# The life of isaac newton and his relations with edmund halley

One of the most famous and consequential meetings in the history of science took place in the summer of 1684 when the young astronomer Edmund Halley paid a visit to Isaac Newton, during which Halley asked Newton what path a planet would follow if it were attracted toward the sun by a force proportional to the reciprocal of the squared distance.

Wren, Hooke, and Halley had discussed the problem at a coffee house following a meeting of the Royal Society in January of 1684, and Wren had offered a cash prize to whoever could provide a derivation of the shape of planetary orbits under the assumption of an inverse-square central force of attraction toward the presumed stationary sun. Hooke had claimed to have a proof that the paths were ellipses, but never provided it.

Against this background, Halley paid a visit to Newton, who later told Abraham De Moivre about the fateful meeting.

After they had been some time together, the Dr asked him 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.

## The life of isaac newton and his relations with edmund halley

Sir Isaac replied immediately that it would be an ellipse. The Doctor, struck with joy and amazement, asked him how he knew it.

Why, saith he, I have calculated it. Whereupon Dr Halley asked him for his calculation without any farther delay. It represents arguably the greatest single advance in human understanding ever achieved in the history of science.

## Isaac Newton: A life scientific

Just two months after its publication, the mathematician David Gregory wrote a letter to Newton, saying Having seen and read your book I think my self obliged to give you my most hearty thanks for having been at the pains to teach the world that which I never expected any man should have known.

For such is the mighty improvement made by you in the geometry, and so unexpectedly successful the application thereof to the physics, that you justly deserve the admiration of the best Geometers and Naturalists, in this and all succeeding ages.

In a careful series of propositions 11 to 13 of Book 1the Principia shows that a planet moving in a conical orbit under the influence of a central force toward one of the foci is undergoing acceleration toward that foci with a magnitude proportional to the reciprocal of the squared distance, and hence is subject to an inverse-square force.

Whether this is a plausible reason for omitting it is debatable. Furthermore, the assertion of obviousness is questionable, in view of the degree of difficulty evident in the demonstration of Proposition 41, in which Newton presented a general construction method for the path of a planet subject to any given central force, essentially by integrating the differential equations of motion.

Thus, letting v0 denote the speed of the planet when it is at the radial position r0, and letting v denote the speed of the planet after it has moved to some lower radial position r, we assert that the change in kinetic energy equals the work done by the gravitational force which Newton had proven in Propositions 39 and 40giving the relation where F r is any given radial dependence of the gravitational force.

The following year, as Roger Cotes was beginning to edit the second edition of Principia, Newton sent him a few additional words to be added to the first corollary of Proposition 13.

### The life of isaac newton and his relations with edmund halley

The entire demonstration then read as follows, with the words added in the second edition shown in italics: From the three last Propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force that is inversely proportional to the square of the distance of the places from the centre, the body will move in one of the conic sections, having its focus in the centre of force; and conversely.

For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body being given; and two orbits, touching one the other, cannot be described by the same centripetal force [and the same velocity].

Each edition contained just the most meager clarification, even though he knew full well that the passage was perceived to be unclear.

- Wren, Hooke, and Halley had discussed the problem at a coffee house following a meeting of the Royal Society in January of 1684, and Wren had offered a cash prize to whoever could provide a derivation of the shape of planetary orbits under the assumption of an inverse-square central force of attraction toward the presumed stationary sun;
- Hooke Robert Hooke, 55ish, a senior Fellow of the Royal Society, a friend of Newton and staunch supporter of his work;
- He eventually returns to the Royal Society and triumphantly presents his results on the great comet of 1681, but only after some horrific nightmares, some harsh lessons about himself and his dealings with other people, and some apologies;;;
- He reflects on the meaning of his bad dreams;
- Furthermore, Newton had shown that all possible initial conditions can be achieved by a conic through a given point with a given focus — assuming an inverse square force;
- Sir isaac newton is easily regarded as one of the most influential if it weren't for his friend edmond halley, newton may never have it is exactly this sort of quarrel that led to a poor relationship between hooke and newton.

Even after learning of the added words in the second edition, Bernoulli was unconvinced. To support his objection, Bernoulli noted that a similar sounding argument, when applied to inverse cube forces, would lead to a wrong conclusion. Furthermore, Newton had shown that all possible initial conditions can be achieved by a conic through a given point with a given focus — assuming an inverse square force.

- In April 1686 Newton presented and dedicated to the Royal Society the first third of his illustrious work;
- However, he would keep it a secret for the time being so that his friends "might know how to value it, when he should make it public.

It follows just as Newton said that all possible solution paths for an inverse square force are conics. In contrast, a similar argument cannot be made for an inverse cube force and logarithmic spirals, because such spirals cannot produce all possible initial conditions. The other possible initial conditions correspond to the other species of paths that satisfy an inverse cube force.

As the historian of science William Whewell wrote in 1847 The ponderous instrument of synthesis, so effective in his hands, has never since been grasped by one who could use it for such purposes; and we gaze at it with an admiring curiosity, as on some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden.

This proposition does indeed give explicit constructions for the path of a planet subjected to a central inverse-square force, but the demonstration explicitly assumes that the path is a conic. Unfortunately, this stipulation is not mentioned in the statement of the proposition itself, so the content of the proposition has sometimes been misconstrued. It gives the elliptical, parabolic, or hyperbolic paths explicitly for a given set of initial conditions, but only on the assumption that the path is a conic, which is why Corollary 1 to Proposition 13 is needed, to establish that the path must be a conic.

Perhaps he was conscious of the need for an explicit use of calculus to do justice to the demonstration, and he had been unwilling at least in the first edition to reveal many of the techniques he possessed.

- Newton's father, a farmer, had died before Isaac's birth and his mother had been forced by Smith to abandon Isaac, 3, on her remarriage.
- But once his creative powers were loosed, there was no checking their momentum.
- Smith Barnabus Smith, Newton's stepfather, an elderly clergyman, set in his ways and beliefs. The other possible initial conditions correspond to the other species of paths that satisfy an inverse cube force.
- Now Smith and mother have unwelcome plans for Isaac's future...
- Newton's childhood was anything but happy, and throughout his life he verged on meanwhile, in the coffeehouses of london, hooke, edmund halley, and after publishing the principia, newton became more involved in public affairs. What kind of curve, Halley finally asked, "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?

One might think this would lead to an even simpler solution, since the equation in this case is homogeneous. However, the coefficient of u in this homogeneous equation is now dependent on the ratio of M to h2, and can be positive, negative, or zero.

As a result, there are multiple cases to consider, in contrast with the differential equation based on an inverse square force, for which the homogeneous equation has constant coefficients of unity. If A is not zero, then with a suitable choice of angular origin the path can be expressed as a hyperbolic spiral for some constant k.