So this might not be the realm of 5-year-olds but I was curious from a mathematical standpoint how Newton came to discover specifically that [F]orce was [m]ass multiplied by [a]cceleration and not some other mathematical operation like some logarithm or polynomial or something (not sure what those are either but that’s another eli5 entirely).
In: Physics
You can come up with the law if you can measure position and time. Changes in position over time give velocity. Changes in velocity over time give acceleration .
Set up a cart pulled by a string attached over a pulley. Hang a weight over the pulley to supply a set amount of gravitational force and vary the mass of the cart by adding weights to it. You will quickly discover that the cart’s acceleration varies inversely with its mass. (Twice the mass is half the acceleration, 3x mass is 1/3 the acceleration).
Next leave the cart alone and instead add weight a to the string over the pulley. You will find that the cart’s acceleration varies directly with the weight pulling it. (Twice the weight, twice the acceleration, 3x the weight, 3x the acceleration).
Putting those observations together, a=f/m, which we normally rearrange to F=ma.
Newton didn’t come up with the relationship, it was known since Galileo. Newton combined it with other observations and equations and gave a complete and cohesive description of the laws of motion, so the equation is known as Newton’s second law.
*tl;dr* He didn’t – not in the way we use it today. But what he did come up with was following on from the work of many other mathematicians and physicists (including Galileo) – adding to what they did.
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Newton published his Laws of Motion in his epic work *[Philosophiæ Naturalis Principia Mathematica](https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica)* (Mathematical Principles of Natural Philosophy). The first edition was published in 1687, when Newton was in his 40s and already the Lucasian Professor of Mathematics at the University of Cambridge (one of the most prestigious maths/physics chairs in the world, since held by the likes of Babbage, Stokes, Dirac, Hawking and – in fiction – Data).
Nowadays we know his 2nd law as:
> F = m a
where F, m and a are defined appropriately (with F and a being vectors if needed), or maybe as:
> F = dp/dt
i.e. force is the rate of change of momentum.
Newton didn’t define them this way, not least because he didn’t have algebra (never mind vectors). Algebra was just taking off in European mathematics at the time, but he did his work with geometry. This is why it isn’t necessarily worth reading his *Principia Mathematica* to learn (mathematical) physics; it is clumsily written because he didn’t have the mathematical tools we have today (although he did help invent some to make it work). Plus the original is in Latin. Borrowing from the [1729 translation into English](https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Axioms,_or_Laws_of_Motion), Newton expressed the law as:
> The alteration of motion is ever proportional to the motive force impress’d; and is made in the direction of the right line in which that force is impress’d.
> If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impress’d altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force) if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joyned, when they are oblique, so as to produce a new motion compounded from the determination of both.
“Motion” was his “[Definition II](https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1729)/Definitions):”
> The Quantity of Motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.
i.e. what we would call momentum (“quantity of matter” being what we would call “mass”). He didn’t feel the need to define “velocity” or go into more detail on these terms.
Putting that all together, that isn’t “F = ma” or “Force = mass * acceleration” – what he is calling motion we would call momentum, and his “impressed force” is what we might call “impulse”:
> Impulse = change in momentum
So where did he get the Second Law from? He generalised from the work of earlier mathematicians and physicists. To quote from his commentary on the laws:
> Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiment. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air.
By my reading of this, he is acknowledging that these laws aren’t really his laws. They were already known and understood by others including Galileo (who died months before Newton was born). What Newton did was put them all together in one place, define some things, and then do a lot of work with them (particularly applying them to gravity).
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So what we see is that Newton didn’t really come up with his Second Law of Motion, and certainly didn’t come up with it in the form we use it today.
If you are interested there is [a more technical summary of the history of Newton’s First and Second Laws here](https://www.eduhk.hk/apfslt/v15_issue1/changwj/page3.htm), with a neat diagram near the bottom, showing their development, along with the concepts of inertia and force.
Note that “force” and “mass” are not something that can be directly observed. Taken by itself, the 2nd law merely *define* what a force is, or what a mass is (depending on your philosophical point of view). We use “force” as a convenience quantity to make it easier to state physical laws. So you can’t really consider the 2nd law by itself. “mass” by itself is a blurry concept, because once again, we can’t observe mass directly, only on what it effects is. The concept of “mass” doesn’t even take shape until the 13th century, where it is suggested that there is something else that measure quantity of matter beside volume and weight.
So it’s not meaningful to talk about the 2nd law by itself. Newton himself had no ideas what force is, constantly mixing up between what we would consider different concepts today. What you can do however, is consider the whole of 3 laws.
The concept of momentum went all the way back to Persian scientists in the 1000, and so is the idea that it’s proportional to quantity of matter and speed. By the time of Descartes, he had expressed the idea that momentum of the universe is conserved, but even before him, other philosophers had expressed similar ideas. But even at this point, “mass” is a fuzzy concept that could mean a few different things, like size or weight. “velocity” itself is not quite defined either, Descartes though that velocity is just a numerical quantity, so his “conservation of momentum” is mathematically different from our modern version, but the idea is there.
However, Galileo made a significant experiment: the resistant to change in movement is proportional to the weight of an object. Or in modern term, inertia mass is directly related to gravitational mass. This idea is adopted by Newton too.
Throughout the years after Newton, there are many arguments about what is “mass” or more precisely what is quantity of matter. It’s a very nebulous topic.
What Newton’s 3 laws expressed is just the fact that the total change in momentum is 0, where momentum is proportional to velocity and inertia (resistant to change of motion). But not just that, he make it more precisely the idea that velocity is a quantity with direction. Total momentum isn’t a numerical quantity, it’s a quantity with direction. Of course, lacking in modern mathematical concept of vector, he expressed this idea by describing force as a quantity with direction, formulate the 3rd law, and describe 2nd law also with direction. However, he had no ideas what “quantity of matter” is, precisely, and didn’t even have unit for mass, so even his 2nd law are different from today’s version. Lacking units for mass, he can only talk about force as being proportional to quantity of matter.
What Newton thought as his 3 laws are actually very different from our modern version of the 3 laws, because the concepts needed for it did not exist back then: vector calculus and mass. The modern version is formulated much later, but we attributed it to Newton anyway. The fact that momentum is proportional to quantity of matter and velocity is not original to him, it had been around since the 1000.
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