THE MODERN GEOMETRY. xv 



tance than that of geometrical views and knowledge ; for when practical 

 calculations are disregarded, and the deduction of general truths alone 

 occupies us, then, first of all, we must exclude from the drawn figure aU 

 special relations that is, strike out of graphical statics the essentially 

 graphical part. A truth comprehended only in the abstract holds good 

 for all figures which can be drawn in accordance with the given condi- 

 tions. 



We place, then, in one line geometry and geometrical statics (mechanics). 

 From geometry we obtain a method of construction, or descriptive geome- 

 try, which finds its practical applications in architecture and machine 

 drawing. From geometrical statics we obtain also a construction method 

 or routine viz., graphical statics which finds its practical applications in 

 the graphical calculation of structures and machines. Both descriptive 

 geometry and graphical statics have still, with reference to these practical 

 ends, to develop and make use of the general relations which subsist be- 

 tween the geometrical constructions to which they give rise, and thus each, 

 according to its means, contribute to the discovery and spread of geo- 

 metrical and mechanical truths. 



From this co-ordination <5f descriptive geometry and graphical statics 

 we must not, however, infer an equal importance ; for, while in geometri- 

 cal drawing we have always to represent an ideal image, and the graphical 

 method is therefore directly suggested, we have for statical calculations 

 the analytical process also at our disposal, and everything depends then 

 upon the relative advantages and disadvantages of the graphical and ana- 

 lytical methods. We have thus noticed all the most important points 

 which occur in a theoretical consideration, and there only remains to make 

 a comparison from a practical standpoint (X.). 



vm. 



THE MODERN GEOMETRY. 



Geometry treats of figures or constructions in space. These figures and 

 their properties are not always regarded and treated in equal extent and 

 generality. 



Geometrical knowledge found its origin in practical needs, and tho 

 ancients confined themselves almost exclusively to special investigations 

 of individual figures and bodies of definite form, such as presented them- 

 selves to the eye. In the phorisms of Euclid (-285), according to Pappus 

 (end of the fourth century), the mutual relations of the circle and straight 

 lines were, indeed, given with a certain degree of completeness, but these 

 have not come down to us. 



Properties thus determined had naturally only a limited significance, 

 and could neither count upon permanence nor give satisfactory conclu- 

 sions. Investigators sought, therefore, assistance where it was best afforded, 

 in analysis. This was, in the sixteenth century, by the algebra of Vieta 

 (1540-1603), notably enriched. 



