Doubt reparametrization of curve in Spivak’s “Elementary Mechanics”

I was reading Spivak book on mechanics, and in the first chapter he reproduces with modern day mathematics Newton's proof of proportionality between mass and weight.

He proceeds to explain the equations of motion of a simple pendulum with radius $$l$$.

And then he states the following: for any $$\alpha > 0$$ the path

$$\gamma(t)= \alpha \ {c} \left(\frac{t}{\sqrt{\alpha}}\right)$$

Follows a circle with radius $$\alpha$$ times the radius of the path c with time reparameterized by the factor $$\frac{1}{\sqrt{\alpha}}$$. He then says the following which i don't get how he derives it:

the angle $$\vartheta(t) = \frac{\theta(t)} {\sqrt{\alpha} }$$

I don't get why the new angle after the reparametrization: $$\vartheta(t)$$ is not equal to $$\theta(t/\sqrt{\alpha})$$

• where in the book is this? Which page? Sep 4, 2020 at 0:51
• The page in the book is 16, and the book is elementary mechanics ftom a mathematician's point of view by spivak Sep 4, 2020 at 0:52
• Welcome to Physics StackExchange! (Hola Christian!) Next time please include this information when you ask the question. Spivak has another book on mechanics. Sep 4, 2020 at 1:05

This was a misprint. He meant $$\vartheta(t)=\theta(t/\sqrt{\alpha})$$