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Education Math

350-Year-Old Newton's Puzzle Solved By 16-Year-Old 414

Posted by samzenpus
from the top-of-the-class dept.
First time accepted submitter johnsnails writes "A German 16-year-old, Shouryya Ray, solved two fundamental particle dynamic theories posed by Sir Isaac Newton, which until recently required the use of powerful computers. He worked out how to calculate exactly the path of a projectile under gravity and subject to air resistance. Shouryya solved the problem while working on a school project. From the article: 'Mr Ray won a research award for his efforts and has been labeled a genius by the German media, but he put it down to "curiosity and schoolboy naivety." "When it was explained to us that the problems had no solutions, I thought to myself, 'well, there's no harm in trying,'" he said.'"
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350-Year-Old Newton's Puzzle Solved By 16-Year-Old

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  • That Moment (Score:5, Interesting)

    by Rie Beam (632299) on Sunday May 27, 2012 @09:38AM (#40127897) Journal

    We all had that moment in school when a teacher would pose an "impossible" problem, thought to ourselves "Well, they've never faced ME before!", spent a few minutes toying with it and finally giving up. This kid...did not.

    Kudos all around! The rest of your life will, unfortunately, now no longer live up to something you accomplished when you were 16.

    • Re:That Moment (Score:5, Interesting)

      by Shavano (2541114) on Sunday May 27, 2012 @09:48AM (#40127957)

      There are two things impressive about this. One is the fact that you mention, that the kid did not give up until he had the solution and was smart enough to solve a problem that stumped every mathemetician for 350 years. The second is that people still try to solve difficult analytic problems at all instead of just turning it into a computing problem.

      I don't know which surprises me more.

      • by Bananatree3 (872975) on Sunday May 27, 2012 @10:27AM (#40128165)
        Andrew Wiles solved Fermat's Last Theorm with paper only, as he despised the use of computers in writing mathematical Proofs. Another famous example is Grigori Perelman who solved the Poincaré Conjecture - with hundreds and hundreds of pages of mind-numbingly dense mathematics vs computer search.
        • by Machtyn (759119)
          It does seem pointless to me to use a computer to create a proof, except when using it to quickly calculate the known and already proven equations.

          Of course, that's coming from a guy who continually messes up a number or sequence here or there.
          • Re: (Score:3, Insightful)

            by Anonymous Coward
            It seems pointless to you because you are totally ignorant of math. A lot of these "hundreds of pages of mind-numbingly dense mathematics" proofs are long but tedious derivations which a computer can grind through in seconds.

            If you're doing a half page proof that square root of 2 is irrational, then a computer would be pointless, but clearly you don't know that math is more complicated than that.

            And to head off potential flames, I completely respect people who want to and are able to work through tho
            • by ais523 (1172701) <ais523(524\)(525)x)@bham.ac.uk> on Sunday May 27, 2012 @07:23PM (#40131035)

              Half a page? If (x/y)^2 = 2, then x^2 = 2y^2, so x is even. Let z = x/2, now we have 2z^2 = y^2, so y is also even. Thus, any fraction that's equal to the square root of 2 cannot be expressed in lowest terms, so cannot exist. That's, what, three lines at most?

              I agree with the main point, though; quite a few of the proofs I do are just boring churning through tens of possible cases. Up to 100 or so it's plausible to do it by hand, although tedious and it's easy to make mistakes; significantly beyond that, though, you're going to want to automate it.

            • by sco08y (615665)

              It seems pointless to you because you are totally ignorant of math. A lot of these "hundreds of pages of mind-numbingly dense mathematics" proofs are long but tedious derivations which a computer can grind through in seconds.

              If you're doing a half page proof that square root of 2 is irrational, then a computer would be pointless, but clearly you don't know that math is more complicated than that.

              And to head off potential flames, I completely respect people who want to and are able to work through those derivations by hand, but to think doing it with a computer is pointless just shows your ignorance.

              Most importantly, if there are hundreds of pages of dense computation to prove X, if I'm writing a function and I have some invariant, I can just write a comment,

              "And invariant Y remains satisfied because of X, see the fun proof at..."

              I don't really give a damn about the details, It Just Works.

        • by Chase Husky (1131573) on Sunday May 27, 2012 @10:54AM (#40128309) Homepage

          Another famous example is Grigori Perelman who solved the Poincaré conjecture - with hundreds and hundreds of pages of mind-numbingly dense mathematics vs computer search.

          Perelman's three primary papers ("The entropy formula for the Ricci flow and its geometric applications" http://arxiv.org/abs/math.DG/0211159 [arxiv.org], "Ricci flow with surgery on three-manifolds" http://arxiv.org/abs/math.DG/0303109 [arxiv.org], and "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds" http://arxiv.org/abs/math.DG/0307245 [arxiv.org]) on modifying Hamilton's Ricci flow program to deal with singularities and proving Thurston's geometrization conjecture only span 68 pages, with the actual proofs/meaningful remarks comprising about 45 pages of that.

      • Re: (Score:3, Insightful)

        by Anonymous Coward

        Computing tends to be a brute force analysis of all the possible inputs. That doesn't work well for NP hard problems and is often impossible with problems dealing with infinity... Not all problems are solvable by computers yet and instead need the analytical approach. Also computers may not find the most elegant solutions, for example there are problems which have been solved but required the invention of a new type of math to do so.

        • Re:That Moment (Score:5, Insightful)

          by K. S. Kyosuke (729550) on Sunday May 27, 2012 @11:53AM (#40128659)

          Computing tends to be a brute force analysis of all the possible inputs.

          Hello? We've had symbolic computing ever since 1960's. There are many software tools today to assist mathematicians with creating and verifying proofs (e.g, Coq is probably the best known one). What's wrong with using them? Not to do that would be like using a pencil and paper instead of typing when you're preparing a publication – I'd think that brain power and time should be used constructively.

          • Business wise I'd type, fiction wise I prefer to use pen and paper.

          • by Anonymous Coward on Sunday May 27, 2012 @01:12PM (#40129095)

            I just wanted to say that I LOVE Coq.

          • Re:That Moment (Score:4, Insightful)

            by ArundelCastle (1581543) on Sunday May 27, 2012 @01:37PM (#40129217)

            What's wrong with using them? Not to do that would be like using a pencil and paper instead of typing when you're preparing a publication – I'd think that brain power and time should be used constructively.

            It's not a matter of the tool is wrong. It's a matter that assuming one tool is always best is wrong.
            Your premise is based on: using a computer is easier and better for 100% of humans. That's not true. Allow me to introduce you to my parents. Allow me to introduce you to senior engineers who can craft new formulas on a whiteboard faster than juniors can wake their laptops.

            Different areas of the brain are involved with the act of handwriting than with touch typing or pecking. Make LCARS speech recognition a reality and we have a winner. Solving problems that stump otherwise intelligent humans for *hundreds* of years, *clearly* requires some creatively alternate use of the brain, and not Microsoft Clippy. ("I see you're trying to solve an unprovable theorem, would you like to Quit without Saving?") I don't even need to cite sources that say poor UIs slow people down. That's how it is. Computers add cruft, otherwise there wouldn't be a market for applications that remove distractions when writing.

            ...like using a pencil and paper instead of typing when you're preparing a publication...

            Poor analogy. Publication implies mass reproduction and distribution. An *author* can write however they want to form their ideas, the result is the same. How the idea gets distributed is irrelevant to the core point. (Also there are such things as shorthand.)

          • by Carewolf (581105)

            What's wrong with using them?

            They are not helpful. Automatic proof, or automatic proof-verification is a research field, and has been so for decades, and has still YET to come up with something helpful to anyone doing real mathematical proofs. They have only barely reached the ability to help with play-thing problems handed to high school students, and even them the computer generated result (or input), is obtuse and stupid - not helpful in any way.

      • Re:That Moment (Score:5, Insightful)

        by Chris Mattern (191822) on Sunday May 27, 2012 @11:11AM (#40128419)

        Analytic solutions are far superior to computed approximations. They are far easier to calculate--computers have made computed approximations far easier, but most of the time that doesn't mean that they're *easy*--only that they're now possible. Being able to obtain the answer in a small fraction of the time is still a big advantage. They are more precise and do not require initial parameters. And they provide much greater understanding and insight into the underlying phenomenon. There is no surprise at all that people are still looking for analytic solutions.

        • by trout007 (975317)

          It depends on what you are trying to do. I'm a mechanical engineer and engineering is all about good enough. You have to economize resources to get a job done. While solving a dynamics problem analytically may give you more understanding into the solution it does take take to work out real world multidimensional problems. Numerical solutions to differential equations are very useful. I would have preferred to spend more time in Diff Eq setting up problems than solving problems analytically.

        • by am 2k (217885)

          Analytic solutions are far superior to computed approximations. They are far easier to calculate--computers have made computed approximations far easier, but most of the time that doesn't mean that they're *easy*--only that they're now possible. Being able to obtain the answer in a small fraction of the time is still a big advantage.

          Have fun with solving the Navier-Stokes equations then ;)

        • Re:That Moment (Score:5, Insightful)

          by Pseudonym (62607) on Sunday May 27, 2012 @07:59PM (#40131195)

          Analytic solutions are not necessarily easier to calculate.

          Analytic solutions tend to involve special functions for which the computer can only compute an approximation anyway. Have you ever tried to write code to evaluate the error function over the entire domain of floating point numbers? (Yes, I know, it's now in the standard library; ten years ago, it wasn't.) That's one of the easier ones.

          Even if there are no special functions, analytic solutions are still often harder to calculate if the problem is big enough. Think of solving systems of linear equations, one of the standard workhorses of numeric programming. We're talking really big ones; hundreds of thousands of equations in hundreds of thousands of unknowns or bigger. In the real world, this problem would almost certainly be solved using successive approximations, even though high school students know how to solve them analytically.

          Finally, and most importantly, the problem statement is usually an approximation. Take the OP as an example. What this kid almost certainly solved was an analytic solution to the problem of a particle in a gravitational field with linear air resistance. Well, air resistance is not linear. At low velocities, and for projectiles with a sufficiently small cross-section, it's close enough. But it's still an approximation.

          The advantages of analytic solutions are almost always not computational. What they buy you is understanding. The methods of obtaining the solution, and the form of the final equations, often reveal some deep insights about the problem. For many situations, that's far more valuable. And it's certainly something that no computer can give you.

      • by iamhassi (659463)

        There are two things impressive about this. One is the fact that you mention, that the kid did not give up until he had the solution and was smart enough to solve a problem that stumped every mathemetician for 350 years. The second is that people still try to solve difficult analytic problems at all instead of just turning it into a computing problem.

        I don't know which surprises me more.

        ^^^^ This.

        I think the most impressive part is that even though we hear all the time about "X-teen year old invents BLAH" we're like "Great!" but secretly think "BIG Deal! Who can't invent something? How is that challenging, really? Oh look their dad's an electrical engineer that works at XYZ... hmmmm....." but this 16-year-old actually solved something that the best mathematicians on Earth haven't been able to solve for 350 years.

        Major kudos kid! Only way that can be topped is if a teen cures canc

      • It's people like this kid who pop-up very rarely in the world that will eventually improve our mathematical understanding and all the technology based on advanced mathematics. That's a positive thing.

    • by mwvdlee (775178) on Sunday May 27, 2012 @09:54AM (#40127983) Homepage

      The rest of your life will, unfortunately, now no longer live up to something you accomplished when you were 16.

      Imagine the freedom of no longer having to live up to anybody's expectations. ;)

    • Re: (Score:3, Interesting)

      by Anonymous Coward
      If he solved it, then WHAT IS THE SOLUTION?! There is no link, no nothing, and we are apparently to trust this lame emotional article with no factual content. I'm surprised nobody else raised this point.
    • And the submitter gave up right while copying the name of the kid from the article to slashdot.
      "Shouryya Ray" became "Shouryya Ra" and samzenpus also let it through without any corrections.

  • terrible article (Score:5, Insightful)

    by Anonymous Coward on Sunday May 27, 2012 @09:39AM (#40127907)

    The article itself is mathless. It doesn't tell you what the solution was, or even present the exact problem that was solved.

    • Re:terrible article (Score:5, Interesting)

      by sco08y (615665) on Sunday May 27, 2012 @09:52AM (#40127977)

      The article itself is mathless. It doesn't tell you what the solution was, or even present the exact problem that was solved.

      And running a search for the kid's name turns up the same article fifty fucking times over. Google did some work on link farms... they need to do some work deduping / despamming press releases.

    • Re: (Score:2, Insightful)

      by Anonymous Coward

      With all due respect to this brilliant student, I wouldn't worry too much about that - the problem isn't actually solved until its been peer- reviewed and thd other mathematicians agree that his approach is correct.

    • The answer was 42 . . . now what was the question?

    • You are right. This article is awful, conveying no sense of the nature of the problem or its complexity, and giving no idea of the solution at all.

      The only equations I'm aware of for a falling particle subject to air resistance take the form

      m v' = -mg -a*v-b*v^2

      which is a constant coefficient Riccati differential equation for the velocity v. I'm reasonably sure this would have an analytic solution.

      Maybe complications arise in the 2D motion case, or perhaps the problem includes a particle which is also spinning. Maybe the drag terms take more complicated forms. I don't know. The article is pretty dreadful to be honest.

  • I did not know that the two problems described were unsolved. I thought that "how to calculate exactly the path of a projectile under gravity and subject to air resistance" was already figured out. I guess "exact path" is the trick here. An the other about an "object striking a wall"...

    Should make for even better gaming physics...
    • Re: (Score:2, Informative)

      by Anonymous Coward

      There is no problem solving the equations numerically. This kid found analytical solutions to the equation of motion (or at least, that's how I read TFA). Punching in the exact solution is faster and more accurate than taking a zillion small but discrete steps, which is what you're stuck doing right now. Well, that depends on the complexity of the solution, but as a general rule...

    • I'm slightly confused as well. In my high school AP calculus-based physics class we did projectile motion with air resistance and gravity at the beginning of the year. In fact, my teacher used that particular topic to "weed out" the students that probably wouldn't be able to handle the remainder of the course. He taught the material way above the actual AP requirement and make the topic exam so hard that a few kids switched into the lower-level physics course afterward.
  • Specifics? (Score:5, Insightful)

    by Rie Beam (632299) on Sunday May 27, 2012 @09:45AM (#40127937) Journal

    Can anyone actually find the problems in question somewhere? I've been scouring Google and the whole thing is very vague -- no story really goes into depth about the actual problem he solved and how.

  • by Anonymous Coward on Sunday May 27, 2012 @09:54AM (#40127993)

    German media praise math geniuses, while american media praise hollywood actors/actresses (read: human rubbish) and reality show weirdos. In the US a "genius" is someone who makes millions, especially with lower education and without being able to do anything. That's "free market economy", and "supply and demand", right?

    "The land of the free and of the brave" (with some fat on the belly).

  • When in Doubt... (Score:5, Informative)

    by Rie Beam (632299) on Sunday May 27, 2012 @10:03AM (#40128041) Journal

    ...go to the source! The German articles I've scoured seem to have a little more information about the problem itself and what he actually accomplished. The oldest one only records that he "claims" to have solved them (earlier this month), but so far no actual data. Close.

    http://www.enso-blog.de/jugend-forscht-drei-arbeiten-aus-ostsachsen-beim-bundeswettbewerb [enso-blog.de]
    http://www.morgenpost.de/vermischtes/article106358144/16-jaehriger-Schueler-loest-uraltes-Mathe-Problem.html [morgenpost.de]

  • Gotcha! (Score:5, Informative)

    by Rie Beam (632299) on Sunday May 27, 2012 @10:08AM (#40128071) Journal

    http://jugend-forscht-sachsen.de/2012/teilnehmer/fachgebiet/id/5 [jugend-for...sachsen.de]

    Text is in German. It all stems from a Youth Research competition he entered into back in March of this year. This is, so far, the best summary I've found -- there is a paper, apparently, but no link just yet.

    'Two problems in classical mechanics have withstood several centuries of mathematical endeavor. The first problem is therefore to calculate the trajectory of a body thrown at an angle in the Earth's gravitational field and Newtonian flow resistance. The underlying power law was discovered by Newton (17th century). The second problem is the objective description of a particle-wall collision under Hertzian collision force and linear damping. The collision energy was derived in 1858 by Hertz, a linear damping force has Stokes (1850) is known. This paper has so far only the analytical solution of this approximate or numerical targets for the problems solved. First, the two problems are solved fully analytically. For the first problem will be investigated further using the analytical solution, the physical behavior of the system and set up outline solutions for generalized models. For the second problem is carried out in order to increase efficiency and convergence control a semi-analytical optimization. Finally, the analytical results are compared with numerical solutions so as to validate accuracy and convergence to numerically."

    • Re:Gotcha! (Score:5, Interesting)

      by Rie Beam (632299) on Sunday May 27, 2012 @10:13AM (#40128097) Journal

      On a sad note, he only placed 2nd in the overall competition :(

    • Re:Gotcha! (Score:5, Informative)

      by bcrowell (177657) on Sunday May 27, 2012 @12:17PM (#40128799) Homepage

      That helps a little, but still doesn't really clarify completely what he did. I'll explain a little about what I know about the projectile problem and what I can figure out about what he might have accomplished here.

      In the Principia [wikisource.org], Newton poses three closely related problems. One is projectile motion under the influence of a frictional force that's proportional to velocity (book II, section I). Next he considers the case where the friction is proportional to the square of the velocity (book II, section II), and finally the case where it's of the form av+bv^2, where a and b are constants (book II, section III). Let's call these cases 1, 2, and 3.

      Case 1 is pretty straightforward. The x and y motions are decoupled, and each of the motions is governed by a first-order, linear, inhomogeneous equation.

      Case 2 is actually of more physical interest than case 1 for most real-world projectiles. For example, when you toss a baseball in air, its Reynolds number is about 10^4 or 10^5, and in that regime, a force proportional to v^2 is a pretty decent approximation. There is a well known closed-form solution for the one-dimensional subcase (I actually had a student a few years back who figured it out for herself, which was impressive), which is y=A ln[cosh(t sqrt(g/A))].

      A hint is that this page [jugend-forscht.de] has a photo of him holding up a large sheet of paper with his closed-form solution on it. The equation is clearly visible, and reads g^2/(2u^2)+(alpha g/2)[v sqrt(u^2+v^2) / u^2 + arsinh |v/u|] = const. The notation isn't explained, but clearly u and v are the components of some vector, probably the velocity vector. If so, then the constant alpha has to have units of inverse meters.

      This makes me think that what he's solved is the full two-dimensional version of case 2. It can't be case 3, because besides g there is only the one constant alpha appearing in his equation. If you write down the equation of motion, a=F/m=(mg-bv^2)/m=g-(b/m)v^2, the constant that naturally occurs is b/m, which has units of inverse meters. It also makes sense that his solution has a hyperbolic trig function in it, since the y(t) for the one-dimensional version of case 2 has a hyperbolic trig function in it.

      If my interpretation is right, then you should get a correct one-dimensional result from his equation when u=0. Unfortunately his equation blows up to infinity in that case, so I'm not sure how to extract any sane interpretation from it. By setting alpha=0, you should also get the case with zero friction. That does sort of make sense, since it says u is a constant, which it should be in that case.

      It would be interesting to see if my interpretation is right by doing a numerical simulation and seeing if his expression really does seem to be a constant of the motion.

      One thing to point out is that he may not have actually solved the full problem as set by Newton. He hasn't found the equation of the trajectory in closed form (which I think was what Newton was most interested in), and he also hasn't found the position in closed form as a function of time. (This is all assuming my interpretation is right.)

      • Re:Gotcha! (Score:4, Informative)

        by Anonymous Coward on Sunday May 27, 2012 @01:51PM (#40129309)

        You're right, he's demonstrated a constant of motion (i.e. a first integral) in the 2D version of Newton's Case 2. The constant alpha in his equation is what you called b. Gravity points in the -v direction.

        You can easily check this by differentiating his equation with respect to time, and then eliminating the derivatives of u and v using the expressions

        du/dt = -b u sqrt(u^2 + v^2)
        dv/dt = -b v sqrt(u^2 + v^2) - g

        His solution can probably be extended to Case 3 quite easily, if anyone feels like a challenge :)

      • Re:Gotcha! (Score:4, Interesting)

        by bcrowell (177657) on Sunday May 27, 2012 @02:18PM (#40129457) Homepage

        Doing a reply-to-self because I checked my interpretation using a numerical simulation. I wrote some python 3 code, which does a reasonably realistic simulation of a baseball being hit for a home run. Slashdot's lameness filter wouldn't let me post it, so I put it here: http://ideone.com/yeP4y [ideone.com]

        The results:

        u= 36.86184199300463 v= 25.810939635797073 Ray= 0.07075915491208162 KE+PE+heat= 147.825
        u= 30.646253624059415 v= 12.467830176777555 Ray= 0.07075939744839914 KE+PE+heat= 147.82340481003814
        u= 26.608846983666997 v= 1.6625489055858707 Ray= 0.07075957710355621 KE+PE+heat= 147.8224303518585
        u= 23.559420165753 v= -7.761841618975968 Ray= 0.08597247439794412 KE+PE+heat= 147.82171310054588
        u= 20.86163826256129 v= -16.094802395195508 Ray= 0.10413207421166563 KE+PE+heat= 147.82115230900214
        range= 120.88936569485678 , vs 194.17117929504738 from theory without air resistance
        u= 18.25141606403427 v= -23.242506129076933 Ray= 0.12066666645699123 KE+PE+heat= 147.8207286473949
        u= 15.70673363979356 v= -29.088976584679852 Ray= 0.1353850869274781 KE+PE+heat= 147.8204307883206
        u= 13.30143766684643 v= -33.65200048062784 Ray= 0.14867603720136566 KE+PE+heat= 147.8202356199746
        u= 11.11267911406159 v= -37.07517115834146 Ray= 0.16096016949002218 KE+PE+heat= 147.8201144079141
        u= 9.186200956690504 v= -39.564763699985484 Ray= 0.17255826567110216 KE+PE+heat= 147.82004160975018

        The notation is that u and v are the x and y components of the velocity vector, "Ray" is the expression that Ray seems to be claiming is a constant of the motion, and the final column is the total energy, which should be conserved.

        I tested my code two ways: (1) Energy is very nearly conserved. (2) If I turn off air friction, the range is very nearly as calculated by theory.

        Let R be the expression that Ray says is a constant, under my interpretation of his variables. Then dR/dt appears to be very nearly zero early on in the simulation. However, later on it starts to drift upward. So I suspect that one of the following is true: (1) Ray is wrong; (2) my interpretation of his notation is wrong; or (3) my simulation doesn't use good enough numerical techniques to demonstrate with good precision that Ray is right.

        Anyone who's got Runge-Kutta, etc., on the tip of their tongue want to try a better simulation of this?

  • Flash journalism (Score:5, Insightful)

    by yoctology (2622527) on Sunday May 27, 2012 @10:36AM (#40128213)
    These stories about overwhelming acts of personal genius, especially stories that lack the details of the alleged act, are, without memorable exception, false. But we all like a good story about an under-caste upsetting gray hairs and the established order of things.

    Think about that for a moment. A story supposedly lionizing science lacking the most basic facts that would permit substantial verification, or falsification, of that science. This is just flash journalism at work.
  • by quax (19371) on Sunday May 27, 2012 @12:23PM (#40128839)

    This longer piece (German) [www.welt.de] quotes him pointing out that he is very weak in Graph theory and Combinatorics. Nevertheless he skipped two classed in school and will be able to start university this fall.

    Won't be the last time we heard form this guy.

  • In the summary he is first named Ra, and then later referred to as "Mr. Ray". Which one is correct?
  • by bcrowell (177657) on Monday May 28, 2012 @07:51AM (#40134077) Homepage

    As often seems to be the case with these news articles about teenage prodigies, this has been overhyped. It turns out that what he did is not new and is not a complete solution to the problem.
    Parker, Am J Phys 45 (1977) 606 [df.uba.ar] has a summary of the preexisting results. The expression immediately after equation 23 is the constant of the motion that Ray rediscovered.

    A reddit user has a nice simple derivation: http://redd.it/u74no [redd.it] (Note that there is an error because he claims to have proved it in general, but it's only valid when v (the vertical velocity) is positive.)

    For more on the history of the problem:

    Synge and Griffith, Principles of Mechanics, p.~154 http://archive.org/details/principlesofmech031468mbp [archive.org]

    Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, p.~229 http://archive.org/details/treatisanalytdyn00whitrich [archive.org]

    According to Whittaker this was first done by D'Alembert in 1744.

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