The first step towards resolving 
everything was gathering precise data and 
the name most commonly associated with 
this new data gathering is Tycho Brahe 
who in 1580 makes very, very detailed 
observations of planetary motion with an 
unprecedented accuracy, and these are 
very useful 
Interestingly Brahe got the brilliant 
equipment and the patronage he needed to 
make these observations by making a 
failed attempt to prove that comets and 
other such unpredictable changes in the 
sky are in fact terrestrial events. 
Things that happen on earth, not far like 
say meteors or shooting stars. 
they're not, but this didn't sit well 
with current world views. 
The important thing for us is that he 
made these very, very precise 
measurements and some of them he shared 
with Johannes Kepler, who came to work 
for him as an assistant. 
And in 1609 after playing with the data 
and playing with the data, this was very 
precise data, close good enough that you 
could take into account the angular 
measurements including the position of 
earth relative to the planets in this 
heliocentric model that we're that Kepler 
was pursuing, 
precise enough that you could use these, 
these angular measurements to actually 
not just make a 3-dimensional model with 
ordering of the planets, but actually 
find predict the ratios of their sizes in 
other words you could tell whether Mars' 
orbit was seven times that of Earth or 
three times that of Earth in radius 
rather than just that it's larger. 
And with this precise data Kepler 
discovers much to his surprise that three 
very, very simple rules describe all of 
the motions of all of the planets and 
these rules are important so let's go 
through them. 
Kepler's First Law says, that the orbits 
of planets, the trajectories that planets 
follow are not circles. 
They're not circles upon circles. 
They're a simple geometric figure, he 
found this very surprising. 
They are ellipses. 
So what is an ellipse? 
Well, an ellipse, is an oval, squashed 
circle type thing, and it's characterized 
by two special points called the focii, 
the focus. 
then plural, the full side of the 
ellipse. 
And an ellipse, is a sum, is a set of all 
the points in the plane such that the sum 
of their distances from the two full 
side, is the same. 
So if I take any point on this ellipse, 
it's distance from this point, plus it's 
distance from that point are always the 
same and if I take these two points to be 
on top of each other then I am thus 
designing a circle because then those are 
just twice the distance from that one 
point, so if the two focal points on a 
top of each other then the ellipse is in 
fact a circle. 
So this is what an ellipses and Kepler 
first law says the orbits of all planets 
or ellipses such that one of the focal 
points is in fact the Sun. 
What's in the other focal point? 
Well, the simulation here gives you a 
hint. 
The other focus is the empty focus. 
There is nothing there. 
It has no physical significance. 
It's just a punked in space. 
It's not even the same for all the 
planets. 
So, the empty focus of Mercury's orbit is 
nowhere near Venus' orbit but they both 
share one focal point and that is where 
the sun is. 
So we need to know a little bit about the 
geometry of an ellipse. 
So again an ellipse is characterized you 
see as I run here by this effect, the sum 
of these two distances is always the same 
and I get some particular size and we 
characterize an ellipse by the size of 
its. 
Semi minor axis to the half of the 
diameter if you will, of the ellipse. 
Perpendicular, aligned through the 
center. 
Perpendicular to the line between the 
focii, that's the shorter axis. 
So that's why it's a minor axis and this 
is half of it. 
And then the semi-major axis, which is 
the longer radius. 
If this is a circle, they'd both be the 
radius. 
The longer one is the semi-major axis. 
That's along the line between the focii. 
And an eclip, ellipse is characterized, 
either by giving these two numbers, or 
alternatively, giving one of them. 
Typically we give the semi-major axis. 
And the parameter called eccentricity 
that measures how different the two 
axises are from each other. 
So eccentricity zero is when the focii 
are on top of each other, and you have a 
circle in fact, an eccentricity one which 
this won't allow me to achieve, is the 
case where the eclip, ellipse is so 
smashed down that it becomes a straight 
line, and the semi minor axis vanishes. 
the eccentricities of, the planets, you 
can reproduce the most eccentric 
planetary orbit is Mercury's, and that it 
has an eccentricity of about 0.2. 
The eccentricity of the earth is closer 
to 02. which means that it's very close 
to a circle indeed. 
This is roughly what the earth's orbit 
looks like. 
So this is Kepler's first law, the orbit 
of any planet is an ellipse, and the sun 
is at one of the focal point of that 
ellipse. 
The second law states that as a planet, 
as a planet that we again exaggerate the 
electricity of this, as a planet orbits 
the Sun Kepler says it sweeps out, the 
line between the planet and the Sun 
sweeps out equal areas in equal times, so 
this will give us a demonstration. 
This turns on a sweep for 
particular amount of time and I can start 
it whenever I want and the point is that 
what this means is the areas of all these 
triangular shapes have to be the same, 
the implication is that when the planet 
is near the Sun, it needs to be 
travelling faster so the stubby triangles 
are abroad whereas when the planet is 
farther from the Sun, it can traverse 
slower because the longer triangles can 
be kind of narrow. 
So not only do Kepler's laws tell you the 
shape of the orbit but also the relative 
speed with which the planet goes, 
it goes fastest when it's nearest the 
sun, this point is called perihelion, the 
point of nearest approach to the sun. 
And it's always along the major axis you 
can see, and at the other end of the 
major axis is the point of farthest 
distance from the sun, aphelion, which is 
where the planet moves most slowly. 
So these are Kepler's first two laws. 
And let's go back and summarize that. 
So, we said Kepler's First Law was that 
the orbit of every planet is an ellipse. 
And these ellipses for the actual planets 
in the Solar System are very close to 
being circles. 
the second law tells us how a, planet, 
moves faster or slower as it orbits the 
sun. 
This is this law of equal areas. 
The implication is that the planet moves 
fastest near perihelion and slowest near 
aphelion when it's far from the sun. 
This is not very dramatic for, say the 
earth. 
For objects that might orbit the sun in 
for more eccentric orbit, which are 
sometimes very near the sun and sometimes 
very far from the sun, say comets this 
effect is much more dramatic. 
So when we get to, more eccentric orbits, 
which exist in the universe, 
we will have to consider this, aspect of 
Kepler's second law more seriously. 
And this is enough to determine all the 
orbits of the planets. 
And this leads to agreement with 
planetary positions to a precision, by 
now there were precise observations, that 
neither the Copernician model as it was 
stated nor the Ptolemaic model could 
achieve. 
And this is a very simple, very 
straightforward geometric construction. 
So Kepler was sure that he was on to 
something. 
Now the third law is different in the 
sense that it does not address the motion 
of a planet. 
It's a relation between the orbits of 
the, the motions of the different 
planets. 
Remember that Kepler by now had enough 
information to measure the actual size of 
the semi-major axes of the planets, at 
least their ratios. 
So one could decide, let us call the 
semi-major axis of earth's ellipse one 
something. 
We call it one astronomical unit. 
So by convention, the definition of 
something called an astronomical unit is 
that the semi-major axis of an ellipse is 
typically called a. 
We will often call this R because 
whenever possible, we'll be talking about 
circles, and for a circle, the semi-major 
axis as we saw, is the radius. 
The semi-major axis of Earth's orbit 
defines a unit of distance called an 
astronomical unit. 
And so we could always measure distances 
in astronomical units, and later figure 
out how many meters are in an 
astronomical unit. 
Something that was difficult for Kepler 
to do, though he had some measurements, 
we'll talk later about how precisely we 
know what an AU is. 
And what Kepler knows is he then knows 
the semi-major axes for all of the 
planets in units of an astronomical unit, 
or at least he knows their ratios. 
And what he also knows, their sidereal 
periods, because he has the motion down, 
and because if you want we can get it 
from the synodic period using the 
Copernican calculation, although since 
things are not circles you have to be a 
little more careful. 
And he notices the following relation. 
The square of the period, for any planet, 
and the cube, of, the semi major axis, 
are related by a constant of 
proportionality. 
Now I can always find a number, such that 
P squared is K times A cubed. 
Just divide P squared by A cubed. 
The point is that if you divide P squared 
by A cubed, for each of the five planets 
that Kepler knows about, you get the same 
answer. 
And this is very important because this 
is then not a property of the planet, 
some of the planet, property of the solar 
system, there is something that all these 
planets share and it's the value of this 
ratio. 
There might be something interesting here 
and it's worth expressing the same 
perhaps a different way. 
So remember that we can write the period 
of a planet that it the, of the planet's 
orbit, the sidereal period can be related 
to let me imagine that we are only doing 
circular orbit so R and a are the same 
thing and I will call AR. 
And so I can write the period, relate the 
period to the radius of the orbit and the 
speed, which I will assume uniform 
because it's a circle with which, the 
planet orbits the sun. 
So the period is simply the circumference 
of the orbit, two pi R divided by the 
speed with which the planet moves. 
So this is my expression for the period 
and A is simply R in this case so 
Keppler's Law becomes, 
well, I'd have to square this thing so 4 
pi squared R squared over V squared is K 
times R cubed. 
And I have some enticing cancellations 
here. 
I can cancel R squared and leave just one 
power of R over here. 
And what I can find is an equation that 
we might find use for later that says, if 
you know how far from the sun a planet 
orbits, from this you can figure out its 
period. 
From this, you can figure out the speed 
with which it's moving multiplying by v 
squared and dividing by KR, I find that V 
squared is 4. 
pi squared over KR, a relation, to which 
we will have a time to return. 
And, what is the numerical value of K? 
Well, K, is a kind of funny physical 
constant the first that, of its kind but 
we'll meet many of them. 
Remember that A is a length in, meters or 
astronomical units or whatever it is, P 
is a length of time, so K is some number 
of perhaps seconds squared per meters 
cubed. 
It is a completely different number of 
hours where per mile cubed, K is the 
physical constant which depends on the 
units you are using, you can always pick 
your units of distance and time to make K 
equal to 1 if you so desire. 
it natural units it, that of that we will 
try to use, seconds and meters it most 
certainly is not one. 
But the important thing for us is that K 
whatever it is, is the same for all the 
planets and that is hinting at some deep 
underlying physics. 
Because it's something that is true for 
all of them. 
Setting this aside for a minute we have 
other progress. 
Galileo about a year after Kepler obtains 
his laws, gets his hands on the newest 
technology, a telescope, a spyglass. 
And we don't know if he was the first to 
do it. 
But he was certainly the first to publish 
this and to become known for it. 
He used a telescope to look up in the 
sky. 
And, he finds many, many, many, many, 
exciting things when he looks up at the 
sky with this newfangled invention. 
the first thing he finds is he finds that 
Venus, the planet Venus, goes through 
phases. 
So it's not surprising to us. 
It's a chunk of rock. 
It's illuminated by the sun. 
It's angled to the sun's changes. 
So Venus goes through phases, but in 
particular the relation between the 
phases of Venus and its position in the 
sky show that sometimes Venus is closer 
to Earth than the Sun is, and we get the 
effect, effectively a new Venus, just 
like the new moon works. 
But on the other hand, unlike the full 
moon, which happens when the moon is on 
the other side of Earth from the sun 
Venus is never that twelve hours away 
from the sun in right ascension, Venus is 
always near the sun remember. 
A full Venus happens when Venus is on the 
other side of the sun from Earth. 
And what this means is this does not fit 
in with the precise parameters of the 
Ptolemaic Model, but it fits very nicely 
into the, Keplerian or even the 
Copernician model. 
Also, he discovers that Jupiter has 
satellites. 
He sees the four Galilean Moons, or as 
he, seeking patronage, calls them, the 
Medician Stars. 
moving back and forth around the image of 
Jupiter and he realizes quickly that what 
he is seeing is something, a collection 
of bodies orbiting Jupiter. 
Well they're clearly not orbiting the 
Earth so this is an important. 
A piece of breaking down the cultural 
objection to the earth. 
Moving, he sees various exciting things, 
he sees mountains on the moon, he sees 
sun spots on the sun, he sees some weird 
structure on Saturn that later Huygens 
discovers are the rings of Saturn. 
so Galileo finds many things with his 
telescope and in particular the evidence 
of the faces of Venus is really the 
smoking gun that 
Dooms the geocentric picture. 
Or the Ptolemaic Geocentric picture as 
we, we will talk about later. 
Whether you think that the whole, the 
Earth is stationary or the whole universe 
is doing a complicated motion, or 
something else. 
At the end of the day, that's very, 
difficult and almost philosophical 
question. 
We'll see what we can have to say about 
why we prefer one description over the 
other, but in particular the fact that 
Venus is sometimes behind the Sun and 
sometimes in front of the Sun, that's now 
a fact. 
and this does not fit with the Ptolemaic 
Model. 
Galileo also, applied himself to 
beginnings of the science of mechanics, 
the study of motion, and is well known 
for, formulating the, the, what is called 
the Principle of Inertia. 
As he put it, an object will retain its 
state of motion unless disturbed 
externally. 
Now, we are all familiar with this in the 
following painful sense. 
If you have a refrigerator that's not 
moving. 
And you want to set it in motion, you 
want to change its state of motion from 
rest to motion, 
you have to expend quite a bit of effort 
to move that refrigerator. 
But Galileo reminds us is but then if you 
let it go the refrigerator will stop. 
What Galileo realized is that this is an 
artifact of rubbing against the floor and 
if you put the refrigerator on ice or on 
roller skates and started it moving it 
would be just as hard to stop it or 
change its direction of motion as it was 
to get it started in the first place. 
And that is an abstraction that is, 
cannot be understated. 
Galileo made many, many, many Brilliant 
deductions. 
But he never really got mechanics right, 
what was missing for Galileo. 
Among many things but the technical 
ingredient that was missing was that the 
mathematics required to discuss mechanics 
correctly hadn't been invented yet. 
real progress await Isaac Newton. 
And Newton also didn't find the right 
mathematics, it hadn't been invented yet 
so Newton just ahead and invented it, 
that was calculus and we will have to 
work around it because we don't use 
calculus. 
So we've made incredible progress. 
Look, the planets have and, and in 
general the heavens have been removed 
from some our spiritual realm of spheres 
and in all have objects that are a given 
distance and a given position in the 
universe, its a 3D world out there, there 
are moons that orbit the planets. 
This is all a physical system. 
We can study it. 
We can observe it. 
Galileo observes the moon. 
As a. 
Moon, something to be observed, not some 
heavenly object. 
we have Kepler's laws, and the 
predictions of Kepler's laws from such 
simplicity, such precision, is extremely 
compelling. 
Moreover, it turns out that these laws 
are a lot more universal than Kepler 
could have imagined. 
In fact they govern orbiting, they govern 
any kind of orbiting system. 
They govern the Solar System. 
They govern the various moons of Jupiter, 
as was discovered not long after he 
published his book. 
They govern every orbiting system, in 
some sense electrons in an atom. 
Each system is characterized perhaps by a 
different Kepler constant K. 
That was a property of our solar system 
but their relation, the fact that objects 
are ellipses and the central objects is 
at one of the focal points. 
And that the semi-major axis and the 
periods are related by P squared equals a 
constant times A cube for object, 
different electrons orbiting a nucleus, 
or different moons orbiting the same 
planet. 
This, it turns out, goes way beyond what 
Kepler put into his model comes out of 
it. 
And in physics, when something is that 
universal, it tells you that there's some 
fundamental law underlying it. 
There's a reason why all of these thing 
share a behavior and again, these 
underlying, 
physical laws of which Kepler was just 
finding the manifestation in the Solar 
System, were, waiting discovery by 
Newton, so we're now ready to discuss Sir 
Isaac.