PIONEERS OF SCIENCE: NEWTON, HOOKE, HALLEY AND THE PRINCIPIA
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Robert Hooke first discussed the subject of planetary motion in May 1666. He proposed that the motion of planets should be thought of in terms of an attractive force exerted by the Sun, holding them in orbit. To visualise this, imagine something like a conker, tied to a bootlace and whirled around your head. Your head represents the Sun, the conker is a planet and the shoelace is the attracting force. This was a rather simple model, but after he completed his work on the re-building of London, Hooke returned to the subject with a refined paper that he presented in 1674. The lecture was titled A System of the World and, among other things, it included three supposed aspects of planetary motion. The first point was that an attractive force held the constituent parts of a planet, cancelling their tendency to fly apart under centrifugal forces. He also proposed that this force extended beyond the planet in an invisible sphere. Any body that fell within this sphere must be dragged inwards. For his next point, Hooke said that a body (the Moon, perhaps) had the tendency to move forever in a straight line once set in motion. However, should another force intervene, the path would be deflected and bent into a curve; possibly a circle or an ellipse. His third argument was that the power of the attractive force fell off at a ratio of one over the distance from the object.
In this lecture, there were some arguments that represented a clear step forward and one argument that was just plain wrong. Where Hooke was in error concerns the rate at which the force of gravity falls off. In his model, an object twice as far from the Earth experiences half as much gravity. This is incorrect, because gravity falls off by the ratio of one over the distance squared. Therefore, an object twice as far from the Earth feels one-fourth gravity. But there were plenty of things that he got right. Hooke’s second point, that an orbit is the result of the tendency to travel in a straight line, plus a single force pulling it inward, would one day become known as Newton’s Second Law of Motion. It’s also interesting to note that Hooke talked about one single force. On the other hand, other scientists (including Newton) still discussed orbits as if they were the result of two forces (centrifugal and gravitational). Hooke was also the one that introduced the idea of action at a distance- the idea that gravity reaches out across empty space to tug on a moon or planet.
In 1679, having completed this paper, Hooke looked for a second opinion and chose Newton’s. He included a letter in which he asked Newton if he had any reason to doubt the proposal that an orbit was a straight line bent by gravity. By giving Newton a copy of his paper, Hooke introduced Newton to the concept of action at a distance, as well as the idea that a straight line bent by gravity equals an orbit. Interestingly, ’action at a distance’ subsequently appears in Newton’s work without further comment.
NEWTON’S REPLY
While Hooke had been engaged in astronomical works, Newton had put such affairs on hold while he dealt with the possible troubles resulting from his unorthodox beliefs. This was probably the main reason why he was so irritated by the argument over light. When he received the letter in 1679, Newton wrote back in reply and claimed to have lost interest in philosophical arguments of this kind. But he did indulge his old rival by including a diagram to illustrate a way to demonstrate the rotation of the Earth. Newton explained that, in the past, it was thought that an object dropped from a tower would fall slightly west of it, since the Earth rotated in an easterly direction during its fall. He then said that this was wrong. What should actually happen is that the object lands slightly east of the tower. Why? Because the top of the tower is further from the Earth and therefore has more circumference to get around. At the start of the object’s fall, the tower would be moving eastwards, but at a greater velocity than objects at the surface and would ’outrun the perpendicular’ and shoot forwards to the east side. In other words, a ball dropped from a tower should land in front of it, not get left behind. But Newton’s diagram did not show a ball impacting with the ground. Instead he drew what he believed would be the path taken by a ball that fell through the Earth, heading for the centre of gravity. According to him, it would spiral toward the centre.
Newton had not wanted to get drawn into a philosophical debate, but his diagram ensured that a philosophical debate is what he got. Hooke realised that the idea of a ball falling through the Earth and spiralling toward the centre was wrong. Assuming his theory of circular motion, the correct diagram should be a shrinking ellipse. In turn, Newton conceded that his first proposal was incorrect, but also claimed to have spotted an error in Hooke’s diagram. The error lay in the idea of a shrinking ellipse. Newton drew a more complex path in which the ball orbited in an indefinite ellipse, its orbit shifting around with the passing of time and therefore tracing a path that looked somewhat like the three-leafed clover. Hooke saw that this assumed a specific force of gravity. He wrote back, saying Newton had assumed a force of attraction that an equal power at all distances from the centre. Hooke suggested that the force was in a duplicate proportion to the distance from the reciprocal.
Or, put in simpler terms, an inverse square law.
This argument was purely philosophical. After all, in reality a ball could never fall through the Earth. Nevertheless, the two scientists were really asking a couple of questions and one gave the answer to the other. The first question: What shape would an orbiting planet trace if gravity adheres to an inverse square law? Answer: An ellipse. Second question: If a planet is orbiting in an ellipse, what law of gravity may we infer from this? Answer: An inverse square law of gravity. Progress was being made, but for Hooke the end of the line had come. He had a strong hunch that an inverse square law of gravity resulted in elliptical orbits, but he lacked the mathematical skills needed to confirm this. But Newton, he knew, did have the skills to solve this puzzle and so he suggested a challenge: Find the mathematical curve of an orbiting body and explain the physical reason for the shape of this path. This time, Newton did not reply.
ENTER EDMUND HALLEY
When I first mentioned Hooke, he was sat in a coffee house in London. Both Halley and Wren had guessed at the existence of an inverse square law, but admitted proving this fact mathematically was beyond them. Hook claimed to have such proof but, as we have seen, this was almost certainly an exaggeration and when Halley offered the prize of a book if he produced his proof within two months, Hooke was not forthcoming. That summer, Halley had to sort out his father’s affairs and this is probably why he travelled to Peterborough- because he had relatives there. As this put him in the same area as Newton, Halley decided to call in on him. This was not the first time they had corresponded, since they had already discussed the comet via letters. When they met face to face in August 1684, Halley asked Newton what shape an orbit would be if one assumed an inverse square law. Newton’s reply was that it would be elliptical in shape and that he had mathematical proof that this was so. Unfortunately, he had misplaced this evidence but promised to re-do it.
Halley waited and in November 1684 he was handed a nine-page treatise called On The Motion of Bodies In Orbit. It showed that all of Kepler’s observations concerning orbital motion could be tied to a centripetal force that was inversely proportional to the square of the distance. This, though, was but a taste of things to come. In 1687, Newton’s masterpiece, the three volume Philosophiae Naturalis Principia Mathematica was published. Earlier, I said that this is regarded as the most important scientific work ever written. You may be wondering why this particular book should be singled out for such high praise. Why is it more significant than, say, Einstein’s Theory of General Relativity or Darwin’s Origin of Species? The reason is that these books carry on a trend that was really started with Newton- that the universe can be understood with the application of logic and reason, rather than appealing to magic or gods. It is appropriate to say it started with him because although thinkers like Kepler were heading in this direction, their ideas were disorganised and not clearly expressed. It was Newton who gathered all the information and organised it, building the foundations of modern physics. He showed that the mathematical framework that governs the orbit of planets are universal laws that apply across the cosmos. This means that he could assume that (all else being equal) a planetary system anywhere in the universe must adhere to the same inverse square law and that we can assume any living creature new to science breathes for the same reason we do.
THE NATURAL UNIVERSE
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So what does this do for the concept of God? Although Newton’s laws revealed that it is unnecessary to appeal to magic or gods in order to explain things like the tides or a solar eclipse, it does not then follow that he had explained away The Creator. Rather, Newton and subsequent scientists presented a god whose creation took care of itself once it was set in motion. What is more, they came to believe that, for all the incredible complexity inherent in nature, underneath it all were mechanical principles that can be understood.
None of this came particularly easily to Newton. While he was engaged in writing the Principia, his obsessive nature revealed itself once more. He rarely slept and wrote standing up at his desk. Conversation with him was impossible, since his mind was utterly focused on the task- often he would be seen promptly changing direction and rushing back to his study. His home was filled with piles of papers, the result of decades-worth of research and he wrote to Flamsteed demanding yet more data. He asked for information on stars, the moons of Jupiter and even the tides. These seemed unconnected to most but as Newton feverishly worked he saw the connection and it was his law of gravity.
SEEKING DEEP ANSWERS
To find the most fundamental answers, one must question the most intuitive, everyday occurrences. Not many people question why the sky is blue, or why blue is blue, but often such questions lead to profound insights. It had already been established that the planets were travelling around the Sun, but Newton now saw a problem with this assumption. It had something to do with relativity. This is a term that will be forever associated with Einstein but it can actually be traced all the way back to Galileo’s time. Recall the experiment in which it was proven you cannot distinguish between a state of rest and a state of uniform motion. It’s all relative. From the perspective of a person on the shore, a ship moves across a motionless sea. From the boat’s perspective, it is still and it is the waters that pass beneath it. When we talk about velocity, we require a frame of reference; it means speed and direction in relation to something. Again, if you were travelling on a train and it was a perfectly smooth ride with no change in velocity and you had no outside reference to show otherwise, you would feel exactly as though you were not moving at all. But acceleration is a different matter altogether. By acceleration we mean a change in direction or speed and this can be detected without external reference. The train slams on its brakes and you are thrust forward. The driver speeds up and you are pushed back into the seat. It turns a corner and you are pulled left or right.
Velocity requires a frame of reference but acceleration does not. Newton wanted to know why. Also, if velocity is speed and direction in respect to a frame of reference, what does it mean to say a planet is moving? Moving in respect to what? To follow this line of reasoning, consider again that conker, whirled around your head in a model of a solar system. The conker is clearly moving but what is the ultimate frame of reference? You might think it’s your head. However, you are moving since you stand on the Earth as it spins on its axis. Pushing the question further out, what is our planet moving relative to? Well, it’s going round the Sun. But the Sun itself is travelling through the galaxy and the galaxy (like all the others) is in motion as well. Again, in reference to what? To Newton, there was only one thing left that could possibly act as the definitive frame of reference and that was Space. Hitherto, it had not been known if such a thing really existed. Think of the alphabet: If you remove all the letters you are left with nothing, since the space between the letters exists only when they do. Similarly, if you remove all matter from the universe are you again left with nothing? Newton argued against this, establishing space and time as independent of the matter contained within the Universe. To say something had velocity was to say it had speed and direction with respect to absolute space. Absolute space became the canvas upon which he crafted his laws.
THE LAWS OF MOTION
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Having set the stage, Newton then defined his Three Laws of Motion. The first law, which is known as The Law of Inertia states: Every body preserves in its state of being at rest or moving uniformly straight forward, except in as far as it is compelled to change its state by forces impressed. This is a refined version of Galileo’s law. As you may remember, Aristotle believed that the natural state of motion for anything was one of rest and any velocity could only come about by applying force. This seems like common sense, because the ever-present force of friction always brings about a change in velocity. Later, Galileo saw that things naturally travel at a uniform speed (and this can be zero) provided no force intervenes. The states of rest and uniform motion can be treated as the same and the Law of Inertia defines the reference frame (known as the inertial reference frame or Galileo’s reference frame) in which the other two laws are valid. It shows that forces are the fundamental cause of motion.
Newton’s second law is the Fundamental Law of Dynamics, which states: A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed. Furthermore it states: The acceleration of an object of constant mass is proportional to the resulting force acting upon it. You can see that the this law expands upon the first. Aristotle believed that a force was necessary if there was to be any velocity, which can be expressed with the equation F=MV. Newton showed that the correct equation was actually F=MA because a force only brings about a change in velocity or, in other words, an acceleration. Force acts as a kind of messenger between two objects, allowing them to interact by exchanging momentum.
The third law, the Law of Reciprocal Actions, is perhaps the most widely known of the three. It is commonly written down as for every action there is an equal and opposite reaction, but its properly written as, whenever one body exerts a force on a second body, the second body exerts an equal and opposite force upon the first body. In simple terms, this law shows that whenever you exert a force upon an object, it will exert the same force on you at the same magnitude but in the opposite direction. A weightlifter does not just pull at his weights, the weights pull at him. A person stepping off of a boat onto the shore finds that as they move toward the shore, the boat typically moves in the opposite direction. The Sun tugs at the planets and they tug at the Sun.
Everything affects everything else. This concept was expanded upon to give the Law of Gravitation: Every object in the universe attracts every other object with a force along the line of centre for the two objects that is proportional to the product of their masses and inversely proportional to the square of the separation between the two objects. With this law, Newton was able to modify Kepler’s Third Law and rid it of its mystical connotations. The mystical aspect of Kepler’s law can be found in the way he treated the Sun as a fixed point upon which the Universe revolved. To him, the Sun was almost a god so he naturally gave it a dominant position. But Newton’s law showed that the Sun is not as special as Kepler thought. A body cannot exert a force upon another body without the body returning the favour and he calculated that the Sun and the planets actually orbit around a common centre of mass. Actually, to be fair to Kepler, the Sun is so much more massive in comparison to the other planets that the centre of mass is very close to it. Picture two people sitting on a seesaw trying to balance it. If one person is heavier than the other, they have to sit closer together.
Newton showed that this force, gravity, was everywhere and that everything contributed to it. But in the argument over how an object would fall through the Earth, gravity had been assumed to be concentrated at a point in the centre of the planet. Newton showed how it was safe to calculate gravity in accordance with this simplified model, since his Law Of Gravitation shows that every particle of matter attracts every other particle and as a result you can calculate gravity as if it works according to the simplified model. Proving that this was so was relatively straight forward using the Calculus. Unfortunately, Newton had to avoid using it, since his readership would not have trusted his calculations unless they were written in the geometry that they were familiar with. It is not known if Newton wrote in Calculus first and then translated it, or if he wrote it all in geometry and did without the Calculus altogether. Either way, it was a monumental achievement.
NEWTON’S HELPERS AND ENEMIES
Newton was a famously introverted character, but even he could not have undertaken a task like completing the Principia alone. The person who most helped the book complete its journey from inspiration to published tome was Edmond Halley. He made sure the practical side of the book (things like reading through the proofs and dealing with publishers) went ahead with as few hitches as possible. In fact, without Halley the Principia may never have seen the light of day, for it was he who paid for it to be published. Originally, the Royal Society had agreed to cover the costs, but ultimately declined to do so. There are two possible reasons for this decision, one of which makes for a good conspiracy theory but is, sadly, the less plausible of the two. As the Principia was being written, Hooke and Newton once more became locked in a bitter dispute. The former scientist had read Newton’s work in progress and once again taken offence at the lack of credit for pointing its author in the right direction. Newton’s response was typically hot-tempered: He went through volume III of the Principia and made sure all reference to Hooke was removed. Meanwhile, the Royal was anxious to seem impartial to this row, and as agreeing to publish the Principia was tantamount to taking sides, the safe option was not to publish it.
The other possibility is more mundane but much more likely true. The Royal did not pay for the book because they could not afford the costs. Most of its budget had already been spent on a book by Francis Willughby called A History of Fishes which had gone on to be singularly un-sellable. This left the Society completely unable to cover the costs of the Principia’s publication, which with hindsight sounds like the ultimate case of backing the wrong horse. Halley took it upon himself to pay for the book’s publication, at a time when his pocket was hardly full.
THE FAMOUS COMET
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After Newton’s masterpiece was published, Halley’s determination in seeing it reach this stage was ultimately rewarded with the immortalisation of his name. Whereas its author largely turned away from science and forged a remarkably successful career at the Royal Mint, Halley went on to become the first post-Newtonian scientist. He made a huge number of contributions to science during this time, but of relevance here are his predictions concerning a certain comet and his realisation that the Universe must be much larger than had been supposed. The revelation concerning the comet came about because Newton had shown these objects have an orbit shaped like a very elongated oval. When Halley took this shape and applied it to the comet of 1682, he found that its orbit was the same as other comets that had appeared in 1607, 1531 and 1456. There was a pattern emerging and Halley soon spotted it. Every 76 years (give or take one or two) a comet put in an appearance. In 1705 Halley, like Flamsteed before him, concluded that this was the same comet, in perpetual orbit around the Sun. In the same year he published a book called A Synopsis of the Astronomy of Comets, in which he famously predicted its return in 1758.
But what of the man who had first suspected that the comets were one and the same? What had Flamsteed been up to all this time? The answer is he was still working on providing accurate star charts. This length of time in completing the job was partly due to the fact that it was always going to take a considerable amount of time, but also because Flamsteed was deliberately sitting on them. This was his way of punishing the Crown for not paying him what he considered to be a fair amount for his work. Newton, in his capacity as the Royal Society’s president, had visited Flamsteed in 1704 and tried to negotiate the papers’ release. That did no good and in the end Queen Ann herself had to order Flamsteed to make his charts available. This was an authority he could not defy and his charts were handed over to Halley, who was charged with putting them in order. The charts were published in 1712 with more refined versions coming out in 1725.
Before the first charts had come out, Halley had the opportunity to study an earlier version. When he compared the star’s positions in this chart with one compiled by Hipparchus in the 2nd century BC, the two generally agreed but a few stars were not in the same place. What is more, the extent to which they were separated was too large to be attributable to primitive techniques, and Halley realised that these stars must have really moved relative to one another. This, in turn, showed that the stars were spread out in three-dimensional space, making the idea that they were suns far enough away to resemble spots of light a bit more plausible.
Halley had already worked out a way of measuring the distance from Earth to the Sun. The inspiration for this idea came from a transit of Venus that he saw in 1691. In 1716, Halley showed how this event, observed from different points around the world, could use certain surveying techniques to find the distance to the Sun. Transits of Venus were rare events, but thanks to the progress made in tracking the planets’ orbits, Halley was able to predict that transits would occur in 1761 and 1769. He never lived to see his predictions validated, dying on 14th January 1742. Sir Isaac Newton died on 28th March 1727 and both men left unfinished business for the scientific world to be getting on with. Newton had shown how gravity worked but could not say what gravity was. This was a puzzle that would be solved in the 20th century, but before then astronomy would begin to grasp the enormity of space…
REFERENCES:
Science: A History by John Gribbin
Newton by James Gleick
Wikipedia entries on Newton, Hooke, and Halley