Introduction to parametrizing a surface with two parameters

WEBVTT
Kind: captions
Language: en

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All the parameterizations
we've done so far have been

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parameterizing a curve
using one parameter.

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What we're going to start doing
this video is parameterizing a

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surface in three dimensions,
using two parameters.

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And we'll start with an
example of a torus.

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A torus, or more commonly
known, as a doughnut shape.

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And we know what a
doughnut looks like.

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Let me draw it in a suitable,
well, I don't have any

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suitable doughnut colors,
so I'll just use green.

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A doughnut looks
something like this.

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It has a hole in the center,
And maybe the other side of

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the doughnut looks something
like that, and we could

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shade it in like that.

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That is what a
doughnut looks like.

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So how do we construct that
using two parameters?

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So what we want to do, is you
can just visualize it, a

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doughnut, if you were to
draw some axes here.

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So that's our doughut.

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Let me draw some axes.

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So let's say I have the z-axis
that goes straight up and down.

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So we have drawn it here, the
doughnut's a little at a

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tilt, so the z-axis, I'll
tilt it a little bit.

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So our z-axis goes straight
through the center

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of the doughnut.

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So that right there, this is
going to be an exercise in

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drawing more than
anything else.

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So that is my z-axis, and then
you can imagine the z-axis goes

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from there, and then this
coming out of here

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will be my x-axis.

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That right there is my x-axis,
and then maybe my y-axis

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comes out like that.

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And the whole reason why I drew
it this way, is that if you

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imagine the cross section of
this doughnut, I'll draw a

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little bit neater, but the
cross section of this doughnut

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in the x-z axis, is going to
look something like this.

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If I were to slice it in the
x-z axis, it would look

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something like that.

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That would be the slice.

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It would trace out, and
we're thinking about not

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a full doughnut, just the
surface of a doughnut.

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So it would trace out
a circle like this.

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If you were to cut the doughnut
in the positive z-y axis, it's

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going to trace out a circle
that looks something

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like that, right there.

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And if you go out here, you're
going to get bunch of circles.

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So if you think about it, it's
a bunch of circles rotated

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around the z-axis.

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If you think of it that way,
it'll give us some good

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intuition for the best way
to parameterize this thing.

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So let's do it that way.

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Let's start off with
just the z-y axis.

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I'll draw it a little bit
neater than I've done here.

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So that is the z-axis, and
that is y, just like that.

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And let's say that the center
of these circles, let's say it

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lies on, you know, it can lie,
when you cross the z-y axis,

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the center sits on the y-axis.

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I didn't draw it that
neatly here, but I think

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you can visualize.

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So it sits right
there on the y-axis.

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And let's say that it is a
distance b away from the center

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of the torus, or from the
z-axis it's a distance of b.

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It's always going to
be a distance of b.

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It's always, if you imagine the
top of the doughnut, let me

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draw the top of the doughnut.

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If you're looking down on a
doughnut, let me draw a

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doughnut right here, if you're
looking down on a doughnut, it

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just looks something like that.

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The z-axis is just going to
be popping straight out.

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The x-axis would come down like
this, and then the y-axis would

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go to the right, like that.

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So you can imagine, I'm
just flying above this.

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I'm sitting on the z-axis
looking down at the doughnut.

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It will look just like this.

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And if you imagine the cross
section, this circle right

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here, the top part of the
circle if you're looking down,

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would look just like that.

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And this distance b is a
distance from the z-axis

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to the center of each
of these circles.

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So this distance, let me draw
it in the same color, from the

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center to the center of these
circles, that is going to be b.

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It's just going to keep
going to the center

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of the circles, b.

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That's going to be b,
that's going to be b.

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That's going to be b.

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From the center of our torus
to the center of our circle

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that defines the torus,
it's a distance of b.

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So this distance right here,
that distance right there is b.

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And from b, we can imagine
we have a radius.

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A radius of length a.

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So these circles have
radius of length a.

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So this distance right here is
a, this distance right here is

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a, this distance right there
is a, that distance

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right there is a.

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If I were look at these
circles, these circles

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have radius a.

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And what we're going to do
is have two parameters.

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One is the angle that this
radius makes with the x-z

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plane, so you can imagine
the x-axis coming out.

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Let me do that in
the same color.

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You can imagine the
x-axis coming out here.

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So this is the x-z plane.

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So one parameter is going to
be the angle between our

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radius and the x-z plane.

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We're going to call that angle,
or that parameter, we're

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going to call that s.

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And so as s goes between 0 and
2 pi, as s goes between 0 in 2

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pi, when the 0 is just going to
be at this point right here,

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and then as it goes to 2 pi,
you're going to trace out a

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circle that looks
just like that.

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Now, we only have
one parameter.

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What we want to do is then
spin this circle around.

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What I just drew is that
circle right there.

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What we want to do is spin
the entire circle around.

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So let's define
another parameter.

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We'll call this one t, and
I'll take the top view again.

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This one's getting a
little bit messy.

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Let me draw another top view.

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As you can see, this is
all about visualization.

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So let's say this is my
x-axis, that is my y-axis.

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And we said we started
here on the z-y plane.

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We are b away from the z-axis,
so that distance is b.

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In this diagram, the z-axis
is just popping out at us.

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It's popping out of the page.

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We're looking straight down.

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It's just like the same
view as right there.

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And what I just drew, when s is
equal to 0 radians, we're going

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to be out here, exactly
one radius further

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along the y-axis.

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And then we're going to rotate.

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As we rotate around, we're
going to rotate and then

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come all the way over here.

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That's when we're right over
there, and then come back down.

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So if you looked on the
top of the circle, it's

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going to look like that.

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Now, to make the doughnut,
we're going to have to

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rotate this whole setup
around the z-axis.

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Remember, the z-axis
is popping out.

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It's looking straight up at us.

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It's coming out of
your video screen.

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Now to rotate it, we're
going to rotate this

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circle around the z-axis.

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And to do that, we'll define a
parameter that tells us how

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much we have rotated it.

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So this is when we've
rotated 0 radians.

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At some point, we're going to
be over here, and we would

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have rotated it, this is b as
well, and our circle is going

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to be looking like this.

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This is maybe this point on
our doughnut, right there.

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At that point, we would
have rotated it,

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let's say p radians.

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So this parameter of how much
have we rotated around the

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z-axis, how much have we gone
around that way, we're

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going to call that t.

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And t is also going to
vary between 0 and 2 pi.

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And I want to make
this very clear.

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Let's actually draw the domain
that we're mapping from to

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our surface, so that we
understand this fully.

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So let me draw some, and then
we'll talk about how we can

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actually parameterize
that into a position

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vector-valued function.

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So right here, let's
call that the t-axis.

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That's, remember, how much
we're rotated around

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the z-axis right there.

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And let's call this
down here our s-axis.

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And I think this will help
things out a good bit.

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So when s is equal to 0, and
we vary just t, so they're

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both going to vary
between 0 and 2 pi.

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So this right here is 0,
this right here is 2 pi.

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Let me do some
things in between.

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This is pi, this would be pi
over 2 obviously, pi over 2,

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this would be 3 pi over 4.

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You do the same thing
on the p-axis.

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It's going to go up to 2 pi.

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Let's do that.

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So we're going to
go up to 2 pi.

00:09:03.820 --> 00:09:06.060
I really want you to visualize
this, because then the

00:09:06.060 --> 00:09:10.190
parameterization, I think, will
be fairly straightforward.

00:09:10.190 --> 00:09:16.240
So that's 2 pi, this is pi,
this is pi over 2, and

00:09:16.240 --> 00:09:20.890
then this is 3 pi over 4.

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So let's think about what it
looks like if you just hold s

00:09:23.930 --> 00:09:28.770
constant at 0, and we just
vary t between zero and 2 pi.

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And let me do that in
magenta, right here.

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So we're holding s constant,
and we're just varying

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the parameter 2 pi.

00:09:37.250 --> 00:09:39.800
So this, if you think about it,
should just form a curve in

00:09:39.800 --> 00:09:41.030
three dimensions,
not a surface.

00:09:41.030 --> 00:09:43.500
Because we're only varying
one parameter right here.

00:09:43.500 --> 00:09:45.370
So let's think about
what this is.

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Remember, s is, let
me draw my axes.

00:09:51.360 --> 00:09:58.330
So that is my x-axis, that
is my y-axis, and then

00:09:58.330 --> 00:10:01.970
this is my, I'm getting
messier and messier.

00:10:01.970 --> 00:10:05.630
That is my z-axis, right--
actually, let me draw it a

00:10:05.630 --> 00:10:06.860
little bit bigger than that.

00:10:06.860 --> 00:10:10.570
I think it will help all
of our visualizations.

00:10:10.570 --> 00:10:11.080
All right.

00:10:11.080 --> 00:10:20.210
So this is my x-axis, that is
my y-axis, and then my z-axis

00:10:20.210 --> 00:10:22.450
goes up like that. z-axis.

00:10:22.450 --> 00:10:25.360
Now remember, when s is equal
to 0, that means we haven't

00:10:25.360 --> 00:10:29.550
rotated around this
circle at all.

00:10:29.550 --> 00:10:31.300
That means we're out here.

00:10:31.300 --> 00:10:34.830
We're going to be b away,
and then a away again.

00:10:34.830 --> 00:10:35.050
Right?

00:10:35.050 --> 00:10:36.990
We haven't rotated
around this at all.

00:10:36.990 --> 00:10:39.630
We're setting s as
equal to zero.

00:10:39.630 --> 00:10:43.650
So essentially, we're going to
be b away, so this is going to

00:10:43.650 --> 00:10:46.800
be a distance of b away, and
then we're going to

00:10:46.800 --> 00:10:48.480
be another a away.

00:10:48.480 --> 00:10:50.310
The b is the center of the
circle, and then we're going

00:10:50.310 --> 00:10:51.300
to be another a away.

00:10:51.300 --> 00:10:52.550
We're going to be
right over there.

00:10:52.550 --> 00:10:54.730
So this is a plus b away.

00:10:54.730 --> 00:10:56.130
And then we're going to vary t.

00:10:56.130 --> 00:10:59.870
Remember, t was how much we've
gone around the z-axis.

00:10:59.870 --> 00:11:01.670
These were top views over here.

00:11:01.670 --> 00:11:07.640
So this line right here, in our
s-t domain, we can say, when we

00:11:07.640 --> 00:11:11.940
map it, or parameterize it,
it'll correspond to the curve

00:11:11.940 --> 00:11:13.980
that's essentially the outer
edge of are doughnut.

00:11:13.980 --> 00:11:16.690
If this is the top view of the
doughnut, it will be the outer

00:11:16.690 --> 00:11:18.800
edge of the doughnut,
just like that.

00:11:18.800 --> 00:11:20.690
So let me draw the outer edge.

00:11:20.690 --> 00:11:22.620
And to do that a little bit
better, let me draw the axes

00:11:22.620 --> 00:11:25.440
in both the positive and
the negative domain.

00:11:25.440 --> 00:11:29.140
It might make my graph a little
bit easier to visualize things.

00:11:29.140 --> 00:11:33.820
Positive and negative domain,
this is negative z right there.

00:11:33.820 --> 00:11:37.480
So this line in our t-s plane,
I guess we could say, this

00:11:37.480 --> 00:11:42.470
magenta line, we hold s at 0
radians and we increase t, this

00:11:42.470 --> 00:11:47.750
is t is zero, this is t is
equal to 2 pi, that's t is

00:11:47.750 --> 00:11:53.290
equal to pi, this is t is equal
to 3 pi over 2, all the way

00:11:53.290 --> 00:11:55.780
back to t is equal to 2 pi.

00:11:55.780 --> 00:12:00.470
This line corresponds to that
line, as we rotate, as we

00:12:00.470 --> 00:12:04.890
increase t and hold
s constant at 0.

00:12:04.890 --> 00:12:06.900
Now let's do another point.

00:12:06.900 --> 00:12:11.750
Let's say when s is at pi,
right, remember, when s is

00:12:11.750 --> 00:12:14.660
at pi, we've gone exactly,
pi is 180 degrees.

00:12:14.660 --> 00:12:17.430
When s is at pi, we've
gone exactly 180 degrees

00:12:17.430 --> 00:12:19.340
around the circle, around
each of these circles.

00:12:19.340 --> 00:12:20.810
So we're right over there.

00:12:20.810 --> 00:12:23.520
And now let's hold it constant
at pi, and then rotate it

00:12:23.520 --> 00:12:24.710
around to form our doughnut.

00:12:24.710 --> 00:12:27.540
So we're going to form the
inside of our doughnut.

00:12:27.540 --> 00:12:32.120
So when s is at pie, and we're
going to take t from 0, so when

00:12:32.120 --> 00:12:35.620
s is pi and t is 0, we're going
to be, this was the center of

00:12:35.620 --> 00:12:37.410
our circle, we're going
to be a below that.

00:12:37.410 --> 00:12:39.930
We're going to be
right over there.

00:12:39.930 --> 00:12:47.430
And then as we vary, as we
increase t, so as we move up

00:12:47.430 --> 00:12:54.620
along, holding s at pi, and we
increase t, we're going to

00:12:54.620 --> 00:13:02.070
trace out the inside of our
doughnut, that will look

00:13:02.070 --> 00:13:04.170
something like that.

00:13:04.170 --> 00:13:06.340
That was my best
shot at drawing it.

00:13:06.340 --> 00:13:08.050
And then we can do
that multiple times.

00:13:08.050 --> 00:13:11.660
When s is pi over 2, I want to
do multiple different colors,

00:13:11.660 --> 00:13:15.960
when s is pi over 2, we've
rotated up here exactly

00:13:15.960 --> 00:13:16.980
90 degrees, right?

00:13:16.980 --> 00:13:18.900
Pi over 2 is 90 degrees
at this point.

00:13:18.900 --> 00:13:21.500
And then if we vary t, we're
essentially tracing out the

00:13:21.500 --> 00:13:23.840
top of the doughnut, right?

00:13:23.840 --> 00:13:25.110
So let me make sure I draw it .

00:13:25.110 --> 00:13:26.990
So the cross section, the top
of the doughnut, we're going

00:13:26.990 --> 00:13:30.690
to start off right over here.

00:13:30.690 --> 00:13:34.910
So when s is pi over 2, and you
vary it right, and then you

00:13:34.910 --> 00:13:39.100
very t, I'm having trouble
drawing straight lines.

00:13:39.100 --> 00:13:42.500
And then you vary t, it's
going to look like this.

00:13:42.500 --> 00:13:44.980
That's the top of that
circle right there.

00:13:44.980 --> 00:13:47.420
The top of this circle is
going to be right there.

00:13:47.420 --> 00:13:50.490
The top of this circle is
going to be right over there.

00:13:50.490 --> 00:13:53.000
Top of the circle is going
to be right over there.

00:13:53.000 --> 00:13:54.300
So then I just
connect the dots.

00:13:54.300 --> 00:13:59.470
It's going to look
something like that.

00:13:59.470 --> 00:14:00.770
That is the top
of our doughnut.

00:14:00.770 --> 00:14:02.750
If I was doing this top view,
it would be the top of the

00:14:02.750 --> 00:14:04.100
doughnut, just like that.

00:14:04.100 --> 00:14:06.750
And if I wanted to do the
bottom of the doughnut, just to

00:14:06.750 --> 00:14:10.530
make the picture clear, if I
were to make the bottom of the

00:14:10.530 --> 00:14:13.350
doughnut, the bottom of the
doughnut would be-- see, if I

00:14:13.350 --> 00:14:17.210
take s as 3 pi over 4 and I
vary t, that's the bottoms

00:14:17.210 --> 00:14:17.850
of our doughnuts.

00:14:17.850 --> 00:14:21.870
So let me draw the circle, it's
right there, the circle is

00:14:21.870 --> 00:14:24.400
right there, you wouldn't be
able to see the whole thing

00:14:24.400 --> 00:14:26.950
if this wasn't transparent.

00:14:26.950 --> 00:14:30.660
So you'd be tracing out the
bottom of the doughnut,

00:14:30.660 --> 00:14:31.350
just like that.

00:14:31.350 --> 00:14:33.860
I know this graph is becoming
a little confusing, but

00:14:33.860 --> 00:14:34.860
hopefully you get the idea.

00:14:34.860 --> 00:14:38.470
When s is 2 pi again, you're
going to be back to the outside

00:14:38.470 --> 00:14:40.220
of the doughnut again.

00:14:40.220 --> 00:14:43.460
That's also going
to be in purple.

00:14:43.460 --> 00:14:46.480
So that's what happens when we
hold the s constant at certain

00:14:46.480 --> 00:14:48.730
values and vary the t.

00:14:48.730 --> 00:14:50.780
Now let's do the opposite.

00:14:50.780 --> 00:14:55.140
What happens if we hold t
at 0, and we very the s?

00:15:00.510 --> 00:15:03.610
So t is 0, that means we
haven't rotated at all yet.

00:15:03.610 --> 00:15:05.160
So we're in the z-y plane.

00:15:05.160 --> 00:15:16.690
So t is 0, and s will start at
0, and it'll go to pi over 2,

00:15:16.690 --> 00:15:18.180
that's that point over there.

00:15:18.180 --> 00:15:20.670
Then it'll go to pi.

00:15:20.670 --> 00:15:23.300
This point is the same
thing as that point.

00:15:23.300 --> 00:15:28.020
Then it will go to 3 pi over
4, then it'll come back

00:15:28.020 --> 00:15:29.550
all the way to 2 pi.

00:15:29.550 --> 00:15:34.080
So this line corresponds to
this circle, right there.

00:15:34.080 --> 00:15:43.270
We could keep doing these if we
pick when t is pi -- let me use

00:15:43.270 --> 00:15:46.290
a different color, that's
not different enough.

00:15:46.290 --> 00:15:50.870
When t is pi over
2, just like that.

00:15:50.870 --> 00:15:53.980
We would have rotated around
the z-axis 90 degrees,

00:15:53.980 --> 00:15:55.280
so now we're over here.

00:15:55.280 --> 00:15:58.710
And now when we vary s, s will
start off over here, and

00:15:58.710 --> 00:16:02.330
it'll go all the way
around like that.

00:16:02.330 --> 00:16:05.025
So this line corresponds
to that circle.

00:16:05.025 --> 00:16:08.180
We could keep doing
it like this.

00:16:08.180 --> 00:16:11.610
When t is equal to pi, that
means we've got all the way

00:16:11.610 --> 00:16:14.670
around the circle like that,
and now when we vary s from 0

00:16:14.670 --> 00:16:18.470
to pi over 2, we're going to
start all the way over here,

00:16:18.470 --> 00:16:20.590
and then we're going to vary,
all the way, we're going to go

00:16:20.590 --> 00:16:24.430
down and hit all those contours
that we talked about before,

00:16:24.430 --> 00:16:26.920
and I'll do one more, just
to kind of make this,

00:16:26.920 --> 00:16:29.510
the scaffold, clear.

00:16:29.510 --> 00:16:31.540
This dark purple,
hopefully you can see it.

00:16:31.540 --> 00:16:34.150
When t is 3 pi over 4,
we've rotated all the way.

00:16:34.150 --> 00:16:38.070
So we're in the x-z plane.

00:16:38.070 --> 00:16:41.900
And then when you vary s, s
will start off over here, and

00:16:41.900 --> 00:16:48.460
as you increase s, you're going
to go around the circle, around

00:16:48.460 --> 00:16:50.950
the circle, just like that.

00:16:50.950 --> 00:16:54.220
And of course, when you get all
the way back full circle, t

00:16:54.220 --> 00:16:55.810
over pi over 2, that's
the same thing.

00:16:55.810 --> 00:16:57.170
You're back over here again.

00:16:57.170 --> 00:17:01.620
So this is going to be, we can
even shade it the same color.

00:17:01.620 --> 00:17:03.000
And hopefully you're
getting a sense now of

00:17:03.000 --> 00:17:04.130
the parameterization.

00:17:04.130 --> 00:17:05.450
I haven't done any math yet.

00:17:05.450 --> 00:17:07.450
I haven't actually showed you
how to mathematically represent

00:17:07.450 --> 00:17:10.360
it as a vector value function,
but hopefully you're getting

00:17:10.360 --> 00:17:15.670
a sense of what it means to
parameterize by two parameters.

00:17:15.670 --> 00:17:23.080
And just to get an idea of what
these areas on our s-t plane

00:17:23.080 --> 00:17:30.350
correspond to onto this
surface, in I guess you could

00:17:30.350 --> 00:17:34.640
say, in R3, this little square
right here, let's see

00:17:34.640 --> 00:17:36.300
what's bounded by.

00:17:36.300 --> 00:17:38.900
This little square, I want to
make sure I picked a square

00:17:38.900 --> 00:17:41.100
that I can draw neatly.

00:17:41.100 --> 00:17:46.755
So this square right here, that
is between, when you look at t,

00:17:46.755 --> 00:17:50.070
it's between 0 and pi over 2.

00:17:55.340 --> 00:17:58.740
And s is between
0 and pi over 2.

00:17:58.740 --> 00:18:04.120
So this right here is
this part of our torus.

00:18:07.080 --> 00:18:11.840
If you're looking at it from
the top, it would look

00:18:11.840 --> 00:18:13.390
like that, right there.

00:18:13.390 --> 00:18:16.020
You can imagine, we've
transformed this square.

00:18:16.020 --> 00:18:18.560
I haven't even shown you how
to do it mathematically yet.

00:18:18.560 --> 00:18:22.130
But we've transformed
this square to this

00:18:22.130 --> 00:18:24.570
part of the doughnut.

00:18:24.570 --> 00:18:27.340
Now, I think we've done about
as much as I can do on

00:18:27.340 --> 00:18:28.530
the visualization side.

00:18:28.530 --> 00:18:30.490
I'll stop this video here.

00:18:30.490 --> 00:18:31.970
In the next video, we're going
to actually talk about, how

00:18:31.970 --> 00:18:35.290
do we actually parameterize
using these two parameters?

00:18:35.290 --> 00:18:39.380
Remember, s takes around each
of these circles, and then t

00:18:39.380 --> 00:18:41.090
takes us around the z-axis.

00:18:41.090 --> 00:18:43.690
And if you take all of the
combinations of s and t, you're

00:18:43.690 --> 00:18:46.050
going to have every point
along the surface of this

00:18:46.050 --> 00:18:47.470
torus or this doughnut.

00:18:47.470 --> 00:18:50.760
How do you actually go from an
s and a t that goes from 0 to

00:18:50.760 --> 00:18:56.030
2 pi, in both cases, and turn
it into a three-dimensional

00:18:56.030 --> 00:19:00.130
position vector-valued function
that would define this surface?

00:19:00.130 --> 00:19:02.280
We're going to do that
in the next video.


Determining a position vector-valued function for a parametrization of two parameters

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:02.830
In the last video, we started
to talk about how to

00:00:02.830 --> 00:00:06.840
parameterize a torus,
or a doughnut shape.

00:00:06.840 --> 00:00:09.670
And the two parameters we were
using, and I spent a lot of

00:00:09.670 --> 00:00:11.750
time trying to visualize
it, because this is all

00:00:11.750 --> 00:00:12.660
about visualization.

00:00:12.660 --> 00:00:14.950
I think this is really the
hard thing to do here.

00:00:14.950 --> 00:00:19.490
But the way we can parameterize
a torus, which is the surface

00:00:19.490 --> 00:00:23.860
of this doughnut, is to say say
hey, let's take a point let's

00:00:23.860 --> 00:00:25.790
rotate it around a circle.

00:00:25.790 --> 00:00:26.580
It could be any circle.

00:00:26.580 --> 00:00:29.000
I picked a circle
in the z-y plane.

00:00:29.000 --> 00:00:32.670
And how far it's gone around
that circle, we'll parameterize

00:00:32.670 --> 00:00:37.320
that by s, and s can go between
0 all the way to 2 pi, and then

00:00:37.320 --> 00:00:39.990
we're going to rotate this
circle around itself.

00:00:39.990 --> 00:00:42.480
Or I guess a better way to say
it, we're going to rotate the

00:00:42.480 --> 00:00:45.860
circle around the z-axis, and
it's all at the center of the

00:00:45.860 --> 00:00:49.290
circle, so we're always going
to keep a distance b away.

00:00:49.290 --> 00:00:51.800
And so these were top
views right there.

00:00:51.800 --> 00:00:55.590
And then we defined our second
parameter t, which tells us how

00:00:55.590 --> 00:00:59.180
far the entire circle has
rotated around the

00:00:59.180 --> 00:00:59.920
z-axis access.

00:00:59.920 --> 00:01:03.150
And those were our two
parameter definitions.

00:01:03.150 --> 00:01:05.780
And then here we tried to
visualize what happens.

00:01:05.780 --> 00:01:08.350
This is kind of the domain
that our parameterization

00:01:08.350 --> 00:01:09.820
is going to be defined on.

00:01:09.820 --> 00:01:14.140
s goes between 0 and 2 pi,
so when t is 0, we haven't

00:01:14.140 --> 00:01:16.870
rotated out of the z-y plane.

00:01:16.870 --> 00:01:20.910
s is at 0, goes all the
way to 2 pi over there.

00:01:20.910 --> 00:01:26.510
Then when t goes to 2 pi, we've
kind of moved our circle.

00:01:26.510 --> 00:01:29.460
We've moved it along,
we've rotated around

00:01:29.460 --> 00:01:30.970
the z-axis a bit.

00:01:30.970 --> 00:01:34.950
And then this line in our s-t
domain corresponds to that

00:01:34.950 --> 00:01:40.490
circle in 3 dimensions,
or in our x-y-z space.

00:01:40.490 --> 00:01:43.650
Now given that, hopefully we
visualize it pretty well.

00:01:43.650 --> 00:01:46.540
Let's think about actually how
to define a position

00:01:46.540 --> 00:01:51.520
vector-valued function that is
essentially this

00:01:51.520 --> 00:01:53.140
parameterization.

00:01:53.140 --> 00:01:55.360
So let's first to do
the z, because that's

00:01:55.360 --> 00:01:57.660
pretty straightforward.

00:01:57.660 --> 00:01:59.860
So let's look at this
view right here.

00:01:59.860 --> 00:02:02.580
What's our z going to
be as a function?

00:02:02.580 --> 00:02:06.240
So our x's, our y's, and
our z's should all be

00:02:06.240 --> 00:02:08.690
a function of s and t.

00:02:08.690 --> 00:02:10.780
That's what it's all about.

00:02:10.780 --> 00:02:13.340
Any position in space should
be a function of picking a

00:02:13.340 --> 00:02:15.300
particular t and
a particular s.

00:02:15.300 --> 00:02:16.800
And we saw that over here.

00:02:16.800 --> 00:02:19.380
This point right here, let
me actually do that with

00:02:19.380 --> 00:02:20.820
a couple of points.

00:02:20.820 --> 00:02:23.690
This point right there,
that corresponds to that

00:02:23.690 --> 00:02:25.790
point, right there.

00:02:25.790 --> 00:02:26.970
Then we pick another one.

00:02:26.970 --> 00:02:30.660
This point right here,
corresponds to this

00:02:30.660 --> 00:02:33.100
point, right over there.

00:02:33.100 --> 00:02:34.770
I can do a few more.

00:02:34.770 --> 00:02:36.180
Let me pick.

00:02:36.180 --> 00:02:40.490
This point right here,
so s is still 0.

00:02:40.490 --> 00:02:44.580
That's going to be this outer
edge, way out over there.

00:02:44.580 --> 00:02:47.770
I'll pick one more, just
to define this square.

00:02:47.770 --> 00:02:50.350
This point right over here,
where we haven't rotated t at

00:02:50.350 --> 00:02:53.800
all, but we've gone a quarter
way around the circle, is

00:02:53.800 --> 00:02:54.920
that point right there.

00:02:54.920 --> 00:02:58.400
So for any s and t we're
mapping it to a point

00:02:58.400 --> 00:03:00.530
in x-y-z space.

00:03:00.530 --> 00:03:03.950
So our z's, our x's, and
our y's should all be

00:03:03.950 --> 00:03:06.910
a function of s and t.

00:03:06.910 --> 00:03:09.690
So the first one to think
about is just the z.

00:03:09.690 --> 00:03:13.020
I think this will be
pretty straightforward.

00:03:13.020 --> 00:03:19.940
So z as a function of s and
t is going to equal what?

00:03:19.940 --> 00:03:25.080
Well, if you take any circle,
remember s is how the angle

00:03:25.080 --> 00:03:28.250
between our radius
and the x-y plane.

00:03:28.250 --> 00:03:29.795
So I can even draw
it over here.

00:03:29.795 --> 00:03:33.350
Let me do it in another color.

00:03:33.350 --> 00:03:34.910
I'm running out of colors.

00:03:34.910 --> 00:03:37.550
So let's say that this is
a radius, right there.

00:03:37.550 --> 00:03:40.750
That angle, we said, that is s.

00:03:40.750 --> 00:03:46.110
So if I were to draw that
circle out, just like

00:03:46.110 --> 00:03:47.990
that, we can do a little
bit of trigonometry.

00:03:50.840 --> 00:03:52.970
The angle is s.

00:03:52.970 --> 00:03:56.950
We know the radius is
a, the radius of our

00:03:56.950 --> 00:03:58.890
circle, we defined that.

00:03:58.890 --> 00:04:04.760
So z is just going to be the
distance above the x-y plane.

00:04:04.760 --> 00:04:06.640
It's going to be this
distance, right there.

00:04:06.640 --> 00:04:09.900
And that's straightforward
trigonometry.

00:04:09.900 --> 00:04:12.275
That's going to be a, I mean,
we can do SOCATOA and all

00:04:12.275 --> 00:04:14.620
of that, you might want
to review the videos.

00:04:14.620 --> 00:04:18.320
But the sine, you can
view it this way.

00:04:18.320 --> 00:04:24.210
So if this is z right there,
you could say that the sine of

00:04:24.210 --> 00:04:28.190
s, SOCATOA is the opposite
over the hypotenuse, is

00:04:28.190 --> 00:04:30.750
equal to a z over a.

00:04:30.750 --> 00:04:37.020
Multiply both sides by a, you
have a sine s is equal to z.

00:04:37.020 --> 00:04:41.790
That tells us how much above
the x-y plane we are.

00:04:41.790 --> 00:04:43.180
Just some simple trigonometry.

00:04:43.180 --> 00:04:46.970
So z of s and t, it's only
going to be a function of s.

00:04:46.970 --> 00:04:52.740
It's going to be a
times the sine of s.

00:04:52.740 --> 00:04:54.110
Not too bad.

00:04:54.110 --> 00:04:57.420
Now see if we can figure out
what x and y are going to be.

00:04:57.420 --> 00:04:58.500
Remember, z doesn't matter.

00:04:58.500 --> 00:05:02.250
Doesn't matter how much we've
rotated around the z-axis.

00:05:02.250 --> 00:05:04.980
What matters is, how much we've
rotated around the circle.

00:05:04.980 --> 00:05:07.820
If s is 0, we're just going
to be in the x-y plane,

00:05:07.820 --> 00:05:11.310
z is going to be zero.

00:05:11.310 --> 00:05:14.140
If s is pi over 2, up here,
then we're going to be

00:05:14.140 --> 00:05:16.190
traveling around the
top of the doughnut.

00:05:16.190 --> 00:05:19.805
And we're going to be exactly
a above the x-y plane, or z

00:05:19.805 --> 00:05:20.700
is going to be equal to a.

00:05:20.700 --> 00:05:24.000
Hopefully that makes
reasonable sense to you.

00:05:24.000 --> 00:05:28.020
Now let's think about what
happens as we rotate around.

00:05:28.020 --> 00:05:29.830
Remember, these two
are top views.

00:05:29.830 --> 00:05:34.210
We are looking down
on this doughnut.

00:05:34.210 --> 00:05:39.090
So the center of each of these
circles is b away from the

00:05:39.090 --> 00:05:43.410
origin, or from the z-axis,
what we're rotating around.

00:05:43.410 --> 00:05:45.220
It's always b away.

00:05:45.220 --> 00:05:48.660
So our x-coordinate, or our x-
and y-coordinate, so if we go

00:05:48.660 --> 00:05:51.240
to the center of the circle
here, we're going to be b away,

00:05:51.240 --> 00:05:56.050
and this is going to be b
away, right over there.

00:05:56.050 --> 00:06:00.000
So let's think about where we
are in the x-y plane, or how

00:06:00.000 --> 00:06:03.850
far the part of our, what
we're, I guess you could

00:06:03.850 --> 00:06:08.280
imagine, if you were to project
our point into the x-y

00:06:08.280 --> 00:06:13.150
plane, how far is that going
to be from our origin?

00:06:13.150 --> 00:06:15.030
Well, it's always going to
be, remember, let's go

00:06:15.030 --> 00:06:16.050
back to this drawing here.

00:06:16.050 --> 00:06:18.720
This might be the
most instructive.

00:06:18.720 --> 00:06:21.440
This is just one particular
circle on the z-y plane, but

00:06:21.440 --> 00:06:22.910
it could be any of them.

00:06:22.910 --> 00:06:30.390
If this is the z-axis, over
here, this distance right here

00:06:30.390 --> 00:06:32.230
is always going to be b.

00:06:32.230 --> 00:06:34.780
We know that for sure.

00:06:34.780 --> 00:06:37.295
And so what is this
distance going to be?

00:06:43.200 --> 00:06:46.110
We're at b to the center, and
then we're going to have some

00:06:46.110 --> 00:06:49.900
angle s, and so depending on
that angle s, this distance

00:06:49.900 --> 00:06:53.650
onto, I guess, the x-y plane,
you know, if we're sitting on

00:06:53.650 --> 00:06:56.310
the x-y plane, how far are
we from the z-axis, or the

00:06:56.310 --> 00:06:57.870
projection onto the x-y plane.

00:06:57.870 --> 00:07:02.300
Or you can, you know, the
x or the y position.

00:07:02.300 --> 00:07:03.850
I'm saying it as many
ways as possible.

00:07:03.850 --> 00:07:05.490
I think you're visualizing it.

00:07:05.490 --> 00:07:09.260
If z is a sine of theta, this
distance right here, this

00:07:09.260 --> 00:07:12.770
little shorter distance right
here, that's going to

00:07:12.770 --> 00:07:14.300
be a cosine if s.

00:07:18.240 --> 00:07:20.640
s is that angle right there.

00:07:20.640 --> 00:07:24.000
This distance right here is
going to be a cosine of s.

00:07:24.000 --> 00:07:28.670
So if we talk about just
straight distance from the

00:07:28.670 --> 00:07:32.910
origin, along the x-y plane,
our distance is always going

00:07:32.910 --> 00:07:41.660
to be b plus a cosine of s.

00:07:41.660 --> 00:07:43.860
When s is out here, then it's
actually going to become a

00:07:43.860 --> 00:07:45.910
negative number, and that makes
sense, because our distance

00:07:45.910 --> 00:07:47.020
is going to be less than b.

00:07:47.020 --> 00:07:49.730
We're going to be at
that point right there.

00:07:49.730 --> 00:07:55.180
So if you look at this top
views over here, no matter

00:07:55.180 --> 00:07:57.510
where we are, that is b.

00:07:57.510 --> 00:07:59.420
And let's say we've
rotated a little bit.

00:07:59.420 --> 00:08:03.090
That distance right here, if
you're looking along the x-y

00:08:03.090 --> 00:08:10.570
plane, that is always going
to be b plus a cosine of s.

00:08:10.570 --> 00:08:14.370
That's what that distance
is to any given point.

00:08:14.370 --> 00:08:16.760
We're depending on
our s's and t's.

00:08:16.760 --> 00:08:21.910
Now, as we rotate around, if
we're at a point here, let's

00:08:21.910 --> 00:08:25.980
say we're at a point there, and
that point, we already said, is

00:08:25.980 --> 00:08:32.890
b plus a cosine of s, away from
the origin, on the xy plane.

00:08:32.890 --> 00:08:35.670
What are the x and y
coordinates of that?

00:08:35.670 --> 00:08:36.240
Well, remember.

00:08:36.240 --> 00:08:37.430
This is a top-down.

00:08:37.430 --> 00:08:40.115
We're sitting on the z-axis
looking straight down

00:08:40.115 --> 00:08:41.470
the x-y plane right now.

00:08:41.470 --> 00:08:43.460
We're looking down
on the doughnut.

00:08:43.460 --> 00:08:46.330
So what are our x's
and y's going to be?

00:08:46.330 --> 00:08:48.690
Well, you draw another
right triangle right here.

00:08:48.690 --> 00:08:49.880
You have another
right triangle.

00:08:49.880 --> 00:08:52.400
This angle right here is t.

00:08:52.400 --> 00:08:55.520
This distance right here
is going to be this times

00:08:55.520 --> 00:08:57.180
the sine of our angle.

00:08:57.180 --> 00:09:03.000
So this right here, which is
essentially our x, this is

00:09:03.000 --> 00:09:11.410
going to be our x-coordinate, x
as a function of s and t, os

00:09:11.410 --> 00:09:18.480
going to be equal to the sine
of t, t is our angle right

00:09:18.480 --> 00:09:20.850
there, times this radius.

00:09:20.850 --> 00:09:23.550
Times, we could write
it either way, times

00:09:23.550 --> 00:09:28.195
b plus a cosine of s.

00:09:28.195 --> 00:09:30.910
Because remember, how far we
are depends on how much around

00:09:30.910 --> 00:09:31.900
the circle we are, right?

00:09:31.900 --> 00:09:34.340
When we're over here,
we're much further away.

00:09:34.340 --> 00:09:36.980
Here we're exactly b away,
if you're looking only

00:09:36.980 --> 00:09:38.580
on the x-y plane.

00:09:38.580 --> 00:09:41.670
And then over here, we're
b minus a away, if

00:09:41.670 --> 00:09:43.900
we're on the x-y plane.

00:09:43.900 --> 00:09:48.790
So that's x as a
function of s and t.

00:09:48.790 --> 00:09:54.790
And actually, the way I defined
it right here, our positive

00:09:54.790 --> 00:09:57.750
x-axis would actually
go in this direction.

00:09:57.750 --> 00:10:02.570
So this is x positive, this is
x in the negative direction.

00:10:02.570 --> 00:10:04.710
I could've flipped the signs,
but hopefully, you know, this

00:10:04.710 --> 00:10:07.350
actually make sense that that
would be the positive x,

00:10:07.350 --> 00:10:08.370
this is the negative x.

00:10:08.370 --> 00:10:10.530
Depends on whether using a
right-handed or left-handed

00:10:10.530 --> 00:10:12.740
coordinate system, but
hopefully that makes sense.

00:10:12.740 --> 00:10:15.770
We're just saying, OK, what is
this distance right here that

00:10:15.770 --> 00:10:18.930
is b plus a cosine of s?

00:10:18.930 --> 00:10:23.080
We got that from this right
here, when we're taking a view,

00:10:23.080 --> 00:10:27.130
just a cut of the torus.

00:10:27.130 --> 00:10:30.420
That's how far we are, in kind
of the x-y direction at any

00:10:30.420 --> 00:10:33.330
point, or kind of radially
out, without thinking

00:10:33.330 --> 00:10:34.420
about the height.

00:10:34.420 --> 00:10:36.920
And then if you want the
x-coordinate, you multiply it

00:10:36.920 --> 00:10:43.170
times the sine of t, the way
I've had it up here, and the

00:10:43.170 --> 00:10:47.140
y-coordinate is going to be
this, right here, the way

00:10:47.140 --> 00:10:49.500
we've set up this triangle.

00:10:49.500 --> 00:10:53.780
So y as a function of s and t
is going to be equal to the

00:10:53.780 --> 00:11:01.260
cosine of t times this radius.

00:11:01.260 --> 00:11:06.270
b plus a cosine of s.

00:11:06.270 --> 00:11:09.030
And so our parameterization,
and you know, just play with

00:11:09.030 --> 00:11:11.170
this triangle, and hopefully
it'll make sense.

00:11:11.170 --> 00:11:14.500
I mean, if you say that this is
our y-coordinate right here,

00:11:14.500 --> 00:11:20.530
you just do SOCATOA, cosine of
t, CA is equal to adjacent,

00:11:20.530 --> 00:11:23.560
which is y, right, this is
the angle right here,

00:11:23.560 --> 00:11:25.000
over the hypotenuse.

00:11:25.000 --> 00:11:29.850
Over b plus a cosine of s.

00:11:29.850 --> 00:11:31.770
Multiply both sides of the
equation times this, and you

00:11:31.770 --> 00:11:36.680
get y of s of t is equal to
cosine of t times this

00:11:36.680 --> 00:11:38.790
thing, right there.

00:11:38.790 --> 00:11:41.010
Let me copy and paste
all of our takeaways.

00:11:46.960 --> 00:11:48.165
And we're done with
our parameterization.

00:11:51.830 --> 00:11:56.040
We could leave it just like
this, but if we want to

00:11:56.040 --> 00:11:59.550
represent it as a position
vector-valued function, we

00:11:59.550 --> 00:12:01.590
can define it like this.

00:12:01.590 --> 00:12:04.950
Find a nice color, maybe pink.

00:12:04.950 --> 00:12:10.010
So let's say our position
vector-valued function is r.

00:12:10.010 --> 00:12:15.580
It's going to be a function of
two parameters, s and t, and

00:12:15.580 --> 00:12:18.530
it's going to be equal
to its x-value.

00:12:18.530 --> 00:12:20.010
Let me do that in
the same color.

00:12:20.010 --> 00:12:22.070
So it's going to be, I'll
do this part first.

00:12:22.070 --> 00:12:31.970
b plus a cosine of s times sine
of t, and that's going to go in

00:12:31.970 --> 00:12:35.510
the x-direction, so we'll
say that's times i.

00:12:35.510 --> 00:12:38.390
And this case, remember,
the way I defined it,

00:12:38.390 --> 00:12:40.450
the positive x-direction
is going to be here.

00:12:40.450 --> 00:12:43.420
So the i-unit vector
will look like that.

00:12:43.420 --> 00:12:47.780
i will go in that direction,
the way I've defined things.

00:12:47.780 --> 00:12:56.710
And then plus our y-value is
going to be b plus a cosine of

00:12:56.710 --> 00:13:04.590
s times cosine of t in the
y-unit vector direction.

00:13:04.590 --> 00:13:08.520
Remember, the j-unit vector
will just go just like that.

00:13:08.520 --> 00:13:10.460
That's our j-unit vector.

00:13:10.460 --> 00:13:13.670
And then, finally, we'll throw
in the z, which was actually

00:13:13.670 --> 00:13:16.330
the most straightforward.

00:13:16.330 --> 00:13:22.800
plus a sine of s times the
k-unit vector, which is the

00:13:22.800 --> 00:13:25.330
unit vector in the z-direction.

00:13:25.330 --> 00:13:27.950
So times the k-unit vector.

00:13:27.950 --> 00:13:34.000
And so you give me, now, any s
and t within this domain right

00:13:34.000 --> 00:13:41.530
here, and you put it into this
position vector-valued

00:13:41.530 --> 00:13:44.240
function, it'll give you the
exact position vector that

00:13:44.240 --> 00:13:47.260
specifies the appropriate
point on the torus.

00:13:47.260 --> 00:13:51.520
So if you pick, let's just
make sure we understand

00:13:51.520 --> 00:13:52.620
what we're doing.

00:13:52.620 --> 00:13:55.860
If you pick that point right
there, where s and t are both

00:13:55.860 --> 00:13:58.430
equal to pi over 2, and you
might even want to go

00:13:58.430 --> 00:13:59.730
through the exercise.

00:13:59.730 --> 00:14:01.870
Take pi over 2 in all of these.

00:14:01.870 --> 00:14:03.860
Actually, let's do it.

00:14:03.860 --> 00:14:10.820
So in that case, so when r of
pi over 2, what do we get?

00:14:10.820 --> 00:14:15.730
It's going to be b plus a
times cosine of pi over 2.

00:14:15.730 --> 00:14:18.060
Cosine of pi over
2 is 0, right?

00:14:18.060 --> 00:14:19.900
Cosine of 90 degrees.

00:14:19.900 --> 00:14:22.940
So it's going to be b, right,
this whole thing is going to be

00:14:22.940 --> 00:14:25.920
0, times sine of pi over 2.

00:14:25.920 --> 00:14:29.030
Sine of pi over 2 is just 1.

00:14:29.030 --> 00:14:34.650
So it's going to be b times i
plus, once again, cosine of pi

00:14:34.650 --> 00:14:40.730
over 2 is 0, so this term right
here is going to be b, and then

00:14:40.730 --> 00:14:46.650
cosine of pi over 2 is 0,
so it's going to be 0 j.

00:14:46.650 --> 00:14:51.110
So it's going to be plus 0 j.

00:14:51.110 --> 00:14:54.970
And then finally, pi over 2,
well, there's no t here,

00:14:54.970 --> 00:14:56.520
sine of pi over 2 is 1.

00:14:56.520 --> 00:14:58.120
So plus a times k.

00:15:02.150 --> 00:15:04.230
So there's actually
no j-direction.

00:15:04.230 --> 00:15:10.950
So this is going to be equal
to b times i plus a times k.

00:15:10.950 --> 00:15:13.550
So the point that it specifies,
according to this

00:15:13.550 --> 00:15:16.010
parameterization, or the vector
[UNINTELLIGIBLE], is b times

00:15:16.010 --> 00:15:17.920
i plus a times k.

00:15:17.920 --> 00:15:24.840
So b times i will get us right
out there, and then a times k

00:15:24.840 --> 00:15:26.530
ill get us right over there.

00:15:26.530 --> 00:15:29.080
So the position of the
vector being specified

00:15:29.080 --> 00:15:31.470
is right over there.

00:15:31.470 --> 00:15:33.170
Just as we predicted.

00:15:33.170 --> 00:15:36.110
That dot, that point right
there, corresponds to that

00:15:36.110 --> 00:15:37.570
point, just like that.

00:15:37.570 --> 00:15:40.080
Of course, I picked points it
was easy to calculate, but this

00:15:40.080 --> 00:15:44.260
whole, when you take every s
and t in this domain right

00:15:44.260 --> 00:15:48.180
here, you're going to
transform it to this surface.

00:15:48.180 --> 00:15:50.930
And this is the
transformation, right here.

00:15:50.930 --> 00:15:56.090
And of course, we have to
specify that s is between, we

00:15:56.090 --> 00:15:58.210
could write it multiple ways.

00:15:58.210 --> 00:16:04.290
s is between 2 pi and 0,
and we could also say t

00:16:04.290 --> 00:16:06.760
is between 2 pi and 0.

00:16:06.760 --> 00:16:09.480
And you could, you know, we're
kind of overlapping one extra

00:16:09.480 --> 00:16:11.860
time at 2 pi, so maybe we can
get rid of one of these equal

00:16:11.860 --> 00:16:14.380
signs if you like, although
that won't necessarily change

00:16:14.380 --> 00:16:16.710
the area any, if you're
taking the surface area.

00:16:16.710 --> 00:16:19.370
But hopefully this gives you at
least a gut sense, or more than

00:16:19.370 --> 00:16:21.770
a gut sense, of how to
parameterize these things, and

00:16:21.770 --> 00:16:23.850
what we're even doing, because
it's going to be really

00:16:23.850 --> 00:16:27.690
important when we start talking
about surface integrals.

00:16:27.690 --> 00:16:29.740
And the hardest thing about
doing all of this is

00:16:29.740 --> 00:16:30.520
just the visualization.


Napoleon forced to abdicate

WEBVTT
Kind: captions
Language: en

00:00:00.460 --> 00:00:04.390
In the last video, we saw
Napoleon's army get decimated,

00:00:04.390 --> 00:00:07.080
especially as they tried
to retreat from Russia.

00:00:07.080 --> 00:00:09.640
The numbers I threw out in that
video as he started off

00:00:09.640 --> 00:00:15.005
were on the order of 450,000
soldiers under his command.

00:00:15.005 --> 00:00:16.470
And then when he retreated,
it was on the

00:00:16.470 --> 00:00:18.920
order of 10,000 soldiers.

00:00:18.920 --> 00:00:21.230
I just want to take a
little aside here.

00:00:21.230 --> 00:00:23.060
These numbers aren't firm.

00:00:23.060 --> 00:00:25.470
Historians aren't even sure
on the exact numbers.

00:00:25.470 --> 00:00:29.750
And sometimes you'll see
accounts of 500,000 to 600,000

00:00:29.750 --> 00:00:33.550
entering and 30,000
to 40,000 leaving.

00:00:33.550 --> 00:00:36.140
I went with these numbers just
because on Minard's map, these

00:00:36.140 --> 00:00:37.450
were pretty close to
those numbers.

00:00:37.450 --> 00:00:39.540
Right here he has 422,000.

00:00:39.540 --> 00:00:42.140
Leaving he has 10,000
right there.

00:00:42.140 --> 00:00:44.510
But I just want to make sure you
realize that these aren't

00:00:44.510 --> 00:00:45.520
exact numbers.

00:00:45.520 --> 00:00:48.750
And depending on, I guess, how
you account for his troops,

00:00:48.750 --> 00:00:50.770
whether you're talking about
troops directly under

00:00:50.770 --> 00:00:53.060
Napoleon's command, whether
they are troops that are

00:00:53.060 --> 00:00:56.460
directly involved in the
offensive invasion of Russia.

00:00:56.460 --> 00:00:58.520
Or they're just troops that
are maybe supporting it.

00:00:58.520 --> 00:00:59.760
You're going to get
different numbers.

00:00:59.760 --> 00:01:03.290
And depending on whether you
view troops and how you count

00:01:03.290 --> 00:01:03.820
the casualties.

00:01:03.820 --> 00:01:05.080
So it's all about
how you count.

00:01:05.080 --> 00:01:09.960
But needless to say, the
French grand army was

00:01:09.960 --> 00:01:12.220
decimated after invading
Russia.

00:01:12.220 --> 00:01:14.630
And I even told you, at the end
of the last video, that

00:01:14.630 --> 00:01:19.220
after that, the rest of Europe,
these other countries

00:01:19.220 --> 00:01:22.080
that kept forming and reforming
coalitions against

00:01:22.080 --> 00:01:24.910
Napoleon, began to
smell blood.

00:01:24.910 --> 00:01:28.670
So you can already view the
invasion of Russia as the

00:01:28.670 --> 00:01:30.370
beginning of the Sixth
Coalition.

00:01:30.370 --> 00:01:32.600
You already had Great Britain.

00:01:32.600 --> 00:01:35.620
Great Britain throughout this
entire time period, throughout

00:01:35.620 --> 00:01:38.940
the Napoleonic Wars, and even
during the French Revolution,

00:01:38.940 --> 00:01:42.670
was in a constant state of war
with the French and Napoleon.

00:01:42.670 --> 00:01:51.140
Obviously, you have Russia now
is also a belligerent, is also

00:01:51.140 --> 00:01:54.350
an antagonist against
Napoleon.

00:01:54.350 --> 00:01:56.840
Just because he retreated
doesn't mean the war with

00:01:56.840 --> 00:01:58.310
Russia is over.

00:01:58.310 --> 00:02:01.650
Prussia had been humiliated
multiple times by Napoleon.

00:02:01.650 --> 00:02:06.560
And now they jump on the
bandwagon after hearing about

00:02:06.560 --> 00:02:09.380
Napoleon getting decimated.

00:02:09.380 --> 00:02:13.000
And so they, a combined Russian
and Prussian forces,

00:02:13.000 --> 00:02:14.920
engage Napoleon.

00:02:14.920 --> 00:02:19.040
But as much as Napoleon's forces
seem to have gotten

00:02:19.040 --> 00:02:19.930
decimated--

00:02:19.930 --> 00:02:22.670
So if you look at that Minard
map up here that we went over

00:02:22.670 --> 00:02:26.220
in some detail in the last
video, that covers right here.

00:02:26.220 --> 00:02:28.320
The invasion of Russia started
right about here, and it went

00:02:28.320 --> 00:02:30.460
right about in that direction
right there.

00:02:30.460 --> 00:02:33.360
So Napoleon, he's now
retreating back.

00:02:33.360 --> 00:02:35.210
You might say, gee, he
only has 10,000 or

00:02:35.210 --> 00:02:36.860
20,000 or 30,000 troops.

00:02:36.860 --> 00:02:37.730
That's nothing.

00:02:37.730 --> 00:02:42.490
He's just going to be put out
very quickly by the combined

00:02:42.490 --> 00:02:43.300
allied forces.

00:02:43.300 --> 00:02:47.220
But Napoleon, as we kind of say,
regardless of what you

00:02:47.220 --> 00:02:50.270
think of him as a person,
he was no dummy.

00:02:50.270 --> 00:02:51.900
And he was good at
raising forces.

00:02:51.900 --> 00:02:55.820
And he was a very, very, very
good military tactition.

00:02:55.820 --> 00:02:59.930
He was very quickly able to
raise, to take his 30,000

00:02:59.930 --> 00:03:02.930
soldier army, get
it to 130,000.

00:03:02.930 --> 00:03:05.800
And eventually he's able
to get it to 400,000.

00:03:05.800 --> 00:03:08.380
Although this force right here,
it's not going to be the

00:03:08.380 --> 00:03:12.030
same quality of 400,000 as the
400,000 that he might have

00:03:12.030 --> 00:03:14.260
had, say, entering the
Russian campaign.

00:03:14.260 --> 00:03:15.600
And there were other
troops also that

00:03:15.600 --> 00:03:16.890
were allied with Napoleon.

00:03:16.890 --> 00:03:19.540
German troops from the
Confederacy of the Rhine that

00:03:19.540 --> 00:03:21.880
were sympathetic to
the French Empire.

00:03:21.880 --> 00:03:24.760
But once again, they weren't
as disciplined or under as

00:03:24.760 --> 00:03:26.230
direct control of Napoleon.

00:03:26.230 --> 00:03:31.700
But, it's needless to say, he
was able to very quickly get

00:03:31.700 --> 00:03:37.720
up to some reasonable force that
could maybe withstand a

00:03:37.720 --> 00:03:38.880
Sixth Coalition.

00:03:38.880 --> 00:03:40.510
But the Sixth Coalition
was much larger.

00:03:40.510 --> 00:03:43.370
It's on the order of, depending
on the account you

00:03:43.370 --> 00:03:50.480
look at, they amassed forces
mainly in what's now Germany--

00:03:50.480 --> 00:03:52.680
but at that time, the
Confederation of the Rhine.

00:03:52.680 --> 00:03:56.390
And before that, the Holy Roman
Empire-- of on the order

00:03:56.390 --> 00:03:59.180
of 1 million troops.

00:03:59.180 --> 00:04:00.350
Not $1 million.

00:04:00.350 --> 00:04:03.860
One million soldiers.

00:04:03.860 --> 00:04:07.360
So even though Napoleon was able
to raise a force, he was

00:04:07.360 --> 00:04:11.620
still out numbered in what's now
Germany, or the Germanic

00:04:11.620 --> 00:04:14.940
kingdoms. He was still out
numbered by a factor of 2:1.

00:04:14.940 --> 00:04:17.940
But despite that, the first
several engagements with the

00:04:17.940 --> 00:04:21.709
combined Russian and Prussian
army, Napoleon was victorious.

00:04:21.709 --> 00:04:24.530
Or I shouldn't say victorious,
because that implies too much.

00:04:24.530 --> 00:04:26.540
He won those battles.

00:04:26.540 --> 00:04:34.080
And these are at Lutzen,
at Bautzen.

00:04:34.080 --> 00:04:36.770
And then at Dresden, he had a
very significant victory.

00:04:39.470 --> 00:04:42.270
And these are all right about
here, just so you have a sense

00:04:42.270 --> 00:04:43.280
of the geography.

00:04:43.280 --> 00:04:46.840
You can actually see Dresden
on the map there.

00:04:46.840 --> 00:04:50.820
And after these defeats, the
Coalition said gee, this

00:04:50.820 --> 00:04:53.170
Napoleon character, even though
we outnumber him, even

00:04:53.170 --> 00:04:56.460
though he had to kind of very
quickly get these troops, he's

00:04:56.460 --> 00:04:57.900
a really good military
tactition.

00:04:57.900 --> 00:05:01.020
And he's still kind of kicking
our butt on the battlefield.

00:05:01.020 --> 00:05:04.140
They then issued what's called
the-- let me write this

00:05:04.140 --> 00:05:08.100
down--the Trachtenberg Plan.

00:05:08.100 --> 00:05:10.510
Although it's not clear that
they really had to.

00:05:10.510 --> 00:05:14.180
The Trachtenberg Plan, which
essentially says, try not to

00:05:14.180 --> 00:05:19.230
engage forces that Napoleon
is in command of directly.

00:05:19.230 --> 00:05:22.340
Try to engage his people
that Napoleon has

00:05:22.340 --> 00:05:23.250
had to delegate to.

00:05:23.250 --> 00:05:25.610
His marshalls, the
other generals.

00:05:25.610 --> 00:05:31.520
So don't engage Napoleon.

00:05:31.520 --> 00:05:34.455
One day I'd like to make more
detailed videos on the actual

00:05:34.455 --> 00:05:38.240
kind of-- you have to say-- the
actual genius of Napoleon

00:05:38.240 --> 00:05:39.190
on the battlefield.

00:05:39.190 --> 00:05:42.350
And go battle by battle and
see why people consider

00:05:42.350 --> 00:05:44.670
Napoleon to be a great
military commander.

00:05:44.670 --> 00:05:47.020
But his enemies definitely
appreciated the fact that,

00:05:47.020 --> 00:05:51.030
despite being outnumbered, he
was very tricky and very wily,

00:05:51.030 --> 00:05:53.930
and was always able
to kind of snatch

00:05:53.930 --> 00:05:55.200
victory from the defeat.

00:05:55.200 --> 00:05:57.690
So they had this plan, let's
just not engage Napoleon.

00:05:57.690 --> 00:06:02.260
We have so many troops, let's
just try to incur losses on

00:06:02.260 --> 00:06:06.110
the French on other commanders
other than Napoleon.

00:06:06.110 --> 00:06:11.360
But needless to say, they hugely
outnumbered the French.

00:06:11.360 --> 00:06:15.750
All of the fighting was
going on in this

00:06:15.750 --> 00:06:18.380
general area at the time.

00:06:18.380 --> 00:06:21.710
And at Leipzig, and just so
you get a sense of time.

00:06:21.710 --> 00:06:26.410
Remember, the invasion of Russia
was at the end of 18--

00:06:26.410 --> 00:06:27.550
So let me write this down.

00:06:27.550 --> 00:06:36.070
So the Russian invasion-- let
me do this in another color.

00:06:36.070 --> 00:06:37.450
Do it right here.

00:06:37.450 --> 00:06:48.020
So the Russian invasion, that
was at the end of 1812.

00:06:48.020 --> 00:06:50.310
That was really kind of the
beginning of winter in 1812

00:06:50.310 --> 00:06:53.660
that really decimated
Napoleon's troops.

00:06:53.660 --> 00:06:59.270
Then we go to 1813, you're
looking at Lutzen and Bautzen.

00:07:05.720 --> 00:07:09.330
That's in May of 1813.

00:07:09.330 --> 00:07:13.140
So after that winter, that gave
Napoleon some time to get

00:07:13.140 --> 00:07:14.100
his troops together.

00:07:14.100 --> 00:07:15.800
And things happened slower
this day and age.

00:07:15.800 --> 00:07:18.260
People didn't have perfect
intelligence in terms of what

00:07:18.260 --> 00:07:19.580
was going on in the
battlefield.

00:07:19.580 --> 00:07:22.450
And it actually took time for
just information to travel,

00:07:22.450 --> 00:07:23.750
and for armies to travel.

00:07:23.750 --> 00:07:27.160
They were traveling mainly
on foot at that time.

00:07:27.160 --> 00:07:35.850
And then August 1813, you had
Napoleon's significant victory

00:07:35.850 --> 00:07:37.100
at Dresden.

00:07:39.100 --> 00:07:47.870
But then finally, 1813, in
October-- and at this point

00:07:47.870 --> 00:07:51.220
the Coalition had convinced
Austria to also join in.

00:07:51.220 --> 00:07:53.360
So this is kind of the
ultimate coalition.

00:07:53.360 --> 00:07:55.530
If you look at most of the
coalitions, they seldom had

00:07:55.530 --> 00:07:58.320
bold Austria and Prussia and
Russia and all of these guys.

00:07:58.320 --> 00:07:59.960
But now they're all piling on.

00:07:59.960 --> 00:08:01.450
So now you have Austria.

00:08:01.450 --> 00:08:03.250
Austria is part of
the Coalition.

00:08:03.250 --> 00:08:05.590
And Sweden.

00:08:05.590 --> 00:08:07.870
And of course, you can't forget
what's happening in the

00:08:07.870 --> 00:08:08.940
Iberian Peninsula.

00:08:08.940 --> 00:08:14.090
You have all of the rebels
in Spain and Portugal.

00:08:14.090 --> 00:08:18.270
So you pretty much have every
major power in Europe is now

00:08:18.270 --> 00:08:20.410
allied against Napoleon.

00:08:20.410 --> 00:08:23.960
And in October of 1813,
especially the Austrians,

00:08:23.960 --> 00:08:26.560
Prussians, and Russians, they
meet him at Leipzig.

00:08:30.450 --> 00:08:33.460
Which is right over here.

00:08:33.460 --> 00:08:36.809
And at Leipzig, they
outnumber him 2:1.

00:08:36.809 --> 00:08:38.610
And they were able to
take care of him and

00:08:38.610 --> 00:08:40.480
force him to retreat.

00:08:40.480 --> 00:08:43.620
So then Leipzig happens, first
really major defeat.

00:08:43.620 --> 00:08:46.350
This is actually an image of
the Battle of Leipzig.

00:08:46.350 --> 00:08:48.230
They call it the Battle
of the Nations.

00:08:53.000 --> 00:08:58.290
Because so many belligerents
were involved in this battle.

00:08:58.290 --> 00:08:59.410
People estimate that there were

00:08:59.410 --> 00:09:03.810
600,000 soldiers involved.

00:09:03.810 --> 00:09:06.630
400,000 thousand on the side
of the Sixth Coalition.

00:09:06.630 --> 00:09:08.890
200,000 on the side
of Napoleon.

00:09:08.890 --> 00:09:14.020
And they estimate on the order
of a 100,000 dead or wounded.

00:09:14.020 --> 00:09:18.640
So this was a major, major,
super bloody battle.

00:09:18.640 --> 00:09:22.430
And it forced Napoleon
to retreat.

00:09:22.430 --> 00:09:23.920
So he had to retreat
from Russia before.

00:09:23.920 --> 00:09:26.060
And now he's retreating from
the Confederation of the

00:09:26.060 --> 00:09:28.710
Rhine, Which was essentially
French-controlled territory.

00:09:28.710 --> 00:09:30.500
So Napoleon is really
on his heels.

00:09:30.500 --> 00:09:33.070
He now is going to
defend France.

00:09:33.070 --> 00:09:35.160
And now remember, at the same
time as this, you had that you

00:09:35.160 --> 00:09:36.940
had all this stuff going
on in the peninsula.

00:09:36.940 --> 00:09:38.870
You had Arthur Wellesley.

00:09:38.870 --> 00:09:40.220
You remember him right here.

00:09:45.700 --> 00:09:48.930
He's the British general
leading, you could view them

00:09:48.930 --> 00:09:52.020
as the rebels in Spain
and Portugal.

00:09:52.020 --> 00:09:55.300
At the end of 1813, he's winning
a series of victories

00:09:55.300 --> 00:09:55.980
against the French.

00:09:55.980 --> 00:09:58.770
And he's pushing into France.

00:09:58.770 --> 00:10:00.520
This is the Pyrenees Mountains
right here.

00:10:00.520 --> 00:10:03.350
He's crossing, he's winning
a series of

00:10:03.350 --> 00:10:04.930
battles in the Pyrenees.

00:10:04.930 --> 00:10:06.960
And then in a last-ditch effort,
Napoleon once again,

00:10:06.960 --> 00:10:08.220
he's hugely outnumbered.

00:10:08.220 --> 00:10:12.180
He engages his enemies in
battles in northeast France,

00:10:12.180 --> 00:10:13.920
right around that area.

00:10:13.920 --> 00:10:17.720
And there's actually a period,
there's this six day campaign.

00:10:17.720 --> 00:10:20.660
And this now, we are in 1814.

00:10:20.660 --> 00:10:23.320
So Leipzig was at
the end of 1813.

00:10:23.320 --> 00:10:24.030
He retreats.

00:10:24.030 --> 00:10:26.320
Now we're in 1814.

00:10:26.320 --> 00:10:32.080
And at the beginning of 1814 in
February, you have the Six

00:10:32.080 --> 00:10:33.330
Day Campaign.

00:10:37.430 --> 00:10:39.910
And this was kind of Napoleon's
final shot at

00:10:39.910 --> 00:10:41.850
really showing his
military genius.

00:10:41.850 --> 00:10:44.800
Despite the fact that he was
hugely outnumbered, there were

00:10:44.800 --> 00:10:48.400
four significant battles where
he was essentially able to, at

00:10:48.400 --> 00:10:51.790
least in those battles, route
the Coalition, despite being

00:10:51.790 --> 00:10:52.910
hugely outnumbered.

00:10:52.910 --> 00:10:55.780
So he kept showing his
military genius.

00:10:55.780 --> 00:10:59.050
But at some point, the numbers
just became overwhelming.

00:10:59.050 --> 00:11:01.340
And the French troops just
couldn't handle the Coalition,

00:11:01.340 --> 00:11:03.870
especially after being
decimated in Russia.

00:11:03.870 --> 00:11:07.380
And then losing significant
troops even in some of their

00:11:07.380 --> 00:11:10.650
victories in what is now
Germany, or the Confederation

00:11:10.650 --> 00:11:12.120
of the Rhine then.

00:11:12.120 --> 00:11:16.920
And then the allies, the
Coalition, eventually in March

00:11:16.920 --> 00:11:25.076
of 1814, the Sixth Coalition,
marches into Paris.

00:11:28.820 --> 00:11:32.270
And this right here is an image
of the Russians marching

00:11:32.270 --> 00:11:43.760
into Paris, around
March of 1814.

00:11:43.760 --> 00:11:45.520
But Napoleon didn't
want to give up.

00:11:45.520 --> 00:11:49.360
In April Napoleon is telling
his generals,

00:11:49.360 --> 00:11:51.470
let's go retake Paris.

00:11:51.470 --> 00:11:55.940
But finally, the generals are
ready to give up on Napoleon,

00:11:55.940 --> 00:11:57.760
and they refused
to follow him.

00:11:57.760 --> 00:11:59.820
And then Napoleon says hey,
then I'm going to tell the

00:11:59.820 --> 00:12:00.610
troops to follow me.

00:12:00.610 --> 00:12:02.200
And they're like, well you could
try, but they're not

00:12:02.200 --> 00:12:03.150
going to follow you.

00:12:03.150 --> 00:12:05.430
So they essentially don't agree
to do anything that

00:12:05.430 --> 00:12:06.630
Napoleon wants.

00:12:06.630 --> 00:12:09.360
Forcing Napoleon, and it was
going to happen one way or the

00:12:09.360 --> 00:12:11.870
other, this way was
just less bloody.

00:12:11.870 --> 00:12:16.600
In April of 1814-- super
important time in history--

00:12:16.600 --> 00:12:18.700
Napoleon is forced
to abdicate.

00:12:25.020 --> 00:12:28.510
And not only does he abdicate--
maybe I'll do a

00:12:28.510 --> 00:12:31.590
future video on it-- but he also
has to rescind any claim

00:12:31.590 --> 00:12:40.250
that any of his son or any of
his future-- not the word

00:12:40.250 --> 00:12:43.560
ancestors, what's the opposite
of ancestors?

00:12:43.560 --> 00:12:47.210
Descendants, any of his future
descendants might have claim

00:12:47.210 --> 00:12:50.740
as Emperor of France
or King of France.

00:12:50.740 --> 00:12:52.630
So he's forced to abdicate.

00:12:52.630 --> 00:12:54.490
Super important time, 1814.

00:12:54.490 --> 00:12:56.450
So if you think about
when Napoleon ruled.

00:12:56.450 --> 00:13:03.140
Remember, he was able to take
power at the end of 1799.

00:13:03.140 --> 00:13:07.110
So it's been on the order of 13
or 14 years where Napoleon

00:13:07.110 --> 00:13:10.230
has just been the absolute
ruler of France.

00:13:10.230 --> 00:13:12.220
And has been able to really
kind of wreak a

00:13:12.220 --> 00:13:13.360
lot of havoc on Europe.

00:13:13.360 --> 00:13:15.370
And now, it all comes
to an end.

00:13:15.370 --> 00:13:16.850
And we'll see, it's kind
of temporary now.

00:13:16.850 --> 00:13:20.720
But we're getting near the end
of hearing about Napoleon.

00:13:20.720 --> 00:13:23.830
They force them to abdicate by
the Treaty of Fontainebleau.

00:13:33.940 --> 00:13:36.590
And they exile him to Elba.

00:13:36.590 --> 00:13:39.060
So this is-- just so you know
where Elba is-- a little

00:13:39.060 --> 00:13:44.140
island off the Italian
coast right here.

00:13:44.140 --> 00:13:46.810
This is where Napoleon
was forced to stay.

00:13:46.810 --> 00:13:49.690
Doesn't seem like that
bad of a place.

00:13:49.690 --> 00:13:51.610
And they actually let him keep
the title of Emperor.

00:13:51.610 --> 00:13:53.000
And they allowed him
to rule over Elba.

00:13:53.000 --> 00:13:55.210
And he was actually able to
do something reasonably

00:13:55.210 --> 00:13:56.770
constructive things
with the island.

00:13:56.770 --> 00:13:58.980
You're going to see he leaves
the island very shortly.

00:13:58.980 --> 00:14:01.050
But it wasn't that bad.

00:14:01.050 --> 00:14:04.170
In modern days, if someone was
indirectly responsible, or

00:14:04.170 --> 00:14:06.700
directly responsible, for
butchering millions of your

00:14:06.700 --> 00:14:10.290
civilians or soldiers, you
wouldn't send him to a place

00:14:10.290 --> 00:14:12.250
with a nice climate and give
him a nice house like this.

00:14:12.250 --> 00:14:14.890
And allow them actually
rule over the island.

00:14:14.890 --> 00:14:18.180
But I think at this time, all
of these generals and kings,

00:14:18.180 --> 00:14:21.830
they all viewed each
other as gentlemen.

00:14:21.830 --> 00:14:25.320
And they never wanted to be
too vicious to each other,

00:14:25.320 --> 00:14:28.380
just in case things
were to come back

00:14:28.380 --> 00:14:30.160
around to them, I guess.

00:14:30.160 --> 00:14:36.930
But he got exiled to Elba in, as
I just said, April of 1814.

00:14:36.930 --> 00:14:41.000
And then, the Coalition, we'll
talk more about this, they put

00:14:41.000 --> 00:14:44.200
Louis XVI-- you remember Louis
XVI, with the whole French

00:14:44.200 --> 00:14:46.770
Revolution and the
Estates-General-- they put his

00:14:46.770 --> 00:14:49.190
younger brother, they restore
him to the crown.

00:14:49.190 --> 00:14:53.770
This is right here
is Louis XVIII.

00:14:57.410 --> 00:14:59.560
This is Louis XVI's
younger brother.

00:15:04.200 --> 00:15:06.010
They make him King of France.

00:15:06.010 --> 00:15:08.080
So after all that business about
the French Revolution,

00:15:08.080 --> 00:15:10.770
and then Napoleon comes to
power, and all the Rights of

00:15:10.770 --> 00:15:11.840
Man and all that.

00:15:11.840 --> 00:15:13.200
When everything is said
and done, they put a

00:15:13.200 --> 00:15:14.910
king back in power.

00:15:14.910 --> 00:15:16.710
This guy, with the satin
robes again.

00:15:16.710 --> 00:15:19.150
Doesn't look too different
than his older brother.

00:15:19.150 --> 00:15:21.150
And just in case you're
wondering, hid older brother

00:15:21.150 --> 00:15:22.510
is Louis XVI.

00:15:22.510 --> 00:15:24.090
He's Louis XVIII.

00:15:24.090 --> 00:15:25.870
Who was Louis XVII?

00:15:25.870 --> 00:15:28.390
This is Louis XVII.

00:15:28.390 --> 00:15:29.640
Right here.

00:15:35.600 --> 00:15:39.560
This is Louis XVI's son, he was
actually next in line to

00:15:39.560 --> 00:15:44.290
the thrown before his uncle,
or Louis XVI's younger

00:15:44.290 --> 00:15:45.770
brother, Louis XVIII.

00:15:45.770 --> 00:15:49.470
But he died while he
was in prison.

00:15:49.470 --> 00:15:51.850
If you were a Royalist, you
would have considered him king

00:15:51.850 --> 00:15:56.250
after the beheading, the
decapitation of his dad.

00:15:56.250 --> 00:15:58.420
And actually, there's a hugely
fascinating story here where

00:15:58.420 --> 00:16:01.300
people say that he
died in 1795.

00:16:01.300 --> 00:16:05.330
But many, many people think that
he escaped and was able

00:16:05.330 --> 00:16:06.340
to live a normal life.

00:16:06.340 --> 00:16:08.220
And who knows what?

00:16:08.220 --> 00:16:10.710
But needless to say, he wasn't
anywhere to be found.

00:16:10.710 --> 00:16:15.140
So they made Louis XVIII-- Louis
XVI's younger brother--

00:16:15.140 --> 00:16:17.050
King of France.

00:16:17.050 --> 00:16:18.250
But, we're going to see,
this isn't the

00:16:18.250 --> 00:16:20.970
last we hear of Napoleon.


French invasion of Russia

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.740
In the last video, I talked
about the two campaigns that

00:00:03.740 --> 00:00:06.860
eventually lead to Napoleon's
downfall.

00:00:06.860 --> 00:00:08.510
One of those was the Peninsular
Campaign.

00:00:13.000 --> 00:00:15.860
Where Napoleon and the French
Empire had to waste a

00:00:15.860 --> 00:00:19.060
significant amount of resources
fighting and trying

00:00:19.060 --> 00:00:20.460
to hold Spain.

00:00:20.460 --> 00:00:25.530
And the other, almost more
direct catalyst for the fall

00:00:25.530 --> 00:00:32.630
of Napoleon, was his
invasion of Russia.

00:00:32.630 --> 00:00:36.110
We're going to see in this is
video why it was so disastrous

00:00:36.110 --> 00:00:36.600
for Napoleon.

00:00:36.600 --> 00:00:40.560
This is easily one of the top
five, I would say, famous

00:00:40.560 --> 00:00:43.480
military campaigns in
all of history.

00:00:43.480 --> 00:00:47.970
So let's remind ourselves the
relationship between Napoleon,

00:00:47.970 --> 00:00:50.110
or between the French
Empire and Russia.

00:00:50.110 --> 00:00:54.740
After the Fourth Coalition, you
might remember the Treaty

00:00:54.740 --> 00:01:03.100
of Tilsit, they're friends.

00:01:05.620 --> 00:01:08.440
Russia was part of the
Continental System, which

00:01:08.440 --> 00:01:11.700
means that they would boycott
or that they would not trade

00:01:11.700 --> 00:01:12.710
with Britain.

00:01:12.710 --> 00:01:15.200
Or allow Britain to use
any of Russia's ports.

00:01:15.200 --> 00:01:18.090
And Russia was allowed to do
anything that they want with

00:01:18.090 --> 00:01:18.990
the Ottoman Empire.

00:01:18.990 --> 00:01:20.730
So everything was good.

00:01:20.730 --> 00:01:24.750
But then we move forward
to 1811.

00:01:24.750 --> 00:01:27.090
We're in 1811 and Russia
starts relaxing the

00:01:27.090 --> 00:01:28.340
Continental System.

00:01:38.750 --> 00:01:41.240
Essentially meaning that they
started to trade a little bit

00:01:41.240 --> 00:01:41.840
with Britain.

00:01:41.840 --> 00:01:48.760
So this all of a sudden begins
to annoy, it angers Napoleon.

00:01:48.760 --> 00:01:52.740
In order for an embargo to
work, in order for the

00:01:52.740 --> 00:01:55.290
Continental System to work,
everyone on the continent has

00:01:55.290 --> 00:01:56.300
to not trade with Britain.

00:01:56.300 --> 00:01:58.310
Because if someone trades with
Britain, Britain could just

00:01:58.310 --> 00:02:00.560
trade with them and then that
trade could circulate through

00:02:00.560 --> 00:02:01.320
the rest of Europe.

00:02:01.320 --> 00:02:02.730
So in order for this to work,
it has to be total.

00:02:02.730 --> 00:02:05.080
So this already annoys
Napoleon.

00:02:05.080 --> 00:02:07.040
Especially because he defeated
the Russians

00:02:07.040 --> 00:02:09.199
in the Fourth Coalition.

00:02:09.199 --> 00:02:10.650
In the Treaty of Tilsit
he thought he was

00:02:10.650 --> 00:02:11.650
being nice to them.

00:02:11.650 --> 00:02:14.110
Just don't do any trade with
Great Britain, we're going to

00:02:14.110 --> 00:02:15.630
be good buddies.

00:02:15.630 --> 00:02:21.750
But then what really got
Napoleon is in 1812, he hears

00:02:21.750 --> 00:02:27.230
that Russia is thinking about
taking the French Empire's

00:02:27.230 --> 00:02:30.600
part of Poland, or the
Duchy of Warsaw.

00:02:30.600 --> 00:02:41.850
So Russia thinking about
taking Poland.

00:02:41.850 --> 00:02:45.360
So Napoleon figures, gee, we're
no longer friends, I

00:02:45.360 --> 00:02:47.690
better attack them
preemptively.

00:02:47.690 --> 00:02:52.330
So against the advice of a lot
of people about actually

00:02:52.330 --> 00:02:55.820
invading Russia, Napoleon
decides to invade.

00:02:55.820 --> 00:03:04.290
So this directly leads
to Napoleon invading.

00:03:04.290 --> 00:03:06.090
And his other option, obviously,
would have been

00:03:06.090 --> 00:03:08.950
just to defend Poland
from Russia.

00:03:08.950 --> 00:03:13.020
But Napoleon, he just says no,
look if they're going to

00:03:13.020 --> 00:03:15.810
double cross me like that, I'm
just going to go full tilt and

00:03:15.810 --> 00:03:16.790
show them who's boss.

00:03:16.790 --> 00:03:25.380
So he decides to invade
in April of 1812.

00:03:25.380 --> 00:03:26.550
And Napoleon is no dummy.

00:03:26.550 --> 00:03:29.980
He realizes that the Russian
winter is really vicious.

00:03:29.980 --> 00:03:33.470
So he says, if I start in the
late spring like this and I

00:03:33.470 --> 00:03:36.210
engage the Russians and I
decisively defeat them,

00:03:36.210 --> 00:03:37.300
everything will be
said and done.

00:03:37.300 --> 00:03:39.140
And I'll be Emperor of
Russia, as well.

00:03:39.140 --> 00:03:42.960
Before Russia even begins
to get cold.

00:03:42.960 --> 00:03:45.330
So what I'm going to do here,
to just understand how

00:03:45.330 --> 00:03:49.660
devastating this campaign ends
up being to Napoleon, I'm

00:03:49.660 --> 00:03:51.750
going to use one of the
most famous charts in

00:03:51.750 --> 00:03:52.780
the history of charts.

00:03:52.780 --> 00:03:56.060
Some people consider this
to be the best, most

00:03:56.060 --> 00:03:58.400
well-designed chart ever.

00:03:58.400 --> 00:04:03.050
And this was created
by Charles-- let

00:04:03.050 --> 00:04:03.910
me use another color.

00:04:03.910 --> 00:04:05.325
I keep using this.

00:04:05.325 --> 00:04:11.310
This was created by Charles
Joseph Minard.

00:04:15.470 --> 00:04:18.279
You can actually get a bigger
version of this on Wikipedia.

00:04:18.279 --> 00:04:19.820
But that was too big
for my screen.

00:04:19.820 --> 00:04:21.440
So this is a smaller version.

00:04:21.440 --> 00:04:22.770
But it's a fascinating chart.

00:04:22.770 --> 00:04:28.600
So what it shows is the size
of Napoleon's army as it

00:04:28.600 --> 00:04:29.860
invades Russia.

00:04:29.860 --> 00:04:33.670
And the size of the army is
shown by the width of this

00:04:33.670 --> 00:04:35.330
brown line right here.

00:04:35.330 --> 00:04:41.900
So when he starts invading
Russia, he has 450,000 men.

00:04:41.900 --> 00:04:44.300
If you could read this number,
that's what it says.

00:04:44.300 --> 00:04:45.990
And you can find zoomed
versions on the web.

00:04:45.990 --> 00:04:48.380
If you just do a search on this
and on the invasion of

00:04:48.380 --> 00:04:50.040
Russia, you'll find
this chart.

00:04:50.040 --> 00:04:52.400
It's very, very famous.

00:04:52.400 --> 00:04:55.370
More than even teaching you the
history, this is also just

00:04:55.370 --> 00:04:56.480
a neat chart to look at.

00:04:56.480 --> 00:04:57.310
And it's something that
you should be

00:04:57.310 --> 00:04:59.030
exposed to at some point.

00:04:59.030 --> 00:05:01.660
But as you can see, and just
so you have some scope of

00:05:01.660 --> 00:05:04.030
what's going on, not only does
it show you the size of his

00:05:04.030 --> 00:05:06.690
forces, it shows the path
that it travels.

00:05:06.690 --> 00:05:10.040
And you can already see that as
the forces are travelling,

00:05:10.040 --> 00:05:14.130
the number of the soldiers in
the force is getting smaller

00:05:14.130 --> 00:05:14.550
and smaller.

00:05:14.550 --> 00:05:16.000
And he keeps updating
the numbers.

00:05:16.000 --> 00:05:18.900
So obviously that width is
smaller than that width, by it

00:05:18.900 --> 00:05:22.690
looks like it's less than half
the forces at this point than

00:05:22.690 --> 00:05:23.300
at this point.

00:05:23.300 --> 00:05:26.490
And just to understand where
this is going on, here is a

00:05:26.490 --> 00:05:30.840
map of Eastern Europe.

00:05:30.840 --> 00:05:35.930
And Charles Joseph Minard's map
starts right around here

00:05:35.930 --> 00:05:37.550
in Lithuania.

00:05:37.550 --> 00:05:40.750
So this is where the Napoleon
forces were strongest, where

00:05:40.750 --> 00:05:42.220
he had 450,000.

00:05:42.220 --> 00:05:43.530
And it kind of goes like this.

00:05:43.530 --> 00:05:44.830
This is Moscow right here.

00:05:44.830 --> 00:05:47.130
So this is kind of
the endpoint.

00:05:47.130 --> 00:05:53.290
And if I were to trace this
shape onto here, it would look

00:05:53.290 --> 00:05:55.650
something like this.

00:05:55.650 --> 00:05:58.030
I'm just trying to
show the path.

00:05:58.030 --> 00:06:00.940
Roughly like this.

00:06:03.740 --> 00:06:04.850
And then you get to Moscow.

00:06:04.850 --> 00:06:07.130
So that line is the same
thing as that line.

00:06:07.130 --> 00:06:09.700
Just so you have some context
within the broader

00:06:09.700 --> 00:06:10.800
geography of Europe.

00:06:10.800 --> 00:06:16.100
So it's starting off, in what's
almost exactly the

00:06:16.100 --> 00:06:20.240
northeastern border of
present-day Poland, or you

00:06:20.240 --> 00:06:21.930
could say the southwestern
border

00:06:21.930 --> 00:06:24.260
of what's now Lithuania.

00:06:24.260 --> 00:06:27.290
Crosses through what
is now Belarus.

00:06:27.290 --> 00:06:29.320
And then goes on into Russia.

00:06:29.320 --> 00:06:32.770
Now, as you can see here, I
already pointed out, that as

00:06:32.770 --> 00:06:37.830
he's traveling, as the forces
are travelling, these 450,000

00:06:37.830 --> 00:06:42.530
men, horses, and armaments, and
cannons, and everything

00:06:42.530 --> 00:06:44.110
else is getting thinner
and thinner.

00:06:44.110 --> 00:06:48.120
And one is, it's just basic
military logistics.

00:06:48.120 --> 00:06:50.100
You have to keep your
supply routes open.

00:06:50.100 --> 00:06:56.310
You have to keep leaving men
behind to essentially guard

00:06:56.310 --> 00:06:57.250
the supply routes.

00:06:57.250 --> 00:07:01.640
So just as the longer your
campaign, and the further into

00:07:01.640 --> 00:07:04.540
enemy territory you have, you
have to keep leaving men

00:07:04.540 --> 00:07:05.800
behind to guard it.

00:07:05.800 --> 00:07:09.320
And this whole time, Napoleon
kept wanting to engage with

00:07:09.320 --> 00:07:10.000
the Russians.

00:07:10.000 --> 00:07:13.850
The whole time that he was
travelling, he kept expecting

00:07:13.850 --> 00:07:15.060
to engage the Russians.

00:07:15.060 --> 00:07:17.460
But just as he was about to
engage the Russians, the

00:07:17.460 --> 00:07:18.860
Russians would back
up a little bit.

00:07:18.860 --> 00:07:19.800
And this kept happening.

00:07:19.800 --> 00:07:22.070
They're were small skirmishes,
so obviously he lost some

00:07:22.070 --> 00:07:23.120
troops here or there.

00:07:23.120 --> 00:07:27.090
But most of this was just to
maintain the supply line.

00:07:27.090 --> 00:07:32.560
Eventually, he is able to have,
I guess we could say, a

00:07:32.560 --> 00:07:36.520
skirmish, I don't want to say
a skirmish, it was actually

00:07:36.520 --> 00:07:38.740
maybe a brief battle
with the Russians

00:07:38.740 --> 00:07:41.420
in August in Smolensk.

00:07:41.420 --> 00:07:46.760
And that is, let me make sure
I get it right on the map,

00:07:46.760 --> 00:07:48.530
right there.

00:07:48.530 --> 00:07:50.660
But once again, Napoleon
was victorious.

00:07:50.660 --> 00:07:53.790
But the Russians retreated, it
wasn't a decisive victory over

00:07:53.790 --> 00:07:55.210
the Russian army.

00:07:55.210 --> 00:07:57.990
Eventually, he just keeps
chasing them, keeps chasing

00:07:57.990 --> 00:08:06.870
them, until we get to
September 7, 1812.

00:08:06.870 --> 00:08:10.360
And now you could imagine that
we're kind of in the fall now.

00:08:10.360 --> 00:08:14.620
And it gets cold quickly
out here in Russia.

00:08:14.620 --> 00:08:18.050
So already, it's getting to be a
kind of precarious position.

00:08:18.050 --> 00:08:19.330
But Napoleon is confident.

00:08:19.330 --> 00:08:20.060
He's like, you know what?

00:08:20.060 --> 00:08:23.300
If I can decisively defeat the
Russians, I can take Moscow.

00:08:23.300 --> 00:08:25.070
Moscow will have food it in.

00:08:25.070 --> 00:08:25.980
And all will be said and done.

00:08:25.980 --> 00:08:29.830
And that's actually brings
up another point.

00:08:29.830 --> 00:08:32.020
You might remember when I talked
about the War of the

00:08:32.020 --> 00:08:35.140
Third Coalition, that Napoleon,
and in general good

00:08:35.140 --> 00:08:37.880
military tactics to invade
during the fall.

00:08:37.880 --> 00:08:39.990
Because of fall is the
harvest season.

00:08:39.990 --> 00:08:42.440
That way you don't have to
have as many supplies

00:08:42.440 --> 00:08:43.179
carried with you.

00:08:43.179 --> 00:08:45.730
You don't have to guard the
supply routes as carefully.

00:08:45.730 --> 00:08:49.600
Because you can just go and take
all of the farmer's crops

00:08:49.600 --> 00:08:50.890
to feed your troops.

00:08:50.890 --> 00:08:53.420
But what the Russians did is
what they call scorched earth

00:08:53.420 --> 00:09:02.620
tactics, which essentially
means exactly

00:09:02.620 --> 00:09:03.830
what it sounds like.

00:09:03.830 --> 00:09:09.050
As the Russians kept retreating
back, they burned

00:09:09.050 --> 00:09:12.790
everything that could be of
benefit to Napoleon's troops.

00:09:12.790 --> 00:09:17.810
So they burned all the fields
as they retreated, so that

00:09:17.810 --> 00:09:20.920
Napoleon's troops couldn't
access the local resources.

00:09:20.920 --> 00:09:24.600
So they had to keep having a
very-- I guess you could say--

00:09:24.600 --> 00:09:28.480
costly supply chain from where
they started, as opposed to

00:09:28.480 --> 00:09:29.910
just feeding off of the land.

00:09:29.910 --> 00:09:31.890
And the Russians just
continued to do this

00:09:31.890 --> 00:09:33.190
on and on and on.

00:09:33.190 --> 00:09:34.540
All the way to Moscow.

00:09:34.540 --> 00:09:39.180
All the way to September 1812,
where we have the Battle of

00:09:39.180 --> 00:09:41.000
Borodino, if I'm saying
it correctly.

00:09:52.345 --> 00:09:59.310
And at this point, Napoleon
only had a 135,000 of his

00:09:59.310 --> 00:10:02.480
original 450,000 men.

00:10:02.480 --> 00:10:05.220
The rest had to be left behind
to guard the supply route.

00:10:05.220 --> 00:10:08.820
Or they deserted because they
were starting to get hungry.

00:10:08.820 --> 00:10:11.880
Or they were lost just with
minor skirmishes that happened

00:10:11.880 --> 00:10:13.580
along the entire way.

00:10:13.580 --> 00:10:17.770
And at the Battle of Borodino,
after a long, protracted,

00:10:17.770 --> 00:10:21.950
super bloody battle, in which--
let me write these

00:10:21.950 --> 00:10:25.920
numbers down-- 44-- and these
are all estimates-- 44,000

00:10:25.920 --> 00:10:28.940
Russian casualties.

00:10:28.940 --> 00:10:31.890
Mostly death, but also
wounded or captured.

00:10:31.890 --> 00:10:36.850
And 35,000 French.

00:10:36.850 --> 00:10:39.410
Many historians think this is
the most of bloody day of

00:10:39.410 --> 00:10:41.940
battle up to this
day in history.

00:10:41.940 --> 00:10:43.960
The French are victorious.

00:10:43.960 --> 00:10:46.060
So we have a French
victory here.

00:10:48.930 --> 00:10:50.900
So Napoleon is feeling
good about himself.

00:10:50.900 --> 00:10:54.660
Borodino is right outside
of Moscow.

00:10:54.660 --> 00:10:55.260
He's like, you know what?

00:10:55.260 --> 00:10:56.850
I'm just going to walk
into Moscow.

00:10:56.850 --> 00:10:59.180
They're going to hand the keys
of the city over to me.

00:10:59.180 --> 00:11:01.760
And then my troops are going to
have a good time in Moscow

00:11:01.760 --> 00:11:03.740
being able to eat all the
food that must be there.

00:11:03.740 --> 00:11:07.380
And I will now be Emperor
of Russia, as well.

00:11:07.380 --> 00:11:09.600
So about a week later, he walks
into Moscow and it's

00:11:09.600 --> 00:11:10.850
completely deserted.

00:11:17.530 --> 00:11:20.390
All of the inhabitants have left
and they've taken all the

00:11:20.390 --> 00:11:21.860
food with them.

00:11:21.860 --> 00:11:24.433
And then on top of that,
Moscow starts burning.

00:11:29.270 --> 00:11:32.420
And there's some debate as
to why Moscow burned.

00:11:32.420 --> 00:11:35.750
Some historians believe it
is because the Russians

00:11:35.750 --> 00:11:38.160
deliberately set fire
to the city.

00:11:38.160 --> 00:11:40.780
Some believe that it was just
a tinder box waiting to burn

00:11:40.780 --> 00:11:44.410
and Napoleon's troops were just
a little irresponsible

00:11:44.410 --> 00:11:45.800
with where they built fires.

00:11:45.800 --> 00:11:46.870
It's not clear.

00:11:46.870 --> 00:11:51.230
But needless to say, it made
Moscow useless to Napoleon.

00:11:51.230 --> 00:11:52.880
And this is kind of
a major change.

00:11:52.880 --> 00:11:56.900
Napoleon just expected look,
if I beat your army

00:11:56.900 --> 00:12:00.300
decisively, like I did it
at Borodino, I won.

00:12:00.300 --> 00:12:01.200
I take the city.

00:12:01.200 --> 00:12:03.730
Now, you should give me
food and shelter.

00:12:03.730 --> 00:12:04.750
And I'm your new Emperor.

00:12:04.750 --> 00:12:06.740
But the Russians had something
different in mind.

00:12:06.740 --> 00:12:09.100
They said, no you will
never take Russia.

00:12:09.100 --> 00:12:10.270
You can have Moscow.

00:12:10.270 --> 00:12:11.650
You can have a burned Moscow.

00:12:11.650 --> 00:12:12.590
There's nothing in it.

00:12:12.590 --> 00:12:17.310
And we'll just like to see you
retreat in the Moscow winter.

00:12:17.310 --> 00:12:22.020
And by Napoleon's own accounts,
this is a quote that

00:12:22.020 --> 00:12:26.370
he gave. The most terrible of
all my battles was the one

00:12:26.370 --> 00:12:27.750
before Moscow.

00:12:27.750 --> 00:12:30.630
The French showed themselves
to be worthy of victory.

00:12:30.630 --> 00:12:31.950
This is at Borodino.

00:12:31.950 --> 00:12:34.940
But the Russians show themselves
worthy of being

00:12:34.940 --> 00:12:37.170
invincible.

00:12:37.170 --> 00:12:40.510
So with no one to recognize
Napoleon or to help him out or

00:12:40.510 --> 00:12:44.440
to feed his troops, he's kind
of in a worthless Moscow.

00:12:44.440 --> 00:12:47.090
He decides to just turn around,
especially because the

00:12:47.090 --> 00:12:48.580
Russian winter is approaching.

00:12:48.580 --> 00:12:50.770
So on this chart you can see
they went all the way to

00:12:50.770 --> 00:12:53.610
Moscow, they had
135,000 troops.

00:12:53.610 --> 00:12:55.610
They lose 35,000 of them.

00:12:55.610 --> 00:12:57.720
So you have 100,000 troops
coming back.

00:12:57.720 --> 00:13:00.730
This black line, right here,
this represents the retreat.

00:13:00.730 --> 00:13:02.970
And then things start
getting cold.

00:13:02.970 --> 00:13:05.780
So it's really interesting
about this

00:13:05.780 --> 00:13:07.970
chart that Minard made.

00:13:07.970 --> 00:13:09.230
It has a couple things.

00:13:09.230 --> 00:13:12.940
You see the route that
Napoleon's army took as it

00:13:12.940 --> 00:13:14.370
went into Russia.

00:13:14.370 --> 00:13:16.130
Then in black, this
is its retreat.

00:13:16.130 --> 00:13:17.660
It gets thinner and thinner
and thinner.

00:13:17.660 --> 00:13:18.940
And you see where they went.

00:13:18.940 --> 00:13:20.770
And on top of that,
he shows the

00:13:20.770 --> 00:13:24.240
temperature as they retreated.

00:13:24.240 --> 00:13:26.470
And this is on something called
a Reaumur scale, but

00:13:26.470 --> 00:13:28.120
you can just multiply these
numbers, if you can

00:13:28.120 --> 00:13:30.230
see them, by 1.25.

00:13:30.230 --> 00:13:33.170
So you can see that it was
already, it was not that bad.

00:13:33.170 --> 00:13:35.190
But over here, we're already
below freezing.

00:13:35.190 --> 00:13:37.340
We're well below freezing
over here.

00:13:37.340 --> 00:13:40.620
At the coldest point, if
Minard's numbers are right,

00:13:40.620 --> 00:13:43.080
and I've seen some historians
contest the numbers, but it

00:13:43.080 --> 00:13:44.490
was cold nonetheless.

00:13:44.490 --> 00:13:48.740
It's negative 37 degrees as
we start approaching into

00:13:48.740 --> 00:13:49.880
November and December.

00:13:49.880 --> 00:13:51.890
So Napoleon's forces, as before,
they had this whole

00:13:51.890 --> 00:13:53.110
supply chain.

00:13:53.110 --> 00:13:53.950
They had scorched earth.

00:13:53.950 --> 00:13:55.020
So there's nothing to eat.

00:13:55.020 --> 00:13:56.110
They still had nothing to eat.

00:13:56.110 --> 00:13:57.600
Then it started getting cold.

00:13:57.600 --> 00:14:00.030
And while it was cold and they
were retreating, some of the

00:14:00.030 --> 00:14:01.150
troops would desert.

00:14:01.150 --> 00:14:04.370
They would be attacked
by peasants.

00:14:04.370 --> 00:14:07.580
Just little bit by little bit
they'd be attacked by various

00:14:07.580 --> 00:14:08.940
Russian forces.

00:14:08.940 --> 00:14:12.950
And then kind of the final blow
was when they were trying

00:14:12.950 --> 00:14:15.710
to cross the Berezina River.

00:14:15.710 --> 00:14:18.130
So this right here is
the Berezina River.

00:14:23.620 --> 00:14:27.050
And you can even see from
Minard's chart that the width

00:14:27.050 --> 00:14:28.440
was respectable.

00:14:28.440 --> 00:14:30.080
But then after trying
to cross that

00:14:30.080 --> 00:14:32.260
river, it just got decimated.

00:14:32.260 --> 00:14:36.770
Almost half of the soldiers
died or disappeared.

00:14:36.770 --> 00:14:39.930
And these are depictions, these
are paintings, of the

00:14:39.930 --> 00:14:42.320
retreat of Napoleon's forces
in the winter.

00:14:42.320 --> 00:14:50.850
This is actually a painting of
the crossing of Berezina.

00:14:54.810 --> 00:14:57.940
And by the time that the forces
got back at the end of

00:14:57.940 --> 00:15:00.700
1812-- all the way over here,
by Minard's map-- you only

00:15:00.700 --> 00:15:04.970
have 10,000 troops of the
original 450,000.

00:15:04.970 --> 00:15:06.930
So only 1 in 45 returned.

00:15:06.930 --> 00:15:08.000
The other were either captured,

00:15:08.000 --> 00:15:11.620
deserted, or just died.

00:15:11.620 --> 00:15:14.210
So when this happened, you could
imagine, not only did

00:15:14.210 --> 00:15:17.290
Napoleon lose so many soldiers,
also lost on the

00:15:17.290 --> 00:15:20.010
order of 200,000 horses.

00:15:20.010 --> 00:15:24.730
Actually, total casualties for
this one campaign, the

00:15:24.730 --> 00:15:28.920
estimates are about 300,000
French troops.

00:15:28.920 --> 00:15:33.310
The estimates I saw had about
200,000 Russian troops.

00:15:33.310 --> 00:15:36.310
And just to be clear, a big
portion of the French troops

00:15:36.310 --> 00:15:40.450
died as they tried to retreat.

00:15:40.450 --> 00:15:44.830
And then there was a huge amount
of Russian civilian

00:15:44.830 --> 00:15:45.320
casualties.

00:15:45.320 --> 00:15:47.460
And there were also soldiers
from other countries

00:15:47.460 --> 00:15:51.440
participating, especially
on the French side.

00:15:51.440 --> 00:15:58.390
When all is said and done,
estimates are over 1 million

00:15:58.390 --> 00:16:01.000
dead from the French invasion.

00:16:01.000 --> 00:16:05.660
But beyond even just the human
casualties, Napoleon lost on

00:16:05.660 --> 00:16:08.820
the order of 200,000 horses.

00:16:08.820 --> 00:16:12.160
I forgot the exact number,
but thousands of canons.

00:16:12.160 --> 00:16:15.250
So you could imagine, that it
was just this huge, huge loss

00:16:15.250 --> 00:16:16.440
to the French military.

00:16:16.440 --> 00:16:19.750
So by the time this was done,
the rest of the powers of

00:16:19.750 --> 00:16:23.910
Europe could kind of finally see
Napoleon being weakened.

00:16:23.910 --> 00:16:26.180
He was already bogged down
in the peninsula.

00:16:26.180 --> 00:16:30.140
He had this huge defeat as he
tried to invade Russia.

00:16:30.140 --> 00:16:33.510
They finally realized that now
is the time that they can form

00:16:33.510 --> 00:16:38.070
maybe a Sixth Coalition and
really take Napoleon out.