Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva

Now, all planets move in circular orbits. The number of degrees which each planet moves in its orbit in the course of a day is fixed. There again, the number of yojana-s moved per day is the same for all planets. For planets which move along smaller orbits, the circle would be completed in a shorter time. For those which move along larger orbits, the circle would be completed only in a longer period. For instance, the Moon would have completely moved through the twelve signs in 28 days, while Saturn will complete it only in 30 years. The length of time taken is proportional to the size of the orbit. The completion of the motion of a planet once in its orbit is called a bhagaṇa of that planet. The number of times that a planet completes its orbit during a catur-yuga is called its yuga-bhagaṇa (revolutions per aeon).

Now is demonstrated the situation and motion of the bhūgola, vāyugola and bhagola. The Earth is a sphere supporting on its entire surface all things, moving and non-moving, maintaining itself (suspended) in the sky at the centre of the celestial sphere (nakṣatra-gola) by its own power, and not depending on any other support. Now, it is the nature of all heavy things to fall on the Earth from all regions of the sky all around. Hence, the Earth, everywhere, is below the sky. Similarly, from all locations on the Earth, the sky is above. Now, the southern half of the Earth-sphere is abundant with regions of water. And, in the northern half, the land region is in profusion and watery region less. Then, with the land of India (Bhārata-khaṇḍa) appearing to be in the upward (northern) direction, at the confluence of the landed and watery division (of the Earth), there is a city known as Laṅkā. Conceive a circular line (vṛttākāra-rekhā) from that place, east-west, cycling round the Earth. On this line are situated four cities (including Laṅkā), to the west Romakapurī, to the other (diametrically opposite) side Siddhapura, and to the east Yavakoṭi.

The seven great circles which are frequently employed in deriving various results in this chapter are listed in Table 10.1. These circles are indicated by solid lines in Figure 10.1. Three more circles which are referred to later in the chapter are indicated by dashed lines. In Table 10.1, the second column gives the names of the circles in Sanskrit. The third column gives their modern equivalents. In the last column we have listed the poles (visible ones with ref. to Figure 10.1) of these great circles.

Now, the method to identify the (four) directions. First prepare a level surface. It should be such that if water falls at its centre, the water should spread in a circle and flow forth on all the sides uniformly. That is the indication for a level surface. On this surface draw a circle (in the following manner): Take a rod slightly bent at both ends and, with one end of the rod fixed at the centre, rotate the other end on all sides (so that a circle will result). The point where the end (of the rod) is fixed is known by the terms kendra and nābhi (centre). The line resulting from the rotation of the other end is called nemi (circumference). Fix (vertically) at the centre a uniformly rounded gnomon (śaṅku). On any morning, observe the point on the circumference where the tip of the shadow of the gnomon graces and enters into the circle and, in the same manner, also the point where the tip of the shadow graces the circumference and goes out of the circle in the afternoon. Mark these two points on the circle with dots. These two points, between themselves, will be almost along the east-west. For this reason, these are termed east and west points. These would have been the exact east and west points if they were the shadow-points of the stars which do not have any north-south motion. The Sun has a north-south motion on account of (its motion between) the solstices, and during the interval from the moment, when the western shadow-point gets marked, to the moment when the eastern shadow-point is formed, if the Sun has moved north due to the change in its declination, then to that extent the tip of the shadow would have moved to the south.

Calculate the gnomon and shadow of the Moon in the above manner. From these calculate the dṛkkarṇa in terms of yojana-s. Using the dṛkkarṇa-yojana calculate the minutes of the corresponding lambana. The minutes of lambana of the Sun and Moon are to be applied, respectively, to the (true longitudes of) Sun and Moon. When the resulting true longitudes of the two are the same, that will indicate the time of the middle of the eclipse.

Next is stated vyatīpātā. Now, if the declinations of the two, Sun and Moon, become equal at some time, when one of them is in an odd quadrant with the declination increasing and the other in an even quadrant with declination decreasing, then at that moment vyatīpāta is said to occur.

Here, the lagna corresponding to the rising and setting of a planet having a latitudinal deflection (vikṣepa), is calculated. The visibility correction (darśana-saṃskāra) is the correction that should be applied to the longitude of the planet to obtain the lagna corresponding to the rising and setting of the planets.

Now is stated the (computation of) the elevation of the cusps of the Moon. For this, first compute the second true hypotenuse (dvitīya-sphuṭa-karṇa) of the Sun and the Moon. Apply also the second true correction (dvitīya-sphuṭa-saṃskāra) for the Moon. Here, the view of (Śrīpati, author of) Siddhāntaśekhara is that when the radius of the ucca and nīca circles have been ascertained, a correction has to be applied to them. The view of Muñjāla, author of Laghumānasa is that the antya-phala of the Moon is to be multiplied by Moon’s manda-karṇa and five and divided by trijyā. These two views are worth consideration. Then, (for the Moon), compute the dṛkkarṇa and apply the corrections of bhū-pṛṣṭha and nati. Then compute the nati for the Sun. Compute and apply the correction of lambana for both the Sun and the Moon. Ascertain also the distance, at the required time, between the centres of the solar and lunar spheres.

In Indian astronomical texts, as a first approximation, the planets are taken to move uniformly along different circular orbits; the linear velocity of all the planets is taken to be a constant. In other words, if R _p be the radius of the planetary orbit (usually given in yojanā-s), and T _p be the sidereal time period, then where C is a constant. Given C, the radius of the planetary orbit is determined, if the time period of a planet is known. The term yuga-bhagaṇa refers to the number of complete revolutions made by the planet in a catur-yuga consisting of 43,20,000 years. This period is also called a Mahā-yuga and consists of four parts namely Kṛta-yuga, Tretā-yuga, Dvāpara-yuga and Kali-yuga.

The chapter commences with a discussion on the three spheres, (i) Bhūgola — the terrestrial sphere, (ii) Vāyugola — the equatorial celestial sphere (described with reference to the celestial equator which is revolving uniformly due to Pravaha-vāyu) and (iii) Bhagola — the zodiacal celestial sphere (described with reference to the ecliptic). This is followed by a discussion on the motion of equinoxes. Then, we find the description of some of the important great circles and their secondaries, which are used as the reference circles for describing the location of a celestial object using different co-ordinates. Finally, there is an elaborate discussion on the determination of the declination of a celestial object with latitude.

Now, towards demonstrating in detail, the above-stated principles, fifteen problems are posed in relation to the divergence between the said seven (great) circles.

Apart from providing the rationale behind different procedures, this chapter also summarizes and synthesizes all the problems related to the diurnal motion of the Sun and shadow measurements carried out with a simple instrument called śaṅku (gnomon).¹ Since a major portion of the chapter deals with the measurement of shadow (chāyā) cast by gnomon, the choice of the title of the chapter, ‘Chāyā-prakaraṇam’ (chapter on gnomic shadow) seems quite natural and appropriate.

The dṛkkarṇa d in yojanā-s is calculated in terms of the gnomon (R cos z), and the shadow (R sin z), as where D is the dvitīya-sphuṭa-yojana-karṇa and R _e is the radius of the Earth (Refer to Figure 11.37 and equation (11.274)). The lambana-s of the Sun and Moon should be applied, to obtain their true longitudes (for the observer). When the true longitudes are the same, it is the mid-eclipse. Now, we had where, we approximate d by D, the true distance from the centre of the Earth in the denominator (essentially ignoring the higher order terms in \(\frac{{{R_e}}}{D}\)).

Next is stated darśana-saṃskāra. This is indicated by that part of the ecliptic which touches the horizon when a planet having vikṣepa rises above the horizon. Consider a set up. in which the northern rāśi-kūṭa is raised and the planet is in one of the first three rāśi-s beginning from Meṣa; let the point of contact of the ecliptic and the rāśi-kūṭa-vṛtta passing through the planet be rising on the horizon. Further, suppose that the planet has vikṣepa towards the northern rāśi-kūṭa. Then, the planet will be raised above the horizon. Therefore, the gnomon of the planet at that time is computed first. When this gnomon is taken as Rcosine, its hypotenuse will be the distance between the planet and the horizon on the vikṣepa-koṭi-vṛtta. Now, the dṛkkṣepa-vṛtta meets the apakrama-vṛtta towards the south at a distance equal to the distance between the zenith and the dṛkkṣepa. In the dṛkkṣpepa-vṛtta itself, at a place north of the horizon, at a height equal to the distance between the horizon and the dṛkkṣepa, is the northern rāśi-kūṭa. The northern vikṣepa is that which moves towards the northern rāśi-kūṭa. Applying the rule of three: If the maximum distance between the horizon and the rāśi-kūṭa (vṛtta) touching the planet is the dṛkkṣepa, how much will be the distance from the horizon to the planet with vikṣepa; the result would give the gnomon of the planet with vikṣepa. Then, the proportion: If for the Draw P’P₁ perpendicular to CP and P₁F₁ perpendicular to OL.

Though the title of this short chapter is candra-śṛṅgonnati or ‘Elevation of the Moon’s Cusps’, it is exclusively devoted to the computation of the distance between the centres of the lunar and solar discs (bimbāntara). The bimbāntara of course figures prominently in the computations of the Moon’s phase and the elevation of its cusps, but these are not discussed in the Text as available.

It is now generally recognised that the Kerala School of Indian astronomy,¹ starting with Mādhava of Saṅgamagrāma in the fourteenth century, made important contributions to mathematical analysis much before this subject developed in Europe. The Kerala astronomers derived infinite series for π, sine and cosine functions and also developed fast convergent approximations to them. Here, we shall discuss how the Kerala School also made equally significant discoveries in astronomy, in particular, planetary theory.