**CRITICAL REMARKS ON THE USE OF MEDIEVAL ECLIPSE RECORDS
FOR THE DETERMINATION OF LONG-TERM CHANGES IN THE EARTH'S ROTATION**

**W. DALMAU**

*Universitat Tubingen, Institut fur Astronomie und Astrophysik,
Abteilungen: Theoretische Astrophysik und Geschichte der Naturwissenschaften,
Auf der Morgenslelle 10, 72076 Tubingen, Germany*

Abstract. The use of some Arabic medieval solar and lunar eclipse records
for the determination of secular changes in the Earth's rotation is critically
reviewed. The published results derived from these data suggest a non-uniform
decrease in the Earth's rotation rate over the last 27 cy. There is, however,
up to this day no sound physical explanation for the deduced 'non-tidal
oscillations', with an apparent period of about 1500 yr and a semi-amplitude
of some 4 ms in the l.o.d., which overlayed to a constant secular tidal
change in the Earth's rotation rate produce a net non-uniform deceleration
of the Earth's rotation. In this paper we discuss a set of observations,
which were executed by professional Arabic astronomers. We show by our
analysis the way in which the non-uniform deceleration of the Earth's rotation
was constructed. A correct reading of the Arabic medieval observations
shows that they do not contradict a secular constant.decrease in the Earth's
rotation rate of nearly -4.6×10^{-22}
rad s^{-2}. This value is in accordance with other similar ones
derived from ancient eclipse records and from satellite tracking data.

**1. Introduction**

Up to this day there is no better approach for the determination
of long-term (or secular) changes in the Earth's rotation than the evaluation
of medieval and ancient observational records of solar and lunar eclipses.
Even accurate telescopic observations from the last three centuries does
not allow the precise determination of such a small change in the length
of the day (l.o.d.) of approximately 1.7 ms cy^{-1}. This secular
effect is made indistinct by much larger decade fluctuations in the Earth's
rotation rate (Kane and Trivedi, 1990).

The conditions for a solar eclipse depend on the precise
location of the observer on the Earth, and on the positions of the Sun
and Moon in space. If we use the uniform Earth's rotation rate implicitly
contained in a day of 86400 SI s in a computation of the eclipse path for
a medieval solar eclipse, the derived eclipse conditions will not correspond
with the observed ones. The derived eclipse path will be found to be shifted
westwards from the place of observation by an amount proportional to the
latitude of the respective path element. The angular difference in geographical
longitude, Dl, for which one has to correct the calculations in order to
make them coincide with the observation, is proportional to Int(w(*t*)*
dt*), where w(*t*)
is the instantaneous sidereal Earth's rotation rate. For a constant secular
deceleration of the Earth's rotation Dl, or
D*T*
= Dl/1.002738 s, will increase quadratically.

In the above argument, we implicitly have assumed that ths Moon's notion, which has a highly sensible influence on the computation of solar eclipses, is known accurately, hi fact, it is as a result of modem measuring techniques like lunar laser ranging, a well known parameter since the middle of the last decade. Studies about the secular change of the Earth's rotation rate before this time were encumbered with the difficulty that one had to separate two highly correlated quantities from a single set of observational data: the Earth's rotation and the secular tidal acceleration of the Moon.

The secular change of the Earth's rotation rate has two
main components. They are often called the *tidal* and the *non-tidal*
secular change. They have opposite signs. The tidal is the larger one by
a factor of ca. 4. The reason for the tidal secular change - in a strongly
idealized model for the net tidal effect - is that tidal dissipation causes
a displacement of the tidal bulge of the Earth relative to the Earth-Moon
direction. The tidal bulge exerts a secular torque, which expands the lunar
orbit while decelerating the Earth's rotation. The tidal secular change
safely can be assumed to be constent for the whole historical period (Brosche
and Sundermann, 1982). An extensive deglaciation event at the end of
the Pleistocene, some 10000 years ago, seems to be responsible for the
non-tidal secular change. Solid ice layers with a thickness of 3 km and
a mass of 10^{19} kg above northern Canada and Scandinavia melted
at this time. The result were a redistribution of water (global sea-level
rise of ~ 80 m) and an enormous discharge of
load in the deglaciated zones. The changing deformation of the Earth, which
was produced in response to this event, caused arid causes a steady variation
of the inertia tensor, which in turn induces a change in the Earth's rotation
rate [Sabadini and Peltier (1981) and Yoder
et al. (1983)].

Though the advances within the field of geophysics have
been enormous, the relevant increase in the understanding of the historical
observational record has been very slight. The editing of the Arabic sources
is difficult and requires strong interdisciplinary collaboration. A review
of the work completed up to date is discouraging and shows very clearly
that a major improvement in this matter will not take place in the near
future. Therefore, astronomers and physicists had to find their own way
without the help of historians of science. But not one of them was really
trained for this demanding task. The major work was done by R.R.
Newton (1970-1984) and F.R. Stephenson together
with L.V. Morrison and S.S.Said (1984-1995). The main difference between
the two approaches was that Newton attempted a separation
of the two highly correlated quantities, the Earth's rotation and the Moon's
motion, while Stephenson et al. determined
the changes in the Earth's rotation rate using a constant value for the
secular tidal acceleration of the Moon (*n ^{Õ}*

**2. The Arab Historical Sources**

One of the most important astronomical Arab sources, *az-Zij
al-Kabir al-Hakimi* from ibn Yunus (d. A.D. 1009) has not met
with a critical edition and a serious commentary. The *zij* exists
as a fragmentary copy. The observations preserved in it, among them the
most important compilation of solar and lunar eclipses of the epoch [Leiden,
MS 1957 (Cod. Or. 143), Chp. 4-6], were used successfully more than two
centuries ago to determine the secular tidal acceleration of the Moon.
Caussin published in 1804 the Arabic text of these
first chapters together with his French translation (Caussin,
1804). The printed Arabic text represents the-only serious preliminary
study, but his French translation is not precise enough and sometimes misleading.
Although this publication is by no means a critical edition, it was the
basis of the modem astronomical and geophysical studies.

On the other hand we have the astronomical treatise *az-Zij
as-Sabi'* from Al-Battani (b. before 858; d. A.D. 929), which, due to
the monumental work done by C.A.Nallino (Nallino,
1899-1907), is the best edited and commented *zij* [Arabic with Latin
translation and notes of the astronomer G. Schiaparelli. For a general
view of this *zij* and many others see Kennedy
(1956) and also Nallino (1944)]. Only one
Arabic copy exists, which according to Nallino
was written about A.D. 1100 (Escorial MS). Al-Battani has preserved in
his *zij* two solar and two lunar eclipse observations. These are
probably the ones that helped Halley around the year 1693 to discover the
secular (tidal) acceleration of the Moon.

In the following section, we will discuss 5 observations
of al-Battani and ibn Yunus. The discussion of the other 36 solar and lunar
eclipses recorded in their *zijes*, has been done in Dalmau
(1993) and Dalmau et al. (1997). The few but
*representative* observations discussed in this paper clearly demonstrate
the unirained and incorrect use of the eclipse records by the authors mentioned
above. It is also the main purpose of our discussion to emphasize the problems
inherent in the interpretation of historical records and their prospective
solution by critical editions.

**3. Eclipse Observations of al-Battani**

Al-Battam was undoubtedly one of the greatest Islamic
medieval astronomers [see Hartner (1970) for an
extensive description of his work and his life]. The theoretical framework
of his *zij* was strongly Ptolemaic. Al-Battani used his eclipse observations
to demonstrate that Ptolemy's statements about the diameters of the Sun
and the Moon, determined by Ptolemy with the help of two lunar eclipses,
were untenable. In the 5^{th} Book, Chp. XIV, of the *Almagest*
we read: "... the Sun's diameter always subtends approximauly the same
angle, there being no noticeable difference due to distance, but that ths
Moon subtends the same angle as the Sun only when it is at its greatest
distance from the Earth ... [Toomer (1984), p. 252]".
According to this passage, annular solar eclipses were not possible.

Ptolemy determined a lower limit for the Moon's diameter
using two lunar eclipse observations, in which the Moon was near its apogee.
The method used by Ptolemy was based on two points: 1. On the knowledge
obtained through the observation of the time and the magnitude of the maximal
eclipse phase (according to him the middle of the eclipse); 2. On the knowledge,
obtained by his lunar theory, of the Moon's distance in latitude at the
middle of the eclipse from the northern limit of the Moon's orbit. This
is obtained for a given time from the tabulated mean parameters of the
Moon's motion (mean position in longitude, mean anomaly, mean northpoint
distance and mean elongation). Ptolemy's proof was correct and convincing
[see Toomer (1984), pp. 252-254]. Therefore, in order
that al-Battani's refutation could also be convincing, it had to rely,
1. on better observations of the time and magnitude for the middle of the
eclipse, and 2. on the mean parameters of the Sun and the Moon's motion
as tabulated in his zij [Al-Battam uses for his proofs, departing from
Ptolemy's argumentation, the latitude of the Moon at the moment of the
apparent conjunction for solar eclipses, and of the latitude at the moment
of opposition for lunar eclipses, moments that he calls "the middle of
the

eclipse"].

The eclipses were observed in the cities *ar-Raqqa*
and *Antakiya*. The first of them was al-Battani's main working site
and also the reference place of his *zij* (*ar-Raqqa* j
= 36°). Al-Battani does not make any comments about how or by whom
the observations were carried out. Also, he does not comment how he determined
the time of the middle of the eclipses. In the following, we present the
results of our calculations of the eclipses according to al-Battani's tabulated
values and motion theory (only for *ar-Raqqa*). The procedures to
calculate the times and the magnitudes of lunar and solar eclipses are
given in Chapters LIII and LIV of al-Battam's *zij*. The procedure
for each eclipse calculation consists of some 35 steps. Starting point
of the calculations are the tabulated values of the mean sun, the solar
anomaly (in Ptolemaic sense), the position of the solar apogee and the
numbers of days elapsed since the reference epoch of the *zij* (Du
l-Qarnayn). Al-Battani's procedure is straightforward leaving no room,
at any step, for any kind of assumptions; that means, a medieval astronomer
wanting to control al-Battani's refutation of Ptolemy's statements about
the diameters of the Sun and the Moon, would have obtained, following al-Battani's
rules, the same results as we have obtained. It must be emphasized that
al-Battam has not reported any of his own calculations regarding the times.
As additional information -just to compare al-Battani's lunar theory with
the modem one implicitly contained in the ephemeris DE102/LE51 of the Jet
Propulsion Laboratory [see: Newhall et al. (1983)]
- we have also computed some eclipse parameters using DE102. In the following
only some important values are given. Al-Battam's descriptions of the observations
are given in quotation marks [see also Nallino (1907),
pp. 56-57]:

1. Lunar eclipse A.D. 883, Jul 23. (*ar-Raqqa*):
The time of the true opposition was 07:55:00 p.m. mean local time. The
Sun was in Leo 3°45'. From al-Battani's

Ref. | A.D. date | Eclipse type | Place of obs. | calculated mid. of ecl. (author) | observed mid. of ecl. (al-Battani) |

1 | 23.7.883 | Lunar | ar-Raqqa |
8^{h}07^{m}16^{s} |
>8^{h} |

2 | 8.8.891 | Solar | ar-Raqqa |
1^{h}03^{m}01^{s} |
~1^{h}00^{m} |

3 | 23.901 | Solar | ar-Raqqa |
8^{h}30^{m}28^{s} |
~8^{h}30^{m} |

4a | 2.8.901 | Lunar | Antakiya |
15^{h}19^{m}05^{s} |
~15^{h}20^{m} |

4b | 2.8.901 | Lunar | ar-Raqqa |
15^{h}34^{m}05^{s} |
~15^{h}35^{m} |

Equation of Time (E.q.T.), we derive for this solar position: 3°4'; or 00:12:16 in time units. The local true time of the true opposition was 08:07:16 p.m. (equinoctial hours). In good agreement with the time given by al-Battam: "by a little amount more than 8 equinoctial hours after noon". According to al-Battani's zij, this lunar eclipse should have been total, with a magnitude of 17.25 digits (digits = twelves of the diameter) [modem computations using DE102 leads to 11.5 digits; according to Ptolemy: 11.5 digits], but from the observational record we read: "little more than a half and a third of the diameter".

2. Solar eclipse A.D. 891, Aug 8. (*ar-Raqqa*): The
time of true conjunction was 00:50:24 p.m. mean local time. The Sun was
in Leo 19ÿ32'. From al-Battani's E.q.T. we derive for this solar position:
3°16'32"; or 00:13:04 in time units. The true local time of the true
conjunction was 01:03:28 p.m. (equinoctial hours). The apparent conjunction
was, as stated by al-Battam, 00:07:30 after the time of the true conjunction,
that is 01:10:58 p.m. (equinoctial hours) or 01:03:01 p.m. in seasonal
(unequal) hours for *ar-Raqqa* (using day-hours of 01:07:34). In good
agreement with the time given by al-Battani: "one seasonal hour after noon".
According to Ptolemy, the magnitude should have been greater than 3/4.

3. Solar eclipse A.D. 901, Jan 23. (ar-Raqqa): The time
of true conjunction was 21:19:29 p.m. mean local time. The Sun was in Aquarius
8°7'. From al-Battam's E.q.T., we derive for this solar position: 0°14'46";
or 00:00:59 in time units. The local true time of the true conjunction
was then 21:20:28 p.m. (equinoctial hours). The apparent conjunction was,
as stated by al-Battam, 50 min before the time of the true conjunction,
that is 20:30:28 p.m. The time is counted from the noon of the previous
day of observation; this means 03:29:32 before the actual noon. In perfect
agreement with the time given by al-Battam: "less than 31/2 hours [equinoctial
hours] before noon". In Figure 1, we have plotted the eclipse path with
lines of equal magnitude. The diagram was calculated using D*T* = 0 s. The
plotted magnitudes are digits of the diameter. The observed magnitude in
Antakiya was little more than the half part of the Sun. In *ar-Raqqa*,
the magnitude was " in the appearance" less than 2/3. From Figure 1, we
can see that the difference of the observed magnitudes is, for any value
of D*T*,
not so high as the records insinuate [the varying of D*T*
means that one has to shift the eclipse path eastwards in longitude with
D*T*
= Dl/1.002738 s]. According to Ptolemy, the
eclipse should have been total.

4b. Lunar eclipse A.D. 901 Aug 2. (*ar-Raqqa*): The
time of the true opposition was 15:21:28 p.m, mean local time. The Sun
was in Leo 14°20'. From al-Battani's E.q.T., we derive for this solar
position: 3°9'20" or 00:12:37 in time units. The local true time of
the true opposition was then 15:34:05 p.m. (equinoctial hours). This is
in perfect agreement with the time given by al-Battani: "15 1/3 + 1/4 equinoctial
hours after noon, approximately". According to al-Battani's tables, this
lunar eclipse should have been total (he observed: ~
12 digits), with a magnitude of 18.0 digits [DE102: 12.8 digits].

Stephenson and Said (1989)
analyzed the observations assuming that the only unknown parameter was
the clock error D*T*
arising from irregularities in the Earth's rotation. For the solar eclipse
of the year A.D. 901 (in ar-Raqqa) they determined D*T*
= 1840 sand for the lunar eclipse of the same year (in ar-Raqqa) they deduced
D*T
=* 420 s. Despite the fact that they have not considered the
unusual definition of al-Battani's "middle of the eclipse", to explain
this large difference in D*T*
for observations done in the same year they have to assume either an observational
error or a scribal mistake in the range of 24 minutes (using the correct
"middle of the eclipse" the error is of some 20 minutes). This means, that
the recorded times in al-Battani's *zij* cannot be correct (if one
does not want to assume the catastrophe theory for D*T*
- change of 20 minutes in 1 year - or an incorrect gravitational theory
in DEI 02).

The agreement between our calculations and al-Battani's
observations (see Table I) is obvious and reveals the true reason for the
large difference in D*T*
found by Stephenson and Said (1989). The tables
in Ptolemy's Almagest and also those in al-Battani's *zij* require
a broad observational foundation, as shown by the chapters leading to their
construction here and there; and the lunar or solar eclipses mentioned
above form no part of this foundation. As we know from ibn al-Qifti, al-Battani
wrote two recensions of his *zij*. The extant Arabic manuscript must
be a copy of the second recension, for it contains the lunar eclipse of
August 2. A.D. 901. The first recension of the zij could not have contained
this eclipse [and probably not the solar eclipse of the same year], because
Thabit ibn Qurra, an elder contemporary of al-Battani, wrote before he
died on February A.D. 901 some comments about one of the last chapters
of the *zij* (Hartner, 1970). The zij was already
published at the time of the eclipse(s). Despite these facts, the observed
times are consistent with al-Battam's solar and lunar theory, suggesting
that the numerical data have been manipulated.

*Figure 1*. Solar eclipse of August 8. A.D.
901,
D*T*
= 0 s.

To gain more clarity, future studies will have to reexamine
the whole numerical context. Further, it is clear from above arguments
that for the time being the observations listed in Table I cannot be used
to determine D*T*
values. Stephenson and Said (1989) by having
used them (all weighted with 1.0) introduced a large bias in their statistical
analysis, because al-Battani's observations are the ones in their data
set, which produce the lowest D*T*-values.

Newton (1984), starting from his general mathematical analysis, discards them and argues that the error must come from recording and not from observation. He concludes that the observations, specially the lunar ones, may have been forced to agree with a preconceived theory.

**4. Eclipse observations of ibn Yunus**

Ibn Yunus observed a partial solar eclipse on December
13th, A.D. 977 in the mosque of *Qarafa *in the vicinity of today's
Cairo. This is the first solar eclipse of his *zij*, that he himself
has witnessed. If we compare it with the other eclipse records contained
in his *zij*, we find something unusual, a list of the names of 8
persons, which precedes the observational part of the record. Five of them
were instructed on the event which soon was to happen, but were not specialized
in astronomical matters. The other three came together with him to the
place of observation. Ibn Yunus does not comment, why he has preserved
those names in an astronomical treatise. Perhaps he wanted them as witnesses
or had another social motive. We give here the translation of the record:

"And the group waited for the beginning of this eclipse, and it sensibly began to appear, when the altitude of the Sun was greater than 15 degrees and less than 16.

And the opinion of those, who were present, was in agreement,
that the eclipsed part of the Sun's diameter was about eight digits, and
this is of the area of its circle less than seven digits. And its clearance
(of the Sun) was complete, when its altitude was greater than 33 degrees,
with about a third of a degree, according to that, what I [Ibn Yunus] have
determined for myself about its altitude. And those, who were present,
were in agreement about the completion of clearance (of the Sun);" [Caussin
(1804), pp. 179_{12-15} + 181_{1-3}; MS 1057, (Cod.
Or. 143), p. 110 _{4-16}]

The description of the observation in the first part of
the record does not allow an accurate interpretation of the observed eclipse
magnitude. The moment when "it sensibly began to appear" was not the astronomical
beginning of the eclipse. Ibn Yunus gives for this moment only a vague
measurement of the altitude of the Sun. This shows clearly that he either
was not interested in making a better description of this eclipse phase,
or that he simply could not make a better measurement of the observed phase
(not of the altitude!). It must be pointed out that he is not saying: the
beginning of the eclipse has occurred at *this* altitude of the Sun.
He is saying: at the moment, *when* "it sensibly began to appear"
the Sun's altitude was "greater than 15 degrees and less than 16". Very
different the description of the end of the eclipse: "and its clearance
was complete". The described moment is the moment of the end or just after
the end. Only for the second determination of the Sun's altitude does he
make a personal statement --using his authority--, that he himself has
determined the altitude to be 33 degrees and (more) by about a third of
a degree.

Stephenson and Said (1989),
without a plausible justification, put the altitude of the Sun at *exactly*
15.°5 for the first geometrical contact (The first geometrical contact
is the moment when the two luminaries are tangential to each other; thus,
at this moment the Sun is by no means eclipsed). Further, they put the
4. geometrical contact at exactly 33.°3 [Newton's
procedure is nearly the same. (Newton, 1984), p.
114]. They then deduce D*T*(l.contact)
= 1820 s and D*T*(4.contact)
= 1980 s and give to both values the same statistical weight of 1.0 in
a least square fit with a parabola. Beyond the fact that it does not make
sense to interpret the observed parameters in that manner, they introduce
a systematic error by doing this. As one can see from Figure 2, the first
and the fourth contact are not statistically independent quantities. D*T
*is
always the same for all circumstances of a particular eclipse, and this
fact can be used to weight and determine a lower limit of error for the
particular eclipse.

We have used, as a working hypothesis, the parabola D*T*
= 31.5s cy^{-2} (*t - t*_{0})^{2}
with *t*_{0}
=A.D. 1800 (w^{Õ}_{secular}=
-4.6×10^{-22}rad s^{-2}) and *n ^{Õ}*

*Figure 2*. Solar eclipse of December 13. A.D. 977.

of the diameter. For the fourth geometrical contact, we find the altitude of the Sun to be 33.ÿ0. For this AT, the maximal eclipse magnitude was 7.5 digits.

The eclipse magnitude was at the moment, when the eclipse
"sensibly began to appear", nearly 1.2 digits, and this is in agreement
with naked-eye observations. The eclipse record says nothing about the
observational technique. It is, however, not necessary to speculate: such
a retarded apprehension is possible. In the solar eclipse from Jan 24.
A.D. 1004, which is the best described in his *zij*, ibn Yunus has
recorded a similar case: the magnitude of the eclipse was at the moment
when it was apprehended = 1.5 digits and the altitude of the Sun was 16.°0;
but for this eclipse ibn Yunus makes an estimate for 'the beginning' to
be as the Sun's altitude was 18.°5.

At the end of this eclipse record, we read that the Sun
and the Moon were each near perigee. In the following two eclipse records,
we find similar statements. Did ibn Yunus use these three eclipses like
al-Battani to determine the diameters of the Sun and the Moon? This question
and many others should be solved first, because the evaluation of al-Battani's
observations clearly indicate that they have been manipulated. In this
case, we must calculate these three eclipses using the *zij* of ibn
Yunus in the manner illustrated in section 3. But it is impossible to do
so as long as there is no critical edition of this important medieval source.
Already now we may safely state that this eclipse record cannot, by any
means (not even a statistical one) be used against the concept of a uniform
change in the Earth's rotation rate.

**5. Summary**

Historical observations of eclipses have been the best
tools for determining secular changes in the Earth's rotation rate, until
today. The analysis of eclipse records given by F.R.Stephenson,
L.V. Morrison, S.S. Said and R.R. Newton suggests
that there is evidence for a non-uniform decrease in the Earth's rotation
rate. In the present paper, w'e have shown on a set of representaive observations
how the supposed irregularities in the Earth's rotation were constructed.
We have limited ourselves to the discussion of the two main Arabic historical
sources written by professional astronomers, because the results obtained
by Stephenson et al. have mainly been derived
from this two sources (88% of their data set). In our opinion, the authors
listed above have not evaluated the Arabic records with the sufficient
care. They have taken the numerical data from the sources without considering
the context. The statistical method used by them meets by no means the
requirements of the subject. The theoretical models and astronomical tables
used by the Arabic astronomers are essential for their work and their observations,
but they have been completely neglected. For the present, it can be stated
that the Arabic medieval observations *do not contradict* a constant
secular change in the rate of the Earth's rotation, its numerical value
being nearly -4.6×10^{-22} rad s^{-2}. The exact
determination of the secular change in the Earth's rotation will depend
on a critical edition with a serious commentary of the *zij* of ibn
Yunus, because it yields the best source of information on the relevant
observations.

**Acknowledgements**

We wish to thank Prof. Dr. Harms Ruder and Prof. Dr. Matthias
Schramm (Institut fur Astronomic und Astrophysik, Universitat Tubingen)
for support and helpful discussions, Mr. E.M. Standish (Jet Propulsion
Laboratory) for providing a copy of DE102/LE51, Prof. Dr. Fuat Sezgin (Institut
fiir Geschichte der Arabisch-Islamischen Wissenschaften, Universitat Frankfurt)
for providing a copy of Leiden MS 1057 (Cod. Or. 143).

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