-
Acadmie royale de Belgique Koninklijke Academie van Belgi
CLASSE DES LETTRES
ET DES SCIENCES MORALES
ET POLITIQUES
MMOIRESCollection in-8 Tome LIX
Fascicule 4.
KLASSE DER LETTEREN EN
DER MORELE EN STAAT-
KUNDIGE WETENSCHAPPEN
VERHANDELINGEN
Verzameling in-8 Boek LIXAl vering 4.
Commmtaryon the
AstroiiomicaJ TreatisePar. gr. 2425
f
PAR
O. NEUGEBAUER
Brown University, Providence, R.I., U.S.A.
BRUXELLES
PALAIS DES ACADMIES
Rue Ducale, i
BRUSSEL
PALEIS DERACADEMIN
Hertogsstraat, I
N0 1819
1969
-
LISTE DES PUBLICATIONS RECENTES DE L'ACADMIE
CLASSE DES LETTRES
ET DES SCIENCES MORALES ET POLITIQUES
Mmoires in-8 2e Srie
Tome XXX
1. 1431. Favresse, F. L'avnement du rgime dmocratique Bruxelles
pendant lemoyen ge (1306-1423) ; 1932 ; 334 p 80
2. 1450. Rochus, L. Lalatinit de Salvien ; 1934 ; 142 p 70
Tome XXXI
1442. De Boom, Ghislalne. Les Ministres plnipotentiaires dans
les Pays-Basautrichiens principalement Cobenzl ; 1932 ; 421 p
100
Tome XXXII
1445. Doutrepont, Georges. Jean Lemaire de Belges et la
Renaissance ; 1934 ;L-442 p 80
XXXIII
1449. Vercauteren, Fernand. tude sur les Civitates de la
Belgique seconde.Contribution l'histoire urbaine du Nord de la
France, de la fin du IIIe la fin du XIe sicle ; 1934 ; 10 cartes, 4
facs., 488 p puls.
Tome XXXIV
1460. Van Werveke, H. De Gentsche financin in de Middeleeuwen ;
1934 ;3 diagr., 423 p 90 >
Tome XXXV
1468. Bonenfant, P. Le problme du pauprisme en Belgique la fin
de l'an-cien rgime ; 1934 ; 579 p 160
Tome XXXVI
1. 1462. Lefvre, J. La Secrtairerie d'tat et de Guerre sous le
rgime espagnol,1594-1711 ; 1934 ; 268 p 60
2. 1481. Velge, H. Y a-t-il lieu de crer en Belgique une Cour du
contentieux admi-nistratif ? Quelles devraient tre sa comptence et
son organisation ?1935; 159 p 40
XXXVII
1. 1483. Puttemans, A. La censure dans les Pays-Bas autrichiens
; 1935; 1 pl. ;376 p 80
2. 1482. Leemans, E.-A. Studie over den Wijsgeer Numenius van
Apamea metuitgave der fragmenten ; 1937 ; III-174 p 80
Tome XXXVIII
1. 1497. Cornll, Georges. Une vision allemande de l'tat travers
l'histoire et laphilosophie ; 1936 ; 198 p 50
2. 1517. Yans, Maurice. Histoire conomique du duch de Limbourg
sous la MaisondeBourgogne. Les forts et les mines ; 1938 ; 1 carte,
278 p 60
Tome XXXIX
1. 1523. Adontz, Nlcolas. SamueH'Armnien, Roi des Bulgares ;
1938 ; 61 p 302. 1524. Delatte, Ivan, La vente des biens nationaux
dans le Dpartement de Jemap-
pes ; 1938 ; 136 p puis.3. 1526. Van Steenberghen, Fernand. Les
uvres et la doctrine de Siger de Bra-
bant ; 1938 ; 195 p 60 4. 1549. P. Peeters, S. J. L'uvre des
Bollandistes ; 1942 ; 128 p 60
Tome XL
1532. Doutrepont, Georges. Les Mises en prose des popes et des
Romans che-valeresquesdu XIVe au XVIe sicles ; 1938 ; 732 p 240
Tome XLI
1534. Laurent H. et Qulcke F. Les origines de l'tat Bourguignon.
L'accessionde la Maison de Bourgogne aux duchs de Brabant et de
Limbourg ; 1940 ;507 p 160
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ACADMIE ROYALE DE BELGIQUE
CLASSE DES LETTRES
MMOIRESCollection in-8. Deuxime srie.
KONINKLIJKE ACADEMIE VAN BELGI
KLASSE DER LETTEREN
ERHANDELINGENVerzameling in-8. Tweede reeks.
BOEK LIX, afl. 4.
BRUXELLES
PALAIS DES ACADMIES
Rue Ducale, 1
BRUSSEL
PALEIS DER ACADEMIN
Hertogsstraat, 1
1969
-
IMPRIMERIE J. DUCULOT
s. a.
GEMBLOUX
-
Commentaryon the
Astronomical Treatise
Par. gr. 2425
PAR
O. NEUGEBAUER
Brown University, Providence, R.I., U.S.A.
Impression dcide le 7 octobre 1968
Lettres. T. LIX fasc. 4.
-
To the memory of F. Cumont and A. Delattewho first recognized
the importance of Par.gr. 2425
Introduction
Par. gr. 2425 was written by a 15th century hand. The text
whichconcerns us here (fol. 232v to the end, fol. 285v) is divided
into 86consecutively numbered sections of very uneven length ( x )
but itis easy to see that they do not form a real unit.The first
three sections are a table of contents, or summary, of
an astrological treatise ascribed to Antiochus and published
byCumont in CCAG 8, 3 p. 111-119.Sections 4 to 27 are astronomical
tables but obviously incomplete.
One finds, e.g., tables for planetary latitudes and visibilities
but nomean motions and equations. These tables contain clear
evidenceof Islamic influence (in particular the values = 23;35,0
for theobliquity of the ecliptic and i = 4;46,0 for the inclination
of thelunar orbit); they are, at least in part, identical with the
tables usedin the computations of the subsequent sections.Sections
28 to 69 can be easily dated from the examples which
they contain. We find three sets of dates:
A.M. 6569 i.e. A.D. 1060/61 (Nos. 28, 30, 35, 36, 45, 49,
50)A.M. 6577 i.e. A.D. 1069 (Nos. 46-48, 53, 57, 58)A.M. 6580 i.e.
A.D. 1072 (Nos. 59, 61).
The last example concerns a solar eclipse which was very
inconspi-cuous in Byzantium. Only 14 years later, in A.D. 1086, the
path ofa total eclipse passed right over the city. This makes it
practicallycertain that our text had been completed before this
event.Apparently our text was compiled over a period of one or
two
decades and this may explain the inconsistency and
repetitiousnessin the arrangement of its topics. It is also clear
that the present order
1 As usual with texts of this type later accretions are found at
the end: the astrolo-gical sections 63 (= Geoponica 1,8) and 64,70
to 86 (fol. 281 r, 8 to 285). In particularNos. 73 to 86 (fol.
285") are only a list of classifications of the zodiacal signs.
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6 COMMENTARY ON THE ASTRONOMICAL TREATISE
is not genuine; sections 49 to 52, e.g., are an intrusion
between Nos. 46to 48 and Nos. 53 to 58 which concern the same
example.The tables might, of course, be much older, though their
Islamic
component makes a date before the middle of the 9th century
unlikely.The above quoted values for and i are first attested in
the tablesof Habash al-Hsib, about A.D. 850 (*), and it is tempting
to identifythe (2), with the zj of Habash al-Hasib. Our textwould
then be a witness for the early transmission of the first,
i.e.Abbasid, period of Muslim astronomy to Byzantium. In the
samedirection point the very close parallels, in particular in the
sectionon eclipses (No. 60 to 62), with a commentary to
al-Khwrizmiby al-Muthann which is preserved in Hebrew and Latin
trans-lations (3) of the llth and I2th centuries.It is of interest
to note that Byzantium is given the latitude = 41
which is characteristic for clima V (4) and which is indeed the
correctlatitude of Constantinople. In the tradition of the " Handy
Tableshowever, Byzantium is placed between clima V and VI at =
43;5much too far to the north (5 ).
28. Length of Seasonal Hours at Daylight
Let () be the oblique ascension for a given geographical
latitude of the point of longitude of the ecliptic. Let be the true
solarlongitude at a given date, determined by means of solar tables
(called" the tables of Khaspa "). Then the length of daylight in
degreesis given by the " day arc i.e. the arc above the horizon
travelledby the sun at the given day:
d = ( + 180) - ().
For an alternate procedure see No. 37, for an example No. 61 (6
).
1 Kennedy, Survey, p. 126 (N 15) and p. 151.2 Fol. 257', 22.3
Cf. Mills Vallicrosa, Bibl. Catedr. de Toledo, p. 192, and the
editions by Goldstein
(1967) and Mills-Vendrell (1963) respectively.4 More accurately
40;56 according to the Almagest.5 Cf., e.g., Halma II p. 58/59.
Incidentally, Halma's heading "eigth climate" is in
all probability his own invention. In Vat. gr. 1291 fol. 5 r the
heading is "climate forthe parallel through Byzantium". The
latitude 43;35 given by Halma is based on amisreading of () as =
30.
6 Below p. 26, I.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 7
Since I o = 0;4 we have for the length of daylight in
equinoctialhours
dh = 0;4'd.
Finally, one seasonal hour of daylight, measured in degrees, is
ofthe length
Vb- = d = 0;5-d.12
Example
[A.M. 6569 (= A.D. 1060)] O Ind. 14 Dec. 29 at
Constantinople'i.e. clima V:
A0 = [3]14;47thus (2)
p(Ao) = 307;20,26 (0 + 180) = 84;44,24
hence
d = 444;44,24 - 307;20,26 137;24and
dh= 0 ;4 137;24= 548min96sec = 9 ;9,36*[briefly : 0;4 2, 17;24=
9 ;9,36]
l1* =0;5 137;24= 687' = 11 ;27 [briefly : 0;5 2,17;24= 11
;27].
29. Length of Seasonal Hours at Night
The length of the night in equinoctial hours is given by means
ofNo. 28:
nh = 24h - db
and the length of one seasonal hour of night in degrees = 30 1*
* of daylight in degrees.Finally, the length of the night measured
in degrees (the " night
arc ") isn = 360 - d
where d is known from No. 28.
1 Cf. e.g., Nos. 35 and 36.2 Theon's "Handy Tables" would give
and 84;35,24 respectively.
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8 COMMENTARY ON THE ASTRONOMICAL TREATISE
30. Noon from Sunrise
From No. 28:
-d = 6sh2
Therefore in the example of No. 28 :
6,ft =6-ll;27 [briefly:6-ll;27 = 1,8;42]
0;468;42 = 274min48sec = 4;34,48 [briefly:0;4-l,8;42 =
4;34,48Jfrom sunrise to noon.
31 to 33. Equinoctial or Solstitial Noon Altitude of the Sun and
Geo-graphical Latitude
If h0 is the noon altitude of the sun at equinox at a locality
of geo-graphical latitude then
For Constantinople: = 41, h0 = 49.If hx is the noon altitude of
the sun at the summer solstice, h2
at the winter solstice, then
where is the obliquity of the ecliptic. The value = 23 ;35 is
com-monly used in Islamic tables (e.g. by Habash, Battn,
Kshyr,Birnx i 1)); the same value is used in the table of solar
declinations,fol. 239v/240v, but not fol. 247r/249v which are based
on = 23 ;5 1,20as in the Almagest or = 23 ;51 as in the Handy
Tables.
34 to 36. Noon Altitude of the Sun in General and Geographical
La-titude
If h is the noon altitude of the sun at a given day, then
h0 = 90 = or = 90 h0 = E0 .
= 90 (ft t ) = 90 (h2 + )
where h0 is the equinoctial noon altitude.
1 Cf. Kennedy, Suivey p. 151-156.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 9
Example
A.M. 6569 (= A.D. 1061) Ind. 14 Febr. 23, at Constantinople.For
this date
A0 =)(ll;15 = 341;15.
The table of declination (fol. 239*) gives (x )
for = 341 |5| = 7;29,4
= 342 I I = 7;6,50
By linear interpolation: 0;22,140;15 = 0;5,33,30 0;5,33,thus
for = 341 ;15 | | = 7;29,4 - 0;5,33 = 7;23,31.
Since for Constantinople h0 = 49 (cf. No. 31), we find for the
givendate
/i = 49 7;23,31 =41;36,29
a result slightly garbled in the text.
Alternate Method for the Same Date
There must have existed a table (not extant in our MS) whichgave
to every degree of solar longitude the corresponding noonaltitude h
of the sun (2) of course computed for the given , in ourcase = 41.
From our text we can restore the entries
= )(11 h = 41 ;31
X 12 41 ;54
Thus by interpolation for = )( 1 1 ;15 :
h = 41 ;31 + 0;15 * 0;23 = 41;31 + 0;5,45 = 41;36,45
i.e. slightly more than with the more accurate tables.
37. Length of Daylight
In order to find the " day arc " (cf. No. 28) one can also
proceedas follows: from the tables of right ascensions one can find
to the
1 In the tables fol. 247' one finds, however, 7;33,58 and
7;10,46 i.e. 0;0,1* more thanin the Almagest (I, 1S) which is based
on < = 23 ;5 1,20. The tables fol. 239" agree withNo. 33 in
assuming e = 23;35,0.
2 The text seems to call these tables "for rising times" which
is certainly incorrect.Probably the tables for h were combined with
tables for ().
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10 COMMENTARY ON THE ASTRONOMICAL TREATISE
solar longitude Q for the given day the right ascension ()
andfor the given clima also the oblique ascension (). Then
d = 2(90 ( I () () | )) if the sun is f the equator.
The correctness of this procedure is evident from fig. 1. The
arcin question is SMzl. Its half is measured on the equator by the
rightangle CE plus ET = p; this proves the above-given rule.
Hor.
Fig. 1.
The process is unnecessarily complicated since the method ofNo.
28 requires only the table of oblique ascensions for the
givenclimate. Here one has to have also a table of right ascensions
(which,incidentally, is not the table of " normed right ascensions
" foundon fol. 238v/239r, or in the Handy Tables, reckoned from z
0)
Nos. 38 to 41. Trigonometric Functions
For a circle of radius R = 60 we use the following notation
Sin = R sin CosO = R cos0 Vers = R Cos.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 11
In the text only SinO and VersO are named (evOeia and evOela
respectively). The angles are correspondingly distin-guished as
nepi^epeia and respectively. CoSappears only in the form Sin(90 0).
The table called here v is not, as the name seems to indicate, a
tableof chords (as in Almagest I, 1 1) but a table of Sines. Such a
tableis found on fol. 239v/240v but no table of Vers 0 is given in
ourMS.
The rules given for VersO in Nos. 40 and 41 are illustrated in
fig. 2:
R/r
/0vers a
Fig. 2.
Find Vers : if < 90 Vers = R Sin(90 )
if > 90 Vers = R + Sin(0 - 90)
Find : if Vers 0 < R find ' = arcSin( Vers 0)
then = 90 - 0'
if Vers > R find ' = arcSin(Vers 0 R)
then = 90 + 0'.
42, 43. Time since Sunrise from Solar Altitude
The rules of the text can be formulated as follows: if h is the
noon
altitude of the sun, h' the altitude at t after sunrise, then t
can befound from
RSinh'Sin t =
Sin/i(1)
forpositions of the sun before noon and 0;4-/represents the
seasonalhours elapsed since sunrise. For positions after noon the
time sincesunrise is 12 0 ;4 / seasonal hours, t (in degrees) being
obtainedfrom (1).Obviously these rules cannot be generally correct.
Multiplication
by 0;4 can only result in equinoctial hours when t is given in
degrees.
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12 COMMENTARY ON THE ASTRONOMICAL TREATISE
Also (1) is only correct if the sun is in the equator, i.e. for
h = h0the equinoctial noon shadow. Then indeed (cf. fig. 3)
Z
. , sin h'0sinn0 =
sini
If, however, the sun is not in the equator, thus h h0, then the
pro-blem is not determined by h and h' alone (cf. fig. 4) since the
positionof the small circle RXM travelled by the sun depends on the
solardeclination CM. The correct relation is given in No. 65
0).
Z
Also the final transformation of units cannot be correct in
the
form it is expressed. Apparently the length of daylight seems
nowto be assumed as known.
1 Below p. 41.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 13
44, 45. Daily and Hourly Motion
If the longitudes of a celestial body on three consecutive
days(at noon i1)) were n _ u , +1 respectively, then the daily
velocityvo/ on day n is given by A_j or by +1 without ourbeing told
which value has to be accepted in case they are different.The
corresponding hourly velocity v/h is to be found from
po/* = io/d. 0;52
which is equivalent to the trivial v0/h = ^ vo/d.As an example
is used the solar motion for A.M. 6569 (= A.D.
1061) Ind. 14 and
=219;15 on Febr. 1
_! =18;14 on Jan. 30 (sic!)
thus vo/ d = 1;1 and v'h = ;5 ' 1;1 = 0;2,32,30/ft .2
The text mentions the (now meaning apogee, not altitude?)of the
sun and declares it to be (meaning?) withoutthese data possibly
being of influence on the determination of thedaily motion. It is
perhaps a mistaken rendering of some statementconcerning the
distinction between direct and retrograde motionof a celestial
body.
46 to 48. Solar Longitude at Sunrise, Sunset and Midnight
The solar longitude is considered to be known for noon of
thegiven day and for the preceding and following days.
Consequentlythe daily and the hourly motion is known. Similarly it
is assumedthat the length of one seasonal hour for the given day is
given. Thusthe solar motion during + 6s h or during 1 2 can be
computed, thehourly motion being considered constant.
1 This is to be expected in the tradition of the Almagest and is
confirmed by the threesubsequent sections.
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14 COMMENTARY ON THE ASTRONOMICAL TREATISE
Examples
[A.M. 6577 (= A.D. 1069)] Ind. 7 Febr. 20 day 6 (= Friday) 0)[at
noon:] XQ = )( 8;11.Also known: the daily motion vld = l"la thus
v,h = 0;2,30o/ft
and furthermore the length of the seasonal hour: l s'1 =
13;45,1,40.Consequently
6S" = 1,22;30,10 = 5;30,0,40" 5;.
The solar motion during 6s h is therefore
= 5\0 -0-,2,1 = 0;13,45.
Hence the solar longitude on Febr. 20
at sunrise: )( 8 ;1 1 0;13,45 = )( 7;57,15
at sunset: )( 8 ;1 1 + 0 ; 1 3,45 = )( 8;24,45.
For the solar motion during 12" one finds, of course, 120;2,30
=0;30 thus for the solar longitude at midnight to Febr. 21
)( 8 ;1 1 + 0;30 = )( 8;41.
The multiplications required by these steps are performed in
thetext very clumsily because each sexagesimal digit is multiplied
sepa-rately in decimal fashion, e.g. 0;2,305;30 is computed as
follows
0;2,305 = 10' + 150"0;2,300;30 = 60" + 900"
total = 10' + 210" + 900" = 0;13,45.
49, 50. Place and Time of Conjunction
Assume that at noon near conjunction the longitudes of sun
andnoon are and Ac respectively and A = . Since the dailymotion of
the moon is about 13 /velocity 12o/d, we know that the time until
conjunction is ~ = 0;5
. The place of the conjunction is therefore
Aq + 0;5 = + + 0;5 .
A more refined procedure is given in No. 59.
1 The same date occurs explicity in No. 57 and again without
year in Nos. 53 and 58.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 15
Example
A.M. 6569 (= A.D. 1061) Ind. 14 Febr. 22 [noon]:
A0 = X 10;16 = )(4;10 thus = 6;6
and 0;5 = 0;30,30. Thus the place of conjunction is
0 + 0;30,30 = )(10;36,30
lc + 6;6 + 0;30,30 = )( 10;36,30.
For the determination of the time of conjunction a more
accuratemethod is followed, in which the rough estimate of a
relative velocityof 120,d is replaced by = vc ve for the given day.
Thereforethe time required to cover the elongation is computed as =
/ and the resulting hours are transformed into seasonalhours.
Consequently we have in continuation of the preceding example
vc = 0;34,2,300/* [thus 13;370/'1, assumed to be known]
v0 = 0;2,27,30 [thus 0;59 , assumed to be known]
hence
= 0;31,350/.
Since we had = 6;6 we find = / = 1 1 ;35,18fc 1) forthe time of
the conjunction, reckoned in equinoctial hours sincenoon of Febr.
22.
The half length of daylight on Febr. 22 is assumed to be knownas
5;32" (2); therefore the conjunction fell 1 1 ;35,18 5;32 = 6
;3,18ftafter sunset. Since it was assumed that half of the daylight
is 5;32",half of the night would be 6;28\ and therefore 6 ;3,18 =
6;3,18 6/6 ;28 5;37,55 of night.In the text this transformation is
carried out in an unnecessarily
complicated way and is furthermore not quite accurate. First
theequinoctial hours are changed to time degrees and expressed
asseconds :
^"^ = 90;49,30 = 326970".
Then it is stated that one seasonal hour of the night amounts to
16;9= 58140". This is not correct since 6 ;28ft = 1,37, thus one
seasonal
1 Rounded from 11 ;35,18,12,...2 Cf. No. 46 where it was found
that on Febr. 20 6" = 5;30\
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16 COMMENTARY ON THE ASTRONOMICAL TREATISE
hour of night is 16 ;10 not 16;9. But operating with the latter
valuethe text finds the quotient 90;49,30/16;9 in the following
fashion:
Thus the result is 5;37,25 seasonal hours of night.
51, 52. Seasonal and Equinoctial Hours
The transformation of equinoctial hours into seasonal hours
andvice versa requires the knowledge either of the length of
daylightor the equivalent in degrees of one seasonal hour. The
examplesof the text are based on the assumption that 13;24" = 12s"
or onthe equivalent relation P'1 = 16;45. Indeed 13 ;24- 15/12 = 16
;45.
53. Ascendant from Solar Altitude
For a given geographical latitude and a given date, the
longitude and the altitude h of the sun at noon can be found (No.
35), aswell as the length of the day arc (No. 28 or No. 37) and the
corres-ponding length of one seasonal hour. Consequently also the
solar
326970" _ 5 ! 3627058140" 58140
36270-60 = 2176200
2176200 _ 37 2502058140 58140
25020-60 = 1501200
Hor. at
\
\ c
Hor.
Ecl.
Fig. 5.
-
COMMENTARY ON THE ASTRONOMICAL TREATISE 17
longitude 1R at sunrise can be considered known (No. 46) and
fromthe tables of oblique ascensions for the given climate one then
canfind the right ascension aR of the point of the equator which is
inthe horizon simultaneously with AR (cf. the schematic
representationin fig. 5).Let us assume that the sun is observed
before noon of the given day
at an altitude h'. Then one can find the time t which has
elapsed sincesunrise (No. 42, or rather No. 65), and hence the
equator arc whichhas risen in the time from sunrise to the
observation. By adding thisarc to aR we find the right ascension of
the point E of the equatorwhich is in the horizon at the moment of
observation. The table
of oblique ascensions will then give the longitude of the point
Hof the ecliptic which rises simultaneously with E at the moment
ofobservation.
Example
[A.M. 6577 (= A.D. 1069)] Ind. 7 Febr. 20 day 6 (= Friday) ( x
)[at Constantinople, = 41 (No. 31)]
observed solar altitude: h' = 36, thus the time since
sunrise
t = 3;56,40,40a = 59;10,10 = 4;181 . (2)
According to No. 46 the solar longitude at sunrise was
AR = )(7;57,17
and therefore the simultaneously rising point of the equator
aR = 347; 12,9
(the Handy Tables for clima V would give 347;19,21). Thus we
findfor the equator arc risen since sunrise the endpoint E of right
ascension
347; 12,9 + 59;10,10 = 406;22,19 = 46;22,19 =
to which corresponds the ecliptic point H of longitude
= 8;27,41
(the Handy Tables for clima V would give 8;28,3). Thus 8;27,41is
rising when the sun at )( 8 ;1 1 has reached the altitude of
36.
1 Cf. p. 14 note 1.2 The value for t in seasonal hours is not
needed for the following. The given value
does not agree with the Iength of the day arc at the same date
in No. 46. Accordingto No. 46 one has * = 13;45,1,40 thus d/2 =
1,22; 30, 10 and 59;10,10 = 4;18,8''1 Using d/2 = 1,22;30,10 and
the sine tables of fol. 239" ff. I find with the methodof No. 65
the value t = 57;38, 13,31 ~ 3;50,33*.
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18 COMMENTARY ON THE ASTRONOMICAL TREATISE
54. Midheaven from Ascendant
If p(H) is the oblique ascension of the ascendant H at the
givenclimate, a' = a + 90 the normed right ascension (i.e. right
ascensionreckoned from z 0), then a'(M) of the culminating point M
of theecliptic is given by ( x )
a'(M) = p(H).
Continuation of the Example from No. 53
We know that = 46 ;22, 19 46 ;22 = p(H) = a'(M). Inter-polation
in the tables of the normed right ascensions gives
M = s13;50
(cf. fig. 5) (2 ). Thus at the moment of observation ss 13;50
wasculminating.
55. The Loci
The " Loci " are 12 consecutive ecliptical arcs, counted in
thesense of increasing longitudes, such that locus 1 begins at the
ascen-dant H, locus 4 at the lower culmination M, locus 7 at the
settingpoint , locus 10 at midheaven M (cf. fig. 6). The three loci
of eachquadrant are constructed in such a fashion that the
endpoints haveconstant right ascensional diiferences.
Continuation of the Example from No. 53 and 54
Loci 7, 8 and 9. We want to divide the ecliptic arc zlM in
threesections such that the right ascensional differences of the
endpointsare equal, i.e. (in fig. 6) such that F7F8 = F8F9 = F9C =
. Because
F7C = 3 = 90 - WF7 = 90 - (a(H) - p(H))
= p(H) - (a(H) - 180 + 90) = p(H) - (a(J) + 90)
= p(H) - '(A)
where a'(zl) is the normed right ascension (3) of the setting
point ,we have
= 1(()-'()).
1 For a proof see, e.g., Almagest II, 9.2 The result is the same
for the Handy Tables or for our tables (fol. 238").3 Cf. No.
54.
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COMMENTARY ON THE ASTRONOMICAL TREATISE
In No. 53 it was found that
H 8;27 p(H) = 406;22,19 = 46;22,19 mod 360
19
thus
and therefore
thus
= t 8;27 a'(d) = 336;38,48 1 )
5 0
T0*
' \
"
\/ \ Fs / \/ \\c/^ \/ \ \/ \ v
W
/ \ v
k \\() /*** rf *"^
ipfHlV
EcLA
Eou.
Fig. 6.
1 Exactly agreeing with the Handy Tables, whereas fol. 239 r
would give 336;1 1,48. The scribe of our MS repeatedly misread the
final 8 as 2.
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20 COMMENTARY ON THE ASTRONOMICAL TREATISE
Hence we have the following normed right ascensions for the
end-points of loci 7 to 9:
fts F7 : a' = 336;38,48
F8 : 359;53,18
F9 : 23;7,48
M C : 46;22,18 = p(H)
with constant difference t. The diametrically opposite points
definethe endpoints of the loci 1 to 3.
1For the loci 10 to 12 and 4 to 6 the constant difference is -
(180
3) = 60 = 36;45,30. Therefore
M C : a' = 46;22,18 = p(H)
Fn : 83;7,48
F12 : 119;53,18
H Fj : 156;38,48 = a'(J) - 180.
The points of the ecliptic which correspond to the points F8 ,
F9 ,Fu , F12 , etc. can be found directly from the tables of normed
rightascensions. E.g. a'(F8) = 359 ;53,18 is found to be the normed
rightascension of = f 29;54 (^).
56. The Loci. Alternative Method
If the sun were in H (cf. fig. 6) the time measured by the
equatorarc F^C would bring the sun from the horizon to the
meridian. ThusF^C is the half length of daylight l/2 d or 6
seasonal hours at thetime of the year when the solar longitude is
A(H). The Handy Tableslist for each climate for every degree of not
only the correspondingoblique ascension () but also the length t of
one seasonal hourof daylight. Since FtF12 = F^F^ = FtlC = 21 one
can find thenormed right ascension of the boundaries of the loci 10
to 12 byadding the amount of 21 or 41 respectively to a'(M) = p(H).
Thisbrings us to a'(H). To this value we add 2t' and 41' where 21'
is thelength 60 21 of two seasonal hours of night for the solar
longitudeA(H). The results are a'(F2) and a'(F3) = a'(M)
respectively.
1 Both from the Handy Tables and fol. 239'.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 21
Continuation of the Example from No. 53 and 54
In No. 53 it was found that H n 8 ;27. For this solar
longitudeone finds for one seasonal hour the length t = 18 ;23 (x
). Thus 21= 36;46 is the length of the loci 10 to 12 (and 4 to 6).
For the loci1 to 3 (and 7 to 9) the length is consequently 21' = 60
21 = 23;14.In No. 55 the corresponding values were slightly more
accurate,namely, 36;45,30 and 23; 14,30 respectively. Consequently
thereappear small deviations between the results obtained in No. 55
andNo. 56.
57. Ascendant from Midheaven
The relation from No. 54
a'(M) = p(H)
is now used in the opposite direction.
Continuation of the Example from Nos. 46 to 48, 53 to 56
For A.M. 6577 (= A.D. 1069) Ind. 7 Febr. 20 day 6 (= Friday)noon
it has been found that = )( 8 ; 1 1 . In other words we knowthat M
= X 8 ; 1 1 . From the tables (fol. 238v) (2) it follows that
a'()(8;ll) = 69;53,16.
The tables for oblique ascension are said to give for p(H) = 69
;53,16the argument H = as 1 ;55 (3 ). This is the ascendant at noon
of theday of observation.
58. Positions of Sun and Moon at Hours Different from Noon
Positions found from tables refer to noon of the given day.
Formoments difTerent from noon one has to find the necessary
correctionas product of time difference and hourly velocity.
Example, continued from Nos. 45 to 48 and 63 to 57
[A.M. 6577 (= A.D. 1069)] Ind. 7 Febr. 20. At the moment
ofobservation (of a solar altitude of 36) the time elapsed since
sunrise
1 The Handy Tables give forll 8 the value t = 18;24, forn 8;27 /
~ 18;52.2 Same value from the Handy Tables.3 The Handy Tables would
give H = 23 2;8.
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22 COMMENTARY ON THE ASTRONOMICAL TREATISE
is t = 3 ;56,40,40 (as in No. 53). Half length of daylight: djl
=5;30,0,40'1 (as in No. 46). Therefore the time before noon:
t = 1 ;33,20".
Since the hourly velocity of the sun is 0;2,30o/'1 (or l 0/d)
(^) the cor-responding motion is
= 1 ;33,200;2,30 0;3,53 0;4.
The position of the sun at noon was found to be )( 8 ;1 1 (No.
46-No. 57), therefore at the moment of observation )( 8 ;1 1 0;4 =X
8;7.For the moon an hourly motion of 0 ;33,32/ is assumed (2).
What
follows contains several errors. The time difference is taken to
be
1 ;35,20" (instead of 1 ;33,20) which leads to a motion of
= 1 ;35,200;33,32,30 = 0;53,17,38,20.
The longitude at noon is given as 17 ;36 (modern computation
showsthat it must be in z) and therefore at the time of observation
17 ;36 0;53 = 16 ;43 (in z) which is changed in the text to the
senselessvalue 36;43.
59. Accurate Longitude of Conjunction
In No. 49 it has been described how one can find the longitudeof
a syzygy from given positions of sun and moon at the precedingnoon.
This procedure is now refined insofar as the noon positionsare used
not only preceding but also following the conjunction.The rules of
the text (which are reminiscent of Almagest VI, 5) andthe
subsequent example are easiest understood if one discusses
theproblem beforehand in modern terms.
1. Notation. We assume the following longitudes as known:
at noon of day n: , ; 0 = >0
at of day + 1: , ' ; ' ^ = ' > 0
~' = > '- = Vc, Vc Vq = .
1 Same assumption in Nos. 46 to 48; the example in No. 50
belongs to another yearbut assumes 0;59"ld for the solar velocity
on Febr. 22.
2 Corresponding to a daily motion of l3;25/d, no derivation
given.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 23
2. Accurate determination of the longitude of the
conjunctionunder the assumption of constant velocities of sun and
moon during
Fig. 7.
From
A q lcvo vc
it follows that
= A0J - Xc vf- = (A0 - Ac)^ - Xc(Vf - = + ov v \ vj
and similar from 1
thus
= ' +
= + ^ = ^-'^ (1)
which solves our problem.
3. Successive approximation. If we assume convenient roundvalues
for and vc, e.g. = l/d, vc = I3' d thus = \2/dand
SW (2)
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24 COMMENTARY ON THE ASTRONOMICAL TREATISE
then the two equivalent components in (1) will give in general
diffe-rent results, namely
a = Ac + cAk (3a)
b = 'c - ' (3b)such that
a b 0. (4)
The values a and b are easy to compute but we now need a
secondstep to find corrections which amount to a or b. In orderto
determine these corrections we form with (1) and (3a).
a = (c + 41^ - (c + cX) = (cv - rc).\ vj ov
But from our initial definitions and (3a), (3b), and (4) it
follows that
cv vc = ( ac A'q + Aq ) ' + Xc
= (Ac + cJ) ( cJA') = a b = thus
= - (5)
and similarly
^ + '- (6)
4. The rules of the text require one to compute first the values
aand b from (3a) and (3b) respectively, using c = 1;5 = 13/12,
andthen to find the correct value by means of (5) and (6) which
mustgive identical results.It is obvious that this procedure has no
advantages over the direct
use of the accurate relation (1). It is true that it is easy to
find the
approximate solutions a and b, but the term in (5) and (6) is
exactly
as inconvenient to compute as the term ^ m (1). Thus the
determi-nation of the values (3a), (3b) and (4) is superfluous and
apt to intro-duce unnecessary rounding errors.
5. Example: A.M. 6580 (= A.D. 1072) Ind. 10 May 20 day 1(=
Sunday). The following noon positions are given (^):
1 Cf. also No. 61 (below p. 26).
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COMMENTARY ON THE ASTRONOMICAL TREATISE
May 20 (day ri) May21 (day n + 1)
= 5;9,30 X'c = n 16 ;39,2 vc = l3;3T,13 0'd
Ac = n 3 ;1 ,49 '0 = n 6;6,47 vQ = 0;57,17/
= 2;7,41 JA' = 10;32,15 = 12;39,560/J
Procedure of the ext '
12JA' = 2;7,41 1;5 2; 18, 19
= 3;1,49
a = n 5;20,8.
Thus from (3a):
' = 10;32,15 - 1 ;5 11;24,5612
k'c = 16;39,2
thus from (3b): b = n 5;14,6
and from (4): = a b = 0;6,2.
Consequently
= 0;0,28,34,56 12;39,56
and therefore
= 2;7,41 0;0,28,34,56 0;1,0,49 0;1,1
a = 5;20,8
thus from (5): = n 5;19,7.The same value is obtained if one uses
(6).
1;4,31,22
Direct procedure:vc _ 13;37,13 12;39,56 '
and therefore
= 2;7,411;4,31,22 2;17,18
= 3;1,49
thus from (1): = 5 ;19,7.
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26 COMMENTARY ON THE ASTRONOMICAL TREATISE
60 to 62. Solar Eclipses
In No. 60 we find the rules for the computation of the
circumstancesof a solar eclipse whereas Nos. 61 and 62 contain the
practical appli-cation to the case of the eclipse of A.D. 1072 May
20 (*). Since thedetails of the procedure are rather complex the
description in termsof general formulae would become very unwieldy.
It therefore seemspreferable to divide our commentary into separate
sections, eachof which combines the theoretical introduction with
the numerical
example.
1. Elements at the true conjunction
The following continues and amplifies the data given in the
exampleofNo. 59 (2) concerning the solar eclipse of A.M. 6580 (=
A.D. 1072)Ind. 10 May 20. The given data are in agreement with No.
59:
and
Sun
May 20May 21
Moon
May 20May 21
1
64;26,4665 ;25,540;59,8
62;53,776; 3,4213 ;10,35
3; 1,49 16;39, 2
13 ;37,13
5;9,30 6;6,47
0;57,17
mean
anomaly95 ;33,16108 ;37, 1013; 3,54
mean
elongation358 ;26,2110;37,4812 ;1 1 ,27
ascending node
motion position
294; 4, 8 n 5;55,52294; 7,19 n 5;52,410; 3,11 0; 3,11
For Constantinople the oblique ascension of 0 on May 20 and ofA0
+ 180 are
p( 5;9,30) = 43 ;31,14 p(f 5;9,30) = 262;57,43
1 Oppolzer, Canon No. 5411.2 Cf. above p. 24 5.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 27
thus the day arc
d = 219;26,29 14;37,46
and noon falls 7 ; 1 8,53h after sunrise, while the seasonal
hour amountsto
1*" 18;17,12.
The noon altitude of the sun on May 20 is
h = 70;17,35 ( x ).
Also for May 20 we have = 0 = 2;7,41, = vc Vq = 12;39,560/
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28 COMMENTARY ON THE ASTRONOMICAL TREATISE
of the parallax is a vector pointing away from the meridian.
Itsamount can then be expressed in terms of time reckoned in
equinoctialhours. Similarly, the position of the point at which we
wish to findthe parallax can be defined by its hour angle H. Then
we know thatpx = 0 for H = 0 and px = max = poX for H = 90. For
points Hbetween 0 and 90 it is simply assumed that
Px = ^S1S0 H = poismH R = 150. (2)K
In our special case it is assumed that = 1 ;36 . Hence
Pox _ 4R 2,30 6,15
and therefore
pJ = -^-Sin150 H (3)0,1j
In the explanatory text (fol. 270v, 18 etc.) the denominator
6,15 isincorrectly replaced by 5 and the factor is said to be H
instead ofSin150 H. The actual computation, however, always
correctly uses6,15 and Sin H.If the true conjunction occurs at an
hour angle H0, parallax increases
4H0 according to our hypothesis by an amount of Sin150 H0 .
This point, however, is nearer to the horizon and therefore
belongsto a region of higher parallax than at H0 . Indeed, the
increment
4 4
ofH0 would now be ^ Sin150 Hj where Hj = H0 + Sin150 H0 .Again,
the new position would produce a longitudinal parallax H0
4 4+ Sin150 H2 where H2 = H0 + Sin150 Ht . It s easy
to see that this iteration process is rapidly converging. In our
examplefive steps are computed, at which level no more changes
occur withinthe seconds of time. The numerical values are shown in
the followingtable:
ti
H, = 15 f,Sin 150 H|4 Sin150 H,/6,15i, +1 = f + ...
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COMMENTARY ON THE ASTRONOMICAL TREATISE 29
i = 0
4;1,57*60;29,15130;31,56
1 ;25,32" O5;27,29
i = 1
5;27,2981 ;52,15148 ;29,25
1 ;35,25 ;36,59"
i = 2
5;36,59*84; 14,45149; 14,24
1;35,31*5;37,28"
= 3
5;37,28"84;22,0149;16,13
1 ;35,32"5;37,29ft
z = 4
5;39,29"84;22,15149;16, 17
1 ;35,325 ;37,29"
Here we see that the longitudinal parallax moves the momentof
the true conjunction (at 4 ; 1 ,57ft p.m.) to a later moment of
apparentconjunction (at 5;39,29).
3. Fol. 275', 7 to 23
I do not understand this section which speaks about a "
horoscope "located at n 4;52,2 which, however, is near the setting
point ofthe ecliptic (exactly 20;0,0 ahead of it). Fortunately no
use is madeof this point in the following.
4. Latitudinal Parallax
The Hindu method for determining the latitudinal parallax
restson the fact that this component is approximately constant
alongthe ecliptic for a fixed zenith distance of the nonagesimal
point Vof the ecliptic (2). The determination of this distance is
thereforethe next step.We know that the longitude of the true sun
at noon of May 20
was n 5;9,30 and we have found that the apparent
conjunctionoccurs at an hour angle of 84;22,15 to the West of the
meridian.The normed right ascension of the noon position of the sun
is (3)
'( 5;9,30) = 153;8,59.
If we add to it the hour angle of 84 ;22,1 5 we obtain the
normedright ascension
a'(M) = 237 ;3 1,1 4
of the point M of the ecliptic which is in the meridian at the
momentof the apparent conjunction, i.e. at the middle of the
eclipse. Conse-quently we have for the rising point at the middle
of the eclipse (4)
p(H) = 237 ;3 1 ,14
1 This is an error for 1 ;23,32. AU that follows is based on
this incorrect figure.2 Cf. Neugebauer, Al-Khwrizm p. 122 ff.3
According to the table fol. 238".* Cf. No. 54.
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30 COMMENTARY ON THE ASTRONOMICAL TREATISE
thus H = \ 15;7,58 = T 225;7,58. Therefore the longitude of
thenonagesimal V of the ecliptic
A(V) = 225 ;7,58 - 90 = 135;7,58 = Q 15;7,58.
The declination of this point is (')
y =
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COMMENTARY ON THE ASTRONOMICAL TREATISE 31
The rules given next assume that the latitude of a point of
thelunar orbit of distance from the node and with an inclination
i
of the lunar orbit with respect to the ecliptic is given by
= i sin = Sin .H R
The text considers two possibilities for the norm of the
trigonometricfunctions: either R = 150 or R = 60 is used for Sin .
Also for the
inclination
~ -,
two cases are considered, namely / = 4;30 (*) and i = 5;0.
Con-sequently the following rules are given
= -Sin 150 ro = -Sin60 g) ifi = 4;305 2 (4)
In our case i = 4;30 is assumed and therefore one finds for the
lati-tude at V
= ? Sin150 = ? Sin 150 69;12,38 = 1 140;13,50 = 4;12,25This
amount is now added to the declination
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32 COMMENTARY ON THE ASTRONOMICAL TREATISE
was assumed, exactly as in al-Khwrizm's tables C 1 ). The text
says(fol. 276r, 23) incorrectly only 0;48.In the actual computation
of , following (5), the value z' =
20;54,38 is mentioned, followed by 1;50,40 of unkown meaningand
it is said that
Sin a5O 20;54,48 = 52;16,59.
Actually, however the result agrees much better with the
correctvalue Sin, 50 20;23,59. At any rate, the result is
=- 52;16,59 - 11;19 '40'4- - 0;17 40 40
This is the " adjusted parallax " according to the text (fol.
216',22) which means that a fixed amount for the solar parallax is
includedin the value of pmax .
5. Elements at the Middle of an Eclipse
It has been found that the longitudinal parallax amounts to At=
1 ;35,32. The lunar velocity during May 20/21 is 13;37,130/d
.Therefore the moon moves in consequence of parallax in longitudeby
1 ;35,32 1 3 ;37, 13/24 0;54,13. Since the longitude of the
trueconjunction is n 5 ; 19,7 the apparent conjunction occurs at n
5 ; 19,7+ 0;54,13 = 6;13,20.Similarly for the position of the node
which moves 0 ;3,1 1 o/
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COMMENTARY ON THE ASTRONOMICAL TREATISE 33
thus the argument of latitude = 0;18,13. Assuming as
inclinationof the lunar orbit i = 4;30 we would find according to
(4) p. 31.
= |sin 15O 0;18,13 ^0;47,41,32 0;1,25,51.Subtracting from it the
latitudinal parallax of 0;17 we find for theapparent latitude 0 =
0;15,34. The final result of the text is0 = - 0;15,36.The details
of the computation in the text are as follows. First
is correctly given as 0;18,13. Then (fol. 277r, 5) has Sin = 0;1
5,4,36,50 which must be the result of a copyist's confusion of 5 =
and 9 = 0 because Sin60 = 0;19,4,36,50. According to (4) the
9 10
use of R = 60 would imply the use of either ^ Sin or Sin 9
but not of j Sin which requires R = 150. In the next step 9 Sin
is said to be 422,0,45 which means Sin = 0;46,53,25. The divisionof
9 Sin by 5 supposedly gives 1,25,51,24 which would mean that9 Sin =
7,9,17,0 = 429,17,0 and not 422,0,45. Nevertheless 0; 1,25,51 is
the correct value but in subtracting 0;17 from it thetext makes the
final error of calling the result 0;15,36 insteadof - 0;15,34.
6. Eclipse Magnitude; Linear Digits
Following a method known from Hindu astronomy (J ) one
canexpress the apparent angular diameters of sun and moon as
linearfunctions of their daily velocity
dQ = vo/d dc = (6a)20 247
Considering the lunar latitude constant for the duration of
theeclipse we can say that no eclipse will occur when the apparent
lati-tude 0 computed for the moment of the apparent conjunction
exceedsthe value
= ^(do + c) (6b)1 Khanda-Khdyaka I, 31. The same formulae, e.g.
in Hugo Sanctallensis (Arch.
Seld. B 34 fol. 45% 19 ff. and 45", 7 ff.) or in Ibn Ezra (Mills
Vallicrosa, Tablas astron.p. 166), both based on al-Muthann's
commentary to al-Khwrizm. Cf. also Goldsteinp. 226 f.
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34 COMMENTARY ON THE ASTRONOMICAL TREATISE
If, however, \ 0 \ < cl then the eclipse magnitude, expressed
in lineardigits of the solar diameter, is given by (cf. fig. 8)
m = (3-p0). (7)"
Fig. 8.
In the case of the eclipse under discussion we have vQ =
0;57,17o/d,vc = 13;37,13o/d from which one obtains by (6)
de = 0;31,30 dc = 0;33,5 3 = 0;32, 17,30
hence from (7) with \ 0 \ = 0;15,36 for the magnitude
m". 6 ;21 ,31
measured in digits, 12 of which correspond to the apparent
solardiameter.
The diameter of the moon, measured in the same units, is given
by
dl =
dp,(8)
that is, in the case of our example
12-0-33 5iz ,, = 12 ;36,H.0;31,30
7. Eclipse Magnitudes. Area Digits
Ptolemy says in the Almagest that the majority of those
whoobserve eclipse prognostics " (') are accustomed to measure
eclipse
1 VI, 7 p. 512, 8 Heib. : ot r kXci, . Ithink that the
translation of Manitius "Finsternisphasen" (p. 384,14) is
incorrect;, is the technical term for prognostics; similar Heib. I
p. 536, 21. Cf. e.g.,Ptolemy's Phaseis (opera II p. 10 ff.) and the
index Ptol. opera III, 2 (ed. Boer).
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COMMENTARY ON THE ASTRONOMICAL TREATISE 35
magnitudes by the area of obscuration instead of by the
obscuredfraction of the diameter. For this reason Ptolemy gives a
table ofconversion from linear digits to area digits ('), a table
which is alsocommonly found in mediaeval works. Ptolemy explains by
meansof two numerical examples (2) how area digits can be
computedfrom given rQ , rc , and but his method does not agree in
the detailswith the method followed in our text. A variant of
Ptolemy's methodis described by al-Brun in a treatise on chords (3
). Here we comea little nearer to our text, which agrees with Birun
also in the useof the approximation 22/7 for whereas Ptolemy uses
3;8,30.
A
Fig. 9.
1 VI, 8 p. 522 Heiberg.2 Heiberg p. 513, 6-516, 3 for a solar
eclipse.3 Translated by H. Suter in Bibliotheca Mathem. ser. 3, 11
(1960) p. 46-48. Suter
did not realize that area digits were Brim's goal and therefore
introduced severalincorrect emendations into the text.
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36 COMMENTARY ON THE ASTRONOMICAL TREATISE
Practically identical with all the steps of our procedure,
however,is the commentary of al-Muthann to al-Khwarizmi (*). It is
on thebasis of this close parallelism that it is possible to
understand therules given on fol. 272 of our text. This is very
fortunate since thenumerical example (fol. 277r, 12 to 277v, 19) is
incomplete, coveringonly the first step of the procedure.In
following the rules of our text we make use of fig. 9 (2). The
first step consists in the determination of the parts x and y
into whichthe obscured part m of the diameter of the sun is divided
by thechord EF. It is at the determinat'ion of y that our numerical
examplebreaks off.
The rule of the text for finding x and y, called " axis of the
solarsphere " and " axis of the lunar sphere " respectively, can be
formu-lated as follows: Let all distances assumed to be measured in
digitsof the solar disk (thus dQ = 12). Then we are asked to
computethe quantity (called )
= dc + 12 2m (1)
which yields the desired parts in the form
(12 - m)my - - x = m y. (2)
For the proof of (2) we have only to remark that, for k = 1/2
EF,
k2 = (dc - y)y = (dQ - x)xthus
hence
or
d -y _ = doj^ _
d ~(x + y) , t _ X ^ d0 ~(x + y)
dc + d0 - 2(x + y) = X + dQ -(x + y) y
This relation is the equivalent of (1) and (2) (3) since dQ = 12
andX + y = m.
1 Cf. above p. 6 n. 3, cf. Goldstein p. 239 f.2 There is no
figure given in our text but the Latin version contains drawings
which
are, however, marred by countless discrepancies in lettering.3
It is easy to see that 2.
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COMMENTARY ON THE ASTRONOMICAL TREATISE 37
For our numerical example we have
= 12 ;36,1 1 + 12 - 2-6;21,31 = 11;53,9and
(12 - m)m = (12 - 6 ;21 ,31) 6 ;21 ,31 35;52,17.Hence
y = 35;52,17 ^ 3;1 4 (!)11 ;53,9
and therefore x = 3;20,27 which is, however, no longer
computed.The next step consists in finding the angle 0 (cf. fig. 9)
from
0 = arcSin^ 50 (25^/(12 x)x). (3)With 0, found in degrees, one
computes
Ao =0 (4)and with it
E0 = A0 - (6 - x)J{\2 - x)x (5)which is the area of the segment
EFB.To prove this statement we remark first that
k = yj{d0 x)x = r0 sin 0 = SinR 0
and thus, with rQ = 6 and R = 150
150Sin 15O 0 = k = 25^(12 x)x
which proves (3). If then A0 represents the area of the sector
ESFBwe have (4), using 22/7:
. 20 2 ., 22 0 22 A0 = 0 36 = 0. 360 7 180 35
From this area we substract the triangle
ESF = (r0 x)k = (6 x) ^/(12 x)x
which leaves us with (5) for the area of the segment EFB.The
third step repeats this procedure for the lunar segment ECF.
1 Actually 3;1,5 would be better since the quotient has the
value 3;1,4,46,.
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38 COMMENTARY ON THE ASTRONOMICAL TREATISE
One finds the angle from
= arcSin^- ^/(12 (6)and with it
Finally
Ac = -^d-. (7)2520 W
Ec = Ac - (rc - y) J{\2 - x)x. (8)To prove (6) one has to
realize that in this part of the procedure
(for no good reason) a sine-table is used which is based on R =
60,not on R = 150 as in (3). Then, indeed
Sinr = R = yj(12 x)x.rc rc
For the area of the sector ETFC one finds
. 2vj dc 22 j2 1 1 j2Ac = - X d&] - d&i.
360 4 180-4-7 2520
It would have been more consistent to operate here with rc
insteadof dc and thus come to the more convenient formula
A _ 11 2 rcw630
but also the Latin text uses (7).From Ac we have to subtract the
triangle
ETF = (rc - y)k = (rc - y)J{\-2 - x)xin order to find the area
of the segment (8).Finally we have to compute the lense-shaped area
EBFC
E = EG + Ec (9)
using (5) and (8). Then
M = ^E (10)oo
is the eclipse magnitude in " area-digits " of which the solar
diskcontains 12. Indeed, with this definition
-
COMMENTARY ON THE ASTRONOMICAL TREATISE 39
8. Duration
Fig. 10 shows that under the assumption of unchanging latitudeof
the moon during the progress of an eclipse the distance from
the
first contact to mid-eclipse or from mid-eclipse to last contact
isgiven by
Vo + rcY - therefore the duration by
= + ^ V = ^c _ V^'d'
In our text the computation of At is not given but the first
respec-tively last contact is said to occur at
12;56,22 0;53,33 = 12;3,49" after sunrise
12;56,22 + 0;53,33 = 13 ;49,55ft after sunrise
while t0 = 1 2 ;56,22'1 after sunrise had been found for the
middleof the eclipse. Thus At = 0;53,33ft should be the
half-duration ofthe eclipse. Substituting in the above formula the
values
r0 + rc = 0;32,17,30 0 = - 0;15,36 = 12;ll,270/d
one finds y/(rQ + rc)2 = ^/;1 3,1 9,23 0;28,19 and finallyAt
0;55,44,...'1 i.e. an error of the text of only 0;2A .The
above-given moments for first and last contact are finally
converted to seasonal hours on the basis that it has been found
that
1.*. = 18;17,12 t0 = 1,24;22,15 after noon.
For the half-duration one finds
= 0;43,51,31,.18;17,12
-
40 COMMENTARY ON THE ASTRONOMICAL TREATISE
and for the middle of the eclipse
1 JA-17 I s= ,^,^, 6 11 36 49 38 s.A. after sunrise
18;17, 12
therefore for first and last contact
10 ;36,49,38 0;43,55,31 = 9;52,54s'1 after sunrise
10;36,49,38 + 0;43,55,31 = 1 1 ;20,45s ft - after sunrise
respectively.The eclipse computation ends with the statement
that the solar
altitude at first contact was 18, at last contact 8. No
computationis given; probably the tables mentioned in No. 66 had
been used (').
63, 64. Astrologica
No. 63 is identical with Geoponica I, 8 for which see
Bidez-Cumont,Mages hell. II p. 179. A fragmentary demotic text
which deals withthe rising of Sothis in combination with moon and
planets was pu-blished by George R. Hughes, JNES 10 (1951) p.
256-264. Accordingto Hephaistion XXIII the heliacal rising of
Sirius should take placeon Epiphi 25 which corresponds, however, to
July 19, not to July 20of the Geoponica.
65, 66. Hour Angle from Solar Altitude
Consider the following data be known:
1/2 d half-length of day arc of the sun for the given dayh0 noon
altitude of the sun at the given dayh observed altitude of the
sun.
Then the following rule is given for finding the hour angle H,
i.e.the distance of the sun from the meridian, reckoned in
equatorialdegrees
TT d Vers f/2-Sin/iVers H = Yers (1)
2 Sin/i 0
H is here measured in degrees. Since the length of one
seasonalhour is also known, namely s = d/ 12, we find for the
distance of
1 Using the method of No. 65 I find, however, altitudes of about
28;4 and 8;49"respectively, i.e., a decrease of about 19 ;1 5 in
altitude during 1 ;47\ This seems moreplausible than a decrease of
only 10.
-
COMMENTARY ON THE ASTRONOMICAL TREATISE 41
the sun from noon the seasonal hours H/s; the equinoctial
hoursare H/15 = 0;4 H.
Fig. 11.
The correctness of (1) follows from fig. 11 which represents
themeridian section of the celestial sphere and the day-circle of
the sun rotated into the same plane. Then
AM _ MM' _ sin h0AB ' BB' sin/i
and
* . d sinAAM = vers - hence AB = vers 2 2 sin h0
But
BM = vers H = AM AB
which proves (1).Computation with the above formula was made
unnecessary by
the construction of tables (for given geographical latitude )
withd
double entry, namely for h and h which give the time ^ H,
i.e.the time elapsed since sunrise. Such a table is not preserved
in ourtext but is known from Persian sources, computed for = 36,
i.e.probably for Raqqah, al-Battn?s place of observation (*).An
incorrect, or only approximately correct, solution for the
same problem was given here in No. 42 (cf. above p. 12).
1 1 owe this information to Prof. E. S. Kennedy.
-
42 COMMENTARY ON THE ASTRONOMICAL TREATISE
67. Solar Longitudes from Tables
The procedure corresponds to the solar theory of the Almagest,in
particular in the use of a table for the equation of the type Alm.
111,6.
68. Lunar Longitudes from Tables
The underlying theory is based on the refined lunar model of
theAlmagest. The corresponding table to which our text refers,
whichis, however, not preserved in our MS, is certainly of the
structureof the table in Alm. V, 8 but with a different arrangement
and coun-ting of columns:
here : common numbers Alm. V,8 : columns 1 and 2column 1 column
3
column 2 column 6
column 3 column 4
column 4 column 5
The tables of mean motions also not preserved seem to beless
elaborate than the tables in Alm. IV,4 since we are told to
com-pute the double elongation from the mean motions of sun and
moon.In the Almagest at least the elongation itself is tabulated,
and theHandy Tables give directly the double elongation.The
procedure for finding the true longitude consists in forming
first the double mean elongation
2 = 2(Xc ^)
and finding in the tables with 2 as argument the values of Cj
and c2 .From the mean anomaly one finds the true anomaly
= +
and with as argument the equations c3 and c4 , tabulated for
themaximum respectively minimum distance of the epicycle (i.e.
atsyzygy or quadrature respectively). Then the true longitude is
givenby
= Xc + C3 + C2C4
69. Planetary Longitudes from Tables
The wording of the text is not easy to understand but
neverthelessit is clear that the procedure follows essentially the
pattern of Alm. XI,11,12 except for a different arrangement of the
columns:
-
COMMENTARY ON THE ASTRONOMICAL TREATISE 43
here: column 1 Alm. ,: columns 3 + 4column 2 column 8
column 3 column 5
column 4 column 6
column 5 column 7.
Our text proceeds as follows. We consider to be given the
longitude of the apogee of the planet in question (cf. fig. 12) as
well as the
Fig. 12.
mean longitude 1 for the given moment. For the outer planets
amean anomaly can be computed as = 1 1q, for an innerplanet is
independent of the sun. In the Almagest is tabulatedfor both cases
(IX,4).One now forms = 1 and enters with it the tables which
give in column 1 the correction c^c) by means of which one
obtains
K = K + C(k) < 180if |
= Cj (k) [ > 180.
-
44 COMMENTARY ON THE ASTRONOMICAL TREATISE
With K as argument one finds the coefficient of interpolation
2()which has the value 1 at the apogee, + 1 at the perigee and 0at
mean distance of the center C of the epicycle from the observer
O.With as argument one finds the epicyclic equation c4(a),
whichwould be valid for C at the mean distance from O, and the
incrementsc3(a) and c5(a) which correspond to a position of C at A
P respec-tively. Then
c3(a)c'4(a) = c4(a) + c2(k)
cs(a)
is the equated epicyclic equation in general position.
Obviously
c'a(ol) 0 if ^ 180.
Finally the true longitude of the planet is given by
= + + c'A(a).
70 to 86. Astrologica
No. 70 discusses the astrological significance of the aspects
(con-junction, sextile, quartile, trine, and opposition) between
the moonand the planet (including the sun).No. 71 gives the Spia
according to the " Egyptian " system and
according to Ptolemy. For each sign is given (a) the length of
eachsection, (b) the ruling planets, (c) the summation of the
intervalslisted in (a).No. 72 is a list of the houses, exaltations
and depressions, decans
and different types of triangle rulers.Fol. 285v counts a list
of 14 lines enumerating types of zodiacal
signs (from " male " to " human-shaped ") as 14 sections, from73
to 86.
Appendix. The Tables of foll. 238v to 256w
In the following I give a short summary of the tables which
precedethe text discussed here.
No. 4 (fol. 238v, 239r): normed right ascensions.No. 5 (fol.
239v-240v): table of sines (R = 60), lunar latitude (maxi-
mum 4;46,0), solar declinations ( = 23 ;35,0) ; cf., however,
No. 8.Nos. 6 to 8 ( x ) (fol. 241 r-249v): tables for the planetary
latitudes.
1 With some errors in the numbering of the tables.
-
COMMENTARY ON THE ASTRONOMICAL TREATISE 45
In No. 8 (fol. 247r-249v) is added another table for the solar
decli-nations, but with = 23 ;51 (cf. No. 5).Nos. 9-13 (fol.
250r-252r): planetary stations.Nos. 14-27 (fol. 252v-256v): tables
for the visibility of the planets
for the climata 2 to 6. These tables contain many errors and
mal-arrangements; in particular the Nos. 24-27 (fol. 256r, 256v)
belongbetween No. 17 (fol. 253r) and No. 18 (fol. 253v). Nos. 22
and 23(fol. 255r, 255v) give the planetary phases as in the
Almagest XIII, 10.
Bibliographical Abbreviations
Almagest: Ptolemaeus, Syntaxis mathematica, ed. Heiberg, Leipzig
1898.Bidez-Cumont; Mages hell.: Les Mages hellniss, 2 vols., Paris
1938.CCAG: Catalogus codicum astrologorum graecorum.Goldstein,
Bernard R.: Ibn al-Muthann's Commentary on the Astronomical
Tables of al-Khwrizml. Yale University Press, 1967.Halma:
Commentaire de Thon d'Alexandrie... Tables Manuelles... 3 vols.,
Paris
1822, 1823, 1825.Handy Tables : cf. Halma and Ptolemaeus, Opera
astronomica minora, ed. Heiberg,Leipzig 1907 p. 157 ff.
JNES: Journal of Near Eastern Studies.Kennedy , Survey: A Survey
of Islamic Astronomical Tables. Trans. Am. Philos.Soc., N.S. 46,2
(1956) p. 121-177.
Kennedy, Parallax : Parallax Theory in Islamic Astronomy. Osiris
47 (1956) p. 33-53.Mills Vallicrosa , Bibl. Toledo: Las
traducciones orientales en los manuscritosde la Biblioteca Catedral
de Toledo. Madrid 1942.
Mills Vallicrosa , Tablas astron. : El libro de los fundamentos
de las Tablasastronomicas de R. Abraham Ibn "Ezra. Madrid-Barcelona
1947.
Mills Vendrell , Eduardo: El comentario de Ibn al-Mutann a las
TablasAstro-nmicas de al-Jwarizmi, Madrid-Barcelona, 1963.
Neugebauer , Al-Khwrizm: The Astronomical Tables ofAl-Khwarizmi.
DanskeVidensk. Selsk., Hist.-filos. Skrifter 4,2 (1962).
Neugebauer, Byz. Astr. : Studies in Byzantine Astronomical
Terminology. Trans.Am. Philos. Soc., N .S. 50,2 (1960) p. 1-45.
-
XLII
1547. Doutrepont, G. La littrature et la Socit ; 1942 ; LII-688
p 280 Tome XLIII
1. 1553. Wodon, L. Considrations sur la Sparation et la Dlgation
des Pouvoirsen Droit Public Belge ; 1942 ; 71 p 40
2. 1566. Willaert, L. Les origines du Jansnisme dans les
Pays-Bas catholiques ;1948 ; 439 p 150
Tome XLIV
1. 1571. Lonard, J. Le bonheur chez Aristote ; 1948 ; IV-224p
802. 1584. Kerremans, Ch. tude sur les circonscriptions judiciaires
et administratives
du Brabant et les officiers placs leur tte par les Ducs,
antrieurement l'avnement de la Maison de Bourgogne (1406) ; 1949 ;
2 cartes, 436 p 150
Tome XLV
1. 1596. Grgoire, H., Goossens, R. et Mathieu, M., Asklpios,
Apollon Smintheuset Rudra ; 1949 ; 11 fig. et 2 cartes ; 204 p
80
2. 1598. Stengers, J. Les Juifs dans les Pays-Bas au Moyen Age ;
1950; 1 carte,190 p 75
3. 1595. Dechesne,L. L'avenir de notre civilisation ; 1949 ; 124
p 504. 1601. Piron, Maurice. Tchantchs et son volution dans la
tradition ligeoise ;
1950 ; 9 pl., 120 p 60Tome XLVI
1. 1600. Grgolre, H., Orgels, P., Moreau, J. et Maricq, A. Les
perscutionsdans l'Empire romain ; 1951 ; 176 p puls.
2. 1607. Honlgmann, Ernest. The lost end of Menander's
Epitrepontes ; 1950 ; 43 p. 253. 1608. Haesaert, J. Pralablesdu
Droit International public ; 1950 ; 93 p 504. 1620. Hoebanx, J. J.
L'Abbaye de Nivelles des Origines au XIVe sicle ; 1952 ;
11 cartes ; 511 p 200 Tome XLVII
1. 1621. Dereine, Ch. Les Chanoines rguliers au diocse de Lige
avant saintNorbert ; 1952 ; 1 pl. ; IV-282 p 120
2. 1633. Cornil, Suzanne. Ins de Castro. Contribution l'tude du
dveloppementlittraire du thme dans les littratures romanes ; 1952 ;
153 p 75
3. 1634. Honigmann, E. Pierre l'Ibrien et les crits du
pseudo-Denys l'Aropa-gite ; 1952 ; 60 p 40
4. 1640. Honigmann, E. et Maricq, A. Recherches sur les Res
Gestae divi Saporis ;1953 ; 4 planches hors-texte ; 1 carte ; 204 p
100
Tome XLVIII
1. 1645. Govaert, Marcel. La langue et le style de Marnix de
Sainte-Aldegonde danssone Tableau des Differensdela Religion ; 1953
; 312 p 150
2. 1647. Hyart, Charles. Les origines du style indirect latin et
son emploi jusqu'l'poque de Csar ; 1954 ; 223 p 100
3. 1648. Martens, Mina. L'administration du domaine ducal en
Brabant au MoyenAge (1250-1406) ; 1954 ; 4 pl. ; 2 cartes ; 608 p
400
Tome XLIX
1. 1650. Van Ooteghem, J. Pompe le Grand, btisseur d'Empire ;
1954 ; 56 fig.,665 p 400
Tome L
1. 1654. Spilman, Reine. Sens et Porte de l'volution de la
Responsabilit civiledepuis 1804 ; 1955 ; 132 p 80
2. 1658. Bartier, John. Lgistes et gens de finances au XIVe
sicle ; 1955 ; 4 pl. ;452 p 300
2b. 1658bis. Idem : index-additions et corrections ; 1957 ; 76 p
40 Tome LI
1. 1662. Finet, Andr. L'Accadien des Lettres de Mari ; 1956 ;
XIV-358 p 200 2. 1669. Mogenet, Joseph. L'introduction l'Almageste
; 1956 ; 52 p 403. 1670. Joly, Robert. Le Thme Philosophique des
Genres de vie dans l'Antiquit
Classique ; 1956 ; 202 p 120 4. 1674. Mortier, Roland. Les
Archives Littraires de l'Europe (1804-1808) et
le Cosmopolitisme Littraire sous le Premier Empire ; 1957 ; 252
p 140 Tome LII
1. 1675. Delatte.Louis. Un office byzantin d'exorcisme ; 1957 ;
VIII-166 p 1002. 1676. Lejeune, Albert. Recherches sur la
Catoptrique grecque d'aprs les sources
antiques et mdivales ; 1957 ; 53 fig. ; 200 p 150 3. 1683.
Wanty, mile. LeMilieu Militair belge de 1831 1914 ; 1957 ; 280 p
1404. 1686. Bonenfant, Paul. Du meurtre de Monterau au trait de
Troyes ; 1958 ;
XVI-282 p 300
-
LIXI
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et le Contrle de la Population
Ox\Thynchus au III' sicle de notre re ; 1958 ; 1 h.-t. ; XX-170
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(1596) ; 1959 ;
132 p 80 4. 1699. Van Ooteghem, J. Lucius Licinius Lucullus ;
1959 ; 27 fig., 233 p 1605. 1704. Henry H. Frost, Jr. The
functional sociology of Emile Waxweiler; 1960;
244 p 150 >Tome LIV
1. 1707. Lemerle, Paul. Prolgomnes une dition critique et
commente des Conseils et Rcits de Kkaumnos ; 1960 ; 120 p 80
2. 1714. Moraux.Paul. Une dfixion judiciaire au Muse d'Istanbul
; 1960 ; 62p. .. 503. 1717. Dabln, Jean. Droit subjectif et
Prrogatives juridiques. Examen des thses
de M. Paul Roubier ; 1960 ; 68 p 50 4. 1720. Delatte, Armand.
Herbarius. Recherches sur le crmonial usit chez les
Anciens pour la cueillette des simples et des plantes magiques ;
1961 ;16 fig., 223 p 240
5. 1721. Peeters, Paul. L'ceuvre des Bollandistes ; 1961 ; 209
pages ; 2 h.-texte 1406. 1723. Honigmann, Ernest. Trois mmoires
posthumes d'histoire et de gographie
de l'Orient chrtien ; 1961 ; 2 pl., 216 p 200 Tome LV
1. 1725. Kupper, Jean-Robert. L'Iconographie du dieu Amurru dans
la glyptiquede la l re dynastie babylonienne ; 1961 ; 96 p. ; 9 pl
80
2. 1728. Duprel, E. LaConsistance et la Probabilit Constructive
; 1961 ; 39 p 303. 1730. Van Ooteghem, J. Lucius Marcius Philippus
et sa famille ; 1961 ; 10 fig.,
200 p 1604. 1737. Goossens, Roger. Euripide et Athnes ; 1962 ;
772 p 450
Tome LVI
1. 1738. Slmon, A. Position pliilosophiquedu Cardinal Mercier ;
1962 ; 120 p 802. 1740. Severyns, A. Texte et Apparat. Histoire
critique d 'une tradition imprime ;
1962 ; I-XII ; 374 p. ; 5 dpliants 3203. 1749. Simon, A.
Rencontres Mennaisiennes en Belgique ; 1963 ; 266 p 200 4. 1750.
tienne, Hlin. La dmographie de Lige aux XVIIe et XVIIIe sicles
;
1963 ; 282 p 260 5. 1753. Grgoire, Henri. Les perscutions dans
l'Empire romain (2e d.) ; 1964 ;
267 p. 200 6. 1755. Van Ooteghem, J. Caius Marius ; 1964 ; 338
p. ; 20 fig. 260
Tome LVII
1. 1757. Lenger, Marle-Thrse. Corpus des Ordonnances des Ptolmes
; 1964 ;368 p. 2 pl 260
2. 1760. Lallemand, Jacqueline. L'administration civile de
l'gypte de l'avne-ment de Diocltien la cration du diocse (284-382)
; 1964 ; 342 p. ; 3 fig. 260
3. 1761. Jeanjot, Paul. Les Concours annuels de la Classe des
Lettres et des Sciencesmorales et politiques de l'Acadmie royale de
Belgique. Programmeset rsultats des Concours (1817-1967) ; 1964 ;
234 p 150
4. 1765. Dumzil, Georges. Notes sur le parler d'un Armnien
musulman deHemsin ; 1964 ; 52 p 50
5. 1767. Maline, Marie. Nicolas Gumilev, pote et critique
acmiste ; 1964 ; 380 p. 300 6. 1770. Salmon, Pierre. La politique
gyptienne d'Athnes : 1965; xxxn-276 p. 240
Tome LVIII
la. 1784. Derchain, Philippe. Le papyrus Salt 825, rituel pour
la conservation de lavie en gypte ; 1965 ; Vol. 1 ; 216 p. ; 10 fig
450
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Monnaies et colonisation dans l'Occident grec ; 1965 ;
178 p. ; 12 pl 220 3. 1789. Jacques, Xavier et Van Ooteghem, J.
Index de Pline le Jeune ; 1965 ;
XX-975 p 720 4. 1799. Leleux , Fernand. Charles Van Hulthem
1764-1832 ; 1965 ; 574 p. ; 1 pl 500
Tome LIX
1. 1807. J. Van Ooteghem, S. J. Les Caecilii Metelli de la
Rpublique ; 1967 ; 349p. ; 14 pl 320
2. 1812. O. Bouquiaux-Simon, Les lectures homriques de Lucien ;
1968; 414 p. 4003. 1813. Gaier, Claude. Art et organisation
militaires dans la principaut de Lige
et dans le comt de Looz au Moyen Age ; 1968 ; 393 p. ; 16 fig.
360 4. 1819. O. Neugebauer, Commentary on the Astronomical Treatise
Par. gr. 2425 ;
1969 ; 45 p. ; 12 fig 100
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