for indications of self-restraint. The result of personal inquiries
among adults, quite independent of and prior to these, gave me the
proportion of 1 in 30 as a provisional result for adults. This is as
well confirmed by the present returns of 1 in 21 among boys and
youths as I could have expected.
I have not a sufficient number of returns from girls for useful
comparison with the above, though I am much indebted to Miss Lewis
for 33 reports, to Miss Cooper of Edgbaston for 10 reports from the
female teachers at her school, and to a few other schoolmistresses,
such as Miss Stones of Carmarthen, whose returns I have utilised in
other ways. The tendency to see Number-Forms is certainly higher in
girls than in boys.
Fig. 20 is the Form of Mr. George Bidder, Q.C.; it is of much
interest to myself, because it was, as I have already mentioned,
through the receipt of it and an accompanying explanation that my
attention was first drawn to the subject. Mr. G. Bidder is son of
the late well-known engineer, the famous "calculating boy" of the
bygone generation, whose marvellous feats in mental arithmetic were
a standing wonder. The faculty is hereditary. Mr. G. Bidder himself
has multiplied mentally fifteen figures by another fifteen figures,
but with less facility than his father. It has been again transmitted,
though in an again reduced degree, to the third generation. He says:
--
"One of the most curious peculiarities in my own case is the
arrangement of the arithmetical numerals. I have sketched this to
the best of my ability. Every number (at least within the first
thousand, and afterwards thousands take the place of units) is
always thought of by me in its own definite place in the series,
where it has, if I may say so, a home and an individuality. I should,
however, qualify this by saying that when I am multiplying together
two large numbers, my mind is engrossed in the operation, and the
idea of locality in the series for the moment sinks out of prominence."
Fig. 21 is that of Prof. Schuster, F.R.S., whose visualising powers
are of a very high order, and who has given me valuable information,
but want of space compels me to extract very briefly. He says to the
effect:--
"The diagram of numerals which I usually see has roughly the shape
of a horse-shoe, lying on a slightly inclined plane, with the open
end towards me. It always comes into view in front of me, a little
to the left, so that the right hand branch of the horse-shoe, at the
bottom of which I place 0, is in front of my left eye. When I move
my eyes without moving my head, the diagram remains fixed in space
and does not follow the movement of my eye. When I move the head the
diagram unconsciously follows the movement, but I can, by an effort,
keep it fixed in space as before. I can also shift it from one part
of the field to the other, and even turn it upside down. I use the
diagram as a resting-place for the memory, placing a number on it
and finding it again when wanted. A remarkable property of the
diagram is a sort of elasticity which enables me to join the two
ends of the horse-shoe together when I want to connect 100 with 0.
The same elasticity causes me to see that part of the diagram on
which I fix my attention larger than the rest."
Mr. Schuster makes occasional use of a simpler form of diagram,
which is little more than a straight line variously divided, and
which I need not describe in detail.
Fig. 22 is by Colonel Yule, C.B.; it is simpler than the others, and
he has found it to become sensibly weaker in later years; it is now
faint and hard to fix.
Fig. 23. Mr. Woodd Smith:--
"Above 200 the form becomes vague and is soon lost, except that 999
is always in a corner like 99. My own position in regard to it is
generally nearly opposite my own age, which is fifty now, at which
point I can face either towards 7-12, or towards 12-20, or 20-7, but
never (I think) with my back to 12-20."
Fig. 24. Mr. Roget. He writes to the effect that the first twelve
are clearly derived from the spots in dominoes. After 100 there is
nothing clear but 108. The form is so deeply engraven in his mind
that a strong effort of the will was required to substitute any
artificial arrangement in its place. His father, the late Dr. Roget
(well known for many years as secretary of the Royal Society), had
trained him in his childhood to the use of the _memoria technica_ of
Feinagle, in which each year has its special place in the walls of a
particular room, and the rooms of a house represent successive
centuries, but he never could locate them in that way. They _would_
go to what seemed their natural homes in the arrangement shown in
the figure, which had come to him from some unknown source.
The remaining Figs., 25-28, in Plate I., sufficiently express
themselves. The last belongs to one of the Charterhouse boys, the
others respectively to a musical critic, to a clergyman, and to a
gentleman who is, I believe, now a barrister.
DESCRIPTION OF PLATE II.
Plate II. contains examples of more complicated Forms, which
severally require so much minuteness of description that I am in
despair of being able to do justice to them separately, and must
leave most of them to tell their own story.
Fig. 34 is that of Mr. Flinders Petrie, to which I have already
referred (p. 66).
Fig. 37 is by Professor Herbert McLeod, F.R.S. I will quote his
letter almost in full, as it is a very good example:--
"When your first article on visualised numerals appeared in _Nature_,
I thought of writing to tell you of my own case, of which I had
never previously spoken to any one, and which I never contemplated
putting on paper. It becomes now a duty to me to do so, for it is a
fourth case of the influence of the clock-face. [In my article I had
spoken of only three cases known to me.--F. G.] The enclosed paper
will give you a rough notion of the apparent positions of numbers in
my mind. That it is due to learning the clock is, I think, proved by
my being able to tell the clock certainly before I was four, and
probably when little more than three, but my mother cannot tell me
the exact date. I had a habit of arranging my spoon and fork on my
plate to indicate the positions of the hands, and I well remember
being astonished at seeing an old watch of my grandmother's which
had ordinary numerals in place of Roman ones. All this happened
before I could read, and I have no recollection of learning the
numbers unless it was by seeing numbers stencilled on the barrels in
my father's brewery.
"When learning the numbers from 12 to 20, they appeared to be
vertically above the 12 of the clock, and you will see from the
enclosed sketch that the most prominent numbers which I have
underlined all occur in the multiplication table. Those doubly
underlined are the most prominent [the lithographer has not rendered
these correctly.--F. G.], and just now I caught myself doing what I
did not anticipate--after doubly underlining some of the numbers, I
found that all the multiples of 12 except 84 are so marked. In the
sketch I have written in all the numbers up to 30; the others are
not added merely for want of space; they appear in their
corresponding positions. You will see that 21 is curiously placed,
probably to get a fresh start for the next 10. The loops gradually
diminish in size as the numbers rise, and it seems rather curious
that the numbers from 100 to 120 resemble in form those from 1 to 20.
Beyond 144 the arrangement is less marked, and beyond 200 they
entirely vanish, although there is some hazy recollection of a
futile attempt to learn the multiplication table up to 20 times 20."
[Illustration: PLATE II. _Examples of Number Forms_.]
"Neither my mother nor my sister is conscious of any mental
arrangement of numerals. I have not found any idea of this kind
among any of my colleagues to whom I have spoken on the subject, and
several of them have ridiculed the notion, and possibly think me a
lunatic for having any such feeling. I was showing the scheme to G.,
shortly after your first article appeared, on the piece of paper I
enclose, and he changed the diagram to a sea-serpent [most amusingly
and grotesquely drawn.--F. G.], with the remark, 'If you were a rich
man, and I knew I was mentioned in your will, I should destroy that
piece of paper, in case it should be brought forward as an evidence
of insanity!' I mention this in connection with a paragraph in your
article."
Fig. 40 is, I think, the most complicated form I possess. It was
communicated to me by Mr. Woodd Smith as that of Miss L. K., a lady
who was governess in a family, whom he had closely questioned both
with inquiries of his own and by submitting others subsequently sent
by myself. It is impossible to convey its full meaning briefly, and
I am not sure that I understand much of the principle of it myself.
A shows part only (I have not room for more) of the series 2, 3, 5, 7,
10, 11, 13, 14, 17, 18, 19, each as two sides of a square,--that is,
larger or smaller according to the magnitude of the number; 1 does
not appear anywhere. C similarly shows part of the series (all
divisible by 3) of 6, 9, 15, 21, 27, 30, 33, 39, 60, 63, 66, 69, 90,
93, 96. B shows the way in which most numbers divisible by 4 appear.
D shows the form of the numbers 17, 18, 19, 21, 22, 23, 25, 26, 27,
29, 41, 42-49, 81-83, 85-87, 89, 101-103, 105-107, and 109. E shows
that of 31, 33-35, 37-39. The other numbers are not clear, viz. 50,
51, 53-55, 57-59. Beyond 100 the arrangement becomes hazy, except
that the hundreds and thousands go on again in complete, consecutive,
and proportional squares indefinitely. The groups of figures are not
seen together, but one or other starts up as the number is thought of.
The form has no background, and is always seen _in front_. No Arabic
or other figures are seen with it. Experiments were made as to the
time required to get these images well in the mental view, by
reading to the lady a series of numbers as fast as she could
visualise them. The first series consisted of twenty numbers of two
figures each--thus, 17, 28, 13, 52, etc.; these were gone through on
the first trial in 22 seconds, on the second in 16, and on the third
in 26. The second series was more varied, containing numbers of one,
two, and three figures--thus 121, 117, 345, 187, 13, 6, 25, etc.,
and these were gone through in three trials in 25, 25, and 22 seconds
respectively, forming a general result of 23 seconds for twenty
numbers, or 2-1/3 seconds per number. A noticeable feature in this
case is the strict accordance of the scale of the image with the
magnitude of the number, and the geometric regularity of the figures.
Some that I drew, and sent for the lady to see, did not at all
satisfy her eye as to their correctness.
I should say that not a few mental calculators work by bulks rather
than by numerals; they arrange concrete magnitudes symmetrically in
rank and file like battalions, and march these about. I have one
case where each number in a Form seems to bear its own _weight_.
Fig. 45 is a curious instance of a French Member of the Institute,
communicated to me by M. Antoine d'Abbadie (whose own Number-Form is
shown in Fig. 44):--
"He was asked, why he puts 4 in so conspicuous a place; he replied,
'You see that such a part of my name (which he wishes to withhold)
means 4 in the south of France, which is the cradle of my family;
consequently _quatre est ma raison d'être_.'"
Subsequently, in 1880, M. d'Abbadie wrote:--
"I mentioned the case of a philosopher whose, 4, 14, 24, etc., all
step out of the rank in his mind's eye. He had a haze in his mind
from 60, I believe [it was 50.--F.G.], up to 80; but latterly 80 has
sprung out, not like the sergeants 4, 14, 24, but like a captain,
farther out still, and five or six times as large as the privates 1,
2, 3, 5, 6, etc. 'Were I superstitious,' said he, 'I should
conclude that my death would occur in the 80th year of the century.'
The growth of 80 was _sudden_, and has remained constant ever since."
This is the only case known to me of a new stage in the development
of a Number-Form being suddenly attained.
DESCRIPTION OF PLATE III.
Plate III. is intended to exhibit some instances of heredity. I have
no less than twenty-two families in which this curious tendency is
hereditary, and there may be many more of which I am still ignorant.
I have found it to extend in at least eight of these beyond the near
degrees of parent and child, and brother and sister. Considering that
the occurrence is so rare as to exist in only about one in every
twenty-five or thirty males, these results are very remarkable, and
their trustworthiness is increased by the fact that the hereditary
tendency is on the whole the strongest in those cases where the
Number-Forms are the most defined and elaborate. I give four
instances in which the hereditary tendency is found, not only in
having a Form at all, but also in some degree in the shape of the
Form.
Figs. 46-49 are those of various members of the Henslow family,
where the brothers, sisters, and some children of a sister have the
peculiarity.
Figs. 53-54 are those of a master of Cheltenham College and his
sister.
Figs. 55-56 are those of a father and son; 57 and 58 belong to the
same family.
Figs. 59-60 are those of a brother and sister.
The lower half of the Plate explains itself. The last figure of all,
Fig. 65, is of interest, because it was drawn for an intelligent
little girl of only 11 years old, after she had been closely
questioned by the father, and it was accompanied by elaborate
coloured illustrations of months and days of the week. I thought
this would be a good test case, so I let the matter drop for two
years, and then begged the father to question the child casually,
and to send me a fresh account. I asked at the same time if any
notes had been kept of the previous letter. Nothing could have come
out more satisfactorily. No notes had been kept; the subject
had passed out of mind, but the imagery remained the same, with some
trifling and very interesting metamorphoses of details.
[Illustration: PLATE III. _Examples of an Hereditary Tendency to see
Number-Forms_, _4 Instances where the Number Forms in same family
are alike_ _3 Instances where the Number-Forms in same family are
unlike_]
DESCRIPTION OF PLATE IV.
I can find room in Plate IV. for only two instances of coloured
Number-Forms, though others are described in Plate III. Fig. 64 is
by Miss Rose G. Kingsley, daughter of the late eminent writer the Rev.
Charles Kingsley, and herself an authoress. She says:--
"Up to 30 I see the numbers in clear white; to 40 in gray; 40-50 in
flaming orange; 50-60 in green; 60-70 in dark blue; 70 I am not sure
about; 80 is reddish, I think; and 90 is yellow; but these latter
divisions are very indistinct in my mind's eye."
She subsequently writes:--
"I now enclose my diagram; it is very roughly done, I am afraid, not
nearly as well as I should have liked to have done it. My great fear,
has been that in thinking it over I might be led to write down
something more than what I actually see, but I hope I have avoided
this."
Fig. 65 is an attempt at reproducing the form sent by Mr. George F.
Smythe of Ohio, an American correspondent who has contributed much
of interest. He says:--