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Hi Don,

I dont care what they knew or what they didnt know, i am just telling you the simple facts of the data, and what it can fit. It does not change the a priori assumption of yours that it had to be an oval, when clearly other functions can fit. You seem to believe that they knew about second order equations, since the ellipse and the circle are a form of? So why not a cubic equation, which is what a third order polynomial is? You believe that they were mathematically advanced enough to be able to compute the shapes of ovals and ellipses, be able to calculate square roots and surds as well as Pi and Phi to great accuracy, but that they could not compute cubic equations. You seem to be a great authority on what they did and didnt know, so please show me irrefutable evidence that they did not know what a cubic equation was?

Even so, Lets assume that they had no clue what a cubic or higher order polynomial was. If they could do geometry with circles and ellipses (and hence ovals), they must have been familiar with squared terms, and a squared term in an equation makes it a second order polynomial, and thus a quadratic. The equations for circles and ellipses (and hence ovals) are based upon these types of equations.

circle



ellipse



quadratic




Which is also the equation of the parabola, a related conic section to the ellipse. It is not much of a stretch to imagine that IF they had the ability to compute an ellipse, they could also compute a parabola, and it is not much more of a step to introduce higher order terms to make it cubic or quartic in nature (you just multiply everything by x and add a constant after all). In the modern age, all this is assumed to be of the same ability level for high school kids (at least in the UK A-level system), since it is all related.

The fact is this, you believe they were advanced enough in geometry to play with ellipses and curves, and polynomials describe curves. They may not have known or though of quadratics or cubic polynomials, and they may not have even known about ellipses and circles. Either way, i don't loose sleep over it, and it is no skin of nose. You can believe what you believe, but dont use me bolster your credibility and prop up your stance.

One last thing, a quick bit of fitting shows that one can fit a quadratic (i.e. parabola), through the data with equation y = -0.0058x2 + 0.1283x + 96.929 and an R-squared of = 0.998. If you had asked what parabola fits the data, then that would be a likely one. So here we have another model that it can fit. So what should one do? Decide which one it should be upon what we think the ancients had knowledge of, or find more unique data?

Thats the point I am making, in that you cannot call it irrefutable proof that the data forms an ellipse, given that it is limited to 6 data points in one quadrant, and given that other functions fit that same data equally as well. If we knew it was an ellipse then fine,all is well, but we dont, and we must always approach data with the least amount of assumptions possible, and let the data talk for itself, and not let our prejudices bias the analysis or conclusions. After all, you yourself were convinced it had to have a specific size and ratio before you convinced yourself that it now fits with some other observation.

You may be right, and you may be wrong with your assumption of an ellipse, but I approach things with no preconceptions of what the data is meant to be, and only fitted an ellipse because you asked what the best oval to fit the data was. If I were presented the data blind, my first instinct would be to see what order of polynomial it fitted, and keeping things simple, using what I call "The Principle of Least Astonishment", I would have most likely concluded tat the data was quadratic in nature, barring any further evidence to the contrary. I may have tried to fit it to a circle and an ellipse, and been surprised to see it fits a ellispe very well, but I would still want more data to confirm the shape, and I am afraid there doesnt seem to be any more data.

Jonny