Newton and the Counterfeiter Read Online Free PDF Page B

discussed the complexity of the problem of discovering "heavenly motions upon philosophical principles." Accordingly, Wren would not take a claim of a solution on faith. Instead, he offered a prize, a book worth forty shillings, to the man who could solve the problem within two months. Hooke puffed up, declaring that he would hold his work back so that "others triing and failing, might know how to value it." But two months passed, and then several weeks more, and Hooke revealed nothing. Halley, diplomatically, did not write that Hooke had failed, but that "I do not yet find that in that particular he has been as good as his word."
    There the matter rested, until Halley put Wren's question to Newton: "what he thought the Curve would be that would be described by the Planets supposing the force of attraction towards the Sun to be reciprocal to the square of their distance from it." Newton immediately replied that it would be an ellipse. Halley, "struck with joy & amazement," asked how he could be so sure, and Newton replied, "Why ... I have calculated it."
    Halley asked at once to see the calculation, but, according to the story he later told, Newton could not find it when he rummaged through his papers. Giving up for the moment, he promised Halley that he would "renew it & send it to him."
    While Halley waited in London, Newton tried to re-create his old work—and failed. He had made an error in one of his diagrams in the prior attempt, and his elegant geometric argument collapsed with the mistake. He labored on, however, and by November he had worked it out.
    In his new calculation, Newton analyzed the motions of the planets using a branch of geometry concerned with conic sections. Conic sections are the curves made when a plane slices through a cone. Depending on the angle and location of the cut, you get a circle (if the plane intersects either cone at a ninety-degree angle), an ellipse (if the plane bisects one cone at an angle other than ninety degrees), a parabola (if the curve cuts through the side of the cones but does not slice all the way through its circumference), or the symmetrical double curve called a hyperbola (produced only if there are two identical cones laid tip to tip).
    As he calculated, Newton was able to show that for an object in a system of two bodies bound by an inverse square attraction, the only closed path available is an ellipse, with the more massive body at one focus. Depending on the distance, the speed, and the ratio of masses of the two bodies, such ellipses can be very nearly circular—as is the case for the earth, whose orbit deviates by less than two percent from a perfect circle. As the force acting on two bodies weakens with distance, more elongated ellipses and open-ended trajectories (parabolas or hyperbolas) become valid solutions for the equations of motion that describe the path of a body moving under the influence of an inverse square force. To the practical matter at hand, Newton had proved that in the case of two bodies, one orbiting the other, an inverse square relationship for the attraction of gravity produces an orbit that traces a conic section, which becomes the closed path of an ellipse in the case of our sun's planets.
    QED.
    Newton wrote up the work in a nine-page manuscript titled
De motu corporum in gyrum
—"On the Motion of Bodies in Orbit." He let Halley know the work was done, and then presumably settled back into his usual routine.
    That peace could not last, not if Halley had anything to do with it. He grasped the significance of
De Motu
immediately. This was no mere set-piece response to an after-dinner challenge. Rather, it was the foundation of a revolution of the entire science of motion. He raced back to Cambridge in November, copied Newton's paper in his own hand, and in December was able to tell the Royal Society that he had permission to publish the work in the register of the Royal Society as soon as Newton revised it.
    And then ...