A nice overview of the Principia can be found here, which partially addresses your question: http://plato.stanford.edu/entries/newton-principia/#NewLawMot

Newton expressed the law in words and geometric demonstration, as was the custom at the time, rather than as a mathematical formula of second derivatives, and therefore has priority on the idea. Acceleration, as the second derivative of position with respect to time, is a concept expressed using calculus, which was invented around the time Newton was writing the Principia (and Newton is credited with some of the development of calculus). Hermann's later work made greater use of calculus and proved the law using derivatives, more in keeping with our modern conception, but he only formalized Newton's work and, as far as I can see, gave full credit to Newton for the law.

A nice overview of the Principia can be found here, which partially addresses your question: http://plato.stanford.edu/entries/newton-principia/#NewLawMot

Newton expressed the law in words and geometric demonstration, as was the custom at the time, rather than as a mathematical formula of second derivatives, and therefore has priority on the idea. Acceleration, as the second derivative of position with respect to time, is a concept expressed using calculus, which was invented around the time Newton was writing the Principia (and Newton is credited with some of the development of calculus). Hermann's later work made greater use of calculus and proved the law using derivatives, more in keeping with our modern conception, but he only formalized Newton's work and, as far as I can see, gave full credit to Newton for the law.

The mathematicians are rewriting history to play the central role, but that's only because they don't understand what is an experimental science, and because they form a powerful lobby. They'll call Newton's laws "a vague metaphore," which then can't be credited.

The whole work of Newton have been credited to a mathematician, same thing happend with relativity, and there are other examples. There is a powerful lobby of mathematicians rewriting history.

Newton didn't write the mathematical expression of the law, since the multiplication is already know from the Babylonians. What did Newton is to define mass, and more importantly, force. He observed that the law is satisfied if the gravitational mass is taken, then the force can be deduced if an absolute space is postulated. Then from assuming that gravitation is universal, and so is valid for celestial bodies, and giving it a particular expression, he could find the Kepler laws again, for the exact law of motion of the planet around the sun were already known, but Kepler didn't develop mechanics, it was only empiric laws (just like the Lorentz tranform before relativity and even after Poincaré.)

There is therefore no mathematics in the physical work of Newton, just physics, and no wonder.

Math without experiment and without interpretation, that is defining what we are speaking about, is not physics. Newton didn't make mathematics, save of course when he was developing mathematical techniques, in which case he didn't make physics. The deduction of the Kepler's law is only the verification of the hypothesis about gravitation, which is physics and not mathematics, no more than physics is information technology when QCD is verified by a numerical calculation on a network.

Computer science is able to tell whether the result is reliable or not. there is no and can't be a one to one correspondance between mathematics and physics, since physics is reality and never change, while theories and methods change. Newton's method doesn't work for the orbit of mercury.

It is the math of computing, not the math of physics. If the assumptions are wrong, even if the math is correct, the result is wrong, and more certainly with better math. Math can only falsify a theory by making predictions, not build one. Physics even with math may be nothing better than the science of story books. Poincaré had the good formula, the right math, but no assumption at all, he failed. No mathematician ever did advances in physics, and now they try and steal the work of others.

Dear Claude Masse, I will be very thankful, if you elaborate or give reference of " Poincaré had the good formula, the right math, but no assumption at all",

Like mathematics, a computer code can be checked by someone else, there is no fundamental difference, save that it is numeric and not symbolic, and in more and more domains it becomes essential.

For about only thirty years, important discoveries that was credited to physicists, sometimes for centuries, has been attributed to mathematicians who never even claimed the paternity. At the beginning, relativity was even credited to Einstein's wife, misknowing other major contributors. That is enough to show that there is a will to steal the work of the physicists. In the same time, the purly mathematical approach to physics, embodied in superstring theory, has been a flop. There is not even testable predictions, a blatant demonstration that they don't know what they are talking about. The discovery of the positron didn't happen at all like it is told, it was much more complicated, just look at the relevant literature. Relativity isn't the Lorentz transformation, it is a minimal set of physical assumptions that has it as a consequence, and that is applicable to all of phyics and not only electromagnetism.

Yet, il the same span of time, there have been considerable advance in mathematics, in particular in non linearity, which has been in part inspired by physical topics. The today's system, where scientists are consistently evaluated by their peers with the attached threat on the reputation, have made a powerful lobby out of the mathematical community, that speaks with a single voice. The perversion of the system favors rigor and checkability against creativity and boldness. It it a conservative one that stifle any progress. That becomes besides more and more a concern before the fifty-yearly vacuum of advances in physics.

Back in time, scientific research was not an industry and there wasn't so much useless junk about nearly everything. It was possible for an individual to know virtually all, including both in physics and in mathematics. There is no point of comparision with now.

If Newton lived today, he would have lost any reputation because of his musing about alchemistry, relationship between the colors of the rainbow, the alexandrine, the planets and the musical scale, and so on. He would have no position in an academic institution. Yes, he was not only a mathematician and a physicist, but this story have been repressed or scorned.

I think you''ve posed the question wrong. "m" represents mass. "a" represents acceleration. So what you are REALLY asking is, "Who DEFINED F as m times a?" The answer might be: the same person who defined "m" and "a".

"Not Even Wrong" is another bashing of the work of the physicists from a mathematician. But he missed the point, if there is some trouble in string theory, it is precisely that it rests on "reine Vernunft" (pure reason,) advocated by the mathematicians. Lack of rigor is not the major issue, it has always be so in the whole history. Most of the time, physics inpires mathematics that in turn provides a more rigorous and general framework. This example only shows that his voice is more heard than the one of the physicists.

"mathematics vs. physics vs. reason" is an interesting topic. A classic example I like to raise is the following: What is the sum of all positive natural numbers up to infinity? To a rational person, the answer is either infinity or something similar, but to a mathematician or to a physicist, the answer defies logic! (By the way, the answer is -1/12. If you want to see the proof, Google it.)

How is it not a proof? If we are referring to the same thing, what I've seen (and followed) began by asking What is the sum of all natural numbers? Then, after a number of steps, the answer: -1/12 was revealed, which, by the way, corresponds to a value used by string theorists every day. As they say... If it looks like a proof, and barks like a proof, it's a proof. It's NOT a definition, from what I know about "definitions."

I didn't think my response would be controversial, so I've decided to post the proof in video form.

http://www.youtube.com/watch?v=w-I6XTVZXww

Physics is not mathematics, there is no problem. But that a mathematician complains that not mathematics is not mathematics IS the problem. That it is heard on mathematical grounds is still a bigger problem. The philosophical and physical critisism, we don't need his, many physicists, including me, have already done it a long time ago.

There is no mathematical validity already in standard physics. The Dirac equation isn't rigorously workable to get physical predictions, it must describe several particles, and lead to paradoxes, where negative energy solutions must be included, coming from nowhere. The Dirac equation can't even predict the positron, since before that, it has to address the negative energy sea, which since being described by the equation, should have an infinite electrical charge. Feynman's interpretation is not much better. The positron was just observed, then heuristically classified with the existing equation, period. Orthodox quantum mechanics has no logically consistent interpretation, and even mathematically doesn't have a consistent set of axioms. In spite of all that, there are procedures that allow to describe and predict a lot of physical behavior with an incredible precision.

If one thinks it is because they are only tentative theories, and soon we'll have a theory of everything perfectly mathematically valid, it is an illusion. Most probably some issues will be solved, but giving birth to still more difficulties and questions. So is physics, from its inception to our last days. Evaluating it from a mathematical perspective is as ludicrous as evaluating mathematics from a butcher's perspective.

Charles, since you are from Jesus College, I have the suspicion that your negative feelings toward string theory are not so much its apparent conflict with traditional mathematics, but rather your dislike of those aspects of string theory (such as the possibility that matter can come out of nothing) which run contrary to your personal religious beliefs.

Regarding string theory being "garbage" because it runs contrary to logic, I suspect that some held that same viewpoint when calculus was invented, and astrophysics for that matter. Nine plants and an oblate spheroid earth must have blown their minds since they, too, ran contrary to currently-held religious beliefs. Let's not even talk about evolution.

There is not a consistent and/or complete set of axioms of quantum mechanics. There is the axiom of unitary evolution, and the axiom of projection that are logically incompatible and/or there lacks an axiom defining what is a measurement and what is not. No mathematician on earth can do anything to that.

They can't do for another reason. Mathematics is on its own, while physics is constraint by experiment. Until we have a complete description of reality, there can't be a mathematical sound theory. And to have a complete description, we must have a infinite set of data, which is impossible. In mathematics there are infinities, which is sort of a logical closure, in physics there can't be.

As an example, in mathematics there are cases where derivatives can't be swapped. Those cases never occur in physics, so it isn't sloppy to swapped them without having checked the conditions. That's because a derivative involves an infinity. But that also means that a derivative is meaningless in physics, from the point of view of mathematics. Indeed, at the Planck scale, as all our theories use derivatives, they all are logically inconsistent. Even the notion of particle we use everyday is nonsensical.

String theory doesn't attemp to replace quantum field theory, it builds upon it and aims to have it as a consequence. It is then impossible that it be mathematically valid, since QFT, like every physical theory, is not. It is almost impossible to do any mathematics without infinities, apart for some trivial cases. Physics must manage it without any, and do it rather well. The physicists don't need the approval of the mathematicians, they know their job, and are capable of logical thinking as well.

Dear Claude Masse, I will be very thankful, if you elaborate or give reference of " Poincaré had the good formula, the right math, but no assumption at all",

The references are the original papers:

H.Poincaré "Sur la Dynamique de l'Electron" Comptes Rendus de l'Académie des Sciences 140(1905)1504 (in French)

A.Einstein "Zur Elektrodynamik Bewegter Körper" Ann. Phys. (Leipzig) 17(1905)891

http://gallica.bnf.fr/ark:/12148/bpt6k2094597/f896.image.r=annalen%20der%20physik.langFR (In German, I can read both.)

Poincaré showed that the Lorentz transform is a symmetry of the Maxwell equations, and forms a group with the rotations of space, but the bodies are contracted in space which remain conventional.

Einstein gave an operational definition of distance and time through comparision with rods and synchonisation of remote clocks with light rays, and by postulating that the speed of light is independant of the motion of the source and the observer, he derived the Lorentz transform. The framework was now applicable to every physical process, which allow to show for example that energy and momentum transform like time and space coordinates.

Poincaré alludes to Galilean relativity, he speaks about space and not space-time. Physical ideas often have a very ancient origin, but it is not straightforward to apply them to new facts in a complete and consistent system, that is, in a theory.

Dear Dr. Claude Massé and Dr. Charles Francis, please accept my sincere thanks for enlightening.

Certainly Newton. However, according to history, arabian scientists in 600 AD and greek philosophers 500 BD and other civilizations i.e. chinese, egyptian and Iranians in ancient era investigated kinematics and invented static and dynamic instruments based on first and second laws.

On the other hand, the calculus and differntiation concepts were invented by liepnitz,newton and others. Ancient scientists focused on geometry and trigonometry that were essential for architecture and astronomy.

Just a word of caution, don't expect neat, precise equations in ancient books, but endless long winding explanations from which you may derive such an equation. By the way, these laws are NOT invented, rather they are observed and developed over some time by several researchers, ending up with name of one of them.

Dear Ramanand Jha

Your question has already been answered, but Feynman gave a series of (taped) lectures at Cornell, and in the first he gave a good explanation of how Newton must have thought: in these days people thought that the moon travels around the earth because an angel pulls it, in a tangential way, but using his intuition of fluxions (which were not yet expressed as derivatives) Newton understood that the force is towards the earth (I suppose that he used the wrong assumption that the trajectory is a circle, because knowing that it is an ellipse and the velocity changes according to one of Kepler's law, it is not so easy to "deduce" that). Then, seeing an apple fall, he would have deduced that an angel pulls the apple toward the earth, hence the "angel" pulling the moon is called gravitation.

Newton could have deduced that the force is in the inverse of the square of the distance as a consequence of one of Kepler's law, which followed from the precise observations of Brahe, and of guesses, since the supposed distances in the solar system at their time were completely wrong. A good value for the size of the earth orbit "around the sun" (although it is approximately around the center of gravity of the solar system) was only obtained in the 1670s by Cassini (in Paris, with his assistant in Cayenne), or by Flamsteed (alone in Greenwich). Instead, Newton got the correct force by an hand-waving argument, that the strength at distance r is equally distributed along the surface of the sphere of radius r. What is this strength that he was thinking about? Nobody knows, because no one yet understands what gravitation is, or just what mass is!

Of course, mass is a form of energy, and Poincaré had obtained the formula e = m c^{2} in 1900, by an argument concerning the "Lorentz force" (which appeared 30 years before Lorentz, in an article by Maxwell): since there is an action, there must be a reaction. In other words, using an analogy with sailing, when a boat sails upwind, it must make waves in the sea and in the air, since there are physical conservation laws to obey, although one tends to keep an eye on the boat. In the same way, experimental physicists follow the trace of a charged "particle" turning in a magnetic field, without taking into account all the waves which are created.

Newton seems to have used an idea of Barrow. On another occasion, he used an idea of Hooke, acknowledged in a letter to him, where Newton said that if he had seen further, it was because he was on the shoulders of giants, probably a way to make fun of Hooke (who was not tall), but the saying goes back to John of Salisbury, who attributed it to Bernard de Chartres.

The argument of Feynman suggests that Newton had a quite good intuition of the law f = m a, even though he did not write it this way!

Luc TARTAR, mathematician

Dear Bill Streifer

I followed the link you gave for seeing the "proof" that the sum of integers is -1/12, and the first one was very silly, but I found a second which was less silly.

No mathematician should write a formula like the one you use, either with an arrow sign or with an equal sign, without telling in what topology the series is considered.

Euler wrote formally that 1 + 2 + 4 + 8 + ... is -1, and it was not a mathematical statement. Said otherwise, he was writing that 1+x+x^{2}+... is 1/(1-x), and there are (at least) two ways to give a mathematical meaning to the formula, but in both ways replacing x by -1 is forbidden!

A first one is analysis, saying that the sequence S_{n} = 1+x+...+x^{n} converges (as n tends to infinity) to 1/(1-x) (in the usual convergence for real numbers), and it is valid if -1 < x < +1 (or for complex numbers if |x| 1, zeta(s) is the sum of n^{-s} from n = 1 to infinity. Again, Euler wrote formally that zeta(-1) = -1/12, but he had no idea what his formal computation could mean.

Then came Cauchy, who did the theory of power series in the complex plane, defining a radius of convergence, and 1+x+x^{2}+.. converges for x complex of modulus < 1, and the sum is 1/(1-x) in the unit complex disc.

The zeta function is not a power series but a Dirichlet series and for s complex it has a meaning for s of real part >1.

Then Cauchy explained what analytic continuation is: starting at a point x_{0} one may consider the Taylor series at x_{0}, which converges in some disc, and one may repeat the operation and find the domain of analyticity (but there are difficulties in some cases).

For example, the analytic extension of 1+x+x^{2}+.. in the unit disc is the function 1/(1-x) defined in the whole complex plane except at x = 1, but it is not given by the power series!

Riemann showed that the zeta function has an analytic extension in the whole complex plane except at x = 1, but it is not given by the (Dirichlet) series: this extension indeed has zeta(-1) = -1/12.

Luc TARTAR, mathematician

PS: For each prime p, there is an interesting metric (for number theorists) on the field of rational number, introduced by Hensel at the end of the 19th century, and for p = 2, the sequence S_{n} = 1+2+...+2^{n} converges to -1.

Dear Luc Tartar, thanks for your interesting enlightenment.

Others have already given useful answers, but one thing I should like to add is that Newton's work is concerned with inertia, p, and how it changes over time; as such it does not lead to F=ma, but rather the more general F = dp/dt (the difference being important when the mass of the object changes over time).

Surely that only applies to the rest mass m0? It's been a while since I worried about such things. I was actually thinking primarily of the quantum mechanical regime, where "p" is a rather better defined concept than "a".

Philip, the general equation F = dp/dt does lead to F = ma in Newtonian Mechanics where mass is virtually constant. Consider the following:

F = dp/dt = d(mv)/dt = mdv/dt = ma.

We experience mass variation in Relativistic-Quantum Mechanics.

As I previously posted here, Newton has predicted the solar system motions

using F= ma in polar coordinates using calculus and differential equations which he invented. I believe that each scientific achievement comes from the collective attempts of many researchers. Hence, Newton is famous for those mechanics laws and Maxwell is known foe electromagnetic laws, even though Gauss , faraday and ampere have greatly contributed to develop those laws. Similarly, F=ma is attributed to Newton.

Newton is the greatest scientist of his time and even today.

Dear Parviz Parvin

You wrote that Newton predicted the solar system motions using F= ma and differential equations which he invented, but I was told that Newton only pointed out to the importance of studying differential equations, and others did that, maybe by following the more efficient approach of Leibniz.

You write that Maxwell is known for electromagnetic laws, although Gauss, Faraday and Ampere contributed, but what Maxwell added was to unify electricity and magnetism, which were considered different questions before. However, he developed a mechanistic approach, because of beliefs of physicists concerning "aether", and it was Heaviside who simplified his work and wrote the system which one uses now, which I call the Maxwell-Heaviside equation for this reason.

You are right that each scientific achievement comes from the collective attempts of many researchers, but you are wrong in writing that Newton is the greatest scientist of his time and even today. Maybe you write this because others consider Einstein a genius, but Einstein made some errors from a physical point of view, like calling "Brownian motion" something resembling to the work of Bachelier, i.e. confusing jumps in position and jumps in velocity (observed under his microscope by R. Brown). Actually, jumps in velocity contradict the law of conservation of linear momentum, so that one prefers to say that they result from collision with very small particles, forgetting that it could just be particles interacting with waves!

Luc TARTAR, mathematician

Dear Luc

I am not an expert in science history, however teaching general Physics we can feel Newton and Maxwell sovereign in classical Physics. Before Newton the most important law in Physics was Hook's law! Most of us still think Newtonian or compare modern physics with Newtonian concepts. I attest The influence of Newton in calculus, differentiation and integral as an inevitable basic tool in modern science. However, the most Newton's proceeding is the scientific investigation based on experiments. It is worth to read Rutherford writing about Newton to get acquainted further with Newton.

Finally! Newton was living 3 centuries ago and he was really 300 years ahead of his time: Hubble space telescope was simply designed based on the concept of Newton telescope!

In addition, Leonhard Euler propose F=ma as a generalization of the Newton's Second Law, in 1750... The post Newtonian Physicist are very important in the modern interpretation of Newton's Mechanics, and most of our interpretation about Newton's work began with that scientists, like: Euler, Mach, etc! For example, Euler proposes the use of Vectors as forces, velocities or acelerations entities; the notion of Aceleration as a kinematics magnitude defined in every kind of curves; and the concept of Pontual Mass!!

Number of French scientists - Jean Buridan (1300 – 1358), Albert of Saxony (1316-1390) etc (see http://en.wikipedia.org/wiki/Inertia), have introduced concepts of inertia and mass at XIV century.

Newton's second Axiom of motion states that

Change of motion is proportional to the motive force impressed and takes place in the direction of the right line in which it was impressed.

Let F be the measure of the motive force.

From Descartes onwards, the quantity of motion that a body has was considered to be the product of its speed and quantity m of matter (nowadays 'mass'). Huygens refined the definition to mV, where V is the speed of motion in a given direction.

Let (mV) be the quantity of motion of a body in the direction of its own motion, and delta mV the change of motion that a body undergoes as a result of an impressed force F, which is not necessarily in the direction of V.

Then, by axiom 2, delta (mV) = F.

The quantity of matter of a body was universally considered to be fixed for a given body. Therefore, we have

F = m(delta V),

which, of course, is the time integral form of f = ma,

where f = dF/dt

and a = dV/dt.

Newton knew this, of course, but the calculus was still in its infancy, and Newton had only a crude form of it (his theory of fluxions).

I puzzled about this too. The formula f = ma is to be found in a book by the English mathematician McLaurin in the early eighteenth century. But I do not know whether he was the first to use it. Maybe Jacob Hermann was.

Brian Ellis

Laws of nature are discovered, not invented. Issac Newton is usually given credit for the equation F = mass x acceleration. Since Einstein, the concept of force has been replaced by the concept of energy, which is interchangable with mass (rest-energy).

Precisely, David. That was my original point. Newton didn't "invent" F = m*a any more than Einstein "invented" E=mc^2. If Einstein had died at childbirth, some other brilliant physicist would have eventually discovered the relationship between mass and energy. But it might have happened too late for its practical application during WWII.

Its definitely the Euler who first wrote f=ma in order to explain the rotational phenomenon.