# F = ma -- How was did we come to understand that this compact form expressed what Newton said in words?

My understanding is, Newton in the 17th century did not use this formula but rather said, in words basically that if you apply a force it will cause a mass to accelerate in the direction of that force.

How did we move from descriptive words to using multiplication to express the same idea? Many people must have said, What does multiplication have to do with exerting a force? (Indeed, I ask that same question today in the 21st century.)

• Principia, Definition II:"The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly... in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple." Nobody familiar with how proportionality works had to ask why proportionality to velocity and "quantity of matter" meant proportionality to their product. Law II then declared force to be proportional to its "alteration" (derivative). Mar 11 at 1:34
• You are right; at Newton's time the algebraic symbolism was quite new and its application to dynamics was still to be done. See here for Euler. And see also this one. Mar 11 at 7:49
• Regarding Jakob Hermann, see Niccolò Guicciardini, An Episode in the History of Dynamics: Jakob Hermann’s Proof of Proposition 1 Book 1 of Newton’s Principia (Hist.Math. 1996) Mar 11 at 7:55
• See Hermann's Phoronomia (1716) Mar 11 at 8:59
• See Book I, Coroll.I to Prop.I: " so that if the body were $C$, and the action of gravity, by which its individual elements may be affected, may be called $G$, the quantity or the product $C.G$ establishes the absolute weight of the body $C$, on understanding by this letter $C$ to be the mass of this body." Mar 11 at 9:06

## 1 Answer

Sometimes you just have to Wikipedia-it:

Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the Principia, he had already developed calculus (which he called "the science of fluxions"), but in the Principia he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of Euclid to be more rigorous. Consequently, the Principia does not express acceleration as the second derivative of position, and so it does not give the second law as $$F=ma$$. This form of the second law was written (for the special case of constant force) at least as early as 1716, by Jakob Hermann.

Note that Newton says that $$F$$ is proportional to $$\dot p$$ (the derivative/change of momentum $$p= m v$$). No need to have a product of anything.

Looking at Hermann's book (Phoronomia), he writes directly $$G=M \mathrm dV/\mathrm dt$$, $$G$$ is the weight, $$M$$ the mass and $$V$$ the speed.

Leonhard Euler wrote independently in Novi Commentarii academiae scientiarum Petropolitanae that $$F=M \mathrm d^2x/\mathrm d t^2$$, in 1775.

Futher reading: https://plato.stanford.edu/entries/newton-principia/

• I do not post for fame and profit: I am asking because I do not understand how multiplication became a shorthand for something that I do not think is clearly expressed by the equation. Newton's wording from 350 years ago is understandable, the equation is not. Mar 10 at 21:47
• @releseabe The "multiplication" is hidden in the word "proportional" with the constant of proportionality (assuming the mass is constant) being the mass. Thus, the multiplication is implicit in Newton's wording (again, assuming the mass is constant). Mar 11 at 0:48
• @releseabe can you be more specific? multiplication between what and what? In Newton's account he says it is proportional meaning that an increase in force is also a linear increase in the change of momentum. Mar 11 at 9:21
• f = ma -- multiplication seems to mean applying acceleration -- so the physical act of impelling an object becomes mathematical multiplication, quite a conceptual leap, it seems to me. Mar 11 at 9:53
• @releseabe Multiplication between acceleration and what? Mass? again Newton did not use multiplication. Newton used momentum $p=mv$, so then your question is why is momentum $m$ times $v$? Mar 11 at 10:12