Newton

Born Dec 1642 (Old calendar) (the same year Galileo died) he was raised by a stepfather and his mother. He want to King's school in Grantham. When his stepfather died his mother wanted him to take over the farm, but he hated it, and the head of King's school persuaded his mother to send him back to school. He want to Trinity College Cambr. in 1661. In 1665 the U was closed for almost 2 years due to the plague, and Newton spent the time at home thinking and doing experiments in Alchemy, and optics, and thinking about motion that he had learned by reading Descartes and Galileo (and also Kepler). One of the famous stories about him during this time is the apple story. The apple story was written by a friend of Newton -- William Stukeley-- 25 years after Newton's death. Stukeley told of their conversation in 1726:

After dinner, the weather being warm, we went into the garden, & drank thea under the shade of some appletrees, only he, & myself. Amidst other discourse, he told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. "Why should that apple always descend perpendicularly to the ground," thought he to him self: occasion'd by the fall of an apple, as he sat in a contemplative mood: "Why should it not go sideways, or upwards? But constantly to the earths centre? assuredly, the reason is, that the earth draws it. There must be a drawing power in matter. & the sum of the drawing power in the matter of the earth must be in the earths centre, not in any side of the earth. Therefore dos this apple fall perpendicularly, or toward the centre. If matter thus draws matter; it must be in proportion of its quantity. Therefore the apple draws the earth, as well as the earth draws the apple.

That there is a power like that we here call gravity which extends its self thro' the universe & thus by degrees, he began to apply this property of gravitation to the motion of the earth, & of the heavenly bodys: to consider thir distances, their magnitudes, thir periodical revolutions: to find out, that this property, conjointly with a progressive motion impressed on them in the beginning, perfectly solv'd thir circular courses; kept the planets from falling upon one another, or dropping all together into one centre. & thus he unfolded the Universe. This was the birth of those amazing discoverys, whereby he built philosophy on a solid foundation, to the astonishment of all Europe

Some of the key features here are a)That he thought of a "drawing power" directly of the earth on the apple-- Descarte and Huygens found this kind of idea repugnant. Things were affected through contact with some agent-- collisions of particles, or vorticies in some plenum. Newton here was postulating a force which did not operate via any mechanism but directly from the earth to the apple. b) He guessed that this drawing power was proportional to the mass of the body, rather than to any other feature. c) And he guessed it was to the centre of the earth, or apple, not to any other part. d) If the earth drew the apply then the apply must also draw the earth (this was to be his third law of motion.)

He wrote, in a paper which he never published, a description of the motion of a canon ball or a stone flung horizontally from a mountain (and assuming as Galileo did) that the whole thing were in the void. If shot the canon ball sufficiently rapidly, it would fall, but the fall would be sufficiently far that the earth's surface had also fallen away at the same rate, and the ball would stay at a fixed height above the surface. The ball would orbit the earth.

And at the same time, came the idea that perhaps this explained the moon and the planets as well. Just as Huygens in his thoughts on centrifugal force, saw that to keep something moving on a circle, there must be some force on that object to keep it going in a circle. Perhaps it was the same force as caused the apple to fall. And perhaps it was the same force as kept the planets orbiting the sun.

He knew of Kepler's laws, especially the third law which stated that the the radius of the orbit cubed divided by the square of the time of orbit is the same for all of the planets. Since the distance travelled in one revolution is 2 pi times the radius, and since he distance travelled in one orbit divided by the time of one orbit is the velocity, Kepler's relation could also be written that the square of the velocity of the planet times the radius of the orbit is the same for all planets.

But as Huygens had shown ( Newton came to the same conclusion independently before Huygens finally published his thoughts), the acceleration needed to keep the planet in its orbit is velocity squared divided by the radius, so the acceleration is the (velocity squared times the radius) (which is the same for all planets) divided by the radius squared. Ie, the acceleration required to keep the planets in orbit falls off as the radius squared for the planets and the sun. Why not the same for the earth and the moon? Galileo had found the acceleration that brought the apple to the earth and Huygens and later Newton had greatly improved the accuracy by examining the oscillations of a pendulum (about 10m per second squared in modern unis), and if the acceleration to the moon fell off as the inverse square, and since the moon was about 60 earth radii away, the centrifugal acceleration of the moon must be about 60 squared or 3600 times smaller than the apple's if these ideas were right.

Knowing the radius of the moon's orbit, and the time for one revolution, he could calculate the moon's velocity and centrifugal acceleration. The answer he found was close, but not close enough for him (his value for the radius of the earth was about 18% too small, and thus his values for the moon's radius and velocity were too small, giving a centrifugal force which was too small by 18%). He stuck away the calculation for a few years.

He was an incredibly eclectic intellectual. He was constantly carrying out experiments. At one time, trying to figure out how the eye works, he would insert small spoons into his own eyes between the eyeball and the eye-socket, to put pressure on the eyeball to see what that did to his vision. He was an ardent experimental alchemist, spending a large fraction of his time throughout his life on alchemical experiments. He held heretical theological views, especially on the nature of the Trinity (He was a Unitarian-- God is one, not three), which he wisely kept quiet about (except to his notebooks) . He believed the Egyptians had had a deep wellspring of knowledge about the world that had been lost, and he tried to recover it.

In 1679, Robert Hooke wrote him briefly arguing that a body in motion would continue in motion unless some outside influences acted on it (the idea that Huygens had also had) and also argued that there should be an inverse squared dependence of the acceleration of the planets, but he clearly did not know how to calculate the implications of this (Hooke, Christopher Wren, . Newton recognized the importance of this, and abandoned his research into the ether, and the plenum. He immediately went back to his apple-moon ideas, and with better data on the size of the earth, found it worked spectacularly well. He asked what the orbit of the planets would be in such a theory, and discovered that they would be ellipses. But, in the words ascribed to Mary in the Luke "he hid all these things in his heart" and never communicated them.

Hooke always felt that Newton had stolen the idea of inertia and of the inverse squared law from him, and Newton grew to detest Hooke. Later in life Newton said "If I have seen further that others, it is because I have stood on the shoulders of giants." Hooke was a dwarf.

In 1684 Halley, a member of the Royal Society of London, came to Cambridge. Robert Hooke, who had been Secretary of the Society, and arranging for demonstrations of new discoveries for the Society, said that he know that the the inverse square law of accelerations for the planets implied that the orbits were ellipses, but would not reveal his proof until it was clear noone else could prove it. Halley, on his next trip to Cambridge, dropped in on Newton, and asked him the question. Newton replied that he had proved that 5 years earlier, but had never published it. Halley pushed him to publish, and Newton spent the next two years writing his , which Halley got published (by putting up some of his own money to pay for the publication.)

Here is an online version of the translation into English of the Principia

He started out by listing three fundamental laws which governed all motion.

• A body in motion will remain in straight line motion forever with constant velocity unless acted upon by other forces. (Principle of Inertia) He got this law in part from Descartes, but more importantly, also from Galileo in the sense of compounding of motions.
• The acceleration of body times the mass of that body is equal to the force acting on the body from outside. The idea of forces, had come from his own considerations of the collisions of bodies. What was especially important was the notion of the compounding of motions (which he got from Galileo), that the forces and accelerations were "vector" quantities-- ie that they had a direction associated with them, as well as a magnitude. Thus acceleration was not only a change of speed, but was also a change in the direction of the velocity.
• If one body acts on another body with some force, then that other body likewise acts on the first with an equal and opposite force. This seems to have arisen from his own study in trying to understand collisions of bodies.

He had entirely upset the world view, since changes in motion were not the result of contact with agents, but were entirely encapsulated in these mysterious forces. These forces were also not just magnitudes but also had a direction, just as the acceleration did.

The final law encapsulated the observations of Huygens and himself on the collisions of bodies. If one regards the whole system of the two balls as one object, denoted for example by its centre of mass, then the forces on the two balls would cancel (they are equal and opposite) if one regarded the two balls as one object. This composite system would then travel with constant velocity, since there was no net force acting on the system.

As mentioned, this concept of force which was not a contact force was anathema for many. It felt too much like astrology, where the stars and planets were supposed to exert a mysterious influence on the earth. It was for many a seeming retreat back to superstition. But the huge difference between Newton's forces and those other mysterious occult forces was that former could be and was mathematised. It could be treated logically and precisely.

Much of the book treated the one subject, which he called gravitation. This was the mysterious drawing power of the earth to the apple and to the moon, and the drawing power of the sun to the planets. But more importantly, it was a force which every particle exerted on any other object. If the earth attracted the apple, the apply also attracted the earth. If the sun attracted Jupiter, Jupiter also attracted the sun, and all of the other planets. The third law also said that the apple also exerted a drawing power to the earth, the moon to the earth, and the planets to sun, and to each other.

The book was a "tour de force" of the highest magnitude. Newton showed that the assumption of an inverse squared law of gravity (the force of gravity decreases as the square of the distance away from the body) gave all of Kepler's laws. The elliptical orbits followed from the inverse square law. See this discussion of that derivation.The area law (the radius from the sun to the planet sweeps out equal area in equal time) was also an immediate consequence but only of the fact that the force always points directly toward the sun from the planet. The form of the force did not matter. See this derivation from Kepler's second law

Like Galileo he attacked the problem of the tides. While the gravitational attraction of the moon on the earth exactly compensated the orbital centrifugal force of the earth orbiting around the centre of mass of the earth moon system, that is only true at the centre of mass of the earth. For parts of the earth closer to the moon, the gravitational attraction of the moon is larger than the centrifugal force and those portions of the earth are attracted in net to the moon. For those parts of the earth further away than the centre of mass, it is the centrifugal force that is larger, and those parts of the earth thus go further away, with some of the earths own attraction providing the force to compensate for the lessened gravity. At the sides of he earth, the moon's attraction is not in exactly the same direction as the attraction at the centre of the earth. It has a small inward component. Thus it is the attraction of the moon for all the parts of the earth, in balance or otherwise of the circular motion of the earth, that creates the stretching/squeezing of the earth that make the tides. The sun does something similar and also causes tides on the earth, about 30% smaller than the effect of the moon. And since the two objects, the sun and moon, have the same angular size in the sky, it turns out that the tides depend only on the average density of the bodies, which implies that the Sun has an average density about 1/3 that of the moon (and 1/5 that of the earth it turns out).

With three simple laws, and one assumption of the nature of the force, Newton replaced two and a half thousand years of work trying to understand the motion of the stars and planets and sun and moon. He answered the question as to whether the earth rotates, since having the stars rotate would require a HUGE force which increases with distance to hold them on their circular obit daily around the earth, a clear absurdity. He answer whether the earth goes around the sun or the sun around the earth, since the sun is so much more massive than the earth (as seen in the expression for Kepler's third law applied to the planets that do go around the sun and the moon around the earth) and that means, despite Galileo's relativity, that the earth going around the sun is far more reasonable. Collapsing 2.5 thousand years of technical endeavour into one simple expression truly makes Newton's achievement a revolution. Not a violent one, even intellectually, unless you were an ardent Aristotelian, although they had been dying out for centuries already. The Catholic Church kept the ban on Copernicus' and Galileo's writings for another couple of hundred years, but noone bothered to even notice the bans or worry about them. Their main effect was to keep the Jesuits, one of the most intellectual orders of the Catholic church, from participating the scientific revolution that followed.

I have clearly vastly oversimplified the route Newton took to these insights. Whole books have been written to try to disentangle all of the influences on Newton, which led him to his discoveries-- especially the importance of Galileo, Descartes, Kepler and Hooke on his thought processes.

Thus, in the course of the 17th century, The whole old worldview on dynamics was overthrown. The separation between the sublunary world and the celestial world was gone. The heavens behaved in exactly the same way as the earthly realm did. The natural motions became somewhat more heavenly, in that continuous motion was natural, but not on circles but on a straight line. This applied in the heavens and on the earth (in the void where other stuff would not interfere). The void became accepted. (Torticelli argued that if one inverted a closed water pipe over 10 meters high, what you got in the top was a void, a region devoid of air and experiments followed in seeing what the effects of the void were. For example, pendulums took much longer to slow down in such a vacuum than in air, supporting Galileo's position that slowing down was due to friction, interaction with outside things). Using mercury, that height was reduced to about 76cm in height. Forces could act through the void. Forces were not necessarily transmitted only by contact. Since constant velocity (movement in a straight line with constant speed) was natural, the affect of forces was not on the velocity but on the acceleration (the change in velocity, whether speed or direction). And all this was mathematized. One could study motion in detail and with accuracy, and one could predict what would happen in new situations. The world could be imagined in radically different ways, and those imaginations shown to actually correspond to the way the world behaved in new situations. Pythagoras was right. The world is mathematical at heart, and in a far more comprehensive way than he ever could have imagined.

Motion was compounded. One could separate motion out into parts which acted essentially independently of each other. Motion in the "x" direction was governed by forces acting in that same direction. Motion in the "y" direction, perpendicular to "x", acted independently. (This is called the vector property of forces and of motion>).

And finally the motion of the heavens, and some of the motions on earth--like falling, were governed by gravity, a force which acts across voids, and in which every piece of matter affected every other piece of matter. It was a universal type of force which all matter engaged in. Of course small amounts of matter produced small amounts of force.

For some this mathematization of nature removed the soul from the world. How could one see beauty in a sunset, purpose in a flower if all it was was these mathematical forces and motions in play. It produced a profound disquiet

For others, it opened up a grand challenge. Finally we could understand the world around us, understand both how and why things happened. How to carry out that understanding into more and more fields of human endeavour became the challenge and opportunity for many, not just to understand but also to control and allow us to use the world around us. We have understood the language of the world, and that language is mathematics.

Galileo, Huygens, Kepler

Galileo
• Combination of motion: Motion can be broken up into motion in the three perpendicular directions in space. Galileo broke it into the two horizontal directions and the one vertical. Newton into any three perpendicular directions
• Vertical Fall: The distinguishing feature of motion near the earth is the vertical acceleration (not velocity) down to the earth.
• Galilean Relativity: Was true of the motions in Newton's theory but was not an essential feature of his dynamics

Kepler:
• 2nd law (equal areas). Used this to show that the acceleration of the planets always points to the Sun, and the moon to the earth.
• 3rd Law: (period of revolution squared proportional to distance cubed)the centrifugal acceleration of the planets falls off as the inverse of the distance to the planet squared.
• 1st law (ellipse with Sun at focus): Showed that this is a direct consequence of the inverse squared law for the acceleration of the planets.

Huygens
• The centrifugal acceleration law (velocity squared over radius).
• Law of inertia-- objects travel in straight lines with constant speed unless caused to deviate by external "Force". (Also owes something to Descartes, and to Hooke)

Hooke
• Law on Inertia?: Pointed out to Newton by Hooke in letter of 1679. Embarrassing mistake on Newton's part as to what the orbit would be of a body which fell from a high mountain. Because the velocity would be larger higher up due to rotation of the earth, the object would hit the earth sightly to the east of the base. How would it keep travelling if the earth were transparent to the body? Newton said it would spiral in to the centre of the earth. Hooke pointed out that this could not happen, but there would be some complicated orbit. Newton realised his mistake, but hated Hooke thereafter.
  Note that Descartes and Huygens had also suggested the law of inertia.
• Inverse square law for planets. Hooke mentioned this in his letter. He always claimed Newton stole this idea from him. The actual facts are unclear. Newton apple story-- he claimed he had already realised the inverse square law in the plague year when he thought that maybe there was some relation between the apple falling and the centrifugal acceleration of the moon around the earth. He probably did calculate the centrifugal force of the moon and knew the Galilean acceleration of the apple, and compared them, but whether inverse square law occurred to him then is unclear (the apple story only surfaced many many years after it happened in conversations with friends).
  Differences
  • Emphasis on forces.
    • Collisions -- forces between bodies -- equal and opposite forces on the two bodies.
    • Accelerations: In dynamics it is the accelerations of bodies that need to be explained by external source, not their velocities.
    • "Universal" Gravity: all bodies exert forces on all other bodies even in the void. No aether/plenum/atoms to transmit the force proportional to the masses of the two bodies and inversely proportional to the square of the distance. Sun exerts forces on planets, but planets do on sun as well (Equal and opposite forces). Planets revolve around "sun" but actually around the centre of mass of the planets/sun.
  • Masses: All bodies have an internal attribute which is their mass. Modifies the action of the forces on the body. Mass directly related to weight of body (as opposed to Descartes's size/volume/shape...)
  • Accelerations: What is important about the body is its motion, and in particular its acceleration. Uniform motion needs no explanation. Acceleration caused by forces.
  • Tides: Galileo's tidal explanation violated Galilean relativity. Newton explained them by competition between attraction of moon and sun on earth, and earth's centrifugal force as earth orbits around the centre of mass of the earth-moon and earth-sun system. (On near side to either, attraction wins out, on far side centrifugal acceleration wins out, giving bulges on either side. On sides, inward component of the attraction wins out-- giving 2 tides a day.
  • Rotational bulge of earth.

  copyright W Unruh (2018)