The Computation of π
by Archimedes
Bill McKeeman
Dartmouth College
2012.02.15
Abstract
It is famously known that
Archimedes approximated π by
computing the perimeters of manysided regular polygons, one polygon
inside the circle and one outside.
This presentation recapitulates
Archimedes' computation, using
MATLAB instead of hand, ink and
papyrus.
The surprise to me was how many
"tweaks" Archimedes applied at
various stages of an otherwise
systematic approach.
follow along…
http://www.mathworks.com/matlabcentral/fileexchange/29504the-computation-of-pi-byarchimedes/content/html/ComputationOfPiByArchimedes.html
MATLAB publication is a sort of self-checking paper. This URL
takes you to the MATLAB version of the talk. Grey background is
the MATLAB code; typeset material is MATLAB comments,
everything else is MATLAB output.
Earlier that 200BCE
Archimedes lived in Sicily, Euclid in Egypt, Pythagoras in Turkey and Italy
Archimedes did other things
Syracuse was often under
attack. Archimedes
invented weapons to
defend his city. This iron
ship-dumper may be a
myth. Other weapons such
as fireballs were not.
When a Roman sneak attack finally conquered Syracuse,
Archimedes was considered by the commanding general as
the most important “catch.” Unfortunately a Roman soldier
killed old Archimedes for not being instantly obedient. Too
bad for the Roman soldier.
A polygon inside the
circle and a polygon
outside the circle.
The perimeters of the
polygons bound the
circumference of the
circle which (by
definition) is 2πR.
The inner hexagon
has perimeter 6R
which implies
6R<2πR, or 3<π.
Increasing the
number of edges
tightens the bounds.
Thus thought
Archimedes.
The Plan
Greek Numbers (1)
Greeks used uppercase letters to represent decimal
digits (lower case is used here for readability when
mixed with English). For example ˏαυκδ′ means 1424.
The prefix ˏ turns α into 1000. The postfix ′ ends the
number (identifying it amongst Greek text).
Greek Numbers (2)
• In case you noticed it, 27 letters are needed for this scheme,
so 3 ancient letters were used in addition to the 24 we know.
• Greeks had no digit 0.
• Greeks had no decimal point.
• Greeks did have rational numbers.
For example: γ′ζ″ means 3⅟₇
• Greek numbers evolved over the centuries.
Greek Math
Finally, algebra was unavailable, so words were used to describe
computations and algorithms. The flavor of ancient presentation
is provided by Archimedes' statement
"The surface of any sphere is four times its greatest circle."
We would write
A = 4πR².
The Computation of π
The result presented by Archimedes translates to
3¹⁰/₇₁ < π < 3⅟₇
This is what we want to compute in MATLAB using only the tools available
to Archimedes.
(I will compromise by using decimal integers, fractions and algebraic
notation. You will thank me.)
The Outer N-gon
The first equation is a consequence of similar triangles. The second
and third equations are algebraic manipulations of the first. The
fourth equation is an application of the Pythagorean Theorem.
The Iteration
The last two formulas on the previous page are
combined into an iteration.
So, if an is known, so is en, and therefore the halfperimeter nen, and, finally, an upper bound on π.
The Computation
Archimedes started with the hexagon. That is:
R = 1; e₆ = R√(4/3);
a₆ = R√(1/3)
It’s easy for MATLAB to check the iteration formula
is converging to 3.141592653589793…:
%
a
π
a₆ = R/e₆
% 1.73205 3.4641
a₁₂ = a₆ + √(a₆² + 1)
% 3.73205 3.31539
a₂₄ = a₁₂ + √(a₁₂² + 1)
% 7.59575 3.15966
a₄₈ = a₂₄ + √(a₂₄² + 1)
% 15.2571 3.14609
a₉₆ = a₄₈ + √(a₄₈² + 1)
% 30.5468 3.14271
e₉₆ = R/a₉₆
Hand Calculation
Unfortunately for Archimedes, he did not have double precision
floating point so he, and therefore we, still have some work to
do. It is simpler for hand calculation if intermediate results are
improper fractions with a large integer part. To make this
happen, the iteration formula is multiplied by an arbitrary
integer constant c (to be chosen later).
MATLAB Fractions
• Switch to “Greek Mode” (rational arithmetic)
• Need to take square roots.
– Archimedes picked approximations that led to an elegant result.
– Nobody knows how Archimedes did square roots.
– So, I’ll skip my speculations here.
• Archimedes stated without explanation:
Its easy to check in MATLAB
assert((265/153)^2 < 3 && 3 < (1351/780)^2)
The Upper Bound (1)
For the upper bound computation, Archimedes
chose the following starting values:
c = 153
a₆ = 256 (a little less than 265.0038)
b₆ = 306
Therefore a₉₆ will be less than the fully accurate
irrational value and e₉₆ will be greater than the
fully accurate half-side of the 96-gon, and finally
96*e₉₆ is greater than the half-perimeter and
therefore greater than π.
The Upper Bound (2)
c = 153
a₆ = 265 b₆ = 306
a₁₂ = 571
Archimedes (very carefully ) chose b₁₂
Note that the difference between the true irrational value
591.143 and Archimedes choice 591.125 is very small
The Upper Bound (3)
c = 153
a₆ = 265
a₁₂ = 571
a₂₄ = 1162⅟₈
a₄₈ = 2334⅟₄
a₉₆ = 4673⅟₂
b₆ = 306
b₁₂ = 591
b₂₄ = 1172⅟₈
b₄₈ = 2339⅟₄
…continuing, Archimedes chose again…
…and again…
The Upper Bound (4)
c = 153
a₆ = 265
b₆ = 306
a₁₂ = 571
b₁₂ = 591
a₂₄ = 1162⅟₈ b₂₄ = 1172⅟₈
a₄₈ = 2334⅟₄ b₄₈ = 2339⅟₄
a₉₆ = 4673⅟₂
Using e₉₆ = c/a₉₆
hi = 96c/a₉₆
= 96*153/4673⅟₂
= 3¹³³⁵/₉₃₄₇
Then Archimedes made another choice, picking the elegant 3¹/₇ because
π < 3¹³³⁵/₉₃₄₇ < 3¹/₇
This establishes the upper bound on π.
The Lower Bound
The lower bound computation is similar. The starting values and
numbers are different. The final result is
a₆ = 1351
b₆ = 1560
c = 780
a₁₂ = 2911
b₁₂ = 3013³/₄
a₂₄ = 5924³/₄ b₂₄ = 1838⁹/₁₁ c = 240
a₄₈ = 3661⁹/₁₁ b₄₈ = 1009⅟₆
a₉₆ = 2016⅟₆ b₉₆ = 2017⅟₄ c = 66
Using e₉₆ = c/a₉₆
lo = 96c/a₉₆
= 96*66/2017⅟₄
= 3¹¹³⁷/₈₀₆₉
Then Archimedes made his final choice, picking 3¹⁰/₇₁
The Result
The values Archimedes provided:
3¹⁰/₇₁ < 3¹¹³⁷/₈₀₆₉ < π < 3¹³³⁵/₉₃₄₇ < 3¹/₇
established the bounds on π. Archimedes wrote
ΟΕΔ
meaning ὅπερ ἔδει δεῖξαι
These days we say “what was required to be proved” or
QED.
Summary Calculation