Go to Fun_Math Content Table Three Famous Problems

Willebrord van Royen Snell (1580 - 1626),
who is today known for his discovery of the law of reflection and refraction, was a student of
Ludolph van Ceulen (1540 - 1610)
at the University of Leyden.

Ludolph took the same method Archimedes used almost 19 centuries ago, and computed the value of p up tp 35-th decimal place
using 2^{62} -sided polygon ,inscribed and circumscribed.

Snell searched for better lower and upper bounds so that the value of p can be computed using less number of sides of polygon.
And he found the following sets. Although he could not prove the proposition (which was later done by
Christiaan Huygens (1629 - 1695))
, he used this result to verify Ludolph's 35-th decimal place using only 2^{30}-sided polygon.

*************** Snell_Huyg_lb.dwg** *************
*************** Snell_Huyg_ub.dwg** ************

**To create these drawings: **
** Load pi_approximation.lsp (load "pi_approximation")**

Then from command line, type **Snell_Huygens_LB ** for lower bound drawing.

And from command line, type **Snell_Huygens_UB ** for upper bound drawing.

Referenced drawing: Snell_Huygens_lb.dwg BG1 : representing the half the side-length of the circumscribing polygon BG1 = 3r tan b (1) In triangle EOF: Sine law : EF/(sin(p-q) = r / sin b (2) Cosine law: EF |

Referenced drawing: Snell_Huygens_ub.dwg Length ED is set equal to r. Note that angle DEA = q / 3, and this is exactly the same configuration Archimedes used for trisecting the given angle q. EO = 2r cos(q/3) |

What Snell found and later proved rigorously by Huygens is summarized as follows: |

But he was also made great contributions to the progress of mathematics.

At the age of 25, he published a book "De circuli magnitude inventa".

In his book he not only gave a rigorous proof of two bounds found by Snell, but also proved many relations among perimeters and areas of circle and its inscribing and circumscribing polygons.

Some of Huygens result to be added here. |

René Descartes

********************* Descartes_circle_square_desc.dwg** *******************

For detail, go to the section René Descartes.

Kochansky published this result in 1685.

OA = OB = r, and AD = 3r

BC is paralllel to AD, and angle BOC = 30 degrees.

Then CD = 3.141533..

***************** Kochansky.dwg** ***************

**To create this drawing : **
** Load pi_construction.lsp (load "pi_construction")**

Then from command line, type ** Kochansky **

Jacob de Gelder (1765- 1848) published this result in 1849.

This is based on the approximation 355/113 = 3 + 4^{2}/(7^{2} + 8^{2}).

This decimal portion is to be constructed.

CD = 1, CE = 7/8, AF = 1/2

FG is parallel to CD, and HF is parallel to GE.

Then AH = 4^{2}/(7^{2} + 8^{2}).

****************** Gelder.dwg** *****************

**To create this drawing : **
** Load pi_construction.lsp (load "pi_construction")**

Then from command line, type ** Gelder **

Hobson published this result in 1913.
He constructed the approximate value of square root of pi,which is the true "circle squaring".
p^{1/2} = 1.77245...
His constructed length is 1.77246...,which is very,very close!!

OA = 1.0, OD = 3/5, OF = 3/2, and OE = 1/2

Draw the semicircles DGE , AHF with DE and AF as diameters.

The perpendicular to AB through O intersects These semicircles at G & H.

Then GH = 1.77246...

****************** Hobson.dwg** *****************

**To create this drawing : **
** Load pi_construction.lsp (load "pi_construction")**

Then from command line, type ** Hobson **

Go to Fun_Math Content Table Three Famous Problems

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Feb 12, 2007

Copyright 2006 Takaya Iwamoto All rights reserved. .