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."
"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."