Four house systems, one birth moment

A house system is a rule for placing twelve boundaries on the ecliptic, cutting it into twelve arcs called bhāvas (भाव), houses. It needs no new observation and no extra constant; it divides numbers already there.

Four such rules run on Indian charts: equal houses, Śrīpati (श्रीपति), Placidus and Krishnamurti Paddhati. This page works the twelve boundaries under each, for a hypothetical birth at 9:12 on the morning of 22 July 2009 at Ujjain, 23° 11′ north. All four take the ascendant as given, raised by hand in an earlier essay as 162° 52′ 07″, and none recomputes it. Three print it as the edge of the first house; the fourth puts it in the middle of that house, which is the first sign of what is coming.

Every rule below starts from the same two directions. One is the ascendant. The other is the meridian, the circle running from due north over the point overhead and down to due south, which cuts the ecliptic at the midheaven. That crossing costs one line, and both quantities it needs are on the table already: sidereal time θ = 71° 18′ 45″ and obliquity ε = 23° 26′ 17″.

tan(MC) = tan θ ÷ cos ε

tan 71° 18′ 45″ is 2.95649 and cos ε is 0.91749, so the ratio is 3.22237. The angle with that tangent, taken in the same quadrant as the sidereal time, is 72° 45′ 33″: 12° 45′ of Gemini, counted from the equinox.

Two directions, then, and their opposites half a circle away: four candidate boundaries out of twelve.

Four rules

Equal. Take the ascendant and add 30 degrees, 11 times. That is the whole rule. The midheaven is set aside, so the tenth boundary lands at 72° 52′ 07″ and the real midheaven falls 6′ 33″ short of it, inside the ninth house.

Śrīpati. Named for the eleventh-century astronomer from what is now Maharashtra whose Jātakapaddhati (जातकपद्धति) became a standard handbook, this rule runs in two steps. First, cut each quadrant between the angles into three equal parts. The arc from the midheaven forward to the ascendant is 90° 06′ 33″, so its thirds are 30° 02′ 11″; the arc on from the ascendant to the lower meridian is 89° 53′ 27″, so its thirds are 29° 57′ 49″. That much is the division Porphyry described. Second, and this is the Indian part, treat those twelve points not as edges but as centres. Each becomes a bhāvamadhya (भावमध्य), the middle of its house, and the working boundary, the bhāvasandhi (भावसंधि), falls halfway between one centre and the next. Every boundary therefore sits about 15 degrees behind the point it came from, and the ascendant stops being an edge.

Placidus. Devised by Placidus de Titis (1603 to 1668), an Olivetan monk who taught mathematics at Pavia, this rule divides time rather than arc. Follow a point of the ecliptic from the moment it rises to the moment it crosses the meridian. The sidereal time that climb takes is its semi-diurnal arc, and its length depends on the point's declination, so no two points get the same one. Placidus cuts the climb into thirds. The eleventh boundary holds the points with two thirds of it behind them, the twelfth those with one third. Below the horizon the same trisection is applied to the rise from the lowest point of the sky to the eastern horizon, giving the second and third.

That is a condition, not a formula, and it will not rearrange into one. Longitude sets the declination, declination sets the arc, and the arc sets the longitude again, so the answer sits inside its own definition. You guess, recompute, and repeat until the number stops moving. For the eleventh boundary, starting from the equal-house value as a first guess, the rounds run 102° 52′ 07″, then 103° 36′ 02″, then 103° 35′ 21″, then 103° 35′ 21.4″, and the fourth round changes nothing. Two rounds bring it inside an arcminute and three inside an arcsecond; the semi-diurnal arc there settles at 100° 20′ 33″. The second boundary oscillates either side instead of closing in, and needs about six rounds to reach 190° 57′ 24″.

Krishnamurti Paddhati. KP takes its house division from Placidus unaltered. Its own contributions lie elsewhere, and one matters here: it uses a different ayanāṁśa (अयनांश), the offset between the zodiac counted from the equinox and the one counted from the stars. Swiss Ephemeris, under a great deal of chart software, puts the Lahiri and Krishnamurti values 5′ 49″ apart that morning.

The twelve

Longitudes below are counted from the equinox, the frame the trigonometry uses.

Two of those columns are the same column, and that is the result rather than a slip. Subtract the ayanamsha to reach the sidereal figures an Indian chart prints and the same amount comes off every line of every column, so none of the differences change; the ascendant becomes 18° 52′ of Siṁha (सिंह), Leo, as the earlier essay showed. KP is the exception, its zero being 5′ 49″ short of Lahiri's. That moves no planet out of its house, since planet and boundary shift together, but it renames all twelve, and KP reads a boundary's exact degree.

How far apart

Read straight down the rows, the largest gap between any two systems is 15° 57′ 06″, at the sixth and twelfth boundaries, the smallest 15° 01′ 06″, at the first, second, seventh and eighth. Sripati is one end of every gap. That is not four systems disagreeing about the sky. It is one counting from house middles while three count from house edges, and the 15 degrees is bookkeeping.

Line the conventions up, Sripati's centres against the others' edges, and the real disagreement appears. It is smaller, and uneven:

- First and seventh: exact agreement, all three at 162° 52′ 07″ and 342° 52′ 07″. - Fourth and tenth: 6′ 33″, all of it between equal houses and the midheaven. - Fifth and eleventh: 47′ 37″. - Sixth and twelfth: 56′ 00″. - Third and ninth: 1° 21′ 47″. - Second and eighth: 1° 54′ 42″, the widest.

Equal and Sripati never part by more than 6′ 33″ here. Nearly the whole disagreement is Placidus pulling away from both, which is what dividing time instead of arc does. The house widths say the same: equal gives twelve of exactly 30 degrees, Sripati 29° 57′ 49″ and 30° 02′ 11″, Placidus anything from 28° 05′ 18″ at the first house to 31° 15′ 14″ at the third.

Three checks. Swiss Ephemeris, given the actual instant instead of my rounded inputs, returns every cusp 13 to 14 arcseconds further along. That residue is not the house rules. It is the sidereal time, carried to the nearest second in the earlier essay, and one second of clock moves the ascendant about 14 arcseconds.

Second, the near-agreement of equal and Sripati is local. The two quadrants here are 90° 06′ 33″ and 89° 53′ 27″, within 7 arcminutes of a right angle, so trisecting them gives almost exactly 30 degrees. Move to Srinagar at 34° 05′ north, same sidereal time, and they open to 91° 31′ 10″ and 88° 28′ 50″. Equal and Sripati then part by up to 1° 00′ 46″, equal and Placidus by 3° 58′ 34″. Those 11 degrees of latitude roughly double the spread.

Third, Placidus has a limit. The semi-diurnal arc stops existing once the tangent of the latitude times the tangent of the declination passes one: past that, the point never rises or never sets. For ecliptic points the limit falls at latitude 66° 34′, the polar circle. Everywhere in India the method is defined. Inside the Arctic it is not, and software falls back on something else without always saying so.

Which of the four is correct is not a question this arithmetic answers, and I have no vote to cast. What it does say is where the choice bites. Take a planet anywhere from 177° 51′ to 190° 57′, a band 13 degrees and 6 arcminutes wide: Sripati puts it in the second house, the other three keep it in the first. Take one in the narrow band just past that, 190° 57′ to 192° 52′, 1 degree 55 minutes across: now Sripati, Placidus and KP all read the second house and only equal houses still read the first. Subtract the ayanamsha and both bands fall inside Kanyā (कन्या), Virgo.

Twelve such bands ring the circle, one at each boundary, none touching its neighbours. Added up they come to 183° 53′ 27″. Slightly more than half this zodiac is ground on which the four rules do not agree about which house a planet stands in.