Defining Force

Issac Newton not only introduced the concepts of velocity and acceleration, which can be measured by measuring distances and time, but also introduced the concepts of mass and force. So, technically, $F=ma$ is the oldest equation in Physics. However, putting the definition of mass across, the other day a friend asked how did Newton know that exactly $F=ma$, and why did he not just define the force to be $F=m+a$. So I have made this thread to see what others think.

Isaac Newton did not come out with $F=ma$.

The modern F=ma form of Newton's second law nowhere occurs in any edition of the Principia even though he had seen his second law formulated in this way in print during the interval between the second and third editions in Jacob Hermann's Phoronomia of 1716. Instead, it has the following formulation in all three editions: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. In the body of the Principia this law is applied both to discrete cases, in which an instantaneous impulse such as from impact is effecting the change in motion, and to continuously acting cases, such as the change in motion in the continuous deceleration of a body moving in a resisting medium. Newton thus appears to have intended his second law to be neutral between discrete forces (that is, what we now call impulses) and continuous forces. (His stating the law in terms of proportions rather than equality bypasses what seems to us an inconsistency of units in treating the law as neutral between these two.)

The above paragraph is taken from https://plato.stanford.edu/entries/newton-principia/#NewLawMot

If I don’t remember wrongly, F = ma is from Euler.

https://en.wikipedia.org/wiki/Euler%27s_laws_of_motion

Hope it helps.

Issac Newton not only introduced the concepts of velocity and acceleration, which can be measured by measuring distances and time, but also introduced the concepts of mass and force. So, technically, $F=ma$ is the oldest equation in Physics. However, putting the definition of mass across, the other day a friend asked how did Newton know that exactly $F=ma$, and why did he not just define the force to be $F=m+a$. So I have made this thread to see what others think.

Thank you for your response! I always took for granted that F=ma was credited to Newton, but it's also interesting that what he said is his Principia as a law of motion i assume could be written as $F \propto a$, ...

… but Euler seems to have derived F=ma from first principles.

Actually,for constant mass,accelaration is always exactly in direct relation with the force applied.Thus,Newton took F=ma instead of F=m+a.

I had not read Principia, so I don't really know if the assumption is valid.

To my limited knowledge, I am not sure did Newton "define" motion in Principia and it seems that Newton never presented the “three laws of motion” in the form of equations.

Again, I am not sure about this. If you can support it with reference(s), I would be glad. I would really like to know how did Euler arrive F =
ma. Thanks.

Actually Newton wrote:
F = dp/dt
Given p = mv and a = dv/dt
--> F = d (mv)/dt
F = m dv/dt
F = ma

Well, in both cases, acceleration is directly proportional to the force.....

... Well, I thought the link to Wikipedia you provided in your first post also contained derivations of Euler's laws, ...

... but I really cannot make sense of them.