The Laws of Motion and the Universe

Imagine a man who loved order, but saw a world full of things falling, swinging, spinning, and rushing through darkness. He wanted to know why. Not with guesses and secret forces nobody could measure, but with rules anyone could test. That is how Newton begins his work. He promises a path from what we see to what lies behind, and back again. He says: let geometry measure, let mechanics tell how forces create motions, and how motions reveal forces. He admits the path is hard. He asks us to judge gently where something falters, and points forward: if you can finish where I stop, do so. He wanted to wait until he could show everything together – the Moon wobbling, many bodies pulling each other at once, comets coming in at an angle and leaving again, motions through air and water. He held back. Then came Halley. He asked, begged, came back, and finally said: I will pay for the printing. Thus the work was pushed into the light. Between the lines we also hear a friendly promise: this will not be a book that only says 'it is so'. It will show how we can find out. If you have ever seen a stone in a sling or a spinning bucket of water, you are already near the core. And if you have ever seen a comet's tail stretch across the sky on a winter morning, you are in company with Newton the night he decided to calculate it all. This is the preface and the agreement: fewer secrets, more measurable words, and a map that leads from real traces to invisible forces.

Before the story can run, it needs names for things. Mass is how much stuff is in a body. Put two balls on a scale: the one that presses more has more mass. Quantity of motion is mass times speed. A heavy ball moving slowly can have as much motion as a light ball moving fast. A body has its own inertia – a stubborn will to continue as before. If it stands still, it will be still. If it is moving, it will keep moving. Only a force from outside can change that. Some forces always point toward a center. That is how a hand holds a stone in a sling. That is how gravity holds a stone you throw, so its path is curved. Such forces are called centripetal, 'center-seeking'. Newton also needs clear words about time and space. Clocks and calendars can run a little differently. Yet there is a steady time that flows by itself, he believes. And even though we measure places by things around us, there is a space that does not change. What is this good for? To know what real motion is. Two signs reveal it. First, the bucket experiment: Set a bucket spinning. When the water also spins, its surface becomes concave. That does not happen because the water rubs against the bucket, but because it truly rotates. Second, two balls tied by a cord in empty space: if the cord becomes taut, the balls really go in a circle around their common center. With this in place come three simple, great laws: Everything continues in a straight line at constant speed unless something pushes it. Change in motion happens in the direction of the force, and is larger if the force is larger. And for every action there is an equal and opposite reaction. From this follow many small wonders, like that the whole group of bodies has a common center of gravity that moves smoothly when nothing outside pulls.

How do we connect forces and curved paths? Newton builds a quiet ladder of small steps. He divides areas under curves into thin strips and lets them become so narrow that the outside and inside eventually become equal. He lets short chords, arcs, and tangents meet in an almost-zero point. He shows that if a force acts little by little, what we add grows with the square of the time at the start. That sounds dry, but see what he wins: If a body sweeps out equal triangle areas in equal times around a fixed point, then a force always acts toward that point. And the other way: If such a force always acts, then the body sweeps out equal areas in equal times. This is the heart of a whole new compass: We can look at planets and moons, measure how they sweep out areas around something, and know that a force points there. If we see the planets sweeping equal areas around the Sun, a force points toward the Sun. If we see moons sweeping equal areas around their planets, a force points toward them. From this place in the story, everything is decided by two questions: How strong is the pull, and how does the strength change with distance? When we follow these two lines to the end, we get the shape of the paths, the speed along them, and finally a law that binds together an apple, the Moon, and a comet. Thus small triangle areas with a tip at a center become a big key: an invisible hand that always points inward.

Kepler had seen that the planets move in ellipses around the Sun, and that the time they take is related to the size of the orbit in a certain way. Newton asks: what kind of force law gives such times? He points to a simple rule: If the period in orbit grows so that the square of the period follows the cube of the average distance, then the holding force must weaken with the square of the distance. Twice as far away, a quarter as strong force. This is the famous inverse-square law. It is not just pretty. It creates specific paths: Bound motion becomes an ellipse with the central force at one focus. Escaping paths become parabolas or hyperbolas, also with the focus where the force comes from. Newton goes both ways to be sure. Put a body in such a law – 1 divided by the distance squared – and the path becomes a conic section: ellipse, parabola, or hyperbola. And if you already know that the path is one of these, you can calculate back to find that the force must weaken as the square of the distance. Kepler's three laws now do not fall from the sky as nice patterns, but appear as necessary consequences of a simple force law and the area law. It is like hearing two melodies merge and realizing they are the same song. This insight, supported by the small triangle areas, becomes the very mountain of the book. From up there Newton no longer sees just paths as drawings on paper. He sees a mechanics that can explain what will happen to a planet, a moon, or a comet when we give it a small push or when a distant neighbor pulls weakly on it.

What if we want to replay the movie from a moment in the path? We know a point, know which way the tangent points and how fast the body is going right there. Under a known force law, what is the whole path? Or the other way: If we have seen an arc in the sky and know roughly where the center lies, what kind of force profile is needed for that to happen? Newton builds a toolbox for such questions. He finds ways to construct the center from speed measurements along a curve. He shows how to draw an ellipse when one focus is known, or without any focus given, if you have the right set of points and tangents. He uses clever changes of the picture, where parallel lines become families of lines pointing to a distant point, and difficult shapes become simpler when seen from another angle. All this is like a handbook for drawing and measuring the universe's paths with ruler, compass, and patience. You do not need to see the whole path at once. If you have the right pieces, you can build the whole. Thus you can start in the dark with a small strip of a comet's track and find the way to the Sun, and back out into the night. And if the path turns out to be a conic section, you already know what law was working in the background. This part of the story is not about big words, but about carpentry for the mind: line by line, angle by angle, and steadily closer to a full figure. Thus mechanics becomes useful, not just true.

When a body moves in an ellipse, the speed changes along the way. It hurries near the focus, where the central force is strongest, and slows when it moves away. Newton finds a simple relationship between the speed at a point and how that point lies relative to the tangent and the focus. In a parabola, the rule is even sharper: the speed grows like the inverse square root of the distance to the focus. He also compares speeds for orbits with an imaginary circle at the same distance from the center. With such comparisons he can bring out times for falling straight toward the center. When the orbit shrinks into a straight line, as if a conic section had collapsed, he can use area equalities between circle sectors, parabolas, and rectangular hyperbolas to read fall distances directly as areas under a helper construction. It sounds odd to read a length as an area, but it works because time and force are woven together in these figures. If you remember that the areas are like small clocks that tick equally, it becomes clear why: equal areas mean equal times. So when the speed at a point on a curve can be fixed by a simple length, and the time between points by an area, we suddenly have a pocket almanac for motions. It lets us say something about where and when a body will be, without calculating everything with numbers each time. In the background the inverse-square law keeps everything in check, while the area law hands out time with a steady hand.

What happens when nature almost follows the inverse-square law, but not quite? Then the apsides start to rotate. The apsides are the line that goes through the nearest and farthest points of an orbit. If the force law differs a little, this line glides around. Newton compares a real, slowly rotating ellipse with a completely fixed ellipse and looks at the difference in force at similar points. He adjusts continually, as if tuning a lens until the image is sharp. Under a force that is proportional to distance, like a spring, the shift between upper and lower apsis becomes a quarter turn. Under a force that follows 1 divided by r, the shift is a little over a third of a turn. When the law is nearly inverse-square but with small additions or subtractions, the shift is also small per revolution, but real. The Earth's Moon shows it clearly: its nearest point creeps forward slowly. In this analysis Newton finds that the extra force that distinguishes a completely fixed ellipse from one that slowly turns is of the order 1 divided by r to the third. That means tiny additions with a steeper law than the gravity law itself can explain small rotations of the ellipse axis without overturning the whole orbit. This is important, because nature is seldom perfectly tidy. Small grains of extra forces almost always exist: a third body that pulls a little, or a body that is not perfectly point-shaped. Then it is good to know how an ellipse responds: it gives a little and lets the line of apsides glide.

A force that points to a point in space can always be projected onto any plane. The projected force then points to the projection of the point, and the path in that plane is the same as if everything happened freely in the plane. This means motions in surfaces that cut through a rotating shape can be treated as pure plane problems. Newton uses this to understand beautiful curves, like the sphero-cycloid that a wheel rim draws when rolling on a sphere. Even more famous is the cycloid for a pendulum. When a pendulum bob is forced to swing in a perfect cycloid, the time for each swing is the same regardless of the size of the swing. This is called isochronous oscillation. It is as if a clock that is pulled further to the side still does not come home late. Newton finds times and speeds for such motions by comparing with a companion circle and using simple proportions between arc lengths and time. This part of the story may seem like a detour, but it shows a big idea. When you cannot directly solve a motion, you can find another you know and tie them together through a picture. You look at a difficult oscillation, and in the shadow you see a simple one. The goal is not to grind numbers for numbers, but to learn the taste of the universe: how it likes to do similar things in different places. Thus you recognize it when you meet it again.