A Concise History of Science in India c6.s4.t1: The Sphere and the Coordinates, Direction, Place and Time

A Concise History of Science in India c6.s4.t1

Language: English

The Sphere and the Coordinates, Direction, Place and Time

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In all the texts the earth is assumed to be an immovable sphere at the centre of the universe. The asterisms with the sun, the moon and other planets have a retrograde motion, that is from east to west, completing the circle in a day; the sun, the moon and the planets have a direct motion of their own, that is from west to east, as a result of which they are constantly beaten by the asterisms and fall behind them. As we have noticed previously, Āryabhaṭa showed that the apparent daily retrograde motion can be explained by the direct rotation of the earth, a view opposed by other Hindu astronomers.

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The Sphere and some Important Great Circles

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To represent the positions and motions of the planets, stars and heavenly bodies, a celestial sphere of an arbitrary radius *R* is imagined, at the centre of which lies the earth or the observer. A vertical line through the observer meets the celestial sphere above and below at points *Z*, *Z'* (Fig. 2.1) called the zenith (*ūrdhva svastika*) and the nadir (*adha svastika*). The plane passing through the observer *O* at right angles to the vertical line *ZZ'* cuts the celestial sphere in a great circle *NESW*, called the observer's celestial horizon (*kṣitija*). The horizon passes through the four cardinal points, north, east, south and west. Thus Varāhamihira in his *Pañcasiddhāntikā*a says: 'That (circle) in which the sky is joined as it were to the earth is called "horizon"; in it are drawn east-west and north-south lines.' Varāha used the term *harija* for horizon.

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The east-west and the north-south lines are easily determined by observing the points where the shadow of the gnomon touches a circle drawn on the horizontal plane in the forenoon and in the afternoon. The line joining these two points is parallel to the east-west line passing through *O*; the north-south line passing through *O* is perpendicular to the former. All Hindu texts give clear rules for finding these cardinal directions.

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a *PSi.*, xiv. 17.

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The great circle *NZSZ'* passing through the north-south points and the zenith and the nadir is the meridian (*yāmyottara-vṛtta*). The great circle *EZWZ'* through the east-west points and the zenith and the nadir is called the prime vertical (*sama-maṇḍala*). Brahmagupta, in his *Brāhmasphuṭa-siddhānta*,a defines these circles as follows: ‘One circle called the *sama-maṇḍala* has its plane stretching east and west; another lying north and south is called the *yāmyottara-vṛtta*; another termed the *kṣitija* encircles the other two like a girdle. At the common centre of these circles is situated the observer on the earth.’

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FIG. 2.1 The celestial sphere and some important great circles.

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In Fig. 2.1, the great circle *QELW* represents the celestial equator (*viṣuvanmaṇḍala*) of which the poles are *P*, *P'*. The angular distance of the pole from the horizon equals the latitude φ of the observer, which is equal to the zenith distance of *Q*, the point where the celestial equator meets the meridian. Brahmagupta defines the equator as the circle which meets the horizon at the east-west points and of which the top lies to the south of the prime vertical by the latitude of the observer, and the bottom as much to the north of it. Varāhamihira defines the interval between the pole and the horizon as the terrestrial latitude of the observer. Diurnal motions of planets and stars are represented by small circles parallel to the equator.

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a *Br.Sp.Si.* *Gola.*, 48.

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Of the hour circles or secondaries to the equator special importance attaches to the 6 o'clock circle passing through the pole and the east-west points. In Hindu astronomy this is called *unmaṇḍala* which, as Āryabhaṭa says, is above and below the horizon by the amount of the observer's latitude. The increase and the decrease of the day and night are measured on this circle.

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Finally, we have the ecliptic (*apamaṇḍala*) which is inclined to the equator at an angle of 24°, meeting later at the first point of Aries (*Meṣa*) and the first point of Libra (*Tulā*). Āryabhaṭa states that the northern half of the ecliptic is from the beginning of *Meṣa* to the end of *Kanyā* and its southern half is from the beginning of *Taulya* to the end of *Mīna*.a

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FIG. 2.2

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*Coordinates*

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In representing the positions of celestial bodies, the horizontal, the equatorial and the ecliptic systems are used. In the horizontal system, the altitude (*unnata*) or the zenith distance (*nata*) is defined as in modern astronomy, but the azimuth is reckoned from the prime vertical either from the east or the west point on the horizon, but never exceeding 90°. The azimuth circle is called the *dṛgmaṇḍala* and the azimuth the *digaṃśa*. In the equatorial system, the two coordinates are the declination or *krānti* and the hour angle. *Krānti* is the angular distance of the body from the equator measured along its secondary through the body. Hour angle is measured in two ways, either by the *unnata-ghaṭī* (time-altitude) or by *nata-ghaṭī*. *Unnata-ghaṭī* is the hour angle between the declination circle through the body and that through its rising point on the horizon; *nata-ghaṭī* is the hour angle between the declination circle through the body and the meridian. In the ecliptic system, the position of a body is given by its latitude (*vikṣepa*) and its longitude (*sphuṭa*) measured from the fixed point marked by Zeta Piscium (*Revatī*).

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a *Ā. Gola.*, 1.

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*Latitude, Zenith Distance and Declination*

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When the sun is at either of the equinoxes, its meridional zenith distance is equal to the latitude of the observer. This is easily determined by measuring the equinoctial shadow (*viṣuvatprabhā*, *palabhā*, etc.) *AH*, cast by the gnomon *OA* (Fig. 2.2).

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Let *OA* = *g*, *AH* = S_e, *OH* = *h*

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∴ h = √(g² + S_e²).

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The latitude (*akṣa*), φ, and the colatitude (*lamba*), 90—φ, are given by

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R sin φ = RS_e/(√(g²+S_e²)); R sin (90-φ) = Rg/(√(g²+S_e²)).

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Also

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S_e = (g sin φ)/(sin (90-φ)).

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The *Sūrya-siddhānta*a gives the above rules as follows: 'Radius multiplied respectively by the equinoctial shadow and the gnomon and then divided by the equinoctial hypotenuse gives the sine of latitude and the sine of colatitude.'

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For any declination δ₁ north of the equator or δ₂ south of the equator, the meridional zenith distances Z₁ and Z₂ are given byb

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R sin Z₁ = RS₁/h₁

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R sin Z₂ = RS₂/h₂

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where S₁, S₂ are the shadows and h₁, h₂ are the corresponding hypotenuses of the gnomon. From Fig. 2.2,

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φ = Z₁ + δ₁

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φ = Z₂ - δ₂

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that is,

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φ = Z ± δ.

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Thus the declination of the sun can be found from the latitude of the place and the meridional zenith distance. In this way the maximum declination or the obliquity of the ecliptic with the equator, given in Hindu astronomical texts as 24°, must have been determined.

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*Relation between the Declination, the Longitude and the Obliquity of the Ecliptic with the Equator*

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The *Sūrya-siddhānta*c gives the relation between declination, the longitude and the obliquity of the ecliptic with the equator as follows: 'The sine of the greatest declination is thirteen hundred and ninety-seven;

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a *Sū. Si.*, iii. 13, 14, 16, 17.

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b *Sū. Si.*, iii. 15.

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c *Sū. Si.*, ii. 28.

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by this multiply any sine, and divide by radius; the arc corresponding to the result is said to be the declination.’ According to Bhāskara I,a the *R* sine of the given longitude, multiplied by 1397 and divided by the radius, yields the *R* sine of the declination for that instant of time.

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The relationship may be expressed as follows: $R sin δ = (1397 . R sin λ)/R = (R sin ε . R sin λ)/R$

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where δ = declination of the sun; ε = obliquity of the ecliptic with the equator = 24°; R sin ε = 1397'; λ = longitude of the sun, that is the sun’s angular distance from the first point of Aries along the ecliptic; R = radius of the sphere = 3438'. The above formula follows readily from the properties of the spherical triangle as also from the geometry of the similarity of triangles (Fig. 2.3).

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FIG. 2.3

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Υ MQ = equator; Υ SC = ecliptic; S = the sun; Υ S = λ = longitude of the sun; SY M = CQ = ε = obliquity; SM = δ = declination of the sun. Δs COK, Snm are similar. $∴ Sm/Sn = CK/OC$ $∴ (R sin δ)/(R sin λ) = (R sin ε)/R$ $∴ R sin δ = (R sin ε . R sin λ)/R$

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Since ε is known, this formula can be applied to find out the longitude of the sun from its declination.

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The Ascensional Difference or Cara

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The formula for finding the ascensional difference or *cara* affords yet another example of the skilful use of the celestial sphere and its circles. The ascensional difference is variously defined as the difference between the right ascension and oblique ascension of a rising point on the ecliptic or the difference between the times of the rising of the sun on the horizon and the 6 o'clock circle (*unmaṇḍala*). In Fig. 2.4(*a*), H is the rising point on the ecliptic Υ HC, Υ is the vernal equinox, Υ EQ is the equator and E the east point. The ascensional difference FE is clearly the difference between the oblique ascension Υ E and the right ascension Υ F of the rising point H. FE is also equal to the arc HG or HBG (Fig. 2.4(*b*)) which represents the time difference of the sun’s risings on the horizon and the 6 o'clock circle.

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a *MBh.*, iii. 6. Bhattacharyya and Sen, pp. 144–155.

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In Fig. 2.4(*b*) (*see also* Fig. 2.4(*c*)), *NES* is the horizon, *EQ* the equator, *P* the pole, *Z* the zenith, *HGR* the diurnal circle of the sun (or star), *PGE* the 6 o'clock circle, PON = the latitude of the place, φ, and ROQ = BRO = the declination of the sun (or star), δ. The two important elements in the above constructions are the radius *RB* of the diurnal circle, which is called the ‘day radius’ (*dinavyāsadala*) and the

Linked texts (5)

Sūryasiddhānta — Maya, Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Mahābhāskarīya — Bhāskara I, Sanskrit
Pañcasiddhāntikā — Varāhamihira, Sanskrit