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Appendix III

THE METHOD USED IN CHECKING RECORDED ECLIPSES

For the checking of the eclipses recorded in Chinese records with
astronomical computations there is the monumental Canon der Finsternisse,
by Th. von Oppolzer, published in Vienna, in 1887, in which are
calculated all the eclipses from 1208 B.C. to A.D. 2161. This book of
tables gives the day, hour, and minute when eclipses occurred, the
longitude of the sun at the time of the conjunction, and charts the paths
of the central eclipses. Since and during Oppolzer's time, improvements
in astronomical computations have shown that his calculations
may be slightly in error. In the case of some Chinese eclipses, corrections
of Oppolzer may be found in F. K. Ginzel, Spezieller Kanon der Sonnenund
Mondfinsternisse für das Ländergebiet der klassischen Altertumswis-senschaften
und den Zeitraum von 900 vor Chr. bis 600 nach Chr., Berlin,
1899. Ginzel has again been corrected. The most recent work is P. V.
Neugebauer, Astronomische Chronologie, Berlin, 1929. For Han times,
Oppolzer is correct to within about half an hour.[1] As no one since
Oppolzer has produced a set of tables and charts covering Chinese territory,
so that his book is the only one convenient to use, and as so little
correction of his results is necessary, we may take Oppolzer's calculations
as being as reliable now as ever, except at regions near the limit
of visibility and when great exactness is necessary.

Oppolzer charts only the paths of umbral (i.e., total, annular, and
annular-total) eclipses (more exactly, only central eclipses). In most
cases, the Chinese however observed the sun as only partially eclipsed.
The region in which an umbral eclipse may be viewed as partial extends
in all directions from the umbral path. Oppolzer did not calculate this
area. We have allowed, on Oppolzer's charts, 15/16 to 1 inch at right
angles to the path of an umbral eclipse and ⅜ of an inch from the ends
in the direction of that path, as the area in which an umbral eclipse may
be seen as partial, remembering however that when we approach those
limits, we may need to make allowance for Oppolzer's errors and also
that the region of partiality may extend for much greater distances.[2]

More than one-third of the total number of solar eclipses—about 35.3
per cent—are nowhere umbral. These "partial eclipses" should be distinguished
from umbral eclipses which are viewed from some points as

partial. Since in partial eclipses the moon's umbra does not touch the
earth, Oppolzer has not indicated them on his charts. Yet they figure
among the eclipses recorded in Chinese history.

Astronomical chronology has sometimes neglected partial eclipses; J.
Fr. Schroeter's Spezieller Kanon der zentralen Sonnen- und Mondfinsternisse,
Kristiania, 1923, which claims to chart all eclipses visible in Europe
from 600 to 1800 A.D., omits partial eclipses entirely, even though many
such were visible within that region. Such eclipses are sometimes quite
conspicuous, sometimes being visible even from the polar circle as far
south as five degrees to the other side of the equator, so that historians
have recorded them. In dealing with these partial eclipses, we are aided
by certain empirical principles, known to the Greeks, but not much
used in modern times. Eclipses recur at intervals of 6585.3 days (18
years and 10 or 11 days), which period is called a saros. At intervals of
three saroi, 19,756 days (approximately 54 years 1 month), called by
Ptolemy an exeligmos, solar eclipses recur at approximately the same
place. If the eclipses occurring at intervals of an exeligmos are plotted
on a map, it will be found that they form a series which has certain
definite characteristics. Such a series invariably begins with a run of
partial eclipses visible in the neighborhood of one of the earth's poles,
followed by a long run of umbral eclipses, which gradually shift to the
other pole of the earth, and ends with a run of partial eclipses.[3] A solar
exeligmos series covers a period of from 1244 to 1532 years or 23 to 28
exeligmoi, with an initial and terminal run of 2 to 8 exeligmoi during
which only partial eclipses occur. If the noon points of the umbral
eclipses are plotted for such a series, the resulting curves[4] show periods
of shifting and periods of quietude, in the latter of which the curve
usually describes a loop. In periods of shifting, the noon points during
a period of 6 exeligmoi may shift as much as 180 degrees of longitude and
90 degrees of latitude; whereas, in a period of quiescence, at a node, the
noon points during as much as 8 exeligmoi may be located within an
area of 60° of longitude and 30° of latitude or less.

Since partial eclipses are those in which the moon's umbra (or its prolongation)
passes over one of the earth's poles, and since successive
eclipses in the same exeligmos series are located close to each other, it is
possible to determine definitely that a partial solar eclipse was not visible

in China by counting backwards or forwards by exeligmoi to the first
umbral eclipse. If that umbral eclipse was visible in the south polar
regions, the partial eclipse cannot have been visible in China. In this
manner half of the partial eclipses may be disposed of. A simpler
method of discovering whether a partial eclipse is visible in China is by
noting Oppolzer's calculated value of γ for that eclipse. If it is negative
and between -1.0 and -1.6, the eclipse was partial and visible only in
the southern hemisphere. By the above methods the regions of visibility
for about five-sixths of all eclipses may be approximated without
calculation.

If the nearest umbral eclipse in the same exeligmos series was visible
near the north pole, we cannot say with any certainty by this method
whether the partial eclipses in that series were visible in China. Less
than one-third of such partial eclipses are visible there, since only the
eclipse at one of the three saroi in an exeligmos can usually be visible in
China, and since the farther partial eclipses are from the nearest umbral
eclipse in the same exeligmos series, the more they tend to move northwards,
outside of Chinese latitudes. Because of the shifting in longitude
shown by some series, a determination of the approximate circumstances
of a partial eclipse from umbral eclipses in the same exeligmos series is
not reliable.

The visibility of partial eclipses in the northern hemisphere and of
umbral eclipses that lie near the limits of visibility must be calculated.
For that purpose the method used in Neugebauer, Astronomische Chronologie,
has been used. A useful preliminary step is to use the elements
given by Oppolzer with Neugebauer's tables as the latter directs (Op.
cit. I, p. 103). In this way little more than a mere inspection of Neugebauer's
tables is necessary. But the results thus obtained are only
approximate, and are useful only to eliminate eclipses that are plainly
invisible in China. In all cases where exact results are required, the computation
should be carried through as directed by Neugebauer. The elements
given in Oppolzer furnish a useful check upon such computations.