# The origin of F=ma

Authors writing about history of physics describe that before the writing of the Principia several scholars were aware that if the orbits of the planets would be circular (which they knew wasn't the case) then Kepler's third law was consistent with an inverse square law for gravity. These scholars included figures such as Christopher Wren, John Hooke, and Edmond Halley.

It seems to me that this implies that during that time the relation F=ma was already generally accepted as valid. Since without F=ma you cannot even evaluate the case of circular motion. However, I cannot find descriptions of how that came to be.

I wonder: was F=ma blindly assumed? It seems so. Of course, since F=ma is in fact correct assuming it was justified, but I find it odd that I cannot find any history of how F=ma came to be established.

To my knowledge the earliest attempts at experimental investigation of the relation between acceleration and force was by Galilei. To my recollection historians describe that Galilei had small balls rolling down inclines. A small vessel with a smaal hole near the bottom released a slow stream of water. Galilei (according to the description) would use several different lengths of inclines. He would allow the water to start flowing on release, and stop the flowing when the ball reached the end of the incline. the weight difference between start and end would then be a proportional measure of the time it took the ball to roll down the incline.

This experiment may have actually been performed by Galilei, or possibly it was only a description of something that one could do (similar to the description of simultaneous dropping of two different weights from a high tower that was described as possible demonstration, with later authors erroneously describing that as something that Galilei had actually done.)

I tried looking up information about the views of Pierre Gassendi, who, as I understand it, is recognized as the first to formulate the modern notion of what today is referred to as 'Newton's first law'. But I didn't find description of Gassendi proposing F=ma

Speaking in general:
If you do not assume F=ma then it seems to me you cannot make any progress when it comes to formulating a theory of mechanics. And of course when you want to do science you must believe that you are in a position where progress is possible.

Once newtonian mechanics became established the law F=ma became knows as Newton's second law, and I suppose the vast majority of authors simply assume that Newton was to first to formulate it.

I argue that cannot be, as earlier established work is dependent on the presupposition of F=ma.

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– Danu
Commented May 13, 2020 at 14:20

The following is not a self-written answer; it's a comment. For obvious reasons here on stackexchange comment space is limited. This comment is large.

Descartes struggled at formulating a set of laws of collision, he didn't get things right, that illustrates how opaque the problem is.

As I mentioned in an a comment, the kind of experiment that was readily doable at the time was to construct dual pendulums, with the bobs colliding with each other at the lowest point. Newton describes that kind of setup in the 'Scholium' to the 'Axioms, or laws of motion'.

About collisions: As long as you formulate all problems in dynamics in terms of collisions taking place you can get by without a committed position on how to think about force.

Novel in Newton's exposition in the Principia, it seems, is that Newton went all in on putting static force and dynamic force on equal footing.

One way to illustrate this is to consider how Newton is motivating the third law. Newton is switching between cases of static force (tension in a rope tied to something immovable) and cases of dynamic force (objects exerting a force upon each other, accelerating each other)

Here is what I didn't appreciate when I submitted the question:
In Newton's time, when you consider the possibility that the planets move according to a inverse square law, it isn't necessary to think of the planets being pulled by a force. You can think of the planets as subject to some accelerating influence, not otherwise specified.

Using the inverse square rule for the magnitude of the acceleration, combined with knowledge of the required centripetal force to sustain circular motion you can see the tantalizing consistency of an inverse square law with Keplers third law.

Newton's novel approach:
Newton is committed to thinking of gravity as a force. To my understanding: Newton regards the Sun's gravity as literally tugging at the planets, just like in daily life rope is used for tugging things.

To treat gravity as literally tugging you do need a law that relates force and acceleration. F=ma may have been proposed before, but Newton needs it.

Still, it is odd to me that in the Principia Newton doesn't work particularly hard to convince the reader that F=ma holds good. One rather gets the impression that Newton is confident that the reader will readily accept F=ma.

Comparison: the case for gravitational mass being equal to inertial mass.
If gravitational mass would not be equal to inertial mass then in order to calculate the orbit of a planet you have to know its mass.

Conversely, if gravitational mass is equal to inertial mass then every object, independent of its mass, will follow the same orbit.

So it was necessary for Newton to present a strong case that gravitational mass is equal to inertial mass. One part of that is to assure the readers that that pendulums of unequal mass but with the same string length have the same period of oscillation. In the Principia Newton also considers that the other planets may have a composition that is different from the Earth. Therefore Newton describes experiments (pendulum-type setup) he conducted to address that issue.

So: in the case of whether gravitational mass is equal to inertial mass we do see that Newton puts in effort to convince the reader.

So why no effort to convince the reader of F=ma?

One possible hypothesis (but I think it hardly plausible):
Newton suggests to the readers that the laws of motion he presents arise from prior work by other scholars: Christopher Wren, Christiaan Huygens, John Wallis. However, that prior work was not as advanced as Newton suggests it was. (Conifold describes this as 'Newton assigning credit retroactively') Possibly Newton sougth to smuggle some ideas past his readers. I don't know.

• I think you are onto something with your last suggestion. The difference between $F=ma$ and the equivalence of masses or the inverse square law is that the latter can be presented as "derived" from measurements (directly or via the Kepler's laws). Newton must have realized that what to call $F$, $mv$, $ma$, $mv'''$, etc., is not so derivable. The difference, really, is that with $F=ma$, and not others, one can find simple independent descriptions of $F$ in specific examples. But that can not be rolled into a measurement or an airtight deductive argument like the one he gave for the gravity law Commented Jul 4, 2019 at 3:50
• Galileo and Huygens measured the "strength" of gravity with the distance traveled in free fall in a unit of time, but that does not distinguish among the above $F$-s. How to say that it is just that many examples work out best with his $F$? Imagine the fun a bishop Berkeley would have had with the edifice of Principia resting on affirming the consequent. There was no hypothetico-deductive philosophy to lean on, or the realization that theoretical laws really cannot be derived from observations. Euclid was still the official standard. Commented Jul 4, 2019 at 4:17
• So Newton did what he did. He stated the laws without trying to derive them, laid out the known examples they can nicely explain in the Scholium, and offered a "bribe" to the illustrious predecessors: you pretend these laws are already established, and I pretend you are the ones who established them. Share the credit, spread the risk. That is my speculation. And it could have been otherwise if Leibniz, say, prevailed. We'd have a Hamilton-style dynamics with $F$ treated as an intermediate mathematical convenience. Predictively equivalent, but with different talk and imagery attached. Commented Jul 4, 2019 at 4:18

Buridan introduced p = mv, called it “impetus” and stated that it did not change if no force was applied. Buridan’s Inertia Law is known as Newton’s First Law. If one reads Buridan carefully, one sees he asserts: Force = Deviation from Trajectory, and dp/dt = Force.

Newton needed this law: dp/dt = F (where F is the Force, by definition). It’s an axiom. (Weirdly the Second Law implies the First…)

Maybe one should establish historical priority and call it Buridan's law? That was more than three centuries before Newton... And some of Buridan's students would establish some of the first theorems of calculus (through graphical methods they invented).

• For further reading: Why don't we learn Buridan's laws of motion? (Yes, I know you know about this question--one of the answers is by you--but others here now, or those coming to this question years from now might not know about it.) Commented May 12, 2020 at 6:55