Go to Fun_Math Content Table Three Famous Problems Greek Circle Squarer Later Squarer

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 π up to 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 π 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.

**lower bound:** Refer to the drawing below-left.

Make EA equal to the radius ( r ) ,where E is on AB extended.

Select any point F on the circle, and let G _{1} be the intersection of EF and the tangent at point B. Then

arc BF > BG_{1}

As mentiond above , this is the approximation of π derived by Cardinal Nicholas Cusanus(1401-1464).

**upper bound:** Refer to the drawing below-right

Choose a point D on a circle. Make DE equal to its radius r, where E is on AB extended.

The extension of DE intersects the circle at point F, and at point G_{2} with tangent erected at B. Then

arc BF < BG_{2}

Note here that we started from point D, then find out point F as an intersection DE and the referenced circle..

If we started at F, then the finding point D becomes a "Trisection" problem..

*************** 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(π-θ) = r / sin β (2) Cosine law: EF |

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

What Snell found and were 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. |

Go to Fun_Math Content Table Three Famous Problems Greek Circle Squarer Later Squarer

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Last Updated July 15, 2012

Copyright 2006 Takaya Iwamoto All rights reserved. .