Spherical astronomy of the Indian Classical Astronomy（インド古典天文学の球面天文学）
Yukio Ôhashi （大橋由紀夫）
3-5-26, Hiroo, Shibuya-ku, Tokyo 150-0012, Japane-mail:I.

yukio-ohashi@chorus.ocn.ne.jp

Introduction
I have presented two papers on the history of Indian astronomy in a conference and a seminar atNAOJ, namely, Ôhashi (2011a) on the Vedāṅga astronomy, and Ôhashi (2011b) on the planetary
theory of the Classical Siddhānta astronomy.

The present paper is a kind of continuation of thesepapers. I would like to discuss the spherical astronomy of the Classical Siddhānta astronomy,
where orthographic projection and plane trigonometry are used.
The history of Indian astronomy can roughly be summarised as follows. [For an overview ofIndian astronomy, see Ôhashi (1998) in Japanese or more detailed Ôhashi (2009) in English.](i)

Indus civilisation period (ca.2600BCE – ca.1900 BCE).

(ii) Vedic period (ca.1500 BCE – ca.500 BCE).
(iii) Vedāṅga astronomy period (From sometime between the 6th and 4th centuries BCE up to
sometime between the 3rd and 5th centuries CE?).
(iv) Period of the introduction of Greek astrology and astronomy (Sometime around the 3rd and4th century CE?).
(v) Classical Siddhānta period (Classical Hindu astronomy period). (From the end of the 5thcentury up to the 12th century).
(vi) Coexistent period of the Hindu astronomy and Islamic astronomy (From the 13/14th century
up to the 18/19th century).
(vii) Modern period (Coexistent period of the modern astronomy and traditional astronomy).
(From the 18/19th century onwards).
Around the 3rd (?) century CE, Greek horoscopic astrology was introduced into India, and aroundthe 4th (?) century CE, Greek mathematical astronomy seems to have been introduced into India. In
the Classical Hindu Astronomy period (Classical Siddhānta period) (from the end of the 5th century
to the 12th century), Indian astronomy did not receive apparent foreign influence, and developedindividually.

The Classical Hindu Astronomy period produced several famous astronomers, such as, Āryabhaṭa(b.476), Varāhamihira (6th century), Bhāskara I (fl.629), Brahmagupta (b.598), Lalla (ca.8th or 9thcentury), Vaṭeśvara (b.880), Mañjula (fl.932), Śrīpati (fl.1039/1056), Bhāskara II (b.1114) etc. And
also the anonymous Sūrya-siddhānta (ca.10th or 11th century) is a very popular Sanskrit astronomical
text of this period.

Some of these works are still considered to be authoritative by modern

traditional Hindu calendar makers etc. The period during which these classical astronomical works
were composed can be called Classical Siddhānta period or Classical Hindu Astronomy period.
The “Siddhānta” is the fundamental treatise of mathematical astronomy in Sanskrit.

II.

Three problems ----- Indian spherical astronomy(II.1) Introduction
The Siddhāntas usually consist of two parts, namely, the section of the calculation of planetary
position, and the section of spherics.

The section of the calculation of planetary position issubdivided into the chapter on the mean motion, the chapter on the true motion, the chapter on the“three problems” (direction, place and time), the sections of lunar and solar eclipses, the sections ofthe conjunction of planets and stars, the section of heliacal rising and setting, the section of lunar
phase etc.
What I am going to present here is the spherical astronomy in the chapter on the “three problems”(chap.III) in the Sūrya-siddhānta (ca.10th or 11th century), one of the most popular Sanskrit work onastronomy. There are some versions of the Sūrya-siddhānta, and I followed the most popularRaṅganātha’s version.

Raṅganātha (son of Ballāla) flourished around 1603 AD at Kāśī (=Varanasi=Banaras) in India. He was born in a family of astronomers. He wrote a very popular Sanskritcommentary (1603 AD) on the Sūrya-siddhānta. Raṅganātha’s son Munīśvara (b.1603 AD) wasalso an astronomer, and composed a Sanskrit astronomical treatise Siddhānta-sārva-bhauma (1646AD) etc. There are some other astronomers in their relatives also.
Some texts of the Sūrya-siddhānta have 51 verses in its chap.III, while some texts have 50 verses
in its chap.III, where verse no.32 occurs twice. For example:
The Súrya-siddhánta, an ancient system of Hindu atronomy; with Ranganátha’s exposition, theGúḍhártha-prakáśaka, edited by FitsEdward Hall with the assistance of Bápú Deva Śástrin,(Bibliotheca Indica), Calcutta, Asiatic Socirty of Bengal, 1859, has 50 verses in its chap.III.
The Súrya-siddhánta, an ancient system of Hindu atronomy; with Ranganátha’s exposition, theGúḍhártha-prakáśaka, Calcutta, Sangbada Jnanaratnakara Press, 1871, has 51 verses in its chap.III.
There are two English translations of the Sūrya-siddhānta as follows. The both have detailed
English commentary.

Bápú Deva Sástri and Lancelot Wilkinson (tr.): The Súrya Siddhánta, or an Ancient System ofHindu Astronomy, followed by the Siddhánta Śiromani, translated into English, with extensive
explanatory notes, Bibliotheca Indica Vol.32, Calcutta, Asiatic Society of Bengal, 1861; Reprinted:
Amsterdam, Philo Press, 1974.

This translation has 50 verses in its chap.III.

Ebenezer Burgess (with the help of the Committee of Publication, particularly W.D. Whitney)(tr.): The Sûrya Siddhânta, a text-book of Hindu astronomy, (originally published in the Journal ofthe American Oriental Society, 6(2), 1860, 141 ~ 498), Reprint edited by Phanindralal Gangooly
with an introduction by Prabodhchandra Sengupta, Calcutta, University of Calcutta, 1935;
Reprinted: Delhi, Motilal Banarsidass, 1989. This translation has 51 verses in its chap.III.

(II.2) Construction of the gnomon
The Sūrya-siddhānta (III.1-4) reads as follows.
“1. On a water-levelled even stone-surface or hard cement, draw a circle with a radius of a desired
number of digits (aṅgulas) of the gnomon (śaṅku).”“2. At its centre, a gnomon with a height of fixed 12 digits should be erected. The tip of its
shadow will touch the circle in the forenoon and afternoon. ” (See Fig.1 (a).)“3. On the circle, mark the two points which are called former and latter. Between them, draw
the south-north line by a “fish figure”.” (See Fig.1 (b).) (The “two points” are the points Fp andAp in Fig.1 (a) and (b).)“4. Between the south and north directions, draw the east-west line by a “fish figure”. By fishfigures” between the cardinal points, intermediate directions may be determined likewise.”Fig.1 (c).)

Fig.1, Construction of the gnomon(See(II.3) An observed shadow and the celestial sphere
The Sūrya-siddhānta (III.5-7) reads as follows.
“5. Draw a circumscribed rectangular by the lines going out from the centre. The desiredshadow (prabhā) is known to be on the line of the “base” (bhuja).” (Here, the “desired shadow” isthe segment CB in Fig.2 (a), and the “line of the base” is the line BeBw in Fig.2 (a).)“6. The (vertical) line (on the celestial sphere) passing through the east and west points is called
prime vertical (sama-maṇḍala). And also, the six o’clock line (un-maṇḍala) and the meridian(viṣuvan-maṇḍala) are known.”(See Fig.2 (b).)“7. Establish an east-west line which passes through the tip of the equinoctial midday shadow.
The distance between the (tip of) desired shadow and the equinoctial line (established as above) iscalled “amplitude” (agrā).” (The “equinoctial midday shadow” is the segment Cm in Fig.2, and the“east-west line which passes through the tip of the equinoctial midday shadow” is the line MeMw inFig.2 (a).

The “amplitude” is the distance between the lines MeMw and BeBw in Fig.2 (a). Its

corresponding segment Gh (in Fig.2 (b)) on the celestial sphere is also called “amplitude”.)
The observed shadow changes at every moment, and it corresponds to the hypotenuse of theshadow. The hypotenuse is obtained by the Pythagorean theorem:(Shadow-hypotenuse)2 = (Gnomon-height)2 + (Shadow-length)2.
The gnomon-height is usually considered to be 12 digits.
The Radius (= R) of the celestial sphere is usually considered to be 3438, because 360×60 minutes
divided by 2π are about 3438 minutes.

Fig.2, Observed shadow and celestial sphere(II.4) Midday shadows
The Sūrya-siddhānta (III.12cd-13ab) tells to obtain the “equinoctial midday shadow”(viṣuvat-prabhā) (the segment Cm in Fig.3 (b)) at the observer’s place.
The Sūrya-siddhānta (III.13cd-14ab) tells to convert the “equinoctial midday shadow” (thesegment Cm) and the “gnomon height” (the segment CG in Fig.3 (b)) to the celestial sphere, using
shadow hypotenuse (the segment Gm in Fig.3 (b)), and obtain the co-latitude (90°－φ) and latitude(φ) of the observer as follow.
× Gnomon-height (CG) = Rsine of the co-latitude.

× Shadow-length (Cm) = Rsine of the latitude.
They are converted into the arcs of co-latitude and latitude.
The Sūrya-siddhānta (III.14cd-16) tells the relationship between the desired midday shadow,called “base” (bhuja) (the segment Cd in Fig.3 (b)), the Sun’s midday zenith distance (ζ) and its
declination (δ).

The “shadow-hypotenuse” is the segment Gd in Fig.3 (b).
× “base” (Cd) = Rsine of the Sun’s zenith distance (= Rsinζ).

It is converted into the arc of the Sun’s midday zenith distance (ζ).
If the Sun’s midday zenith distance and declination are in the opposite directions, φ =ζ + δ.
If the Sun’s midday zenith distance and declination are in the same direction, φ =|ζ－δ|.

Fig.3, Midday shadows(II.5) Solar longitude
The Sūrya-siddhānta (III.17cd-19) tells to obtain the Sun’s longitude (λ) from its declination (δ),
when the obliquity of the ecliptic (ε) is known.

In the case of the first quadrant, it is as follows.

Rsinλ.
This formula can be understood as follows. The segment gKc in Fig.4 (a), which is projected as
the segment GKc in Fig.4 (b), is equal to Rsinλ. And also, the segment KqKc is equal to Rsinδ.
As the triangles gKcKq and GLcLq are similar, and the segment LqLc is equal to Rsinε, the above
equation is obtained.

Fig.4, Solar longitude(II.6) Time and shadow
The “ascensional difference” (cara) corresponds to the arc between the point of sunrise and the sixo’clock line. The “Rsine of the ascensional difference” (cara-jyā) (the segment GAe in Fig.5 (c)) isobtained as follows. (The angle AGB is equal to the observer’s latitude (φ).)
GB = Rsinδ,AB =(= Rsinδ×tanφ).

“Cara-jyā” (GAe) =AB,

where r is the rdius (BD) of the diurnal circle, that is r = Rcosδ.
The Sūrya-siddhānta (III.34cd-36 in the texts of 51 verses) tells that the sum or difference of theRadius and the “cara-jyā“ is the “day-measure“ (antyā) (the segment AeM in Fig.5 (c)):
When the Sun’s declination is north: “Day-measure” (antyā) (AeM) = R + “cara-jyā” (GAe).
When the sun’s declination is south: “Day-measure” (antyā) (AeM) = R－“cara-jyā” (GAe).
The “day-measure“ (antyā) (AeM) diminished by the “Rversed-sine (utkrama-jyā) of the hourangle (nata) of the Sun” (the segment MHe in Fig.5 (c)), then multiplied by the “day-radius”(ahorātra-ardha) (r = BD), and divided by the Radius (R = GM) is the “divisor” (cheda) (thesegment AHn in Fig.5 (c)):“Divisor” (cheda) (AHn) =(“day-measure”－“Rversed-sine of the hour angle”).

Then, the “divisor” (the segment AHn in Fig.5 (c) or AHm in Fig.5 (b)) multiplied by the “Rsineof co-latitude” (lamba-jyā) (the segment MMg), and divided by the Radius (the segment GM) is the“Rsine of the Sun’s altitude” (śaṅku) (the segment HnTn in Fig.5 (c) or HmTm in Fig.5 (b)):
Rsine of the Sun’s altitude (HnTn or HmTm) =By the Pythagorean theorem:(Radius)2－(Rsine of the Sun’s altitude)2 = (“Rsine of the Sun’s zenith distance” (dṛg-jyā))2.
The actual gnomon-shadow can be calculated from this value.
The Sūrya-siddhānta (III.37-39 in the texts of 51 verses) explains the method to obtain time fromthe observed shadow. The method is just opposite to the above procedure.

Fig.5, Time and shadw

(II.7) Zodiacal signs
The Sūrya-siddhānta (III.42 in the texts of 51 verses) explains the method to calculate the rightascension (“laṅkā-udaya”, which means the ascension at Laṅkā which is supposed to be on theequator) of the first points of the zodiacal signs. The method is:.
From its result, the arcs of the right ascensions of the first points of zodiacal signs can becalculated.
This equation can be understood as follows. In the Fig.6 (a), the segment Q1q1 is the Rsine ofthe right ascension (α) of the first point of Taurus. The segment GC1 is Rcosine of the declination(δ) of the first point of Taurus. Now, the segment GC3 is the Rcosine of the obliquity of ecliptic (ε).
The segment C1g1 is the Rsine of the longitude (λ) of the first point of Taurus (λ = 30°) projected tothe plane of the equator, that is:
The segment C1g1 =.

Considering the above equations,
Rsinα (the segment Q1q1) = C1g1×Radius÷GC1, that is:.

Fig.6, Zodiacal signs

The Sūrya-siddhānta also tells related topics conceding the oblique ascension of the observer’s
latitude, the rising point of the ecliptic called “lagna”, etc.

In Fig.6 (b), the point Ve is the firstpoint of Aries, the point C1 is the first point of Taurus, etc., and the segment VeQ1 corresponts to theright ascension of the first point of Taurus, etc., and the segment VeD1 corresponds to the obliqueascension of the first point of Taurus at the latitude of observer, etc.

III. Conclusion
We have seen that several problems of spherical astronomy can be solved by plane trigonometry.
They can be understood easily, and sometime they will be useful in astronomy education.

References:Ôhashi, Yukio （ 大 橋 由 紀 夫 ） (1998): “Indo-no dentō-tenmongaku nitsuite – tokuni
kansoku-tenmongaku-shi nitsuite” (インドの伝統天文学について ― 特に観測天文学史について, Traditional Astronomy in India – with Special Reference to the History of ObservationalAstronomy), Tenmon-geppō (天文月報, The Astronomical Herald), 91(nos. 8, 9 and 10), pp. 358 –364, 419 – 425 and 491 – 498 [in Japanese].
Ôhashi, Yukio (2009): “The Mathematical and Observational Astronomy in Traditional India”, inJ.V. Narlikar (ed.): Science in India (History of Science, Philosophy and Culture in IndianCivilization, Volume XIII, Part 8), New Delhi, Viva Books, pp.1 – 88.
Ôhashi, Yukio (2011a): “On Vedāṅga astronomy: The Earliest Systematic Indian Astronomy”, inNakamura, Orchiston, Sôma and Strom (eds.): Mapping the Oriental Sky. Proceedings of the SeventhInternational Conference on Oriental Astronomy, Tokyo, National Astronomical Observatory ofJapan,pp.164 – 170.
Ôhashi, Yukio (2011b): “Planetary theory of the Indian Classical Astronomy”, in Sôma andTanikawa (eds.): Proceedings of the Fourth Symposium on “History of Astronomy” (「第４回天文学史研究会」集録), Tokyo, National Astronomical Observatory of Japan,pp. 20 – 30.