TITLE: How to draw Archimedean-Galileo spiral?
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: 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.