Navamsha as a division problem

A navāṁśa (नवांश) is a sign name got by dividing one number twice, and neither division needs a table. Here is the whole of it for a planet standing at 4° 26′ of Aries:

266 ÷ 200 = 1, remainder 66.

Then 1 ÷ 12 = 0, remainder 1.

The answer is that second remainder. It is one, and counting from zero at Aries, one is Taurus. The rest of this page is an account of what those numbers were.

266 is that position on the zodiac, written in arcminutes. A circle holds 360 degrees and a degree holds 60 arcminutes, so the ring is 21,600 arcminutes around, counted from zero at the start of Aries. Our position is 266 of them, which is the 4 degrees 26 arcminutes above: a short way into the first sign.

200 is the width of one navamsha. The word means ninth part. Each rāśi (राशि), each of the twelve signs of 30 degrees, is cut into nine equal pieces of 3 degrees 20 minutes. A sign is 1,800 arcminutes and a ninth of it is 200. That is why I work in arcminutes here: every quantity stays a whole number, and nothing in the calculation needs a decimal point.

12 is the number of signs, and it is the last thing the calculation uses. There is no other input. Not the birth time, not the place, not the other planets. One longitude in, one sign out.

So here is the rule, in two steps.

Divide the longitude in arcminutes by 200 and keep the whole part. Divide that by 12 and keep the remainder. The remainder is the navamsha sign, with Aries as zero.

The first division counts how many complete ninths lie behind the point. There are 108 ninths in the ring, numbered 0 to 107, because 21,600 divided by 200 is 108. The second division turns a number in that range into one of twelve names.

Three positions

The three longitudes below are chosen rather than observed: one in a movable sign, one in a dual sign, one in a fixed sign, because the classical rule for the navamsha has three cases and those are the three.

4° 26′ is the one at the top of the page. It sits in Aries, which is a movable sign, and its navamsha is Taurus.

260° 00′. In arcminutes, 15,600. Divide by 200: 78 exactly, nothing over, so the point stands on a boundary and opens the 79th cell. Divide 78 by 12: six twelves are 72, remainder 6. Libra. The longitude sits at 20 degrees of Sagittarius, a dual sign.

317° 05′. In arcminutes, 19,025. Divide by 200: 95 whole ninths, 25 arcminutes over. Divide 95 by 12: seven twelves are 84, remainder 11. Pisces. The longitude sits at 17° 05′ of Aquarius, a fixed sign.

Each row is one longitude, converted once and divided twice. Anybody with a sheet of paper can add a row.

The three cases, and why there are not three rules

Most students meet this not as a division at all but as a set of instructions about where to start counting. In a movable sign, count the ninths from that sign onward. In a fixed sign, start from the ninth sign from it. In a dual sign, start from the fifth. I am paraphrasing the Parashara corpus, whose chapter on the divisions sets it out this way, and printed editions number that chapter differently.

Take the third longitude through those instructions. Aquarius is fixed, so the count starts at the ninth sign from it, which is Libra. Within Aquarius the point stands at 17° 05′, or 1,025 arcminutes, and 1,025 divided by 200 is 5 with 25 over, so it lies in the sixth ninth of the sign. Five signs on from Libra is Pisces: the same answer the two divisions gave, by a longer road.

It works every time, and the reason is worth seeing.

Number the signs with Aries as zero. Sign number s opens at ninth number 9s, since nine ninths fill each sign. Reduce 9s by twelves and you get the navamsha standing at the head of each sign: 0 for Aries, 9 for Taurus, 6 for Gemini, 3 for Cancer, then 0 again for Leo. Four values, repeating every fourth sign.

Now measure the same thing as a distance rather than a place. The offset from a sign to its own first navamsha is 9s − s, which is 8s, and reduced by twelves that gives 0, 8, 4, 0, 8, 4 and so on. Three values, repeating every third sign. Nought means the sign itself, eight signs on means the ninth from it, four signs on means the fifth from it. Movable, fixed, dual.

The classical rule has three cases because 8 × 3 = 24, exactly two full turns of twelve. Count position instead of distance and the same pattern gives four cases, because 9 × 4 = 36, exactly three full turns. Both are shadows of one continuous count, and neither needs a rule of its own.

Varahamihira states it the second way. In the first chapter of the Bṛhat Jātaka he says that the signs, beginning from Aries, commence respectively with the navamshas of Aries, Capricorn, Libra and Cancer: four openings shared out among twelve signs, which is that repeat of four exactly. I am reading N. Chidambaram Iyer's English translation of 1885, whose note on the verse puts the business more plainly than anything else I have found. I would paraphrase that note like this: the 108 navamshas, counted from the first point of Aries, simply take the twelve sign names over and over again.

That note also records an equality worth a line of its own. Twelve signs of nine parts make 108. Twenty-seven nakṣatras (नक्षत्र) of four parts make 108 as well, since a nakshatra is 13° 20′ wide and a quarter of it, a pāda (पाद), is 3° 20′. The navamsha and the pada are therefore the same cell, reached from two directions. The ninth number our first division produced is also the pada number, counted round the whole ring rather than within one nakshatra.

The arithmetic is the sturdy half. The input is not.

The longitude has to be a sidereal one, measured from a zero fixed against the stars. Two programs can agree about where the Moon is to a fraction of an arcsecond and still hand you longitudes 5 arcminutes apart, because they place that zero differently, which is a decision about the third century that most software does not print. Five arcminutes inside a cell 200 arcminutes wide puts roughly one position in 40 on the far side of a navamsha boundary. In a sign, 1,800 arcminutes wide, the same disagreement moves about one position in 360. The finer the cell, the more of the convention shows through.

Here is that in numbers. Take 99° 57′: 5,997 arcminutes, 29 whole ninths, and 29 on 12 leaves 5, which is Virgo. Take 100° 02′, 5 arcminutes further on: 6,002 arcminutes, 30 whole ninths, and 30 on 12 leaves 6, which is Libra. Both positions sit in Cancer. Both sit in the same nakshatra, Pushya. The navamsha is the only thing on the page that moves.

The birth time does the same work by another route. The lagna (लग्न), the degree of the ecliptic rising in the east, moves at about 15 arcminutes for every minute of clock time on average, so it clears a 200-arcminute cell in a little over 13 minutes. That is why a birth time rounded to the nearest ten minutes leaves the sign alone and unsettles the navamsha three times in four, as the ten-minute accounting worked out. The Moon covers about 790 arcminutes of the ecliptic a day, so it changes navamsha roughly every six hours. The Sun covers about 59 arcminutes a day, so it takes a little over three.

One line on the wider family. Every varga (वर्ग) has this shape: pick a number, make the cells 30 degrees divided by it, count cells from Aries, map the cell number to a sign. The counting never changes. The map does, and it does not always run on around the ring the way the navamsha's does. Varahamihira gives the three drekkāṇas (द्रेक्काण) of 10 degrees to the sign itself, the fifth from it and the ninth from it, starting afresh inside every sign, and he records Garga's school assigning them another way. The triṁśāṁśa (त्रिंशांश) is not equal parts at all: five stretches of 5, 5, 8, 7 and 5 degrees, reversed in the even signs. All sixteen want a page of their own and will get one.

Two divisions deliver a cell number and a sign name, and they are stubborn about it: same longitude, same stated convention, same answer, from me or from anyone, today and in fifty years.

The last cell in the ring is number 107. Twelve into 107 leaves 11, so it is a Pisces navamsha, and it is also the ninth ninth of Pisces itself. One mark further on the count is back at nothing, which is Aries in both rings at once. The ring closes because 108 is nine twelves, and the count that walks it needs nothing but a place to start and a number to divide by.

A place to start is the one thing the divisions cannot supply. Two programs put it 5 arcminutes apart, a cell is 200 arcminutes wide, and about one position in 40 therefore falls on the far side of a line. That is the figure I would want printed beside any navamsha, because it puts a question to whoever reads the chart next: how much of the reading survives if the cell is the neighbouring one?