neugebauer, commentary on the astronomical treatise par. gr. 2425

Acadmie royale de Belgique Koninklijke Academie van Belgi

CLASSE DES LETTRES

ET DES SCIENCES MORALES

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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

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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.

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.

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.


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.


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 ().


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.


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.


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.


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.


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.


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\


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.


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*.


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.

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.


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'.


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.


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.


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)


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).

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.


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.


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/


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 + ...


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.


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 =


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


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/


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.


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).


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.


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.


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,.


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


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


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.


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.


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:


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.


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.


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

1. 1687. Delatte, Armand. Les Portulans grecs II. Complments ; 1958 ; 85 p 1202. 1688. Mertens, Paul. Les Services de l'tat Civil et le Contrle de la Population

Ox\Thynchus au III' sicle de notre re ; 1958 ; 1 h.-t. ; XX-170 p 120 3. 1690. Mortler, Roland. Le Hochepot ou Salmigondi des Folz (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

lb. 1784bis. Idem. Vol. 2 ; 72 p. ; 18 pl2. 1788. Lacroix, Lon. 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

J. DUCULOT, imprimeur de l'Acadmie royale de Belgique, Gembloux.Printed in Belglum.

neugebauer, commentary on the astronomical treatise par. gr. 2425

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