424 BEPOET— 1892. 



or at least in geometry of situation, if the student could treat the 

 first named as a special application, to a given case of the latter ; how 

 man}' superfluous definitions and demonstrations would be spared to the 

 lecturer if they had already been grafted into the student's mind, so as 

 to be completely mastered by him, as parts of a more comprehensive 

 whole ! ' 



This subject is thus shown to be one woi'thy of discussion, and th.e 

 question seems to turn on two points : — 



1. The real intrinsic value of purely geometrical methods. 



2. The practical value of such methods to the ordinary engineer. 



1. That projective geometry leads to wide and important generalisa- 

 tion there can be no doubt, but that the benefit is so great as to make it 

 inexpedient to apply the analytic process to arrive at geometric results 

 is quite another question. 



Culmann, not altogether unnaturally, criticises those who employ his 

 results, but arrive at the understanding of them by other channels than 

 his own ; but can the statement in the preface to the French translation of 

 his work be sustained in which he says — 



' Our publication was followed by a great number of elementary 

 statics, in which, by reproducing even the most simple of our construc- 

 tions, mostly without changing anything in them, the authors have 

 striven to give analytical demonstrations of them. 



' We imagine that the truth is not there — that one will never succeed 

 in tracing the lines of a construction, and execute simultaneously alge- 

 braical operations which carry the explanation of that construction. 

 Neither can one easily understand the signification of each line, and re- 

 present to themselves statical relations, if one limits oneself to translate 

 the formula whose developments are no longer clear to the memory.' 



If there is any truth in the preliminary remarks in this report, it is, 

 after all, merely a question of regarding either the actual position of lines 

 or the measurement of that position as representing the same thing to 

 the mind, and it is certain that much that is useful in even Culmann's 

 works has absolutely no necessary connection with projective geometry ; 

 while some of his admirers, who quote his views at the beginning of their 

 works on graphical statics with some amount of favour, actually employ 

 largely the analytical methods, and, indeed, scarcely employ the methods 

 of higher geometry at all. Professor Jay Du Bois ^ may be quoted as a 

 good example of this. 



But, after all, the practical problems of engineers lie almost entirely in 

 two dimensions of space, and it is perfectly possible to arrive at an under- 

 standing of the ordinary problems which arise, by a due harmony of 

 analytical and graphical methods. 



2. The second point is one which can only be answered by reference 

 to the actual education of the engineer. Culmann's preface to the German 

 edition of 'Die Graphische Statik,' speaking of the professors of South 

 Germany, says : — 



' They were, indeed, totally opposed to the idea of regarding bodies in 

 which forces are in equilibrium and lineal forms which represent these 

 forces in magnitude and direction as correlative geometrical representa- 

 tions, and then applying to these the geometry of position, in which 

 the relations of such allied representation lay completely worked out. 



' Graphical Statics, 2iid ed., by Jay Du Bois. J. Wiley & Sons, New York 



