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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

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What does this has to do with logic? –  Emil Jeřábek Feb 4 '13 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 '13 at 20:34
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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 '13 at 5:07
    
Most likely, Archimedes or anyone else did not even think of the "acceleration" of anything. Aristotle's physics has no use of it AFAIK. However, there is little doubt that since Archimedes could figure out the tangent line to his spiral by splitting the motion into the radial and the circular parts, he would have no problem figuring out the rate of change of the distance from the origin to the point moving along a line at distance 1 from the origin at constant speed (i.e., of finding $\frac d{dr}\sqrt{r^2+1}$). Also there may be a huge gap between what he knew and what we know he knew ;). –  fedja Aug 14 '13 at 13:05
    
Or he could just recognize it as a hyperbola and merely use the work of Apollonius, with which he was most likely acquainted (though I have a vague feeling that here we may have an ancient version of the infamous Gauss-Bolyai story) and which covers the tangent line problem for conic sections in detail. –  fedja Aug 14 '13 at 13:21
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Archimedes might have tried to bound the arclength of the $i$th whorl of the spiral between the circumferences of two circles (of radii $ki$ and $k(i+1)$). If you consistently take, say, the lower bound, or the upper bound, or any sensible weighted average of the two, then you're adding a constant amount of arclength with each whorl. The particle can traverse this extra length in the same time by accelerating uniformly.

Of course, this isn't completely accurate, as mentioned by Aaron Meyerowitz in the comments. But as the angle between the tangent vectors to the spiral and to the circle decreases, the error approaches zero. Archimedes was fairly comfortable handling the error intuitively in limiting arguments like this, as far as I know.

But this is a boring story. I prefer to imagine that one day Archmedes carefully cut an unrolled papyrus scroll in half along one of its diagonals (either on a whim, or out of frustration with the math that he'd been trying to write on it). When the halves are rolled back up, they're approximately parabolic in cross-section. He would have instantly recognized the parabola, after all that time he spent cutting cones in half. I don't know whether Archimedes knew that the parabola is the curve of constant acceleration, though.

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