## Ancient method to study Archimedean spiral

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 eg 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 tool:
$S(\phi) = \frac{k}{2} \left[ \phi \sqrt{1 + \phi^2} + \ln \left( \phi + \sqrt{1 + \phi^2}\right) \right]$, $lim_{\phi \to \infty} S’’(\phi)=k$ 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

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 What does this has to do with logic? – Emil Jeřábek Feb 4 at 19:20 With just a modest amount of being charitable Emil, I see implicit in the question something similar to "What is a weak first-order theory or other logical theory that doesn't need calculus or modern machinery to do what I ask about and that is close to something Archimedes could have used?" . A logical perspective on formal systems that model reasoning in ancient times might help with this and similar questions of interest. Gerhard "That's How I See It" Paseman, 2013.02.04 – Gerhard Paseman Feb 4 at 20:34 I'm not sure what you mean by uniform acceleration since the acceleration is $S''(\phi)=\frac{k\phi}{\sqrt{\phi^2+1}}$ which does, as you say, approach $k$ quickly, but never is exactly uniform. An interesting comment is that the arc length of the parabola $y=\frac{kx^2}{2}$ has nearly the same behavior. – Aaron Meyerowitz Feb 8 at 5:07