830 Prof. Twining on the Parallelogram of Forces. 
greater than a right angle, which is impossible. But, if the two 
are not commensurable, there may be taken two multiples, nz 
and n+1z, between which DAB shall be intermediate. There- 
fore the effect of AB, at that angle, must be intermediate to its 
effects at nz and m-+1z; so that the angle whose cosine repre- 
sents the effect of AB, at DAB, must be intermediate to nz’ and 
n+1z’; and, as z and 2/ 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 isa 
right angle,—therefore AB canuot 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- 
elude 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 opp0- 
site forces, and that the sum of the former is represented by the 
eee 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 pel 
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 25. 
symbols of force, may suggest a very simple and expeditious pre 
cess for deriving the formulas for the cosines of multiple ang 
from the cosine of the unit angle. 
