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SPICA, 1964, vol. 3, nš 3, p. 3-10
THE TOPOCENTRIC SYSTEM OF HOUSES
by
Wendel Polich & A. P. Nelson Page
(Buenos Aires, Argentina)

First and foremost we must point out the following: we did not construct the Topocentric System of Houses – we discovered it.

In our search for the true Houses, we did not set out from any preconceived idea or preliminary postulate, but sought to locate them purely and exclusively by means of the actual events of life.

There are many systems of Houses that are built around a given theory or exigency, such as the trisection of a given arc, a certain kind of pole, etc., which approximate reality with greater or less accuracy, but a system such as the one we are about to describe, determined exclusively on the basis of actual facts, without any 'a priori' condition – not even the classical trisection - was never achieved in the past.

Origins

'We started out from the following reasoning: if it is possible to establish with total exactitude the precise moment of a birth by means of the important events in the life of a given person (rectifications we have often achieved), then why not determine the circle of a House by means of the events of the same nature as that House? To do this, we had only to invert the procedure employed in PD's: start off from the directional ecliptic point that produced the event, add to it the arc of age (Naibod), and measure the arc upon the respective parallel from the ecliptic point and thus obtain a real point belonging to the circle of the House which we sought!

If we repeated this procedure with other events, characteristic of that House, we would obtain various points upon the respective parallels, which, all together, would perforce have to coincide with a segment of the real circle of the House. Uniting these points in a curve, we would obtain the circle built up out of the actual events in a given life.

The idea was put to the test. We collected series of events, related to a given Intermediate House, of people whose time of birth was known very precisely. The first effective House circle we obtained was of the 9th, by means of the dates of a series of long voyages (see Fig. 1).

FIG. 1 DETERMINATION OF HOUSE-CIRCLE WITH AID OF EVENTS.

We submitted the curve we thus established to mathematical analysis to see to what pole it belonged, so as to be able to reproduce it mathematically. And here we had our first surprise: we found that the curve did not belong to any great circle! This discovery pointed to the fact that none of the known systems of Houses could be right, since all of them were based on great circles, thus destroying our expectations that one of the known systems would triumphantly pass our test as being the only one.

Originally we had no topocentric solution in mind, but now we were forced to examine analytically the curve obtained from the point of view of the place of birth as the centre of a topocentric sphere. And here we had our second surprise: this curve seen from the place of birth proved to be a straight line! This straight line was, for us, for a long time, an incomprehensible Chinese puzzle. A circle determines a plane, but not a straight line! And what could a straight line mean in spherical trigonometry? The problem became more complex because now we needed a pole that determined a curve, which seen from the place of birth was a straight line, but, from the centre of the earth, a curve. But it must not constitute the segment of a great circle. We checked and rechecked all our data; we repeated the experiment for the other intermediate Houses and invariably the same mysterious curve turned up. In vain we sought for the solution upon the sphere, or upon the cylinder elevated perpendicularly from the Equator. All was useless.

Only when we observed the apparent rotation of the firmament in its ascension and descension around the place of birth and around the 'local axis' (which passes through the geographic spot, parallel to the axis of the world), from the point of view of the geographical spot as a fixed point, were we able to see light and understand the reason for the straight line, since this, in conjunction with the local axis first mentioned, determines the plane-of-an-hour angle! And now with this plane we were at last able to continue our mathematical analysis, only to be confronted with a third surprise – this time most marvellous – for behold: every single point of that line was isochronous, each trisecting with absolute precision the semi-arc cut, and all seen topocentrically!

Is the reader aware of the importance of this finding which is in reality a great discovery?! Observe that this trisection was not a preliminary condition to be fulfilled; it was an 'a posteriori' and effective result obtained by a scientific process that proved beyond doubt the truth of the astrological principles underlying Houses, a procedure which can be repeated by anyone who can handle mathematics and has an understanding of PD's. Observe that it was not the trisection of one semi-arc which was obtained, but the exact tri-section of all possible semi-arcs, from the Equator to the Poles.

This discovery radically changed all our concepts on Houses, since it was evident that they were not to be found upon the sphere as it is generally believed, but upon the plane-of-hour angles in function of ascension, with centre at the place of birth. If the tri-section of the semi-arcs mentioned gives the exact division of time. the hour planes ensure the exact division of space around the local axis. In this way, the Topocentric System of Houses contains the joint solution of the two premises held by the two most important astrological schools of thought which exact – as a condition – one, the tri-section of time (Placidus), and the other, the tri-section of space (Campanus).

Ascension of Planets

If the reader desires to form a clear idea of the physical reality of the Topocentric Houses, it is essential that he follow our explanations step by step since this system differs essentially from all others.

Let the reader imagine that he, himself, is situated at the place of birth upon the local axis. Let him raise his eyes to any planet in the sky and, in his imagination, trace a straight line from the planet to himself. This straight line we shall call a "temporal line". This is the straight line which caused as so much trouble. This line is of enormous importance to the native. It is in reality the line of ascension of the planet. The angle that this line forms with the plane of the Prime Vertical is the angle of ascension which we call the "topocentric pole".

All that has been said is to be understood in its relation to the place of birth as topocentre. But the positions that are given in the ephemeris are not given for the topocentric sphere, but for the geocentric as seen from the centre of the earth. To be able to compare the purely topocentric data with the data given in the ephemeris, we have but two alternatives: either we transform the positions given in the ephemerides, or we express the topocentric data in geocentric measures. It is evident that the second alternative is the more practical.

Because of this we must also relate this temporal line and its pole with the centre of the earth and this may be obtained if we work with the geocentric horizon (see Fig. 2) which passes through it. The plane of this horizon (perpendicular to the drawing) cuts the plane of the Meridian along a straight line which we call the "temporal line of the horizon" (commonly known as 'the horizontal polar axis'). This line cuts the local axis at an angular point (H1) which is common to both the geocentric and the topocentric systems; therefore, if from the given planet (B) we draw a straight line to this angular point, we have its geocentric expression. And the pole (j) will be the angle that this new line forms with the plane of the local axis. The formula for the topocentric pole, given further on, already gives the topocentric position of the planet in relation to the centre of the earth.


FIG. 2 TOPOCENTRIC SPHERE WITH CENTRE IN C.

Physical View of the Houses

Let the reader imagine a star situated exactly on the "temporal line of the horizon", in other words, upon the line of intersection of the planes of the horizon and meridian. Observing this star in its apparent rotation around the local axis, which, in reality, is the motion of ascension, the reader will see that the star moves upon its parallel and so also does the temporal line of the horizon, and as a lineal generatrix it describes a cone of rotation with vertex at the angular point (H1), which is fixed.

In 2, 4 and 6 hours, that star and the line will describe exactly 30, 60 and 90° around the local axis, and the straight line at those points of tri-section will be – in function of the real rotation – the temporal lines of the 8th, 9th and 10th houses, and the angle that the temporal line of the house forms at those points of tri-section, with the plane that passes vertically through the local axis, will be its pole. If the reader imagines planes passing through those temporal lines of the houses and the local axis, he will have a vision of the topocentric houses in function of ascension and of-the-hour angles. Where the temporal line of a house cuts the ecliptic, there is to be found the cusp, which automatically trisects its semi-arcs (time), and the quadrant between the meridian and horizon (space). This is illustrated in Fig. 3. Since in the celestial sphere the local axis may be coincident with the axis of the world, they are placed together in the figure. Starting from the 12h. meridian, the 14-, 16- and 18-hour circles are seen; also, the temporal lines of ascension of 12, 14, 16 and 18 hs. which coincide with the hour planes determining houses 10, 11, 12 and Asc.


FIG. 3 HOUSES AND CUSPS (DIURNAL)

Experiences

Since the Topocentric System was constructed exclusively upon the real events of life, naturally the directions and transits calculated with it reflect the events of life with a certainty and exactitude hitherto unknown.

The reader cannot imagine the formidable power and importance that the topocentric poles have in the physical realization of the events. We can now anticipate (this we have experimentally proven) that all events, major and minor alike, are realized directly through the agency of these poles.

The ecliptic aspects, as sources of energy and reflections of the internal potentials of the native, are the causes of the events, which act upon the inner life of the individual within an orb, i.e. either before or after the exact aspect. But their physical, realization is always produced by an ascension transit (mundane), that has no orb and which is discharged exactly at the instant the planet in the sky arrives in its ascensional motion (apparent rotation around the local axis), at the same height of another radical planet or cusp. that is to say the same radical pole (conjunctions and oppositions in OA).

So potent a factor as ascension has been completely ignored in the preceding systems, in which only the horizon and meridian have ascensional value, i.e. power of realization. In the Topocentric System, the circles of all the planets and of all the houses are of astronomical ascension and hence the physical power of realization of all of them!

Practical Procedure

The non-mathematically minded astrologer should not be put off by the seeming complexity of the formulae that follow. He need not understand their essence. They have been included only for the benefit of those who will.

To arrive at the exact cusps of those charts where the exact time of birth is known – as indeed he must if he is to reckon transits and directions to and from them, by far the most important and personal factors in any chart – all he requires are tables of common (decimal) and trigonometric logarithms and of ascendants or of oblique ascension, such as those recommended hereunder. The rest is a simple matter of addition, subtraction, and rule of three, requiring no other specialized knowledge as he will soon see if he experiments with our example.

To calculate the House Cusps, their poles are needed. The general formula for the topocentric pole is:

where j is the topocentric pole and F the geographical latitude.

This formula is as valid for the stars and planets as it is for the cusps. It obtains the real position of the stars and cusps in purely temporal measures, since declination, an expression of the curvature of the sphere, has been eliminated and substituted by arcs of time.

It must be realized that the houses stem from the Equator and that consequently the SA will be 90° and the MD 30° and 60°, according to the intermediate house. Thus:

(for houses 11/5 and 3/9)

(for houses 12/6 and 2/8)

These poles can be obtained with the aid of logarithms, as follows:

Example 1

We have chosen the latitude corresponding to the heart of London (51°32 N) and the poles obtained will invariably be the same for that parallel.

House Poles

Pole House
  log tan F 0,09   991 51°32 ASC
– log of 3 0,47   712
= log tan j  11  9,62   279 22°46 11/5 and 3/9
+ log of 2 0,30   103
= log tan j  12 9,92   382 40°00 12/6 and 2/8

(The pole of the 10th is always 0°00').

The cusps may be calculated directly or extracted from a Table of Ascendants or Oblique Ascensions under the Topocentric Pole of the House. For this purpose, H. J. Gouchon's "TABLE D'ASCENDANCE" is of inestimable value, since it is necessary only to subtract six hours from the S.T. of birth to obtain the "Table" S.T. of the 10th House.

If two hours are added successively to this "T" S.T. for each House, as we have done in the example, the reader will immediately be ready to extract the cusps sought from the Tables, under the respective topocentric poles. It is necessary, of course, to effect the corresponding interpolations between Poles and Sidereal Times, given in Tables, to seconds for precision. If one prefers to work directly with OA, which is often more practical, nothing better can be recommended than E. K. Kuhr's magnificent "PDT" and "AO TABELLEN".

As a check, let him compare results with Placidus' cusps. These will always be well within one degree of the topocentrics.

Example 2

Given S.T. of B:     3h 50m 00s = RAMC 57°30'     Lat. 51°32 N

 

House

OA

Pole

Cusp

Table
S.T.

X

57°30'

0°00'

29°42'

    Taurus

21h 50m

XI

87°30'

22°46'

7°15'

    Cancer

23h 50m

XII

117°30'

40°00'

10°30'

    Leo

1h 50m

I

147°30'

51°32'

7°09'

    Virgo

3h 50m

S.T of B

II

177°30'

40°00'

28°00'

    Virgo

5h 50m

III

207°30'

22°46'

25°13'

    Libra

7h 50m

Example 3

Pole and OA or OD of a Planet or Star

Moon Long.

24°27'

Cancer Lat

–  2°39'

RA

115°52'

Decl

+18°37'


    RA Moon

115°52'

-   RAMC

57°30'

=   MD

58°22'

(58,367°)

______


       Log tan F

51°32'

0,09 991

+       "   tan decl.

+18°37'

9,52 745

=       "   sin AD

9,62 736

25°05'

=======

90°00'

+  25°05'

            Diurnal SA

115°05

(115,083°)

=====


Decim. log MD

58,367°

1,76 617

–  "     "   SA

115,083°

2,06 101

Difference

9,70 516

+  log tan F 51°32'

0,09 991

=  "    "   POLE Moon

9,80 507

32°33

Pole Moon
+  "    "   Decl 18°37'

9,52 745

====

=  "   sin  AD

9,33 252

12°25

AD

======


RA Moon

115°52'

–     AD       

12°25'

=   OA Moon

103°27'

=====

This is the true Topocentric Pole and OA to be used in Primary Directions and Mundane Transits.

NOTE

Observe that although the Moon in this example is in close ecliptic square to the cusps of Houses 3/9 (an ever-present inner potential inherent in the emotional and mental make-up of this native), it is too far from the mundane square

207°30' – 103°27'=104°03'

for it to take form or have an outlet in the physical circumstances and events in his life. But .... "As a man thinketh in his heart, so is he."

December 1963