The system that is named after Regiomontaus is based on the equator. The celestial equator is the extension in space of the earthly equator. The equator is based on the daily axial rotation of the earth: it is the circle that is perpendicular to that axis of the earth. The intersections of the equator with the horizon are the Eastpoint and the Westpoint, the North- and Southpoint are perpendicular to those positions.

The Regiomontanus system divides the equator into twelve equal parts of each 30 degrees. You can start at the horizon of at the Meridian (the circle through north and south), in both cases, this results in the same parts because horizon and meridian are 90 degrees apart.

You end up with a division of the equator but you need to calculate cusps that are situated on the ecliptic. To do so, you draw circles through these division points and through both the Northpoint and the Southpoint. You will find the longitude of the cusps at the intersections of these circles with the ecliptic

Regiomontanus is a quadrant system; ascendant and descendant are the same as cusps 1 and 7, MC and IC are the same as cusps 10 and 4.

In the drawing, you see the horizon as a yellow plane. The red circle is the equator, it intersects the horizon in the East (point E) and West (point W). The blue circle is the Meridian, it intersects the equator at its highest point and touches the horizon in the North (point N) and South (point S).

In the drawing, the ecliptic is a green circle that has an angle of about 23.5 degrees with the equator. Then ascendant is the intersection of the ecliptic with the horizon and the MC is the intersection of the ecliptic with the Meridian.

The drawing shows, as an example, two position circles for the houses 11 and 12. These position circles pass through the Northpoint and the Southpoint, divide the equator into equal parts (the drawing is not fully at scale…) and intersect the ecliptic at the longitude positions for cusps 11 and 12.

Please note that both the horizon and meridian act like a position circle, passing through north and south.

## History

Regiomontanus described his system in 1467. His text was printed in 1490 in his *Tabula Directionum*. The description by Regiomontanus would lead to the most important house system until the start of the twentieth century when the Placidus system gained popularity.

According to Helmut Grössing, Regiomontanus was born on June 6, 1436 (Julian date) at 16:40 in Kongingsbergen. His real name was Johannes Müller, but he named himself after his city of birth.

The Regiomontanus system is also called the *Modus Rationalis*. The system is not invented by Regiomontanus himself but has been described earlier by Abraham Ibn Ezra (born in 1090).

Pico della Mirandola even blamed Regiomontanus of plagiarism. John North suspects patriotic motives for this blaming because the German Regiomontanus had criticized the Italian Campanus. (North p. 44). Walter Koch and Wilhelm Knappich both write that Regiomontanus never claimed he was the inventor of this system. (Koch p. 74, Knappich p. 32). Knappich mentions several other descriptions of the Modus Rationalis from long before Regiomontanus popularized this system (Knappich, p. 28, 29). We do know now that almost all important house systems have not been named after the original inventer.

## Arguments pro and contra

Many consider the use of the equator as a logic choice. Houses are based on a daily rhythm and so is the equator. You will find this argument made by William Holden (Holden, p. 75) and many others.

Several critics, among others Max Duval, make objections against the inequality of size of the Regiomontanus houses. (Duval p. 31,32) . He does divide the equator into equal parts but this results in segments of unequal size.

A second point of critic is the problem of high latitudes. In particular, Morin de Villefranche makes objections against the fact that you cannot always calculate Regiomontanus cusps in polar regions. That is why he developed his own house system. However, in his astrological practice, Morin kept using Regiomontanus.

## Formula

Use the following formula’s to calculate Regiomontanus cusps:

R = ATAN ((sin H . tan GL) / (cos (RAMC + H))) L = ATAN (( cos R . tan (RAMC + H)) / (cos (R + E)))

For H use the following values:

- cusp 11: 30
- cusp 12: 60
- cusp 2: 120
- cusp 3: 150

**GL** is the geographic latitude

**RAMC** is the Right Ascension of the MC

**E** is the oblique angle of the ecliptic

**R** is a help variable

**L** is the longitude of the cusp, you might need to correct this for the correct quadrant.

These formulae are by Geoffrey Dean (Dean, p. 185).

Michael Munkasey also published formulas to calculate Regiomontanus (Munkasey p. 442). However, he gives a wrong definition of Regiomontanus (p. 425) According to Munkasey, Regiomontanus divides the equator starting with 0 degrees Aries. That is not correct. In his formulas, he does use the correct definition.

## Example calculation

We use the same place and location as in our other examples

Enschede, Netherlands (52º 13′ N en 6º 54′ E), decimal value for geographic latitude: 52.2166666666667

November 2, 2016 (Gregorian calendar), 21:17:30 UT.

Sidereal time 0:35:23.6 (decimal 0.5899018653)

Angle E 23° 26′ 13.56586091” (decimal 23.437101628).

MC: 9.62989868323 Converted to degrrees and minutes: 9°37′48″ Arries

Asc: 123.507983345667 = 3°30’28” Leo

RAMC = sidereal time * 15 = 8.8485279795

### Cusp 11

For cusp 11 we use H = 30

We calculate:

R = ATAN ( sin H . tan GL / cos (RAMC + H)

Fill in the values:

R = ATAN ( sin 30 . tan 52.2166666666667 / cos (8.8485279795 + 30)

Evaluate the goniometric functions:

R = ATAN ( 0.5 . 1.28996687154847 / 0.7788069689372) R = ATAN (0.82816854689219) R = 39.63048532170068

And calculate L:

L = ATAN (cos R . tan (RAMC + H) / cos (R + E) )

Fill in the values:

L = ATAN (cos 39.63048532170068 . tan (8.8485279795 + 30) / cos (39.63048532170068 + 23.437101628) )

This results in :

L = ATAN (0.7701739800305 . 0.80541609125717 / 0.45293913963538 ) L = ATAN (1.36952288354568) L = 53.8637799095727

Cusp 11 is 23º51’50” Taurus

### Cusp 12

For cusp 12 we use H = 60

R = ATAN ( sin H . tan GL / cos (RAMC + H)

Fill in the values:

R = ATAN ( sin 60 . tan 52.2166666666667 / cos (8.8485279795 + 60)

Evaluate the goniometric functions:

R = ATAN ( 0.8660254037844 . 1.28996687154847 / 0.36083478736137) R = ATAN (3.0959988336226) R = 72.09967114648910

Calculate L:

L = ATAN (cos R . tan (RAMC + H) / cos (R + E) )

Fill in the values:

L = ATAN (cos 72.09967114648910 . tan (8.8485279795 + 60) / cos (72.09967114648910 + 23.437101628) )

This results in :

L = ATAN (0.3073620795436 . 2.58464478950656) / -0.09648458397868) L = ATAN (-8.23366557252098) L = -83.07519498440457

Correction for quadrant

-83.07519498440457 + 180 = 96.92480501559543

Cusp 12 is 6º55’29” Cancer

### Cusp 2

For cusp 2 we use H = 120

R = ATAN ( sin H . tan GL / cos (RAMC + H)

Fill in the values:

R = ATAN ( sin 120 . tan 52.2166666666667 / cos (8.8485279795 + 120)

Evaluate the goniometric functions:

R = ATAN ( 0.8660254037844 . 1.28996687154847 / -0.62726366476528) R = ATAN (-1.7809800623782) R = -60.68628609490792

Proceed with the calculation of L:

L = ATAN (cos R . tan (RAMC + H) / cos (R + E) )

Fill in the values:

L = ATAN (cos -60.68628609490792 . tan (8.8485279795 + 120) / cos (-60.68628609490792 + c) )

This results in :

L = ATAN (0.4895911699482 . -1.24159426519416) / 0.79601061804979 ) L = ATAN (-0.7636501010334) L = 142.63283341742343

Correction for quadrant:

142.63283341742343 + 180 = 142.63283341742343

Cusp 2 is 22º37’58” Leo

### Cusp 3

For cusp 3 we use H = 150

R = ATAN ( sin H . tan GL / cos (RAMC + H)

Fill in the values:

R = ATAN ( sin 150 . tan 52.2166666666667 / cos (8.8485279795 + 150)

Evaluate the goniometric functions:

R = ATAN ( 0.5 . 1.28996687154847 /-0.93262975302629) R = ATAN (-0.69157501535988) R = -34.66676579493056

Now calculate L:

L = ATAN (cos R . tan (RAMC + H) / cos (R + E) )

Fill in the values:

L = ATAN (cos -34.66676579493056 . tan (8.8485279795 + 150) / cos (-34.66676579493056 + -34.66676579493056) )

This results in :

L = ATAN (0.8224741111883 . -0.38690036018099 / 0.9808544614955 ) L = ATAN (-0.3244268567358) L = -17.9744551316352

Correction for quadrant

-17.9744551316352 + 180 = 162.0255448683648

Cusp 3 is 12º01’32” Virgo

## References

**Dean, Geoffrey**en**Arthur Mather**–*Recent Advances in Natal Astrology: A critical review 1900-1976*. The Astrological Association, Bromley, 1977.**Duval, Max**–*La domification et les transits*. Paris, 1984. (French).**Grössing, Helmut**–*Regiomontanus*als Astrologe. Qualität der Zeit, 1979. A biography of Regiomontanus. (German).**Hentges, Ernst**–*Regiomontanus: Biografische Skizze*. Zenit 6, 1934. Online via http://astrotexte.ch/sources/hent02.html. A biography of Regiomontanus. (German).**Holden, Ralph William**–*The elements of house division*. Essex, 1977.**Kampherbeek, Jan**–*Het huizensysteem van Regiomontanus*. In Spica vol. 4 no. 3, October 1980. (Dutch).**Knappich, Wilhelm**–*Entwicklung der Horoskoptechnik vom Altertum bis zur Gegenwart*. In Qualität der Zeit nr. 38/39. Wenen, september 1978. (German).**Koch, Walter A.**–*Regiomontanus und das Häusersystem des Geburtsortes.*Göppingen/Fils, 1960. (German).**Morin, Jean-Baptiste**. Transl.**James Herschel Holden**–*Astrologia Gallica. Book 17: The Astrological Houses*. Tempe, AZ, 2008.**Munkasey, Michael**–*The astrological thesaurus. Book 1*. St. Paul, 1993.**North, John D**, –*Horoscopes and History*. London, 1986