# Why is Newton's First Law not considered a vacuous statement? [duplicate]

How did Newton come to postulate the law of inertia when every inertial body that he was considering was (by his own other hypotheses) being acted on by the force of universal gravitation.

In other words, he simultaneously posits the law of Inertia and then theorizes that the contrapositive of his statement is not testable ("If no forces act upon a body at rest, then it will stay at rest", but there is never a situation in which no forces are acting upon a body).

Did he get to this by considering the case of how the inertial velocity of something like a ball being swung on a string in a circle takes off in the direction of the tangent to the circle when the centripetal force is removed, or what were the observations that led to postulate this law? That's the only immediately testable example that I have been able to think of so far, as the same experiment with planets orbiting the sun is not/was not feasible to test in his day.

I'm guessing this law may have some historical context related to the philosophical question of "what makes things move" in Aristotle, and Newton used his law to resolve paradoxes that I am not familiar with.

Or is it something entirely different than what I'm thinking, in that what he actually meant was "it takes less force to keep a body in motion than it does to start a body in motion"?

• The first law is just a special case of the second law. (If F is zero, then a is zero.) Feb 18 '20 at 14:44
• @R.W. Bird This is not entirely true. The first law is technically an independent axiom. Feb 18 '20 at 14:47
• My question is really about the phenomenological context for how he got to this axiom -- what were the examples that he was considering? (I understand that he was considering Kepler's laws on the scale of large bodies, but it seems like the "Universal" part of his laws would mean that there was at least one example of how he observed this law in smaller bodies? Feb 18 '20 at 14:50
• You should be considering net force, right? Feb 18 '20 at 14:51
• If no forces act upon a body at rest, then it will stay at rest This is not the first law at all... can you please cite your sources? Feb 18 '20 at 14:59

## 2 Answers

Newton's First Law is not defined the way probably you have understood. It is as follows:

If the net force on a body is zero, then in the frame of this body, it is possible to choose a reference frame which is either at rest or in uniform motion, such a reference frame is defined as "Inertial Reference Frame".

And, the reference frame which is not at rest or in uniform motion, w.r.t. a body on which the net force is zero, is called a "Non-Inertial Reference Frame".

Newton's First Law actually defines the condition in which other Newton's Laws would be applicable, which is "Inertial Reference Frame" in this case. Newton's Laws are applicable only in inertial reference frames, they are not applicable in Non-Inertial Reference frames.

Now, let's come back to your question.

The problem comes when we take the definition of Newton's First Law as you have proposed. In that case, it seems like a Tautology or Vacuous, and you are right in saying that.

When we take the definition I mentioned, then it makes more sense.

I am assuming that you know that Newton's Laws of Motion are not applicable in Non-Inertial Reference Frames, so when we define them, we first have to define the condition under which they would be applicable and Newton's First Law is essentially the condition under which the other laws would be applicable.

If we take your definition of First Law then it seems like an implication of second law itself, which is not true. Science would never put the same thing twice in different languages.

• I guess the fundamental idea is that Newton postulated that rectilinear motion is the default of large bodies, and hence the force upon the body can actually be measured as some kind of difference between the position that is predicted by the inertial velocity at a given point and the actual position of the body at a later point in time. (Eg. this is the actual intuition behind the "differential" from single-variable calculus?) Feb 18 '20 at 22:05

Physics from the time of Newton to now is the discipline where mathematical differential equations are used , whose solutions fit the experimental data points and are predictive of new data. In order to do this , a subset of the possible mathematical solutions of the differential equations is picked by use of postulates/laws/principles , the axioms that relate data to solutions of the mathematical equations.

That is the purpose of Newtons laws of motion, they are the axioms of the kinematic theory distilled from observations . Nothing is redundant. Note that the concept of Force is defined in the second law. He took words used in everyday language and postulated a mathematical meaning for them, and set up the axioms. His theory has been validated innumerable times, though it had to be modified for the very high masses and velocities ( general and special relativity developed for these regions) and at the quantum dimensions.