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):--