Kanga Wrote:
-------------------------------------------------------
> "...if the AEs knew the square root of 3, then
> they would have seen it as something like 1 + 2/3
> + 1/15."
>
> Before we get to this, we have to look at the
> evidence I have pointed to, namely the slope of
> the core pyramid of the Bent pyramid, which has a
> slope of 60*. The question we must ask first is -
> what simple numerical ratio would be needed to
> produce a pyramid that had this slope?

Hello Kanga,

Oh dear, we disagree at the off!
IMO the first question we must ask is: what seked is the closest equivalent to our measure of 60 degrees.
To which the answer is: 4 – i.e. 1 royal cubit rise to 4 palms run – which produces angle 60:15:18.43.

Taking half the length of the sides of the bent pyramid as 310 feet, at 60 degrees a projected height would equal 536.94 feet, whereas at 60:15:18.43 or seked 4 it would equal 542.5 feet.
I fail to see how one can reliably conclude from this that the initial planned gradient of the Bent Pyramid was seen by its designer as anything other than 1 royal cubit rise to 4 palms run or, possibly, 7 palms rise to 4 palms run.
That the cotangent of 1 royal cubit (or 7 palms) rise to 4 palms is 1.75 and that the square root of 3 is 1.732 is, IMO, neither here nor there.
We have tentative evidence for the use of the seked in the 4th Dyn, but – as far as I am aware - no evidence at all for square roots being used for what we today know as the tangent of a gradient.
So why introduce them into 4th Dyn. Egyptian maths?


> Only numerical ratios are involved here. The idea
> that they realized that these ratios, when
> converted to fractions, corresponded to sqrt 3 is
> only theoretical at this stage.

IMO, totally speculative not theoretical.


> There is, however, some evidence for the knowledge
> of sqrt 3 as 97/56 at Saqqara, as I have
> mentioned. Supposing, as I mentioned previously,
> that the mathematician thought that the average of
> 7/4 and 12/7 (= 97/56) was a good approximation to
> sqrt 3. To verify this, he would square it, thus:
>
> 97*97/56*56 = 3 + 1/3136. That's extremely close!

But, again, there is no evidence of AE square root calculations, which implies that they had no knowledge of them or knew them but did not consider them to be of any particular interest.
And, also again, why introduce anything other than the perfectly straightforward seked?


> Now, we find that the dimensions of the enclosure
> around the Step Pyramid are exactly 1040 cubits
> long by 528 cubits wide, creating a perimeter
> length of 3136 cubits. To my mind this is quite an
> amazing (though quirky) confirmation of the
> remainder in the equation above.


Some researchers see the perimeter as 3.143 (3 1/7) x 1000 = 3143 royal cubits
I suppose it comes down to which set of measurements of the perimeter one wants to believe.

> In addition to this, we find that the height of
> each of the steps in the Step Pyramid diminishes
> slightly each step up, such that the height of the
> 5th step is 97 cubits. This appears to be a
> celebration of the numerator in 97/56.

And what, pray, of the heights of the other four steps?
Unless there is a discernable pattern to the changes in the heights I suggest that we are probably looking at nothing more than a coincidence.

Regards,

MJ



Edited 1 time(s). Last edit at 04/20/2008 04:58PM by MJ Thomas.