Go to Fun_Math Content Table Three Famous Problems Greek Circle Squarer

Conon of Samos (about 280BC - about 220 BC)
invented a spiral,the polar equation of which is written as r = a q.

This curve was used by
Archimedes (287BC - 212 BC)
for squaring the circle.

Nowadays this curve is called **Archimedean Spiral**.

In his book " |

***********

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

Then from command line, type **spiral_a **

This will create a Archimedean spiral with incremental angle of 0.001 radian.

Explode the polyline into many small segements, and

extend these small line segements to draw tangent lines
at point P and T.

Open spiral_subtangent.dwg and find out the lenghts of OR,OU, and arc length using LIST command.

From the argument above, it is now possible to get a line segment with length p.

These are the steps to follow:

(1) Draw an Archimedean spiral,starting from point O (at x = y = 0).

Note:for convenience, choose constant "a" such that x-value will be 1.0 when x = 2p

(2) After one turn, the curve cuts x-axsi at point Q.(OQ = 1)

(3) At point Q, draw a tangent line.

(4) This tangent line intersects y-axis at point P.

Length of OP is p

This is equal to the perimeter of a circle with radius OQ.

Drawing a tangent line needs some explanation.

Theorectically speaking, tangent line is defined as a limit of angle q approaching zero.

In computer programming, this is accomplished by zooming to the point (1,0),

and making the angle q smaller and smaller.

**************** tangnet_line.dwg** ***************

**************** Archimedes_circle_squaring_desc.dwg** ***************

You can see the process in animation.

**To create this drawing and animation: **
** Load sqr_circle.lsp (load "sqr_circle")**

Then from command line, type **sqr_circle **

To create jpg files for animation,type **make_anim_jpg**

***** Archimedes_circle_squaring_result.dwg** ****

- Heath,Thomas L. :"History of Greek Mathematics Vol. I From Thales to Euclid" Dover 1981
- Heath,Thomas L. :"A Manual of Greek Mathematics" Dover 1963 original 1931
- Heath,Thomas L. :"A History of Greek Mathematics Vol. II" Dover 1981 original in 1921
- Heath,Thomas L. :"The Works of Archimedes Dover" 2002 original in 1912
- Knorr,Wilber Richard :"The Ancient Tradition of Geometric Problems" Dover 1993
- Knorr,Wilber Richard :"The Textual Studies in Ancient and Medieval Geometry",Birkhauser, 1989

Go to Fun_Math Content Table Three Famous Problems Greek Circle Squarer

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Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved. .