W. Skey. — On FranklancV s Paper on Tivo Dimensions, 107 



once, and so enclosing but one space, is founded upon Lobatchewsky's 

 conception of what parallel straight lines are capable of. The application 

 of this conception to the case in point is not explained by the author, but, 

 if I understand him here aright, a figure is by this means " analytically 

 constructed " — a cross between an ellipsoid and a sphere, with a strain of 

 something undeterminable ; a figure so large that its geodesic lines stand 

 clear of each other at one pole ; a figure, in fact, of the same rare genus as 

 the burstable triangles of Lobatchewsky. If this is the interpretation of the 

 author, and Lobatchewsky's conception is the groundwork of the structure 

 in question, all I need do, in answer to it, is to refer you back to my 

 criticism on the ingenious method employed by this geometrician to raise 

 this very fertile conception. 



But, in doing so, I must insist upon Mr. Fraukland adhering to the 

 limitation which Lobatchewsky has imposed or submitted to in respect to 

 the angle at which his geodesic lines are to incline unto each other, that is, 

 it is to be of a finite value ; not that this is necessary to insist upon for my 

 argument, but that in a way which is authorized by this geometrician it 

 excludes from consideration here geodesic lines which incline to each other 

 at angles which are infinitely small, a labour which I feel fully persuaded 

 would result in nothing, although it has a promising appearance. 



Summing up these results of mine upon the subject of Mr. Frankland's 

 paper, it is now, I think abundantly evident that the analytical conception 

 of a surface such as the one which has been worked upon for the discoveries 

 therein communicated, is not in reality valid, and that though possibly not 

 self- contradictory, as Mr. Frankland urges, it requires premises which are 

 of this nature ; that, in fine, this conception, and the whole of the assump- 

 tions which have been formed upon it, are based upon fallacious reasoning. 

 As a consequence of this, therefore, it remains to us that the simplest 

 surface of finite extent which is even analytically conceivable only, (or 

 as Mr. Frankland puts it, "the simplest continuous manifoldness of two 

 dimensions and of finite extent"), is that of a sphere. 



All now which I desire to do further in this matter is to make a few 

 remarks upon the quotation from Professor Clifford's "Postulates of Space," 

 with which Mr. Frankland closes his paper, as not to do this would be to 

 leave unchallenged (that is in a direct manner), the very remarkable con- 

 ception which it is evident Mr. Frankland has all along been preparing 

 us for — a conception, indeed, which I am fain to consider has a value, but 

 this only in showing to what lengths theories of the kind described lead us 

 when indulged in without stint. Magnificently suggestive as the Pro- 

 fessor is here, he is only so by stultifying the Universe to us — defaming it as 

 it were — levelling it down to our own plane. Evidently referring to the 



