---------------------------------

RUDOLF II Emperor

18 Jul 1552 JUL    CAL
monday JUL
 | lat 48° 13' 0" | N 16°22' E
Vienna, Austria
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natal 19h 0' 0"
lmt 17h 54' 31"
tu 17h 54' 31"
tsn 15h 24' 52"
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timezone
Equation of time 0h 5' 35"
ΔT 0h 2' 28"
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source = https://www.astro.com/astro-databank/Rudolf_II,_Holy_Roman_Emperor

Pierre Brind’Amour, Nostradamus astrophile (Ottawa, n.d.), pp.377–80, 452 [Nostradamus astrophile. Les astres et l’astrologie dans la vie et l’œuvre de Nostradamus, Paris, Klincksieck, 1993 in chapter 11, note 1, the Magic Circle of Rudolf II, Peter Marshall, Bloomsbury, 2006]. The German manuscript of Rudolf’s horoscope, completed in 1564, is held at the Kungliga Biblioteke, Stockholm, v., J.1695 ; In the central cartouche of the horoscope drawing [fol. 2r] the time is given as “H. VI. M. XLV” (6 hours, 45 minutes) [http://nostredame.chez-alice.fr/nerg1.html#ref3]
Note : Tycho Brahe indicates the day of birth: July 18 (but not the time of birth) in Opera omnia, tome 3, edidit I.L.E. Dreyer, p. 382, 1913

Rudolf II (18 July 1552 – 20 January 1612) was Holy Roman Emperor (1576–1612), King of Hungary and Croatia (as Rudolf I, 1572–1608), King of Bohemia (1575–1608/1611) and Archduke of Austria (1576–1608). He was a member of the House of Habsburg.
Rudolf died in 1612, nine months after he had been stripped of all effective power by his younger brother, except the empty title of Holy Roman Emperor, to which Matthias was elected five months later.

Natal chart

THEME

SU is Ru with a [14] score
MO is Ru with a [7] score
VE is P  with a [-11] score
JU is Fa cb with a [1] score

MA is Fa with a [-1] score

SA is Fa with a [-6] score


we see below the list of ZODIACAL aspects :
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                                     SA 120 MO Oc                                JU 60 MA Or        
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The best aspect is  [best :mo 120° (5,9) sa]  and the worst aspect is  [worst :ma 60° (0,38) ju]


The traditionnel almuten (Omar, Ibn Ezra) is MA
we see below the list of dignities for MA :
---------------------------------------
[ term 1 tri 0 rul 0 exn 2 fac 1 ]
[ su 0 mo 2 asc 1 syg 1 pof 0 ]
---------------------------------------
Note 1 : the ‘almuten figuris’ is the lord of the chart, but its determination obeys somewhat different rules according to the schools. The tradition is based above all on the zodiacal dignities. (see p,e,  Alcabitius, Introduction, 59-61, 117 and Avenezra, Nativites, 101) – almuten = al-mu’tazz (arabic term)
[7] As for the governor which is the <planet> predominating (al-mubtazz) over the birth from which one indicates the conditions of the native after the haylāğ and the kadhudāh,n it is the planet having the most leadership in the ascendant, the position<s> of the two luminaries, the position of the Lot of Fortune and the position of the degree of the conjunction or opposition which precedes the birth. When a planet has mastery over two, three or four positions by the abundance of its shares in them, it is the governor and the predominant <planet> (al-mubtazz) and the indicator after the haylāğ and the kadhudāh. From it one indicates the conditions of the native. Some people use it instead of the kadhudāh in giving life.  [Al-Qabisi , Charles Burnett, Keji Yamamoto, Michio Yano, The Introduction to Astrology, IV, 7, p, 117, Warburg, 2004]
Note 2 : There are at least 4 systems for determining the almuten depending on whether the combinations of triplicities and terms are used: the Ptolemaic almuten (followed by Lilly) with Ptolemaic terms ; the same with Egyptian terms; the almuten of Dorotheus with Ptolemaic terms ; the same with Egyptian terms, knowing that one can embellish the whole thing with different weighting system (like Lilly or not using weights like Montanus) [cf. Temperament: Astrology's Forgotten Key, p. 79, Dorian Gieseler Greenbaum 2005]

The Lilly (Ptolemaïc) almuten is MA

In our experience, it seems that Ptolemy's almuten allows one to first appreciate the static side of the natal chart and that the Lilly-type elaboration allows one to deepen the more ‘temporary’ or ‘dynamic ‘ relationships (cf, Shlomo Sela, Ibn Ezra, on Nativities and Continuous Horoscopy, appendix 6, quot 2  ; Horary astrology p, 458, Brill, 2014)

Hyleg - Alchocoden

It is rare that in a theme we immediately find the figure of the hyleg. It is also rare that one does not have the choice between several possibilities. Precisely, in this theme which turns out to be complex in this respect, there are three possibilities which are offered. We will analyse them.

1)- hyleg = SA, dominante planet and MO alchocoden in ZODIACAL system

The search for the ruling planet is explained by Ptolemy in Tetrabiblos III,10. This planet must have at least three dignities out of five [cf Ptolemy, Tetrabiblos, tr. Robbins F.E., Heinemann, London, 1940, p. 277]

The table below gives the synthesis of all the elements in our possession.

DOMINANCE	Diurnal (*)	Nocturnal (*)	House (**)		aspect D	aspect N	Σ D		score Lilly	dignities (***)
	0	2	6	0	2	1	2	ME	1	Fa -- r
SA	2	2	6	0	0	0	2	VE	4	P --
	2	3	5	0	0	0	2	MA	13	Fa --
	2	2	7	1	0	0	3	JU	0	Fa cb --
	4	2	1	1	0	1	5	SA	6	Fa -- r

(*) if diurnal, dignities for SU, ASC, sygyzy ; if nocturnal, dignities for MO, POF, sygyzy
(**) houses : 1, 7, 9, 10
(***) (P)eregrine – (D)etrimental – (F)all – (Ru)ler – (E)xalt – (Ori)entation – (R)etrograde - (T)rip - (Fa)ce – (te)rm

JU is combust ; JU and SA are FA ; SA is r.

'The greatest debility, which can happen to a pianet, is combustion, although both retrogradation and several other (conditions] are very much wont to impede other planets, as you will plainly see in what follows. The combustion of the planets is recognised by moiety of their orbs. And indeed when they are distant from the Sun less then the moiety of their orbs, or precisely that much before and after, the
[condition] is called combust and a complete weakness of power. This is the moiety of orb; SA & JU 9 degrees; MA 8 VE9 & ME, 7 degrees; MO 12 degrees: SU 15 degrees.'

[Schoner, Joannis Schoneri Carolostadii 'Opusculum astrologicum', Canon XVII, The Second Part of the Introduction Treats Compendiously of Astrology., tr Robert Hand, Project Hindsight, p. 13]

We highlight that the planet which has the most dignities is VE, VE which is peregrine... There is a kind of paradox there, like Rudolf II who was not made to be an emperor but an alchemist or an astrologer ... In the same vein we find JU combust even though SU is in his domicile but "only" in house VII; VE is in house VI as well as MO also in his domicile (VE and MO are both cadent). Sa is the true ruling planet in house I, in zodiacal trine with MO.

Anyway, only SA is likely to fulfil the conditions to represent the ruling planet. The table below is the summary of hyleg and alchocoden validation :

A word about continuity correction regarding MO, having to do with being within 5° (in the mundane REGIOMONTANUS system) of the nearest cusp (cadent).

To do this, the procedure is not unequivocal but one of the most logical seems to me to be the one mentioned by Auger Ferrier in 'Jugements astronomiques sur les nativités', Rouen, 1583 (pp, 39-51 and notably pp, 43-48)
the years of life are identified for the cad and succ houses relative to the alchocoden.
cad = 25
ang = 66
we take the difference = -41
take the 1/5 of this difference = -8,2
then take the difference between 5 and the actual position of the point = -23,4 (28,4)
take the rule of three = -38,38
Then we add the cad Y 25 and -38,38 = 63,38
We must add [1-(dom MO (151,6) - cusp (150)/5)]/ x [ cusp ang (25) - cusp succ 66) ]
So, we add to Y : Δ = 27,88

Note that this Δ changes according to the used system of houses. With CAMPANUS, Δ = 0 (MO in VI > 5°), and with PLACIDUS, there is a zodiacal conjunction between the cusp of VI and MO (Mo in V > 5°).

2)- now, we can try SU as hyleg and MA as alchocoden (in MUNDANE system).

Unlike the previous chart, here we see the MUNDANE aspects. They are marked by the presence of a sextil between SU and MA. We note that in this particular case, MA has no dignity over SU; it is SU which has two proper dignities over itself (TRI and RUL).

As I have observed many times before, the MUNDANE system alternative looks interesting; the main problem comes from the fact that the different results according to the house systems oblige to make a choice which must be maintained over a long series of chart ; I consider that the REGIOMONTANUS system represents a middle choice between the CAMPANUS system and the PLACIDUS system.

Primary directions

1)- SU conj square SA (true converse)

	Lat	Dec	AR	MD	SA	HA
SU	-	19 N	127,62	103,61 D	112,67 D	9,06 W
□SA	0 S	21,69 N	66,21	-165 D	116,43 D	281,43 W

MD = meridian distance (from MC if SA f SU  is diurnal or IC if Sa f  is nocturnal) – SA = semi-arc (if f is diurnal, SA f SU is D and all MD’s and SA’s are D, otherwise N – HA = horizontal distance (from the nearest horizon W or E for f SU and m □SA)    

DIRECTION : □SA conj SU
---------------------------------
We must take into account an important element: the ascensional difference (DA); it can be observed on the  graph in a dotted line (measured between the horizon and the axis of the pole). This is the difference between Right Ascension (AR) and Oblique Ascension (OA). This is the difference between Right Ascension (AR) and Oblique Ascension (OA). DA is always calculated in absolute value |DA| and it is added or subtracted from 90° (SA = 90° corresponds to a point on the equator cut by the horizon; depending on whether a star approaches or moves away from the line of horizon, SA is > 90° or < 90°, i,e, (+) depending on whether it is diurnal and northern ; or nocturnal and southern ; (-) depending on whether it is diurnal and southern ; or nocturnal and northern.

sin(DA) = -tan(φ)tan(δ)
φ = latitude 48,22 N
δ SU = 19 +
DA-SU = 22,67°
δ □SA =21,69 +
DA-□SA =26,43°

We will first use the Placidus system of mundane directions. The simplest system is that of Choisnard-Fomalhaut. First you need to retrieve the data from the SA (semi-arc) and the DM (meridian distance) of the diurnal point because the altitude of SU is 5,42°. important note: the SA and DM of the two points are always counted diurnal if the first point (here SU) is above the horizon even if the second is below. They are counted nightly if the first point (SU) is below the horizon regardless of the position of the second point.
For DMs, they are counted in AR from the diurnal meridian if the fixed point SU is diurnal, and from the nocturnal meridian if it is nocturnal.

diurnal meridian MC = 231,22°
AR SU = 127,61°
AR □SA = 66,21°

SA D (d+) SU=112,67°
DM D  SU=283,61°

For the  significator  □SA altitude (h) =-18,81°. so :

SA D (δ+) □SA=116,43°
DM D □SA=-165°

Then we compute Saf/DMf (so : SA f [112,67°] / DM f [283,61°])

Sa f / DM f =0,4

and the angle x = SAm x DM f/SA f, so : SA m [116,43°] x DM f [283,61°]/SA f [112,67°]

 x = 107,08°

We find the direction by DMm - x, so : DM m [-165°] ± x [107,08]
We must now have regard to the double ± sign of the last expression; in the case where f (SU) and m (□SA) are on either side of the meridian, the direction arc is obtained by taking the sum (instead of the difference) of the two quantities DMm and x. This is not the case here, so sign = (+)
the computation of the arc requires, depending on the case, a reduction of 360° (so arc modulo 360°)
---------------------------------
arc D =57,92°
---------------------------------
in the technical sense, It is a direct direction but in the astrological sense, it is a true converse direction since it is an aspect considered as a promissor which goes towards the significator. ; so the m point is an aspect (here □SA) and the f point is a planet or an axis, (here SU)

We can now compute the converse direction : point f is directed towards point m, i.e. the star is directed towards the aspect. This is where the problem of the orientation of the primum mobile arises because it is not concevable to rotate the local sphere in both directions… It does not seem convenient to postulate that the arc of direction is counted in the order of the signs of the zodiac (when it is direct, i.e. when one directs a promissor towards a significator): indeed, the ecliptic has nothing to do with a direction since this one depends only on the diurnal movement ( primum mobile). It is therefore otherwise that we must pass judgment on this.

That time, we compute Sa m / DM m (so : SA m [116,43°] / DM f [-165°])

Sa m / DM m =4,24

and the angle x = SA f x DM m/SA m, so : SA f [112,67°] x DM m [-165°] / SA m [116,43°]

x = 15,88°

We find the direction by DM f - x, so : DM f [283,61°] ± x [15,88°]
We must now have regard to the double ± sign of the last expression; in the case where m (□SA) and f (SU) are on either side of the meridian, the direction arc is obtained by taking the sum (instead of the difference) of the two quantities DM f and x. This is not the case here, so sign = (-)
---------------------------------
arc C =60,5°
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Now we can study the same direction with the Regiomontanus system. To obtain the arc of direction between two signifying points (planets in body, aspect versus planet, planet versus axis) one must find AO (oblique ascent) of f and of m, calculated under the pole of f.
The formulas to use can be found either in the Dictionnaire astrologique of Henri Joseph Gouchon (Dervy Livres, 1937) pp. 266-267, or in his Horoscope annuel simplifié (Dervy, 1973) p.181. Other formulas can be found in Les moyens de pronostic en astrologie, Max Duval (editions traditionnelles, 1986) and Domification et transits (Editions traditionnelles, 1985). We can also cite by André Boudineau : Les bases scientifiques de l’astrologie (Chacornac, 1937) These are references in French but there are many other references in English or German of a less obvious but equally valid use.

First, compute the ascensional difference under f (SU) : cot DAP f = (cot de f x cot lat) / in DM f ± cot DM f, i,e, :  cot (DAP f) = (Cot dec f[19°] x Cot Lat [48,22°]) /sin DM f [283,61°] ± cot DM f  [283,61°]

DAPf = 161,05°

We find the pole of f (SU) by formula : tan(pole f) = sin (DAP f) x cot (dec f) i,e, tan(pole f) = tan f [22,39°] x cot f [19°]

pole SU regio  =47,89°

(1) We need now the DAP of m (□SA) under the pole of f, sin (DAP m) = tan (pole f) x tan (DEC m), i,e, : (SU) : sin (DAPm/f) = tan [47,88°] x tan [21,69°]

DAP m/f = 26,1°

then we find for the points located in the eastern part of the chart : AO f = AR f± DAP f ; sign (+) if Dec f boreal or sign (–) if Dec f Austral ; so : AO f SU = 150° and AO m = AR m ± DAP m ; idem for sign ; so  AO m□SA = 92,32°

---------------------------------
arc D Regio = 57,68°
---------------------------------
We are now going to compute the converse Regiomontanus direction corresponding to the arc  f SU / p □SA

First, compute the ascensional difference under m (□SA) : cot DAPm = (cot dec m x cot lat)/sin DM m ± cot DM m, i,e, :  Cot(DAP m) = (Cot decm[21,69°] x Cot Lat [48,22°]) / Sin DM f [15°] ± Cot DM m [15°]

DAP m = 4,61°

We find the pole of m (□SA) by formula : Tan(pole m) = Sin (DAP m) x Cot (dec m) i,e, Tan(pole m) = Sin m [168,58°] x Cot m [21,69°]

pole □SA regio  =26,47°

We need now the DAP of f (SU) under the pole of m, Sin (DAP f) = Tan (pole m) x Tan (DEC f), i,e, : (□SA) : Sin (DAP f/m) = Tan[26,47°] x Tan [19°]

DAP f/m = 9,87°

then we find for the points located in the eastern part of the chart : AO m = AR m ± DAP m ; sign (+) if Dec m boreal or sign (–) if Dec m Austral ; so : AO m □SA = 78° and AO f = AR f ± DAP f ; idem for sign ; so  AO f SU = 137,48°

---------------------------------
arc C Regio = 59,84°
---------------------------------
H.J. Gouchon [l’Horoscope Annuel simplifié, Dervy, 1973, p, 181-182 and Dictionnaire astrologique, p, 277, 1937-1942, Gouchon ed., 1975, Dervy, but be careful because in DAP's equation, the double sign ± was mistakenly replaced by the sign (-) ] advises to avoid errors, to always place the star A1 (for us f, i.e. SU) in the eastern houses; in fact it is enough to change the registration number of the house based on the transformation (IV-V-VI) -> (X-XI-XII) and (VII-VIII-IX) -> (I-II-III ) to adapt the double sign ± in the calculation of DAP f or DAP m; moreover, this sign must be reversed if |DM| > 90°.

For the Regiomontanus directions, there is another mode of computing, mentioned by Gouchon (Dictionnaire astrologique, op. cit., p. 276) and especially Martin Gansten (Primary directions, pp. 155-157, the Wessex Astrologer, 2009)
This method consists at computing first 3 auxiliary angles before  the pole. It then joins the other method. Contrary to what Gouchon says, I find it easier than the previous one because we avoid the double sign ± in the determination of DAP f.

So, initially, we have A => Tan f = tan dec f [19°] / cos DM f [-103,61°]

A = 124,35°

Then : B = Lat [48,22°] + A [-124,35°]

B = -76,13°

And, Tang C = Cot DM f [-103,61°] x Cos B [-76,13°] / Cos A [-124,35°]

C = 174,13°

Then, we have Sin pole f = Cos C [174,13°] x  Sin LG [48,22°]
---------------------------------
So, pole SU regio = -47,88°
---------------------------------
Now go back to (1)

For m □SA; we have : A => Tan m = tan dec m [21,69°] / cos DM m [15°]

A = 22,38°

Then : B = Lat [48,22°] + A [-22,38°]

B = 70,6°

And, Tang C = Cot DM m [-165°] x Cos B [70,6°] / Cos A [-22,38°]

C = 53,29°

Then, we have Sin pole m = Cos C [53,29°] x  Sin LG [48,22°]
---------------------------------
So, pole □SA regio = 26,47°
---------------------------------
Now go back to (1)

2) SA conj MA

This direction between the two 'mali and infortunae' may seem surprising to consider; however, I remind you that if we consider SA as Hyleg (see above) or MA as alchocoden, there is nothing illogical.

	Lat	Dec	AR	MD	SA	HA
MA	-0,11 S	22,46 N	72,94	21,72 N	62,44 N	40,72 W
CSA	-1,78 S	-12,32 S	334,99	-76,23 N	104,15 N	180,38 E

MD = meridian distance (from MC if SA f SU  is diurnal or IC if Sa f  is nocturnal) – SA = semi-arc (if f is diurnal, SA f SU is D and all MD’s and SA’s are D, otherwise N – HA = horizontal distance (from the nearest horizon W or E for f SU and m □SA) 

DIRECTION : CSA conj MA
---------------------------------
We must take into account an important element: the ascensional difference (DA); it can be observed on the  graph in a dotted line (measured between the horizon and the axis of the pole). This is the difference between Right Ascension (AR) and Oblique Ascension (OA). This is the difference between Right Ascension (AR) and Oblique Ascension (OA). DA is always calculated in absolute value |DA| and it is added or subtracted from 90° (SA = 90° corresponds to a point on the equator cut by the horizon; depending on whether a star approaches or moves away from the line of horizon, SA is > 90° or < 90°, i,e, (+) depending on whether it is diurnal and northern ; or nocturnal and southern ; (-) depending on whether it is diurnal and southern ; or nocturnal and northern.

sin(DA) = -tan(φ)tan(δ)
φ = latitude 48,22 N
δ MA = 22,46 +
DA-MA = 27,56°
δ CSA =-12,32 -
DA-CSA =14,15°

We will first use the Placidus system of mundane directions. The simplest system is that of Choisnard-Fomalhaut. First you need to retrieve the data from the SA (semi-arc) and the DM (meridian distance) of the nocturnal point because the altitude of MA is -16,69°. important note: the SA and DM of the two points are always counted diurnal if the first point (here MA) is above the horizon even if the second is below. They are counted nightly if the first point (MA) is below the horizon regardless of the position of the second point.
For DMs, they are counted in AR from the diurnal meridian if the fixed point MA is diurnal, and from the nocturnal meridian if it is nocturnal.

nocturnal meridian MC = 51,22°
AR MA = 72,94°
AR CSA = 334,99°

SA N (d+) MA = 62,44°
DM N  MA = -21,72°

For the  significator  CSA altitude (h) =-18,3°. so :

 = 104,15°
DM N  CSA = -76,23°

Then we compute Saf/DMf (so : SA f [ 62,44°] / DM f [ -21,72°])

Sa f / DM f =2,87

and the angle x = SAm x DM f/SA f, so : SA m [ 104,15°] x DM f [ -21,72°]/SA f [ 62,44°]

 x = 36,23°

We find the direction by DMm - x, so : DM m [ -76,23°] ± x [36,23]
We must now have regard to the double ± sign of the last expression; in the case where f (MA) and m (CSA) are on either side of the meridian, the direction arc is obtained by taking the sum (instead of the difference) of the two quantities DMm and x. this is the case here so sign = (-)
the computation of the arc requires, depending on the case, a reduction of 360° (so arc modulo 360°)
---------------------------------
arc D =112,46°
---------------------------------
in the technical sense, It is a direct direction but in the astrological sense, it is a true converse direction since it is an aspect considered as a promissor which goes towards the significator. ; so the m point is an aspect (here CSA) and the f point is a planet or an axis, (here MA)

We can now compute the converse direction : point f is directed towards point m, i.e. the star is directed towards the aspect. This is where the problem of the orientation of the primum mobile arises because it is not concevable to rotate the local sphere in both directions… It does not seem convenient to postulate that the arc of direction is counted in the order of the signs of the zodiac (when it is direct, i.e. when one directs a promissor towards a significator): indeed, the ecliptic has nothing to do with a direction since this one depends only on the diurnal movement ( primum mobile). It is therefore otherwise that we must pass judgment on this.

That time, we compute Sa m / DM m (so : SA m [ 104,15°] / DM f [ -76,23°])

Sa m / DM m =1

and the angle x = SA f x DM m/SA m, so : SA f [ 62,44°] x DM m [ -76,23°] / SA m [ 104,15°]

x = -45,7°

We find the direction by DM f - x, so : DM f [ -21,72°] ± x [-45,7°]
We must now have regard to the double ± sign of the last expression; in the case where m (CSA) and f (MA) are on either side of the meridian, the direction arc is obtained by taking the sum (instead of the difference) of the two quantities DM f and x. this is the case here : so, signe = (+)
---------------------------------
arc C =67,43°
---------------------------------
Now we can study the same direction with the Regiomontanus system. To obtain the arc of direction between two signifying points (planets in body, aspect versus planet, planet versus axis) one must find AO (oblique ascent) of f and of m, calculated under the pole of f.
The formulas to use can be found either in the Dictionnaire astrologique of Henri Joseph Gouchon (Dervy Livres, 1937) pp. 266-267, or in his Horoscope annuel simplifié (Dervy, 1973) p.181. Other formulas can be found in Les moyens de pronostic en astrologie, Max Duval (editions traditionnelles, 1986) and Domification et transits (Editions traditionnelles, 1985). We can also cite by André Boudineau : Les bases scientifiques de l’astrologie (Chacornac, 1937) These are references in French but there are many other references in English or German of a less obvious but equally valid use.

First, compute the ascensional difference under f (MA) : cot DAP f = (cot de f x cot lat) / in DM f ± cot DM f, i,e, :  cot (DAP f) = (Cot dec f[22,46°] x Cot Lat [48,22°]) /sin DM f [21,72°] ± cot DM f  [21,72°]

DAPf = 6,83°

We find the pole of f (MA) by formula : tan(pole f) = sin (DAP f) x cot (dec f) i,e, tan(pole f) = tan f [163,28°] x cot f [22,46°]

pole MA regio  =34,83°

(1) We need now the DAP of m (CSA) under the pole of f, sin (DAP m) = tan (pole f) x tan (DEC m), i,e, : (MA) : sin (DAPm/f) = tan [34,83°] x tan [-12,32°]

DAP m/f = -8,74°

then we find for the points located in the eastern part of the chart : AO f = AR f± DAP f ; sign (+) if Dec f boreal or sign (–) if Dec f Austral ; so : AO f MA = 89,66° and AO m = AR m ± DAP m ; idem for sign ; so  AO mCSA = 343,73°

---------------------------------
arc D Regio = 123,41°
---------------------------------
We are now going to compute the converse Regiomontanus direction corresponding to the arc  f MA / p CSA

First, compute the ascensional difference under m (CSA) : cot DAPm = (cot dec m x cot lat)/sin DM m ± cot DM m, i,e, :  Cot(DAP m) = (Cot decm[-12,32°] x Cot Lat [48,22°]) / Sin DM f [103,77°] ± Cot DM m [283,77°]

DAP m = 14,15°

We find the pole of m (CSA) by formula : Tan(pole m) = Sin (DAP m) x Cot (dec m) i,e, Tan(pole m) = Sin m [167,36°] x Cot m [-12,32°]

pole CSA regio  =-45,06°

We need now the DAP of f (MA) under the pole of m, Sin (DAP f) = Tan (pole m) x Tan (DEC f), i,e, : (CSA) : Sin (DAP f/m) = Tan[45,06°] x Tan [22,46°]

DAP f/m = 24,48°

then we find for the points located in the eastern part of the chart : AO m = AR m ± DAP m ; sign (+) if Dec m boreal or sign (–) if Dec m Austral ; so : AO m CSA = 348° and AO f = AR f ± DAP f ; idem for sign ; so  AO f MA = 97,42°

---------------------------------
arc C Regio = 60,83°
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H.J. Gouchon [l’Horoscope Annuel simplifié, Dervy, 1973, p, 181-182 and Dictionnaire astrologique, p, 277, 1937-1942, Gouchon ed., 1975, Dervy, but be careful because in DAP's equation, the double sign ± was mistakenly replaced by the sign (-) ] advises to avoid errors, to always place the star A1 (for us f, i.e. SU) in the eastern houses; in fact it is enough to change the registration number of the house based on the transformation (IV-V-VI) -> (X-XI-XII) and (VII-VIII-IX) -> (I-II-III ) to adapt the double sign ± in the calculation of DAP f or DAP m; moreover, this sign must be reversed if |DM| > 90°.

For the Regiomontanus directions, there is another mode of computing, mentioned by Gouchon (Dictionnaire astrologique, op. cit., p. 276) and especially Martin Gansten (Primary directions, pp. 155-157, the Wessex Astrologer, 2009)
This method consists at computing first 3 auxiliary angles before  the pole. It then joins the other method. Contrary to what Gouchon says, I find it easier than the previous one because we avoid the double sign ± in the determination of DAP f.

So, initially, we have A => Tan f = tan dec f [22,46°] / cos DM f [21,72°]

A = 23,99°

Then : B = Lat [48,22°] + A [-23,99°]

B = 72,21°

And, Tang C = Cot DM f [21,72°] x Cos B [72,21°] / Cos A [-23,99°]

C = -40,01°

Then, we have Sin pole f = Cos C [-40,01°] x  Sin LG [48,22°]
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So, pole MA regio = 34,83°
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Now go back to (1)

For m CSA; we have : A => Tan m = tan dec m [-12,32°] / cos DM m [283,77°]

A = -42,53°

Then : B = Lat [48,22°] + A [42,53°]

B = 5,69°

And, Tang C = Cot DM m [-76,23°] x Cos B [5,69°] / Cos A [42,53°]

C = -18,31°

Then, we have Sin pole m = Cos C [-18,31°] x  Sin LG [48,22°]
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So, pole CSA regio = 45,07°
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Now go back to (1)