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One thing was intentionally not covered in the OP.

Without knowledge of the area formula pi.(r squared) how would the AEs have worked out that a circle with an area of pi square units has a diameter of (approximately) 2 units? My reasoning is as follows:

They would likely square the circle from pi = 19/6 by making the denominator a square (81) thus making the numerator a square (256). The sides of the square would be 16/9.

The diameter of a circle with an area of pi square units would have to be greater than the side of the square (16/9) and less that the diagonal (16/9 x SQR2).

Using 17/12 as SQR2 the diameter would lie between 48 and 68 27ths, that is 58/27 or 19/9.

Using 7/5 as SQR2 the diameter would be between 70 and 102 45ths, that is, 86/45 or 17/9.

The average is 2.

So, the ratio between the diameter of a circle with area of pi square units and the area of the same circle squared is in the region of 2 : (16/9 squared), or 1 : (8/9ths squared). So, subtract a ninth from the diameter and square.

Using this formula and pi as 256/81 it follows that, for every circle, the area is the square of the radius multiplied by pi (as 256/81). I imagine that someone would have spotted this at some time.

Edited 1 time(s). Last edit at 18-Jan-19 23:41 by gjb.

Subject Views Written By Posted
Area and the Rhind Papyrus 1025 gjb 09-Jan-19 05:13
Re: Area and the Rhind Papyrus 234 engbren 09-Jan-19 06:03
Re: Area and the Rhind Papyrus 223 gjb 09-Jan-19 06:10
Re: Area and the Rhind Papyrus 191 engbren 09-Jan-19 06:30
Re: Area and the Rhind Papyrus 224 DavidK 09-Jan-19 10:06
Re: Area and the Rhind Papyrus 199 gjb 09-Jan-19 13:32
Re: Area and the Rhind Papyrus 203 Sirfiroth 09-Jan-19 15:04
Re: Area and the Rhind Papyrus 185 DavidK 09-Jan-19 20:02
Re: Area and the Rhind Papyrus 179 Sirfiroth 09-Jan-19 21:06
Re: Area and the Rhind Papyrus 226 engbren 10-Jan-19 02:02
Re: Area and the Rhind Papyrus 196 Sirfiroth 10-Jan-19 04:15
Re: Area and the Rhind Papyrus 235 engbren 10-Jan-19 05:53
Re: Area and the Rhind Papyrus 201 engbren 15-Jan-19 04:50
Re: Area and the Rhind Papyrus 212 Lee McGiffen 10-Jan-19 10:56
Re: Area and the Rhind Papyrus 249 ScottÂ Creighton 10-Jan-19 12:33
Re: Area and the Rhind Papyrus 242 Martin Stower 12-Jan-19 18:23
Re: Area and the Rhind Papyrus 160 Warwick 14-Jan-19 21:04
Re: Area and the Rhind Papyrus 211 DavidK 10-Jan-19 13:57
Re: Area and the Rhind Papyrus 246 ScottÂ Creighton 10-Jan-19 14:16
Re: Area and the Rhind Papyrus 228 DavidK 10-Jan-19 17:10
Re: Area and the Rhind Papyrus 334 Sirfiroth 10-Jan-19 17:58
Re: Area and the Rhind Papyrus 216 engbren 16-Jan-19 10:24
Re: Area and the Rhind Papyrus 219 gjb 18-Jan-19 23:28
Re: Area and the Rhind Papyrus 230 DavidK 19-Jan-19 00:28
Re: Area and the Rhind Papyrus 229 engbren 06-Feb-19 12:56

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