A Cone Balancing a Cylinder

I am wanting to balance a cone with a cylinder, in the way that Archimedes did in `The Method' with other objects. The plan is to find the volume of the cone using the volume of the cylinder. However, I seem to be having trouble finding a relationship between the sections of the cone and the sections of the cylinder I am balancing it with.

I know that the centre of gravity of the cone is 1/3 of the way along it, and the centre of gravity of the cylinder is half way along it. Having thought about this problem for a while I was convinced that if we take some proportion $\displaystyle \alpha \in [0, 1]$ then the sections to the left and to the right of the respective centres of gravity by this proportion would balance (so the sections at $\displaystyle \frac{\alpha}{2}h$ and $\displaystyle (1-\frac{\alpha}{2})h$ of the cylinder would balance with the sections $\displaystyle \frac{\alpha}{3}h$ and $\displaystyle (1-\frac{2\alpha}{3})h$ of the cone).

I could not get this to work, and in retrospect my guess was clearly wrong (just looking at the extremities gives a contradiction!)

Does anyone have any ideas?

Note that as I know what the volume of a cone is, I was attempting to use a cylinder of radius $\displaystyle \frac{\sqrt{2}}{3}$. This should, I believe, work.