At the end of last semester we talked
about the beginnings of modern science in Europe, in the 16th
and 17th centuries, and we saw that they can be characterized
as a separation of physics from metaphysics. Remember that metaphysics
meant those doctrines which dealt with the inner, active principles
in things, principles which are usually hidden. Thus, for example,
it was thought that celestial bodies had an inner, active principle
in them, which made those bodies move in circles, epicycles, etc.,
whereas the bodies down here, on earth, have an inner, active
principle (gravity), which makes them move down towards the center
of the earth. The principle of light is to illuminate, to make
things visible. Modern physics tries to do away with all those
inner, active principles, and tries to explain everything, every
phenomenon, whether on earth or in the heavens, by an analysis
of a single phenomenon: motion. Furthermore, modern physics seeks
to explain all phenomena on the basis of the motions of atomic
particles: like geometry, it starts from something called points,

whose only inner principle is a non-active one: to be indivisible,
that is, to be as small as possible.

The purpose of modern physics is to find the laws of all motions, and these laws must be written in the language of mathematics. As Galileo put it, "The book of Nature is written in mathematical language." This was as true in the 17th century as it is today. But to find the laws of motion we have to make sure that we can define the basic numerical properties of motion in a meaningful way. These properties are:

- the position of a particle or point at a certain time;
- the velocity of a particle;
- the acceleration of a particle.

Let us look closely at these properties one by one.

Regarding (1), the position of a particle, we saw last semester that Descartes provided us with a method to specify it, the so-called Cartesian coordinates. Let's start with the simple situation where a particle moves along a straight line, which we call the x axis. We pick a point on it and call it 0, "the origin," then pick a unit of measurement and mark it on the axis as the segment from 0 to 1. Thus, as Descartes observed, each point on our straight line will correspond to a "real number": those points on the side of 0 where our 1 is will be positive numbers, those on the opposite side will be negative. Now suppose that a particle is moving along this line: at a given time (call it t) it will be located on a certain point corresponding to a number x, and to make sure we understand this, we'll say that at that time t the particle is located at the point x(t). What we get here is a function, a correspondence: to each value of the variable t, which represents time, there corresponds a number x(t) which tells us where on the line the particle is located. It all seems very simple (although it really isn't, but we'll ignore the complications involved in concepts such as "real number," "instant of time," etc.) If our particle doesn't move on a straight line, but rather on a plane, then we will need a Cartesian system of two coordinates, two axes, x and y, instead of just one, and so we will need two functions of time, x(t) and y(t), to describe the motion of the particle. Similarly, in three-dimensional space we will need three functions, x(t), y(t) and z(t).

The problem starts when we try to define
(2), the velocity of the moving particle at a given time. Because
most of us drive cars and there's a speedometer in our car, we
think we know what velocity means. The problem, though, is not
that simple. Imagine you are driving and a cop stops you: "You
were going at 80 miles an hour!" he tells you. Well, you
might reply, "How can it be? I left home just 15 minutes
ago; I haven't been driving for an hour, so how could I be going
at 80 miles an hour?" At this point the cop may become quite
mad, even though your question was a good one. What does it mean
that you were going, when the cop spotted you, at 80 miles and
hour? One can say: it means that if you were to drive for an
hour at that same speed, you would cover 80 miles. Yes, but you
weren't going at the same speed all the time; you were not going
at a *uniform* speed; you went slow, then you went faster,
then... So, how are we going to define that statement: "At
12:35 (say) you were going at 80 miles per hour"? This needs
some Calculus, which was one of the great inventions of modern
science.

We might start by saying: that you were
going at 80 miles/hour at 12:35 means that between 12:30 and 12:35
you covered 80 miles divided by 12 (because 5 minutes is 1/12
of and hour), that is, 6.66 miles. But this is not good enough,
for in those five minutes, between 12:30 and the time the cop
spotted you at 12:35, you didn't go at a uniform speed; maybe
you were stepping on the gas, going faster and faster. So we
might say, okay, let's take the time interval between 12:34 and
12:35: what the cop claims is that in that one minute you covered
80/60 = 1.33 miles. Still, this is not good enough, for, again,
during that minute you were not going at a uniform speed. What
we need is a definition of *instantaneous* speed, the speed
you were going at exactly 12:35. And the way to define it correctly
is as follows: we take the distance covered in a *small*
time interval, call it h, before 12:35; this distance will be
x(12:35), your location at time 12:35, minus x(12:35-h), your
location at time 12:35-h. This distance, then, is

x(12:35) - x(12:35-h).

Now we divide this distance by the time you took to travel it, namely h, and we get the average velocity of your car during that interval of time, between 12:35-h and 12:35. To get the instantaneous speed at exactly 12:35, we must look at that ratio,

{x(12:35) - x(12:35-h)} / h

and make h smaller and smaller, without however ever letting h
be 0. This operation is called *taking the limit* of our
quotient as h goes to 0. The result is the instantaneous speed
at 12:35, by definition.

To give a concrete example, suppose
that instead of a car we consider a falling body, let's say a
ball which we drop from a height of 5 feet. If we take convenient
units of length (height) and time, the position of the ball at
time t will be given by the function x(t) = (1/2)t^{2}, one half
of time squared. Now we want to compute the instantaneous speed
of the falling ball at, say, time t = t_{o}. Proceeding as before,
we look at the quotient

{x(t_{o}-h) - x(t_{o})} / h = 1/2 {(t_{o}-h)^{2} - t_{o}^{2}} / h

Using high-school algebra, and dividing by h throughout, this is equal to:

1/2 {t_{o}^{2} - 2t_{o}h + h^{2} - t_{o}^{2}} / h = 1/2 {2t_{o} - h}

We said we would let h become smaller and smaller, and look at
what happens to our function. Well, this one is quite easy: as
h becomes closer and closer to 0, our function becomes closer
and closer to t_{o} (after canceling the 2's). So that's the instantaneous
speed of the falling body at time t_{o}: it is simply equal to t_{o}!
Of course, this is what Galileo discovered: the speed of a falling
body is proportional to the time elapsed since it was dropped.

This operation of looking at a quotient
(it's called the incremental quotient of a function) and letting
the quantity h approach 0, is called in Calculus taking the derivative
of the function. What we just did is to prove that the derivative
of the position function of the falling ball, x(t) = (1/2)t^{2},
is the velocity function v(t) = t. This is always the case: velocity
is the derivative of the position function. When we work in two
(or three) dimensions, though, as I said earlier there are two
(or three) position functions, x(t), y(t) and z(t). To get the
velocity we take the derivative of each of them, thus obtaining
two (or three) functions: x'(t), y'(t) and z'(t). These two or
three functions are called the *velocity vector*. In more
than one dimension, velocity is a vector!

Now it is easy to define property (3), the acceleration, or instantaneous rate of change of velocity. It is simply the derivative of the velocity. And again, in two or three dimensions the acceleration is a vector. Calculus doesn't only study derivatives, but also integrals. To integrate is the inverse operation of taking derivatives, much as subtraction is the inverse operation of adding. So by integrating the velocity function we can get the position function, and by integrating the acceleration we get the velocity.

We are in a position now to state Newton's
laws of motion. They were discovered by Isaac Newton (1642-1727)
in the years 1665-67, when he was about 24 years old, and they
involve two other concepts, that of *force* and that of *mass*,
which, as we'll discuss a little later, may look suspiciously
like metaphysical entities. But let's us first state Newton's
three laws of motion:

- The first law (also called the principle of inertia) describes what happens to a particle in the absence of forces: it will move with uniform (unchanging) velocity. This means more than meets the eye. In the first place, since in three dimensions velocity is a vector, to say that it is always the same means in particular that it has always the same direction, so the particle will be moving on a straight line and with uniform speed. Also, the case of a particle at rest is a particular case: "rest" just means that the velocity is 0.
- The second law tells us what happens when there are forces acting on the particle, and it brings up the concept of mass of the particle. It says that the force acting on the particle is proportional to the particle's acceleration. Again, this says more than what's apparent at first sight. In three dimensions (or two), both forces and accelerations are vectors, so saying that the force is proportional to the acceleration means, in particular, that they have the same direction! The constant of proportionality is called the mass of the particle. In math notation we have: F = ma. We must remember that F and a (acceleration) are vectors, but m (mass) is just a number.
- The third law (or principle of action and reaction) says that whenever a force F acts on a particle, there's another force, -F, equal in magnitude and direction but in the opposite sense. So, when I kick a stone, I feel in my foot a force equal to the one I impart on the stone, but in the opposite sense. Similarly, when I attach the stone to a string and swing it around, the acceleration vector of the moving stone points toward my hand (because the velocity vector moves toward the inside of the circle), and so therefore, by Newton's second law, the force too points toward my hand (centripetal force); this third law says that there is a force in the opposite direction, the centrifugal force, which I certainly feel.

In those same 18 months, while Newton
was in the countryside because the plague raged in the big cities,
he made another historical discovery. As if discovering Calculus
and the three laws of motion wasn't enough! This new discovery
was the law of gravitation. It states that whenever we have two
particles, with masses m_{1} and m_{2} respectively, and which are located
at two points with a distance r in between, there is a force on
each of the particles in the direction of the other (thus they
are attracted), and whose intensity is proportional to the product
of the two masses and inversely proportional to r squared. In
math notation, magnitude of F = Gm_{1}m_{2}/r^{2}, where G is a constant
(the gravitational constant): its numerical value, of course,
depends on the units of measurement that we choose.

Two remarks on this: first, it's not
at all clear that the masses we are using when we state the law
of gravitation are the same as those which appear in the statement
of Newton's second law; still, experimental evidence shows that
they are the same. Second, when we attribute this strange attractive
property to massive particles, aren't we indulging in metaphysics?
For we are saying, indeed, that matter has a inner, active principle:
matter attracts matter. At the time, physicists (who called themselves
"natural philosophers") accused Newton of doing exactly
that, indulging in metaphysics, and the followers of Descartes
(mostly in France) couldn't stomach the law of gravitation. What
can we say in Newton's defense? Well, surely he was indulging
in metaphysics, but with a difference: he wasn't just saying,
like others had been doing for centuries, that things have an
inner, active principle and leaving it at that; he gave a mathematical
law for that inner, active principle. That made a lot of difference.
He abstained from answering the metaphysical question, "What
*is* this attractive force?" Rather, he just gave a
mathematical formula for it. Still, the main reason for the acceptance
of Newton's gravitation was its tremendous success. As the saying
goes, nothing succeeds like success.

Let's explain the success of Newton's
gravitation. First, he succeeded in going from particles or mass
points attracting each other to big bodies such as the earth and
the moon. To do this he used integration, the Calculus operation
mentioned before, which allowed him to add up the contributions
of very small (infinitesimal) portions of matter; the net result
is that a body like the earth exerts a gravitational attraction
on other outside bodies which is as if all the mass of the earth
were concentrated at its center of gravity (in a spherical and
homogeneous body that would be its geometrical center). The details
of this proof are too technical for this lecture. Then, again
using Calculus, he derived all of Kepler's laws for planetary
motion (which we saw last semester) from the law of gravitation:
so he proved that planets move in ellipses (or, more generally,
conic sections) with the sun at one focus, that the area velocity
is constant, and that the square of the periods of revolution
of the planets are proportional to the cube of their mean distance
from the sun. Newton also managed to explain the tides as a consequence
of the gravitational pull of the moon on the waters of the earth
oceans. This is not as simple as it seems: you should try to
explain, yourselves, why there's a high tide not only on that
portion of the earth facing the moon, but also on the opposite
side. In any case, all kinds of phenomena, including the pendulum
studied earlier by Galileo, were for the first time derived mathematically
from simple formulas: the law of gravitation plus Newton's three
laws of motion. That was no mean achievement, and the world couldn't
help but take notice. Many thinkers, including Descartes, had
maintained it is thanks to God that the universe keeps going;
after Newton, the idea became more and more popular that if God
was necessary, it was only to give the universe its initial impulse;
after that, it kept on going according to the laws of gravitation
and motion. The most usual metaphor was a clock: God was the
clockmaker, and the one who wound up the mechanism, but his further
intervention was not needed. By around 1800, however, even the
initial divine intervention had been abandoned: when Laplace,
a great French astronomer and mathematician, presented his *Celestial
Mechanics* to Napoleon, the ruler asked, "Where is God
in all this?", and the scientist replied, "Sire, I had
no need of that hypothesis."

For reasons we cannot get into, Newton
did not publish his discoveries until much later, in 1687, in
a book titled (in Latin) *Philosophia Naturalis Principia Mathematica*,
or Mathematical Principles of Natural Philosophy, probably the
greatest book in the history of science. Meanwhile, another great
scientist, the German Gottfried W. Leibniz (1646-1716) had (whether
independently from Newton or not is still a matter of bitter contention)
discovered the essentials of Calculus. Leibniz' Calculus had
one advantage over Newton's: notation. This may seem a small
matter, but giving good names and symbols to mathematical concepts
can be of great help in thought and computations. In any case,
the fight over the priority in the discovery embittered both men,
and had a great influence over the subsequent history of math.
There is another aspect of the difference between Newton's and
Leibniz' thought which left deep traces, becoming an important
difference between English and Continental (European) science
and philosophy. Newton was content to demonstrate that phenomena
occur according to laws formulated in math language, without inquiring
too much into what is force, what is mass, and especially how
can two masses attract each other at a distance, without contact.
Leibniz, instead, was one of the greatest metaphysicians, who
could not accept an explanation unless he understood those thorny
questions: "What is it?" and "How?" So, for
example, he maintained that gravitation is caused by the motions
of a very subtle substance, rather like light, which he called
"ether." This ether was to have an agitated life among
physicists, until, by the end of the 19th century, it was discovered
experimentally that we cannot say much about it; we cannot even
measure its velocity. And so ether was dropped, probably for
good. We will return to that by the end of the semester.

Newton studied the phenomenon of light,
and published his *Optics*, at about the same time he was
occupied with the invention of Calculus and the laws of motion.
Among other experiments, he decomposed a ray of white light into
the colors of the rainbow by making it pass through a glass prism,
then he recomposed the different colors back into white light.
In relation to light, we must add that before Newton, the French
mathematician Pierre de Fermat (1601-1665) had stated the principle
that bears his name: light travels always on a path which minimizes
the time it takes to go from a point P to a point Q. Naturally,
if light travels on a homogeneous medium without any obstacles,
it will go in a straight line (assuming space is Euclidean); but
when we have different media, for example first air then glass,
then air or another type of glass, etc., Fermat's principle lets
us derive the laws of reflection and refraction about which we
talked in the first semester. This principle of Fermat is quite
puzzling if we put ourselves in a metaphysical frame of mind:
when light starts from P, how does it know which of the infinitely
many paths will take it to Q in the minimum possible time? Yet
that's the way things happen.

Finally, let's go back to the motion
of an object on a straight line. As we saw last semester, Zeno
of Elea formulated the first paradoxes about motion. One of them
is the so-called Achilles and the turtle. Remember that Achilles
runs a foot race with the turtle, but he of course runs much faster
than the animal, so he gives it a headstart. Lets us say, to
simplify matters, that Achilles starts one mile behind the turtle,
and that his speed is constant, 2 miles per hour, while the turtle
runs at a constant speed of 1 mile per hour. Let us compute the
times. It takes Achilles 1/2 hour to get to the point where the
turtle started, meanwhile the turtle will have advanced 1/2 mile
and is now located at the point T_{1}, 1/2 mile ahead of Achilles.
Then it takes Achilles 1/4 of an hour to get to that point T_{1},
but the turtle will then be located at a point 1/4 of a mile ahead
of him, at the point T_{2}. It takes Achilles 1/8 of an hour to
get to that point, meanwhile the turtle... and so on. If we add
up all those times we get the following: 1/2 + 1/4 + 1/8 + ...
The problem is that the sum goes on and on: although the terms
becomes smaller and smaller, there are infinitely many of them.
One of the great achievements of the Calculus invented by Newton
and Leibniz is that it allows us to compute such an infinite sum,
which is called an infinite series.

How do we do it? First, observe that
each term is obtained from the previous one by multiplying by
1/2: this is called a geometric series. Now, one learns in high
school (we won't do it here, but you may look it up or ask) that
when we have a *finite* geometric series such as: c + c^{2}
+ c^{3} + ... + c^{n}, this sum is equal to: c(1 - c^{n})/(1 -c). In
the case we are considering c is 1/2, so if we add the first n
items we have, that is, if we add our terms beginning with 1/2
and stopping at 1/2 to the power n, we get: 1/2(1 - 1/2^{n})/(1 -1/2).
Since 1 - 1/2 = 1/2, simplifying we obtain simply: 1 - 1/2^{n}.
But remember, we are only adding n terms, and n may be very big,
but this is still a finite sum. How do we get the *infinite*
sum? Again, the solution is what's called *taking the limit*.
We ask, what happens to our expression, 1 - 1/2^{n}, as n gets bigger
and bigger? And the answer is: as n gets bigger and bigger, 1/2
to that power gets smaller and smaller, that is, closer and closer
to 0. The limit of our expression is therefore equal to 1. In
other words, our infinite sum, starting with 1/2 and adding all
the powers of 1/2, is equal to 1! Or, yet in other words, it
takes Achilles exactly one hour to overtake the turtle. Leibniz
was a master at adding infinite series quite harder than the one
we just did. Next lecture we will have more to say about such
infinite series.

Chapters 7, 8 and 9 of Richard Feynman's *Lectures on Physics*,
Volume 1.