It is well-known the properties of Archimedean spiral ($\rho = k\phi$) which is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. For example, any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance. The properties were studied by Archimedes by means of math available at the time.

However, let's look at the length of Archimedean spiral which requires more advanced tools:

$S(\phi) = \frac{k}{2} \left[ \phi \sqrt{1 + \phi^2} + \ln \left( \phi + \sqrt{1 + \phi^2}\right) \right]$,
$\lim_{\phi \to \infty} S’’(\phi)=k$
In other words we observe a movement with uniform acceleration along the Archimedean spiral (the junction point of the line which rotates with constant angular velocity and Archimedean spiral moves along Archimedean spiral with uniform acceleration).

So, my question is to understand if there is a simple way to figure “uniform acceleration” out with a tool available at the times?

Now, we can calculate the limit, but does that mean that Archimedes failed to discover the property?

I searched in Archimedes Palimpsest, but it is unclear if Archimedes has a tool to realise the uniform acceleration which appears for the curve he studied. http://en.wikipedia.org/wiki/Archimedes_Palimpsest