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Placido and the semi-arc method of house division
     by Michael Whackford

The following article was first published in The Traditional Astrologer magazine (Issue 7, Winter 1994). Limited magazine space meant that one detail of semi-arc house division was left undeveloped. Following recent forum discussions on the limitations of houses in Polar Regions, Mike has kindly agreed to expand the original article to include an explanation of why the Placidian method of division does not fail near the Earth's Poles.

It is more commonly recognised today that Placidus did not design the method of house division to which his name has been attached. Both he and, 500 years before him, Abraham Ibn Ezra ascribed that method to Ptolemy. It appeared on an astrolabe in 1305 and Magini had devised ingenious tables for it in 1604, while Placido was still a toddler. Some adherents of Equal House or Regiomontanus have also claimed Ptolemaic authority for their favoured systems, but an unprejudiced appraisal of the Ashmand and Loeb translations of the Tetrabiblos shows that Placido and Abraham have by far the stronger case, despite the comments of the translator in the Loeb edition.

The survival of 'semi-arc' houses, not to mention the name 'Placidus,' has been ungraciously attributed to their being 'publisher-friendly' and therefore a convenient choice for Raphael's Ephemeris. Whilst it is true that a table of houses may be constructed for most latitudes with little recourse to calculations, it is unfair to both the man and his approach to dismiss either on these grounds. The houses in question are inextricably linked to both the idea behind Planetary Hours and the most 'natural' system of Primary Directions. To use either whilst adopting a different house system is to rather miss the point.[1]

"I Desire no other Guides but Ptolemy & Reason" [2]

It was to Ptolemy's astronomical tome, Almagest, that Placido owed much. Although Ptolemy's 'geocentric' model of the universe had been overturned, its description of the motion of heaven remains correct from our earthly point of view and Placido, recognising this, adhered to the principles outlined by the great astronomer. He was not, however, averse to any new-fangled ideas provided they appealed to his sense of reason: Kepler had recently presented a sound a case for some new aspects, the quintile and biquintile. Meteorological evidence had been advanced for the quintile and Placido took both on board, whilst rejecting seriously minor aspects.[3]

The relative inaccessibility of the language used in the last English translation of his Primum Mobile seems to have led many to overlook the fact that Placido also advocated Tertiary Directions, some 300 years prior to Herr Troinski's claim to have discovered them. While it is astrological cowardice to have, simultaneously, several directional techniques at one's disposal, Placido was particular as to how and where these were used, urging caution with regard to their application. This is typical of his common-sense approach to most matters. Not so sensible, perhaps, were his prescriptions for handling the timing of events and planetary latitude, in primary directions. Both appear to have sprung from the understandable desire to reconcile primaries with day-for-a-year progressions, but both, being questionable, have caused perennial and perhaps unnecessary debate.

Unfortunately, the standard definitions of the house system that bears his name have often been repeated incomprehensibly (or wrongly), and some authors, in misleading themselves as well as their readers, have caused several corrupt variations to gain currency. We will describe it again here but with a different approach:[4]

Imagine yourself standing outside on a cloudless night with a clear, uncluttered horizon stretching around you. If you face due south, stars will ascend from the horizon to your left, arch up into the sky, and then set to your right. When a single star is due south of you it is:
  • at the highest point of its journey;

  • conjunct the imaginary vertical line that divides east from west and yields the MC in a horoscope;

  • exactly half-way through its journey, having spent half the time that it is going to spend in the visible sky.
So far we have divided the sky into two: left and right; but the amount of time the star took to get from horizon to halfway point may also be divided, this time by three. For the first third of its journey our star, following ascension, has been in the 12th house, the next third represents its transit through the 11th house, and the last third belongs to the 10th. The remaining (equal) times are spent in the 9th, 8th and 7th houses, until the star sets upon the westerly horizon.

Obviously, a star that rises from a position close to due south will not rise very high nor be visible for very long: it will rise, 'culminate' at low altitude and then set, all in a relatively short 'space of time.' However that passage may yet be trisected, on either side of the culmination point and all stars, regardless of where they rise and set, may have their journeys divided into six equal parts. Each point derived from this division represents the cusp of a house.

Connecting those cuspal points, each derived from countless semi-arcs, generates the system's house boundaries. These house-lines commence together at the South Point of the horizon and then fan out across the sky, up and over the observer to his left and right, with the North/South Meridian passing directly overhead. These lines (or "lunes"), together with local horizon, form the real boundaries that separate any house from its neighbour. Ecliptic house cusps, formed where each line passes across the Zodiac, are therefore merely a rough guide to planetary house position and should be treated as such. This is not to belittle their role in determining house rulership, etc. but they will not always reflect accurately the true house position of any planet or star with much Ecliptic latitude.

This is the concept by which the houses are formed and, although we achieve the division by dividing time, it would be a serious error to assume that we are not also dividing space. In the Placidian system all the space that rises and sets is carved up according to the principles outlined and it is as 'spatial' as any other method. Unlike other systems, however, its intermediate dividing lines are not circular so cannot be described by spherical trigonometry, (though the horizon remains circular by definition and, in this system, the MC's line is circular by default). Thus only the 1st, 7th, 10th and 4th cusps are calculable by trigonometric means and calculating exactly where the intermediates' meet the zodiac is indeed a tedious business. Consequently, Placido did not calculate exact cusps; rather, he used trigonometry to approximate their positions (exactitude was not required because he directed planets to house boundaries according to those planets' own passages across the sky, and the arithmetic for this is very simple indeed). It is regrettable that many professors have subsequently presumed that trigonometric approximations of the system were in fact the system itself. They are not, and a list of commentators who thought otherwise makes for some alarming reading.

As a system of house division 'Placidus' has this to commend it; because it is a product of both the rotation of the Earth and the geographical position of the observer, the method truly reflects the motion of the heavens from that observer's point of view. Whilst most other systems are geometric contrivances, this one remains, in Placido's word: 'natural'.

The circumpolar skies

In the foregoing, we discussed the sky as it appears to observer's south. The reader is now invited to turn about and imagine the heavens to the north. At British latitudes, stars contained within a vast area of the northerly skies do not rise or set. Rather, they rotate in what might appear to be a flat disk, as though in orbit around the Pole Star. Many come to upper culmination above the observer, slightly to the south, before descending to the north-west. The lower culmination of these stars ensues, just above the northerly horizon's North Point. Alternatively, imagine somebody standing before you with an open umbrella, its shaft resting at a shallow angle across their shoulder. With thumb and forefinger they slowly rotate the handle in what is to you an anti-clockwise direction. The brolly's open carapace then approximates the shape and motion of the circumpolar skies.

Definitions of the semi-arc system are routinely couched in terms of rising and setting, which seems to limit this method to the region of sky that does rise and set. Indeed the foregoing description followed that precedent but it is just that: a convenient precedent. If we accept at face value the astrological tenet that houses 1- 6 are found entirely below the observer's horizon and that houses 7-12 are always above, it follows that 'circumpolar' stars - those that never set - must in principle reside permanently in houses 7-12.

Under the Placidian method a star's true house position is determined, as we have seen, by its location along its own path across the sky. This path is called a diurnal arc, when the star is above the horizon, and a nocturnal arc, when it is below. These arcs are each divided into six, which, when added together, give each and every star - or planet or degree - its 12 house cusps. A star that cannot set will have no nocturnal arc but its 24-hour per day diurnal arc may yet be divided by six.

In practice this means that, commencing from its lower culmination due North, the star ascends upward through the 12th, just as it would had it risen in the East in the 'normal' way. After 60 degrees of arc (rather than the 30 degrees that separate houses at the Equator), it leaves the 12th, continuing up through the 11th and 10th. Following upper culmination, it passes downward through the 9th, 8th and 7th houses. On leaving the 7th, having returned to lower culmination, the star passes not to the 6th but straight back into the circumpolar 12th, where the cycle recommences. However unorthodox this may seem at first, it is a faithful reflection of the real sky that concurs also with the astrological tenet re-stated above.[5] In principle this scotches once and for all the oft-repeated axiom that "Placidus cannot be applied above the Arctic Circle." It also provides for circumpolar planetary hours, Gauquelin sectors and, theoretically at least, circumpolar pre-natal epochs. In horoscopes however it presents a number of choices and challenges that take us into uncharted waters.

Placido may never even have considered the circumpolar problem perceived by later generations. He never had to. But the very idea that his favoured system can itself provide the solution, simply by taking account of the differing conditions presented by differing local skies, underlines yet again his assessment of the method: 'natural.'

Notes & References:

  1 ] Students of horary who retain the traditional use of planetary hours whilst following William Lilly in his choice of Regiomontanus houses may wish to reflect upon the inconsistency of this.
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  2 ] Placidus de Titis, Primum Mobile, J. Cooper translation (reprinted: Bromley, Kent, 1983). p.47
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  3 ] Placido followed Ptolemy in his use of the conjunction, sextile, square, trine and opposition aspects. To these he added the parallel, which he considered an aspect in its own right, and the 'new' aspects of 72° (quintile), 135° (Sesquiquadrate) and 144° (bi-quintile). A later thesis by Kepler included 30° (semi-sextile) and 45° (semi-square) but Placido specifically rejected them. The eight geometric aspects he used are related to ratios found in prominent musical resonances. It has recently come to light that Placido also composed music.
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  4 ] Usual apologies to readers south of the equator - please make the usual modifications: - for 'north,' read 'south', and for 'left,' read 'right'.
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  5 ] Diagrams illustrating this phenomenon and more may be found throughout the writer's Correlation series:

“The Polar Horoscope”, Correlation, volume 19, no. 2, 2000/2001. - d/w reversing Ascendants.

“Polar Houses”, Correlation, volume 21, no. 1, 2002/2003a. - takes down most house methods.

“The Polar Arcs”, Correlation, volume 21, no. 1, 2002/2003b. - introduces circumpolar Placidus.

“Polar Meridians”, Correlation, volume 21, no. 2, 2003. - concludes C/polar Plac. and d/w seasonal-vs-diurnal cycles.

“Global Horoscopes”, Correlation, volume 23, no. 1, 2005. d/w C/polar EH, junks Campanus (sadly) and draws conclusions.

For details of these publications contact the Astrological Association of Great Britain; see
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Michael Wackford has studied astrology since 1976. For 11 years he considered the fundamentals of hoscopy with the late Neil Gillings, a little known yet well-respected technical astrologer who was often consulted by his peers, including Charles E. O. Carter. The author has contributed articles to the AA's Journal and The Traditional Astrologer and has advised other astrologers, including Robert Zoller.

Enquiries about this article should be addressed to the author at: 29 Davigdor Rd, Hove, BN3 1QB, UK

© Mike Wackford, 2006.

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