How did we get the 'modern' law of motion?

Question

Newton, in his famous book Principia, stated second law of motion as:

The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

A more mordern version of second law states, on wikipedia, as

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

$$\mathbf{F}=\frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=\frac{\mathrm{d}(m \mathbf{v})}{\mathrm{d} t}$$

From Newton's law the best I can do to make it mathematical is is to take $F$ as the force applied and $h$ be the 'change in motion'. Then according to the Newton's law: $$ F \propto h $$

If I compare 'my' mathematical expression to that of Wikipedia's I find that $$ h=\frac{d P}{d t} $$

So my question is how do we know that $ h=\frac{d P}{d t} $?
Or more clearly how do we know that what Newton meant by 'change in motion' is 'change of momentum'? And why did Newton not stated the second law as that of Wikipedia's. After all he was 'Newton'. He had invented calculus!

Answer 1

Newton defined "quantity of motion" at the beginning of Book I in Definition II.

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

Here, "arising from the velocity and quantity of matter conjunctly" means it is the result of multiplying the velocity and mass.

A comment to another question may answer why Newton didn't express his laws more mathematically:

I'm pretty sure that Newton never wrote his law of gravitation in algebraic form, nor thought in terms of a gravitational constant. In fact the Principia looks more like geometry than algebra. Algebra was not the trusted universal tool that it is today. Even as late as the 1790s, Cavendish's lead balls experiment was described as 'weighing [finding the mass of] the Earth', rather than as determining the gravitational constant.

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Addendum:

I've been reading some sections of Newton's Principia more closely and I think Newton is using an ambiguous definition for force. Take the following passage from the Scholium following the Three Laws of Motion:

When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time.

The first use of the word "force" in "the uniform force of its gravity" would seem to be the usual definition of force that causes acceleration. The second use, "impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities," seems to refer to a force acting over time that causes a finite change in velocity. Modern physicists would refer to the latter usage as "impulse," which would be expressed mathematically as $J = F\Delta t$ or $J = \int Fdt$ for varying forces.

Given the novelty of calculus at the time, perhaps Newton could not speak confidently of instantaneous actions over infinitesimal quantities of time, so he only spoke of finite intervals, which would also yield to more geometric arguments of motion through space. If Newton had written more mathematically rather than geometrically, he might have expressed the Second Law as either $$F = \frac{\Delta(mv)}{\Delta t}$$ or $$F\Delta t = \Delta(mv)$$ with both of the left-hand sides of the equations being referred to as force.

Answer 2

Newton stated his law for constant mass, to day we can think of rockets for example with changing mass. For constant mass i.e. dm/dt=0 the two laws are the same.

Answer 3

In Newton's time, with the means available, it would have been very hard to present specific experimental evidence for the Second Law.

For the moment assume that at ground level gravity is uniform. Then if the second law holds good a projectile will move along a parabola.

To my knowledge it was only many years after Newton that someone actually set up a specific experiment. The setup allowed an object to be released from some height, with a very repeatable horizontal velocity. Therefore as it was accelerated down by gravity the trajectory should be in the form of an inverted parabola. Nails were hammered into a board, along the trajectory. The object would fall right along that curved row of nails each time. To within measuring accuracy, that trace trajectory was an inverted parabola. Of course, this was still a very crude method of corroboration.

In Newton's time the second law was plausible of course.

In the Principia Newton demonstrated that if you use the second law as universal law of acceleration and you use the inverse square law of gravity as universal law of gravity, then you can derive that all celestial bodies will move along Kepler orbits.

That demonstration was tremendously powerful of course. Kepler's laws were known to hold good. The fact that Newton's law of Universal Gravity reproduces Kepler's laws is overwhelming evidence that Newton's law of Universal Gravity holds good.

In a wider sense that applies for everything that went into the calculation of the orbits of the celestial bodies. If a certain set of conceptual building blocks leads to reproducing Kepler's laws, then that corroborates that entire set of conceptual building blocks.

Specifically, if Newton's Second Law would not hold good, then the combination of the Second Law and the inverse square law of gravity would not reproduce the actual celestial motions.

This answer originates from an exchange that I had in the History of Science and Mathematics stackexchange after submitting a question about the origin of f=ma