32 Prof. Tait on Listing's Topologie. 



yet been rescued from their most undeserved obscurity, and 

 published in an English dress, especially when so much that 

 is comparatively worthless, or at least not so worthy, has 

 already secured these honours. I was altogether ignorant of 

 the existence of the Vorstudien till it was pointed out to me by 

 Clerk-Maxwell, after I had sent him one of my earlier papers 

 on Knots; and I had to seek, in the Cambridge University 

 Library, what was perhaps the only then accessible copy. 



(1) Down and Up are at once given us by gravity. They 

 are defined as the direction in which a stone falls, or in which 

 a plummet hangs, and its reverse. Even below-decks, when 

 the vessel is lying over under a steady breeze, and we " have 

 our sea-legs on," we instinctively keep our bodies vertical, 

 without any thought of setting ourselves perpendicular to the 

 cabin-floor. And this definition holds in every region of space 

 where the earth's attraction is the paramount force. In an 

 imaginary cavity at the earth's centre the terms would cease 

 to have any meaning. 



East, in the sense of "Orient," is the quarter in which the 

 sun rises; and this distinction is correct all over the earth 

 except at the poles, where it has no meaning. But if we 

 were to define South as the region in which the sun is seen at 

 midday, our definition would be always wrong if we were 

 placed beyond the tropic of Capricorn, and at particular seasons 

 even if we were merely beyond that of Cancer. Still there is 

 a certain consensus of opinion which enables all to understand 

 what is meant by South without the need of any formal defi- 

 nition. 



But the distinction between Right and Left is still more dif- 

 ficult to define. We must employ some such artifice as "A 

 man's right side is that which is turned eastwards, when he 

 lies on his face with his head to the north." For, in the lapse 

 of ages of development, one may perhaps be right in saying 

 with Molicre's physician, "Nous avons changd tout cela" and 

 men's hearts may have migrated by degrees to the other side 

 of their bodies, as does one of the eyes of a growing flounder. 

 Or some hitherto unsuspected superiority of left-handed men 

 may lead to their sole survival; and then the definition of the 

 right hand, as that which the majority of men habitually em- 

 ploy most often, would be false. 



I will not speak further of these things, which I have intro- 

 duced merely to show how difficult it sometimes is to formulate 

 precisely in words what every one in his senses knows per- 

 fectly well; and thus to prepare you to expect difficulties of a 

 higher order, even in the very elements of matters not much 

 more recondite. 



