> Yes, and you only find this because you assume a trajectory first, and then find force as a result of the trajectory.

Sure, the choice of solution is arbitrary. No one (not even the author) argues otherwise. In fact it's the whole point.

> The force isn't causing the movement, you first assume on your trajectory that the ball stops being at rest immediately after T.

If the gravity were not there, this could not be a valid solution. The net force from gravity causes the acceleration. It is only a necessary condition and not sufficient, but the fact remains that gravity acts on the particle whenever it sets in motion.

> Here is Wikipedia's translation of the first law: > Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

So there are naturally multiple translations of Principia, but I just cracked open the first free one I could find on Google[1] and it doesn't mention anything about compelling, merely endeavoring. Even if we generously accept that it is "compel," we must then reckon with the fact that in the solution where it sets in motion, the net force has in fact compelled it to do so. I wouldn't hang my hat on the exact wording implying anything about determinism.

> Newton's first law is literally that an object stays at rest in its inertial reference frame unless a force causes it to leave its state of rest, yet you have to assume that immediately after T it leaves rest without any valid physical reason.

It leaves rest for the perfectly valid physical reason that there's a net force. You are trying to pick apart the timeline to find a moment where the particle was impelled to move with no net force, but no such time exists. This argument absolutely conflates nondeterminism (the particle could leave at any time) with acausality (nothing caused the particle to move).

[1] https://redlightrobber.com/red/links_pdf/Isaac-Newton-Princi... The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may be considered both as resistance and impulse ; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly are taken to be so.