## Sunday, April 14, 2013

### History of Science -- Part Four: Isaac Newton

Isaac Newton was born in 1642, the year that Galileo died. With the wide acceptance of the experimental method, there was a sense of scientific progress, though Aristotle’s erroneous physics was still being taught. The Royal Society of London, a major scientific organization, was founded in 1660. Its motto, Nullis in verba, translates loosely as, “Take nobody’s word for it.” It would have delighted Galileo.

Newton, a handy fellow, was supposed to take over the family farm. But more interested in books than plows, he managed to go to Cambridge University by working at menial tasks to help pay his way. He did not shine as a student, but science fascinated him. In those days it was called “natural philosophy.” When the Great Plague forced the university to close, Newton returned to the farm for a year and one-half. And what a time it was, for Newton and the rest of the world.

Young Newton understood Galileo’s teaching that on a perfectly smooth horizontal surface a block, once moving, would slide forever. A force is needed only to overcome friction. With a greater force, the block would speed up, it would “accelerate.” Galileo, however, accepted the Aristotelian concept that falling was “natural” and needed no force. He also had planets moving “naturally” in circles without force. Galileo just ignored the ellipses discovered by his contemporary, Kepler. To conceive his universal laws of motion and gravity, Newton had to move beyond Galileo’s acceptance of Aristotelian “naturalness.”

Newton tells that his inspiration came as he watched an apple fall. He likely asked himself: Since a force was needed for horizontal acceleration, why not a force for vertical acceleration? And if there’s a downward force on an apple, why not on the moon? If so, why doesn’t the moon fall to Earth like the apple?

His solution was very simple. Here’s a picture. Assume you have a cannon on a mountain. If you drop a cannon ball, it will fall straight down. If you shoot it out horizontally, it will travel a great distance, but still fall to the ground. In fact, it will hit the ground at the same time as the dropped ball if you are shooting it short distances and assume the Earth is flat. But what if you shoot it for a longer distance? Newton knew, as we do, that the Earth isn’t flat. So if you shoot the cannon ball far enough, it will actually fall a ways around the round Earth. If fact, if you shoot the ball fast enough, it will actually miss the Earth, falling around it the whole time. It will move horizontally around the Earth, continuing to fall, and actually be in orbit. In fact, it will come all the way back around and hit the cannonier in the back of the head.

Of course, on the surface of the Earth, air resistance would rob the cannon ball of velocity and it would eventually fall to Earth. But on a planet without air, like the moon, this would actually work. Here on Earth we just shoot our satellites up above the atmosphere with enough velocity that they fall around the Earth in a stable orbit.

The moon doesn’t crash to Earth only because it, like that fast cannon ball, has a velocity perpendicular to Earth’s radius. Newton realized what no one had before: The moon is falling.

Galileo thought that uniform motion without force applied only to motion that was parallel to the surface of the Earth, in a circle about the Earth’s center. Newton corrected this to say that a force is needed to make a body deviate from a constant speed in a straight line.

This is Newton’s first law of motion, often called “Inertia”: A body at rest will remain at rest unless acted upon by an outside force, and a body in motion will remain in motion (in a straight line) unless acted upon by an outside force. So the moon would keep going in a straight line if it were not for the force drawing it to the Earth: gravity.

Further, Newton figured out just how much force is needed. The more massive the body, the more force required to accelerate it. Newton speculated that the force needed was just the mass of the body times the acceleration produced or F = ma. That is Newton’s second law. Force equals mass times acceleration.

If you measure the mass in kilograms and the acceleration in meters per second per second (I’ll explain later), then the force is one Newton in honor of Sir Isaac. This is Newton’s universal law of motion. Note the first law is really a statement of the second law when F = 0.

Let’s explore a little mathematics and define some terms that may not be clear. Let’s start with “velocity,” or as it is commonly known “speed.” Velocity is the rate of change of location or distance. On your car speedometer you read “miles per hour.” At 60 mph you are traveling 60 miles in an hour.

Here is a precise way to calculate that. You might be on a highway with mile markers and you have a watch with a second hand. So you note the starting mile post and the time. Call the milepost x0 and the time t0. The zero is the start time and start distance. Drive for some time and then note the new milepost (x1) and time (t1). You can now calculate the average speed or velocity by calculating the distance traveled and the time it took. The total distance is the final milepost minus the initial milepost and the time is calculated the same way. You then divide the distance by the total time. If the distance is in miles and the time is in hours, you will get “miles per hour.”

$v = \frac{x_{1} - x_{0}}{t_{1} - t_{0}}$

Where
v = velocity
x = distance
t = time

So velocity (or speed) is the rate of change of distance. The calculation shown here is the average velocity over a period of time. Newton explored the instantaneous velocity. If you travel at a constant speed, the average and instantaneous is the same value. But what if speed is increasing, like a drag racer that starts from standing still and accelerates over the quarter mile? Newton had to invent a new kind of mathematics to perform these instantaneous calculations. We now call that new math “The Calculus.” In Calculus, the rate of change is called the “derivative.” Newton would show the derivative with respect to time by putting a dot over the variable letter.

$v = \dot{x}$

In Calculus “speak” we say that velocity is the first derivative of distance or location. Acceleration is the rate of change of velocity, so acceleration, then, is the second derivative of distance.

If you are measuring velocity in feet per second, then acceleration would be “feet per second per second” or "per second squared."

Newton put two dots for this second derivative.

$a = \ddot{x} = \dot{v}$

So it was a busy year and one-half for Newton as he worked out the universal laws of motion, gravity, lunar orbits, and created the mathematics that would allow him to calculate all these values.

When the plague subsided, Newton returned to Cambridge. Isaac Barrow, then the Lucasian Professor of Mathematics, was soon so impressed with his one-time student that he resigned to allow Newton to take the Lucasian chair. The quiet boy became a reclusive bachelor. Newton was reserved and moody and was often angered by well-intended criticism. You’d rather spend an evening with Galileo.

Newton’s ideas needed testing. However, his force of gravity between objects that he could move about on the Earth was far too small for him to measure. So he looked to the heavens. Using his equation of motion and his law of gravity, he derived a simple formula. A chill, no doubt, ran down his spine when he saw that his formula was precisely the unexplained rule Kepler had noted decades earlier of the time it took each planet to orbit the sun.

Newton could also calculate that the orbital period of the moon was consistent with a falling object gaining a speed of ten meters per second each second — something experimentally shown by Galileo. His equations of motion and gravity governed apples as well as the moon — on Earth as it is in Heaven. Newton’s equations were universal.

Newton realized the significance of his discoveries, but controversy over the first paper he ever wrote had seriously upset him. The idea of publishing now terrified him.

Some twenty years after his insights on the farm, Newton was visited by the young astronomer Edmund Halley (of the famous comet). Knowing others were speculating on a law of gravity that would yield Kepler’s elliptical orbits for the planets, Halley asked Newton what orbits his law of gravity would predict. Newton immediately answered, “ellipses.” Impressed by the quick response, Halley asked to see the calculations. Newton could not find his notes. “While others were still seeking a law of gravity, Newton had already lost it.”

After Halley warned him that others might scoop him, Newton spent a furious eighteen months producing Philophiae Naturalis Principia Mathematica. What is now just referred to as Principia was published in 1687 with Halley footing the bill. Newton’s fears of criticism were realized; some even claimed he stole their work.

Though Principia was widely recognized as the profound revelation of Nature’s laws, being mathematically rigorous and in Latin, it was little read. But popularized versions soon appeared. Newtonianism for Ladies was a best seller. Voltaire, aided by his more scientifically talented companion, Madame du Chatelet, in his Elements of Newton claimed to “reduce this giant to the measure of the nincompoops who are my colleagues.”

The revealed rationality of Nature was revolutionary. It seemed to imply, in principle at least, that the world should be understandable as the mechanism of clocks. This was later dramatically demonstrated by Halley’s accurate predictions of the return of a comet. Until then, comets were commonly thought to foretell the death of kings.

Principia ignited the intellectual movement known as the Enlightenment. Society would no longer look to the Golden Age of Greece for wisdom. Alexander Pope captured the mood when he wrote, “Nature and Nature’s laws lay hid in night. / God said, Let Newton be! And all was light.”

When he needed better mathematics, Newton invented Calculus. His studies of light transformed the field of optics. He held a chair in Parliament then returned to Cambridge. He became Director of the Mint and took the position seriously. In his later years, Sir Isaac — the first scientist ever knighted — was perhaps the most respected person in the Western world. Paradoxically, Newton was also a mystic, immersing himself in supernatural alchemy and interpretations of Biblical prophecies.

Although there were some scary moments when mathematicians realized some subtleties in The Calculus such as “continuity requirements for Integration,” but all worked out in the end as an army of somewhat lesser minds filled in the missing spaces in Newton’s mathematical creation.

Newton did have a rival in the invention of Calculus. Recall that Newton didn’t publish his work for twenty years, and, during that time, a European named Gottfied Leibniz independently invented The Calculus. Unfortunately, that ultimately led to a feud between the two men and separated the scientific communities of England and the Continent for many years as each championed their “home boy.”

Sadly, Leibniz had created a better notation for Calculus that was more flexible and easier to use than the dot notation Newton favored. Since the English scientists adopted the “dot” in defense of Newton, it set English science back for nearly one hunderd years and advancements in Calculus exploded on the Continent.

The advance of science is not only dependent on brilliant ideas, but how well those ideas can be taught and spread to the public, and good notation always helps.

A similar occurrence will be discussed later when I describe the beginnings of Quantum Mechanics and how the original work in an awkward and little known mathematical method called “Matrices” was overshadowed by later work that used a simpler formulation and math. Science and the advance of science is often tied to its acceptance and understanding by scientists and the general public alike. Calling the Higg’s Boson “The God Particle” is part of the modern hype that I hope to reduce. But we’ve got to climb a few more foothills before we get to the top of that mountain.

My next chapter will expand on Newton’s legacy and the beginning of what is now called “Classical Physics” in deference to “Relativistic” and “Quantum Physics,” the twentieth century inventions. But first we need to cover the preceding three centuries as Kepler's, Galileo's, and Newton’s Genesis is brought to fruition.

#### 1 comment:

1. By the way, there is a third derivative of distance. That is the rate of change of acceleration. It is called "jerk."

Acceleration in an automobile pushes you steadily back into your seat. If the acceleration changes, such as taking your foot off the "accelerator" and then flooring it again, occupants will perceive "jerk."

If the driver slams on the breaks, occupants will perceive negative acceleration. Ain't physics grand.