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
*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.
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