ÿþ<html> <head> <left><title>Approximation</title> </head> <body bgcolor="eeffee"> <hr size="4" color="#0000FF"> Go to <A HREF="./../../Fun_Math_by_CAD.html"> Fun_Math Content Table </A> &nbsp;<A HREF="./../Famous_Problems.html"> Three Famous Problems </A> &nbsp;<A HREF="./../Sqr_Circle/Greek_Sqr.html"> Greek Circle Squarer </A> &nbsp;<A HREF="./../Sqr_Circle/Later_Squarer.html"> Later Squarer </A> <hr size="4" color="#0000FF"> <h2>Approximation </h2> <h3>1. Kochansky </h3> <P> Kochansky published this result in 1685. <p> <IMG SRC="Kochansky.jpg" width="400" height="400" align=left> <P> <h4>Construction steps </h4> <P>OA = OB = r, and AD = 3r <p>BC is paralllel to AD, and angle BOC = 30 degrees. <p>Then CD = 3.141533.. <br clear=left> ***************<b> <a href="Kochansky.dwg">Kochansky.dwg</a></b> *************** <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="pi_construction.lsp">pi_construction.lsp</a> &nbsp;&nbsp; (load "pi_construction")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Kochansky </b> <h3>2. Gelder </h3> <P>Jacob de Gelder (1765- 1848) published this result in 1849. <br>This is based on the approximation 355/113 = 3 + 4<sup>2</sup>/(7<sup>2</sup> + 8<sup>2</sup>). <br>This decimal portion is to be constructed. <p> <IMG SRC="Gelder.jpg" width="400" height="400" align=left> <P> <h4>Construction steps </h4> <P>CD = 1, CE = 7/8, AF = 1/2 <p>FG is parallel to CD, and HF is parallel to GE. <p>Then AH = 4<sup>2</sup>/(7<sup>2</sup> + 8<sup>2</sup>). <p> <br clear=left> ****************<b> <a href="Gelder.dwg">Gelder.dwg</a></b> ***************** <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="pi_construction.lsp">pi_construction.lsp</a> &nbsp;&nbsp; (load "pi_construction")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Gelder </b> <h3>3. <a href = "http://www-history.mcs.st-and.ac.uk/history/Biographies/Hobson.html">Ernest William Hobson</a> (1856 - 1933)</h3> <P> Hobson published this result in 1913. He constructed the approximate value of square root of pi,which is the true "circle squaring". p<sup>1/2</sup> = 1.77245... His constructed length is 1.77246...,which is very,very close!! <p> <IMG SRC="Hobson.jpg" width="400" height="400" align=left> <P> <h4>Construction steps </h4> <P>OA = 1.0, OD = 3/5, OF = 3/2, and OE = 1/2 <p>Draw the semicircles DGE , AHF with DE and AF as diameters. <p>The perpendicular to AB through O intersects These semicircles at G & H. <p>Then GH = 1.77246... <p> <br clear=left> ****************<b> <a href="Hobson.dwg">Hobson.dwg</a></b> ***************** <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="pi_construction.lsp">pi_construction.lsp</a> &nbsp;&nbsp; (load "pi_construction")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Hobson </b> <p>More topics to be added. <h4>References </h4> <p> <br> <hr size="4" color="#0000FF"> Go to &nbsp; <A HREF="./../../Fun_Math_by_CAD.html"> Fun_Math Content Table </A> &nbsp;<A HREF="./../Famous_Problems.html"> Three Famous Problems </A> &nbsp;<A HREF="./../Sqr_Circle/Greek_Sqr.html"> Greek Circle Squarer </A> &nbsp;<A HREF="./../Sqr_Circle/Later_Squarer.html"> Later Squarer </A> <hr size="4" color="#0000FF"> <p align="center"></b>All questions/suggestions should be sent to <a href="mailto:takaya.iwamoto@comcast.net">Takaya Iwamoto</a></p> <p align="center">Last Updated Nov 22, 2006</p> </Body>. </body></html>