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It is known that some plane curves can be drawn with a tool. For instance, I heard at a web site that Archimedes created his spiral in the third century B.C. by fooling around with a compass and others.

Let’s however look at the spiral defined by the equation: $r'(\theta)^2+r(\theta)^2=\theta^2$, $r(\theta=0)=0$

I am looking for a method ( a tool) which could help to plot the spiral on paper ( I named it as Archimedean-Galileo spiral. For large $\theta$, the curve represents Archimedean spiral: $r=\theta$. When $\theta$ is small it transforms in Galileo spiral $r=\theta^2$) .

The spiral has a property that the junction point of the curve and the ray uniformly rotated in the origin coordinates when the junction point moves with uniform acceleration.

Do you think that there is a way to draw it without computer, but with other special curves (tools)?

I thought about the spiral of Theodorus, but I am not sure how the spiral of Theodurus is connected with the equation.

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Since the differential equation doesn't appear to have closed-form solutions (Maple doesn't find any), this seems quite unlikely (depending, of course, on your definition of "tool"). –  Robert Israel Jan 4 '13 at 19:33
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up vote 8 down vote accepted

This is not a complete solution, but a suggestion on what you can try. First, note that the "planimeter" will "compute" for you integrals $$ F(x)=\int ydx, $$ where $y=f(x)$ is the given curve. This "computation" is in the form of a numerical output measured by a rotating wheel, but, using Paucellier-Lipkin inversor you can convert this to a linear motion which will lead to a geometric drawing of the graph of $F(x)$.

On the other hand, "Kempe's universality theorem" allows you to trace arbitrary planar algebraic curves using mechanical linkages. Original Kempe's proof suffered from a number of deficiencies (its description could be found e.g. here). The deficiencies were fixed here. (Note that David Khudaverdyan has a computer-aided implementation of a curve-drawing algorithm for some planar curves, see here.) Combination of planimeter and Kempe-like linkages, allows you to "compute integrals" of algebraic functions using a mechanical device. By playing with these constructions, maybe you can construct a device (impractical, of course), which will trace the curve you are interested in.

An interesting question coming out of this is if there is a generalization of "Kempe's universality" for 1-st order algebraic ODEs (or even more general ODEs/PDEs), e.g.:

Suppose that $P$ is, say, a polynomial, function of several variables and you have an ODE of the form: $$ P(u', u, t)=0, $$
where $u=u(t)$ is an unknown function ${\mathbb R}\to {\mathbb R}^2$. Is there a "mechanical device" (where one would allow both mechanical linkages and "planimeter-like" gadgets, yet to be specified) which will trace solution curves of this ODE?

Edit: Indeed, Shannon in "Mathematical theory of the differential analyzer" J. Math. Phys. Mass. Inst. Tech. 20 (1941), 337-354, proved that one can "draw" solutions of arbitrary algebraic differential equations using mechanical devices ("analog computers"). A gap in Shannon's proof was noted and fixed by M.B. Pour-el, see here.

On the practical side, there are real differential analyzers (like the one by V. Bush), see here, built up to 1950s, which were used to solve algebraic differential equations.

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You have the answer: the spiral has a property that the junction point of the curve and the ray uniformly rotated in the origin coordinates when the junction point moves with uniform acceleration.

Take a ruler and fix one of its extremities to a motor, so that the ruler materialize the ray uniformly rotated.

Make a hole on the table just below the motor axis. Then pass a string trough this hole, attach a weight at the string under the table, and manage the other extremity of the string to be able to slide along the ruler.

Fix a pencil somewhere on the string along the ruler. When you drop the weight, it will fall down with a uniform acceleration, dragging the pencil along the curve you want to plot.

I wouldn't be surprised if, in facts, Archimedes was not thinking about such a device when he invented the curve.

Note that, with the ancient Greek mentality, to draw the curve with a computer would have been an achievement much superior to build the mechanical device.

They were right after all: along the way Archimedes would have, at least invented, modern algebra, differential calculus, the planimeter and the computer.

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Did Archimedes know (or even expect) that the acceleration of a falling object is uniform? –  Andreas Blass Mar 17 '13 at 19:14
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