IKS/ Part 3· Astronomy Chapter 9 of 14
Part 3 · Chapter 9

Astronomy

ज्योतिषम्

A set of numbers written down around 499 CE gives the length of the sidereal day as 23 h 56 m 04.10 s. The modern value is 23 h 56 m 04.09 s. This chapter is about how such numbers were obtained, what they were for, and the calendar they still drive.

Indian astronomy is unusual in never having been a specialist pursuit. Its output — the pañcāṅga — was consulted daily by ordinary households, and still is.

1What it was forकालनिर्णयः

The tradition states its own purpose in one word: कालनिर्णय, kāla-nirṇaya — the determination of time.

That is a narrower brief than modern astronomy's and it explains almost everything about the shape of the subject. The questions are: when is the new moon? when does the sun enter this constellation? when is the solstice? when will the moon be eclipsed, and for how long? These are computational questions with numerical answers, and they must be answered in advance, because rites, agriculture and civic life are scheduled by them.

Not remote

Celestial bodies are treated as continuous with earthly life rather than as distant objects. Consulting the pañcāṅga is a daily domestic act, not a scientific one, and it is why the tradition survived the loss of its institutions.

Observation, then computation

The method is systematic: observe, record, codify, look for periodicity, build a model that predicts, and revise the model when prediction and observation part company. Several Indian texts explicitly instruct the astronomer to correct the parameters against observation.

The engine of the mathematics

Every mathematical advance in chapter 8 — trigonometry, indeterminate equations, series, the beginnings of calculus — was driven by a problem in this chapter. The two subjects are not neighbours; they are one subject.

2The celestial sphereगोलः

The celestial sphere with equator, ecliptic, poles, zenith and nadir The earth at the centre, the celestial equator, the ecliptic tilted at 23 and a half degrees, north and south celestial poles, and the observer's zenith and nadir. earth north celestial pole south celestial pole celestial equator ecliptic — the sun's yearly path 23½° Celestial equator The earth's equator projected onto the sky. Fixed relative to the observer's horizon; the reference for declination. Ecliptic The apparent yearly path of the sun through the stars. Tilted 23½° to the equator — which is why there are seasons. Zenith and nadir The zenith is the point directly above a given place; the nadir is 180° from it, directly below. Both are local — they depend on where the observer is standing, which is what makes spherical astronomy necessary. local vertical
Two great circles at 23½° to each other, and everything follows. The two points where they cross are the equinoxes; the two points furthest apart are the solstices. The Sanskrit terms are krānti-vṛtta for the ecliptic and viṣuvat-vṛtta for the equator.

3Solstices and equinoxesअयनम् · विषुवत्

The four cardinal points of the sun's yearly path The two equinoxes where the ecliptic crosses the equator and the two solstices at its extremes, and the northern and southern courses of the sun. celestial equator SUMMER SOLSTICE northernmost point WINTER SOLSTICE southernmost point EQUINOX EQUINOX viṣuvat — day equals night uttarāyaṇa — the sun moving north dakṣiṇāyana — the sun moving south Where these appear in the texts The equinoctial day, viṣuvat, is named in the Aitareya-brāhmaṇa (18.4). The Vedāṅga-jyotiṣa gives a method for computing the date of the nth equinox — that is, an algorithm, not an observation. The Brahmāṇḍa-purāṇa (chapter 4) gives the changing day length across the two courses in muhūrtas. Uttarāyaṇa is still the basis of Makara Saṅkrānti, Pongal and Bihu.

4The sky as a clockपुरातत्त्वज्योतिषम्

Because the earth's axis precesses with a period of about 25,800 years, the equinox drifts steadily backwards through the constellations — roughly one degree every seventy-two years. A text that records where the equinox stood therefore carries a date.

Astronomical statements in Vedic texts and the dates they imply Three statements — the Krttikas rising due east, the winter solstice in Sravistha, and the solstice at the start of Sravistha — implying dates around 2950, 1660 and 1300 BCE. c. 2950 BCE Śatapatha-brāhmaṇa 2.1.2.3 “The Kṛttikās do not swerve from the east.” The Pleiades rise due east only when they sit on the celestial equator. c. 1660 BCE Maitrāyaṇīya-brāhmaṇa-upaniṣad 6.14 Winter solstice at the mid-point of Śraviṣṭhā; summer solstice at the beginning of Māgha. c. 1300 BCE Vedāṅga-jyotiṣa Winter solstice at the beginning of Śraviṣṭhā; summer solstice at the mid-point of Āśleṣā. EARLIERLATER Note the internal consistency: the three statements place the solstice at successively later points, in the order the precession requires, and the intervals between the implied dates match the intervals between the positions. That coherence is the real force of the argument, more than any single date.
Dates follow Subhash Kak's compilation. Chapter 1's caution applies in full: the method is sound, and the disagreement among specialists is about whether these statements are observational or prescriptive — whether the text is recording where the equinox was, or stipulating where it should be reckoned. Both readings are held by serious scholars.
A second, quieter piece of evidence

The oldest lists of the nakṣatras — Taittirīya-saṃhitā 4.4.10, and parallels in the Atharvaveda — begin with Kṛttikā. Every modern list begins with Aśvinī, two positions earlier. Two nakṣatras is 26° 40′ of arc; at one degree per seventy-two years, that is about nineteen centuries of precession. The shift in where the list starts is itself a fossil of the sky moving, and it is visible without any calculation at all — simply by comparing two lists.

5The twenty-seven nakṣatrasनक्षत्राणि

The distinctively Indian division of the zodiac is not by twelve but by twenty-seven — because the unit being tracked is the moon, not the sun.

The moon's sidereal period is 27.32 days. Divide the ecliptic into twenty-seven equal arcs and the moon occupies almost exactly one of them per night. Each arc is 360°/27 = 13° 20′ — that is, 800 minutes of arc — and each is named after a conspicuous star within it.

The twenty-seven nakshatras mapped onto the twelve rasis Two concentric rings: an outer ring of twenty-seven equal nakshatras and an inner ring of twelve rasis, each rasi spanning two and a quarter nakshatras. Aśvinī Bharaṇī Kṛttikā Rohiṇī Mṛgaśīrṣa Ārdrā Punarvasu Puṣya Āśleṣā Maghā P.Phalgunī U.Phalgunī Hasta Citrā Svātī Viśākhā Anurādhā Jyeṣṭhā Mūla P.Āṣāḍhā U.Āṣāḍhā Śravaṇa Dhaniṣṭhā Śatabhiṣak P.Bhādra U.Bhādra Revatī Meṣa Vṛṣabha Mithuna Karka Siṃha Kanyā Tulā Vṛścika Dhanus Makara Kumbha Mīna ecliptic 360° OUTER RING — 27 NAKṢATRAS Each spans 13° 20′ = 800 arc-minutes. Tracks the moon, which crosses about one per night (sidereal month 27.32 days). The list in Taittirīya-saṃhitā 4.4.10 begins with Kṛttikā; later lists begin with Aśvinī — the precession of §4, visible in a word list. INNER RING — 12 RĀŚIS Each spans 30°. Tracks the sun, which crosses one per solar month. 27 ÷ 12 = 2¼, so each rāśi contains exactly two and a quarter nakṣatras — and the boundaries do not coincide except every fourth rāśi. Both grids are used at once.
Two independent partitions of the same circle, one lunar and one solar, maintained simultaneously. Almost every complication in the Indian calendar arises from having to reconcile them.

6Five kinds of yearपञ्च संवत्सराः

A year can be defined in at least five ways, depending on which body you follow. The Vedic corpus names all five and keeps them distinct — which is the mark of people who have noticed that they disagree.

Five definitions of a year in the Vedic corpus Samvatsara the solar year of 366 days, Idavatsara the civil year of 360, Anuvatsara the lunar year of 354, Vatsara measured by the moon among the nakshatras, and Parivatsara by Jupiter. Saṃvatsaraसंवत्सर the solar year — the sun's passage through the twelve rāśis ≈ 366 days the basis of the solar calendars of Tamil Nadu, Kerala, Bengal, Assam, Odisha, Tripura and partly Punjab Iḍāvatsaraइडावत्सर the civil (sāvana) year — twelve months of thirty days 360 days Ṛgveda 1.164.11: “the wheel of time, formed with twelve spokes, revolves without wearing out; on it are 720 sons” — 360 days and 360 nights Anuvatsaraअनुवत्सर the lunar year — twelve months, each ending at new moon ≈ 354 days used everywhere for fixing festivals and auspicious times, whatever the local civil calendar Vatsaraवत्सर twelve sidereal lunar cycles of 27.32 days — the moon through the nakṣatras ≈ 328 days Parivatsaraपरिवत्सर the time Jupiter takes to move from one rāśi to the next ≈ 361 days — the basis of the sixty-year Jovian cycle Jupiter's twelve-year orbit maps neatly onto the twelve rāśis, which is why it acquired a calendrical role
The intercalation problem, and its Vedic solution

Twelve lunar months come to about 354 days; the solar year to about 365¼. The two drift apart by eleven days a year, and after three years the lunar calendar is a month behind the seasons. The Ṛgvedic authors were aware of this, and the solution is an intercalary month — an extra thirteenth month inserted periodically. The Yajurveda refers to the ekādaśa-rātra ceremony as the mechanism for reconciling the two counts, bringing the year to 365 days and leaving the residual quarter-day to be handled separately. A five-year cycle, the yuga of 60 solar / 61 civil / 62 lunar months, appears in the Taittirīya and Vājasaneyī saṃhitās.

7Months and fortnightsमासः · पक्षः

The lunar month divided into a bright and a dark fortnight Sukla-paksa runs from new moon to full moon; krsna-paksa from full moon back to new moon. sun earth amāvāsyā new moon pūrṇimā full moon śukla-pakṣa — the bright fortnight, waxing kṛṣṇa-pakṣa — the dark fortnight, waning A lunar month is new moon to new moon (or full to full): about 29.5 days, longer than the 27.32-day sidereal month because the sun has moved too.

The twelve lunar months are Caitra, Vaiśākha, Jyeṣṭha, Āṣāḍha, Śrāvaṇa, Bhādrapada, Āśvayuja, Kārtika, Mārgaśīrṣa, Puṣya, Māgha, Phālguna — and each is named for the nakṣatra the moon occupies at the full moon of that month. In most Caitra months the full moon stands near Citrā (Spica); in most Vaiśākha months near Viśākhā. The naming scheme is a mnemonic for a fact about the sky.

8Tithiतिथिः

The tithi is the unit that catches most newcomers out, because it is not a day. It is an angle.

The tithi defined as a twelve degree increment in the moon-sun elongation Successive positions of the moon and sun as seen from the earth, with each tithi being a twelve degree increase in their angular separation. earth S₀ sun M₀ moon — coincident: new moon, 0° after 1 tithi — separation 12° after 2 tithis — 24° after 7½ tithis — 90° after 15 tithis — 180°, full moon 12° Definition A tithi is the time in which the angle between the moon and the sun, seen from the earth, increases by 12° — a thirtieth of the circle. tithi = (θ_moon − θ_sun) / 12° Why a tithi is not a day The moon's speed varies through its orbit, so the time to gain 12° varies too — from about 21.5 to 26 hours. Consequences you can see in any pañcāṅga: a tithi can span two sunrises (a repeated tithi) or fall entirely between two (a skipped tithi, kṣaya-tithi). Neither is an error.
Thirty tithis make a lunar month: fifteen in the bright fortnight, fifteen in the dark. Because the definition is angular and the moon's motion is not uniform, the calendar has irregularities built into it that no amount of tidying can remove — they are properties of the moon.

9The siddhāntic traditionसिद्धान्तज्योतिषम्

Work or authorPeriodContribution
Sūrya-siddhāntabefore the 6th c. CEAuthor unknown, several recensions. An early version is summarised by Varāhamihira; a later one is still used by traditional calendar makers today
Varāhamihira — Pañca-siddhāntikā6th c. CEA critical summary of five earlier astronomical systems — and therefore our main evidence for what those systems said
Āryabhaṭa — Āryabhaṭīyab. 476 CE, KusumapuraThe rotation of the earth; accurate planetary parameters; the sine function and its table; the physical explanation of eclipses
Bhāskara I — Āryabhaṭīya-bhāṣya, Mahā-bhāskarīya7th c. CEThe indispensable commentary; develops the Āryabhaṭan system in his own texts
Brahmagupta — Brāhmasphuṭa-siddhānta, Khaṇḍakhādyaka7th c. CEA complete computational system; a practical handbook; and the algebra of chapter 8
Lalla — Śiṣyadhīvṛddhida-tantra8th–9th c. CEA textbook of the Āryabhaṭan system with new algorithms
Mañjulācārya — Laghu-mānasa10th c. CEAn explicit second correction to the moon's longitude; the derivative of the sine function and instantaneous velocity of sun and moon
Śrīpati — Siddhānta-śekhara11th c. CEWidely quoted by later astronomers
Bhāskarācārya II — Siddhānta-śiromaṇi, Karaṇa-kutūhalab. 1114 CEThe standard algorithms corrected and generalised; a table-based computation manual; a chapter on instruments
Work or authorPeriodContribution
Mādhava — Veṇvāroha, Sphuṭa-candrāpti1340–1425 CEInfinite series for π, sine and cosine; a method of computing the moon's position at short intervals
Parameśvara — Dṛggaṇita1360–1455 CEFifty-five years of recorded eclipse observations, used to correct the parameters. An observational programme, stated as such
Nīlakaṇṭha Somayājī — Tantra-saṅgraha1444–1550 CEA revised planetary model in which the inner planets orbit the sun; systematic spherical trigonometry; improved eclipse theory
Jyeṣṭhadeva — Gaṇita-yukti-bhāṣā16th c. CEFull demonstrations of the school's results, in Malayalam prose
Gaṇeśa Daivajña — Graha-lāghavab. 1507 CESimplified planetary computation without trigonometric tables — still used for almanac work
Kamalākara — Siddhānta-tattva-vivekab. 1616 CEIndian parameters with elements of Ptolemy's system incorporated
Rājā Sawai Jai Singh — Zij Muhammad Shahi1688–1743 CEFive masonry observatories; a new set of tables
Candraśekhara Sāmanta — Siddhānta-darpaṇab. 1835 CERevised lunar theory from his own naked-eye observations; simple instruments of his own design; reform of the Odia calendar
The last of the line

Candraśekhara Sāmanta is worth pausing on. Working in rural Odisha in the second half of the nineteenth century, with instruments he built from bamboo, he detected and corrected errors in the lunar theory by observation — including the moon's second inequality — apparently without access to European astronomy. He is the demonstration that this was still a living investigative tradition when it ended, not a fossil.

10The Āryabhaṭīyaआर्यभटीयम्

One hundred and twenty-one verses, in four sections, written when the author was twenty-three. It is the founding text of Indian mathematical astronomy.

The four sections of the Aryabhatiya Gitikapada with 13 verses, Ganitapada with 33, Kalakriyapada with 25, and Golapada with 50. Gītikā-pādaगीतिकापाद 13 VERSES The letter-numeral notation for very large numbers; the kalpa and mahā-yuga; the revolution numbers of the planets and their parameters. Gaṇita-pādaगणितपाद 33 VERSES Square and cube roots; areas and volumes; the value of π; geometric construction of sines and the sine table; arithmetic progressions; sums of the first n numbers, of their sums, and of their squares and cubes; the kuṭṭaka; relative velocity; an interest problem. Kālakriyā-pādaकालक्रियापाद 25 VERSES The reckoning of time; calendrical concepts; the planetary models — epicycle and eccentric circle; explicit procedures for computing positions. Gola-pādaगोलपाद 50 VERSES Spherical astronomy at different latitudes; the shape and situation of the earth; parallax; the causes of lunar and solar eclipses.

The earth turns

Āryabhaṭa holds that the apparent daily rotation of the heavens is caused by the earth's own rotation, and gives the image that makes it plausible: a person in a moving boat sees the fixed objects on the bank moving backwards. The Sanskrit is anulomagatir nausthaḥ…, Gola-pāda 9.

Later Indian astronomers, including Brahmagupta, rejected this. It is an instructive case: the tradition contained the idea, argued about it, and did not converge.

Eclipses have physical causes

Gola-pāda states that the moon is eclipsed by entering the earth's shadow, and the sun by the moon coming between. The geometry — the shadow cone of chapter 8's gnomon figure — is worked out, and the durations computed.

This is stated in a culture where the received explanation was the demon Rāhu. Āryabhaṭa's successors keep the name Rāhu as a label for the lunar node — the point where the orbits cross — which is exactly where an eclipse can occur. The mythological term is retained and given a geometric meaning.

11The parameters, checkedभगणाः

The Gītikā-pāda gives, for each body, the number of complete revolutions it makes in a mahā-yuga of 4,320,000 years. Dividing the number of civil days in the yuga by that figure gives the body's period. Here is what happens when you do the division.

The two numbers everything depends on

Civil days in a mahā-yuga (yuga-sāvana-dina): 1,577,917,500. Sidereal rotations of the earth in the same period: 1,582,237,500. The difference between them is exactly 4,320,000 — one extra rotation per year, which is precisely what the geometry requires and a good internal check on the system.

Aryabhata's periods computed from his revolution numbers, against modern values A table comparing derived periods for the sidereal day, month, year and five planets with modern values and the error in each. BODYREVOLUTIONS PER YUGAPERIOD DERIVEDMODERN VALUEERROR Sidereal day1 582 237 500 rotations 23 h 56 m 04.10 s23 h 56 m 04.09 s0.01 s Sidereal month57 753 336 (moon) 27.321 668 d27.321 661 d0.6 s Sidereal year4 320 000 (sun) 365.258 68 d365.256 36 d3 m 20 s Mercury17 937 020 (śīghrocca) 87.969 88 d87.969 1 d1.1 m Venus7 022 388 (śīghrocca) 224.698 14 d224.701 d4 m Mars2 296 824 686.999 7 d686.980 d28 m Jupiter364 224 4 332.27 d4 332.59 d7.6 h Saturn146 564 10 766.06 d10 759.22 d6.8 d Read the errors, not the agreements. The moon and the earth's rotation — the two things most often observed — are right to seven significant figures. The outer planets, which move slowly and are hardest to time, are worst: Saturn is out by a week in thirty years. That pattern is exactly what a genuine observational programme produces.
Derived by dividing 1,577,917,500 civil days by each revolution count. For Mercury and Venus the figure given is the śīghrocca — in effect the heliocentric period — which is why those two rows come out as the planets' true orbital periods rather than their apparent ones.
Why a mahā-yuga at all?

Stating periods as “so many revolutions in 4,320,000 years” looks like cosmological ornament. It is a computational device. Working in integers avoids fractions entirely: to find where a planet is on a given day, you multiply the elapsed days by the revolution count and divide by the yuga's days, all in whole numbers. The very large yuga also lets the periods be chosen so that everything returns to a common starting configuration — which gives the system a defined epoch. Āryabhaṭa's own epoch is the start of the current Kali-yuga: Friday, 18 February 3102 BCE.

Note also that Āryabhaṭa divides the mahā-yuga into four equal quarters of 1,080,000 years, rather than the 4:3:2:1 proportion of the Purāṇas. On this, as on the earth's rotation, he differs from the tradition around him.

12Nīlakaṇṭha's revisionतन्त्रसंग्रहः

Around 1500 CE, a Kerala astronomer noticed that the standard procedure for Mercury and Venus did not work properly, diagnosed why, and changed the model.

Nilakantha's revised planetary model The five planets orbit the sun in eccentric orbits, while the sun orbits the earth. earth sun Mercury Venus Mars Jupiter Saturn What the model says All five planets move in eccentric orbits around the sun — the centres slightly displaced from it. The sun in turn orbits the earth. A geo-heliocentric system. How to state its significance The geometry of the planets' motion relative to the sun is correct, and the equation of centre for Mercury and Venus is applied to the right centre — which is what the old model got wrong. It is not heliocentrism, and Nīlakaṇṭha does not claim it is. The earth stays put. Tycho Brahe proposed a similar arrangement around 1588; Kepler's elliptical orbits, the next correction, come in 1609.
The interest here is methodological. A discrepancy between prediction and observation was traced to a specific defect in the model, and the model was changed to fix it — with the change published, argued for, and adopted by the school. That is a scientific tradition operating normally.

13The five limbsपञ्चाङ्गम्

Pañcāṅga means “five limbs”. The Indian almanac states five quantities for each day, and every one of them is computed from the true longitudes of the sun and the moon.

The five components of the pancanga and their formulas Tithi, karana, nakshatra, yoga and vara, with the arithmetic that produces each. Tithiतिथि the moon–sun elongation, in units of 12° (θ_M − θ_S) / 12° 30 per lunar month Karaṇaकरण half a tithi — the same angle in units of 6° (θ_M − θ_S) / 6° 60 per lunar month Nakṣatraनक्षत्र which of the 27 arcs the moon is in θ_M in minutes / 800 the moon alone; 800′ = 13° 20′ Yogaयोग the sum of the two longitudes, in units of 13° 20′ (θ_M + θ_S) / 800′ 27 of them, named Vāraवार the weekday, from the running count of days since the epoch ahargaṇa mod 7 the only one not derived from longitudes — see §15
Notice the economy. Two observed quantities — where the sun is, where the moon is — and five derived ones, obtained by three subtractions, one addition and four divisions. Everything in the almanac follows from the planetary theory of §11.

14Computing a pañcāṅgaगणितम्

Two worked instants, each carried through all four longitude-derived limbs.

LimbWorkingResult
Tithiθ_M − θ_S = 60° 12′ − 19° 7′ = 41° 5′. 41° 5′ ÷ 12° = 3 61⁄1443 elapsed → the 4th tithi, Caturthī of the bright fortnight
Karaṇa41° 5′ ÷ 6° = 6 61⁄726 elapsed → the 7th karaṇa
Nakṣatraθ_M = 60° 12′ = 3,612′. 3,612 ÷ 800 = 4 103⁄2004 elapsed → the 5th, Mṛgaśīrṣa
Yogaθ_M + θ_S = 79° 19′ = 4,759′. 4,759 ÷ 800 = 5 759⁄8005 elapsed → the 6th yoga

The fractional parts matter too: the 759⁄800 in the last row says the current yoga is 95% elapsed and will end shortly. An almanac gives the clock time at which each limb changes, which is what makes it usable.

LimbWorkingResult
Tithiθ_M < θ_S, so add 360°: 561° 2′ − 219° 17′ = 341° 45′. ÷ 12° = 28 23⁄4828 elapsed → the 29th tithi, Caturdaśī of the dark fortnight
Karaṇa341° 45′ ÷ 6° = 56 23⁄2456 elapsed → the 57th karaṇa
Nakṣatraθ_M = 201° 2′ = 12,062′. 12,062 ÷ 800 = 15 31⁄40015 elapsed → the 16th, Viśākhā
Yogaθ_M + θ_S = 420° 19′; subtract 360° → 60° 19′ = 3,619′. ÷ 800 = 4 419⁄8004 elapsed → the 5th yoga
Two conventions to keep straight

Longitudes are measured on a circle, so a difference can come out negative — add 360° and continue. A sum can exceed 360° — subtract it. Both appear in this example, and forgetting either is the commonest error in doing these calculations by hand.

15Ahargaṇaअहर्गणः

The fifth limb needs something none of the others do: a continuous count of days that does not care about months or years.

Āryabhaṭa introduces exactly that, and it appears to be the first such scheme anywhere. अहर्गण, ahargaṇa, means simply “heap of days” — the number of days elapsed since the epoch. Because calendars differ between regions and eras, and because month lengths vary, a running day-count is the only stable way to identify a date and the only convenient way to compute a weekday.

Computing the weekday from the ahargana Dividing the day count by seven and reading the remainder against a table beginning at Friday. THE EPOCH Friday, 18 February 3102 BCE The start of the current Kali-yuga, and ahargaṇa 0. Every day since has a number. WORKED EXAMPLE ahargaṇa A = 1 870 348 1 870 348 ÷ 7 = 267 192, remainder 4 remainder 4 → Tuesday REMAINDER TABLE — COUNTING FROM THE EPOCH DAY 0Friday 1Saturday 2Sunday 3Monday 4Tuesday 5Wednesday 6Thursday
Compare the Julian Day Number, introduced by Joseph Scaliger in 1583 and used by astronomers today for exactly the same reason. The ahargaṇa is the same idea, with a different epoch, roughly a thousand years earlier.

16Finding north with a stickदिङ्निर्णयः

Before any observation can be made, the observer needs a true meridian. The construction requires a vertical rod, a piece of string and one clear day.

Determining the cardinal directions from a gnomon's shadow A circle is drawn around the foot of a vertical rod; the two points where the shadow tip crosses it, in the morning and afternoon, define the east-west line. X — the tip O Z — the zenith W′ — forenoon shadow tip E′ — afternoon shadow tip the east–west line the meridian — north–south The procedure 1 · Set the gnomon OX exactly vertical on level ground, so its tip points at the zenith. 2 · Draw a circle of convenient radius centred on O. 3 · In the forenoon, mark W′ where the shadow's tip crosses the circle; in the afternoon, mark E′. 4 · W′E′ is the east–west line, because equal shadows fall symmetrically about the meridian.
No compass, no clock, no prior knowledge of the site. The symmetry argument in step 4 is the whole content: two shadows of equal length must lie at equal angles either side of the north–south line, whatever the latitude and whatever the date.

17The instrument catalogueयन्त्राणि

Bhāskara II devotes a chapter of the Siddhānta-śiromaṇi to instruments. The list is a working laboratory.

Gola-yantra गोलयन्त्र

The armillary sphere: nested rings representing the fixed and movable great circles of the celestial sphere. Does the work of an astrolabe, and doubles as the teaching instrument for spherical astronomy.

Cakra-yantra चक्रयन्त्र

A graduated wheel of wood or metal with a needle at the centre. Illuminated by the sun, the needle's shadow gives the sun's altitude; used more generally for the longitudes and latitudes of planets. Cāpa-yantra is half of it, Turīya-yantra a quadrant.

Nāḍīvalaya नाडीवलय

A disc set in the plane of the equator, divided into 60 ghaṭikās and also into the twelve signs with arcs proportional to their rising times at that latitude. Read directly, it gives the time elapsed since sunrise and the rising sign.

Ghaṭī-yantra घटीयन्त्र

The sinking-bowl water clock of chapter 6 — a pierced copper bowl floated on water, which fills and sinks in one unit of time.

Śaṅku शङ्कु

The gnomon, made of ivory or metal. The simplest instrument and the most used: direction, latitude, time of day and the sun's declination all follow from its shadow.

Phalaka-yantra फलकयन्त्र

A board with a circle of 30 aṅgulas radius, graduated in ghaṭikās and degrees, from which the zenith distance can be read directly. Dhī-yantra, a stick with a plumb line, gives heights and distances.

18Jantar Mantarयन्त्रमन्दिरम्

Between 1724 and 1735, Sawai Jai Singh II built five masonry observatories — at Delhi, Jaipur, Ujjain, Mathura and Varanasi — on a diagnosis that is worth understanding, because it is an engineering argument rather than an astronomical one.

Why Jai Singh built instruments out of masonry Small metal instruments suffer from graduation error, flexure and wear; very large fixed masonry instruments reduce all three. THE DIAGNOSIS The brass instruments then available in Europe and India were small, and small instruments have a floor on their accuracy: · graduations can only be cut so finely · metal flexes under its own weight · bearings wear, and weather shifts the alignment THE RESPONSE Build the same instruments enormously larger, in stone and lime mortar, fixed to the ground. · a bigger arc means finer readable divisions · masonry does not flex or wear appreciably · nothing moves, so nothing goes out of alignment The trade: no portability, and one latitude each. The Samrāṭ Yantra at Jaipur — the largest of them gnomon graduated quadrant A right triangle whose hypotenuse is parallel to the earth's axis — that is, a sundial's gnomon — built 27 metres high, with quadrants flanking it graduated to read the shadow's position. At that scale the shadow's edge moves about a millimetre a second, and the instrument resolves time to roughly two seconds.
Jai Singh's programme also included commissioning Sanskrit translations of Euclid, Ptolemy and Arabic and Latin astronomical works, and sending an embassy to Portugal for European tables. It is the last great synthetic effort of the tradition before the colonial reorganisation of Indian education.

19Self-checkपरीक्षा

Check your reading

1 · Why is the zodiac divided into twenty-seven parts as well as twelve?

Two grids over one circle, one lunar and one solar. Each rāśi contains exactly 2¼ nakṣatras.

2 · A tithi is —

Because the definition is angular and the moon's speed varies, a tithi runs from about 21.5 to 26 hours — which is why tithis are sometimes repeated and sometimes skipped.

3 · What does the pattern of errors in Āryabhaṭa's planetary periods indicate?

Saturn is out by nearly a week; the sidereal day by a hundredth of a second. Theory-derived numbers would not have that shape of error.

4 · What did Nīlakaṇṭha's revised model change, and what did it not?

A geo-heliocentric system, and Nīlakaṇṭha does not claim more than that. The ellipses are Kepler's, a century later.

5 · Why is the ahargaṇa necessary?

The same reasoning behind the Julian Day Number used by astronomers today — introduced about a thousand years later.

Questions worth arguing about

Why did Āryabhaṭa's rotating earth not prevail?

Brahmagupta and others rejected it on physical grounds that were reasonable at the time: if the earth turned, things not attached to it would be left behind, and there was no theory of inertia to answer that objection. Note also that rotation makes no difference to the computations — the predicted positions are identical either way — so the question could be set aside as idle. A tradition organised around predictive accuracy has no lever to settle a question that does not affect predictions.

Is the mahā-yuga framework a scientific device or a cosmological doctrine?

Both at once, and separating them is anachronistic. Computationally it is a device for integer arithmetic with a defined epoch; cosmologically it is a claim about cycles of world-ages. Āryabhaṭa's willingness to alter the doctrine — equal quarters instead of 4:3:2:1 — where computation required it suggests he treated the computational role as primary.

What was lost when the pañcāṅga tradition became purely almanac-making?

The calculations survived, and are still performed. What thinned out was the observational side that corrects the parameters — Parameśvara's fifty-five years of eclipse records, Sāmanta's bamboo instruments. A computational tradition without an observational one drifts, slowly and invisibly, and several modern pañcāṅgas still use parameters that observation would have corrected long ago.

20Glossaryशब्दकोशः

IASTDevanāgarīSense
ahargaṇaअहर्गण“Heap of days”: the continuous count of days since the epoch.
amāvāsyā / pūrṇimāअमावास्या / पूर्णिमाNew moon and full moon.
ayanaअयनA solar half-year: uttarāyaṇa (northern) or dakṣiṇāyana (southern).
ghaṭikāघटिकाA time unit; sixty make a day and night.
golaगोलThe sphere; spherical astronomy.
kāla-nirṇayaकालनिर्णयThe determination of time — the tradition's statement of its own purpose.
karaṇaकरणHalf a tithi; also, a genre of practical computation manual.
krānti-vṛttaक्रान्तिवृत्तThe ecliptic.
mahā-yugaमहायुग4,320,000 years; the period over which revolution numbers are stated.
nakṣatraनक्षत्रOne of twenty-seven arcs of 13° 20′; also the star naming it.
pakṣaपक्षA fortnight: śukla (bright, waxing) or kṛṣṇa (dark, waning).
pañcāṅgaपञ्चाङ्ग“Five limbs”: the almanac, and the five quantities it states.
rāśiराशिOne of twelve 30° signs of the zodiac.
śaṅkuशङ्कुThe gnomon.
śīghroccaशीघ्रोच्चThe “fast apex” — a parameter which for the inner planets amounts to the heliocentric period.
siddhāntaसिद्धान्तA complete astronomical system; the genre of the major treatises.
tithiतिथिA 12° increment of the moon–sun elongation.
viṣuvatविषुवत्The equinox.
yantraयन्त्रAn instrument.
yogaयोगA pañcāṅga limb: the sum of the two longitudes in units of 13° 20′.
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