THE GRAPHICAL CALCULUS. XXX VI i 



IV. 



THE GRAPHICAL CALCULUS. 



The most extended applications of statics are in the field of engineering. 

 Here, not only general properties of form and position are required, but in 

 a large number of cases numerical relations are also necessary. General 

 results of the latter character can, as we have seen, only be embraced by 

 algebraic formulae (I.). The pure graphical theory of construction is there- 

 fore iu this respect lacking in completeness, as it is unable to furnish gen- 

 eral metrical relations. 



The practical engineer has almost always, however, to do with special 

 problems ; dimensions and acting forces are numerically given. Geometry 

 in such cases could give no general relations, because the results desired 

 are the consequences of the special proportions of the figure. In any de- 

 terminate case, however, we may obtain a result holding good for that case, 

 and it only remains to show how generally to obtain such a result. The 

 graphical calculus treats of such methods, and so, although not exclusively, 

 does graphical statics. As soou now as practical use is made of the actual 

 proportions of the figure, everything depends upon the exactness of the 

 drawing. One condition for the application of the graphical method is. 

 therefore, skill in geometrical drawing a requisition, indeed, which the 

 practical engineer can most readily meet. 



The idea at bottom of the graphical calculus is simple. The modifica- 

 tions of numbers in numerical calculations correspond always to similar 

 modifications of the quantities represented by these numbers. The measure 

 of a quantity can be as well given by a line as a number, by putting in 

 place of the numerical the linear unit. In order for a graphical calculus, 

 then, the modifications of lines answering to corresponding numerical 

 operations are necessary, and these are furnished by geometry. They con- 

 sist of graphical constructions, and rest upon the known properties of 

 geometrical figures. The scale furnishes the means of converting directly 

 any numerical quantity into its corresponding linear representation, and 

 inversely any graphically obtained result can be at once transformed into 

 numbers. 



The graphical determination of desired or computable numbers is natu- 

 rally nothing new. From the " Traite de Gnomonique " of de la Hire 

 (1682) to the " Geometric descriptive' 1 '' of Monge (1788), many examples 

 are to be found. The graphical calculus, however, goes further than this. 

 It aims to found a method, a routine, which shall not only apply to bodies 

 in space, but which shall also, like the arithmetical or algebraic calculus, 

 be independent of concrete relations and of general application. It seeks 

 further to obtain its results (products and powers) in the shape of lines 

 convertible by scale into numbers. fHence the important part which area 



