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’etre.’”

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.