Jan. 15, 18 So] 



NA TURE 



255 



2,000. 



right. 



The millions are in a vague, bright distance to the 



One of the sisters writes : — 



18. Figures present themselves to me in lines [as in the an- 

 nexed diagram]. Tbey are about a quarter of an inch in length, 



and of ordinary type. They are black on a white ground. 200 

 generally takes the place of 100 and obliterates it. There is no 

 light or shade, and the picture is invariable. 



Another sister gives a picture in which the numbers 

 form a vertical line from 1, opposite to the eye, up to 100, 

 at which point the scale appears to recede from her. 



The third sister writes : — 



19. Figure?, alu ays stand out di-tinctly in Arabic numerals ; 

 they are black on a white ground, of this size [the specimen was 



Fig. 7. 



clear and round, and in rather large ordinary handwriting], but 

 the numeral 19 is smaller than the rest. 



It is curious that the lines of most of the diagrams I 

 have thus far given should be so feeble and, to appear- 

 ance, wandering, although as a matter of fact they are 

 firmly fixed. Artists speak of the " leading lines " in a 

 picture, and commend pictures in which the leading lines 

 are graceful. I have little doubt that one of the reasons 

 why minds vary in artistic power is that the leading 

 channels in the blank schedules of their minds vary in 

 character. I should expect that natural artists might be 

 found whose habit was to visualise numerals not in shaky 

 lines, but in bold and beautiful curves. In the instances 

 I am about to give, especially in the first of them, there 

 is more tendency to geometric precision, and I should be 

 most curious to learn (by actual and careful test) whether 

 or no such cases are generally correlated with a true eye 

 to straightness, squareness, and symmetry. 



In the following example the numbers are not associated 

 with visual figures, but with points on an ascending and 

 descending scale, which is a pure line having neither 

 breadth nor colour. It is described as perfectly flexible 

 and extensible, much, I suppose, as if it were printed on 

 a strip of india-rubber sheeting, and it is applicable to the 

 measurement of large distances or small ones, to frac- 

 tions, and to straight lines or curves. A very curious 

 description is given in detail, which I will not here repro- 

 duce, of the way in which the scale is used in mental 

 arithmetic. The writer adds : — 



20. The accompanying figure lies in a vertical plane, and is 

 the picture seen in counting. The zero point never moves, 

 it is in my mind ; it is that point of space known as "here," 



while all other points are outside or "there." When I was a 

 child the zero point began the curve ; now it is a fixed point in 

 an infinite circle ... I have had the curious bending from o to 





1M 



V-'. 



Fie. 3. 



30 as long as I can remember, and imagine each bend must mark 

 a stage in early calculation. It is absent from the negative side 

 of the scale, which has been added since childhood. 



Another correspondent sees figures in a circle, having o 

 at the right hand of its horizontal diameter and 100 at the 

 left hand. Positive numbers are reckoned from o to 100 

 from the right, over the top to the left, and negative 

 numbers the other way. The same takes place with 

 figures between 100 and 200, 200 and 300, &c. 



Another correspondent sees them for the most part in 

 a regular row like park palings. The description and 

 sketch are as follows : — 



21. As far as 12 the numerals appear to be concealed in black 

 shadow ; from 12 to 20 is illuminated space, in which I can 

 distinguish no divisions. This I cannot illustrate, because it is 

 simply '.ark and light spaa, but with a tolerably sharp line of 

 division at 1 2. From 20 to 100 the numerals present themselves 

 as follows, but less distinctly : — 



Nihil 1 1 



igiiii 



An account is appended of the way in which simple 

 mental arithmetic is effected by this arrangement, which 

 at present I pass over. 



I will conclude my list with a statement written by a 

 mathematical astronomer of rapidly rising reputation, 

 whose "practice of working arithmetic" mentioned m 

 the concluding paragraph must be understood to signify 

 " performing masses of laborious calculations " :— 



22. The numbers 1, 2, 3, 4, &c. are » n a straight row, and I 



