Misunderstanding about Simon Stevinus's proof I came across this proof in my textbook, but I am not sure exactly what it's proving. Could someone provide an explanation as to why this proof proves that forces combine in the same way as displacements?
Simon Stevinus's proof.

 A: They're proving that if something sits on an incline, it's weight along the line parallel to the incline is $mg \sin \theta$. The reason they say this is significant is because you can get the same exact result by treating forces as vectors and adding up the normal forces and gravitational forces.
To restate his argument more clearly:

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*If we put a loop of chain around an incline, it'll eventually come to an equilibrium (obviously it can't be a perpetual motion machine and move forever!)

*Once we have it at an equilibrium, we can remove the part that hangs below as it is perfectly symmetrical and will not affect the equilibrium of each side

*Now, we know, since the system is in equilibrium, that the weight of side $a$ must equal the force that side $b$ exerts. If we assume constant mass density, $\lambda$, than the weight of side $a$ is $(a\lambda)g$, and therefore the force needed to support side $b$ is $(a\lambda)g$. However, we can rewrite $a$ in terms of $b$; namely, $a=b\sin \theta$. Therefore, the force needed to support side $b$ is $(b\lambda \sin \theta)g$, which is $mg\sin \theta$, where $m$ is the mass of side $b$.

You can find more information here: Inclined Plane.
