An Astrological House Formulary

Michael P. Munkasey

OVERVIEW

Astrological house systems are often thought of as some form of incomprehensible entity -- almost

as an evil that lurks behind the chart. In reality, it is not the houses themselves that are recondite,

but more correctly the mathematical description of the houses. Once an astrological practitioner

realizes that the mathematics of how a house system is created can be ignored, then the

application of the meanings of the houses and their usage in astrological practice are concentrated

upon. This article is not about the meanings of houses in a keyword sense. This article is about the

other parts of houses: their technical descriptions and mathematical formulations. Early in my

astrological studies I was given a wonderful opportunity by circumstances, which at the time were

not perceived as pleasant, to derive the mathematics of the various house systems. My sense of

curiosity compelled me to bring some order to an area where I could find no reasonable or

consistent sets of information. Then, unlike today, there were few ready reference books on the

mathematics of house systems, and I perceived the need to create understanding and order for all

interested parties then and in the future.

This article is intended both for the education of general NCGR members and also to provide

specific information to persons who wish to calculate house cusps for any known house system.

Provided herein, for the first time ever in one place, to my knowledge, is a short written description

of what all house systems and sensitive points which I could find described in the popular

astrological literature are in an astronomical sense. The word descriptions of the house systems

are purposely kept short, but each house system where mathematics are less than straight forward

is rigidly and accurately described in a way which would allow any reader with a reasonable

interest in pursuing this subject to calculate all house cusps. All that is needed is a hand calculator

or computer, some persistence, and a knowledge of how to read and use the types of

mathematical notations shown. Only a high school level of mathematics is required. For those

persons wishing to pursue this adventure I offer good calculating. To the other readers I offer in

one place, for the first time, a compendium of the mathematics of house systems and sensitive

points heretofore not available except privately.

MY HISTORY OF INVOLVEMENT

Contrary to the opinions of many who know me I was not well equipped in an educational sense to

do this type of work. While I liked mathematics as a subject when I was in school, the form of

mathematics needed for this study was not one where I had much expertise. So, I had to teach

myself forms of arithmetic which I had either long ago conveniently forgotten, or had never learned

formally in school. My purpose and drive behind this study was manifold and centered around

wanting to understand: why astrologers used different house systems; which house systems was

best to use from a mathematical sense; what the difference was among the various house

systems; what houses are in space and what role they perform in astrology; and last but not least,

to find short cut paths for computing various positions of the house cusps. This last was particularly

important, because this work was being done in that era before personal computers or

programmable calculators, and computing house cusps using logarithms for systems not

commonly available was a long and arduous task, and one which I did not like to do. So, necessity

being the mother of invention, I had to forge into areas where others had not adequately left

instructions for proceeding. (Note: definitions for most technical terms used in this article appear at

the end.)

After several months of effort I had managed to gather up word and other descriptions of the

various house systems. Over the years I found 22 different house systems described in the

astrological literature. What I did when I located these was to draw astronomical globes in diagram

after diagram showing what these houses systems did in space to divide the areas so that

astrologers could use house cusps in their practice. What I found is that there are two general

approaches for creating a house system: one thought is to take a point on the ecliptic and divide

the ecliptic according to a systematic scheme. This would then give the houses. The Equal House

System is one such example: the ascendant (ASC) is taken as a starting point and thirty degree

increments are added to the ASC for each subsequent house cusp. This is an easy system to use,

requires only a minimum of mathematics, and works anywhere there is an ecliptic visible. Having

an ecliptic visible becomes an important consideration, especially in the far polar regions where

distortions in space around the ecliptic are magnified, and an ecliptic may not always be

mathematically or actually present. The second system requires using one of the other great circles

of astronomy, dividing that in some acceptable manner, and then projecting that division onto the

ecliptic. These are the projection house systems, like Placidian and Regiomontanus. The Koch

house system is a variation on the projection house system methodology, and I sometimes classify

by itself while terming its approach to the mathematics of houses 'intersection' as opposed to

'projection'.

WHAT HOUSES ARE

Houses are the divisions of space around an event. An event which we wish to consider

astrologically occurs. It may be a birth, it may be a mundane event, it may be a horary question, or

it may be something other than these. An event occurs and astrologers wish to examine a

horoscope for that event. The next step is to determine the placement of that event in the hierarchy

of space. To do this we need to introduce some order to the space we are going to diagram -- and

diagram is exactly what we will do with the space. So, an event occurs, and now we must create a

frame for picturing the planets within our portion of space. Doing this poses certain important

questions which must be answered. How do we divide this space? What is our starting point for

this division? How do we translate how we see the planets from Earth with where they are in their

orbits around the Sun? What about the parts of the sky we can not see -- like those parts that are

on the other side of the earth? How do we represent those 'hidden' spaces? Since space is rather

'plastic', and if we can change our view of the event depending where in space we are viewing

from, then what constant boundaries can we create for classifying the space around our event?

These questions have a serious philosophical leaning, and are essentially solved when the

diagram of the horoscope is produced. That is, all questions such as stated above are solved when

the horoscope is drawn except the one question of which house system to use. One horoscope

produced usually implies that one house system was chosen for dividing space. Another

horoscope of the same event may show a completely different house system. But the nagging side

issue question always remains: which house system is best to use?

There is no simple or direct answer to that question. However, I can give you two good thoughts on

the subject: use that house system which divides space in such a way that the planets fall into

houses which describe their function in the nature of the event; and, use that house system which

gives cusps against which you can time events. That is, if the Moon function of this event is

described well by a Moon in the eighth house, then the house system you choose should not place

the Moon in the seventh or ninth, or some house other than the eighth house. Also, if subsequent

events can not be timed to the house cusps derived mathematically and plotted on the horoscope,

then choose some other house system. In general, I find that for natal events the Placidian system

works well and fulfills these two guidelines. Why it works well I do not know. It may well be that the

thought form created by a mass of astrologers using the Placidian system is all that lies behind its

functioning. I can not explain why it works, but my practical side allows me to see that it does work

well and consistently for natal events where it is used and applied. For events other than natal,

other house systems seem to work better. Some house systems also work well with natal events,

besides Placidian. The type of experimentation and conjecture of which house system to use

where and when, however, is beyond the scope and intention of this article.

HOUSES VS. THE PERSONAL SENSITIVE POINTS

One confusing issue which is often blurred over is the difference between house cusps, and the

personal sensitive points. The primary difference between houses and the personal sensitive

points is that houses show wedges or divisions of space while the personal sensitive points

represent a freezing of time in a horoscope. When one looks at the astronomy surrounding an

event it becomes quickly evident that there is an astronomically defined framework provided for

dividing the various portions of space, and there are points in space where these divisions intersect

the ecliptic. There are four levels of definition in the astronomical space surrounding an event, and

we can call these levels: ecliptic, equatorial, horizon, and lunar. Each of these levels allows us to

bring some mathematical order to the placement of the planets in space. Each of the levels allows

us to divide space and time in ways that are meaningful to us as astrologers. More importantly,

though, the mathematical intersections of these levels, or astronomical or mathematical planes as

they are also called, gives us eight rigidly defined sets of places where peaks, or points, which are

able to gather energy and potential are formed. Astrologers have generally named these points the

personal sensitive points, and specifically refer to them as: the ascendant (ASC), the MC (or

Medium Coeli), the equatorial ascendant (EQA) [popularly, but incorrectly, called 'the east point'],

the vertex (VTX), the co-ascendant (CAS), the polar ascendant (PAS), the Aries point (ARI) , and

the Moon's node (NOD).

Each of these points is really a pair of points, and one part of the pair can not be thought of without

also considering its companion: with the ascendant there is a descendant, with the MC there is an

IC, etc. The pairings of the points are unchangeable -- because one side exists by definition, so

does the other. The points and their pair names are listed at the end of this article. Further, each

personal sensitive point is formed when some great astronomical circle from one of the four levels

or planes mentioned above intersects the ecliptic. There are only eight ways this can be done, and

there are only eight sets of personal sensitive points. Each is astrologically equal to the other in

strength and function -- but not all are equally well defined in popular astrological literature. Since it

is the ecliptic that we plot on the outside of a horoscope, then places where the ecliptic is

intersected in a significant manner must be important astrologically.

In other places I have written extensively about the meanings of each of the personal sensitive

points. This is not the place to elaborate on their meanings, only their mathematical definitions. In

reality, they are quite simple to compute, and all that is required is the local sidereal time (LST) of

an event (which is the number gotten when the time of an event is combined with the terrestrial

longitude of the event, and Greenwich sidereal time taken from an ephemeris, along with a few

minor corrections), and a table of ascendants and MC's. Armed with this, a person can with just a

little practice calculate all of the personal sensitive points for an event, except the Moon's nodes, in

less than two minutes. Given the formulations, which follow shortly, the calculations can be placed

in a computing machine, and derived without the need for any tables.

ASTRONOMICAL CONSIDERATIONS

In the astronomy defining an event, there are great circles which divide the sphere of space into

logical parts. These great circles are called: meridian, horizon, prime vertical, celestial equator,

polar axis circle, ecliptic, and horizon. The Moon's nodes are a special case, where the plane of the

Moon's motion defines these where it cuts the ecliptic. All of the houses and sensitive points used

in astrology are a result of these circles or mathematical derivatives from these circles where these

circles or their derivatives cut the ecliptic. You may wish to use the notes and definitions at the end

of this article for a reference.

House systems use the great circles of astronomy as starting points for their definitions and also

for projection purposes. Having a good understanding of the astronomy of the horoscope does help

in being able to visualize the construction of the individual house systems. In this sense, the guides

and definitions at the end of this article are of help. Please note that certain house systems are

known by more than one name. While I have looked diligently to compile a list of all known and

described house systems in popular literature, I may have accidentally overlooked some. Do not

dwell too strongly on the word definitions which, admittedly, are short. It is the mathematics which

really define the procedure for calculating the various houses systems. I tried to derive the

mathematics clearly and directly, but I may have inadvertently made some errors. I would

appreciate any corrections and/or additions to this list to be sent to me thru NCGR. Also, I wish to

thank Gary Duncan, Neil Michelsen, and George Noonan for their assistance with this effort when it

was in its formative stages.

WRITTEN DEFINITIONS OF THE INDIVIDUAL HOUSE SYSTEMS

The Alcibitius Declination House System

(Sometimes spelled 'Alcabitius'). The diurnal arc of the ascendant is tri-sected and projected

by hour circles onto the ecliptic to form the house cusps.

The Alcibitius Semi-Arc House System

The diurnal arc of the ascendant is tri-sected and projected by vertical circles onto the

ecliptic to form the house cusps.

The Arcturan House System

The horizon circle is cut at thirty degree intervals starting at the east point of the horizon,

and these points are projected onto the ecliptic using longitude circles.

The Campanus House System

The prime vertical is cut at thirty degree intervals starting at the east point of the horizon,

and these points are projected onto the ecliptic using house circles.

The Classical House System

The diurnal arc of the ascendant is tri-sected and projected by hour circles onto the ecliptic

to form the house cusps, but five degrees is subtracted from the ASC to form the first house

cusp. This slight correction to the Alcibitius Declination house system described above was

used in antiquity to correct for observational effects due to atmospheric refraction close to

the horizon.

The Earth House System

Zero degrees of Libra is taken as the first house cusp and each house cusp is thirty degrees

farther along in the zodiac.

The Equal House System

The ascendant is taken as the cusp of the first house and each house is thirty degrees

further along in the zodiac. Note that in this house system the MC is not necessarily the

cusp of the tenth house.

The Horizontal House System

The horizon circle is cut at thirty degree intervals starting at the east point of the horizon,

and these points are projected onto the ecliptic using vertical circles.

The Koch House System

The diurnal arc of the ascendant is tri-sected and projected by ascendant arcs onto the

ecliptic to form the house cusps.

The MC House System

The midheaven (MC) is taken as the cusp of the tenth house and each house is thirty

degrees further along in the zodiac. Note that in this house system the ascendant (ASC) is

not necessarily the cusp of the first house.

The Meridian House System

The celestial equator circle is cut at thirty degree intervals starting at the Aries point, and

these points are projected onto the ecliptic using hour circles.

The Moon House System

The Moon is taken as the tenth house cusp and each house cusp is thirty degrees farther

along in the zodiac. Note that in this house system neither the ascendant nor the MC are

necessarily house cusps.

The Morinus House System

The celestial equator circle is cut at thirty degree intervals starting at the Aries point, and

these points are projected onto the ecliptic using longitude circles.

The Natural Graduation House System

A complicated mathematical variation of the Porphyry House System, as described on pp.

46- 47 in "New Waite's Compendium" by Colin Evans.

The Natural Hours House System

The times of sunrise and sunset are noted for the location and date of the horoscope. The

degrees of the ASC at Sun rise and Sun set give the degrees of the ascendant and

descendant respectively. The hemispheres between the ASC and DSC are divided into six

sectors, each representing two 'hours' of time. These sectors also give the house cusps.

Note that the ascendant and descendant are no longer tied together as a pair in this system,

and the MC may fall in any house.

The Octopos House System

The prime vertical is cut at forty-five degree intervals starting at the east point of the horizon,

and these points are projected onto the ecliptic using house circles. This produces eight

houses instead of twelve as by most other systems, and these are then numbered starting at

the descendant and going counter-clockwise, so that the house placed at the seventh house

is called the first house, and the house normally near the ninth house is the second house,

etc. For those persons with a fear of the twelfth house, this is the one to use.

The Placidian House System

The celestial equator circle is cut at thirty degree intervals starting at the Aries point, and

these points are projected onto the ecliptic using house circles. The original cusps are then

recalculated in a complicated adjustment cycle which continues until no further cuspal

movement is perceived.

The Porphyry House System

The arc between the MC and the ascendant is measured and tri-sected, with the ecliptic

locations becoming the eleventh and twelfth houses. Then the arc between the ASC and IC

is treated similarly for the cusps of the second and third houses. The ASC and MC are the

cusps of the first and tenth houses, and the other houses are opposite paired (e.g., the 8th

cusp is opposite the 2nd).

The Radiant House System

Same as the Horizontal House System.

Regiomontanus House System

The celestial equator circle is cut at thirty degree intervals starting at the Aries point, and

these points are projected onto the ecliptic using house circles. This is similar to the

Placidian house system, but without the complicated adjustment algorithm required by that

method.

The Solar House System

The position of the Sun is taken as the first house cusp and each house cusp is thirty

degrees farther along in the zodiac. Note that in this house system neither the ascendant

nor the MC are necessarily house cusps. This system is commonly used when the

ascendant and MC are not known.

The Sun House System

The Sun is taken as the fourth house cusp and each house cusp is thirty degrees farther

along in the zodiac. Note that in this house system neither the ascendant nor the MC are

necessarily house cusps.

The Topocentric House System

This is a very slight mathematical variation of the Placidian algorithm, which, supposedly,

allows for a more accurate calculation of the intermediate house cusps in the polar regions.

The Zariel House System

Identical to the Meridian House system, except in name.

CALCULATION CONVENTIONS

The following standard abbreviations are used in the mathematics which follow:

e represents the obliquity of the ecliptic

f represents the terrestrial latitude

ASC is the ascendant

MC is the MC

RAMC is the Right Ascension of the MC

F, G, J, K, and L are working terms, unimportant astronomically

+. -, x (or, *), ÷, = represent their normal arithmetic functions

SIN, COS, TAN, COT, etc. represent the trigonometric functions

For calculator purposes: COT = (1 ÷ TAN ) and vice-versa, etc.

ARCSIN, ARCCOS, ARCTAN, etc. represent the trig inverses

H11, etc. stands for the offset to compute the cusp of house eleven, etc.

C11, etc. stands for the value of the cusp of house eleven, etc.

Standard computer notation parenthesis nesting conventions are used throughout the

formulations. That is, three left parenthesis must be balanced by three right parenthesis.

Calculations are always performed within the inner parenthesis first, and then outward to the

outer parenthesis. Persons attempting the mathematics herein should refer to reasonable

reference books if they are unfamiliar with trigonometric procedures. Particularly, the

process of adjusting house cusp calculations for the correct trigonometric quadrant can be

somewhat tricky if not performed with care. House cusps which are over 360o or under 0o

should be converted to lie between 0o and 360o . That is, if you compute a house cusp as

being 372o then this should be changed to 12 Aries. Add 360o to any negative values or

results. House cusps with values between 0o and 29.99o lie in Aries; between 30o and

59.99o in Taurus; between 60o and 89.99o degrees in Gemini, and so forth around the

zodiac and through the signs.

PRELIMINARY CALCULATIONS AND THE PERSONAL SENSITIVE POINTS

1. The RAMC (the right ascension of the midheaven) is computed from Local Sidereal Time (LST)

by converting time units to degree units. An example of this calculation follows:

Given an LST of 12H 15M 00S, then first convert this to a decimal form of time, or 12.25

hours. 12.25 x 15 = 183.75o which is the RAMC.

Given an LST of 6H 27M 14S, then convert this to a decimal form of time, or 6.453889

hours. 6.453889 x 15 = 96.808333o .

2. MC = ARCTAN ( TAN (RAMC) ÷ COS e )

3. ASC = ARCCOT (- ( (TAN f x SIN e) + (SIN RAMC x COS e) ) ÷ COS RAMC)

4. EQA = ARCCOT ( - ( TAN RAMC x COS e) )

5. VTX = ARCCOT (- ( (COT f x SIN e) - (SIN RAMC x COS e) ) ÷ COS RAMC)

6. CAS = ARCCOT (- ( (COT f x SIN e) + (SIN RAMC x COS e) ) ÷ COS RAMC)

7. PAS = ARCCOT ( ( (TAN f x SIN e) - (SIN RAMC x COS e) ) ÷ COS RAMC)

8. ARI The Aries Point is always zero of Aries.

9. The declination of any point on the ecliptic can be calculated from:

declination = ARCSIN ( SIN (zodiacal longitude of point ) x SIN e)

10. The obliquity of the ecliptic, for any date in modern times, is calculated by:

e = 23o 27' 08.26" - 46.845" x T - .0059" x T2 + .00181" x T3

where T is in fractions of a century starting from Jan 1, 1900

THE ALCIBITIUS DECLINATION HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the number of zodiacal degrees between the ASC and MC:

L = ASC - MC

3. Determine the Diurnal and Nocturnal Semi-arcs:

D = ARCTAN ( TAN L x COS e )

P = 180o - D

4. Determine these intermediate working values:

F = D ÷ 3 J = P ÷ 3

G = F x 2 K = J x 2

5. Compute the house cusp intervals as follows:

H11 = ARCTAN ( TAN F ÷ COS e )

H12 = ARCTAN ( TAN G ÷ COS e )

H2 = ARCTAN ( TAN K ÷ COS e )

H3 = ARCTAN ( TAN J ÷ COS e )

6. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = MC + H11 C5 = 180o + C11

C12 = MC + H12 C6 = 180o + C12

C1 = ASC C7 = 180o + C1

C2 = MC + H2 C8 = 180o + C2

C3 = MC + H3 C9 = 180o + C3

THE ALCIBITIUS SEMI-ARC HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the right ascension of the ASC (RASC):

RASC = ARCTAN (TAN (ASC) x COS e )

3. Compute the RAMC to RASC interval:

T = RASC - RAMC (Add 360o if negative)

4. Compute the trisections of the diurnal and nocturnal arcs:

D = T ÷ 3

P = (T - 180o ) ÷ 3

5. Compute the RA of each of the house cusps:

RA11 = RAMC + D

RA12 = RA11 + D

RA2 = RASC + P

RA3 = RA2 + P

6. Compute the house cusp intervals as follows:

H11 = ARCCOT ( - ( ( TAN f x COS e ) + (SIN RA11 x COS e) ) ÷ COS RA11)

H12 = ARCCOT ( - ( ( TAN f x COS e ) + (SIN RA12 x COS e) ) ÷ COS RA12)

H2 = ARCCOT ( - ( ( TAN f x COS e ) + (SIN RA2 x COS e) ) ÷ COS RA2)

H3 = ARCCOT ( - ( ( TAN f x COS e ) + (SIN RA3 x COS e) ) ÷ COS RA3)

7. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = MC + H11 C5 = 180o + C11

C12 = MC + H12 C6 = 180o + C12

C1 = ASC C7 = 180o + C1

C2 = ASC + H2 C8 = 180o + C2

C3 = ASC + H3 C9 = 180o + C3

THE ARCTURAN HOUSE SYSTEM FORMULATION

This house system works very well in polar areas, but gives erratic and unpredictable results in the

tropical regions.

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the following constants for later usage:

Compute the Decl. of the ASC: D = ARCSIN (SIN ASC x SIN e )

Oblique Ascension of the East Point: J = RAMC + 90o

G, the angle between the ecliptic and the horizon:

G = ARCCOS ( ( SIN f x COS e ) + (COS f x SIN e x COS J) )

K, the arc in degrees from the East Point to the ASC:

K = ARCSIN ( SIN D ÷ COS f )

3. Determine the house cusp intervals:

H10 = K + 90o H1 = K + 0o (or, 180o)

H11 = K + 60o H2 = K + 150o

H12 = K + 30o H3 = K + 120o

4. Determine the ecliptic to cusp angle:

R10 = ARCTAN ( COS G x TAN H10 )

R11 = ARCTAN ( COS G x TAN H11 )

R12 = ARCTAN ( COS G x TAN H12 )

R1 = ARCTAN ( COS G x TAN H1 )

R2 = ARCTAN ( COS G x TAN H2 )

R3 = ARCTAN ( COS G x TAN H3 )

5. Compute the individual house cusps as follows:

C10 = ASC - R10 C4 = 180o + C10

C11 = ASC - R11 C5 = 180o + C11

C12 = ASC - R12 C6 = 180o + C12

C1 = ASC - R1 C7 = 180o + C1

C2 = ASC - R2 C8 = 180o + C2

C3 = ASC - R3 C9 = 180o + C3

The R value when added to the ASC may give you the opposite side house cusp as a result. Add

180o to your answer if this occurs.

THE CAMPANUS HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the following house cusp intervals:

H11 = 30o H2 = 120o

H12 = 60o H3 = 150o

3. Compute an intermediate number:

J11 = ARCCOT ( COS f x TAN H11 ) J2 = ARCCOT ( COS f x TAN H2 )

J12 = ARCCOT ( COS f x TAN H12 ) J3 = ARCCOT ( COS f x TAN H3 )

4. Compute the Prime Vertical interval:

F11 = RAMC + 90o - J11 F2 = RAMC + 90o - J2

F12 = RAMC + 90o - J12 F3 = RAMC + 90o - J3

5. Compute the house cusp positions as follows:

P11 = ARCSIN ( SIN H11 x SIN f ) P2 = ARCSIN ( SIN H2 x SIN f )

P12 = ARCSIN ( SIN H12 x SIN f ) P3 = ARCSIN ( SIN H3 x SIN f )

6. Compute the associate angles as follows:

M11 = ARCTAN ( TAN P11 ÷ COS F11 ) M12 = ARCTAN ( TAN P12 ÷ COS F12 )

M2 = ARCTAN ( TAN P2 ÷ COS F2 ) M3 = ARCTAN ( TAN P3 ÷ COS F3 )

7. Compute the ecliptic intervals:

R11 = ARCTAN ( ( TAN F11 x COS M11 ) ÷ COS ( M11 + e) )

R12 = ARCTAN ( ( TAN F12 x COS M12 ) ÷ COS ( M12 + e) )

R2 = ARCTAN ( ( TAN F2 x COS M2 ) ÷ COS ( M2 + e) )

R3 = ARCTAN ( ( TAN F3 x COS M3 ) ÷ COS ( M3 + e) )

8. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = MC + R11 C5 = 180o + C11

C12 = MC + R12 C6 = 180o + C12

C1 = ASC C7 = 180o + C1

C2 = MC + R2 C8 = 180o + C2

C3 = MC + R3 C9 = 180o + C3

THE CLASSICAL HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the First Cusp Locus Longitude:

L1 = ASC - 5o

3. Determine Right Ascension of the First Locus:

A1 = RASC = ARCTAN (TAN L1 x COS e)

4. Determine the local hour angle of the MC:

t = ARCCOS ( - ( ( TAN f x TAN e ) x SIN (A1 ) )

5. Tri-sect the hour angle arcs:

g = t ÷ 3

h = ( 180o - t ) ÷ 3

6. Determine the right ascension of the tenth house cusp:

A10 = A1 - t

7. Compute the house cusp positions as follows:

A11 = A10 + g

A12 = A11 + g

A2 = A1 + h

A3 = A2 + h

8. Compute the individual house cusps as follows:

C10 = ARCTAN (TAN A10 ÷ COS e) C4 = 180o + C10

C11 = ARCTAN (TAN A11 ÷ COS e) C5 = 180o + C11

C12 = ARCTAN (TAN A12 ÷ COS e) C6 = 180o + C12

C1 = ARCTAN (TAN A1 ÷ COS e) C7 = 180o + C1

C2 = ARCTAN (TAN A2 ÷ COS e) C8 = 180o + C2

C3 = ARCTAN (TAN A3 ÷ COS e) C9 = 180o + C3

THE HORIZONTAL HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Compute the P point:

P = ARCTAN ( TAN RAMC x COS e )

3. Compute the angle between the horizon and the ecliptic:

G = ARCSIN ( COS f x SIN (RAMC + 90o ) ) ÷ SIN ASC )

4. Compute:

J = ASC - P

K = ARCTAN ( TAN J x COS G )

5. Assign the house cusp intervals:

H10 = 90o H2 = 0o

H11 = 60o H2 = -30o

H12 = 30o H3 = -60o

6. Compute an intermediate result:

M10 = H10 + K M1 = H1 + K

M11 = H11 + K M2 = H2 + K

M12 = H12 + K M3 = H3 + K

7. Compute the next intermediate result as follows:

R10 = ARCTAN ( TAN M10 ÷ COS G )

R11 = ARCTAN ( TAN M11 ÷ COS G )

R12 = ARCTAN ( TAN M12 ÷ COS G )

R1 = ARCTAN ( TAN M1 ÷ COS G )

R2 = ARCTAN ( TAN M2 ÷ COS G )

R3 = ARCTAN ( TAN M3 ÷ COS G )

8. Compute the individual house cusps as follows:

C10 = ASC - R10 C4 = 180o + C10

C11 = ASC - R11 C5 = 180o + C11

C12 = ASC - R12 C6 = 180o + C12

C1 = ASC - R1 C7 = 180o + C1

C2 = ASC - R2 C8 = 180o + C2

C3 = ASC - R3 C9 = 180o + C3

THE KOCH HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner. Use the MC as the cusp of the tenth

house.

2. Calculate the declination of the MC:

D = ARCSIN ( SIN MC x SIN e )

3. Calculate the ascensional difference of the MC:

J = ARCSIN ( TAN D x TAN f )

4. Calculate the oblique ascension of the MC:

OAMC = RAMC - J

5. Calculate the general house cusp displacement interval:

DX = ( ( RAMC + 90o ) - OAMC ) ÷ 3 )

(This should be a positive number; add 360o to any negative DX number)

6. Compute the house cusp positions as follows:

H11 = ( OAMC + DX - 90o )

H12 = H11 + DX

H1 = H12 + DX

H2 = H1 + DX

H3 = H2 + DX

7. Calculate the individual house cusps:

C11 = ARCCOT (- ( (TAN f x SIN e) + (SIN H11 x COS e) ) ÷ COS H11)

C12 = ARCCOT (- ( (TAN f x SIN e) + (SIN H12 x COS e) ) ÷ COS H12)

C1 = ARCCOT (- ( (TAN f x SIN e) + (SIN H1 x COS e) ) ÷ COS H1)

C2 = ARCCOT (- ( (TAN f x SIN e) + (SIN H2 x COS e) ) ÷ COS H2)

C3 = ARCCOT (- ( (TAN f x SIN e) + (SIN H3 x COS e) ) ÷ COS H3)

8. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = C11 C5 = 180o + C11

C12 = C12 C6 = 180o + C12

C1 = C1 C7 = 180o + C1

C2 = C2 C8 = 180o + C2

C3 = C3 C9 = 180o + C3

THE MERIDIAN HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the house cusp intervals:

H10 = 0o H1 = 90o

H11 = 30o H2 = 120o

H12 = 60o H3 = 150o

3. Determine the ecliptic interval point for each cusp:

F10 = RAMC + H10

F11 = RAMC + H11

F12 = RAMC + H12

F1 = RAMC + H1

F2 = RAMC + H2

F3 = RAMC + H3

4. Compute the house cusps:

C10 = ARCTAN ( TAN F10 ÷ COS e )

C11 = ARCTAN ( TAN F11 ÷ COS e )

C12 = ARCTAN ( TAN F12 ÷ COS e )

C1 = ARCTAN ( TAN F1 ÷ COS e )

C2 = ARCTAN ( TAN F2 ÷ COS e )

C3 = ARCTAN ( TAN F3 ÷ COS e )

5. Determine the individual house cusps as follows:

C10 = C10 C4 = 180o + C10

C11 = C11 C5 = 180o + C11

C12 = C12 C6 = 180o + C12

C1 = C1 C7 = 180o + C1

C2 = C2 C8 = 180o + C2

C3 = C3 C9 = 180o + C3

THE MORINUS HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the house cusp intervals:

H10 = 0o H1 = 90o

H11 = 30o H2 = 120o

H12 = 60o H3 = 150o

3. Determine the ecliptic interval point for each cusp:

F10 = RAMC + H10

F11 = RAMC + H11

F12 = RAMC + H12

F1 = RAMC + H1

F2 = RAMC + H2

F3 = RAMC + H3

4. Compute the house cusps:

C10 = ARCTAN ( TAN F10 x COS e )

C11 = ARCTAN ( TAN F11 x COS e )

C12 = ARCTAN ( TAN F12 x COS e )

C1 = ARCTAN ( TAN F1 x COS e )

C2 = ARCTAN ( TAN F2 x COS e )

C3 = ARCTAN ( TAN F3 x COS e )

5. Determine the individual house cusps as follows:

C10 = C10 C4 = 180o + C10

C11 = C11 C5 = 180o + C11

C12 = C12 C6 = 180o + C12

C1 = C1 C7 = 180o + C1

C2 = C2 C8 = 180o + C2

C3 = C3 C9 = 180o + C3

THE NATURAL HOURS HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner.

2. Determine the times of sunrise and sunset for the location of the chart. You can consult your

local newspaper, tables in the "Nautical Almanac", or using a table of ascendants and MC's

determine what clock time during the day the Sun's degree would be conjunct the ASC and the

DSC. Finally, if you wish to calculate that information astronomically, you can also do that, but I am

not providing those formulas or methods here because they can become too tricky, especially in

polar latitudes where the Sun doesn't rise or set during every 24 hour period.

3. Suppose you determine that there are twelve hours and thirty-six minutes of daylight, and, thus

eleven hours and twenty-four minutes of night for the day of the event at your event location. Then

convert these hours and minutes to arcs of a circle as follows:

12H 36M = 12.6 Hrs ( ( 12.6 ÷ 24 ) x 360o ) = 189o of daylight arc

360 - 189 = 171o of night time arc

4. Compute the cuspal increments:

D = 189o ÷ 6 = 31o 30' for the daylight increment

N = 171o ÷ 6 = 28o 30' for the night time increment

5. Compute the house cusps:

C12 = ASC - D

C11 = C12 - D

C10 = C11 - D

C9 = C10 - D

C8 = C9 - D

C7 = C8 - D

C1 = ASC

C2 = ASC + N

C3 = C2 + N

C4 = C3 + N

C5 = C4 + N

C6 = C5 + N

THE PLACIDIAN HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner. Use the MC as the cusp of the tenth

house and the ASC as the cusp of the first house. This is a very fast converging algorithm adapted

from a work by M. Vijayaraghavulu.

2. Determine the following house cusp intervals:

H11 = RAMC + 30o H2 = RAMC + 120o

H12 = RAMC + 60o H3 = RAMC + 150o

3. Set the Semi-arc ratios:

F11 = 1 ÷ 3 F2 = 2 ÷ 3

F12 = 2 ÷ 3 F3 = 1 ÷ 3

4. Compute the cuspal declinations:

D11 = ARCSIN ( SIN e x SIN H11 ) D2 = ARCSIN ( SIN e x SIN H2 )

D12 = ARCSIN ( SIN e x SIN H12 ) D3 = ARCSIN ( SIN e x SIN H3 )

5. Compute the first intermediate values:

A11 = F11 x ( ARCSIN ( TAN f x TAN D11 ) )

A12 = F12 x ( ARCSIN ( TAN f x TAN D12 ) )

A2 = F2 x ( ARCSIN ( TAN f x TAN D2 ) )

A3 = F3 x ( ARCSIN ( TAN f x TAN D3) )

6. Compute the house cusp positions as follows:

M11 = ARCTAN ( SIN A11 ÷ ( COS H11 x TAN D11) )

M12 = ARCTAN ( SIN A12 ÷ ( COS H12 x TAN D12) )

M2 = ARCTAN ( SIN A2 ÷ ( COS H2 x TAN D2) )

M3 = ARCTAN ( SIN A3 ÷ ( COS H3 x TAN D3) )

7. Compute the intermediate house cusps:

R11 = ARCTAN ( ( TAN H11 x COS M11 ) ÷ COS ( M11 + e) )

R12 = ARCTAN ( ( TAN H12 x COS M12 ) ÷ COS ( M12 + e) )

R2 = ARCTAN ( ( TAN H2 x COS M2 ) ÷ COS ( M2 + e) )

R3 = ARCTAN ( ( TAN H3 x COS M3 ) ÷ COS ( M3 + e) )

8. Substitute: D11 = R11; D12 = R12; D2 = R2; and D3 = R3. Then repeat steps 5 thru 8 again.

Substitute the R's for the D's a third time and repeat steps 5 thru 8. The answer for R on the third

try is the cusp you desire.

9. Compute the individual house cusps as follows:

C11 = R11 C5 = 180o + C11

C12 = R12 C6 = 180o + C12

C2 = R2 C8 = 180o + C2

C3 = R3 C9 = 180o + C3

THE REGIOMONTANUS HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner. Use the MC as the cusp of the tenth

house and the ASC as the cusp of the first house.

2. Determine the following house cusp intervals:

H11 = 30o H2 = 120o

H12 = 60o H3 = 150o

3. Set the equatorial intervals:

F11 = RAMC + H11 F2 = RAMC + H2

F12 = RAMC + H12 F3 = RAMC + H3

4. Compute the house poles:

P11 = ARCTAN ( TAN f x SIN H11 ) P2 = ARCTAN ( SIN f x SIN H2 )

P12 = ARCTAN ( TAN f x SIN H12 ) P3 = ARCTAN ( SIN f x SIN H3 )

5. Compute the first intermediate values:

M11 = ARCTAN ( TAN P11 ÷ COS F11 )

M12 = ARCTAN ( TAN P12 ÷ COS F12 )

M2 = ARCTAN ( TAN P2 ÷ COS F2 )

M3 = ARCTAN ( TAN P3 ÷ COS F3)

6. Compute the intermediate house cusps:

R11 = ARCTAN ( ( TAN F11 x COS M11 ) ÷ COS ( M11 + e) )

R12 = ARCTAN ( ( TAN F12 x COS M12 ) ÷ COS ( M12 + e) )

R2 = ARCTAN ( ( TAN F2 x COS M2 ) ÷ COS ( M2 + e) )

R3 = ARCTAN ( ( TAN F3 x COS M3 ) ÷ COS ( M3 + e) )

7. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = R11 C5 = 180o + C11

C12 = R12 C6 = 180o + C12

C1 = ASC C7 = 180o + C1

C2 = R2 C8 = 180o + C2

C3 = R3 C9 = 180o + C3

THE TOPOCENTRIC HOUSE SYSTEM FORMULATION

1. Compute the RAMC, MC, and ASC in the normal manner. Use the MC as the cusp of the tenth

house and the ASC as the cusp of the first house.

2. Determine the following house cusp intervals:

H11 = RAMC + 30o H2 = RAMC + 120o

H12 = RAMC + 60o H3 = RAMC + 150o

3. Set the Semi-arc ratios:

P11 = ARCTAN ( TAN f ÷ 3 )

P12 = ARCTAN ( 2 x ( TAN f ÷ 3 ) )

P2 = ARCTAN ( 2 x ( TAN f ÷ 3 ) )

P3 = ARCTAN ( TAN f ÷ 3 )

4. Compute the intermediate angle M:

M11 = ARCTAN (TAN P11 ÷ COS H11 )

M12 = ARCTAN (TAN P12 ÷ COS H12 )

M2 = ARCTAN (TAN P2 ÷ COS H2 )

M3 = ARCTAN (TAN P3 ÷ COS H3 )

5. Compute the intermediate house cusps:

R11 = ARCTAN ( ( TAN H11 x COS M11 ) ÷ COS ( M11 + e) )

R12 = ARCTAN ( ( TAN H12 x COS M12 ) ÷ COS ( M12 + e) )

R2 = ARCTAN ( ( TAN H2 x COS M2 ) ÷ COS ( M2 + e) )

R3 = ARCTAN ( ( TAN H3 x COS M3 ) ÷ COS ( M3 + e) )

6. Compute the individual house cusps as follows:

C10 = MC C4 = 180o + C10

C11 = R11 C5 = 180o + C11

C12 = R12 C6 = 180o + C12

C1 = ASC C7 = 180o + C1

C2 = R2 C8 = 180o + C2

C3 = R3 C9 = 180o + C3

A variation of this system, devised by Alexander Marr of Germany to give 8 cusps instead of the 12

normally used, requires changing steps 2 and 3 above as follows:

Substitute: Ha = RAMC + 45o Hb = RAMC + 135o

Pa = Pb = ARCTAN ( TAN f ÷ 2 ) & then recalculate

This gives a cusp with mid-point like qualities which Mr. Marr claims produces interesting

correspondences. Your investigation is recommended.

Definition of Terms and Abbreviations

BODY: A planet, a star, or some similar object which exists in space and time.

CELESTIAL EQUATOR: A great circle denoted by an extension of the Earth's equator infinitely

projected into space. This is the circle along which the measurement of right ascension is made.

CELESTIAL SPHERE: That sphere which would be formed if one were to infinitely extend the

'sphere' of Earth outward into space.

CO-EQUATOR: The mirror image of the Earth's equator. The equator mathematically associated

with the co-latitude of a place on Earth.

CO-LATITUDE: The number obtained when the terrestrial latitude is subtracted from ninety

degrees. For the city of Philadelphia, located at forty degrees north terrestrial latitude, the colatitude,

or angular distance of Philadelphia from the Earth's North Pole, is fifty degrees.

CO-POLAR AXIS CIRCLE: The great circle formed when the mathematics used to derive the polar

axis circle is mirrored from the Earth's poles, rather than from the Earth's equator.

ECLIPTIC: That great circle of the celestial sphere which the Sun traces, when seen from the

Earth, in its yearly travels against the backdrop of the sky.

ECLIPTIC PLANE or SYSTEM: The mathematical plane which contains the Solar System, with the

Sun as its center and its planets at the center of their motions. A sphere of space using the ecliptic

as its equator.

EQUATORIAL PLANE: The mathematical plane represented by infinitely extending the Earth's

equator into space.

EQUATORIAL SYSTEM: A sphere of space using the celestial equator as its main central circle or

equator.

GREAT CIRCLE: A circle contained within the celestial sphere which has as its center the center

point of the celestial sphere.

HORIZON: A great circle, for which there are actually four associated terms:Visible, Rational,

Sensible, and Celestial. In the way that we use these terms, the Visible Horizon is our view of

where the earth and the sky meet off in the distance from where we stand on or near the earth. The

Celestial Horizon is the horizon we use mathematically as our starting point to calculate houses

and sensitive points, and it is the visible horizon as if that horizon were starting at the center of the

earth (as opposed to where we are located on or near the surface of the earth) and was extended

infinitely into space.

HORIZON PLANE or SYSTEM: The plane which contains the horizon. The same as the celestial

horizon. A sphere of space, with the Celestial Horizon serving as its equator. See also: Horizon.

HOUR CIRCLE: A great circle which is perpendicular to the Celestial Equator and which passes

through a particular body in space.

HOUSE CIRCLE: A great circle which has as its poles the North and South points of the Horizon,

and which is perpendicular to the Prime Vertical.

LOCAL SIDEREAL TIME: The time calculated for a horoscope when a time of event is added to

the longitude correction, the time zone correction, the acceleration, the delta T correction, and the

sidereal time from an ephemeris.

LONGITUDE CIRCLE: A great circle which starts at the pole of the ecliptic and travels around the

Celestial Sphere perpendicular to the ecliptic. It is like a circle of longitude, but in the ecliptic

system, as opposed to on a globe of the earth.

MERIDIAN: A great circle of the Horizon system which passes through the Zenith, the nadir, and

the North and South points of the horizon.

NADIR: The South Pole of the horizon system. Opposed to the Zenith.

OBLIQUITY: The angle in space formed between the ecliptic and the celestial equator. Presently it

is about twenty-three and a half degrees and decreasing slowly with time.

PERPENDICULAR: Ninety degrees. Circles which meet at ninety degree angles.

POLAR AXIS CIRCLE: A great circle which passes through the North and South Poles of the Earth

and the East and West points of the horizon.

POLE: When describing three or four dimensional space (using time as a fourth dimension) a pole

is a mathematical point that is ninety degrees everywhere from a circle. For instance, the earth's

North or South Poles are ninety degrees from all points on the earth's equator.

PRIME VERTICAL: A great circle which passes through the Zenith, the Nadir, and the East and

West points of the horizon. It is ninety degrees from the meridian, and vice-versa.

VERTICAL CIRCLE: A great circle which is perpendicular to the horizon and passes through the

Zenith and the Nadir.

ZENITH: The North Pole of the horizon system. The point in the horizon system which is over your

head. Opposed to the nadir.

ZODIAC: A small portion of the celestial sphere which is about eight degrees on either side of the

ecliptic circle.