More about Placidean circles of positions (under the pole) and Monteregio method

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 - 

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

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)

---------------------------------------------------------------------------------------------------------------------------
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/((sin(SAD-90)+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 ascensional difference (AR) of S while TV is the ascensional  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° ♑.

2)- PLACIDUS 1558

Placidus calculates in his system thus:
 

 The direction of the Moon in the place of opposition of Saturn by converse motion is calculated as follows: the distance of Saturn counted from the IC (immum coeli) is 3.59° because its right ascension (RA) is 43.67°, the height at the pole of the 5th house and the 11th is 23.4°, the semi-nocturnal arc (SA N) of Saturn is 70.95° of which the third is 23.65°, that is for Saturn a polar height of almost 6° (in fact the pole is ≈ 4.92°). Note that the mundane position of the opposition of SA is PM = 8.36 (this is the ratio of 90° to the nocturnal semi-arc of 70.95° at the meridian distance of  6.69°). In any case, at this pole, the oblique ascension (OA) of the position opposite to SA is 224.97° (the ascensional difference DA of SA under the pole of MO is -1.3°) (Placidus finds for opposite to SA 227°21') and that of the MO at this same place is 279.99° (Placidus finds 280°19') ; by subtraction we obtain the arc of direction 55.02° (Placidus obtains 52°58'). To find the equivalence in years, we add to this arc of 55.02° (52°58') the right ascension of the SU of 345°39' and we obtain 42.88° (i.e. 12°52' ♉, on April 24, 1500), position where the SU, counting from the day and hour of birth, arrives in 59.5 days (Placidus finds 58 days) which means as many years. [adapted from Primum mobile, Placidus de Titis, trad John Cooper, 1814, p. 133]
The important point to consider is that the ascensional difference of the promissor as of the significator is taken under the pole of SA, i.e. of the promissor.

SIN(DA ☍♄) = TAN(p♄)*TAN(𝛅♄) and SIN(DA ☌☽) = TAN(p♄)*TAN(𝛅☽)

with p♄ = 4.97° and 𝛅♄ = -14.78°

Then, Placidus retains the date of 1519 which coincides with the coronation of Charles V (June 28, 1519) :

'In his 19th year, when he was chosen emperor, the MO had arrived at the cusp of the twelfth, and VE at the second ; therefore the medium coeli was directed to the * of the MO and TRI of VE, and they were both in parallel by rapt motion: the MO also came to the * of VE in zodiac, near 26° CAP, and to the quintile in the world [mundane quintile] by converse motion. But the most important was, the SU to parallel of JU in the zodiac, near 25° of ARI, where he acquires the same declination as JU' [Primum mobile, op cit, p. 134]

We note that Placidus does not hesitate to mix zodiacal and mundane directions, which considerably (and mistakenly) amplifies the chances of success...

In 1519, in mundo, we find this direction which coincides with the event:

directio conversa : [C] ∆☽ ☌☉

speculum	Lat	Dec	AR	MD	SA	HA
SU	-	-6,26 S	345,40	54,69 N	97,8 N	43,11 W
∆MO	-2,12 S	5,92 N	19,36	20,72 N	82,63 N	61,91 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 ∆MO)
under bracket [] the fixed point, (here SU)
- Lat ∆MO -2,12 S and lat MO : -2° 7' 4"

3)- PLACIDUS 1519
Below we see an old method of obtaining position circles, practically contemporary with Placidus. It can be used to illustrate the writings of Placidus and contemporary authors such as Argoli and Coley (see bibliography)

	CUSPAL DIST		∆ MO	SU	PLAC DIR – POLE
	OA ASC	310,08	signif	promissor
	OA ∆ MO	26,73
	dist horiz	61,91
	house ∆ MO	3	∆ MO [SU] 97,8 N
	horary time	13,77	82,63	90	pole ∆ MO
	double ht	27,54	20,72	22,57	17,28
houses		32,55	DM	PM
3	0,33	10,85
4	0,00	0,00		DA SU / ∆ MO	OA SU / ∆ MO
	pole 3	23,40	23,40	4,11	15,25
	dec ∆ MO	5,92			OA MO
	DA ∆ MO /3	2,57	2,57	-4,35	349,75
	RA ∆ MO	19,36		1,29	334,50
	OA	21,93	21,93		25,50
	dist ∆ MO cusp / 3	4,80	22,75		dir

after Sepharial (Walter Gorn Old), cuspal distances, debatable ground, the horoscope, 2, 5, 19-23, Oct 1903 and Sepharial, directional astrology, p, 72,1921

The direction of the SU in the place of  ∆ MO by converse motion is calculated as follows:
the distance of ∆ MO counted from the IC (Immum Coeli) is 20,72° ; because its right ascension (RA) is 19,36°, the height at the pole of the III house and the IX is 23,4°, the semi-nocturnal arc (SA N) of ∆ MO is 82,63° of which ⅓ (double horary time) is 27,54°, that is for ∆ MO a polar height of ≈ 17,28°
Note that the mundane position of the ∆ MO is PM = 22,57 (this is the ratio of 90° to the nocturnal semi-arc of 82,63° at the meridian distance of  20,72°).
If we wanted to determine the Placidus domitude, it would be sufficient, depending on the height h (actual h =-30,31), to compute : (h<0) 270-PM or (h>0) 90-PM, or in the present case dom = 67,43, or : 7,43 [ d, III ],
In any case, at this pole, the oblique ascension (OA) of the position ∆ MO is 349,75° (the ascensional difference DA of ∆ MO under the pole of SU is DA = -4,35°) and that of the SU at this same place is OA [SU] = 15,25 ° with its own declination ; by subtraction we obtain the arc of direction |-25,5|°.

REGIOMONTANUS 1519

DP REGIOMONTANUS (5)			DP REGIO-CAMPA D		DP REGIO-CAMPA C
DIRECTIO CONVERSA	A2 ∆MO	A1 SU	A1 SU	A2 ∆MO	A2 ∆MO	A1 SU
Tan A	tan dec/cos dm		-10,75		6,33
B (1)	+LG-A or -LG+A		-40,30		57,38
Tan C	cot DM.cos B/cos A		28,80		55,11
Sin pole (2)	Cos C.sin LG		42,96		26,42
Sin DA (3)	Tan pole A1.Tan Dec A2 Tan pole A2. Tan Dec A1	DAP (6)	-5,86	5,54	2,95	-3,12
AO (4)	AR ± DA		351,25	13,82	16,41	348,51
arc	AO1 – AO2			-22,57		27,90
				DIRECT		CONVERS

(1) B must be treated as positive number
(2) sign of pole has the same sens of LG for DA Here, DA = DA/pole A
(3) sign [-] if pole and Dec have the opposite sign – sign [+] if planet located in western half, sign [-] if planet located in eastern half ; Signs [+] and [-] must be reversed for births in the southern hemisphere
(4) to find AO of a star A2 under the pole of A1, we calculate the  DA of A2 under the pole A1 ex: tan pôleA1.tan DecA2=sin DA A2/poleA1
(5) algorithm and lessons from : a)- Gouchon (‘Dictionnaire astrologique’, Dervy, 1946, 1975, p, 276, attributed to H. Selva) ; b)- Martin Gansten (‘Primary directions’, pp, 155-157, 2009, Wessex Astrologer) - instructions for use only appear in Gansten – c)- Astrologia gallica, Morin de Villefranche, trad Holden (appendix 5, pp, 151-153) ; d)- Henry Coley, Clavis astrologiae elimata, 1676, pp. 609-648

arc = 22.57 Y.

and this direction we must add two others : *☉ ☌☽ and *♃☌☽ which are "directio directa".

speculum	Lat	Dec	AR	MD	SA	HA
MO	-2,12 S	-25,43 S	277,79	57,71 D	53,97 D	-3,74 W
*SU	0,00	-23,25 S	261,06	40,97 D	57,89 D	16,92 W

– MD = meridian distance (from MC if SA f [MO]  is diurnal or IC if Sa f  is nocturnal)
– SA = semi-arc (if f is diurnal, SA f [MO] 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 [MO] and m *SU)
under bracket [] the fixed point, (here MO)
- Lat *SU 0 and lat SU : 0° 0' 0"
------------------------------------------------------------------------
PLACIDUS

	CUSPAL DIST	(*)	2/3 SU	MO	PLAC DIR – POLE
	OA ASC	310,08	signif	promissor	2/3 SU [E] | MO [E]
	OA 2/3 SU	228,95	OD	OA	asc or desc
	dist horiz	81,14	1	4	2/3 SU above earth | MO under earth
	house 2/3 SU	12	2/3 SU [MO 126,03 N]
	horary time (ht)	9,65	57,89	90	pole 2/3 SU
	double ht	19,30	40,97	63,69	41,96
houses		32,55	DM	PM
12	0,67	21,70
1	1,00	32,55		DA MO / 2/3 SU	OD MO / 2/3 SU
	pole 12	40,37	51,05	-31,05	292,10
	dec 2/3 SU	-23,25			OA MO
	DA 2/3 SU /12	-21,42	-32,11	-34,81	312,60
	RA 2/3 SU	261,06		17,40	20,50
	OA	239,63	228,95		dir
	dist 2/3 SU cusp / 1	10,68	8,61

(*) 2/3 means sextil or *

The direction of the MO in the place of  2/3 SU by converse motion is calculated as follows:
the distance of 2/3 SU counted from the MC (medium coeli) is 40,97° ; because its right ascension (RA) is 261,06°, the height at the pole of the XII house and the I is 40,37°, the semi-diurnal arc (SA D) of 2/3 SU is 57,89° of which ⅓ (double horary time) is 19,3°, that is for 2/3 SU a polar height of ≈ 41,96°
Note that the mundane position of the 2/3 SU is PM = 63,69 (this is the ratio of 90° to the diurnal semi-arc of 57,89° at the meridian distance of  40,97°).
If we wanted to determine the Placidus domitude, it would be sufficient, depending on the height h (actual h =7,42), to compute : (h<0) 270-PM or (h>0) 90-PM, or in the present case dom = 333,69, or : 3,69 [ d, XII ],
In any case, at this pole, the oblique descension (OD) of the position 2/3 SU is 312,6° (the ascensional difference DA of 2/3 SU under the pole of MO is DA = -34,81°) and that of the MO at this same place is OA [MO] = 292,1 ° with its own declination ; by subtraction we obtain the arc of direction |20,5|°,

REGIOMONTANUS

	2/3 SU	MO
	37,04	49,77
	-60,36	131,67
	-4,67	-24,88
	-71,17	-86,34	PM
	341,17	3,66	dom regio
	12	1
	11,17 [ d, XII ]	3,66 [ d, I ]	domitude
	49,50	50,99	pole regio
A	coA=tanp/tanλ	3,66	∠ meridian↑ p/λ
B	57,97	cosB=-tanp tanδm	SA D 2/3 SU / pole 50,99
Arm	261,06		AR 2/3 SU
TS	220,084		AR MC
4	-20,65	(ARm-TS) ± A-B	pf Or D
quadrant MO	arc		orient pf

arc = 20.26 Y