CHAP, l'] SUPPLEMENT TO CHAP. Vn. 99 



demonstrations intelligently, and to comprehend perfectly and oven apply 

 readily the method for themselves. 



We cannot, indeed, make the reader familiar with all the principles of 

 the calculus, but all these principles are by no means needed. Its funda- 

 mental idea, a few of its terms and applications, are all that he need be 

 familiar with in order to perform the simple integrations we shall encoun- 

 ter, as readily as the most skilled mathematician. This portion of the 

 present Supplement may, perhaps, be considered by many as unnecessary and 

 superfluous. "We are, indeed, justified in assuming such knowledge. But 

 as we believe our plan practicable, we cannot resist the desire of making 

 our development intelligible to all, and thus rendering our treatment of the 

 simple girder at least complete. 



The practical man as well as the mathematician may thus have at his 

 disposal the powerful aid of the calculus, so far at least as his purposes 

 require it, and be able to deduce for himself the formulae which hitherto 

 he has accepted " upon faith." It may also not be improbable that here 

 and there one may be found who, pleased with the simplicity of the prin- 

 ciples and the f ruitf ulness of their application, may be led to further prose- 

 cute the study for his own satisfaction. 



"We shall first, then, notice briefly the two methods of calculation above 

 referred to ; then devote a few pages to the development of those prin- 

 ciples and rules of the calculus of which we shall make use, and finally 

 apply these principles to the discussion of the curve of deflection of loaded 

 beams. 



2. Hitter's method. This method is referred to in Art. 14. It 

 rests simply upon the principle of the lever, or the law of statical moments ; 

 requires no previous knowledge, and converts the most difficult cases of 

 strain determination into the most elementary problems of mechanics. 

 Hitter, in his " Theorie eiserner Dach- und Briicken-Constructionen," has 

 applied this simple principle in such detail and fullness, and so clearly set 

 forth its elegance and simplicity, that it very generally, and justly, goes by 

 his name. 



"Its results are clear and sharp as the results of Geometry, and of direct 

 practical application. There is hardly another branch of engineering 

 mechanics which, for such a small amount of previous study, offers such 

 satisfactory results, and which is so suited to engage the interest of the 

 beginner." 



We have given in the Appendix to Chap. I. (Arts. 6, 9, 10) detailed ex- 

 amples of its application. Throughout this work similar illustrations of 

 its use will be met with, so that it is only necessary here to state more fully 

 than in the text its general principle. 



If any structure holds in equilibrium outer forces, it does so by virtue of 

 the strains or inner forces which these outer forces produce. Now the 

 outer forces being always given, we wish to find the interior forces or 

 strains. If, then, the structure is framed, and we conceive it cut entirely 

 through, the strains in the pieces thus cut must hold in equilibrium all the 

 outer forces acting between the section and either end. Thus, in Pig. 6, 

 PI. 2, a section cutting D, 7 and H completely severs the truss. Then the 



