Byrd Wrote:
-------------------------------------------------------
> And moving onward...
>
>
> engbren Wrote:
> -------------------------------------------------------
> > There is very little by way of advanced
> > mathematics claimed - simply knowledge of
> > arithmetic progressions, which is demonstrated
> in
> > the RMP examples presented. There are also
> > examples in the Lahun Papyrus fragments that I
> > have not discussed but have very similar
> content
> > to the examples discussed in my paper. Our
> > knowledge of Egyptian mathematics comes to us
> from
> > a very limited set of extant sources relative
> to
> > the Babylonian sources.
>
> You mention calculating via Rhind, but don't
> establish how you know that the Egyptians used
> this method to calculate hours instead of simply
> observing and measuring (over thousands of years)
> via water clock and dividing to get results or
> simply importing it from the Babylonians (I note
> from Wikipedia and other sources that evidence for
> the Babylonian water clocks is older than the
> Egyptian ones by about 500 years.)
>
It is actually unclear - certainly they had the mathematics of arithmetic progression. This is one of Neugebauer’s pre-requisites for the development of mathematical astronomy. It is thought that the scales on the Karnak water clock represent the attempt to measure equinoctial hours per an inscription. Per Clagett, p70:
“The first document bearing on water clocks is an inscription from a ruined tomb near the top of the hill of Sheikh Abdel-Gurna in Western Thebes. This was the tomb of the official mentioned above, Amenhemhet, who lived under the reigns of the first three kings of the 18th dynasty … in the much worn inscription, the deceased tells us (lines 7-9) that “while reading in all the books of the divine word” (i.e. the whole of Egyptian literature he found “[the (longest) night of wintertime] to be 14 [hours long] when the shortest night of the summertime is 12 hours [long] and that hour lengths from month to month increase and then decrease.”

Whether this was divided using an arithmetic progression, or whether it was simply modelled practically by marking out the minimum and maximums and assuming a straight line between them. This does fit in well with the classical evidence that their modelling of the planetary phenomena was graphical.

Clagett, p71 also outlines the investigations by Borchardt and Sloley, who ultimately determine the constant difference in scale height of 1/3 finger. If this was based on observation, it would be more sinusoidal and we find that in fact the Egyptians in later water clocks made attempts to correct by using a 1:2:3 ratio for refer Clagett p74.

Clagett here refers to “Ancient Egyptian Science, Volume II, Calendars, Clocks and Astronomy” by Marshall Clagett

>
> > I am unsure of why this is an objection - the
> only
> > model which is presented in my paper represents
> an
> > early attempt to determine equinoctial hours.
> It
> > models the length of the nighttime hours from
> > solstice to solstice, which is modelled with a
> > simple linear regression from minimum to
> maximum
> > and back, hence a zig zag function. The focus
> on
> > this modelling thus far has been on the
> accuracy
> > of their time measurement, or on the accuracy
> of
> > the ratio of minimum to maximum nighttime
> hours.
> > However, what I have focused on is the
> technique
> > to model irrespective of the accuracy of the
> > model.
>
> You don't make it clear why they would develop
> math to do this when they already established and
> standardized (we know from their water clocks) the
> information via observation. If I understand the
> evidence correctly, the drive for the Babylonians
> to switch calculation of hours to math was the
> development of horoscopes (Van der Waerden, B. L.
> "The date of invention of Babylonian planetary
> theory." Archive for History of Exact Sciences 5.1
> (1968): 70-78.) -- but I don't see you explaining
> a reason for why the Egyptians would have gone
> from their millennia-old observational clocks to a
> calculated clock.
>
>
I don’t know why, but we have an inscription to say that they did per the above
> > ... - it is known that
> > Babylonians also had mathematical exercises
> which
> > dealt with arithmetic progressions, although I
> > have only found somewhat throw-away statements
> > that these can be found approximately 1,000
> years
> > prior to the development of the Babylonian
> > mathematical astronomy texts which apply the
> > knowledge to astronomy circa 400 BCE. This
> would
> > place the development of arithmetic progression
> > knowledge from Babylonian sources at
> approximately
> > the same time as the RMP. What is different is
> the
> > Karnak Clepsydra applies arithmetic progression
> to
> > model a time varying process connected to
> > astronomy which is much earlier than the use of
> > similar models found in astronomical texts from
> > approximately 400 BCE in Babylonian sources.
>
> You have not explained how you know it's an
> arithmetic progression instead of the cumulation
> of a lot of observations. Pogo (Pogo, Alexander.
> "Egyptian water clocks." Isis 25.2 (1936):
> 403-425.) describes deriving it from constructing
> what he calls "a prism" - which wouldn't involve
> any calculations. I don't find his method
> particularly convincing, but it's cited in a
> number of scholarly publications(so others with
> better knowledge than mine found it plausible at
> some point.) Whether or not you wish to compare
> your version versus his is up to you.
>
If it were observational data, it would not take the form of a zig zag function.

> >The graphical form of the System B zig zag
> function exhibited in the Karnak Clepsydra
> > example is consistent with the attestation from
> Theon Smyrnaeus . The view from scholars, is
> > that the references to the Egyptians in these
> texts largely refers to Egyptians of a later
> period
> > after the advent of Babylonian mathematical
> astronomy, or of Petosiris and Nepchepso
>
>
> Theron says that (your summary) "Babylonians and
> Egyptians succeeded in predicting the celestial
> phenomena. He indicates the Egyptians undertook
> this by graphical methods" This seems far more
> consistent with Pogo's graphs than with your math
> -- and in any case, no real prediction is being
> made by the "zig zag function" and Theron (so you
> say) mentions "predicting."
>
> You need to present some sort of evidence that
> this "zig zag" function in Egypt was used to
> predict an astronomical phenomena that they could
> not have observed -- like the position of a planet
> for a future event. Otherwise you can't knock
> over the assertion that the clock markings were
> imported from older clocks and those were
> determined by simple observations.
It is actually thought that the markings are imported from an earlier water clock, about 100 years prior to this one. It is known that the Babylonians ultimately arrived at a better ratio of nighttime to daytime hours, to which they also applied a System B zig zag function.

Per my earlier note, I will make an update to the paper with the above to address some of the concerns you have raised and publish in the next week or so.

Thanks & Regards,
Brendan