Mean and True Positions of Planets as Described in Gaṇitagannaḍi – A Karaṇa Text on Siddhāntic Astronomy in Kannaḍa

The unpublished seventeenth-century Kannaḍa-language mathematical work Gaṇitagannaḍi is transmitted in a single palm-leaf manuscript.  It was composed by Śaṅkaranārāyaṇa Jōisaru of Śṛṅgeri and is a karaṇa text cast as a commentary on the Vārṣikatantrasaṅgraha by Viddaṇācārya. Gaṇitagannaḍi's unique procedures for calculations wer introduced in an earlier paper in volume 8 (2020) of this journal.  In the present paper the procedures for calculations of the mean and true positions of planets are described.


Mean and True Positions of Planets as Described in Gaṇitagannaḍi -A Karaṇa Text on Siddhāntic
Astronomy in Kannaḍa B. S. Shylaja and Seetharam Javagal Jawaharlal Nehru Planetarium, Bengaluru, and Independent

I N T RO D U C T I O N
T HIS IS THE SECOND PAPER published in this journal concerning an astronomical manual (Sanskrit karaṇa) of 1604 CE in Kannaḍa, named Gaṇitagannaḍi, that is a commentary on Vārṣikatantra of Viddaṇācārya, written by Śaṅkaranārāyaṇa Jōisaru. 1 The earlier paper discussed the first chapter which was on the exact instant when the sun enters the sidereal Aries at the beginning of a given solar year (Sanskrit meṣasaṅkrānti) and the mean longitudes of all the planets at that instant. The present paper is on the mean positions of the planets corresponding to a count of civil days from the epoch (ahargaṇa) specified by the lunar phase (tithi) and lunar month of a given year. This is part of the first chapter itself. We include the second chapter giving the true positions which are obtained after the application of the first equation (manda) correction and the second equation (śīghra) correction for the five planets which look like stars (tārāgraha), i.e., Mercury, Venus, Mars, Jupiter and Saturn.
In the first part of this paper we provide a description of the procedure highlighting the technique used in the text. This will be followed by a diplomatic transcription of the text. The palm leaf manuscript includes both Sanskrit and Kannaḍa languages and is written in the archaic script called Nandināgarī. We provide a translation of the commentary from Kannaḍa. The suggestions on An unpublished draft of the present paper was included in the book Shylaja and Javagal 2021. teaching young students who are not yet competent [with the basics], the method of expression of the meanings of the words becomes very important. Scholars should not mistake that there is a misinterpretation or that the grammatical rules for gender and number (vacana) and case (vibhakti) are violated.
In this paper we have tried to follow a similar rationale, so that it will be understandable for present-day students and scholars who may not have previously been exposed to the texts and methods of teaching in the medieval period. The mathematical treatment of concepts is given priority. We have provided the original text with the translation so that any doubt on possible deviation from the original can be inspected immediately. We believe that this will be useful for readers who wish to understand the mathematical techniques. As we shall see later, the crisp and short phrases require a lengthy explanation, even to a person conversant with the tools of mathematics. Here is an example: the commentary for verse number 2.3 states, The śīghrahara 10 [vyomendavaḥ] should be added to koṭiphala, if śīghrakendra is mrigādi and subtracted if it is karkyādi.
It is implied that a number 10, (called śīghrahara for a specific reason) should be added to the result obtained and called koṭiphala if the angle (called "centre" or kendra) with which we started the entire scheme of correction, is between 0 and 180; it should be subtracted if the angle is between 180 and 360 degrees. Thus, the English rendering requires a higher number of longer sentences. The difficulty posed by the absence of relevant diagrams in the original manuscript transmission is addressed by their introduction in this paper, in order to facilitate the mathematical treatment.

T H E M E A N P OS I T I O N S
G ENERALLY, ALL THE TEXTS (Siddhānta or Karaṇa) start with the calculation of mean positions starting from the value of the ahargaṇa itself. Gaṇitagannaḍi too starts with a modified ahargaṇa or dyugaṇa which corresponds to the count from the midnight before the meṣa saṅkrānti (the date of entry of the sun in to Aries). For the date of calculation which is referred to as desired date (iṣṭa dina) and the average lunar day decided by the phase of moon (tithi) are known. The calculations are done to fix the tithi of the phase of the moon corresponding to the date of entry of sun in to Aries (saṅkrānti). Since the year is reckoned on the first day after new moon before the meṣa saṅkrānti, given as caitra śuddha pratipat, the meṣa saṅkrānti need not coincide with this. It is here that the method differs from that of other texts such as Karaṇakutūhala (Balachandra Rao and Uma 2008) HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 by Bhāskarācārya, where the ahargaṇa count is directly used to get the mean longitudes. In a later text, the Grahalāghava by Gaṇeśa Daivajña, the total number of civil days is regrouped in to cakras of 11 years and a modified number is used for deriving the mean longitudes of all planets (Balachandra Rao and Uma 2006).
For the Moon, from the described procedure, it is clear that the mean motion is taken to be 12 + 12 68 + 1 degrees per day. The procedure requires that the longitude of the moon obtained in units of rāśi, degrees, minutes and seconds, be converted in to one unit namely degrees. Since rāśi is 30 degrees, its count is multiplied by 30 and added to the degrees count. For example, if the longitude of the moon is 1 rāśi and 10 degrees, it is equivalent of 40 degrees. To get the tithi we have to divide this number by the motion of the moon 12 + 12 68 + 1 per day. The procedure states that division by 12 should suffice and the quotient is not needed. It should be noted that the words bhāga, aṃśa and bhāgi are used interchangeably for degrees. Thus conversion to degrees and division by 12 provides the saṅkrānti tithi, which is used for the exact calculation of dyugaṇa. This provides the count of the month since the solar month starts from saṅkrānti tithi. Days from caitra śuddha pratipat to the date of interest are counted including the intercalary month if applicable. The number corresponding to that of saṅkrānti tithi is then subtracted. Here the idea of ṛtu (can be understood as season, a year has six ṛtus) is introduced to avoid one step of calculation. (Each ṛtu means 2 months). Therefore, dyugaṇa count from the meṣa saṅkrānti is obtained.
Let us take an example. In the year śaka 1069 (corresponding to 1147 CE) the meṣa saṅkrānti occurred on 5 th day after full moon in the month of Caitra based on a stone inscription (Shylaja and Geetha, 2016). This corresponds to March 24 as verified by another inscription recording a solar eclipse of the same year. Thus there is a difference of 20 days, which will be the carried on as the difference between dyugaṇa and ahargaṇa (which starts from Caitra śu 1) counts.
After getting the number of dyugaṇa, its verification is done by the week day by dividing by 7. The remainder zero corresponds to Thursday, 1 corresponds to Friday and so on. The difference between dyugaṇa and sāvana dhruva (it is the longitude for the beginning of the year for the planet as explained in the earlier paper) is called pada and is expressed in ghaḷige and vighaḷige. The subtraction is explained step by step. Thus pada defines the number of days to the desired date as counted from the meṣa saṅkrānti, (defined as the First point of Aries in the current usage of spherical astronomy text books) effectively, the longitude expressed in units of days.
A quantity called pada was used in Vaśiṣṭasiddhānta as described by Shukla (2016, p502). It was coined as 1/248 th part of the motion of the moon, equivalent to 1/9 th of a day. Similar definitions existed for Jupiter and Saturn too, to derive the longitude. It referred to unequal divisions of the planets' motion in a sidereal revolution. Here, in Vārṣiktantra, the definition itself is different. Lalla also has defined similar divisions in Śiṣyadhīvṛddhidatantra (Chatterjee 1981). But that definition also is very different -the word used is pāda, which means a quarter. In the conventional methods (example Karaṇakutūhala) the ahargaṇa count is converted to the dhruvaka (longitude) and added to the dhruvāṃśa obtained earlier.
Here the same procedure is adopted to get the dhruvaka using the quantity pada.
Therefore as a first step, pada is converted to units of degrees, arc minutes and arc seconds by dividing by 70. All the steps for this are explained -the first division gives degrees. The remainder is multiplied by 60 and then divided by 70 to get ghaḷige. The remainder of this is again multiplied by 60 and divided by 70 to get vighaḷige. This gives the mean longitude of the sun (stated as Ravi). The rationale for this is explained in the next sentence -the mean gati (daily motion) of Ravi 59 ′ (arcmin) 8 ″ (arcsec), which is expressed as 1 deg − 52 arcsec = 1 − 52/3600 deg Now, 52/3600 is very close to 1/70. Hence the motion of the sun is taken to be (1 − 1/70) degree per day. So, when the pada (in days, ghaḷige and vighaḷige) is divided by 70 and the ratio is subtracted from itself, the result would be the mean longitude of the sun in degrees, arcminutes and arcseconds, as the mean longitude is zero at the meṣa saṅkrānti.
For Mars, Mercury, Jupiter, Venus, Saturn, Moon's nodes and the Moon's apogee, the mean daily motions can be inferred to be: 4/229, (4/30 + 1/325), 1/361, 40/749, 1/897, 1/566 and 3/808 rāśis, respectively. The values in degrees per day are found by multiplying these by 30. The mean longitudes for any ahargaṇa would be dhruvāṃśa + (pada × gati) Thus for the remaining part of the verse gati is expressed as a ratio with the values of multiplier and divisor defined in the bhūtasankhya system for all planets. In case of Mars, the conversion in to units of degree is explained. If the pada is | | , is divided by 60 and added to , the sum is divided by 60 and added to . Thus the final value of pada is expressed in units of days. The mean motion of Mars is 4 rāśis or 120 degrees in 229 days here. Hence, numerator is 4 (in units of rāśis), expressed as kṛti, and denominator is 229 expressed as nidhi pakṣa netra here, and (pada) multiplied by the ratio is the mean motion of Mars during a pada.
For Budha śīghrocca, the multiplier is not stated explicitly. Using the idea that a plural has been used for the multiplier, the same number as for the previous one (Mars) is employed. The divisor is 30. Apart from this, to this, one has to add 1 divided by 325. Thus the correction is in 2 steps.
For the Moon, from the described procedure, it is clear that the mean motion   Finally another correction for only the sun and the moon is specified. That is to add the result of pada divided by 150; the rationale for this is not explained here but is covered in the chapter called Chāyādhikāra.
The mean values for all planets are for the midnight of Laṅkā (equator). Here the central meridian is described as passing through Laṅkā, Ujjain (Avanti), Rohtak, Mānasa Sarovar and another place called Svāmimale mountain (which is not mentioned in the original Vārṣiktantra). The correction for location of the observer requires the viṣuvadchāyā, which is the shadow length of a 12 aṅgula (inches) gnomon on the day of equinox. The lambajyā ( cosine) of this is multiplied by 5060 and divided by 120 to get the correction called yojanaphala.
The rationale is derived from Sūryasiddhānta (Bapu Deva Sastri 1861: vv. 1-59). The radius of earth is taken as 800 yojanas. Therefore its circumference is 5060 yojanas. Here the value for the ratio the circumference to the diameter of a circle, π, is taken as square root of 10. This calculation is needed to find the time difference between the observer's place and the standard meridian just defined. The daily motion of each planet is different and therefore the time differences will have to be calculated individually. However, the observer is not on the equator but a certain latitude φ. Therefore the circumference will be along a circle parallel to the equator, which is obtained by multiplying the radius by cos φ as shown in Figure 1. Here we have the value of latitude from the gnomon shadow on equinoctial day. Therefore to get the cosine of that we have to use the sine tables (provided in the next chapter on true values) for an angle (90 − φ). This works out to be 116|27. The number 5060 corresponding to the equator, is multiplied by 116|27 and divided by 120 so that we svadeśabhūparidhi, (the circumference of the small circle at the latitude of the observer) for the given place.
The viṣuvad chāyā is 3 aṅgula; the lambajyā is 116|27, can be understood as latitude, φ = tan −1 (3/12) = 14 ∘ |2 ′ |11 ″ cos φ = 116|27 From the Figure 1, the circumference of the earth at this latitude is bhūparidhi = 5060 × cos φ/120 Here, 5060, = 2 × 800 × √ 10, is taken from the Sūryasiddhānta The sines are to be obtained from the sine tables provided in the next chapter with the value of as 120. This latitude of 14 ∘ 2 ′ 11 ″ refers to a location north of Śṛṅgeri (latitude 13 ∘ 25 ′ ). However, since the author mentions the name Śṛṅgeri in the next chapter the Chāyādhikāra, this small difference may be attributed to his location in the outskirts of the town. The next step is to get the mean values for the time of the day. This is achieved by taking the difference from midnight of the same day or the previous day. This is multiplied by the gati (or the daily motion) of the individual planets.
Thus we see that the technique offers a different approach as compared to the conventional methods (like those of the Karaṇakutūhala) in the determination of the mean positions. The multipliers for deriving the dhruvakās (longitudes) of planets have been modified suitably.

T RU E PO S IT I O N S
T HE PROCEDURE is based on the Sūryasiddhānta but many details are not explicitly mentioned. After getting the mean positions as explained in Section 2, the corrections to derive the true positions are performed in two steps. The first correction is called the manda correction and the second one is called śīghra. The very first verse introduces the reference points needed for the second correction, called śīghrocca. The farthest point on the epicycle created for this correction also has the same name.
From the second verse onwards the procedure for the manda correction is described.
The positions of the mandocca (apogee) for all the planets are given. Then there is an explanation for how these numbers have been arrived at. As per the definitions provided in Sūryasiddhānta (1-41 and 42), the number of years since the epoch is multiplied by the number of revolutions in a mahāyuga or kalpa and is divided by the number of years in that period to get the mandocca in revolutions. The fractional part multiplied by 360 corresponds to the position on the ecliptic in degrees for the required date. He further states there can be an error of 1 or 2 liptis (arc minutes) from the epoch specified by the Ācārya and therefore he has added 1 degree to account for such small deviations.
The word kendra is used to indicate the angle between mandocca (or śighrocca) and the mean position (or position after manda correction). They are referred to by abbreviations manda (or śīghra). The corrections (as shown in the following discussion) derived using these are called mandaphala (or śīghraphala). It should be noted that word mṛgādi is used here. All along the discussion used the zodiacal signs -here it becomes luni-solar Mṛga corresponding to the month Mārgaśira.
This correction can be understood with the help of Figure 2. The basic idea of the manda correction is to account for the elliptical orbit, which is achieved with another smaller circle moving along the mean circular orbit. (Bapu Deva Sastri 1861). In Figure 3, at the apogee A, the planet is farthest and at B it is the closest. The planet moves along the small circle so that the distance difference is achieved over half the orbit. There is always a phase difference between the true position (shown in red colour) and the mean position of the planet (shown in black).
Since the projection of the position on the radius vector is needed for the calculation we have to get the sine of the angle called mandakendra, shown in Figure 1. The correction is indicated by the dashed line in Figure 3.
The next verses describe how to get the sine values. In all the astronomical texts the trigonometric sine ratio is treated as the arc sine (angle), called as jyā.
In this text the word jīva is also used. Here is taken as 120. The arc itself is expressed in units of degree (bhāga) arc minutes (kalā) and arc seconds (vikalā). A table is provided for the calculation of sine of any angle. Every 10 degrees is termed a khaṇḍa (section) and the value of the differences of sine is provided.  The author proceeds to explain how to get the sine for any angle. The angle should be divided by 10 to identify the khaṇḍa. All values preceding it are added up. The jyā difference corresponding to the remainder after dividing by 10 is obtained between the successive khaṇḍa and added to the earlier sum. This procedure will be clear with an example. If we want find the sine for 34 degrees, we look up the value for number 3, since 34/10 has quotient 3 (khaṇḍa number) and the remainder is 4. The sum of all jyā values preceding 3 is 21 + 20 + 19 = 60. Now we have to interpolate between khaṇḍas 3 and 4 for the remainder 4, as ( 17 10 × 4) = 6 and remainder is 8. This is added to 60 as 66 and remainder 8 is multiplied by 60 to get 48 ′ . Thus the sine of 34 is 66 ∘ 48 ′ .
The text also gives the same numbers in the reverse order as

2|5|9|12|15|17|19|20|21|
The sums of all the preceding values of jyā, are provided in the next verse in the bhūtasankhya system. These are termed pinḍīkŗta jīva. Another interesting part introduced by the author is the table of utkramajyā. This trigonometric ratio (1 − cos) is not included along with the other three in the text books of today, although it has been named versine.

|2|7|16|28|43|60|79|99|120|
For example if the angle is 60, its utkramajyā is R (1 − cos 60) which is 60. This is the number in the 6 th khaṇḍa. The next verse gives the values of divisors for manda corrections for the planets. Here the author follows a technique that is different from others, for example, Karaṇakutūhala. In most Indian texts on astronomy including Sūryasiddhānta, the computation of mandaphala is based on an epicycle model ( Figure 4).
where and are the radii of the epicycle and the deferent, respectively.
HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 From the descriptive procedure we understand that the mandaphala is given as sin × 60 where is the mandakendra, and the denominator is called the corrected mandaccheda. Here and are specified for each planet. For instance, for the Sun, = 3230 (vyomāgnidanta) and = 90 (khāṅka). The first term in the denominator is much larger than the second term.
Therefore, sin × 60 is approximated as sin × 60 In the Sūryasiddhānta, is of the form ′ − ′ sin ′ 1 − ′ ′ sin and = 360. For instance, for the Sun, ′ = 14 and ′ = 1 3 . In Table 3 the values for mandaphala from the two texts are compared. Table 3 shows that the Ganitagannaḍi expression for the mandaphala would give very nearly the same results as the ones following from Sūryasiddhānt rules. For all planets the divisors are derived and provided using the siddhanatic vaues of the peripheries. The values from Karaṇakutūhala (Balachandra Rao and Uma 2008) are compared in Table 4.
The correction is explained in the next verse. It is called tātkālika, which can be interpreted as "as applicable for that instant." The procedure to apply this correction appears to be have been devised by the author himself. The sine of the mandakendra is converted to lipti (arcminutes) by multiplying the value in degrees by 60. Then the liptis are added so that the entire value is in liptis. These are to be divided by different numbers for each planet specified by the verse beginning with khāṅkaiḥ, 90 for the sun and so on. The result is again added to the numbers specified earlier in the verse beginning with vyomāgnidantaḥ, to get the divisors. Dividing the sine of mandakendra by this corrected divisor gives the mandaphala. The numbers are 90 (khāṅka), 490 (khatāna), 300, (viyadabhrarāma), 70 (khāsva) 170 (kha śailendu), 21 (indupakṣa), 380 (khāṣṭāgni). Thus the correction extends to the fraction of a degree.   The final value after the correction is called mandasphuṭa (corrected for manda).
The next verse provides similar divisors śīghracheda for the second correction. Although the author declares it is on the same lines as done for manda correction, the procedure is not very clear as can be seen later. Prior to the discussion on śīghra correction, he summarises the procedure for manda in a single sentence, whose translation was also difficult. Here is the summary: • Get the mandakendra, difference between mandocca, the apogee and the mean • Get the sine of mandakendra and koṭi ( cosine) also. (Koṭi is not needed for manda correction) • Convert the sine in to arc minute by multiply by 60 and adding to the lipti component. • Divide it by the appropriate number as given by the sequence stated in the verse starting with 90. • Add the result to corresponding numbers provided by the sequence starting with 3230. • Divide the product of 6 and sine of mandakendra by the corrected divisor. This last step, namely dividing it by 6, was not specified in the procedure earlier. This division is necessary because while, converting it into lipti we had multiplied it by 60. Essentially, the procedure can be written in the modern notations as an equation, where, corrected divisor = number 3230 + sin (in liptis) Correction factor 90 for the sun.
Similar devisors are derived for other planets.
The śīghra correction takes the manda corrected position as the reference. Let us first see how the correction is achieved.
The procedure for śīghraphala, which has been very aptly clarified and compared with the procedure in Sūryasiddhānta by the referee is being reproduced here. 2 The śīghrocca for the planets is the sun itself. In the Figure 2.4, the sun, the planet and the earth are represented by S, E and P. The relevant angles are marked as θ ms mandasphuṭa, θ s śīghra, , radius of śīghra epicycle and , the radius of deferent.
2 The anonymous referee of has kindly provided a critical analysis and compar-ison of this formula with the one in Sūryasiddhānta.

IN THE GAṆITAGANNAḌI
Now, let us see the procedure in Gaṇitagannaḍi.
The epicycle of the śīghra is rather large although Figure 5 represents it as a small circle of radius . ′ is the direction of śīghroccha, the conjunction of the planet with the sun. The manda corrected position is . By the time the mean position has changed to from conjunction the projection on the epicycle would have moved from ′ to . The corresponding shift on the orbit takes it to the point as the true position. The śīghraphala, s is expressed as (Somayaji, 1971) sin θ = sin where is called the calabāṇa, , the distance of the planet from earth at the desired instant. (Bapu Deva Sastri 1861). The word caladbāṇa also is used. From the properties of similar triangles we can show that The procedure requires that sin and sin cos be determined, these are termed bhujaphala and koṭiphala respectively. The author has used a different technique to compute , the calabāṇa. The term in the expression + cos has been fixed to 10. Accordingly koṭiphala is added to 10 and its square is added to the square of bhujaphala, essentially getting the value of 2 . Its square root is the divisor for bhujaphala again to get śīghraphala.
Then the value of a is adjusted as per the ratio / . For, example for Mars, the ratio is known to be 1.5 (given as the ratio of radii of peripheries with 360). If d is 10 the value of will /1.5. However the ratio / will not change. It is to be noted that the coefficients of sin and cos are same. By this adjustment the coefficient of numerator in (3) also will be the same. To take care of the trijyā, multiplication by 120 also is necessary. Let us call the ratio of / as . Since is fixed at 10 the value of is 10/ . The śīghracheda is 720 which we can write as A. bhujaphala = sin 60 śīghracheda = 120 sin 60 10 10 = sin (5) Similarly the coefficient of cos also is adjusted by dividing by A. This looks very tricky but we can see that it is devised to get rid of several steps such as division by 60 and 120. Thus the same Bhujaphala and Koṭiphala (with = 120) are Here, the divisor, śīghracheda is provided for all planets (eg., for Mars it is 1110) The hypotenuse calabāṇa is defined as calabāṇa = [{10 + } 2 + 2 ] 2 śīghraphala = ∆θ, is given by sin ∆θ = 120 × × /calabāṇa this is same as equation (3) Thus if we identify / of (3) with 720/śīghracheda of (4); they are identical. Table 6 lists the numbers and the implied ratios, which is in agreement with the values currently in use. Thus Gaṇitagannaḍi (GG) has the same procedure from the Sūryasiddhānta (SS) to aid calculations. (śīghrakendra as , can have any value from 0 to 90). The agreement to second decimal place implies 6 ′ . The values of the ratios of the radii of the planets are concealed in these numbers (A) provided as śīghracheda.  This procedure has a great advantage in computations, since only bhujaphala and koṭiphala are to be read out from the sine tables and the constants take care of the conversions. It can be summarized as follows: 1. Calculate the calabāṇa and śīghrakendra for the individual planet 2. Get the bhujaphala and koṭiphala putting the corresponding śīghracheda 3. Calculate śīghraphala putting using (6) Thus if we identify / of (4) with 720/śīghracheda of (3); they are identical. The comparison of the ratios are in the Table 6   The next step is to get the sine inverse form the same sine tables which is quite straight forward and explained already.
The next verse describes the procedure for getting the sphuṭagati, the true motion of the planet. We will see that the concept of calabāṇa has been utilized here also to lessen the steps of calculations. It is assumed that the reader is aware of the procedure and the steps are mentioned very briefly. The average value of the gati obtained as an average for one revolution is called the mean. The first step of mandasphuṭa correction uses the value of koṭiphala arrived above. This procedure is not discussed here.
The correction in the second step requires the śīghra corrected value and the calabāṇa, earth planet distance, to get the sphuṭagati or the true motion. In case of the sun and the moon the second step is not needed. Here only the second step is explained. The planet earth distance which was termed calabāṇa is being used again here.
The difference between the gati of the śīghrocca (U) and that of the planet (V) is multiplied by a quantity which we shall call , defined as the difference of the śīghrahara as per catuḥpratinyāya. This phrase is not explained and the meaning is not very clear. But we try to understand the procedure and interpret. After multiplication it is divided by the same śīghrahara used for getting calabāṇa. This is added to or subtracted from the gati obtained after manda sphuṭa correction.
The sphuṭagati consists of three components -the mean motion of the planet, the mean motion after the manda correction and the mean motion after the śīghra correction. The last quantity is given by where θ is the śīghraphala and caladbāṇa, is the earth-planet distance. Here represents the mean motion of śīghrocca and is the mean motion of the planet. In the case of planets śīghrocca is the sun itself. Therefore ( − ) is a measure of the difference in speeds of sun and planet. The difference between the two becomes substantial as the planet -earth distance and the sun -planet distances have a larger range as compared to the sun or the moon. The statement above can be expressed as an equation as given in the text as Thus we can interpret that the quantity is cos θ. The meaning of catuhpratinyāya perhaps is discussed elsewhere and assumed to be known to the reader.
The calabāṇa is converted to arc seconds and subtracted from mandasphuṭagati if calabāṇa is smaller; that gives the sphuṭagati.
The equation (8) also shows the effect of the difference of speeds as seen from the earth. The projection of difference of speeds in the line of sight is achieved by the multiplication by cos θ. If is negative the difference implies the vakragati -the apparent reversal in the direction of motion. This idea is used to fix the onset of retrograde motion for these five planets.
The next verse mentions a correction to be done for the sun and the moon. This is described in the Sūryasiddhānta as per the verse quoted in the text. (This HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 verse is included in the appendix) This is called the bhujāntara correction and is needed because of the non-uniform motion of the sun. As can be guessed this is a direct consequence of the elliptical orbit.
All these computations are for midnight at Ujjain. The time difference will be determined with reference to a uniform motion of 360 degrees a day or 21600 arcminutes per day. The word cakralipti, number of liptis (arcminutes) in a cakra (circle), is used for 21600. This is the only place where kaṭapayādi system has been used to denote this as anantapura.
The procedure here is as follows:-the sine of the sun is converted to lipti and divided by 27; the result in liptis is added to the sun and the moon. Addition or subtraction is decided by bhujaphala (as positive or negative). That gives Ravibhu-jasaṁskṛtacandrawhich means moon corrected for Ravibhuja. This correction is to be done for all planets. But for all the others it is quite small and therefore the author states that he specifically applies it for the moon. This correction should be done for all planets. This is as per the verse in Sūryasiddhānta (II -46) -the gati of planets should be multiplied by Ravibhujaphala in kala and divided by 21600; the result is positive or negative as is the case for the Sun. The mean gati of the Moon is 791. Dividing 21600 by 791 gives 27. Therefore the author gives the rule as divide by 27.
This completes the second chapter called Grahasphuṭādhikāra. The colophon is identical to the one for the first chapter, with identical adjectives.
This chapter for calculation of true positions of planets has used procedures which render computations easy and simplified. The rationale for the procedures has been explained. The ratios of planetary distances and the epicycle radii are compared with those given in Karaṇakutūhala. The constants used here have been modified by the author himself and small corrections also have been incorporated.
Finally a note on the colophon: the author has attributes "like a full moon for the ocean of nectar, and, who, to the ignorant astronomers is like Garuḍa (Brahminy kite, the mythological enemy for snakes) to snakes." The corresponding translation can have two interpretations in the absence of the specific case endings: • Dēmaṇajyotiṣāgragaṇya-sudhārṇava-pūrṇacandra -can be a single phrase meaning "like the full moon for the nectar ocean of Dēmaṇa who is an expert astronomer." • Vāsavguru Dēmaṇa -can be one phrase comparing Dēmaṇa to the guru of the Gods. Now agragaṇya gets attributed to the ocean so that the implied meaning is "like the full moon for the nectar ocean of expert astronomers." Generally the ocean and full moon metaphor is used to signify happiness -akin to the high tides associated with full moon. Here the ambiguity arises with the word dēmaṇajyotisagragaṇya leading to the above two possibilities. This is by treating the expression as a descriptive compound (karmadhāraya) as suggested by the referee and K. R. Ganesha (personal communication).
There is yet another interpretation as provided by Mahesh and Seetharama Javagal (2020) in the context of edition of Karaṇābharaṇa by the same author, Śaṅkaranārāyaṇa Joyisa. The translation of the same phrase reads Composed by Śaṅkaranārāyaṇa Joyisa who is "a falcon to the serpents of unaccomplished astronomers," and the full moon emerged from the nectar-ocean of the foremost astronomer Dēmaṇa Joyisa, the one who is equivalent to the guru of Indra. This is based on the mythological story that the moon was churned out of the ocean (amṛta manthana). The simile classified as rupakālankāra describes the happiness of the father provided by the genius of the son. Here the fact that Dēmaṇa is the father has been utilized although not specified and Garuḍa is translated as falcon.
These titles are not found in the earlier works of Śaṅkaranārāyaṇa Joyisa, namely Tantradarpaṇa (1601 CE) and Karaṇābharaṇam (1603 CE) where it reads …composed by Śaṅkaranārāyaṇa Joyisa, the son of Dēmaṇa Joyisa, the astronomer, who is equivalent to the guru of Indra, a resident of Śṛṅgapurī. (Mahesh and Seetharama Javagal 2020) Perhaps he was bestowed with the titles in 1604 CE. Or, did he crown himself, or, did he become more poetic?

GRA H A M A D H YĀ D H IK Ā R A TR A N S L AT I O N
T HIS CHAPTER, a continuation of the verses explained in our earlier paper (Shylaja and Javagal 2020), explains getting the mean positions of all planets. Since there is an ending note stating that Dhruvādhikāra is concluded and Grahamadhyādhikāra is commencing, we may consider this as a sub-section of the first chapter. For degrees the words bhāga and bhāgi are used interchangeably. We have retained the usage in English as well.
The text is provided in the next section as is given in the manuscript which has the text in both the languages, Sanskrit and Kannaḍa. As mentioned earlier the script is Nandināgarī and here we have put both languages in Kannaḍa script. Translation and the verses from 1 to 10 of Dhruvādhikāra have been already provided in the earlier paper. It is to be noted that the translation is provided only for the ṭīke or commentary in Kannaḍa not for the mūla, the original Sanskrit verses. Very long phrases have been split to shorter sentences.

GRAHAMADHYĀDHIKĀRA
Now the procedure to derive the mean values for the required date will be explained from the number of dyugaṇas, whose derivation, based on the parameters like saṅkrānti and tithi is explained.

VERSE || 11 ||
[This is to get the tithi of saṅkrānti.] The dhruvāṃśa, obtained earlier [for the beginning of the year], of the moon is considered. The rāśi part is multiplied by 30 and added to the degrees part so that we have it expressed in degrees. This is divided by 12 to get the quotient as the saṅkrānti tithi. The remainder is of no consequence here.
Saṅkrānti tithi is defined as the number of civil days intervening from caitra śuddha pratipat to meṣa saṅkrānti for the sun [to cover it] with its mean daily motion.
[Now the calculation of dyugaṇa] Number of days from caitra śuddha pratipat is counted -this should include the intercalary month if needed. The number of saṅkrānti tithis is subtracted. Every ṛtu, season, has two months. The number of ṛtus elapsed are subtracted to get the dyurāśi. The dyugaṇa is obtained by removal of saṅkrānti tithis and ŗtus; this is the number of tithis from the beginning of the solar year [meṣa saṅkrānti].

VERSE || 12 ||
As per [the verse starting with] "nagāpta śiṣta", the dyugaṇa thus obtained is divided by 7. The remainder is added to the vāra of sāvanadhruva. 7 is subtracted from it if it is more than 7.
As per [the verse starting with] "vārapatirniśīthe", the remainder obtained is the week day number starting from Friday. If the number is 1 it is midnight of Friday; 2 implies midnight of Saturday [and so on].

FIRST LINE OF VERSE || 13 ||
This method gives the week day correction of one day more, or one day less which can be applied to the dyugaṇa-count (so that the calculated and the actual week-day are the same). The sāvana dhruva should be subtracted from the corrected dyugaṇa in units of ghaḷige and vighaḷige. This is how you can do it. Take out one dyugaṇa [which is equal to 60 ghaĪige] and [write it] as 59 ghaḷighe. Place the dyugaṇa and take one from it and bring (it as) 60 (ghaḷige). From this, with (1 ghaḷige further written as) 59 (ghaḷige and 60 vighaḷige), the sāvanadhruva with ghaḷige and vighaḷige is subtracted. The result is called a pada expressed in units of day, ghaḷige and vghaḷige. ("pada" is the time-interval between the meṣa saṅkrānti (beginning of the solar year) and the beginning of the desired day) [This is the number] to be used for all the planets. Now consider the pada twice -as per [the verse starting with] "khāgamśa hīnena phalam", divide the second one by 70 to get degrees. The remainder is multiplied by 60 and divided by 70 to get lipti and similarly vilipti. These values are subtracted from the (first) pada to get the mean sun in bhāga units. That should be divided by 30 to get raśi. The remainder is bhāga, thus you get the mean sun for the midnight of the desired day in units of bhāga, lipti and vilipti.
The rule applied is -for one day the mean gati is 59 lipti 8 vilipti. The mathematical explanation of this is that the value gets lesser by 1 bhāgi in 70 days.

SECOND LINE OF VERSE || 13 ||
[The same pada is used now for the moon.] It is multiplied by 12 specified as arkanighnam (arka 12, nighnam, multiplication) in the verse. Two copies of the product are kept; the lower copy is divided by 68 to get the product in units of bhāga, and added to the upper copy. This is added to one pada. Divide it by 30; if the result is more than 12 divide it by 12; discarding the quotient, the remainder is the rāśi, and the lower units are bhāga, lipti and vilipti. As per the verse "dhruveṣu yojya", this quantity in rāśi and other units, is added to the dhruva for the beginning of the year (obtained earlier) for the moon to get the mean moon for midnight of the desired day.

VERSES || 14 || AND || 15 ||
To get the mean Kuja (Mars): All the three copies of the pada are multiplied by 4 as per (the verse) "kṛtaghnāt." The lower two are divided by 60 and (and the square of 60 respectively) and added back. The sum is divided by nidhi, pakṣa, netra that is 229, to get bhāgi. The reminder is multiplied by 60 and again divided by 229 to get lipti and same way to get vilipti. The rāśi and sub units, obtained this way is added to the dhruva of Kuja (obtained earlier) to get mean Kuja.
Here kŗti [corresponds to] 4 rāśi and nidhi, pakșa, netra [corresponds to] 229 days, arrived at as completion of 4 rāśi s by the mean Kuja. Therefore the rule of three used is [as follows] 229 days correspond to 4 rāśi -therefore how many rāśi for the desired number of days? It is the same procedure for all the planets [hereafter]. Now the (determination of) śīghroccha, higher apsis of the epicycle, of Budha (Mercury).
The divisor has been specified as 30, from "jñaḥ khāgnibhiḥ", in plural, but not the multiplier. Based on the context we consider the multiplier guṇaka is the same as that for Mars namely 4. The pada is multiplied by 4 as before and divided by 30 and expressed as rāśi which is kept aside. Multiplying the pada by one [you HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 get back] the same pada. This is divided by 325 (paṇcaradaiḥ) to get the rāśi units and added to the earlier obtained rāśi. This is finally added to varṣa dhruva to get Budha śīghroccha. Now the procedure for mean Guru (Jupiter) -the multiplier is 1, specified by bhu. This is to be divided by 361 to get Guru. This [converted to units of rāśi] is added to dhruva for the year to get mean Guru. Now the śigrocca of Śukra (Venus) -pada is multiplied by 40 and divided by 749. The result [converted to rāśi units] is added to the dhruva obtained earlier.
To get the mean Śani (Saturn) -pada is mulitpiled by 1, and divided by 897; the product [converted to units of rāśi] is added to the dhruva to get mean Śani.
The multiplier for, Rāhu (Moon's ascending node) is 1. It is divided by 566. This is subtracted from the Dhruva as per tamasaḥ pratipa, to get mean Rāhu. Adding 6 rāśis will fetch Ketu.
For getting the candrocca, (Moon's apogee) pada is multiplied by 3 and divided by 808; the result [converted to rāśis] is added to previously obtained dhruva.

VERSE || 16 ||
Dyugaṇa is divided by 150; the result expressed in lipti, vilipti is subtracted from the mean values for the sun and the moon. Thus all the mean positons of all planets are obtained for the midnight of Laṇka. Laṇka is to be understood as the south of mahāmeru.

VERSE || 17 ||
Getting the lambajyā of the place of observation (svadeśa) is explained later in the chapter chāyādhyāya; this is multiplied by 5060 and divided by the trijyā 120, which is defined in sphuṭādhyaya. This is the svadeśabhūparidhi, the circumference of the small circle at the observer's latitude. For a place with an equinoctial shadow of 3 aṅgula, the lambajyā is 116/27. The derivation of 5060 is explained as per Sūryasiddhānta in this verse.

Quotation from Sūryasiddhānta
Multiply the square of the earth's diameter (1600) by 10 and its square root is the circumference in yojanas.
The bhūmadhyarekhā stretches from Laṇka to Meru Mountain. Rouhitaka country, Svamimale, Avanti is Ujjaini, Amarādri sāra is Mānasa Sarovara. The north south axis, sutra, passes through these and is called bhūmadhyarekhā.

VERSE || 18 ||
The distance in yojana of the place of observation from the bhūmadhyarekhā to the east or west is to be determined. This number is multiplied by the mean gati [in lipti] of all the planets and divided by the circumference, svadeśabhūparidhi obtained earlier. This is subtracted from the mean values of the respective planets, if svadeśa, the place of observation, is to the east, or added [if it is] to the west. This is the correction [called] yojanasanskāra. Now the procedure for getting all the mean planets at midnight of the iśṭakāla desired date.

VERSE || 19 ||
One has to get the time interval in ghaḷige vighaḷige, ahead of or lagging behind, from midnight; this time interval is multiplied by the madhya gati (mean rate of motion) in lipti, vilipti and divided by 60. If the iśṭakāla (time of interest) is before midnight, the values in [lipti, vilipti] have to be subtracted; if it ls after midnight [they have] to be added. The mean positions of all planets are now available for the desired time.
This completes the first chapter called Grahamadhyādhikāra of the book called Gaṇitagannaḍi, a commentary of Vārṣiktantra in the language of Karṇāṭa written by Śaṅkaranārāyaṇa Jyōtiṣi, who, is like Garuḍa (mythological enemy of snakes, the Brahminy kite) to snake-like ignorant astronomers, and who, akin to a full moon for the ocean of nectar [and] of Bṛhaspatilike Dēmaṇa, an eminent astronomer and a resident of Śṛṅgapura.

VERSE || 2 ||
The śīghrocca was told first; now mandocca is being told. For the sun it is 78 (vasvādri), for Maṅgala, 130 (khaviśva), for Budha, 221 (rūpākṛti), for Guru, 172 (dvadrindavaḥ), for Śukra, 80 (khāṣṭa) and for Śani, 237 (agāgnidasrāḥ). This is HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 obtained from the mandocca bhagaṇas [number of revolutions of the apogee, mandocca] stated in the Sūryasiddhānta in verses (I: 41 and 42) starting with "prāggate sūryamandasya" up to "gognayaḥ śani mandasya", [these numbers are] multiplied by the number of years up to the desired year, and divided by the number of years in a kalpa. The current mandocca will be deficient by a few liptis from the date provided by Ācārya and one bhāgi has been added to account for this as Dhruva (constants).

VERSE || 3 ||
Kendra of a planet is obtained in raśi [and its subunits] after subtracting the śīghrocca or mandocca from the mean planet. If the kendra is tulādi (starting with tulā, the angle is between 180 and 360 degrees) the calculated śīghrabhujaphala or mandabhujaphala should be added to the mean planet. If the kendra is meśādi (the angle is between 0 and 180 degrees) it should be subtracted. Later the method of getting the śīghraphala will be told [where] the śīghrahara 10 [vyomendavaḥ] should be added to koṭiphala, if śīghrakendra is mrigādi and subtracted if it is karkyādi.

VERSE || 4 ||
Now the procedure to get bhuja and koṭi.* (In a right angled triangle, bhuja is the opposite side of the right-triangle and koṭi is the adjacent side of the righttriangle).In the odd quadrant Bhujā is determined by the current angle. Koṭi is (yet to be) covered. In the even (yugma) quadrant it is the opposite of this. That means -the angle to be covered determines the bāhu and the angle covered determines the koṭi. The bhuja and koṭi are for three rāśis (90 deg). For 12 rāśis there are four padas; there are two odd (oja) quadrants. For rāśis 0, 1 and 2 are the same as for raśis 6, 7 and 8. Here bhuja is determined by angle covered and koṭi by the angle yet to be covered. For rāśis 3, 4 and 5 and also for 9, 10, 11 which are even quadrants koṭi is determined by angle covered and bhuja by angle to be covered. Thus the bhuja and koṭi (found) for three raśis repeat for the others. From bhuja, koṭi can be determined by subtracting by 3 rāśis.
(* We are thankful to the anonymous referee for pointing out the confusion in the original work itself. The corrections as per convention have been incorporated here.) VERSE || 5 || Now to get the sine for bhuja and koṭi: The rāśi number is multiplied by 30 and added to bhāgi [degrees], divide this total [in degrees] bhagi by 10. The quotient is the number of the khaṅḍajīvā covered all ready. The corresponding jīvā is written down. The value of the next khaṅḍaj jīvā is divided by 10 and multiplied by the remainder whose lipti and vilipti have been converted to bhāga and added back to the bhāga value. The result (quotient) is added to the earlier obtained jīvā. The remainder (in this step) is multiplied by 60 and divided by 10, converted to lipti, vilipti and added to the jīvā. This is the jīvā or koṭi derived for the desired angle. It should be noted that this is in [units of] bhāgādi (degrees).

VERSE || 9 ||
To get the correction for the instant, sine is multiplied by 60 and added to lipti. The sum is divided by the numbers 90 and others for the respective planets as prescribed in the verse starting with khāṅkaiḥ. (2.9) The result is added to the numbers mentioned earlier as vyomāgnidanta and so on {(490 (khatāna), 300 (viyad abhrarāma), 70 (khāśva), 170 (khaśailendu), 21(indupakṣa), 380(khāṣṭāgni)} to get the corrected divisor sphuṭamandacheda. Now the śīghra cheda for the five planets starting from Kuja. VERSE || 10 || vised similar to mandacheda as specified in [Sūrya]siddhānta.

VERSE || 11 ||
To get the mandaphala and śīghraphala, [one needs] bhujaphala and koṭiphala. The bhujā and koṭijīvas are kept in two places; multiply them by 60 and add to the litpi in lower place. Then they are divided by the respective manda cheda and śīghra cheda to get bhujaphala and koṭiphala in degrees etc. As stated in the verse the manda arises because of only bhujaphala and therefore there is no need of koṭiphala for the manda correction. The jyā of bhuja is multiplied by 60 and added to the lipti part and divided by the divisors as specified by vyomāgni etc. which are made true by correcting with khankaih etc., for the planets beginning with the Sun for the respective planets.

VERSES || 12 ||, || 13 || AND || 14 ||
The mean motion madhyagati of the planets are being told in lipti, vilipti. For Ravi it is 59|8, for the moon 790|35 for Kuja 31|26, for Budha 245|32, for Guru 5|0, for Śukra 96|8, for Śani 2|0, for Rāhu 3|11, for candrocca 6|41. While making the mandasphuṭa correction, for getting the sine, the khaṅḍa corresponding to the eṣya [to be covered part], is multiplied by the madhyagati in lipti, vilipti and multiplied by 6 (rasaghna) and divided by the appropriate divisor and added if it is karkyādi, subtracted for makarādi. This gives mandasphuṭagati. The derivation of the (mean motion) madhyagati is done by dividing the bhagaṇa (number of revolutions) as specified in Sūryasiddhānta by the bhūsāvanadina (number of days). It is [done] like this. Number of revolutions of Ravi is 4320000. The number of sāvana days are 1577917828. When this is divided [by number of revolutions] we get 0 rāśi, 0 bhāga, lipti 59 and vilipti 8. This is done for all planets. The meaning of madhyagati is the number of liptis covered in a day.
Although the procedure for mandasphuṭa is explained, I am summarising it again. It is like this. After obtaining the mean planets, take the difference with respective mandoccas, get the sine by using the rule as bhāgāstayoḥ kenduhṛta, (verse number 2.5 above) multiply by 60 and add the lipti. Consider the numbers specified by the verse khāṅka and so on, added to the original divisors specified by vyomāgni, and divide the jyā, which is already multiplied by 60 by the revised divisor, and take the result in bhāga. When the mean planet is corrected with this, by subtraction, if it is meṣādi, and by addition, if it is tulādi, the mandasphuṭa is obtained. Here it should be remembered that the sun and the moon are sphuṭa by this correction. The five planets Kujādi will be spaṣṭa after the two corrections, namely, manda and śīghra. Thus after completing the explanation for manda, I proceed to explain śīghraphala.

VERSE || 15 ||
Now, the procedure for śīghraphala for the planets starting from Kuja. The śīghraocca subtracted from the mandasphuṭa corrected planet is the kendra. Both the bhuja ( sine) and koṭi ( cosine) are obtained. As per the verse do koṭiḥ jīve kharasaiḥ nihatyāt, (verse 2.11 above) the bhujājīva and koṭijīva are multiplied by 60, the remainder is added back. These are divided by the appropriate divisors as specified by the verse starting with digīśvara. (verse 2.10 above) The result from bhuja is bhujaphala; the result from koṭi is koṭiphala. Vyomendu 10, is the śīghrahara. The koṭiphala obtained is added to or subtracted from this hara (10) as per mrigādi or karkādi. The square of this sum or difference is obtained. Next, as stated by dorjyaphala varga yogāt, the square of bhujaphala is obtained. The two squares are added and the square root is the phala called the calabāṇa.

VERSE || 16 ||
The bhujaphala is multiplied by the trijya 120, divided by calabāṇa. The inverse sine, cāpa, of this is the śīghraphala in bhāga (degrees). For Kuja, Budha, Guru, Śukra and Śani this is applied as positive or negative as mentioned earlier. Thus we get all the true planets.

VERSE || 17 ||
The procedure to get the inverse sine, cāpa (the arc of the angle). The arc has to be obtained (from the jyā). Subtract as many khaṇḍajīvās as possible from the jīvā. Keep aside the number of khaṇḍajīvās subtracted. The remainder is multiplied by 10 and divided by the khaṇḍajīvā which is the khaṇḍa which comes after the subtracted ones. When this is added to 10 times the number (of khaṇḍajīvās) kept aside [earlier], that (sum) is the desired arc in (degrees). The result is the inverse sine, cāpa.

VERSE || 18 ||
The gati of the mandasphuṭa is subtracted from the gati of the difference of śīghrocca and graha. What remains is multiplied by the difference between the śīghrahara and the calabāṇa as per the [rule of] catuḥpratinyāya. This is divided by the calabāṇa and the result in liptadi [sundivisions of arcminutes] is added to the mandasphuṭa if the bāṇa is greater than the hara, and subtracted from it if the bāṇa is HISTORY OF SCIENCE IN SOUTH ASIA 9 (2021) 232-272 less than the hara. The result is the sphuṭagati (true rate of motion). If the (earlier) result is greater than the mandagati, the mandagati is subtracted from the (earlier) result, and what remains is the vakragati (retrograde rate of motion).

VERSE || 19 ||
The sin of the sun (Ravi) is converted to lipti as per the verse uṣṇāṁśu dorjaphalam and divided by 27. The result in liptis is added to or subtracted from the moon, Candra, as per the correction to the sun. If the bhujaphala is positive it is added to the moon. If it is negative for the sun it should be subtracted from the moon also. That gives Ravibhujasaṁskṛtacandra -moon corrected for Ravibhuja. This correction is to be done for all planets as stated in the [Sūrya]siddhānta. But for all the others it is quite small and therefore I told it specifically for the moon. This correction should be done for all planets.  This is as per the statement in [Sūrya]siddhānta -the gati of planets should be multiplied by Ravibhujaphala in kala (arc minutes) and divided by cakralipti, that is 21600; the result is positive or negative as is the case for the sun. The mean gati of the moon is 791 lipti. Adripakṣa, 27 is the result when 21600 is divided by this. Therefore I made the rule for division by 27.
This completes the second chapter called Grahasphuṭādhikāra of the book called Gaṇitagannaḍi, a commentary of Vārṣiktantra in the language of Karṇāṭa written by Śaṅkaranārāyaṇa Jyōtiṣi, who, to the ignorant astronomers, is like Garuḍa to snakes and who, akin to a full moon for the ocean of nectar [and] of Bṛhaspati -like Dēmaṇa, an eminent astronomer and a resident of Śṛṅgapura.
6 T E X T S H ERE WE GIVE THE TEXT from the original palm leaf manuscript for the second half of first chapter and the second chapter (covered in this paper) which has the verses in Sanskṛt and commentary in Kannaḍa. As mentioned earlier the script is Nandināgarī and here we have put both languages in Kannaḍa script.