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I am asking how did Galileo Galilei derive the equation for height as a function of time, which we now write as $y=\frac{1}{2}gt^2$. We now know that the equation is the solution of Newton's second law when the force is a constant $mg$, which can be found by integration. However, Galileo lived between 1564 and 1642, he died before the birth of Newton and Leibniz, which are credited for the discovery of calculus, so he could not have used it. I have heard that he used the so-called Merton rule for deriving it, but how exactly did he used it? Can someone give a modern version of what Galileo did?

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    $\begingroup$ One does not need calculus to find the area of a triangle. And that is all needed here given that velocity is a linear function of time in the uniformly accelerated motion. Oresme and Oxford calculators did this two centuries before Galileo (the "Merton rule"). Galileo's contribution was experimentally confirming that free fall was uniformly accelerated through clever experiments with inclined planes. And nowhere in kinematics does one need Newton's second law. $\endgroup$
    – Conifold
    Nov 10 '20 at 2:06
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Galileo did not derive this law. He discovered it from his experiments with inclined planes. And the law for falling body would make possible to discover Newton's second law, not other way around.

But the fact that constant acceleration (i.e. gaining equal amounts of velocity in equal amounts of time) meant that path is proportional to time squared was known even to Babylonian. You don't need full calculus for that. You can see it by summing arithmetic progression. Babylonian approximate movement of Son and planets by a motion with piecewise linear speed. E.g. they assumed that a Sun moves $1^\circ$ per day for a half year and a little bit less for another half, so it will go $360^\circ$ for 365 days. And they knew formula for calculating position of the Sun in a given day.

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    $\begingroup$ One does not "discover" anything from experiments, they only confirm or disconfirm what was already discovered (hypothetically). And one does need to derive the consequences to do the confirmations. In this case, the law was derived two centuries earlier by Oresme and Oxford Calculators, who expressed it using mean speeds, from the assumption of constant acceleration. Galileo derived a more straightforward version, and confirmed it by experiments with inclined planes. $\endgroup$
    – Conifold
    Nov 10 '20 at 2:10
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    $\begingroup$ @Conifold That's word-smithing to some extent, since lots of things were discovered via experiment, if you allow, say, observation of Jupiter leading to finding out it has moons; or any number of unexpected phase transitions in materials as pressure-temperature experiments were run. $\endgroup$ Nov 10 '20 at 14:16
  • $\begingroup$ @CarlWitthoft You are right, I should have said that laws are not discovered from experiments. $\endgroup$
    – Conifold
    Nov 10 '20 at 18:42
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You don't need calculus to show the relationship that you are pointing out. One only needs to plot distance wrt time.

Of course one needs instruments that can accurately measure time which is perhaps where Galileo was helped by his discovery that the pendulum can act as an excellent clock.

Calculus is required to demonstrate that relationship follows from Newtons laws of motion, most of which, I should wasn't discovered by Newton, but his predecessors, as he freely acknowledged.

I should add that it's not neccessary that calculus is 'required' for these demonstrations as Newton himself did without it in his Principia. In this, Newton was also preceded by Archimedes who also presented geometrical proofs of results which were derived by him through his 'mechanical method', as the geometry was the lingua fraca of mathematicians then, as it is now.

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