# 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.

• Your puzzlement is common because we take the Newtonian notion of force for granted today. It is true that $F=ma$ has to be postulated to build mechanics Newton's way, but it is not true that it can be assumed. For that Newtonian "force" has to have a pre-existing meaning, which it did not. The second law is an implicit definition of what "force" means, and experiments to "test" it would be circular. It meant something else before Newton, see How did Newton measure forces in his experiments to establish the laws of motion? – Conifold Jun 29 '19 at 21:36
• @Conifold I am familiar with the concept that any law of nature is both a law of nature and a operational definition of the concept that it makes a statement on. For example Ohm's law $V=IR$ is an operational definition of what constitutes 'resistance to current'. You cannot first define the concept of 'resistance to current' and then discover Ohm's law; Ohm's law defines what constitutes 'resistance to current'. However, in this particular case I think the historical assessment still has to be that in Newton's time F=ma was already firmly established. – Cleonis Jun 30 '19 at 6:31
• What would be the meaning of "firmly established" when the concept itself was not available? Explanations of experiments given before Newton can be nicely embedded into his formalism, but also into others, which do not use his concept of force. Huygens, and especially Leibniz, favored alternatives based on what came to be called energy and momentum, for example. Later it became part of the vis viva controversy over the basic concepts of mechanics, which was not settled until mid-18th century. – Conifold Jun 30 '19 at 7:29
• @Conifold Yeah; subtleties. The kind of experiments that were feasible to, say, Huygens, was to have dual pendulums, with the bobs colliding with each other at the lowest point. That kind of works around a necessity to provide an unambiguous definition of force; since in the feasible experiments the transfer of motion can be treated as instantaneous. I'm aware of course that in the Principia Newton treats acceleration as the cumulative effect of many transfers of momentum. In the limit of infinitely many small transfers in proportionally smaller time intervals: apparent smooth acceleration. – Cleonis Jun 30 '19 at 8:15
• Newton himself was quite happy to assign credit for the laws of motion retroactively. In the Axioms section of Principia he writes:"By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time... By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies". However, the inspection of cited works shows that they did not discover or determine things that way. – Conifold Jun 30 '19 at 21:07

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 – Conifold Jul 4 '19 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. – Conifold Jul 4 '19 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. – Conifold Jul 4 '19 at 4:18