July 2012. Dedicated to all Circle Squarers past and present and for the attention of most mathematicians, maths teachers, students and others interested in recreational maths:
A while ago I noted 'squaring a circle' under 'famous problems" in the Oxford Mathematics Study Dictionary. Having dealt with the Moons of Hippocrates it got my attention and, ignoring being told about the attempts over 20 centuries and the impossibility of doing it, using only a pen, compass and a straight edge I also tried my luck and have since spent many frustrating hours and a few elated minutes on it. Checking the Web I saw a temptation to call Later Century Circle Squarers, Morons! Elsewhere, I seem to faintly recall, was a similar urge to call Nitpicking Mathematicians, Snobs!
Some squarers, I believe, for all practical purposes succeeded; but not really as the issue of pi not having repeating decimals apparently does not allow such a claim. Anyhow -
SWANNIES METHODS COVERS ALL REGULAR POLYGONS . In the examples given the equilateral triangle, the square, and the regular pentagon are called RP's. Thus follow the steps in the opposite diagram for the circling of a triangle.
METHOD I: Circling an RP.
1. Draw an RP AB... with centre C.
2. Draw an outer semicircle on AB.
4. Draw the circumcircle circle of the RP.
5. Set compass to length of AD and from E mark F on the circumcircle.
6. Let CF intersect AB in G.
7. With C as centre and radius CG describe the required circle equal in area to the RP.
Note: Measurements from the screen are of no use. The diagrams are somewhat flattened vertically.
Opposite is shown the essencial parts of a diagram wherein square AB.. is first circled using Metod I and this construction then used to square a given circle using Method II.
METHOD II: Completing an RP equal in area to a given Circle.
1. Use Method I to complete all the steps towards finding the required circle for an RP larger than the given circle.
2. Draw the given circle with with centre C and let it intersect CF in H.
3. Draw JHI//AB with J on CB and I on CD.
4. Extend JI its own length to K.
5. By simple construction complete the required RP with side KJ.
Have I qualified? For now, in addition to the above, I can only offer "circumstancial proof". However, I feel that within the limits that follows the methods are good. I'm now waiting for anyone using a school type compass and ruler to show that the methods are not true! With the methods given pi only features after completion of the construction (Note: The Moons of Hippocrates proof where the square root of two, squared is a main feature in the proof , bypassing the 'unacceptable' value of pi, is of the kind Mathematicians accept. See the Net.)
NOTE: RP's range from the triangle through the range to the infinite sided RP where the circumscribed circle and inscribed circle merges with the RP. Throughout the range and within the above mentioned tools limits ( granted they 'soon' become impractical to use) I believe the methods are applicable.
On the right -
On the right: Both methods applied for Pentagons.
So the above is not going to bring me fame. However, I would still like to qualify as a Circle Squarer (-
To the likely delight of some(one) I shall admit that, like a true squarer?, I bought the ruler, compass and pencils I used to generate my Methods from the 'shop around the corner' in the small town of Lady Grey (SA) where was visiting my wife.
Since my return to Port Elizabeth I have made myself a compass of the old blackboard type size and spent some more time on the quest to approach the impossible. I won't spoil anyones fun by saying more.
Now all -
Now all milli-
(( The story up till now is all that I originally wanted to present. However someone really intent on using the drawing equipment mentioned in the previous paragraph needs to contend with some more circumstancial proof in Method 1: Let Step 4 read as follows: extend ED its own length to C1 (wich is the same as the centre C in the case of the square). With C1A as radius draw arc AB. With centre E and radius ED draw an arc to intersect arc AB in F. Now draw an arc with centre B and radius BF, extending it as required by the next step some distance on either side of F. Draw CF1, a tangent to the arc, with F1 on the arc. Proceed with Step 6 of the the method using F1 to now replace F in all further steps of the methods.))
Copyright: See Home Page. Contact Particulars: See Personal