Mark Heaton Wrote:
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> The slope of the Grand Gallery is different from
> the slope of the Ascending Passage, and it is
> clear that the builder intended a difference, or
> used a different template.
>
> We can break the slope of both the gallery and the
> ascending passage into sections because Petrie
> gives us his offsets from the mean axis of each.
>
> The offset at the bottom of the gallery is greater
> than the offset at the top of the gallery which
> means that the the overall slope is steeper than
> the mean axis. Petrie was well aware of this and
> remarked that for the most of the length of the
> gallery the mean axis is close to the real slope,
> and picked this out as the intended slope by eye,
> being well aware of the fact that Smyth had proved
> that the slopes were different by dozens of
> measurements.
>
> The angle between the end of the Ascending Passage
> and the resumption of the gallery floor (the first
> section is missing) is close to a 1 in 2 (26
> degrees 34 minutes). It is as if the architect
> emphasised that the slope of the Grand Gallery is
> different from the slope of the Ascending Passage
> by a steeper section at the start.
>
> Other sections of both are similar to their
> respective mean angles.
>
> Lets look at the theory of slopes. Suppose AE
> wanted to build a slope with a rise of 500 digits
> and a run of 1000 digits, then the intended slope
> is approximately 26 degrees 34 minutes in our
> system of angular measurement.
>
> But lets suppose that the builders introduce a
> build error anywhere between plus one part in a
> thousand and minus one part in a thousand, for
> lengths of over say 450 digits. It is possible
> that the horizontal would be as long as 1001
> digits and the the vertical as short as 499 1/2
> digits. This reduces the angle to approximately 26
> degrees 31 minutes.
>
> We know that Petrie's measurements agree with
> Smyth's measurements within 4 arc minutes. Smyth
> was approximately 4 arc minutes more for the
> ascending passage and Petrie was approximately 4
> arc minutes more for the grand gallery.
>
> I think I can rule out any theoretical angle that
> is not within 7 arc minutes of the mean of
> Petrie's and Smyth's measurements, and I would
> hope the angle would be within 3 arc minutes.
>
> Grand Gallery: From memory
> Petrie 26 degrees 21 minutes, Smyth 26 degrees 17
> minutes, mean 26 degrees 19 minutes
> Theoretical angle to square the circle 26 degrees
> 18 1/2 minutes
> Difference of approximately 1 1/2 arc minutes from
> mean of both measurements
> If you insist on using the axis from which Petrie
> measured his perpendicular offsets then the
> difference is still less than 2 arc minutes
>
> Ascending Passage: From memory
> Petrie 26 degrees 2 1/2 minutes, Smyth 26 degrees
> 6 minutes, mean 26 degrees 4 1/4 minutes
> Theoretical angle for 360/2pi for pi = 22/7 26
> degrees 3 1/4 minutes
> Difference of approximately 1 arc minute from the
> mean of both measurements

I will repeat the Ancient Egyptians did not use degrees, they used seked, 360 / (45/22) = 176 and 360 / 176 = 45/22 which is a seked of 3 + 1/3 + 1/15 + 1/45 ((2×7×11)/(3^2×5)) for the ascending passageway, and a seked of 3 + 1/3 + 1/9 + 1/81 ((2^3×5×7)/(3^4)) for the Grand Gallery.

>
> Entrance Passage to junction of Ascending Passage:
> Just checked data
> Smyth used several different methods and estimated
> 26 degrees 27 minutes
> Petrie 26 degrees 26 minutes 42 seconds plus or
> minus 20 seconds say 26 degrees 27 minutes
> The slope length of the Entrance Passage is 5625
> digits
>
> (From my notes it is 1512 digits from the entrance
> to a projection of the ascending passage onto the
> floor of the entrance passage and 4113 digits from
> this point to the bottom of the passage. I used
> 20.61 inches per royal cubit from Cole's survey of
> base square, but if you use Petrie's 20.632 inches
> you will get 6 digits less, still tenable as a
> build error of 1 part in 1000, as Petrie's
> estimate of his own measurement error adds plus or
> minus 2 digits.)
>
> A vertical rise of 28 digits for a horizontal run
> of 5625/1000 digits is 26 degrees 27 minutes 47
> seconds or approximately 26 degrees 28 minutes
> Difference of approximately 1 arc minute from the
> mean of both measurements
>
> Entrance Passage from junction of Ascending
> Passage to bottom of Entrance Passage
> Only Petrie surveyed.
> Looking at the offsets it is very close indeed to
> axis used of 26 degrees 31 minutes 23 seconds.
> A rise of 1 for a run of 2 is 26 degrees 33
> minutes 54 seconds
> Difference of approximately 2 1/2 arc minutes
> It may have been more difficult to tunnel into
> bedrock at a precise angle and cross-section than
> it was to build a passage.
>
> The difference between the mean for the Grand
> Gallery of 26 degrees 19 minutes and the mean for
> the ascending passage of 26 degrees 4 1/4 minutes
> is over 14 arc minutes.
>
> Therefore there is no theoretical seked which
> would account for both the slope of the grand
> gallery and the slope of the ascending passage as
> the sum of build error and measurement error from
> the same intended slope. You have to argue for
> subsidence. The uniformity of all the slopes rules
> out poor workmanship from a widely different
> angle.

I have made no argument for or against subsidence, and it makes no difference what Smyth or Petrie measured other than to confirm what is calculated with seked. As I stated previously there are 2 seperate and distinct angles one derived using 2 dimensional geometry and one derived using 3 dimensional geometry. The ascending passage based on 2d 45/22, then (360 / (45/22) = 176 and the Grand Gallery being based on 3d utilizing a seked of 280/81, then 360 / (280/81) = 729/7 that is (9^3)/7, I see no problem with the two different angles! If you need further clarification let me know?

Regards,
Jacob