I'm very confused by contradicting accounts of a supposed solution by Newton to the problem of finding the trajectory of a projectile moving under uniform gravity against resistance that is proportional to the square of the projectile's speed. Some sources state that he did solve it in the second edition of the Principia (not the first!), but did not solve the more general problem, when an arbitary power law is assumed for the resistance. According to them, this remarkably general problem was solved by John Bernoulli. Other sources claim that Newton did not even solve the quadratic case.

So did Newton find the quadratic law trajectory or not? And if he didn't, what kind of propositions are there in Section 2 of Book 2 of the Principia? I assume that Book 2 of the Principia marks the beginning of modern ballistics; that's why I consider this question important.

Also, i have another question, which is too concerned with the later part of Newton's life. The question is about the problem of orthogonal trajectories which newton solved in 1716 by deriving the general differential equation which the orthogonal family satisfies. I want to ask what is so important about this question ; it doesn't look hard - seriously, even i can solve it. So why so many books attach importance to this solution?, i understand the important connection with potential theory (field lines and equipotenial lines), but why several books declare that Newton's solution to this problem was one of the ingenius achievements of his later life?


2 Answers 2


No, he did not. Not in the first (1687), not in the second (1713), and not in the third (1726) edition, below I quote from Cajori's edition of Motte's translation of the latter. Since he was unable to find the trajectory analytically he tried to approximate it by pre-chosen curves. In the process he apparently messed up with the semi-circles in the first edition of Principia, which Johann (=John) Bernoulli pointed out in 1712. Newton fixed the error in the next year's second edition, without mentioning Bernoulli. The problem was offered as a challenge in 1717, when Bernoulli solved it (in a parametric form) as a particular case of the problem with resistance proportional to an arbitrary power of speed.

Newton did find the trajectory for the linear resistance law, suggested by Galileo, in Section 1 of Book II. But he argued that the square law is correct for rare media like the air, and even that it can be derived from his laws of motion! Section 2 examines this case, first without gravity, and then for the fall in a straight line. But the closest Newton comes to the general trajectory is when he studies the inverse problem under Proposition X of Book II:

"Suppose the uniform force of gravity to tend directly to the plane of the horizon, and the resistance to be as the product of the density of the medium and the square of the velocity: it is proposed to find the density of the medium in each place, which shall make the body move in any given curved line, the velocity of the body, and the resistance of the medium in each place."

He considers four types of trajectories - semi-circle, parabola, hyperbola with oblique and vertical asymptote, and generalized hyperbola with similar asymptotes - and reasons what kind of medium distribution would produce them. His heuristic conclusion for the uniformly dense medium is this:

...the line which a projectile describes in an uniformly resisting medium approaches nearer to these hyperbolas than to a parabola. That line is certainly of the hyperbolic kind, but about the vertex it is more distant from the asymptotes, and in the parts remote from the vertex draws nearer to them than these hyperbolas here described.

Newton's consideration of the semi-circle and hyperbolic cases appears to be influenced by Tartaglia's Nova Scientia (1537), one of the first books applying mathematics to ballistics. Hackborn's On Motion in a Resisting Medium: a Historical Perspective details the Bernoulli affair:

"A “serious error” in Newton's analysis for the semi-circular trajectory in Proposition 10 of the first edition of Principia had unexpected consequences. Johann Bernoulli discovered the error and communicated it to Newton through his nephew, Nikolaus Bernoulli, who was visiting London in September 1712. Newton then tailored a long paste-up correction for the second (1713) edition, already printed, without citing Bernoulli's help... Eventually, responding to a 1717 challenge from Oxford professor John Keill to “[f]ind the curve which a projectile describes” subject to gravity and fluid drag varying as the square of its speed, Johann Bernoulli found an expression for this curve in the case of drag varying as an arbitrary power of speed".

  • $\begingroup$ Conifold, O.K i accept your answer. By the way, i have another question concerning the later part of Newton's life. Perhaps i should ask it in another post. The question is about the problem of orthogonal trajectories which newton solved in 1716 by deriving the general differential equation which the orthogonal family satisfies. I want to ask what is so important about this question ; it doesn't look hard - seriously, even i can solve it. So why so many books attach importance to this solution? i understand the connection with potential theory (field lines and equipotenial lines). $\endgroup$
    – user2554
    Sep 4, 2016 at 10:58
  • $\begingroup$ I also need to clarify and understand what exactly newton found on quadratic-drag trajectories. What did newton mean when he said this curve is of the hyperbolic kind? $\endgroup$
    – user2554
    Sep 4, 2016 at 13:57
  • $\begingroup$ Conifold@ perhaps you haven't seen yet my questions. So i'll repeat them. 1. Why is newton's solution to the orthogonal trajectories problem considered significant? 2. What did newton mean when said the curve is of the hyperbolic kind (certainly it's not an hyperbola)? 3. How did newton approximate these trajectories by hyperbolas? $\endgroup$
    – user2554
    Sep 4, 2016 at 15:40
  • $\begingroup$ @user2554 By "hyperbolic kind" he meant that both the ascending and the descending arcs of the trajectory appear to have linear asymptotes, like hyperbolas but unlike parabolas. When Newton found the distributions that would produce his (generalized) hyperbolic trajectories he could set parameters in them to approximate the uniform distribution, he then took the resulting shapes as approximating the trajectory for it. Orthogonal trajectories were in response to Leibniz's challenge, but it is a separate issue, see Kline p.475 books.google.com/… $\endgroup$
    – Conifold
    Sep 6, 2016 at 2:24
  • $\begingroup$ Thank you Conifold, now the general picture of newton's work on ballistics is clear to me. As for orthogonal trajectories, i'm going to ask about it in another post. $\endgroup$
    – user2554
    Sep 7, 2016 at 11:51

According to Whittaker this problem was reduced to quadrature by D'Alembert in 1744: Whittaker, A treatise on the analytical dynamics of particles and rigid bodies, p. 229 http://archive.org/details/treatisanalytdyn00whitrich

Conifold's answer says Bernoulli solved it in a more general form in 1717. Presumably there is some difference between the types of solutions achieved by D'Alembert and Bernoulli.

I would be surprised if anyone could have accomplished anything on this problem as early as Newton's lifetime, given the primitive state of the art in math and physics at the time. Today, we describe the solution in terms of conserved quantities (there is a constant of the motion in terms of $v_x$ and $v_y$) and write out the analytic solution for the one-dimensional case using logarithms. Doing calculus with logarithms was a brand new thing at that time, and I don't know if the concept of a conserved quantity even existed.

  • 1
    $\begingroup$ Yes, Bernoulli does not give a quadrature but a parametric representation of position and time as functions of the slope of the projectile's trajectory. Hackborn derives his solution using modern notation in the linked paper. $\endgroup$
    – Conifold
    Sep 6, 2016 at 2:29

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