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On Math Stack Exchange, I am impressed by users' skill at finding closed form expressions for definite integrals. For example:

  • Example 1: $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx=4\pi\operatorname{arccot \sqrt{\phi}}$
  • Example 2: $\int_0^{\pi/3}\arccos(2\sin^2 x-\cos x)\mathrm dx=\frac{\pi^2}{5}$
  • Example 3: $\int_0^\infty \frac{\sin^2{(x\sin x)}}{x^2}dx=1$

Out of curiosity, how good was Newton at definite integration? More specifically:

What are some examples of non-trivial definite integrals that Newton evaluated?

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    $\begingroup$ This question is certainly weird in motivation (Newton invented integrals and you asking what was his game?), but it would be nice to see some integrals that Newton calculated. $\endgroup$
    – Mauricio
    Commented Jan 17 at 8:23
  • $\begingroup$ @Mauricio Do you reckon Newton could have evaluated these example integrals easily? $\endgroup$
    – Dan
    Commented Jan 17 at 8:57
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    $\begingroup$ Not sure what you mean. I do not know what was the most complex integral that Newton solved but clearly he was not interested in participating in integral competitions (they did not exist). $\endgroup$
    – Mauricio
    Commented Jan 17 at 9:04
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    $\begingroup$ @Mauricio My motivation for this question is just curiosity. How good was the inventor, compared to those that followed? $\endgroup$
    – Dan
    Commented Jan 17 at 9:08
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    $\begingroup$ Newton was using his "fluxional" calculus as a way of solving geometry problems and differential equations, he was not actively finding closed forms to antiderivatives. $\endgroup$
    – Mauricio
    Commented Jan 17 at 9:45

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Newton integrated in the 'Principia' at least three rate-expressions related to features of the motion of the moon. (Newton's convention for denoting such rates was to call them 'hourly motions'):

(a) book 3: propositions 25-29, especially 26, identify the rate of change of the moon's areal velocity, and this rate was then integrated; (it would have been constant under Kepler's 2nd law in the absence of perturbations, but it does suffer periodic perturbation by the Sun, and Newton was interested in determining the resulting variation and succeeded, under certain simplifying approximations adopted to make the problem tractable within the limits of later-17th-century technique);

(b) book 3: propositions 30-33 identify the rate of fluctuation of the regression of the moon's nodes, which is then integrated so as to find the difference between the true nodal motion and the mean due to the considered cause;

(c) book 3: propositions 34-35 identify the rate of fluctuation of the inclination of the moon's orbital plane, and again there is an integration to find the difference between the true inclination and the mean due to the considered cause.

All of these integrations were described in terms of Newton's geometrical formulations of the calculus, without any notation for equations or integrations that would be easily recognisable today (see e.g. How did Newton write his equations?).

They were however reviewed and explained with translation into terms of later notation (much more accessible!) by no less a reviewer than Pierre-Simon Laplace, in the 1825 volume (in French only) of his 'Traite de la Mecanique Celeste', tome V, book XVI, especially chapter 2, beginning at page 367 (though chapter 1, from page 349, is also a highly recommendable read containing many interesting expressions of Laplace's point of view on Newton's work).

Newton in an earlier phase of his work in the 1670s also gave a solution in the form of a power series to Kepler's (original) problem of determining the orbital position after a given time-interval under equal-area motion in an elliptical orbit. This solution seems to have involved power-series expansion of a rate expression that was then integrated term-by-term to give a trigonometrical function of a relevant angle (eccentric anomaly) which can then be further processed with basic trigonometry to give the main wanted result, the position-angle in the orbit after the given time-interval.

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