Nowadays it seems to be popular among physics educators to present Newton's first law as a definition of inertial frames and/or a statement that such frames exist. This is clearly a modern overlay. Here is Newton's original statement of the law (Motte's translation):

Law I. Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

The text then continues:

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motion, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

And then the second law is stated.

There is clearly nothing about frames of reference here. In fact, the discussion is so qualitative and nonmathematical that many modern physics teachers would probably mark it wrong on an exam.

I have a small collection of old physics textbooks, and one of the more historically influential ones is Elements of Physics by Millikan and Gale, 1927. (Millikan wrote a long series of physics textbooks with various titles.) Millikan and Gale give a statement of the first law that reads like an extremely close paraphrase of the Mott translation. There is no mention of frames of reference, inertial or otherwise.

A respected and influential modern textbook, aimed at a much higher level than Millikan's book, is Kleppner and Kolenkow's 1973 Introduction to Mechanics. K&K has this:

...it is always possible to find a coordinate system with respect to which isolated bodies move uniformly. [...] Newton's first law of motion is the assertion that inertial systems exist. Newton's first law is part definition and part experimental fact. Isolated bodies move uniformly in inertial systems by virtue of the definition of an inertial system. In contrast, that inertial systems exist is a statement about the physical world. Newton's first law raises a number of questions, such as what we mean by an 'isolated body,' [...]

There is a paper on this historical/educational topic: Galili and Tseitlin, "Newton's First Law: Text, Translations, Interpretations and Physics Education," Science & Education Volume 12, Number 1, 45-73, DOI: 10.1023/A:1022632600805. I had access to it at one time, and it seemed very relevant. Unfortunately it's paywalled now. The abstract, which is not paywalled, says,

Normally, NFL is interpreted as a special case: a trivial deduction from Newton's Second Law. Some advanced textbooks replace NFL by a modernized claim, which abandons its original meaning.

Question 1: Does anyone know more about when textbooks begain to claim that the first law was a statement of the definition and/or existence of inertial frames?

There seem to be several possible interpretations of the first law:

A. Newton consciously wrote the laws of motion in the style of an axiomatic system, possibly emulating Euclid. However, this is only a matter of style. The first law is clearly a trivial deduction from the second law. Newton presented it as a separate law merely to emphasize that he was working in the framework of Galileo, not medieval scholasticism.

B. Newton's presentation of the first and second laws is logically defective, but Newton wasn't able to do any better because he lacked the notion of inertial and noninertial frames of reference. Modern textbook authors can tell Newton, "there, fixed that for you."

C. It is impossible to give a logically rigorous statement of the physics being described by the first and second laws, since gravity is a long-range force, and, as pointed out by K&K, this raises problems in defining the meaning of an isolated body. The best we can do is that in a given cosmological model, such as the Newtonian picture of an infinite and homogeneous universe full of stars, we can find some frame, such as the frame of the "fixed stars," that we want to call inertial. Other frames moving inertially relative to it are also inertial. But this is contingent on the cosmological model. That frame could later turn out to be noninertial, if, e.g., we learn that our galaxy is free-falling in an external gravitational field created by other masses.

Question 2: Is A supported by the best historical scholarship? For extra points, would anyone like to tell me that I'm an idiot for believing in A and C, or defend some other interpretation on logical or pedagogical grounds?

[EDIT] My current guess is this. I think Ernst Mach's 1919 The Science Of Mechanics gradually began to influence presentations of the first law. Influential textbooks such as Millikan's only slightly postdated Mach's book, and were aimed at an audience that would have been unable to absorb Mach's arguments. Later, texts such as Kleppner, which were aimed at a more elite audience, began to incorporate Mach's criticism and reformulation of Newton. Over time, texts such as Halliday, which were aimed at less elite audiences, began to mimic treatments such as Kleppner's.


3 Answers 3


I did not do more than read Newton, and a few commentators, so my insight on this is probably meager. But I am sure that you are right that the inertial frame interpretation of the first law is only a modern ex-post-facto justification for making it separate from the second law. Newton certainly never used the first law to define an inertial frame, he just assumed you had one in mind, since inertial frames were not the focus of his investigation.

I think that the statements of the laws of motion are unfortunately following Aristotle more than Euclid. Since physics is no longer regarded as philosophy, we value independence of axioms over clarity of philosophical expounding, and this makes the first law redundant. But if you are stating a philosophical position--- that things maintain their state of motion unless acted upon--- Newton's first law is a neat summary of the foundation of the world-system.

Note that Newton does not state it as "a body in linear motion continues moving linearly". He includes rotational motion too, even though this is a different idea. I think he conflates the two to fix in mind the philosophical position that uniform motion is the natural state of all objects. In Aristotle, the natural state of massive stuff like "earth" is to be down at the center of universe, and of light stuff like "fire" to be up in the heavens, leading to gravity and levity. Newton is replacing this notion with a different notion of natural state. Then the second law talks about deviations from the natural state, and is a separate philosophical idea (although not a separate axiom in the mathematical sense).

The influence of Aristotle has (thankfully) declined through the centuries, making Newtons laws a little anachronistic. I think that we don't have to be so slavish to Newton nowadays.

Newton was aware of the importance of linear momentum and angular momentum conservation. One other way of understanding and his first law can be thought of as making the conservation laws primary. This point of view is both closer to Newton's thinking (it is what makes his "natural states" natural), and it is also a better fit with modern understanding. So it might be nice to restate the first law as "linear momentum and angular momentum are conserved".

All this is based on personal speculation, not on sound historical research, so take with a grain of salt.

  • $\begingroup$ I think following Aristotle point by point had a rhetoric value. He was arguing against Aristotle's system that was in place for almost two thousand years. $\endgroup$
    – timur
    Commented Sep 23, 2017 at 21:03

I will argue that A,B, and C, are all wrong on the grounds of logic (and remark on pedagogy).

Firstly, A is wrong because the first law does not follow from the second law. The first law makes the specific testable predictions that in the absence of a net force, that the motion not change. But the second law makes no such claim because it is possible (see Nonuniqueness in the solutions of Newton’s equation of motion by Abhishek Dhar Am. J. Phys. 61, 58 (1993); http://dx.doi.org/10.1119/1.17411 ) to have solutions to $F=ma$ that violate Newton's first law. So it is not redundant.

As for B, I don't see a logical defect with Newton's laws, he posits an isomorphism between solutions to second order differential equations and spatio-temporal observations. Since he explains how to calculate these solutions, it's a method for making predictions that can be tested. And the first law simply decreases the allowed solutions to only contain those that also satisfy the non-redundant first law, as above.

As for C, if you read the first law as quoted by you very closely you'll notice he mentions forces, plural forces. So he isn't merely restricting himself to nonexistant perfectly isolated bodies. He's talking about special cases where the forces in play produce no net force. He's saying that if you had a very hard very level very smooth surface (so eliminating a net force from gravity and the surface) and then you keep dust from building up or winds from blowing at your device then it will slide or spin at a uniform rate. Sure, testing it would never be perfect, so if it was just a first law that would be a problem, but the second law allows us to bound how big the deviations are by the deviations from zero net force. But that then requires a separate theory of force, specifically force laws that postulate particular forces. This is implicit in the first two laws.

The third law is totally different because it actually constrains what kinds of force laws to consider.

So, the third law constrains what force laws you consider. The second law turns these force laws into predictions about motion, thus allowing the force laws to be tested, not just eliminated for violating conservation of momentum. The first law then excludes certain solutions that the second law allowed.

There is a place for all three laws, and they all mean something. Maybe we should teach them in the reverse order, but ... and this is the only point where I'll bring up any history: I assume the order given by Newton can be blamed on Newton.

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    $\begingroup$ +1, Very nice answer to an old question. To me, Newton's first is the most basic of his laws. It isn't so much that Newton's 1st derives from Newton's 2nd so much as Newton's 2nd is consistent with Newton's 1st by necessity because Newton's 1st is the most basic. Newton's 1st doesn't preclude $F=ma^2$, $\sqrt F = ma$, or any other function $a=a(F)$ so long as $a(0)=0$. Secondly, as you note, Newton's 2nd admits non-physical solutions that are precluded by Newton's 1st. Norton's dome is one such example. $\endgroup$ Commented Dec 29, 2014 at 0:45

Henri Poincaré (1854-1912) and Pierre Duhem (1861-1916) maintained that the law of inertia as formulated by Newton is merely a postulate.

Poincaré wrote in La science et l'hypothese (pp. 112-19), addressing how one would even know that a body without any forces imparted keeps moving indefinitely:

No one has ever experimented on a body screened from the influence of every force, or, if he has, how could he know that the body was thus screened?

Sir Arthur Eddington argued that the the law of inertia as formulated by Newton is circular, thus it conflates inertia to preserve motion with inertia to remain at rest. In his The Nature of the Physical World, Eddington gives a harsh criticism of Newton's First Law (my emphases):

Unfortunately in that case [of external forces acting on it] its motion is not uniform and rectilinear [as it would be if the 1st Law applied]; the stone describes a parabola. If you raised that objection you would be told that the projectile was compelled to change its state of uniform motion by an invisible force called gravitation. How do we know that this invisible force exists? Why! Because if the force did not exist the projectile would move uniformly in a straight line. The teacher is not playing fair. He is determined to have his uniform motion in a straight line, and if we point out to him bodies which do not follow his rule he blandly invents a new force to account for the deviation. We can improve on his enunciation of the First Law of Motion. What he really meant was—"Every body continues in its state of rest or uniform motion in a straight line, except insofar as it doesn't." … The suggestion that the body really wanted to go straight but some mysterious agent made it go crooked is picturesque but unscientific.

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    $\begingroup$ I don't like Eddington's comments on this (or on other things, frankly). The point of Newton's first law is to explain that the natural state is motion, motion doesn't go away, momentum is conserved in isolation. Then the second law says that changes in momentum are physical flows from one body to another of momentum, and defines the force as this flow. The third law says it's a flow, so that as much as one body gains, the other loses. This is just a clear but redundant statement of conservation of momentum in a way suited to the philosophical prejudices of the time. $\endgroup$
    – Ron Maimon
    Commented Jun 23, 2012 at 8:32
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    $\begingroup$ This raises another question: Why didn't Newton just propose the 2nd Law ($F=\dot{p}$) and leave it at that? The 2nd Law implicitly contains the first and third laws, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3: "We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances." There are intro textbooks (e.g., this one) that study momentum first and then force $F=\dot{p}$ based on momentum. $\endgroup$
    – Geremia
    Commented Jun 23, 2012 at 22:18
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    $\begingroup$ Newton is separating out the laws because 1. he likes the number three, it's kind of magical 2. he wants to give the natural state argument then deviations from natural state 3. He felt like it! Who cares what Newton said in detail. He does define momentum first, and then says the force is the change in the momentum. The philosophical prejudice of the time is that when you state laws like Aristotle: you say what a body "wants to do", then how external stuff keeps it from doing what it wants. Newton says a body wants to move the same forever, and a force keeps it from doing that. $\endgroup$
    – Ron Maimon
    Commented Jun 24, 2012 at 5:28
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    $\begingroup$ @Geremiah: Eddington got great fame for himself by being first with the eclipse pictures, but he fudged the data to be the first to confirm relativity (his measurements were not quite accurate enough to do this with the required 5 sigma accuracy). He was basically an Einstein bootlicker, since he realized the guy was great. He then browbeat Chandrashekhar into oblivion when Chandra presented collapse arguments, because Einstein disagreed with black holes. He continued with numerological nonsense making the fine structure constant 1/137. He was a self-promoter/politician, less of a scientist. $\endgroup$
    – Ron Maimon
    Commented Jun 24, 2012 at 5:31
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    $\begingroup$ THe second law contains the first, but not the third. The third is contained in the statement that momentum is a conserved quantity, so it isn't created or destroyed. Newton could have said "Hey, momentum is conserved!" But then he wouldn't have demolished Aristotle, because he wouldn't sound snooty enough for the philosophers. So he says "ey-hay, omentum-may is-ay onserved-kay", and suddenly he is the new king. This "laws of motion" stuff is just politics. What makes Newton truly great is the realization of the conservation laws, and the special problems. In this, he is a singular genius. $\endgroup$
    – Ron Maimon
    Commented Jun 24, 2012 at 5:34

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