CaliforniaKid wrote:Some time ago there was a lot of noise about an equation that supposedly proved that the Hor Book of Breathings was originally some 41 feet in length. Even John Gee, who gave the FAIR presentation that highlighted this equation, was surprised by this result, but after checking and rechecking his figures as well as the assumptions on which the equation is based, he claims to be sold. Last night I was up altogether too late listening to Gee's talk and trying to figure out what in the world he's talking about.
I should begin by highlighting the historical evidence with respect to the missing portion of the roll. After our present fragments of the Hor Book of Breathings (from the front end of the Book) were preserved under glass, the end of the Book remained intact as a roll. We have two eyewitness accounts of this remainder that might help us gauge what was left. In the first place, A Nauvoo visitor named Charlotte Haven described it as a "long roll of manuscript" in her journal. Secondly, Gustavus Seyffarth called it "an invocation to the Deity Osirus, in which occurs the name of the person, (Horus,) and a picture of the attendant spirits, introducing the dead to the Judge, Osirus." This appears to describe the end of the Hor[us] Book of Breathings, followed by Facsimile 3, the closing vignette. Michael D. Rhodes estimates in his critical text that we can expect that there are about 60 cm (or roughly two feet) of writing missing from the end of the Book of Breathings. In looking at Rhodes' transcription of Louvre 3284, a copy of the BoB very similar to the Hor text, I think he may be underestimating by ten centimeters or so. But his figure is believable. Certainly a lower limit for the missing text would be one and a half feet, and an upper limit might be two and a half or three feet. Seyffarth's testimony suggests that there was no other text after the Book of Breathings on the remainder of the roll, so this was probably its total length (unless there was unused space at the end). Would a two-foot text explain Charlotte Haven's testimony that Mother Lucy unrolled a "long roll of manuscript"? It might. We would probably expect something at least three feet in length, but someone like Charlotte who is not familiar with papyrus just might describe a two-foot roll as "long," especially since it may have been wrapped almost twenty-five times. (By the number of "wraps", I mean the number of full rotations required to unroll the text.)
In any case, there's certainly no eyewitness testimony that nails down the existence of a missing text at the end of the Hor scroll. So Gee presents us instead with his mathematical equation. In his words, "the circumference of a scroll limits the amount of scroll that can be contained inside it. Thus we can determine by the size of the circumference and the tightness of the winding how much papyrus can be missing at the interior end of a papyrus roll." Notice that Gee is here careful to say that our result will be an upper limit: it tells us how much papyrus "can be" missing from the interior of the Hor roll. He does not continue to use such reserved language, however. The remainder of his presentation treats his result as the actual amount of missing papyrus.
Gee does not give an explicit reference for his equation, but does mention a couple details. He states, for example, that the study was more than a decade old, that it was written by somebody with a Germanish name that sounds like "Fredam Holfman", and that it was created in order to gauge the original length of "Papyrus Spiegelberg". I haven't tried very hard yet to track this reference down, but since I know little about Egyptology and I don't speak German, it may prove difficult do so. Gee unfortunately also does not give the derivation of his equation. Here's the equation he proffers:
E - length of the last/interior extant winding
S - average diff. b/w successive winding lengths
Z - length of the missing portion
Z = (e^2 - 6.25)/(2s) - e
The idea here seems to be that as you get closer to the interior of a roll, your wraps get continually shorter. If we know the length of the missing portion's outer wrap (E) and the difference in length between each successive wrap (S), we can calculate the number of missing wraps and the approximate length of each of them. (S should be a constant as long as the change in radius for each wrap also remains constant-- i.e., as long as the tightness of the wrapping remains the same throughout the roll. There is some reason to doubt that the tightness of the Hor roll's wrappings were uniform over its entire length, but it was probably close enough for our purposes.) I confess I don't know exactly how Holfman/Gee arrived at the above equation. In working this problem out myself, I came up with the following formula that should give the length of the missing portion: Z = [e^2-e*s]/2s. This is the number of missing wraps, e/s, multiplied by the average circumference of the missing wraps, [e-s]/2. But the results Gee's formula produces are very similar to the results produced by mine. The fault in his analysis does not appear to lie with the formula itself.
Gee gives 9.5 as the value of E, and places S somewhere between .033 and .05. He says that inputting these figures gives us between 835 cm and 1250 cm for the length of the missing portion. It would be fair, I think, to ask how he arrived at his values of S and E. Presumably, it was by measuring the distance between matching lacunae, as shown in the image below.
In order to check Gee's figures, I whipped out my copy of By His Own Hand Upon Papyrus and a handy-dandy ruler and came up with some figures of my own. Larson says his images are 78% of actual size, so all of my measurements below have been multiplied by 1.282 in order to compensate for scale. Obviously it would be better to have had access to the originals, but I think we can get reasonable figures from the photographs, anyway. In the extant portion of the Hor BoB, there are 7 wraps. A lacuna in the middle makes it impossible to gauge the length of wraps # 4 and 5, except by extrapolation (thus the brackets). Anyway, here are my figures:
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Lengths of Wraps:
#1: 10.11 cm
#2: 9.98 cm
#3: 9.9 cm
#4: [9.5?]
#5: [9.1?]
#6: 8.7
#7: 8.32
Diffs b/w each wrap:
#1 & 2: .13
#2 & 3: .08
#3 & 4: [.4?]
#4 & 5: [.4?]
#5 & 6: [.4?]
#6 & 7: .38
Note: The three figures in brackets here must add up to 1.2, but may not have been as evenly distributed as I have suggested above.
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My numbers are obviously very different from Gee's. Feel free to do some measurements of your own and report back here if you find I'm way off. It's a little hard to match up the lacunae perfectly, so this isn't an exact science. But I have done my best to be as accurate as possible. Plugging my figures into Gee's equation gives us 79.14 as the length of the missing portion of the roll. (Plugging them into my formula gives us 91.97.) I should emphasize that I made my measurements before I even started playing with equations, so my numbers were not selected in order to give the results I wanted. An 80-centimeter figure (about two and a half feet) is precisely what we would expect based on Michael D. Rhodes' predictions, cited above.
In conclusion, it appears that Gee's vastly inflated length for the missing portion is the result of faulty measurement. Properly considered, the missing-papyrus-equation seems to leave his theory fatally crippled and without a leg to stand on.
Best,
-Chris
I am grateful to Celestial Kingdom for his help in understanding this situation. In fact, we don't really need to use algebraic equations to deal with the problem. Data of this kind are easily handled on an Excel spreadsheet. It works like this.
What we have here is a sequence of decreasing lengths, which are produced, if I understand correctly, by a length of papyrus being rolled up, and then squashed so that there are two folds produced for each complete wrap of the papyrus. These folds result in damage causing the present periodic indentations we see on the papyrus today. Assuming that the tightness of the wrap is more or less constant, it is easy to show that the circumference of each complete wrap will be a certain constant amount less than the wrap outside it. Eventually, as we get to the core, the length of a wrap will become very small.
We have the outside of the scroll, and it is assumed (for textual reasons) that there was once more of the scroll to the right of what we have now, and that this portion was wrapped inside the extant portion. The problem amounts to:
(a) From the extant wrap-lengths, predict the likely lengths of each of the missing sequence of wraps that were inside the scroll.
(b) Find the sum of those lengths, which is the total length of missing scroll.
Try this at home (it takes a while to describe, but it is really easy):
Simply put Celestial Kingdom's sequence of wrap lengths in a column on your spreadsheet, then select them.
Click the Chart wizard. Choose an option to plot a line graph with data points shown, and produce your chart.
When you have your chart, go to the Chart menu and click on 'Add Trendline'.
For 'type' choose 'Linear', so that the best straight line is fitted to the data (that is the choice consistent with the assumption that there is a constant difference in length between successive wraps).
For 'options' click 'display equation on chart' and set the 'Forecast forward' to a value like 100 - that means the system will predict the lengths of 100 further wraps.
Click OK, and you will see a straight line appearing, which eventually hits zero around 34 on the x-axis. That means the length of the 34th wrap is zero - we have reached the end of the scroll.
The equation you will see on screen is
y = -0.3118x + 10.62
Its meaning is simple: Excel is saying that its statistical analysis of the data shows that they best fit a sequence of lengths starting from 10.62 cm, and diminishing by 0.3118 cm per wrap. (Technical note: 10.62 is the 'zeroth wrap' length, the one that would be to the left and hence outside the left hand edge of the present papyrus. The real papyrus starts from wrap number 1)
It is easy to set up Excel so that it produces a column of lengths predicted by these two figures (the actual measured lengths by Celestial Kingdom are shown in brackets; Excel calculates to greater precision than shown here)
10.3 (10.11)
10.0 (9.98)
9.7 (9.90)
9.4 (9.50)
9.1 (9.10)
8.7 (8.70)
8.4 (8.32)
8.1
7.8
7.5
7.2
6.9
6.6
6.3
5.9
5.6
5.3
5.0
4.7
4.4
4.1
3.8
3.4
3.1
2.8
2.5
2.2
1.9
1.6
1.3
1.0
0.6
0.3
0.0
Now all we have to do to find the missing length of papyrus is to make Excel sum the lengths predicted for 'missing' wraps. The answer comes to 110 cm (to 2 significant figures) or 43 inches. This is certainly of the same order of magnitude as the other estimates cited by Celestial Kingdom above:
Plugging my figures into Gee's equation gives us 79.14 as the length of the missing portion of the roll. (Plugging them into my formula gives us 91.97.) I should emphasize that I made my measurements before I even started playing with equations, so my numbers were not selected in order to give the results I wanted. An 80-centimeter figure (about two and a half feet) is precisely what we would expect based on Michael D. Rhodes' predictions, cited above.
Of course the great thing about Excel is that you can easily try 'what-ifs' or goal seeking. I won't bother you with the details, but if you wanted to come up with a figure like Gee's 41 feet (492 inches or 1250 cm), you would have to change the difference between successive wraps to something more like 0.04 cm, less than half a mm, which is utterly implausible.
Since it can easily be shown that in a tightly wrapped scroll the decrease in length from wrap to wrap is 2πt, where t is the thickness of the papyrus, such a tiny value of the decrease would imply that the thickness of the papyrus was only:
0.04 cm/2π = 0.006 cm, or 0.06 mm.
No papyrus is that thin; the value implied by the length step of 0.3118 cm is about half a mm, which is quite reasonable.
Conclusion: If there was once any missing papyrus to the right of what we now have, the length of the missing portion cannot be much different in magnitude from about 100 cm. That prediction in any case depends on the assumption that the original scroll was a full and tight roll, not wound round a wooden core, and without a void in the middle.
Had either of those been the case (which is not unlikely) there could have been NO missing papyrus at all. There is no way the missing portion could have been 41 feet or 1250 cm long.