geodesic is parametrised by a constant multiple of arc length

Let $\displaystyle S\in \mathbb{R}^3$ be a regular surface, and $\displaystyle \gamma: I \rightarrow S$ a geodesic, where $\displaystyle I$ is an open interval in $\displaystyle \mathbb{R}$. We have that $\displaystyle \frac{d}{dt}|\gamma ' (t) | = 0 \implies |\gamma '(t)| = c$, a constant , i.e.

$\displaystyle

\frac{ds}{dt} = | \gamma' (t) | = c$ ..................(1)

where s is the arc length parameter $\displaystyle s(x) = \int_a^x |\gamma'(p)| dp$, $\displaystyle a\in I$. I understand what is happening up to here but then my notes claim that $\displaystyle t=\frac{s}{c}$. However do we not get that upon rearranging (1)

$\displaystyle ds=cdt \implies s = ct + A$ where A is some constant? Why is A zero? Any help would be appreciated