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4. An updated list of culminations ought to begin with Atlas and I have here used my data for Alcyone:
 
  365 days
  (80 + RA / 24h * 365¼)   Day of culmination  
Atlas 'May 16 (136) 228 'December 31 (365) 135
26 -3   +3
Arneb 'June 12 (163) 225 'January 24 (389) 138
Alnilam 'June 13 (164) 225 'January 25 (390) 138
Phakt 'June 14 (165) 225 'January 26 (391) 138
2 0   0
Betelgeuze 'June 17 (168) 225 'January 29 (394) 138
12 0   0
Sirius 'June 30 (181) 225 'February 11 (407) 138
11 0   0
Castor 'July 12 (193) 225 'February 23 (419) 138
1 -1   +1

There are 193 - 135 = 58 RA days from Atlas up to and including the day of Castor. 'July 13 is day 59 from Atlas and evidently in G connected with an ihe tau glyph:

Ga2-18 Ga2-19 Ga2-20 Ga2-21 Ga2-22 Ga2-23 (54)
Gomeisa (111.6), ρ Gemini (112.1) Castor (113.4)   Markab (114.7), Procyon (114.9) σ Gemini (115.7), Pollux (116.2) Azmidiske (117.4)
'July 11 (192) '12 '13 '14 '15 '16
Heka 3 4 5 6 (59) 7 8

Although day 59 in the manzil calendar comes in 'July 14 with Procyon, because its first day is 'May 17. The culmination night of Procyon was the day after 'February 23:

... The leap day was introduced as part of the Julian reform. The day following the Terminalia (February 23) was doubled, forming the 'bis sextum - literally 'double sixth', since February 24 was 'the sixth day before the Kalends of March' using Roman inclusive counting (March 1 was the 'first day') ...

Procyon 'July 14 (195) 224 'February 24 (420) 139
19 +2   -2
Acubens 'August 3 (215) 226 'March 18 (442) 137
6 +1   -1
Alphard 'August 10 (222) 227 'March 26 (450) 136
10 0   0
Regulus 'August 21 (233) 227 'April 6 (461) 136
12 +1   -1
Alkes 'September 3 (246) 228 'April 20 (475) 135
11 +1   -1
Denebola 'September 15 (258) 229 'May 3 (488) 134
6 0   0
Gienah 'September 22 (265) 229 'May 10 (495) 134
2 0   0
Acrux 'September 25 (268) 229 'May 13 (498) 134
3 0   0
Porrima 'September 29 (272) 229 'May 17 (502) 134
2 +1   -1
Cor Caroli 'October 2 (275) 229 'May 20 (505) 134
1 0   0
Vindemiatrix 'October 4 (277) 229 'May 22 (507) 134
15 0   0
Thuban 'October 20 (293) 229 'June 7 (523) 134
1 -1   +1
Arcturus 'October 22 (295) 228 'June 8 (524) 135
6 -1   +1
Yang Mun 'October 29 (302) 227 'June 14 (530) 136
1 +1   -1
Zuben Elgenubi 'October 31 (304) 228 'June 17 (533) 135
11 -1   +1
Gemma 'November 12 (316) 227 'June 28 (544) 136
12 0   0
Antares 'November 25 (329) 227 'July 11 (557) 136
11 0   0
Ras Algethi 'December 7 (341) 227 'July 23 (569) 136
4 0   0
Ras Alhague 'December 12 (346) 227 'July 28 (574) 136
15 -1   +1
Vega 'December 28 (362) 226 'August 12 (589) 137

Vega is close to the end of the Gregorian year and 362 - 194 = 168 days counted from 'July 14.

18 days after Vega comes Altair and in G I have joined it to the first glyph beyond 236 (= 8 * 29½):

Gb1-4 Gb1-5 Gb1-6 (236) Gb1-7 (*300) Gb1-8
  Sham (297.8) Tarazed (299.3) Altair (300.3)  
'January 12 (377) '13 '14 '15 '16
10 11 12 Shaula 13 (244) Al Naam 1

The manzil calendar has its day 245 (Al Naam 1) - as I have arranged it - at Gb1-8, one day later (exactly as at Procyon). 245 - 59 = 186.

17 +2   -2
Altair 'January 15 (380) 228 'September 1 (609) 135
6 0   0
Gredi 'January 22 (387) 229 'September 9 (617) 134
6 +1   -1
Deneb Cygni 'January 29 (394) 229 'September 16 (624) 134
8 +2   -2
Alderamin 'February 7 (403) 231 'September 27 (635) 132
11 0   0
Sadalmelik 'February 19 (415) 231 'October 9 (647) 132
12 +3   -3
Fomalhaut 'March 4 (428) 234 'October 25 (663) 129
17 -1   +1
Sirrah 'March 22 (446) 233 'November 11 (680) 130
7 -1   +1
Schedir 'March 30 (454) 232 'November 18 (687) 131
13 +2   -2
Achernar 'April 13 (468) 234 'December 4 (703) 129
5 -3   +3
Alrisha 'April 19 (474) 231 'December 7 (706) 132
1 +2   -2
Hamal 'April 21 (476) 233 'December 11 (710) 130
13 -4   +4
Menkar 'May 5 (490) 229 'December 21 (720) 134
10 -1   +1

It might be argued that the heliacal data above are false because they describe what would be the case if the orbit of Earth was circular, I have used a spatial map over the sky dome as if it was a time map.

Indeed it is true that the distances between heliacal risings and midnight culminations ought to be different than those in the table above, because only my midnight culminations are distributed along the elliptical time path of Earth.

On the other hand, direct observations of heliacal risings of stars are difficult, and possibly, therefore, the numbers in the table above could have been used instead.

Or indirect observations by looking at the face of the full Moon reflecting light from Sun, from the diametrically opposite side of the sky dome - the nakshatra method (cfr at Camp 5) - could do the task:

... In Hindu legend there was a mother goddess called Aditi, who had seven offspring. She is called 'Mother of the Gods'. Aditi, whose name means 'free, unbounded, infinity' was assigned in the ancient lists of constellations as the regent of the asterism Punarvasu. Punarvasu is dual in form and means 'The Doublegood Pair'. The singular form of this noun is used to refer to the star Pollux. It is not difficult to surmise that the other member of the Doublegood Pair was Castor.

Then the constellation Punarvasu is quite equivalent to our Gemini, the Twins. In far antiquity (5800 B.C.) the spring equinoctial point was predicted by the heliacal rising of the Twins (see fig. 6.6). By 4700 B.C. the equinox lay squarely in Gemini (fig. 6.7).