'SOO Mr. T. S. Davies on Geometry and Geometers. 



plane figure materially different from this? For one line to 

 be 071 another is surely not " coincidence " in the strict sense 

 of the word ; and certainly there is little reason to consider 

 an argument founded upon two lines so situated, as a legiti- 

 mate proof. Yet how few who have read Euclid have any 

 other idea of supraposition ! Euclid does not use the term; 

 and no commentator or teacher who does use it, would pass 

 it by without showing in what sense it is used. This is easily 

 done by recalling the definitions of the point, the line and the 

 plane, to the student's mind. The difficulty of perfect abs- 

 traction of some material properties is of the same nature 

 here as in the case of " solids;" but it is much simpler, and 

 more easily removed from the young mind. 



As regards the legitimacy of the processes of transferring 

 figures, so much has been said on both sides that it might be 

 difficult to propound a view which somebody or other has not 

 taken before. One thing, however, is clear ; that no geometer 

 has been successful in attempting to build up a legitimate 

 system without the aid of this conception — either openly or in 

 disguise. We are forced back, then, upon the necessity of 

 using it; and perhaps our notions of legitimacy may be of 

 too exalted a character to be ever realised. Are we sure, in- 

 deed, that our conceptions of form and the other properties 

 of figure are not so obtained by us initially as to necessarily 

 involve the consideration of transfer ? Many considerations 

 are involved in this question which lie beyond the ordinary 

 range of geometrical discussion. It was evidently Euclid's 

 desideratum to evade it; and yet he was unable to do so in 

 i. 4, i. 7, and Hi. 24 : nor has any one else succeeded in doing 

 so with perfect strictness. 



That this method of transferring figures was amongst the 

 very earliest modes of demonstration^ there can be no room to 

 doubt; and that it was much more used (even in the school 

 of Plato) before the time of Euclid, appears to me extremely 

 probable. Little, perhaps nothing, is likely to be found of a 

 documentary character either to bear out this view or to sweep 

 it away. It is a natural mode of proceeding, and it is in a few 

 cases even yet an inevitable owe. The I'efinements introduced 

 into geometrical reasoning by Euclid, or perhaps partially so be- 

 fore his time, induced the attempt to eliminate all idea of ;«o//ow, 

 as one of the properties of matter. It is on the same ground 

 that corresponding modern refinements have been attempted, 

 in which the principle should be completely carried out. That 

 the conditions which determined the equality of two triangles 

 were not determined originally without the adoption of the 

 principle of transfer is almost certain; and as to many pro- 



