ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 385 



Messeclaglia — which represents results capable of representation by 

 ordinary methods of graphical treatment. 



Graphical methods of representation are thus seen to narrow down to 

 the use of points and lines upon a plane surface, at the same time with 

 an unlimited range of application. 



Regarding the geometrical constructions involved in graphical calcu- 

 lation, it is now clear tliat the whole subject turns on the intersection of 

 lines, that is to say, on their relative position. Hence, it is not surprising 

 that when any given thing can be represented, not merely in magnitude 

 by the length of a line, but also in direction by its position in space, 

 we have a direct application of that branch of geometry -which deals 

 with the position of lines and planes, and their intersection with each 

 other, the relations of which are independent of numerical magnitudes. 

 This kind of geometry, which is called, under the various names of 

 'modern geometry,' 'higher geometry,' 'projective geometry,' or, best 

 of all, ' geometry of position,' has advanced enormously in importance in 

 recent years, and has become a subject of instruction chiefly on the Con- 

 tinent, in the poly technical schools, mainly from the fact that the most 

 important of all numerical quantities in mechanics which can be repre- 

 sented graphically — namely, force — being capable of direct and complete 

 representation by lines, enabled the general applications of the properties 

 of geometry of position to be at once made. A large number of propo- 

 sitions, which previously had little practical interest, thus assumed a 

 physical meaning, which makes them of the greatest value, and the 

 science of graphic statics, which occupies probably the largest place in 

 Continental engineering schools, owes no small measure of the import- 

 ance in which it is held to the fact that it afibrds a high mental trainins' 

 in geometry, which has at the same time direct practical value. 



It is a remarkable fact that the teaching of geometry of position as an 

 object in the curriculum of foreign schools is owing, in a large measure, 

 to the publication of Culmann's treatise on graphic statics. Weyranch, 

 in his ' Ueber die graphische Statik — zur Orientirung,' and quoted by 

 Jay Du Bois in his ' Graphical Statics,' says : — • 



' After von Staudt, the strict geometry of position remained a long 

 time disregarded, while the synthetic geometry of Steiner has enjoyed, 

 without intermission, till the present day a special preference on the part 

 of mathematicians. Culmann gave the impulse to a change in this re- 

 spect. In his "Graphical Statics " he rests directly npon the work of von 

 Staudt, and, with something more than boldness, assumes a knowledge of 

 the geometry of position among all practical men. Such a course was not 

 indispensable for the foundation of his method, and impeded the spread 

 of the graphical statics ; but by it the geometry of position gained. This 

 last had next, of necessity, to be introduced into the Zurich Polytechnic, 

 and thus arose the first, until now, only complete text-book upon the sub- 

 ject, the " Geometric der Lage," by Reye (Hanover, 1868), as the direct 

 result of the " Graphical Statics ", of Culmann. Since then the modern 

 geometry has been introduced into all technical institutions throughout 

 Germany, and thus placed at the disposal of the arts and sciences.' 



This subject is so important that it will be alluded to again in dealing 

 with graphic calculation. At the same time it may here be pointed out 

 that while, as in plane projective geometry, the relative position of lines 

 may be manipulated graphically, their position may perfectly well and 

 completely be represented by measurement from two fixed points, so that 



1892. c c 



