As a rule, I see things with my mind’s eye almost as distinctly as if I were looking at the things themselves; and I thought that everyone could do the same till I read Galton’s _Inquiries into Human Faculty_ and found how greatly people varied as to this. I also see some things with my mind’s eye as symbols for other things that cannot be seen at all, e.g. boot-trees for arguments. They are trees for shoes, without handles, and made of polished wood; and they are on a grey felt floor with an open doorway at the further end. When two arguments lead up to a third, the corresponding boot-trees turn their toes in towards the other’s heel; and I have seen as many as eight or ten boot-trees pointing like this to half the number in a line beyond them, these also pointing to others further on, and finally a boot-tree going through the doorway. I find it very convenient--I see more at a single glance than I could put into a page of print. Galton speaks of numbers being personified, and gives several instances of children doing this. The son of an old friend of mine--an undergraduate now--tells me he did it when a child and sometimes does it still. His views are--“1 and 0 do not count, being inactive. 2, good-natured, always doing its best to please. 3, sometimes kind and condescending, hated by 8 when added, but not when multiplied to make 24: great friend of 9. 4, not very noticeable, but means well: great friend of 8 and 6. 5, much the same as 4, but no special friend except 2: rather meek. 6, inclined to be selfish: no great friend of 3, pals with 4 and 8. 7, unlucky and despised, bad luck in making such numbers as 49 and 63 when multiplied. 8, fat and good-natured, but inclined to be selfish: likes being made up to good round numbers such as 12, 24, 48, &c. 9, friend of 3, disagreeable and a bully, despised for making brutish numbers such as 27, 63, 81, &c.” I now suspect Pythagoras of having done this as a child and then, instead of putting away childish things, making it a basis for much of his philosophy. Thus, amongst other things, he says that 8 is Justice itself, being _isacis isos_ or _bis bina bis_--in other words, it is composed of 4 and 4, and each 4 is composed of 2 and 2, so that there is even balance throughout. This reasoning must surely be an afterthought to justify some childish fancy. Usually, when people think of numbers, they see the Arabic figures with their mind’s eye; and some people can see these figures manœuvring at each stage of a calculation. (I heard this from George Bidder, who was famous as The Calculating Boy a hundred years ago.) Within narrow limits I see this manœuvring myself; but, although they are mere figures, I feel that they are moving like soldiers on parade. And that comes very near personifying them.