330 Prof. Twining on the Parallelogram of Forces. 



greater than a right angle, which is impossible. Bat, if the two 

 are not commensurable, there may be taken two multiples, nz 

 and n + lz, between which DAB shall be intermediate. There- 

 fore the effect of AB, at that angle, must be intermediate to its 

 effects at nz and n-\-\z] so that the angle whose cosine repre- 

 sents the effect of AB, at DAB, must be intermediate to nz' and 

 n-\-lz' ; and, as z and z' may be taken to any required degree 

 of minuteness, it is evident that, in every case, DAB must vary 

 from its corresponding diagonal angle, in the way of excess or 

 defect, as DAB varies from its corresponding diagonal angle, and 

 therefore the two cannot constitute a right angle. But DAC is a 

 right angle, — therefore AB cannot lie out of the direction of the 

 diagonal of the parallelogram whose sides represent its compo- 

 nent forces acting at right angles to each other. And if two 

 forces are represented by two sides of a parallelogram which in- 

 clude any angle whatever, let the diagonal of the parallelogram 

 which divides that angle be drawn, and let the effects of the two 

 forces be taken, in that diagonal and normal to it, what we have 

 already proved will show that the latter two are equal and oppo- 

 site forces, and that the sum of the former is represented by the 

 diagonal of the parallelogram, — which completes the point de- 

 sired. 



Respecting the two methods of proof that compose the body of 

 this article I may be indulged in remarking, that I conceive 

 them to be new, and to make the rationale of the problem of 

 component and resultant forces easy of comprehension, to a per- 

 haps unusual degree. They are dependent upon no ideas whose 

 clear establishment in the mind, presupposes any considerable 

 amount of mathematical study, — upon differentials and integrals, 

 functions, infinitesimal considerations, or even trigonometrical 

 formulas. But the conclusions are derived from the mechanical 

 axioms by the aid, only, of the most elementary ideas of geome- 

 try or common algebra. Before closing I would drop the remark, 

 that an inspection of fig. 5, in the last method of proof, coupled 

 with the reasonings respecting its constituent lines, employed as 

 symbols of force, may suggest a very simple and expeditious pro- 

 cess for deriving the formulas for the cosines of multiple angles 

 from the cosine of the unit ande. 



