More about Regiomontanus directions

Let us take again the Primum mobile of Placidus. The 1st theme is that of Charles V (Quint) which we have already analyzed. Here we will consider the directions in a more traditional way using the direct teachings of Placidus and Magini. On Magini, we have in the History of Astronomy of the Middle Ages, an entire chapter devoted to the calculation proposed by Delambre, the complete details of which will be seen below on the example of the SA opp MO arc. I give again the theme of Charles V, according to the data of Placidus (Primum Mobile, pp. 59-61: Exemplum Primum Caroli V Austriaci Imperatoris).

For comments on the theme itself, see the following 3 posts:

https://primarydirections.blogspot.com/2017/04/charles-quint.html
https://primarydirections.blogspot.com/2017/09/carolus-v.html
https://primarydirections.blogspot.com/2018/02/carolus-v-3.html

IOANNIS ANTONII MAGINI  PRIMI MOBILIS LIBER NONUS QUI AGIT DE DIRECTIONIBUS pp, 214-232 PROBLEMA XVIII p, 230   

We take the direction : ꝏSA C MO.

1) - MAGINI    - REGIOMONTANUS

direct   
modified from [ Delambre, Hist Astron Moyen Âge, Magini, pp, 486-491 Paris 1819]   
    
    Eq for Fig 129 (pl 12)  see fig for location and explanation of spherical triangles

Figure 129 is mixed. It represents a state of the sky at the moment when star A' arrives in a situation similar to star A. The reference element for the observer is the horizon OBH. The arc OAA'H is the position circle (or incident horizon) whose pole is PR. PA is the hour angle counted from midnight. When A' reaches the arc OAA'H, A is no longer on the circle; we must therefore imagine its image on the incident circle. Furthermore, EBQ is the equator. T is the pole of RS (or of A'V') and ET the symmetry axis.
The time when opp SA is on the circle of position of MO is 7h 31 33 TU and the initial time is 3h 52 (corresponding to 58.57 Y for the date of death, the 12 September 1558 (OS), so : 1558.72.

    
    h (altitude of the promissor) - H (latitude of the observer) - D (declination of the promissor) –
    D' (declination of incident horizon = position circle of promissor) - PH=H - 

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PARAMETERS	24 Fev 1500 | JUL |	house = pcd
DIRECTION	PROMISSOR	SIGNIFICATOR
ꝏ SA C MO	ꝏ SA	C MO
long	226,15 N E	279,12 S E
lat	2,023	-2,097
☊ (***)	na	72,218
AR	224,27	280,09
dec	-14,78	-25,28
h	23,87	-4,26
OA	243,32	315,84
Dom Campa	286,71	5,27
Dom Regio	280,69	8,35
Dom Placidus	279,15	6,29
Pôle Regio	12,92	50,75
Pôle Campa	12,92	50,75
Pôle Placidus	7,30	49,26
DM D|N	D |7,216- N |172,784	D |63,03- N |116,97
DA pôle D|N (Placidus)	D |1,937- N |30,18	D |41,544- N |33,257
DA	19,05	35,75
SA D|N	D |70,95- N |109,05	D |54,25- N |125,75
D Horiz (SA-DM) (min E W)	63,74 (E)	8,78 (E)
temporal hour (*) ꝏ SA (HTn) | C MO (HTd)	11,83	9,04
horary distance (**)	0,61	5,58
horary angle (TS – AR) from midnight (W from S)	352,78	296,97
Azimuth (0-360 W from S)	352,37	306,09
ARMC | TSN	217° 3' 19"	14h 28' 13"
latgeo,	51,05	Equ time
ecliptic	23,51	+ 356° 55' 15"
Length SU	1,00	Luna motion
Length MO	14,61	2,34 (for y time + key EQU)

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One can see hereunder : label of angles, left : value for each equation of angle and right formulas.    

PA (horary angle, from midnight)   
172,78    ||  cosP = (sinh-sin Hsin D)/(cosHcosD)
    
PHA (HAO angle)   
-16,71    ||  TanPHA = sinP/((sinD+cosP)cosH)
    
PR (pole HAO)   
-12,919    ||  sinPR=sinPHsinPHA
    
ET (arc meridian/HAO)   
169,31    ||  tanET=cosHtanPHA
    
H'    ||  H' (incident horizon = pole [ꝏ SA]
12,92    ||  tanH'=sinETtanH
    
PAR (position angle)   
13,37    ||  sinPAR=sinH'/cosD
    
QS, HPR (Az PR)   
79,31    ||  cosQS=sinET

ΔAR' (DA/HAO)   
-3,47    ||  sinΔAR'=tanDtangH'
    
OTE, HTQ, ETH (equat/HAO angle)   
77,08    ||  tanOTE=cotH/sinET
    

   Eq for Fig 130 (pl 12) see fig for location and explanation of spherical triangles

Figure 130 resembles a sagittal shot of the sky. The element of symmetry is the ETQ equator. Point S is the significator (point A in figure 129) and point R is the promissor (point A' in figure 129). Point Z is the image of point R at the instant when R reaches a position similar to S. AQC is the observer's horizon; ASZC is the position circle of S (incident horizon). PR, PZ and PS are arcs of position of the points R, Z and S. ZV is the boreal [+] declination of Z and OS the austral [-] declination of S. TO is the ascensionnal difference (AR) of S while TV is the ascensionnal  difference (AR') of Z. They allow the calculation of the distance OV which must be added or subtracted from the arc KO (difference of the right ascensions of S and Z) to obtain the arc of direction whose the common equation is:

KV = KO - OT - TV

If the OS declination is boreal, OT would change sign; if RK is austral, TV would change sign (adapted from Delambre, History of Astronomy of the Middle Ages, Magini article, p. 490, 1819).

KO (≠AR ꝏ SA C MO)
53,23    ||  KO=ARꝏ SA - ARC MO
    
A (AET angle)    ||  EA = 90-H
16,71    ||  sinA=sinETsinOTE/sinEA
    
T (angle horizon/OTH)    ||  pole of PRS (=90-H')
77,08    ||  cosT=sinAETsinH
    
[-] TO ΔAR' ꝏ SA    ||  D boreal --> sign [-] = switch for [+]
-3,47    ||  sinTO=tanDcotT
    
[+] TV ΔAR' C MO    ||  D' boreal-> sign [+] = switch for [-]
-6,26    ||  sinTV=tanVZcotT
-----------------------------------------------------------------   
tasyir    ||  If OS (D) [+], TO ~ sign,  If RK (D') [-], TV ~ sign
56,02    ||  KV=KO – OT – TV
-----------------------------------------------------------------   
VZ    ||  D' (declin of C MO)
-25,43   
    
OS    ||  D (declin of ꝏ SA)
-14,78   
    
ST    distance ST
-15,17    ||  sinST=sinD/sinT
    
TZ    distance TZ
-26,13    ||  sinTZ=sinD'/sinT
    
ZS    ≠dist
10,96    ||  If D/D'<0 [-]
    
EPA (EPV, EV) ± ΔAR'    horary angle of ꝏ SA
172,78    ||  if d is [-] ~ ΔAR'
    
So, the direction is tasyir = 56.02 (Regiomontanus).

we see superimposed the arc of direction with opp SA emanating from SA (at 46.27° ♉), arriving in conjunction with MO at 280° ♑.