W. Skey. — On Prankland's Paper on Two Dimensions. 106 



elbows as they grow, and of the same fraternity as introduced to us further 

 on by their patron ; straight lines of such potency that any two of them can 

 bind a space ; and lastly, as we shall see, a pseudo-spherical surface combin- 

 ing in itself the utmost simplicity with inconceivable complexity ; — all these 

 again, wonderful as they are, sinking to insignificance compared with the 

 grand culminating idea (as more recently developed by this new order of 

 geometricians), — a space of four, five, or even seven dimensions — a space 

 which, to us, I suppose like the seventh heaven of Paul, will ever be both 

 inconceivable and impervious. 



And, now, proceeding with our observations on Mr. Frankland's paper, 

 we find an Euclidian axiom thus disproved, and such tremendous con- 

 ceptions as these projected. Mr. Frankland, under the impression that 

 his enthusiastic belief in this lias infected us, or that the arguments given 

 are convincing, essays thus to speak in our behalf : " We see, therefore, 

 that geometry is only a particular branch of a more general science, and 

 that the conception of space is a particular variety of a wider and more 

 general conception." Well, geometry may ultimately be thus subordinated. 

 However, I cannot see that its time has come yet. 



But a science so capacious — a science which, to us, is transcendental, at 

 least to the less intellectually advanced of us, requires some mark to dis- 

 tinguish it from that which it has developed from, a mark which shall, if 

 possible, indicate some salient or distinguishing feature of it ; consequently 

 this is done. Mr. Frankland says : " This wider conception, of which space 

 and time are particular varieties, it has been proposed to denote by the term 

 manifoldness." 



To me this is like " giving to airy nothings a local habitation and a 

 name." But we naturally ask. How comes time to be here conjoined with 

 space under the term manifoldness ? The idea of time is, to say the least, 

 brought in here very abruptly. The explanation of this term in its applica- 

 tion to space and time separately I thank him for, but the infinitely harder 

 task of explaining its application to the two conjointly is left to us. 



And now, the overthrow of Euclidian geometry being accomplished, a 

 new kind of geometrical science instituted, and a specific feature of it 

 defined and named, Mr. Frankland introduces us to a surface, which, as 

 he says. Professor Clifford has treated as a surface which is taken by him 

 to be "the simplest continuous manifoldness of two dimensions and of finite 

 extent," or, in plainer and shorter English, the simplest surface of limited 

 extent ; and as it is upon a surface of this kind that those discoveries are 

 made which it is a purpose of his paper to disclose, he explains how this 

 surface is got, so that we may place ourselves in a position to intelligently 

 follow him. 



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