Table of Arctic Housesby Ruud van der Putten

DesciptionThe Table of Arctic Houses lists all house cusps according to the system of Placidus, from 60° to 89° northern latitude. The house cusps are calculated for every degree of latitude and every degree of RAMC (Right Ascension of the Medium Coeli.)

The table is constructed in the same way as a traditional table of houses. There is a smaller table for every four minutes of sidereal time. In the header of each of the smaller tables can be found the values of the RAMC, the LST (Local Sidereal Time) and the position of the midheaven. Also there is the value of the obliquity of the ecliptic, which is the same for the entire table. In the row corresponding to a particular geographic latitude, the ascendant and the other cusps are listed from left to right in the order of the zodiac, starting from the midheaven.

Many columns in the table are partly empty: this is because the shape of the horoscope has changed, not because something is missing. The table lists all Placidus cusps, the empty parts of the columns are an artifact of choosing this form of tabulation.

Calculating cusp positions for antarctic latitudes can be done in the same way as calculating southern-hemisphere positions using a normal table of houses: add or subtract twelve hours to or from the sidereal time and take the opposition points of the positions found in the table.

Placidus cusps in the arctic and antarcticThere are three types of cusps possible in an arctic chart, as opposed to one type in a normal chart:

1. normal cusps2. circumpolar cusps3. retrograde cusps

The normal cuspsThese cusps are not retrograde and not circumpolar. They are like any cusp in a normal chart.

The circumpolar cuspsThese cusps are formed in the circumpolar sky, on the parts of the ecliptic that remain above or below the horizon at all times. These ecliptic degrees never cross the horizon and are only seen to go around the celestial north or south pole, hence 'circumpolar'. Circumpolar cusps are in italic face in the table. The midheaven and the intermediate cusps can be circumpolar; the ascendant is never circumpolar.

The retrograde cuspsA retrograde cusp is formed when the ecliptic is tilted the other way as it crosses the mundane system of semi-arcs.This means two things:

– the cusp runs in retrograde motion through the zodiac– the order of the houses is inverted around the cusp

The ascendant is one of the cusps that can be retrograde. If the ascendant is retrograde, it is still the point where the ecliptic is rising over the eastern horizon, but the later degrees of the rising sign will do so first and the earlier degrees will follow later. It also means that the 1st house (which is below the horizon) is above the ascendant in the horoscope drawing and the 12th house (which is above the horizon) is below the ascendant in the horoscope drawing.

In a chart for this situation on www.astro.com you can see the text "below horizon" in the upper part of the chart and the text "above horizon" in the lower part of the chart. Note that www.astro.com defaults to Porphyry houses for a chart beyond the polar circles.The intermediate cusps can be retrograde aswell; the midheaven is never retrograde.

Circumpolar positions on the eclipticFor convenience, I also created a table showing what parts of the ecliptic are circumpolar at a given geographic latitude. It lists the boundaries of the circumpolar sky on the ecliptic for every five arcminutes of latitude. You can find this table further down in this document. The table of circumpolar positions on the ecliptic works as follows:

On latitudes below 66°33'40", no parts of the ecliptic are circumpolar.On the geographic latitude of 66°35', for example, the table returns 27°34'34"`f – 2°25'26"ag.This means that on 66°35' northern latitude, 27°34'34"f until 2°25'26"g is circumpolar below the horizon and 27°34'34"` until 2°25'26"a is circumpolar above the horizon. On 66°35' southern latitude, this is the other way around.

The boundaries of the circumpolar sky on the ecliptic and the exact positions of the midheaven, the ascendant and the other cusps, are dependent on the obliquity of the ecliptic. The obliquity is the space angle between the planes of the ecliptic and the celestial equator and changes only very gradually. In the 20th century it shifted less than one arcminute. The choice for 23°26'20" is rather arbitrary: it is a rounded value for the obliquity for the time period around the start of the 21st century.

The circumpolar skyThe circumpolar sky is the single most important feature of an arctic chart that sets it apart from charts in the temperate and tropical zones of the earth. The circumpolar sky exists at all geographic latitudes (except right on the equator,) but only at the higher latitudes does it grow so large that planets and parts of the ecliptic can be found in it.

A planet's actual position in the sky is usually not the same as its position on the ecliptic. Most planets are not located right on the ecliptic, but have a latitude to the ecliptic. Whether a planet or point is circumpolar or not depends on its declination, not its zodiacal position. It is therefore quite possible that a planet is located in the circumpolar sky while its position in the zodiac is not circumpolar, or the other way around.

Planets and points in the circumpolar sky never cross the horizon in the course of their diurnal cycle. This means that they remain invisible below the horizon for the whole cycle, or they remain suspended in the air, always located above the horizon.One way or another, this should be an important factor in interpretation. How to interpret this feature exactly is a matter for research.

The most important thing to keep in mind is that Placidus houses are derived from Ptolemaic primary directions. The primary directions are part of the mundane system and are not formed in relation to the ecliptic. They are formed on the parallel circles of declination and their divisions: the semi-arcs. These circles and arcs run parallel to the celestial equator. The ecliptic runs through this two-dimensional mundane system of semi-arcs at an angle (the obliquity) and as a result, distorts and apparently destroys the beautiful symmetry. This distortion becomes most evident in the polar regions of the earth.

Therefore, although some arctic charts can look very strange indeed and may seem unusable, it is important to keep in mind that the underlying structure and symmetry of the mundane system of semi-arcs remains the same at all geographic latitudes and through all 24 hours of sidereal time. The only thing that is different at different latitudes is the size of the circumpolar sky.

In order to clarify some of the features unique to an arctic chart, a few examples follow below.

Examples

Example 1

Local sidereal time is 4:40 hours. Geographic latitude is 72° north.This is a fairly normal house division. Almost all house divisions from 0:00 untill 12:00 hours are like this. The circumpolar sky runs 20°58'30"_e – 9°01'30"bh. In the chart drawing, the circumpolar parts of the ecliptic are the grey areas.

We get the following division:Midheaven 11°32'02" ` cpcusp 11 7°35'29" b cpcusp 12 1°56'01" cAscendant 20°41'25" ccusp 2 5°45'44" dcusp 3 29°26'45" d

Cusp 11 is in italic face in the table and therefore circumpolar. Note that the midheaven is also circumpolar.Planets and points that are located in the circumpolar sky do not have a direct connection to the horizon and this affects interpretation.

Example 2

Local sidereal time is 22:48 hours. Geographic latitude is 75° north.This is an example of a chart with one retrograde cusp. Cusp 11 seems to have multiplied itself at the edge of the circumpolar sky. The same applies to cusp 5, of course.The circumpolar sky runs 10°35'33"_e – 19°24'27"bh.

We get the following division:Midheaven 10°29'56" icusp 11 2°10'45" _cusp 11 8°22'38" _ Rxcusp 11 14°27'42" _ cpcusp 12 11°02'07" a cpAscendant 21°37'28" bcusp 2 23°24'48" bcusp 3 27°49'07" b

The ascendant is so close to the circumpolar sky that the house division is extremely skewed, with a 1st and 7th house of less than two degrees. However, more extreme horoscopes are possible with houses of only a few arcseconds wide. This happens just before or after the moment the midheaven conjoins the ascendant right at the edge of the circumpolar sky.

As can be seen in the table, retrograde cusps that are part of these triple cusps, are short-lived and very local. This situation can occur briefly when an intermediate cusp enters or leaves the circumpolar sky.

Example 3

Local sidereal time is 23:20 hours. Geographic latitude is 68° south. (Looking up 11:20 hours LST and taking oppostion points of the cusps for 68° north.)This is an example of an antarctic chart, also with one retrograde cusp. Here cusps 3 and 9 are on the edge of the circumpolar sky. The circumpolar sky runs 10°21'00"`f – 19°39'00"ag.

We get the following division:Midheaven 19°07'16" icusp 11 15°59'00" ^cusp 12 1°12'00" _Ascendant 10°43'39" _cusp 2 4°11'51" `cusp 3 18°27'58" a cpcusp 3 20°41'00" a Rxcusp 3 21°11'35" a

A planet at 21°00'g is technically in the 8th house, but the cusps are so close together in this horoscope that the planet is 'before the cusp' and it will primarily work out in the 9th house. Therefore, the triple cusps of the 3rd and 9th houses in this chart, can be interpreted as one cusp with 2½° width.

Example 4

Local sidereal time is 18:04 hours. Geographic latitude is 67° north.This is an example with a retrograde ascendant.The circumpolar sky runs 19°12'17"`f – 10°47'43"ag.

We get the following division:Midheaven 0°55'03" g cpAscendant 18°31'31" h Rx

At this latitude the ascendant moves very fast through the fast rising signs in retrograde motion. On 18:00h sidereal time it was at 0°00'^. In this example, four minutes later, the ascendant already moved more than 41 degrees. In the same time the midheaven moved less than one degree.

When the midheaven is below the horizon, it is always circumpolar and at the same time the ascendant always runs through the fast rising signs in retrograde motion.

There are no intermediate cusps.The midheaven is circumpolar cusp 1 and the ascendant is retrograde cusp 1.Similarly, the IC is circumpolar cusp 7 and the descendant is retrograde cusp 7.

All this means that the ecliptic is very close to the horizon all around: the zodiac is lying flat on the horizon. Each degree of the ecliptic is about to rise or set, or has just done so. And also the circumpolar degrees are not far enough away from the meridian to reach the next house.

Example 5

Local sidereal time is 18:40 hours. Geographic latitude is 77° north.This is another example with a retrograde ascendant, but more northerly.The circumpolar sky runs 4°26'21"_e – 25°33'39"bh.

We get the following division:Midheaven 9°11'22" g cpAscendant 18°02'05" i Rx

cusp 12 3°27'54" _ Rxcusp 12 12°26'42" _ cp

At this latitude the ascendant moves much slower through the fast rising signs.

The midheaven is a cusp between the circumpolar 6th and 1st houses when it is below the horizon. The mundane sectors of the 9th and 10th houses are actually quite nearby on the other side of the southern horizon. The ecliptic just doesn't run through these sectors.

The retrograde cusps 6 and 12 are of a different type than the short-lived ones in examples 2 and 3. These retrograde cusps can be followed all the way to the pole.

This house division is very close to the chart of Marie "The Snow Baby" Peary.

Example 6

Local sidereal time is 20:08 hours. Geographic latitude is 70° north.This is an example of how strange some arctic charts can be.The circumpolar sky runs 29°17'56"_e – 0°42'04"bh.

We get the following division:Midheaven 29°49'34" g cpAscendant 0°42'44" h Rx

cusp 12 0°43'36" h Rxcusp 11 0°49'39" h Rx

cusp 11 7°43'16" hcusp 12 10°39'12" ^cusp 12 28°16'17" _ Rx

cusp 12 3°59'43" ` cp

In this horoscope cusps straddle the edges of the circumpolar sky on both sides. Cusps of 6 on one side and the midheaven-ascendant near-conjunction (both are cusps of 1) on the other. The effect is bewildering.

Planets in the first degrees of Aquarius are so close to the ascendant that they will probably work out primarily in the first house, but they will possibly have some 12th-house and 11th-house connotations aswell.Interestingly, there is a section of the 10th house totally disconnected from the midheaven. Planets in this strange 10th house are so close to either of the cusps 11 that they will primarily work out in the 11th house.

A few minutes after the time of this horoscope, the midheaven rises over the horizon, at the same time conjoining the ascendant right in the south point of the horizon. At this point in time the retrograde cusps of the 11th and 12th in Aquarius have merged with the ascendant and the ascendant jumps 180° to Leo. As a result, the midheaven has left the circumpolar sky, becoming cusp 10 again and joins the rest of the 10th house that already formed earlier.

Geographic Circumpolar sky 60° – 70° Geographic Circumpolar sky 70° – 80° Geographic Circumpolar sky 80° – 90°Latitude Obliquity = 23°26'20" Latitude Obliquity = 23°26'20" Latitude Obliquity = 23°26'20"60º00' — — 70º00' 29°17'56" _e 0°42'04"bh 80º00' 25°53'03" ^d 4°06'57"ci — — 70º05' 28°54'55" _e 1°05'05"bh 80º05' 25°39'18" ^d 4°20'42"ci

66º30' — — 70º10' 28°32'09" _e 1°27'51"bh 80º10' 25°25'35" ^d 4°34'25"ci66º35' 27°34'34"`f 2°25'26"ag 70º15' 28°09'37" _e 1°50'23"bh 80º15' 25°11'53" ^d 4°48'07"ci66º40' 24°42'54"`f 5°17'06"ag 70º20' 27°47'18" _e 2°12'42"bh 80º20' 24°58'12" ^d 5°01'48"ci66º45' 22°55'38"`f 7°04'22"ag 70º25' 27°25'12" _e 2°34'48"bh 80º25' 24°44'32" ^d 5°15'28"ci66º50' 21°30'20"`f 8°29'40"ag 70º30' 27°03'18" _e 2°56'42"bh 80º30' 24°30'54" ^d 5°29'06"ci66º55' 20°17'16"`f 9°42'44"ag 70º35' 26°41'37" _e 3°18'23"bh 80º35' 24°17'17" ^d 5°42'43"ci67º00' 19°12'17"`f 10°47'43" ag 70º40' 26°20'08" _e 3°39'52"bh 80º40' 24°03'41" ^d 5°56'19"ci67º05' 18°13'09"`f 11°46'51" ag 70º45' 25°58'49" _e 4°01'11"bh 80º45' 23°50'07" ^d 6°09'53"ci67º10' 17°18'30"`f 12°41'30" ag 70º50' 25°37'42" _e 4°22'18"bh 80º50' 23°36'34" ^d 6°23'26"ci67º15' 16°27'26"`f 13°32'34" ag 70º55' 25°16'45" _e 4°43'15"bh 80º55' 23°23'02" ^d 6°36'58"ci67º20' 15°39'19"`f 14°20'41" ag 71º00' 24°55'59" _e 5°04'01"bh 81º00' 23°09'31" ^d 6°50'29"ci67º25' 14°53'40"`f 15°06'20" ag 71º05' 24°35'23" _e 5°24'37"bh 81º05' 22°56'01" ^d 7°03'59"ci67º30' 14°10'08"`f 15°49'52" ag 71º10' 24°14'56" _e 5°45'04"bh 81º10' 22°42'33" ^d 7°17'27"ci67º35' 13°28'27"`f 16°31'33" ag 71º15' 23°54'39" _e 6°05'21"bh 81º15' 22°29'06" ^d 7°30'54"ci67º40' 12°48'22"`f 17°11'38" ag 71º20' 23°34'31" _e 6°25'29"bh 81º20' 22°15'39" ^d 7°44'21"ci67º45' 12°09'43"`f 17°50'17" ag 71º25' 23°14'33" _e 6°45'27"bh 81º25' 22°02'14" ^d 7°57'46"ci67º50' 11°32'22"`f 18°27'38" ag 71º30' 22°54'42" _e 7°05'18"bh 81º30' 21°48'50" ^d 8°11'10"ci67º55' 10°56'10"`f 19°03'50" ag 71º35' 22°35'00" _e 7°25'00"bh 81º35' 21°35'27" ^d 8°24'33"ci68º00' 10°21'00"`f 19°39'00" ag 71º40' 22°15'27" _e 7°44'33"bh 81º40' 21°22'06" ^d 8°37'54"ci68º05' 9°46'49"`f 20°13'11" ag 71º45' 21°56'01" _e 8°03'59"bh 81º45' 21°08'45" ^d 8°51'15"ci68º10' 9°13'30"`f 20°46'30" ag 71º50' 21°36'43" _e 8°23'17"bh 81º50' 20°55'25" ^d 9°04'35"ci68º15' 8°41'00"`f 21°19'00" ag 71º55' 21°17'33" _e 8°42'27"bh 81º55' 20°42'06" ^d 9°17'54"ci68º20' 8°09'15"`f 21°50'45" ag 72º00' 20°58'30" _e 9°01'30"bh 82º00' 20°28'49" ^d 9°31'11"ci68º25' 7°38'12"`f 22°21'48" ag 72º05' 20°39'35" _e 9°20'25"bh 82º05' 20°15'32" ^d 9°44'28"ci68º30' 7°07'48"`f 22°52'12" ag 72º10' 20°20'46" _e 9°39'14"bh 82º10' 20°02'16" ^d 9°57'44"ci68º35' 6°38'00"`f 23°22'00" ag 72º15' 20°02'04" _e 9°57'56"bh 82º15' 19°49'01" ^d 10°10'59"ci68º40' 6°08'46"`f 23°51'14" ag 72º20' 19°43'30" _e 10°16'30"bh 82º20' 19°35'47" ^d 10°24'13"ci68º45' 5°40'05"`f 24°19'55" ag 72º25' 19°25'01" _e 10°34'59"bh 82º25' 19°22'34" ^d 10°37'26"ci68º50' 5°11'54"`f 24°48'06" ag 72º30' 19°06'39" _e 10°53'21"bh 82º30' 19°09'22" ^d 10°50'38"ci68º55' 4°44'11"`f 25°15'49" ag 72º35' 18°48'23" _e 11°11'37"bh 82º35' 18°56'11" ^d 11°03'49"ci69º00' 4°16'55"`f 25°43'05" ag 72º40' 18°30'14" _e 11°29'46"bh 82º40' 18°43'01" ^d 11°16'59"ci69º05' 3°50'05"`f 26°09'55" ag 72º45' 18°12'10" _e 11°47'50"bh 82º45' 18°29'52" ^d 11°30'08"ci69º10' 3°23'40"`f 26°36'20" ag 72º50' 17°54'12" _e 12°05'48"bh 82º50' 18°16'43" ^d 11°43'17"ci69º15' 2°57'37"`f 27°02'23" ag 72º55' 17°36'20" _e 12°23'40"bh 82º55' 18°03'35" ^d 11°56'25"ci69º20' 2°31'56"`f 27°28'04" ag 73º00' 17°18'34" _e 12°41'26"bh 83º00' 17°50'29" ^d 12°09'31"ci69º25' 2°06'37"`f 27°53'23" ag 73º05' 17°00'53" _e 12°59'07"bh 83º05' 17°37'23" ^d 12°22'37"ci69º30' 1°41'37"`f 28°18'23" ag 73º10' 16°43'17" _e 13°16'43"bh 83º10' 17°24'17" ^d 12°35'43"ci69º35' 1°16'57"`f 28°43'03" ag 73º15' 16°25'47" _e 13°34'13"bh 83º15' 17°11'13" ^d 12°48'47"ci69º40' 0°52'35"`f 29°07'25" ag 73º20' 16°08'21" _e 13°51'39"bh 83º20' 16°58'09" ^d 13°01'51"ci69º45' 0°28'31"`f 29°31'29" ag 73º25' 15°51'01" _e 14°08'59"bh 83º25' 16°45'07" ^d 13°14'53"ci69º50' 0°04'43"`f 29°55'17" ag 73º30' 15°33'46" _e 14°26'14"bh 83º30' 16°32'05" ^d 13°27'55"ci69º55' 29°41'12"_e 0°18'48"bh 73º35' 15°16'35" _e 14°43'25"bh 83º35' 16°19'03" ^d 13°40'57"ci70º00' 29°17'56"_e 0°42'04"bh 73º40' 14°59'30" _e 15°00'30"bh 83º40' 16°06'03" ^d 13°53'57"ci

73º45' 14°42'29" _e 15°17'31"bh 83º45' 15°53'03" ^d 14°06'57"ci73º50' 14°25'32" _e 15°34'28"bh 83º50' 15°40'04" ^d 14°19'56"ci73º55' 14°08'40" _e 15°51'20"bh 83º55' 15°27'05" ^d 14°32'55"ci74º00' 13°51'52" _e 16°08'08"bh 84º00' 15°14'08" ^d 14°45'52"ci74º05' 13°35'09" _e 16°24'51"bh 84º05' 15°01'10" ^d 14°58'50"ci74º10' 13°18'30" _e 16°41'30"bh 84º10' 14°48'14" ^d 15°11'46"ci74º15' 13°01'55" _e 16°58'05"bh 84º15' 14°35'18" ^d 15°24'42"ci74º20' 12°45'23" _e 17°14'37"bh 84º20' 14°22'23" ^d 15°37'37"ci74º25' 12°28'56" _e 17°31'04"bh 84º25' 14°09'29" ^d 15°50'31"ci74º30' 12°12'33" _e 17°47'27"bh 84º30' 13°56'35" ^d 16°03'25"ci74º35' 11°56'14" _e 18°03'46"bh 84º35' 13°43'42" ^d 16°16'18"ci74º40' 11°39'58" _e 18°20'02"bh 84º40' 13°30'49" ^d 16°29'11"ci74º45' 11°23'47" _e 18°36'13"bh 84º45' 13°17'57" ^d 16°42'03"ci74º50' 11°07'38" _e 18°52'22"bh 84º50' 13°05'06" ^d 16°54'54"ci74º55' 10°51'34" _e 19°08'26"bh 84º55' 12°52'15" ^d 17°07'45"ci75º00' 10°35'33" _e 19°24'27"bh 85º00' 12°39'24" ^d 17°20'36"ci75º05' 10°19'35" _e 19°40'25"bh 85º05' 12°26'35" ^d 17°33'25"ci75º10' 10°03'41" _e 19°56'19"bh 85º10' 12°13'45" ^d 17°46'15"ci75º15' 9°47'50"_e 20°12'10"bh 85º15' 12°00'57" ^d 17°59'03"ci75º20' 9°32'02"_e 20°27'58"bh 85º20' 11°48'09" ^d 18°11'51"ci75º25' 9°16'18"_e 20°43'42"bh 85º25' 11°35'21" ^d 18°24'39"ci75º30' 9°00'37"_e 20°59'23"bh 85º30' 11°22'34" ^d 18°37'26"ci75º35' 8°44'58"_e 21°15'02"bh 85º35' 11°09'47" ^d 18°50'13"ci75º40' 8°29'23"_e 21°30'37"bh 85º40' 10°57'01" ^d 19°02'59"ci75º45' 8°13'51"_e 21°46'09"bh 85º45' 10°44'15" ^d 19°15'45"ci75º50' 7°58'22"_e 22°01'38"bh 85º50' 10°31'30" ^d 19°28'30"ci75º55' 7°42'56"_e 22°17'04"bh 85º55' 10°18'45" ^d 19°41'15"ci76º00' 7°27'33"_e 22°32'27"bh 86º00' 10°06'00" ^d 19°54'00"ci76º05' 7°12'12"_e 22°47'48"bh 86º05' 9°53'16"^d 20°06'44"ci76º10' 6°56'54"_e 23°03'06"bh 86º10' 9°40'33"^d 20°19'27"ci76º15' 6°41'39"_e 23°18'21"bh 86º15' 9°27'50"^d 20°32'10"ci76º20' 6°26'27"_e 23°33'33"bh 86º20' 9°15'07"^d 20°44'53"ci76º25' 6°11'17"_e 23°48'43"bh 86º25' 9°02'24"^d 20°57'36"ci76º30' 5°56'10"_e 24°03'50"bh 86º30' 8°49'42"^d 21°10'18"ci76º35' 5°41'06"_e 24°18'54"bh 86º35' 8°37'01"^d 21°22'59"ci76º40' 5°26'04"_e 24°33'56"bh 86º40' 8°24'20"^d 21°35'40"ci76º45' 5°11'04"_e 24°48'56"bh 86º45' 8°11'39"^d 21°48'21"ci76º50' 4°56'07"_e 25°03'53"bh 86º50' 7°58'58"^d 22°01'02"ci76º55' 4°41'13"_e 25°18'47"bh 86º55' 7°46'18"^d 22°13'42"ci77º00' 4°26'21"_e 25°33'39"bh 87º00' 7°33'38"^d 22°26'22"ci77º05' 4°11'31"_e 25°48'29"bh 87º05' 7°20'58"^d 22°39'02"ci77º10' 3°56'43"_e 26°03'17"bh 87º10' 7°08'19"^d 22°51'41"ci77º15' 3°41'58"_e 26°18'02"bh 87º15' 6°55'40"^d 23°04'20"ci77º20' 3°27'15"_e 26°32'45"bh 87º20' 6°43'01"^d 23°16'59"ci77º25' 3°12'34"_e 26°47'26"bh 87º25' 6°30'23"^d 23°29'37"ci77º30' 2°57'55"_e 27°02'05"bh 87º30' 6°17'45"^d 23°42'15"ci77º35' 2°43'19"_e 27°16'41"bh 87º35' 6°05'07"^d 23°54'53"ci77º40' 2°28'44"_e 27°31'16"bh 87º40' 5°52'29"^d 24°07'31"ci77º45' 2°14'12"_e 27°45'48"bh 87º45' 5°39'51"^d 24°20'09"ci77º50' 1°59'42"_e 28°00'18"bh 87º50' 5°27'14"^d 24°32'46"ci77º55' 1°45'13"_e 28°14'47"bh 87º55' 5°14'37"^d 24°45'23"ci78º00' 1°30'47"_e 28°29'13"bh 88º00' 5°02'01"^d 24°57'59"ci78º05' 1°16'23"_e 28°43'37"bh 88º05' 4°49'24"^d 25°10'36"ci78º10' 1°02'00"_e 28°58'00"bh 88º10' 4°36'48"^d 25°23'12"ci78º15' 0°47'40"_e 29°12'20"bh 88º15' 4°24'11"^d 25°35'49"ci78º20' 0°33'21"_e 29°26'39"bh 88º20' 4°11'35"^d 25°48'25"ci78º25' 0°19'04"_e 29°40'56"bh 88º25' 3°59'00"^d 26°01'00"ci78º30' 0°04'49"_e 29°55'11"bh 88º30' 3°46'24"^d 26°13'36"ci78º35' 29°50'36" ^d 0°09'24"ci 88º35' 3°33'48"^d 26°26'12"ci78º40' 29°36'24" ^d 0°23'36"ci 88º40' 3°21'13"^d 26°38'47"ci78º45' 29°22'15" ^d 0°37'45"ci 88º45' 3°08'38"^d 26°51'22"ci78º50' 29°08'07" ^d 0°51'53"ci 88º50' 2°56'03"^d 27°03'57"ci78º55' 28°54'01" ^d 1°05'59"ci 88º55' 2°43'28"^d 27°16'32"ci79º00' 28°39'56" ^d 1°20'04"ci 89º00' 2°30'53"^d 27°29'07"ci79º05' 28°25'53" ^d 1°34'07"ci 89º05' 2°18'18"^d 27°41'42"ci79º10' 28°11'52" ^d 1°48'08"ci 89º10' 2°05'43"^d 27°54'17"ci79º15' 27°57'52" ^d 2°02'08"ci 89º15' 1°53'09"^d 28°06'51"ci79º20' 27°43'54" ^d 2°16'06"ci 89º20' 1°40'34"^d 28°19'26"ci79º25' 27°29'57" ^d 2°30'03"ci 89º25' 1°28'00"^d 28°32'00"ci79º30' 27°16'02" ^d 2°43'58"ci 89º30' 1°15'26"^d 28°44'34"ci79º35' 27°02'09" ^d 2°57'51"ci 89º35' 1°02'51"^d 28°57'09"ci79º40' 26°48'17" ^d 3°11'43"ci 89º40' 0°50'17"^d 29°09'43"ci79º45' 26°34'26" ^d 3°25'34"ci 89º45' 0°37'43"^d 29°22'17"ci79º50' 26°20'37" ^d 3°39'23"ci 89º50' 0°25'08"^d 29°34'52"ci79º55' 26°06'49" ^d 3°53'11"ci 89º55' 0°12'34"^d 29°47'26"ci80º00' 25°53'03" ^d 4°06'57"ci 90º00' 0°00'00"^d 29°59'60"ci

Trigonometric formulas

For the technically inclined astrologer, I give below the formulas I used to calculate the tables.N.B. These formulas are designed for points on the ecliptic only.

ra = right ascensionlst = local sidereal timeramc = right ascension of the medium coeli (=15 × lst)long = ecliptic longitudelat = geographic latitude (positive value is northern latitude, negative value is southern latitude)obl = obliquity of the eclipticha = hour angledsa = diurnal semi-arc

1. Midheaven = arctan(tan ramc ÷ cos obl) and add 0°, 180°, or 360° to place the longitude of the midheaven in the same quadrant as the right ascension.

2. Right ascension of a point on the ecliptic:ra = arctan(tan long × cos obl) and add 0°, 180°, or 360° to place the right ascension in the same quadrant as the longitude.

3. Ascendant = arctan( -cos ramc ÷ (tan lat × sin obl + sin ramc × cos obl)) and add 0°, 180°, or 360° to place the ascendant in the eastern hemisphere of the horoscope.

4. Calculating the boundaries of the circumpolar sky on the ecliptic:Circumpolar = arcsin( sin(90° – lat) ÷ sin obl) and find the other ecliptic points by subtracting this position from 180° and taking opposition points.

5. Placidus cusps in the arctic cannot be calculated directly by any trigonometric formula, but they can be found by making a list of ecliptic positions and their corresponding mundane positions on the semi-arc and then find the positions on the ecliptic where the mundane positions are ⅓, ⅔, -⅓, or -⅔.

Calculating mundane positions on the semi-arc for points on the ecliptic:• To standardize calculation, only calculate positions that are above the horizon. The values of the mundane

positions of points below the horizon are the same as those of their opposition points above the horizon.• Calculate diurnal semi-arc:

dsa = 90° + arcsin(sin ra × tan obl × tan lat)When this formula returns an error, the point in question is circumpolar and the dsa defaults to 180°.

• Calculate hour angle with an inverted formula:ha = ra – ramc; The hour angle must be in this range: -180° ≤ ha ≤ 180°. If ha is smaller than -180°, add 360° and if ha is larger than 180°, subtract 360°.

• Calculate the mundane position on the semi-arc, what is called the proportional ascension factor:paf = ha ÷ dsa; this should now be a value between -1 and 1.

Midheaven and IC: paf = 0cusps 11 and 5: paf = ⅓cusps 12 and 6: paf = ⅔Ascendant and Descendant: paf = 1 or -1cusps 8 and 2: paf = -⅔cusps 9 and 3: paf = -⅓

This procedure for calculating relative ascension has been developed by the Dutch astrologer Wim van Dam in his work on primary directions.

Links

Download the Table of Arctic Houses:http://www.astro.com/swisseph/Table_Arctic_Placidus.pdf

For more details and explantations:http://www.astrodienst.com/astrology/in_polar_asc_e.htmhttp://wiki.astro.com/astrowiki/en/Placidus_House_Systemhttp://www.skyscript.co.uk/polar1.html

Chart wheels by: http://apps.flatangle.com/Charts/

Amsterdam, NetherlandsNovember 2017