**Nature and nature’s laws lay hid in night;**

**God said “Let Newton be” and all was light.**

Alexander Pope’s epitaph sets the capstone on the myth of Newton’s achievements that had been under construction since the publication of the *Principia* in 1687. Newton had single-handedly delivered up the core of modern science – mechanics, astronomy/cosmology, optics with a side order of mathematics – all packed up and ready to go, just pay at the cash desk on your way out. We, of course, know (you do know don’t you?) that Pope’s claim is more than somewhat hyperbolic and that Newton’s achievements have, over the centuries since his death, been greatly exaggerated. But what about the mechanics? Surely that is something that Newton delivered up as a finished package in the *Principia*? We all learnt Newtonian physics at school, didn’t we, and that – the three laws of motion, the definition of force and the rest – is all straight out of the *Principia*, isn’t it? Newtonian physics is Newton’s physics, isn’t it? There is a rule in journalism/blogging that if the title of an article/post is in the form of a question then the answer is no. So Newtonian physics is not Newton’s physics, or is it? The answer is actually a qualified yes, Newtonian physics is Newton’s physics, but it’s very qualified.

The differences begin with the mathematics and this is important, after all Newton’s masterwork is *The Mathematical Principles of Natural Philosophy* with the emphasis very much on the mathematical. Newton wanted to differentiate his work, which he considered to be rigorously mathematical, from other versions of natural philosophy, in particular that of Descartes, which he saw as more speculatively philosophical. In this sense the *Principia* is a real change from much that went before and was rejected by some of a more philosophical and literary bent for exactly that reason. However Newton’s mathematics would prove a problem for any modern student learning Newtonian mechanics.

Our student would use calculus in his study of the mechanics writing his work either in Leibniz’s dx/dy notation or the more modern F’(x) = f(x) notation of the French mathematician, Lagrange (1736–1813). He won’t be using the dot notation developed by Newton and against which Babbage, Peacock, Herschel and the Analytical Society campaigned so hard at the beginning of the nineteenth century. In fact if our student turns to the *Principia*, he won’t find Newton’s dot notation calculus there either, as I explained in an earlier post Newton didn’t use calculus when writing the *Principia*, but did all of his mathematics with Euclidian geometry. This makes the *Principia* difficult to read for the modern reader and at times impenetrable. It should also be noted that although both Leibniz and Newton, independently of each other, codified a system of calculus – they didn’t invent it – at the end of the seventeenth century, they didn’t produce a completed system. A lot of the calculus that our student will be using was developed in the eighteenth century by such mathematicians as Pierre Varignon (1654–1722) in France and various Bernoullis as well as Leonard Euler (17071783) in Switzerland. The concept of limits that are so important to our modern student’s calculus proofs was first introduced by Bernard Bolzano (1781–1848), Augustin-Louis Cauchy (1789–1857) and above all Karl Theodor Wilhelm Weierstrass (1815–1897) in the nineteenth century.

Turning from the mathematics to the physics itself, although the core of what we now know as Newtonian mechanics can be found in the *Principia*, what we actually use/ teach today is actually an eighteenth-century synthesis of Newton’s work with elements taken from the works of Descartes and Leibniz; something our Isaac would definitely not have been very happy about, as he nursed a strong aversion to both of them.

A notable example of this synthesis concerns the relationship between mass, velocity and energy and was brought about one of the very few women to be involved in these developments in the eighteenth century, Gabrielle-Émilie Le Tonnelier de Breteuil, Marquise du Châtelet, the French aristocrat, lover of Voltaire and translator of the first French edition of the *Principia*.

One should remember that mechanics doesn’t begin with Newton; Simon Stevin, Galileo Galilei, Giovanni Alfonso Borelli, René Descartes, Christiaan Huygens and others all produced works on mechanics before Newton and a lot of their work flowed into the *Principia*. One of the problems of mechanics discussed in the seventeenth century was the physics of elastic and inelastic collisions, sounds horribly technical but it’s the physics of billiard and snooker for example, which Descartes famously got wrong. Part of the problem is the value of the energy[1] imparted upon impact by an object of mass m travelling at a velocity v upon impact.

Newton believed that the solution was simply mass times velocity, mv and belief is the right term his explanation being surprisingly non-mathematical and rather religious. Leibniz, however, thought that the solution was mass times velocity squared, again with very little scientific justification. The support for the two theories was divided largely along nationalist line, the Germans siding with Leibniz and the British with Newton and it was the French Newtonian Émilie du Châtelet who settled the dispute in favour of Leibniz. Drawing on experimental results produced by the Dutch Newtonian, Willem Jacob ‘s Gravesande (1688–1742), she was able to demonstrate the impact energy is indeed mv^{2}.

The purpose of this brief excurse into eighteenth-century physics is intended to show that contrary to Pope’s epitaph not even the great Isaac Newton can illuminate a whole branch of science in one sweep. He added a strong beam of light to many beacons already ignited by others throughout the seventeenth century but even he left many corners in the shadows for other researchers to find and illuminate in their turn.

[1] The use of the term energy here is of course anachronistic

I’m a bit surprised it was so hard to find the form of “energy”, since the method of dimensions taught to teenagers nowadays shows easily that its mv^2 not mv. But you skate rather vaguely over the discussion.

Part of the problem is the value of the energy imparted upon impact by an object of mass m travelling at a velocity v upon impact. … Newton believed that the solution was simply mass times velocity, mv … Leibniz, however, thought that the solution was mass times velocity squared … Émilie du Châtelet … settled the dispute in favour of Leibniz. …‘Fraid not. As you note in your footnote, the term ‘energy’ is anachronistic, as it was not used by Newton — Newton said that mv was the

quantity of motion. It is abundantly clear from the Principia that he meant the same by this as what we nowadays callmomentum.Momentum enjoys just as prominent a place in modern treatments of Newtonian physics as kinetic energy, and Newton’s statements about it survive unscathed.

Émilie du Châtelet certainly played an important role in disentangling the two key concepts of energy and momentum from one another.

A few references, for those interested in delving into the controversy, which lasted well into the 18th century:

Thomas L. Hankins, “Eighteenth-Century Attempts to Resolve the Vis viva Controversy”,

Isis56:3 (1965) pp.281-297.Carolyn Iltis, “Leibniz and the Vis Viva Controversy”,

Isis62:1 (1971) pp.21-35.L. L. Laudan, “The Vis viva Controversy, a Post-Mortem”,

Isis59:2 (1968) pp.130-143.George E. Smith, “The vis viva dispute: A controversy at the dawn of dynamics”,

Physics TodayOct. 2006 pp.31-36.I should clarify what I object to in Thony’s description. He makes it sound as though Newton simply got it wrong and Leibniz got it right, with mv being a blunder and mv^2 the correct answer. This badly misrepresents the convoluted and fascinating history. Fully clarifying the notions of “force”, “mass”, “momentum”, and “energy” took several generations, and Thony’s larger point (that Newtonian mechanics did not spring full-grown from Newton’s brow) is of course true in spades. (Note that “vis viva” is Latin for “living force”.)

In partial answer to Connolley’s question, let me make a couple of “20-20 hindsight” remarks. Note that mv^2 is (apart from the factor of 1/2) the

kineticenergy, which isnotconserved ininelasticcollisions. However, the vector quantity mvis conserved in both elastic and inelastic collisions, a point that Newton made. The 19th century resolved this issue, thanks to the work of Joule, Thomson, and others, who came to recognize heat as another form of energy. Indeed, the term “energy” in its modern sense emerges from this work.When modern texts introduce kinetic energy, one of the first things they usually do is to show that the integral of

F.dx(i.e., the work done by a force on a point mass) equals the change in kinetic energy. From this seedling eventually grows the law of the conservation of energy. Newton does basically this computation in Prop. 40 of Book I.In this comment, I just want to embellish on a couple of Thony’s points.

Our student would use calculus in his study of the mechanics writing his work either in Leibniz’s dx/dy notation or the more modern F’(x) = f(x) notation of the French mathematician, Lagrange (1736–1813). He won’t be using the dot notation…Actually, Newton’s dots coexist quite comfortably with Leibniz’s dx’s and Lagrange’s f'(x)’s in modern textbooks. For example, both V.I.Arnold’s

Mathematical Methods of Classical Mechanicsand Goldstein’sClassical Mechanics(both grad-level texts) intermix them freely. The Euler-Lagrange equation is almost always written using both the Newtonian dot and the Leibnizian d/dt, all in the same equation. But the use of Newton’s dots is quite restricted.Newton didn’t use calculus when writing the Principia, but did all of his mathematics with Euclidian geometry. … The concept of limits that are so important to our modern student’s calculus proofs was first introduced by Bernard Bolzano (1781–1848), Augustin-Louis Cauchy (1789–1857) and above all Karl Theodor Wilhelm Weierstrass (1815–1897) in the nineteenth century.All correct, provided we interpret “the concept of limits” to mean “the modern, rigorous epsilon-delta concept of limit” (as I think is implied by Thony’s phrasing), and provided we cut Newton some slack vis-a-vis “Euclidean geometry”.

Archimedes would not have found the Principia up to snuff, rigor-wise. Bk I Section 1, titled “Of the method of first and last ratios of quantities”, is nothing less than an informal treatment of limits; in a scholium, he explicitly says he uses “nascent and evanescent” quantities to avoid the tedium of the ancient method of reductio ad absurdum. Infinitestimals, or limits, or whatever you want to call them, pervade the treatment; for an enlightening example, peruse his proof of the area theorem (Bk I Sec. 2 Theorem 1), where he says “now let the number of triangles be augmented, and their breadth diminished,

in infinitum…”. Same early form of integration that Kepler used.Newton’s dot notation for derivatives has indeed been used extensively by both physicists and engineers of the centuries but I’m am not aware of anybody teaching it in school or college level calculus courses. I would be happy if somebody would prove me wrong.

I was curious, and I happen to have a bunch of 1st year calculus texts on hand (review copies), awaiting disposal the next time I get serious about cleaning the basement. A quick perusal bears out your claim; most don’t mention the dot notation at all, or just once for completeness sake. The D_x notation however is mentioned frequently.

It may be a different story for undergrad physics. Here my convenience sample is smaller: 3 of the 4 calculus-based freshman physics books in my possession do use the dot notation on occasion (as I said, in restricted circumstances — r-dot, r-double-dot, v-dot, that sort of thing). However, a quick look failed to turn up

anyoccurrences of Newton’s dots in perhaps the most celebrated physics textbook of the twentieth century, the Feynman Lectures.“Newton wanted to differentiate his work”Pun intentional? In any case, mathematics was an integral part of his work. 🙂

I would pay good money to read a paper exploring the relationship between ephtaph, obituary and the early development of H.O.S.

I suspect it is integral to the development of a distinctive heroic style.

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