Rectilinear motion, in which speed increases proportionally to distance covered From Spivak's Calculus Book

In 1604, at the height of his scientific career, Galileo argued that for a rectilinear motion in which speed increases proportionally to distance covered, the law of motion should be just that ($x = ct^2$) which he had discovered in the investigation of falling bodies. Between 1695 and 1700 not a single one of the monthly issues of Leipzig's Acta Eruditorum was published without articles of Leibniz, the Bernoulli brothers or the Marquis de l'Hopital treating, with notation only slightly different from that which we use today, the most varied problems of differential calculus, integral calculus and the calculus of variations. Thus in the space of almost precisely one century infinitesimal calculus or, as we new call it in English, The Calculus, the calculating tool par excellence, had been forged; and nearly three centuries of constant use have not completely dulled this incomparable instrument.
NICHOLAS BOURBAKI

Edit: I found the quote: https://www.goodreads.com/quotes/9402260-in-1604-at-the-height-of-his-scientific-career-galileo
Fair enough! However, I don't see how the equation follows. Galileo studied falling bodies, but falling bodies don't typically have speed increasing proportionally to distance, they have speed increasing proportionally to time. Where speed increases proportionally to time(const. acceleration), $x=ct^2$. But, at best, it's counter-intuitive that the equation is identical to, say, for speed increasing proportionally to distance.
let $v(0),x(0)$ be the speed and distance at a time $t=0$. then since speed increases proportionally to distance covered;
$v(t) - v(0) = k(x(t)-x(0))$.
Further manipulation results in
$v(t) - kx(t) = v(0) - kx(0)$
and $v(t) = x'(t)$, so it follows that
$x'(t) - kx(t) = v(0) - kx(0)$
These sorts of equations can be solved by isolating the derivative but they
typically have exponential solutions, not ones of the form $x(t)= ct^2$. Why does the latter work?
 A: Almost all that is "commonly known" about Galilei is overstated or covers some sleight-of-hand. This was started by Galilei himself who in a time of mercenary wars and the plague conducted science in a mercenary way, out of necessity and having not the necessary high aristocratic background to get a permanent professorship, thus had to keep the interest of his aristocratic (and ecclesiastic) patrons. Among others, he studied ballistic motions to improve cannons and how to use them predictably. 
He could not have discovered anything in studying falling bodies as that is too fast and measurement of very short times was nearly impossible at the time. He invented the use of a bucket of water with a hole and spigot, and later pendulums as stop-watch. Now imagine doing free-fall experiments with these instruments. 
Even the experiments of rolling a ball down a slope with a groove to guide it were "improved" in their measured results, as manufacturing tolerances for the ball and the straight edges of the groove were not high enough to get even basic accuracy.
So what Galilei published were more-or-less "thought experiments" where several possible models were discussed and admitted or excluded by how much some theoretical conclusions were conform with the practical experience he was able to perform. The remaining models were then elevated by "adjusted" experimental results.
"Speed increases proportional to distance" means $\dot x=cx$ (everything starting at zero) which has $x=x_0e^{ct}$ as solution. If I remember right he discussed that as one of the models to exclude because of unreasonable theoretical consequences. For $x(t)=ct^2$ you get that $\dot x=2ct=2\sqrt{cx}$. In words that would mean that "speed increases proportional to time". Or the "proportional" in the original statement does not mean linear and is to be interpreted as "monotonous", among which class of models one also gets the one with $x=ct^2$.
