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According to the picture of Hevelius the Eridanus River seems to be born not from the Phoenix but from the Toucan and I have no stars listed from that constellation. This should now be remedied.

 
The point of the reversed Toucan's beak is α, not far beyond 22h, and Tucana lies right across 0h. She has 13 stars with Greek letters:
 
α 22h 18m 30.18s 22h 18.503m 338.9
δ 22h 27m 19.87s 22h 27.331m 341.1
ν 22h 33m 00.01s 22h 33.000m 342.5
Fomalhaut 22h 57m 38.83s 22h 57.647m 348.8
γ 23h 17m 25.81s 23h 17.430m 353.8
η 23h 57m 34.97s 23h 57.583m 364.0
ε 23h 59m 54.91s 23h 59.915m 364.6
ζ 00h 20m 01.91s 00h 20.032m 369.7
π 00h 20m 39.04s 00h 20.651m 369.9
β 00h 31m 32.56s 00h 31.543m 372.6
ρ 00h 42m 28.30s 00h 42.472m 375.4
λ 00h 55m 00.30s 00h 55.005m 378.6
ι 01h 07m 18.57s 01h 07.310m 381.7
κ 01h 15m 45.50s 01h 15.758m 383.8

The last of these 13 is κ, not ω. Somewhere I have read a suggestion that the word 'element' (for instance in Euclid's Elementa) may have originated from an ancient alphabet (A, B) which began with the letters L, M, and N.

I think this idea  is wrong. Instead the origin of the set of letters ought to have been closely connected with the phases of the Moon. Thus there should have been a division of the ancient 'alphabets' into pairs of halves ('waxing' and 'waning').

The first such half could begin with α and end with κ (or their equivalents). The strange and hard to grasp closeness between the beginning (α) and the end (ω) of a cycle will then disappear into thin air. There are 2 cycles in the completed whole and when the last element in the first cycle has ended the first element in the second cycle will begin (instead of mysteriously leaping back to the beginning of the first cycle). When Sun has just set it will not immediately be dawn again.

The end of the Toucan is the star κ adjacent to α Eridani. The 2nd cycle goes on in the River.

Counting from A to K we will reach 11. Then follows L and a jump to the beginning of the next line where M becomes 12. But as we are counting ordinal time number 11 at L means we are at the beginning of the 12th hour.

There are 22 letters in all which means (1 + 11) + (1 + 11) = 24 'hours'. 22 letters indicate the completion of 2 π-cycles.

A reflection around such ideas may have put Sadalmelik (The Luck of the King) at 22h:

Saad Al Thabib 8 9 10 (280)
February 18 19 (50) 20
Ca12-18 Ca12-19 Ca12-20 (336)
oho te vae koia ra kua haga hia kua moe
Kae Uh (334.0), Al Kurhah (334.4) 22h (334.8) ζ Cephei (336.2), λ Cephei (336.3)
Sadalmelik (334.6), ι Aquarii (334.7), Alnair (335.1)
August 20 21 22 (234)
An Nathra 4 5 (97) 6
10h (152.2) Regulus (152.7), λ Hydrae (153.2) no star listed
υ² Hydrae (151.8), Al Jabhah (152.4)

February 19 is Gregorian day 50 and 50 * 3.14 = 157 (half π).