In ancient Egypt agricultural land along the Nile was divided into square plots
and they were distributed evenly among the citizens.
Each year the citizens used to pay the king rent, or taxes, to use these plots.
But there was a problem:
The Nile was known to flood annually, damaging in the process portions of some people's land
and making them unusable.
So to keep things fair, after each flood, the king used to send his surveyors known as
rope stretchers, or people who stretched ropes with knots,
to measure the area of the damaged land and reduce taxes accordingly.
And from this practice the art of geometry was born.
This story is according to Herodotus, the ancient Greek historian.
Whether or not it is true we have no evidence.
But we have evidence that the ancient Egyptians were one of the earliest people to ever study geometry.
We know this from a number of papyri that, somehow, have survived the ravages of time
and made it through the millennia to our present day.
Most importantly the Ahmes, or the Rhind, papyrus,
a 3500 year old document that contains
problems and solutions for calculating areas of different geometric fields.
But how is all of this related to calculus?
See in the last video we talked about how the two main processes of calculus
are cutting and stitching.
And the history of calculus is the history of the evolution of these cutting and stitching processes.
Which we can divide into five main methods.
And the first method, dissection,
is what is believed to have been used by the ancient Egyptians
to calculate areas and volumes of different shapes.
So let's see this method in action.
We start with a rectangle.
The ancient Egyptians took it for granted that the area of the rectangle is its height times its width.
Although there is no cutting and stitching involved in this example
the rectangle is an important shape for the dissection method as we will soon see.
It's worth noting that the Ahmes Papyrus doesn't state problems in terms of variables
it rather states them in terms of specific numbers.
And the numbers used for this rectangle
is a height of 2 khet, which is an ancient Egyptian unit,
and the width of 10.
And we can actually see this when we zoom into the papyrus.
Where this is the symbol for the number two, which is basically two vertical bars.
This is the symbol for the number 10.
And these are the symbols for khet.
Another shape mentioned in the papyrus is a triangle
with a height of 10 Khet and a base of 4.
The ancient Egyptians used the same formula we use today to calculate the area of the triangle
which is: Its height times half its base.
But how did the ancient Egyptians come up with this formula?
The Ahmes papyrus doesn't explain this but it offers a hint.
It talks about transforming the triangle into a rectangle.
And an easy way to do so, which is believed to have been used by the ancient Egyptians,
is the dissection method where we cut the triangle in half
and then rearrange and stitch the pieces back together to form the rectangle
which the ancient Egyptians already knew how to calculate its area
obtaining the formula for the area of the triangle.
What we just saw is our first step on our calculus journey.
As this simple act of cutting would evolve over the millennia leading to differential calculus.
And the simple act of stitching would evolve leading to integral calculus.
Another shape mentioned in the papyrus is a trapezoid.
Which again it talks about transforming it into a rectangle
which we can achieve using the dissection method.
It is believed that the ancient Egyptians used the dissection method
to calculate areas and volumes of different shapes.
And although they studied the circle,
their formula for the area of the circle was an approximate and not an exact one
and for the most part their geometry was rectilinear,
meaning the shapes they studied were straight edged shapes.
The same was also true of the geometry of other ancient civilizations
like Babylonia
India
and China.
Curvilinear geometry, or geometry of curved shapes,
had to wait for another civilization:
Ancient Greece.
The ancient Greeks didn't study curves for practical purposes
they studied them for intellectual ones.
But they didn't have many curves to study.
They had only around 12 curves!
Still, they were 11 more than what was known at the time.
You may be wondering:
But how come they only had around 12 curves?
Aren't curves everywhere in nature!
See curves in nature don't have specific mathematical definitions
and therefore it is not possible to study them mathematically.
Compare this to the circle for example
which is defined as the shape that consists of all the points in a plane
that are equidistant from a given point.
Similarly, all the other curves that were studied in ancient Greece
have clear mathematical definitions.
Some of them are defined as the curves that are obtained by slicing solid geometric figures like
a cone, which we will talk about soon, or a torus.
Others are defined as the curves that are obtained by some form of mechanical motion
and they are called mechanical curves.
The ancient Greeks divided their curves into three ranks
based on what they thought of as the naturalness of the curves.
The first rank contains the most natural of them all:
the circle.
The second rank, of medium naturalness, contains: the ellipse, parabola and hyperbola .
The third rank, the least natural, contains all the other curves.
They focused most of their attention on the top two ranks
which are collectively known as the conic sections.
Or the curves that are obtained by slicing a cone at different angles.
But how did the ancient Greeks study those curves?
They did so by using the second method on our list:
The method of exhaustion.
The method of exhaustion is basically taking the cutting and stitching processes to the extreme.
It originated with Antiphon in the 5th century BC
when he was working on the area of the circle.
Antiphon started by inscribing a polygon inside the circle.
Now finding the area of a polygon is easy as we can cut it into triangles
and then sum up the areas of these triangles.
Clearly there is a difference between the area of the polygon and the area of the circle.
But Antiphon argued that if we double the number of the sides of the polygon
and then double the number with the sides of that polygon,
and kept going on, eventually we use up, or "exhaust", the circle.
And although he was vague about it, it seems that he meant that the
area of an infinite-sided polygon would equal the area of the circle.
But how can we find the area of an infinite-sided polygon?
Antiphon didn't provide an answer for this question but his way of thinking
opened up the door to eventually arrive at a solution for the area of the circle.
So let's see how.
What we want to do is to stitch, or integrate, an infinite number of infinitesimal triangles.
Now we can't do this directly and we need to find a work around.
So let's start with a simpler polygon, say a hexagon,
or more accurately a regular hexagon, where all the sides are equal,
and see if we can't find any pattern.
The hexagon is made up of six triangles
and the area of one triangle is, as we saw earlier, its height
times half its base.
Which we can rewrite as 1/2 h b
Since we have six triangles, then the area of the hexagon is 1/2 h b times six.
Now "h" is the height of the triangle
but it is also the distance between the side of the hexagon and its center,
which is known as the apothem of the polygon
and we will write it for short as "a".
"b" is the base of the triangle, (b x 6) is the combination of the basis of all six triangles
which is also the perimeter of the hexagon and we will write it for short as "p".
Therefore the area of a regular hexagon or any regular polygon
is 1/2 its apothem times its perimeter.
Now when we double the number of the sides of the polygon
we notice that its apothem gets closer to the radius of the circle
and its perimeter gets closer to the circumference.
And when we keep doubling, they keep getting closer
and closer.
This suggests that at Infinity the apothem would line up with the radius of the circle
and the perimeter would line up with the circumference.
Therefore, the area of an infinite sided polygon, or the area of the circle,
is 1/2 the radius times the circumference.
And when it was discovered later that the circumference is equal to 2 pi r
with basic simplification we obtain the familiar formula for the area of the circle:
Pi r squared.
On the one hand, this method is intuitive and it actually gives the correct result.
On the other hand, it is based on shaky logic as it uses infinity and infinitesimals
and therefore it was rejected by the ancient Greeks
particularly after Zeno, a contemporary of Antiphon,
devised his famous paradoxes that showed the absurdity of dealing with infinities and infinitesimals.
In fact, infinity became a taboo for the ancient Greeks and they developed what came to be known
as the horror of the infinite.
So the method of Antiphon had to be modified for it to be accepted by the ancient Greeks.
This was accomplished by Eudoxus and later improved by Archimedes.
Eudoxus and Archimedes used solid logical arguments to show that our polygon
need not go all the way to Infinity for us to be able to make assertions about the area of the circle.
I'll talk more about their arguments in a later video in the series:
The Rigorization of Calculus
as they are a precursor to the modern idea of the limit.
Archimedes then used this method of exhaustion, or the modified version of it,
to study the areas of the ellipse, parabola and Archimedean spiral,
and the volume of the paraboloid and his proudest work: the volume of the sphere.
But the method of exhaustion has a massive drawback:
it is extremely difficult to implement.
See, with the circle, it wasn't that difficult
as all what we had to do is to keep doubling the number of the sides of the polygon.
But as the curves get more and more complex, knowing how to cut and stitch them back
becomes increasingly more difficult
That's why Archimedes had another method, one based on "the taboo", on infinitesimals
that he used first to discover the formulas of the areas and the volumes of the shapes that he was studying
and then use the method of exhaustion to prove these formulas.
Unfortunately, Archimedes' infinitesimal method was lost only to be rediscovered in 1906
long after the discovery of calculus.
Interestingly, it is similar to the third method on our list:
The method of indivisibles.
Which was discovered over 2000 years after the method of exhaustion
and only few decades before the discovery of calculus.
So what caused this massive gap?
And what sparked this explosive development in the 17th century
that culminated in the discovery of calculus?
This is an important chapter in the history of calculus and it is the topic of our next video.
See you then :)