Have you ever wondered how pi was calculated before Newton
discovered it. Up until the 250BC, mathematicians knew for a fact that the
value of pi is between 3 and 4. And you may ask that how did they know that,
the answer to that would be that they did that with the help of a hexagon and
square.
A unit circle (area = π)
was inscribed with a regular hexagon of unit sides. And the circle was inscribed in a
square. The hexagon was then further divided into 6 equilateral triangles with
unit sides. Then the perimeter of hexagon was found to be 6 units. Since the
hexagon was inside the circle the circumference of the circle must be a little more than
that of hexagon.
Circumference > perimeter
2 π > 6
π >3
On the other hand, the perimeter of square > circumference of the
circle.
2 π<8
π <4
Hence, it was concluded to be between 3 and 4, i.e, 3< π<4.
Later in the 250BC, Archimedes improved this calculation. He bisected
the hexagon into a dodecagon (12 sides) and inscribed one into the circle and inscribed
the circle in another 12-sided polygon. After calculation, it was found that π
lies between 6.212 and 6.431. This process repeated for centuries until they
reached the 96-sided polygon and found a range of 3.1408< π<3.1429.
In the late 16th century, Francois Viète
doubled a dozen times than Archimedes and computed the perimeters of polygon of
393,216 sides. And by the early 17th century, Ludolph van Caulen surpassed
him by calculating perimeter of polygon of 2^62 sides. He spent 25 years of his
life calculating π upto 35 correct decimal places.
People were used this method of computing pi until Sir Newton came along.
The first step to his discovery of π was building Binomial theorem from the
Pascals triangle. Sooner after he also delt with the theorem being subjected to
incompetency when power of (1+x) was -1, since it would give an infinite
series. But he proved its accuracy by multiplying both sides by (1+x). The left
hand side gave 1 straightaway and when (1+x) was multiplied on the right hand
side, every term got cancelled leaving only one.
Afterward he went for (1+x) with power of ½ to solve for the area of a
unit circle by equation of the circle, which was given by x^2+y^2=1. Hence, the value y=(1-x^2)^1/2 can be calculated by substituting the value of x by -x^2.
Right after this, he also invented calculus. And the integration from 0 to 1
was used to find the area under y=(1-x^2)^1/2 which is the area of the quarter of
circle in the first quadrant. We also know that the area of quarter of circle
is (πr^2)/4. In this case it will be π/4 (since r=1). Upon equating them and calculating we get
the value of π. To make things easier and accurate he decided to integrate from
0 to ½. It gives the area of the circle from the left of y axis to the straight line of
x=1/2. That area can be calculated as area of a sector of 30 degrees, which
will be π/12, and a right triangle of sides 1 and ½ ( we can calculate the
third by Pythagoras).
And upon equating we once again can find the value of π. Not only that
but when you solve up to 5 places of the integration, you get results π for up to 5 decimal places. And it
will take calculation of integration up to 35 terms to get what Ludolph
van Caulen got after 25 years. This is how years of calculation was simplified
by Sir Isaac in a way it took only a few days to learn the value of pi.
SOURCE- Veritasium on youtube
