Sometimes, if you want to make progress,
you gotta learn not to sweat the small stuff.
♪ ("Birth of the People" by Demetori) ♪
After Adelard introduced The Elements back into Western Europe in the 12th century,
it rapidly became all the rage with the 0.1% of people who could,
you know, read.
Soon, these happy literate few were doing all the things you do with books,
namely illuminating them, to copies of Euclid.
Then, at around 1450,
Western Europe figured out this cool thing called printing,
and it was off to the races.
Page-turner that it was,
The Elements rapidly became the second most printed thing after the Bible.
Fast forward to a little over a century later
and a fellow in England does this painstaking retranslation of The Elements,
providing the world with a more accurate version,
and the first printed version in English.
Perhaps more importantly, though,
the great mathematician and imperialist, John Dee,
who, fun fact: was also a wizard,
wrote an extensive preface to the text,
explaining it in detail so that even without a university education,
people of the emerging merchants and trade classes could understand the beauty of its logic.
And you know what's even cooler?
When you got to the section on three-dimensional shapes,
it became a pop-up book!
How rad is that for 16th century printing technology?
So now, with its cool whiz-bang 3D technology,
Euclid became the mathematics of choice
for people with access to absolutely no other mathematics.
This became huge for England,
as there are some jobs, like getting a ship across an ocean,
which it'd be really nice if somebody knew some mathematics for.
And the growing pool of people who could understand Euclid
meant that there was a growing pool of people who could do those jobs.
Which brings us to one of my favorite French philosopher-mercenaries,
René Descartes.
He is going to build off of Euclid to create—or at least get credit for creating—
the first system that thinks about geometry in a different way.
But Descartes' path to getting there isn't all that simple.
You see, he was born into the working nobility.
That sort of minor, minor nobility that confused all of Europe during the 16th and 17th century.
They were people who got rich enough to buy some land, but not so rich that they could just live off of it.
This low-level subclass of nobility baffled the lingering medieval understanding of the world
So, they usually ended up with some minor title thrown at them
just so the truly wealthy folks could make sense of them.
Descartes' family, in that vein, was a family of doctors and lawyers,
and he himself was sent off to get an education,
so that he might enter the legal profession.
He went to a Jesuit school where he was taught Rhetoric,
Philosophy, Physics
(as it was understood at the time)
and of course, Mathematics.
Naturally focusing on Euclid.
Then, he shuffled off to university to complete his law degree,
mostly to satisfy the nagging of his dad.
But the instant he was done getting that degree,
Descartes declared that he would seek only the education that the world could give him.
And he took off looking for adventure.
And what more adventurous thing was there to do at the time,
than to sign up as a mercenary, and help the Dutch throw the Spanish out of the Dutch Republic?
So, he did that.
But, of course, he got there during a period of relative peace.
So, he was assigned to be a military engineer,
and was immediately sent to study some more geometry.
And while I am sure that this was somewhat frustrating for him,
it was kind of a lucky break for the rest of us,
because the guy who invented the Dutch system of military engineering
was a polymath who was on the cutting edge of Physics and Mathematics at the time.
Which meant more math for Descartes to learn.
But Descartes soon got bored of all of this not-war,
and legged it over to the Holy Roman Empire to find his next adventure,
which was, apparently, locking himself in a room with little more than a stove.
It must have been some room though, because in it,
he simultaneously invented a system of deductive reasoning
that would basically be the underpinning of all future science,
and developed the Cartesian coordinate system,
which was a system used to universally convert algebra problems into geometry,
and geometry problems into algebra.
This incredible work turned out to be a major sea change for both math and science.
See, up to this point, much of Western science really just boiled down to
Interpreting or theorizing from ancient texts.
Descartes' protoscientific method, however, demanded proof.
Nothing was to be believed or extrapolated from unless, its veracity was undeniable.
This philosophy meant that scientists should no longer just accept what they read in Aristotle or Galen,
but should instead test it.
And if what they read failed the test, they were to discard it and search for better answers.
That type of reasoning coupled with the Cartesian coordinate system,
which vastly magnified the power and scope of mathematics,
opened the floodgates for the science of the next few centuries.
Now that mathematics wasn't limited to only that which could be constructed using a compass and a straightedge,
now that it had the tools necessary to analyze any shape or pattern we might find in nature,
it was just a matter of time until somebody put it all together
in a way that helped us understand our world in a fundamentally mathematical way.
Enter a plucky young English fellow named Newton.
Where Descartes had used Euclid as the foundation for his system,
in much the same way as al-Khwarizmi had,
Newton had a new thought:
What if you thought of geometric construction not as static, but as alive?
Something that wasn't just a single, snapshot image, but ongoing?
Something that could be thought of as a process, repeating forever?
With this thought, Newton gave us the answer to one of the oldest problems in mathematics:
squaring the circle.
You see, Newton wanted to write a book
about this neat law of the physical world he'd figured out, called gravity.
But in order to deliver a book about the orbits of the planets and the effects of gravity on moving bodies,
he was going to have to deal with a lot of curves and the area that a curve encompasses.
And the problem with that is that it was impossible.
Mathematicians had been trying since Euclid's time to square the circle,
to figure out what the square area of a circle might be.
In fact, they had been trying to do that for so long and with so little success,
that the phrase "squaring the circle" had worked its way into literature
as a sort of shorthand term for confounding oneself with an impossible task.
But Newton had a new approach.
He knew what to do.
He would just have to invent calculus.
Yep, this is when, how, and why we get calculus.
Although, honestly, there is a whole other series we could do about calculus' origins,
and a fellow named Leibniz inventing it simultaneously .
Another day, perhaps.
So, what is calculus really?
Well, it's actually just an extension of Euclid
by people who understood Descartes
and needed to figure out how much space was below a curve.
So, let's do the proposition that Newton used,
to give the world the fundamental building block of calculus:
The integral.
First, he sets up a right angle.
Next, he draws a curve from the ends of each of the lines in the right angle.
Then, he says: Draw any number of rectangles with the same size base under your curve,
so that your top right corner just touches the curve.
Now, draw a rectangle from the top of your line segment where you started the curve,
to the point at which your first rectangle touches the curve.
Do the same thing for each other rectangle,
starting at the point where the previous rectangle touches the curve .
Do all of that and what do we have?
Well, a really blocky approximation of our curve.
And you can see that we're off by whatever parts of the second set of rectangles stick out past our curve.
So, how do we solve this?
Well, so far this has all been pure Euclid.
How he constructed the curve and the rectangles, are all things we could have learned from The Elements.
So, Newton is still using Euclid to build up his proposition,
but then he adds his own great insight.
He says: Try doing the same thing, but make more and thinner rectangles.
You see how much less rectangle ends up out over the curve?
Well, and hold on to your hats, because here's the kicker and here's actually how integration works,
he says: Imagine constructing infinite infinitesimally thin rectangles.
Do that and the area sticking out above the curve is now zero.
Or put another way, the area of all of these rectangles
is equal to the area under the curve
That's it!
That's all there really is to the fundamental theorem of calculus.
Sure, we've cleaned up the idea a little in the few centuries since then, but that's really it.
Integration is just figuring out how many infinitesimally small rectangles fit under a curve.
And with that and with some Cartesian coordinates thrown in, to turn it all into algebra,
we can now map the physical world to a mathematical space.
Thus, with his great work, The Principia, Newton started a revolution in scientific thinking.
Soon, we were mathematizing everything, from atomic movement to the movement of the stars.
Physics took enormous leaps forward.
Two hundred years later, we'd split the atom.
But so far, everything we've talked about is still fundamentally rooted in Euclid.
Algebra? Just an extension of Euclid.
Calculus? Really still just an extension of Euclid.
But as math rushed forward, people began to think about that Postulate Five again,
and what it might mean.
What if it was valid, but what if it wasn't the only valid possibility?
Join us next time as humanity finally leaps into true, Non-Euclidean geometry.
♪ ("Postulate 5" by Sean Kiner, Dean Kiner) ♪