fuckyeahphysica:

How did the Greeks know ?

Greeks had a strong geometric approach towards problems and as a result their methods are very intuitive.

In this post, we will look at the Method of exhaustion formulated by Archimedes that stands out as a milestone in the history of mathematics

Method of Exhaustion – Archimedes

                                                       Source

In order to find the bounds of pi, Archimedes came up with a remarkably elegant ‘algorithm’, which is as follows:

Lower bound

Inscribe
a n-sided polygon in a circle —> Measure its perimeter(p) —>
Measure its diameter(d) —> pi_min = p/d —-> Repeat with  n+1
sides.

Upper bound

Circumscribe a n-sided polygon in a circle —> Measure its perimeter(p)
—> Measure its diameter(d) —> pi_max = p/d —-> Repeat with  n+1 sides.

And by following this procedure one could obtain the upper and lower bounds of pi !

Heres an animation made on geogebra for a circle of diameter 1. Watch how the lower and upper bounds vary.

Archimedes
did this for a 96 sided polygon and found the value of pi  to be
between 3.14103 and 3.1427. This is a good enough approximation for most
of the calculations that we do even today!

Happy Holidays !