>> We're gonna get started
here in just a moment.
Please help yourself to
some more refreshments.
I know that
Jeff has--
we were talking about
this a few minutes ago.
Jeff has quite a
bit to discuss here.
So thank you all for coming
to our last math seminar
of the 2017/2018
academic year.
It's great to have
you here today.
It's great to
be inside.
Jeff came to me-- this
was a few months ago,
saying that he wanted
to give a talk on this.
So I'm really looking
forward to this.
This is something I've known
about for quite a long time
but I have not explored it
in the detail that Jeff has.
Jeff's-- I'll mention
really briefly--
Jeff's a former student,
he's a tutor here right now,
and will be at Western
Michigan in the fall,
studying mathematics.
It's our great pleasure
to have him here today.
Have fun.
>> Thank you, John.
Well, I hope everybody
is gonna have fun today.
So today, I am talking
about Archimedes,
specifically his work,
"The Sand Reckoner."
Now, a lot of people have
come up to me and said,
"Well, what is
'The Sand Reckoner?'
"What is
this thing?"
I said, "Well, you'll have to
come to the talk, right?"
"No, no-- but I
mean what is this?
"What is
this about?"
And I said, "Well, it's a
manuscript that he wrote
"where Archimedes
actually calculates
"how many grains of sand
there are in the universe."
And people are like,
"Well, how did he do that?"
And I'd say, "Well, you
gotta come to the talk."
(scattered laughing)
So thank you for being here,
and actually being able to
see what's gonna happen here.
So let me jump right in-- I do
have quite a bit to go over.
I think you guys are gonna
get your money's worth here.
I should go
the full hour.
But that means that maybe hold
some questions till the end.
I don't want to
discourage any questions,
but if you do have stuff,
feel free to write it down,
ask it at
the end.
I'll answer
anything you need.
But let's jump
right in.
So I think the first thing
we need to talk about
is what time period
was Archimedes from--
what was life
like back then?
Because we've gotta
look at the context
of when this
was written
to really fully appreciate
what was happening here.
So Archimedes is
from Syracuse.
And if you look
at just this map,
this is from
218 BC,
he is stuck in between Rome,
which is rising to power,
and Carthage, which
is at war with Rome
during most
of his life.
More importantly,
though,
since Archimedes was in
the middle of everything,
he didn't live
in some bubble.
He wrote to
other people.
He corresponded
with them
and bounced ideas back
and forth with them.
The center of learning really
was Alexandria down here.
We say Archimedes is Greek,
but being from Syracuse,
he is from the
island of Sicily,
which is now
part of Italy.
But really, when you think
about Greek mathematicians,
we really need to look at
not where they are from,
but the culture
that they come from,
the language
that they spoke,
and the philosophy
that they used.
So Archimedes
is Greek,
even though he is
from now Italy here.
But as a Greek,
you know...
Greeks were more known for
their pure mathematics,
and their appreciation
of thought and thinking
just for the sake of
thought and thinking.
But Archimedes was a
little bit different.
He had a very
practical side to him.
He invented quite
a few things.
One of them being
the water screw here.
Now, when I say "invented," he
really more improved upon it.
It actually was probably
Egyptian in origin.
But basically was able to
raise water from the ground.
Another big invention
that he did was--
this was called
"Archimedes' Claw,"
and he actually used
that as a defense
when Syracuse was under
attack from the Romans.
Essentially his city decided
to pick the wrong side
in the Second
Punic War.
And Syracuse
was under siege,
and Archimedes was called
to help defend the city.
And so, he made
these war machines,
one of them
being the Claw,
which was able to lift
ships out of the water
and then drop
them back down.
So everybody would
kind of spill back out,
ship would break,
carnage would ensue.
Another one is called
his "death ray,"
which I don't
have a picture of
because it may or
may not have existed.
But essentially-- yeah.
(chuckling)
Sounds pretty bad.
(scattered laughing)
If I had to go up against
a death ray, I don't know.
But basically,
it was just--
what was proposed was
a series of mirrors
able to focus the heat of the
sun on ships coming through,
and it would start
a fire on the ships.
Now, there's a lot of
debate whether that's real.
In my opinion, it
probably wasn't.
But that's the mythology
of Archimedes.
Speaking of the mythology,
one of them--
notice I have
the cartoon here,
because really
that's how seriously
we should take the
"eureka" story.
A well-known story
about Archimedes
is that he was
assigned the task
of trying to figure
out if the king's crown
was fully gold, or whether
there was some silver in it.
Well, he essentially
needed to figure out
the density of the
material, in a way.
And he needed to figure
out the volume of it.
And so, sinking
into his bath,
he saw that some of
the water overflowed,
and then he
realized, "Oh.
"That's how I can
figure out the volume."
And so he jumped
out of the bath
and ran naked
through the streets.
Well, this story
doesn't actually appear
in any works
on Archimedes
for several centuries
later, after this death,
so it probably
was not real.
But Archimedes did do a lot
of work with hydrostatics.
We know about
Archimedes' principle.
You'll learn about
that in physics,
with buoyant forces
and all that.
He did work
on those.
So besides his
practical side,
there was a mathematical
side to Archimedes,
which he valued more
than his practical side.
He actually-- well, I
should say Plutarch wrote
that Archimedes considered
his war machines
as just "mere amusements
in geometry,"
and he considered his
mathematics more important.
He actually had this shape
inscribed on his tombstone.
This is a sphere
inscribed in a cylinder.
That was reportedly his
proudest accomplishment,
was understanding that
both the surface area
and volume of the cylinder
bears a ratio of 3:2
with the surface area
and volume of the sphere.
Now, for us it's not
really that hard
to prove something
like that.
But you think about
third century Greece,
third century BC
Greece,
and you think about
how they don't have
just standard
formulas for things,
able to manipulate
numbers like this,
and they didn't
have calculus.
So to be able to figure
out things like this
was very novel.
Another thing
was his spirals.
Those are very famous,
very beautiful shapes.
But he was able
to determine areas.
And I should really say the
ratios of areas to other things,
not necessarily
numerical areas.
But perhaps his most famous
mathematical discovery here
is not really an
approximation,
but his upper
and lower bound
for the circumference
to diameter of a circle.
Now, circumference to
diameter of a circle
is what we
now call "pi."
So Archimedes didn't find
an approximation for pi,
he didn't
estimate pi,
he actually put it in
between two bounds.
And that's a trend
that you'll see
throughout this work that
we're gonna talk about is,
he didn't do things precisely,
he didn't round off anything.
He actually kept it
as inequalities
to essentially smush
things together
and be able to pinpoint
what he really needed.
You can see the method
of exhaustion here,
which is taking an
inscribed shape,
and a circumscribed
shape,
and then essentially trying to
approach what that circle is.
And so, there's
a lot of ideas
that are now seen
in calculus.
And so, some people
say that Archimedes
was the father of calculus,
he maybe invented calculus.
Not really sure
what that means,
but we can say that a
lot of Archimedes' ideas
are very modern,
and maybe we can call him
a grandfather to calculus.
Of course, Archimedes
died in 212 BC.
That was during the
Siege of Syracuse,
where he was defending
his own city.
The story goes-- one
of the story goes,
is that Archimedes-- as the
Romans snuck into the city,
Archimedes was deep
in thought in work
on some geometrical
diagram.
A Roman soldier
approached,
and Archimedes shouted,
"Do not disturb my circles,"
and then the
Roman killed him.
So that may or may
not have happened.
Again, quite a bit of
mythology with him.
These stories are fun, but
that is part of the...
part of the importance of
Archimedes in the culture here.
So he was a figure almost like
how we think of Einstein today.
Although a lot of his
thought was lost
and not developed upon
for a long time.
We can see that in how his
works were passed down.
So I'm gonna talk a little
bit about these codices.
This is how we have
some information
of the writings
of Archimedes.
If you want more
information on it,
there's a great book called
"The Archimedes Codex,"
and that's on the
handout as a resource,
and also at the end
of this PowerPoint.
But there are
three main sources.
Of course, there's
more than this,
but these are what we look
at-- Codex A, B, and C.
You can see Codex A bears
the most works here.
Codex C is the most
interesting one
because it was lost
for a long time.
We knew about
this codex,
but we didn't know
where exactly it was.
It turned
up in 1906,
and then was lost again
for almost another century,
and it was bought, in 1998,
by an unknown buyer
for $2 million.
So, they were able to
then look at this codex
and actually find
two more works
that were not in
the other two.
These were known in 1906, so
they weren't found in 1998,
but they were read
more clearly in 1998.
What you see here is
actually a prayer book.
What happened was the
original codex was scraped off
and then reused
as a prayer book,
in I believe the 13th
century, Greek Orthodox.
But basically,
the codex was taken,
and it was made
of vellum,
so it was a type
of goat's skin,
and they actually
scraped off the writing,
turned it on its side,
and then rewrote over it.
So what you see is the
writing on top goes this way,
but the actual Archimedean
writing goes this way,
down along
the page.
So again, that's about
all I can say about it
because I'm gonna
run out of time.
But this book
is really good.
I would definitely
recommend that.
"The Sand Reckoner," which is
what we're gonna talk about,
appears only
in Codex A.
And you see that in the list
of all these different works,
"The Sand Reckoner"
kind of stands out.
You look at all these-- there's
the "Sphere and the Cylinder,"
"Measurement of a Circle,"
"Conoids and Spheroids."
It's a lot of geometry, and
mathematical-type treatises.
But "The Sand
Reckoner,"
that sounds a little
unique and unusual.
And it was.
"The Sand Reckoner"
was actually written
to the king at the
time, King Gelon.
I hope I'm saying
that right.
I had to look up
how to say this
because I didn't know if it
was "gell-ion," or "jell-ion."
If you look back
at the Greek,
it is a gamma followed
by an epsilon,
so it should make
a "juh" sound.
But anyways.
King Gelon is who
it was written to.
And so, it's
really not--
it's written for a popular
audience, essentially,
in an understandable way.
And so, it's a
perfect introduction
to Archimedes' work, because
it's not really heavy in proofs
and very rigorous
methods,
but it's laid
out in a way
that essentially starts from
these very basic assumptions,
and it works towards
something incredible,
which is essentially showing
the size of the universe,
and how many grains of
sand can fit in there.
So he starts out this
letter to the king.
"There are some,
King Gelon,
"who think the number of the
sand is infinite in multitude."
So, why would
anybody think
that the sand is infinite
and multitude?
Well, we have to look at the
Greek numerical system here.
So the way that the Greek
alphabetic numerals worked
was essentially each letter
represented a number.
And so, this shows
all of the letters
of the modern
Greek alphabet,
but then there are
three extras here
that we no
longer see.
But each of these just
represented a place value.
So I think it's
easily explained
but just by an
example here.
So let's say we need to
do something like 5,432.
Well, 5,432 is just
the same thing
as 5,000 + 400
+ 30 + 2.
So if we know
the alphabet here,
we can then just match up
these different values here.
Well, 5,000-- I don't
have it on this chart,
but you can see up
on the top there.
For any
thousandths place,
you just had to reuse
the 1 through 9 numerals,
and just stick a
little mark on the top.
So 5,000 would be an epsilon
with a little mark.
400 would be
an upsilon.
30 would be
a lambda.
And 2 would
be beta.
Now, if you
notice here,
I'm writing these
all in capitals.
What I found in my research was
sometimes they use lowercase,
and sometimes they
use capitals.
I think it's kind of like
how we write our own numbers.
We have our different ways of
kind of writing our digits
a little bit
differently.
I think it's dependent
upon the author.
So-- but a lot of times, I
was seeing a lot of capitals.
So this kind of poses a problem
because how could you tell
whether this is a number
or this is a word?
Well, they could
differentiate this by--
over their numeral,
they would put a line.
Usually they wouldn't do it
on the thousandths place
because it has
the mark up here.
But for the rest of it, they
would just do a line over top.
And what you see in printed
Greek from the time,
it's a little bit
different,
but handwriting, they would
have written it like this.
So we have some
limitations to this,
because as we
move up the way,
it's really nice
to do just 9,999,
just by looking
at this.
10,000, they went up to
what's called a "myriad."
Well, a myriad is
just a capital mu,
and then they would write how
many myriads there were on top.
So if you have 10,000,
that is just one myriad,
so you're gonna write
a one myriad there.
If you needed to
go up to 100,000,
well, that is
actually 10 myriads,
so you would do a myriad,
but you would do 10 of them.
10 is iota.
Now, this could continue,
up until we would do...
100 million.
Whoops--
in our way.
Well, how
much is that?
You can just count the number
of zeroes in fours here.
So you have the
one myriad,
and you've
got the two.
So you actually have
a myriad myriads.
So you would think that
you would want to go mu
with a mu
on top.
But actually,
what they would do
was write the mu next to it,
and put a line on top.
And beyond that, they
wouldn't do anymore.
That was it.
You could potentially
just keep making
smaller and smaller
little nested things
in a tower
of numbers.
They didn't do
that, though.
This is where
they stopped.
And so, when you think about
a number system like this
that only goes up
to 100 million,
it's no wonder when you
sit and try to count
a bunch of tiny
little things
that they would think,
"Oh, maybe this is actually
"an infinite
quantity,"
or "Maybe there aren't
numbers that go this high,
"if there is just
a finite amount."
So this is what Archimedes
wants to do, is say,
"Actually, it's
probably not infinite.
"What I want to do is
actually make a number system
"that's able to
express numbers
"greater than the amount
of grains of sand
"that could fill the
entire universe."
So not just the Earth,
but the entire universe.
Now, that begs the question,
"How large is the universe?"
So what does that even
mean in a Greek context?
Well, what we're gonna
do is first talk about
how "The Sand Reckoner"
is laid out here.
So there you have "The
Sand Reckoner" in Greek,
and in Latin.
And it's basically broken
down into six different parts.
The main part that we
are gonna concentrate on
that will take the
longest amount of time
is this proof for the
diameter of the sun.
But the way that
this is set up
is Archimedes essentially
builds towards a large climax.
What you're gonna hear
here is probably like,
"Why is Jeff up here
talking about this?
"Why is he doing this in
such a roundabout way?"
Well, that's the way that
Archimedes lays it out.
He really is
a showman.
He actually writes so that
you eventually get to the end,
and you're like, "Well
that came out of nowhere."
And it's really
entertaining,
so that's what I'm gonna
try to do for you guys.
So, let's
jump right in.
First, we need to talk
about what was the cosmos
to a third century BC
person in Greece.
Well, he writes that
the common account--
there it is
in Greek--
was a sphere with a radius
from the center of the Earth
to the center
of the sun.
So it was a
geocentric model
where the sun revolved
around the Earth.
Now, Archimedes says
this is all good,
but there's actually another
idea that I know about.
There's this guy named
Aristarchus of Samos,
and he wrote
a book.
He said, "His hypotheses
are that the fixed stars
"and the sun
remain unmoved,
"and that the Earth
revolves about the sun
"in the circumference
of a circle."
Think about that
for a second.
This is a
heliocentric model.
The sun at the center,
the Earth revolving around.
Think about this-- this
is third century BC.
This is 1,800 years
before Copernicus.
If that's not the most
interesting thing you hear
in this seminar, that's--
(scattered chuckling)
I mean, I think
that's really cool.
Standalone, even without
any of the mathematics here,
this is the
coolest thing.
This is the only
place we have
that mentions Aristarchus
talking about this.
We have one extant work
from Aristarchus,
and he does not
mention this model.
That's probably because
the work that we have
is an earlier work.
But if we did not have
"The Sand Reckoner"
we would not
know about this.
So, Aristarchus says--
he goes on to say that,
"Actually, the fixed stars
are much, much greater
"than what we
really think."
They're way
out there.
What he said was
the fixed stars
are to like the
Earth's orbit
as the surface of a
sphere is to its center.
So think about
that for a second.
If you're talking about
the center of a sphere,
you're talking about
something with no magnitude.
There's nothing there--
it's just a speck.
There's nothing.
So how can you compare the
surface of a sphere to nothing?
Well, what he's
really trying to say
is that it's
so vast
that the orbit of the
Earth is so small,
it's almost
negligible.
But Archimedes
interpreted it another way
to make it
useful to him.
So what he said was
the ratio of the Earth
to the cosmos in
the common account
is the same as the cosmos
is to the fixed stars
in Aristarchus' model.
So this ratio
he's gonna use,
what he's gonna do is he is
going to calculate, essentially,
how big the cosmos is,
and then, at the end,
he's gonna use this
ratio to figure out
how big the
fixed stars are.
So remember
this ratio.
We'll use
it again.
All right, so Archimedes
has to start somewhere,
so he's gonna make a
couple of assumptions.
I have them
written over here,
so that we can
see them as we go,
because we'll come back to
them, and use them again.
But it's really
a Greek method
of stating your axioms,
your postulates,
and then building
off of those.
So what he's gonna do
is first of all say
that the circumference
of the Earth
is less than or equal
to 3 million stadia.
So I have a
P over here,
but it's just for
perimeter of the Earth.
Same as
circumference.
Now what is
a stadia?
That's gonna be a unit that
we're gonna see a couple times.
There's some disagreement of
how long a stadion should be.
I have the word "stadion" up
there instead of "stadium."
Stadium is Latin,
stadion is Greek,
so we're going
Greek today.
But one stadion is 185 meters
as the accepted value.
There's some-- some
people differ on that,
but we'll
go with that.
And so, when you look at
these different estimates
from the
time period,
Aristotle said that
it was 400,000 stadia.
Eratosthenes had a
very famous experiment
that I don't have
time to go into
that he was able to make
a really good estimate.
But he said
250,000 stadia.
Now, Archimedes,
what he says
is he actually wants
to go further.
So he's gonna way
overestimate this
so that he can make his upper
bound as big as possible
so that at the end, he
can have his numbers
be as big
as possible.
So the actual circumference
of the Earth, in meters?
Well, it's about
40 million meters.
So Archimedes,
far above.
Eratosthenes,
pretty close.
Assumption 2-- the
diameter of the moon
is less than the
diameter of the Earth,
which is less than the
diameter of the sun.
He says, "In
this assumption,
"I follow most of the
earlier astronomers."
Essentially saying,
"Eh, it's trivial."
So I think
we know that.
Assumption 3-- the
diameter of the sun
is less than or equal to 30
times the diameter of the moon.
Well, he also states a couple
other people who made estimates.
Eudoxus is known for his
method of exhaustion.
He said about
nine times.
Now, Phidias-- notice
the note I have here--
Archimedes' father.
So, the line in
"The Sand Reckoner" is,
"Now Phidias,
my father."
Well, it's just that little
tiny mention of his father,
and that's the only
thing we know about him.
So "The Sand Reckoner"
is also--
that's the only place
that we have mention
of Archimedes'
father.
We knew that his father was
some kind of astronomer,
and that he said the
diameter of the sun
is about 12 times the
diameter of the moon.
And that's
all we know.
So again, just one
mention there of that.
And Aristarchus gave
one of 18 to 20 times.
So again, Archimedes says,
"I'm gonna go higher than this
"because I want to make my
upper bounds really large."
Well, didn't quite
go large enough,
because the diameter of
the sun is actually greater
than 400 times the
diameter of the moon.
So, you can kinda
see the pic here.
The moon is just a speck
compared to the sun.
But that's okay, because a
lot of his other estimates
are so much larger,
it's gonna be all right.
Assumption 4-- he says that
the diameter of the sun
is greater than the
side of a chiliagon
inscribed in
the cosmos.
So a chiliagon is a
regular 1,000-sided polygon.
"Regular" just meaning
all the side lengths
are the same length,
and 1,000 sides.
Well, when you look up
a picture on Wikipedia
of what a chiliagon
looks like, it is this.
So, to the
naked eye,
it essentially just
looks like a circle.
But if you can imagine 1,000
little tiny, tiny side lengths
all around there,
and then imagine the
diameter of the sun
being a little bit greater
than one of those side lengths.
I have this
over here,
because we're thinking
about shapes being inscribed
in other ones.
So any time you
inscribe a shape,
it is on the inside
of that circle.
So even though that
looks like a circle,
imagine that being
inscribed in a circle.
Okay.
So I have quotation marks
around "Assumption" here,
and I have a star on
Assumption 4 here,
because it's not
really an assumption.
Actually, Archimedes
proves this.
And that's gonna be the
big proof that we do here.
But before we can do a
proof of Assumption 4,
what we're gonna talk about
is the experiment that he does
to figure out the angular
diameter of the sun.
So, his experiment is
essentially to verify
Aristarchus' finding that the
angular diameter of the sun
is about
half a degree.
"Angular diameter"
just meaning
the angle that subtends
the length over here.
So if this
is the sun,
then the angle here is
the angular diameter.
So the known value
that we have today
is right around
half a degree.
So actually, this is gonna
be a pretty good estimate.
What Archimedes
notes, though,
is that neither the eye,
nor the hands,
nor the instruments
for measuring the angle
are reliable enough for
determining it exactly.
So following
the trend,
he's not gonna find
an exact angle.
He's gonna bound it--
upper and lower bounds.
So how does
he do this?
Well, what he does is
he goes out one morning
and he watches
the sun rise.
So imagine this is the sun,
just rising over the horizon.
And what he does...
is he takes a cylinder,
and he has a rod to
measure the length of that.
He puts the cylinder,
and he puts his eye right
along with that cylinder,
and he's gonna
move the cylinder
until his eye can
just barely see
the sides of the
sun on either side.
So what he's doing is,
if his eye is right here,
and he knows
this length,
he can just barely see the sun
along that edge of the cylinder.
It's making essentially
a tangent there.
It's only touching
it at one point.
So what he's
able to do, then,
is if he knows the
radius of the cylinder,
he can then
measure this angle.
And so could we.
You could go out and do
this if you wanted to.
Because you can just
use trig right here.
This is a
right angle.
You know this side,
you know the hypotenuse,
so you can do
opposite hypotenuse,
you can do the
sine of that.
Archimedes didn't
have trigonometry,
so there's some debate on how
he calculated this upper bound.
You can look into this
article by Shapiro here.
He goes through
the mathematics
of what Archimedes
possibly could have done.
But a lot of people
believe that Archimedes
would have just actually
drawn the diagram to scale
and then measured
it just by hand.
So 1 over 1/64
of a right angle
is a little over
half a degree.
Now, for the lower bound, this
was also very interesting.
What he says is, "The eye
does not see from one point
"but from a
certain area."
So Archimedes
actually adjusts
for essentially the
anatomy of your eye.
So what he's saying
is when you look here,
the ray of light
isn't just coming
from a point
at your pupil.
It's actually going
behind the eye.
Now, Archimedes
probably didn't know
the anatomy of an eye
like we do here,
but he knew that the
light is passing through
and being processed somewhere
within your head, essentially.
And so, to
account for that,
he then takes
another cylinder,
which is no bigger than
the width of your pupil,
and he puts it where
the eye would have been.
And then, he measures the angle
that then is tangent to that.
So that gives us a lower bound
of about 1/200 of a right angle.
So again, there's
mathematical ways
on how he could
have done this,
but it's probably
that he drew it out.
So we have a lower
and upper bound.
I have that
written up there.
Angular diameter
of the sun,
because we will definitely be
needing that in just a moment.
So now, we are ready
to prove Assumption 4.
And this is gonna
be the long proof.
I'm gonna go through
it pretty rapidly.
That's just because
I only had an hour.
If I had two hours for this
seminar, that'd be great...
but I know you guys don't
want to sit that long.
So, this is the
scary diagram
that you saw on
the handout,
and this is the scary
diagram that appears
in the Heath translation
of "The Sand Reckoner."
Looking at that, that's
pretty intimidating,
so let me break
it down fro you.
I did this for you guys, so it
wouldn't be as much hard work.
I could have just been like,
"Oh, it's just trivial."
No.
(audience chuckling)
So what we are
gonna do is--
well, let me
back up a sec.
So Archimedes, first of all,
tells you how to draw this.
So the smaller circle
represents the Earth,
larger represents
the sun.
You have the eye where he
would have been standing
as the sun
was rising.
So that's important because
if you're viewing the sun
just above
the horizon,
you are essentially looking
at a tangent right here.
And then, there's a bunch of
other tangents through here.
I put the right
angles in.
So what we need to
do with this proof
is prove
Assumption 4.
Now, we need to prove
that the diameter of a sun
is greater than a chiliagon
inscribed in the cosmos,
which is going-- the great
circle going through AOB.
But you notice that there
isn't actually the diameter
drawn in this diagram,
at least not directly.
So what we're actually
gonna prove first
is that AB is actually equal
to the diameter of the sun.
All right, so let's
get started with this.
So we are going
to prove...
AB is greater than
the side of a 1,000-gon
and I'm just gonna
write "in the cosmos,"
inscribed in
the cosmos.
So the first thing
we need to note here
is that this length,
EO, is less than CO.
And we know
that because
the sun is just coming
over the horizon.
So proof,
to start out,
note that EO
is less than CO.
So, what does
that mean?
Well, I highlighted
those angles for you.
The green
angle PEQ
then needs to be greater
than the angle FCG.
So angle PEQ needs to be
greater than angle FCG.
But what do we
know about PEQ?
Well, PEQ-- this vertex
is where the eye was,
looking at the
rising of the sun,
so we actually
know that.
That's the angular
diameter of the sun.
So PEQ is greater
than a right angle...
but less than 1/164
of a right angle,
"R" just being
"right angle" here.
So if PEQ is
greater than FCG,
and PEQ is actually
less than this,
then that means
that angle FCG
needs to be less than
1/164 of a right angle.
Yup.
And so, what is 164th
of a right angle here?
Well, if we just kind of
look at this on the side,
a right angle is
90 degrees, over 164.
If I were to take the
circumference of a circle
and split it up into 164--
er, 90 over 164 here,
I'm essentially just doing
360 in the circumference,
just doing like a
unit conversion here.
90 goes into 360
four times,
and then 4
times 164
is 656 of the
circumference.
So that would be how many parts
of a circle there would be.
So if we're talking
about the angle here,
that would
have to be--
if the angle is less than
164th of a right angle,
and we're talking about the
side that is inscribed there,
that means that
the side AB...
needs to be less than
the side of a 656-gon,
inscribed in
the cosmos.
And the reason we see that,
I should mention--
I didn't write
that down--
angle FCG is actually
the same as angle ACB.
And obviously ACB,
if you look at A to B,
that is that
red line.
So next, here, we
want to show that AB
is actually less
than 100th of CO.
Well, we know-- we want
to see basically how much
the side of a
656-gon represents.
Well, the side
of a 656-gon...
has to be less than the
circumference of a circle
divided into
656 parts.
The circumference of a circle
is 2 pi r, in our notation,
divided into
656 parts.
But remember what
we said about pi.
Archimedes knew this,
and we know this--
pi has to be less
than 3 and 1/7,
or 22 over 7.
So we can replace
pi right here.
So 2 times
22 over 7...
44 over 7...
over 656, r.
Dividing that out,
we get 11 over 1,148, r.
And it turns out
that 11 over 1,148
is less than
100th, r.
So, if we're talking about
AB being less than that side,
that means AB needs to be
less than 100th of the radius
of whatever circle
it's inscribed in.
Well, it's inscribed
in this circle.
What the
radius there?
The radius there
is just CO.
So, I'm gonna kind of,
just to save board space...
Therefore, AB has to be
less than 100th of not r--
we can actually
just write CO.
Okay, so now, we
need to show that
AB is actually equal to
the diameter of the sun.
So what we need to
see here is that...
Note that...
these two gold lengths
are the same.
They go out from the center
to the edge of the circle.
So CA is
equal to CO.
And we also have
right angles here.
We see a right angle
from OF to CA.
So OF is
perpendicular to CA.
And we also see one
dropping down, AM,
is perpendicular
to CO.
Well, what
does that mean?
Well, if we have these sides
equal and these angles equal,
what we essentially have
are similar triangles.
And so, that
means that OF...
Uh, let me write
that backwards.
I'm actually gonna
write it this way.
AB and OF
are the same.
Excuse me,
not AB-- AM.
The two purple
sides up there.
But what is OF?
We can see here that
OF is actually
the radius
of the sun.
Now, look at how AB
relates to AM here.
AB is just twice--
2 times AM.
So if AM is the radius of
the sun, and you double that,
that means that AB has to
be the diameter of the sun.
Okay, so we now
know that AB
is the same as the
diameter of the sun.
We just now
need to show
that it is greater than
the side of a 1,000-gon
inscribed in
the cosmos.
Well, from our
Assumption number 2,
we know then that the
diameter of the Earth...
is less than the
diameter of the sun.
Whoops.
And we just show that
the diameter of the sun
is the same
as AB.
But we know that AB is
less than 100th of CO.
That's all we need to
show on that slide.
Just connecting
the dots.
Now, this step...
what we're gonna do is break
CO up into different pieces.
So CO is the distance
from the Earth to the sun.
And we see here that
if we take CH
and add it
to KO,
we know that that has to be less
than the diameter of the sun.
So we see
CH plus KO
needs to be less than
the diameter of the sun,
and the diameter of the sun
is less than 100th CO.
Now what about
the rest of that?
HK right here.
If we add that
to CH and KO,
that needs to make
100 out of 100 of CO,
all the
length of it.
So HK needs to take up
the rest of the room here.
It needs to be greater than
the leftover 99 over 100, CO.
But how does that relate
to what we're going for?
To get CF over EQ,
well...
Let's note that, again,
this is because the sun
is just over
the horizon.
EQ has to be
greater than HK.
And CO, going all the way to
the edge of the circle here,
needs to be
greater than CF.
So combining all
this together,
EQ needs to be
greater than HK.
But HK is greater
than 99/100, CO.
And CO is
greater than CF,
so that means
99/100, CO,
needs to be greater
than 99/100, CF.
Now, what we can do is just
go ahead and swing the CF,
just divide
it over here.
We get EQ over CF is
greater than 99/100,
and reciprocating this,
we'll get CF over EQ...
and 100
over 99.
Be careful with
the inequality.
Gotta flip
that, too.
All right?
Now, this step,
we are gonna use that
lemma that's on the board.
So, Archimedes would not have
stated the lemma like this.
But I think to understand
it as a modern audience,
we want to just think
about it in terms of trig.
This would have
been well-known.
This was after-- Archimedes
lived after Euclid.
Euclid would have shown
this in his "Elements."
But what we're gonna do
is say, "By Lemma..."
We're gonna let alpha
be equal to angle OEQ,
and beta be
equal to OCF.
Then, we know that
alpha divided by beta
is less than the
tangent of alpha
divided by the
tangent of beta.
So alpha
over beta...
needs to be less than
the tangent of alpha.
Well, if you've got right
angles here and here.
So alpha is
the green here.
So the tangent,
opposite over adjacent,
that is OQ
divided by EQ.
And then, the
tangent here,
opposite is OF,
adjacent is CF.
But...
these two red
lines-- OQ, OF--
they're the radius
of the sun.
They're equal.
So OQ is
equal to OF.
Therefore, these two
quantities are equal.
Reciprocate
this back up,
and you see that
OEQ over angle OCF
is less than
CF over EQ.
All right, that's
the challenging steps.
We're on the
home stretch now.
You guys
bored yet?
So think about this for a sec
before I go onto this step.
This was all written in words
in "The Sand Reckoner," right?
They didn't have this
kind of manipulation.
So "The Sand Reckoner"
is about 10 pages long.
I wonder how much this
would have actually taken up
in the original Greek.
We are lucky to
have this notation.
So, we are on the
home stretch now.
What we can
do here
is we can actually just
double up both these angles.
So...
2 and 2,
so double up OEQ,
and double up
angle OCF.
We see that if we
double OEQ, we get PEQ.
Well, that's a pretty
familiar angle.
We've worked with
that already.
And angle
OCF here
actually turns into ACB
when we double it.
And that's nice...
because ACB is gonna
be related to AB.
So let's keep
going here.
Well, we then know that
if we double both of these,
it doesn't change the
actual ratio between them,
so that is still
going to be--
do I have that in the
right direction?
Yeah.
Is going to be
less than 100th,
99-- oh, sorry.
CF, yeah.
CF over EQ,
but then CF over EQ is
less than 100 over 99.
You can tell that even
with our notation,
I'm getting lost
in all this.
Now, PEQ, again, is our
angular diameter of the sun.
So we know
that PEQ...
needs to be greater than
a 200th of a right angle.
So, what we can do
is then just say...
Probably don't need to write
that twice, but that's okay.
What we can do is actually
just multiply this up here,
and then say that that is less
than 100 over 99, angle ABC--
ACB, excuse me.
Because what
we can then do
is just take these
two end pieces,
and multiply this, multiply
its reciprocal over here.
And so, angle ACB
needs to be greater
than 99 over 100,
times 1 over 200.
That is 99 over
2 with four zeros
of a right angle.
And it turns out that this
is greater than 1 over 203s
of a right angle.
1 and 203rds.
Hopefully, I'm--
I don't know.
But that's kind of
an unusual number
until we see
the last step.
Again, Archimedes does this
huge reveal at the end.
And so, what is this
1/203rds of a right angle?
Well, we'll do that
unit conversion again.
Think about it as
a circumference
being 360 degrees.
So 90 into 360
is 4.
So you're looking at
1 over 812 of a circumference.
And so, if we're
talking about angle ACB,
and AB right here being
inscribed in that circle,
we see that AB
needs to be greater
than the side of an
812-gon in the cosmos.
And if you have a polygon
that's inscribed in a circle
in the cosmos, if you
then take that same circle
and make it 1,000
sides inscribed,
1,000 is going to be-- each
side is going to be less
than the sides
of the 812-gon.
So AB needs to then
be greater than...
one side of 1,000-gon
in cosmos.
End of proof.
So, hope you guys
are still awake.
I wanted to write that out
in full, not to show off,
but to show you guys the depth
that goes into this work.
This was all
written in words.
There was not actual
algebraic manipulation
like we did here.
And actually, the way that
the Greek was written,
there weren't even
spaces between the words.
So imagine trying
to translate this.
So, from here on out, it's a
little bit easier sailing.
Hopefully, you got
your money's worth
coming to a
math seminar.
We can now talk about
how big the cosmos is.
So, this is the
two-column proof,
and I had originally
done this by hand
but I moved it
to the handout,
so I'm gonna go through
this pretty quickly,
but go ahead and digest
it slowly on the handout.
So we need to prove that
the diameter of the cosmos
is less than
10 billion stadia.
So, using two of our
assumptions here,
we can combine them, and just
say that the diameter of the sun
is less than 30 times the
diameter of the Earth.
And using what we
just proved here,
we can then say that the
perimeter of a 1,000-gon
is less than 1,000 times
the diameter of the sun.
And then, combining
lines 3 and 5 here,
we can combine those,
and say that the perimeter
of a 1,000-gon
is less than 30 times the
diameter of the Earth.
Now, the next line here,
I do need to explain
because I just have the
word "Theorem" here.
I say that 3 times the
diameter of the cosmos
is less than the perimeter
of a 1,000-gon.
Well, what I
mean by that...
is this would have been
well-known to Archimedes
essentially as a
trivial fact.
But if you
take a circle,
and you inscribe just
a hexagon in it...
So this is a
regular hexagon.
We know that the
radius of the circle
is the same thing as each
side of the hexagon here.
So the perimeter
of this hexagon
is equal to 6 times the
radius of the circle,
while the radius of the circle
is just half the diameter.
So...
That is just 3
times the diameter.
Now what if we increase
the sides there?
If I double up
the sides to 12...
Is that perimeter going
to be larger or smaller
than the other
perimeter?
Yeah, we just
increased it.
So that means the
perimeter of a 12-gon
is greater than 3 times
the diameter of the circle.
And if you keep increasing
the sides there,
you're gonna keep
getting bigger and bigger
and greater
than that.
So that's what
we have here.
The perimeter
of a 1,000-gon
is bigger than 3 times
the diameter of the cosmos.
Now, using 6 and 7 here,
combine those together,
you can say that the
diameter of the cosmos
is less than 10,000 times
the diameter of the Earth.
And so, that is something that
we want to consider for later.
That's the second line
on the "to be shown."
So, line 8 is just a
repeat from the last slide.
But using
Assumption 1 here--
well, I should say we're
stating Assumption 1 again.
We knew that
already.
The perimeter of the
Earth is greater
than 3 times
the diameter.
Well, that's because
we know that pi
is greater
than 3, right?
Circumference
of a circle
has to be greater than
3 times the diameter.
Combining
lines 9 and 10,
we see that the
diameter of the Earth
has to be less than
a million stadia.
And then, finally,
combining 8 and 11 here,
all we have
to do is,
since we have the diameter
of the Earth here and here,
squishing these together is kind
of how I see it in my brain.
We just multiply
these out,
and get the diameter
of the cosmos
has to be less than
10 billion stadia.
All right, so remember, we said
that one stadion was 185 meters.
So 10 billion stadia is
almost 2 billion kilometers.
Using what we know
about the distance
to the sun
to the Earth,
the actual diameter
of the "cosmos"
would be about
300 million kilometers.
So again, Archimedes is
not going for accuracy.
He's going for
giant numbers.
Okay?
So, now that we know the
size of the universe,
of the cosmos,
I should say,
we can talk about the
new number system
that he comes
up with.
So this is all just kind
of like nested dolls
is the way that I kind
of think about it.
He starts on
the first level
with just what's
called "orders."
So what he's gonna do
is just count by ones
until he reaches
a myriad myriads.
So remember, a
myriad is 10,000.
That's a 1 with
four zeroes.
So a myriad myriads is
a 1 with eight zeroes.
That's the 100 million
that we saw earlier.
Once you do that, you are then
going to take that last number,
and count by that, until you
reach a myriad myriads of that.
So, what that means is you
are essentially just doing
10 to the 8th
times 10 to the 8th.
That goes to be
10 to the 16th.
Now, you do that again-- you
take that as your new unit,
and you count
by that,
until you have a
myriad myriads of that.
So what's
happening here
is you are just increasing
by a power of 8 each time.
And Archimedes wouldn't
have written it that way,
but he knew about
exponential powers.
One of the things
that I had to cut
from my original presentation
that went way too long
was talking about his work
with geometric progressions.
He would have understood, if
he knew the notation here,
that all you had to do was,
like what I just said,
10 to the 8th
times 10 to the 8th.
You just add
your powers.
He knew that.
He was one of the first to
actually write that down
and prove it, and that was
in "The Sand Reckoner."
Now, we're gonna keep
counting like this
until we reach 10
to the 8th orders.
So, 1, 2, 3, and then
10 to the 8th orders later,
we are gonna get
the last number as
10 to the 8th
TO the 10 to the 8th.
So not TIMES 10 to the 8th,
TO the 10 to the 8th.
And that's a
pretty big number.
But then, we go to
the second level.
What we're gonna
do is say
that all that stuff
that we just did,
that is the
first period.
Now, what we're gonna do is take
the last number that we had
and we're gonna
count by that
until we reach as
many units as that.
And then, we're gonna take
the last number there
and keep doing
the same thing.
So we're gonna keep
counting up, and up, and up,
and we are gonna count until we
get to 10 to the 8th periods,
and then,
we get this.
So this is a very
hard-to-decipher table
when you look at it
in one glance,
but this was also on the
handout to digest more slowly.
I hope you guys can
see the pattern here.
This is all what
we talked about
in those beginnings,
just first level orders
until we get to the
last number there.
And then, we
take this number,
and all we do is we
just multiply by that
what we had in
that first set.
And we're gonna keep
doing that over and over
until we get to the
10 to the 8th period.
And the last number we get here
is P to the 10 to the 8th.
Well, remember
what P was.
10 to the 8th,
to the 10 to the 8th,
and then, to the
10 to the 8th.
It's a very
large number.
So, maybe it's hard
to see in the table,
but we can really appreciate
this number system
by looking at how
large these are.
So the last number in the
first order is 10 to the 8th.
That's
100 million.
Doesn't sound like that
much until you think about,
"Well, what if you had
100 million dollars?"
That'd be pretty nice.
(audience chuckling)
The last number in
the first period
is a 1 with
800 million zeroes.
Now, that's not
800 million.
That's 800 million
zeroes after the 1.
The last number
that he states
is a myriad myriad units of
the myriad myriad-th order
of the myriad
myriad-th period,
and that is a 1 with--
what is this?
Million, billion,
trillion, quadrillion.
80 quadrillion
zeroes.
Again, that's not
80 quadrillion.
That's 80 quadrillion
zeroes.
So, if my
calculations are right,
if you write a
zero every second,
and just sit there and
write zero, zero, zero,
it would take you
2.5 billion years
to write out
this number.
Pretty big
number.
>> But the guy
counting the sand,
who would get
there first?
(all laughing)
>> Right.
All right.
(chuckling)
So-- yeah, it's
a long time.
So what we're
gonna see
is actually we won't
need that many digits,
but that's the power
of this number system.
And conceivably,
you know,
he didn't state that
we could do this,
but you could keep building
those things over and over.
Who knows what the next
level would be called,
but you could just keep
doing that, and doing that.
So now, we can finally state
the number of the sand.
So, the last assumption
that he has to make
is how big is
one grain of sand.
And what he does is he relates
it to the size of a poppy seed.
So you think about
very, very fine sand,
and a poppy seed being something
a little bit more coarse
that you can see
with your eyes.
He says that a poppy seed
contains no more
than 10,000 grains of sand,
and that one finger breadth,
if you measured the
width of your finger,
is no more than
40 poppy seeds across.
So what he actually measures
was 25 poppy seeds,
but again, he makes it
a little bit bigger
so that he can get
those upper bounds.
Now, remember here that we
are talking about a volume
in this first inequality,
and a length in the next one.
So what we need to
say then is that
if we took a sphere that has a
diameter of a finger breadth,
that means that we would be
less than 64,000 poppy seeds.
So, the way to understand that,
if I can do it really quick.
If you look at--
let me draw here.
If you just
have a sphere...
and its diameter is
just a finger breadth,
and you take
a cube...
with a length of
a finger breadth.
And so, obviously, you
can find the exact volume
of that sphere, but again, we're
talking about inequalities.
If you just cube
that length,
the volume of this sphere
has to be less than
the volume
of this cube.
So that's why he's
just using 64,000
and not just the volume
of a sphere here.
And then-- so that's
how many poppy seeds,
so then he has to say how
many grains of sand that is.
Well, if you
know that there's
less than or equal to
10,000 grains of sand,
you just need to
multiply by 10,000.
So, what Archimedes
does is he just starts
at a sphere with a diameter
of a finger breadth.
And we knew that that was
gonna be less than 6.4
times 10
to the 8th.
Now that, obviously, has to
be less than 10 to the 9th.
So what he's gonna do
is he's gonna keep
multiplying his
diameter by 100,
which, in turn, would
multiply this by 100 cubed.
100 cubed is
10 to the sixth,
so this just
goes up by 6.
And so, he keeps increasing
this by 100 each time.
This keeps going up by 6s until
we reach 10 billion stadia.
So this is a sphere with a
diameter of 10 billion stadia.
Well, that's
what we said
was the upper bound
for the cosmos.
So, that means
that the cosmos
should contain less than
10 to the 51 grains of sand.
But that is
only the cosmos.
Right, this is the
common account cosmos,
with the Earth
in the center.
So, remember, we have this
ratio from the beginning.
Aristarchus said-- well, as
Archimedes interpreted it--
that the Earth
is to the cosmos
as the cosmos is
to the fixed stars.
And we know that the
diameter of the cosmos
is less than 10,000 the
diameter of the Earth.
We proved that in our
two-column proof.
All we have to is
just take that ratio.
Well, the cosmos is
to the fixed stars
as the Earth is
to the cosmos.
Now, again, these
are just lengths,
just straight
line lengths.
We needed to talk
about that in volume,
so we need to cube that in
order to go from the diameter
of the cosmos
to the cosmos.
So, remember, we
said that the cosmos
was less than
10 to the 51.
And now, we need to multiply
that by 10,000 cubed.
Well, 10,000 cubed
is 10 to the 4th
to the 3rd.
That's 10
to the 12th.
And so, the universe,
the fixed stars
can hold up to 10 to
the 63rd grains of sand.
So, let's talk about
how big this number is.
Well, it's hard
to find things
in the universe
that are that big,
so sometimes, you have to
kind of relate it to time,
or ways of doing things,
you know, combinatorics.
So, if you take
3.7 billion years,
that's about 4.3 times
10 to the 17th seconds.
Well, what is
13.7 billion years?
It's the age of the
universe, give or take,
probably a
couple years.
That's a
long time.
Archimedes number
is bigger than that.
Now, the diameter of
the observable universe
is about 10 to
the 28th meters.
I'm not an astronomer-- I don't
know how to calculate that.
That's just something
I looked up.
Still, I would imagine
that the diameter
of the observable
universe is pretty large.
And finally, there are
about 10 to the 80th atoms
in the observable
universe.
That's called the
"Eddington number."
Now, some people have
actually shown, give or take,
how many atoms would be in
Archimedes grains of sand,
in just one grain of sand
there, and then said,
"You know, actually, Archimedes
got it about right,
"and his actual estimate
was pretty close
"to the 10 to
the 80th atoms."
That's just
a coincidence.
But still.
Think about-- again,
I keep saying this,
but 3rd century BC
Greece.
He's using this
large of a number
as an estimate
of the universe.
Now, that's only how many
grains of sand he said
that the universe
could contain.
Remember, though, that he had
a lot bigger number there.
Remember the last number
that was stated
was P to the 10
to the 8th?
So that's your 10 with 80
quadrillion zeroes after it.
Well, there's some
pretty big numbers here.
52 factorial is how
many different ways
you can reorder a
standard card deck.
Archimedes' number is
far bigger than that.
There are 10 to the 120th
possible chess games, at least,
so that's the
minimum there.
They call that the
"Shannon number."
And if you've
ever played Go,
we're talking about
a 19 by 19 Go board,
there are over 10 to the 10
to the 48th possible games
that you could
play there.
That's an insanely
huge number.
That's bigger
than that one.
But then, just because
I do want to talk
about large
numbers here,
there's something
called "Graham's number"
that is
inconceivably huge.
I debated whether to even
say anything about this,
but if you're interested
in large, large numbers,
look up this
number.
It set the Guinness
Book of World Records
for the largest number used
in a mathematical proof,
in the '70s,
I think.
It's probably been surpassed
since then, I'm certain.
But this thing has
its own notation
for how large
it gets.
So there's a link
on the handout.
Look it up--
it's amazing.
So if you think
this number is large,
be prepared
to be amazed.
So, Archimedes ends
his paper with,
"I conceive that these
things, King Gelon,
"will appear incredible to
the great majority of people
"who have not
studied mathematics,
"but that to those who
are conversant therewith
"and have given thought
to the question
"of the distances and
sizes of the Earth,
"the sun and the moon,
and the whole universe,
"the proof will
carry conviction.
"And it was for this reason
that I thought the subject
"would not be inappropriate
for you consideration."
And I agree.
That's all I've
got for you.
(applause)
>> Was there any report of
how the king received this?
>> (chuckling)
Uh, that?
No, I didn't find
anything about that.
>> You know, you mentioned
earlier about the fact
that there was no
modern notation.
No algebraic
notation.
The Greeks are not known
for computing with numbers
generally
anyway.
Is there any indication
of how, or what,
he actually
would have done
to come up with a
demonstration like this?
Was it all just
in his head?
Because I know he like to
draw things in the sand.
Okay, well, drawing
this in sand
doesn't seem a very
likely way to do it.
Do we know anything about
his thought processes?
I mean, it's
so long ago.
>> Yeah, your question
is how could Archimedes
have thought of this,
how did he go about it?
Yeah.
Well, I'm not an
expert on it,
but there is the work that
he did called "The Method"
which appeared in
that lost codex.
"The Method" is very interesting
because it gives some insight
into how he processed things
and was able to figure it out.
I think some people
assume that he kind of
would have just played
around with stuff,
and would have worked towards a
more rigorous proof on things,
such as, you know, when he
would find the area of a circle
or something
like that.
He would kind of have an idea
of what it needs to go towards,
now how can he show that
it goes towards this.
As for this,
I have no idea how
he came up with it.
(chuckling)
Um...
it's incredible.
And especially the way
that he presents it, right?
'Cause it's almost
like he's making it up
as he goes
along.
It's kind of like, "Well,
where's he going with this?"
You look at
this proof,
and I know that I was kind of
all over the place with it,
but the mere fact that he could
organize something like that
without the notation
the we have, is amazing.
Any other questions?
You got
nothing for me?
Yeah, Patrick.
>> I'm just curious.
Where did you
find the-- was it--
the proof that
you laid out here,
what source did you use
specifically for that?
>> Yeah, this--
well...
I like to think I
did a lot of work
trying to kind of
help it along,
but a lot of it does come
from the Heath translation.
The problem is,
is Heath wrote around
the beginning of
the 20th century.
His works on Archimedes were
originally published in 1897.
So there's some
dated things in there,
and it can be a
little hard to follow.
So I tried to fill
in a lot of gaps.
But there is a proof laid
out in modern notation there.
I didn't read the
original Greek.
(chuckling)
Yeah.
>> Thank you so
much for that.
That was amazing.
>> Yeah, thank you guys for
coming-- I really appreciate it.
>> If there are questions,
I'm sure Jeff would be happy
to hang around
and answer those.
Thank you all
for coming.
We will have our next
seminar in September.
I'm not sure what the
topic's gonna be yet.
But it's-- I haven't
figured out
what we're gonna
be talking about,
but hopefully something
will come to mind
between now and the
middle of September.
Thanks again
for coming.
Hope to see at least
some of you in September.
And please take food
and drink with you
because we have
a bunch.
