Similarly, the Indian method of converting a rectangle into a square is different to Euclid's.Ĥ. Reversing the Indian method of circling the square is entirely different to this construction. As the construction shown does NOT appear in the Indian Śulbasūtras (Rules of the cord, 600 BCE) which circled the square, it may be assumed the general method of finding a mean proportional had not emerged, either in India, (which made use of the Pythagorean theorem before Pythagoras) or in Greece. Yet before the compass, in both India and Egypt, circles were drawn via 'peg and rope/cord' methods. Squaring the circle cannot be done with compass and straight edge. Thus OP is the side of the square with the same area as the area of the given circle.ģ. (The yellow triangles SOP and POA are similar, so their side lengths are proportional.) In the proportion the product of the extreme (outer) terms = the product of the mean (inner) terms, SO × 1 = OP × OP. 13, “To two given lines to find a mean proportional.” Thus SO is to OP as OP is to 1 and we write this as SO : OP :: OP : 1. The method of constructing the first side OP of the square OPMN is in Descartes' 1637 La Géométrie, via Euclid's Elements, (300 BCE) Book VI Prop. If cutting an onion makes you cry, at least its area is as easy as pi.Ģ. Archimedes converted a circle into a triangle and gave a proof. Given we accept limits and the reals, the concept behind the construction is exact. A single more detailed image of the above, with the notes below, is at NOTES:ġ. I wish you and your family a Happy and Healthy Holiday Season and New Year. We think we know it all, yet here is something special the world appears to have never seen! Others often seem to project modern ideas back through time.
Sometimes I feel like I know what people were thinking about thousands of years ago. Perhaps the world became hypnotised by Euclid's straightedge and compass constructions and abandoned the older methods that might have led to the above construction. From the time of Archimedes, (3rd Century BCE) the above construction should have emerged, yet appears not to have. People tried to square the circle for thousands of years using straight edge and compass alone. The math is correct, yet hand created constructions inevitably compound errors along the way, as the following examples show.
The final result may be a little 'wonky' yet no more 'wonky' than the many published diagrams drawn from precise instructions. This midpoint C will let me create the green dashed circle.ĭrawing the perpendicular dashed black line from O will give me the first square side OP and the rest should follow. Then another string will be used to bisect the distance from the edge of the circle through the center to S. The end of this string will be point S in the diagram.
Then I will cut the string in half (carefully) and extend this from the center of the circle, which will be found from the intersection of perpendicular bisectors of a triangle inscribed in the the circle. Wrapping the string 360° around the CD will be relatively easy and sit above the pencil line around the CD. When I try this by hand, I'm going to square a circle drawn around a music CD. (Image simplified following a suggestion from Geoffrey Cadman.) HOLIDAY FUN!
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