3 articles for "Placidus"

(also de Titus, Latinization of Placido de Titi, pseudonym Didacus Prittus Pelusiensis; 1603–1668)

An Olivetan monk and professor of mathematics, physics and astronomy at the University of Pavia from 1657 until his death. Placidus popularized the system of astrological houses now known as the "Placidian system", current in modern astrology. He did not invent the method; it is acknowledged by the 12th century Hebrew astrologer Abraham Ibn Ezra as the system employed by Ptolemy, an attribution that was accepted by Placidus.

See also:
♦ Placidus House System

Named after Placidus de Titis (1603-1668). When first introduced into England, the Placidian system was universally rejected by astrologers of that period. In 1711, Bishop wrote a severe denouncement in his work, Flagellum Placideanum. Among modern astrologers with strong mathematical training, the Placidian system is generally treated as a close cousin to the Black Plague. Yet, this system is the one most popular among modern astrologers. Critics attribute this popularity to two principal causes:

(1) The availability of the printing press (which came too late for both Campanus and Regiomontanus); and,

(2) the inclusion of Placidian tables in the yearly ephemerides of Raphael and Die Deutsche.

Very few authors understand the Placidian system; and still smaller in number are those writers who present the method as it was formulated by Placidus, its inventor. Most writers assume that once the longitudes of the Placidian House Cusps have been computed for a given chart, that the placement of planets within the Placidian Houses can be determined merely by comparing the zodiacal positions of planets and house cusps. This is not true! Nor is it true for many other house systems, such as the Campanus. Authors who attempt to explain the difficulty write at length about poles of the houses, and about planets 'under their own pole'. Construction of a Placidian Mundoscope will reveal the correct house position of all planets, but its computation is not unlike that of the house cusps and is, therefore, rarely found in printed texts. Computation of Primary Directions is, normally, performed according to the rules advanced by Placidus. But, most such efforts are concerned with the principal angles of the chart (1,4,7,10), and only rarely with the cusps of the other houses. The mathematics required has been given in the writings of M. Vijayaraghavulu and Charles Jayne, but is little understood even among advocates of the Primary Direction methods.

The Placidian House System is often (incorrectly) described as a 'time-based' system (as opposed to a 'space' system, such as Campanus). This classification is derived from the formulae used to compute the house cusps, and is not representative of the resulting houses (the lunes) which ultimately result. Such authors, however, ignore the writings of Placidus (or his followers) who have defined the House Circles, the House Cusps, and the computational procedures. Dalton's Tables are based on selecting the declination of TAU 22°. Raphael's Tables are based upon selecting the declination equal to the Obliquity of the Ecliptic. Such approximations lead to inaccurate results, and are quite unnecessary.

A significant deterrent to wider understanding of the Placidian system is the nature of the mathematical formulae which must be used for computation: It is necessary to solve a transcendental equation by iterative means (Newton-Raphson is usually employed). And, for geographic latitudes within the Arctic and Antarctic Circles (i.e., within 23.45° of the terrestrial poles) strange configurations of house cusps occur: Certain house cusps cease to exist, while others in the same chart appear; this situation leads to charts in which certain houses are are 'missing' altogether. In the Placidian system, the MC and ASC are, respectively, the 10^th and 1^st house cusps. The remaining cusps may be defined as follows: If the cusp exists, then it already has (or subsequently will) cross the Horizon and Meridian as it moves along its diurnal circle. It is desired that the specified cusp shall have completed a specified ratio (1/3 or 2/3rds) of its journey between Horizon and Meridian. But, diurnal circles are small circles of the sphere, and angles are never measured along their arcs. Instead, angles are measured at their poles or, equivalently, on a great circle lying 90° away from said poles. In our application, the poles of all diurnal circles are the North and South Celestial Poles, and the Celestial Equator is the great circle lying 90° from these poles. The appropriate angles, measured along the Celestial Equator, are called the Semi- Diurnal Arc (SDA) for points above the Horizon, and Semi- Nocturnal Arc (SNA) for points below the Horizon. An equivalent definition, suitable for mathematical computation, is that the cusp be located at a trisection point of its Semi-Diurnal or Semi-Nocturnal Arc. To be specific, for the following fractions are required for the various cusps:

Unfortunately, computation of the values of the SDA and SNA require that the declination of the desired house cusp be known. And, the declination depends upon the longitude of the house cusp (that quantity which we seek). This circular dependency may be resolved by iteration, starting with an assumed value for the declination (or some other member of the circular chain). With suitable mathematical techniques (such as Newton-Raphson), this initial guess can be progressively improved until it converges to a stable value with the desired precision.

	   S = sin(LAT)*tan(obl)
	   C = cos(LAT)
	
	For house = 10,11,12,1,2,3:
	   n = house+6, if house <  7
	     = house  , if house > = 7
	   k = (n-10)/3
	   ra = ST+q
	   cos(q) = -sin(ra)S/C
	   tan(L) =  tan(ra)/cos(obl)
	   
	which can be solved for q using Newton-Raphson iteration, if we write:
	   F  =   Ccos(q/CRD) +  kSsin(ra/CRD)
	   F' = (-Csin(q/CRD) + kScos(ra/CRD))/CRD
	   delq = -F/F'
	   q  = q + delq
	   
	The iteration is started with
	   q = 90.0°
	and continued until
	   abs(F) < 0.0001
	Then,
	   y = sin(ra/CRD)
	   x = cos(ra/CRD)*cos(obl/CRD)
	   Ln = atan2(y,x)*CRD+180.0, if house <  7
	      = atan2(y,x)*CRD      , if house > = 7
	and,for n = 4,5,6,7,8,9:
	   Ln = Ln+6 + 180°.
	   
	Sample (11th cusp):
	   Given that:
	     obl  = +23.45°
	     LAT  = +37.00°
	     ST   = 07:32:00 = 113.00°
	     n    = 11
	     k    = 0.333333
	     C    = +0.798636
	     S    = +0.261052
	     q    = 90.0°
	     
	First iteration:
	     ra   = 143.000000°
	     F    = +0.157105
	     F'   = -49.740242
	     delq = +10.368792
	     q    = 100.368792°
	     
	Second iteration:
	     ra   = 146.456264°
	     F    = +0.000509
	     F'   = -49.166597
	     delq = + 0.034016°
	     q    = 100.402809°
	     
	Third iteration:
	     ra   = 146.467603°
	     F    =   0.000000
	     F'   = -49.162243
	And,
	     y    = +0.552408
	     x    = -0.764727
	     L11  = 144.1571°
	     
	For completeness:
	MC =  10: 111.2768 = CAN 21°16.6'
	      11: 144.1571 = LEO 24°09.4'
	      12: 173.7374 = VIR 23°44.2'
	ASC =  1: 198.8521 = LIB 18°51.1'
	       2: 226.6842 = SCO 16°41.1'
	       3: 257.8696 = SAG 17°52.2'

See also:
♦ Mundoscope ♦ House Systems ♦ Albategnius House System ♦ Cusps ♦ Obliquity of the Ecliptic ♦ Ecliptic

One of several house systems.

The Placidus house system is long known in the English-speaking world due to the availability (since the middle of the 19th century) of Raphael's Ephemeris and Raphael's Tables of Houses. There is much argument as to the origin of this system, although it seems generally agreed that it was not the work of the monk Placidus. It appears to have evolved from the Albategnius house system.

The house cusps are formed by points, each of which trisects its own diurnal or nocturnal semi-arc. The points where these complex curves cross the ecliptic are the cusps given in the Placidian Tables of Houses.

See also:
♦ Mundoscope ♦ House Systems ♦ Albategnius House System ♦ Cusps ♦ Obliquity of the Ecliptic ♦ Ecliptic