Venketeswara Pai - Academia.edu (original) (raw)
Papers by Venketeswara Pai
Hindustan Book Agency, 2017
Sources and Studies in the History of Mathematics and Physical Sciences
May the guru, an embodiment of intelligence and bliss, keep ever rising in the space of my heart,... more May the guru, an embodiment of intelligence and bliss, keep ever rising in the space of my heart, like the Sun dispelling the darkness of ignorance.
Sources and Studies in the History of Mathematics and Physical Sciences
The longitudes (sphuṭas) and latitudes (vikṣepas) of the planets have been instructed in various ... more The longitudes (sphuṭas) and latitudes (vikṣepas) of the planets have been instructed in various ways by different scholars. Therefore, they have to be established after examination of their shadows etc. (chāyādi), as observed by the instruments.
Sources and Studies in the History of Mathematics and Physical Sciences
mandādihāraguṇitā bhagaṇā yutonā jñānīndrasaṃguṇadhanādiguṇairguṇāḥ syuḥ | mandādihārahatabhūdiva... more mandādihāraguṇitā bhagaṇā yutonā jñānīndrasaṃguṇadhanādiguṇairguṇāḥ syuḥ | mandādihārahatabhūdivasāśca hārāḥ proktā mahāguṇaharāsta ime'pavartyāḥ || 1 || The product of 200 (jñānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be added to or subtracted from the product of the [śakābda]-hāras [of the planets], beginning with manda (85), and their respective revolution numbers. These are guṇas [of the planets]. The [śakābda]-hāras, beginning with manda, multiplied by the the number of civil days (bhūdina) in a mahāyuga are the hāras. These are the mahāguṇas and mahāhāras, whose apavartana is to be done (they have to be factored by their GCD).
The above verse defines the mandakendrahārakas of the planets. It may be recalled that the mandak... more The above verse defines the mandakendrahārakas of the planets. It may be recalled that the mandakendra is the difference between the planet and its mandocca (apogee). The mandakendra-guṇakāras and the mandakendrahārakas determine the successive approximations to the rate of motion of the mandakendra.
Sources and Studies in the History of Mathematics and Physical Sciences
May the diameter be multiplied by four, kept separately at several places, and divided by the odd... more May the diameter be multiplied by four, kept separately at several places, and divided by the odd numbers 3, 5, 7 etc. [The results] may be sequentially applied negatively and positively to the diameter multiplied by four. Then we obtain a very accurate [value of the] circumference.
In continuation of the compilation of all bright stars in various texts based on the listed coord... more In continuation of the compilation of all bright stars in various texts based on the listed coordinates, we present here the coordinates of the stars in the region of Gemini. Using the 27 nakṣatras on the ecliptic, the coordinates were matched for the epochs of the catalogues to resolve possible ambiguity in the identification of faint stars. In an attempt to cover the stars beyond the zodiac, conversions of coordinates are very useful. Those lists which hint at the actual observational data are chosen for the study. Although the region specified is Gemini, it covers all Declinations within the Right Ascension 4 hours to 6 hours.
Sources and Studies in the History of Mathematics and Physical Sciences
Let the heap of days (dharādinaugha), and the difference in the revolutions of the Moon and its a... more Let the heap of days (dharādinaugha), and the difference in the revolutions of the Moon and its apogee be mutually divided. Then, from these results, the guṇakāras and hārakas associated with the Moon’s anomaly (candra-kendra) are to be obtained as stated earlier.
In his Tantrasaṅgraha, Nilakan. t.ha Somayāji has given a method for determining the arc (cāpa) c... more In his Tantrasaṅgraha, Nilakan. t.ha Somayāji has given a method for determining the arc (cāpa) corresponding to a given Rsine (bhujā), when both are small, using an iterative procedure. Nilakan. t.ha also gives a method for finding the arc length for small Rsines, when the difference between the arc and the Rsine (bhujācāpāntara) is equal to an integral number of seconds of arc. These are described in greater detail in Laghuvivr . ti and Yuktidipikā— commentaries on Tantrasaṅgraha—and also in Karan. apaddhati of Putumana Somayāji. In this paper, we discuss these methods of finding the (small) arc, given the Rsine.
Journal of Astrophysics and Astronomy
Based on a thorough examination of the position of the planets at the time of eclipses, planetary... more Based on a thorough examination of the position of the planets at the time of eclipses, planetary conjunctions, and so on, the revolution numbers etc. [of the planets] in a kalpa have to be conceived of for achieving concordance with observations.
The Growth and Development of Astronomy and Astrophysics in India and the Asia-Pacific Region
Based on their listed coordinates, we have compiled a catalogue of more than 100 bright stars inc... more Based on their listed coordinates, we have compiled a catalogue of more than 100 bright stars included in various texts from the Surya Siddhanta to the Siddhanta Darpana by Chandrashekhara Samanta in the nineteenth century. Using the 27 nakshatras on the ecliptic, which fix the position of the Solar System bodies, the coordinates were matched for the epochs of the catalogues. This resolved some ambiguity in respect of the identification of faint stars and provided a means to extend the method to other stars outside the zodiac. We have specifically chosen those lists that are characterized by observations, which are highlighted in the discussion. Our study reveals that a scale similar to the magnitude scale of brightness (currently in use) was in vogue in ancient Indian astronomy. Stars used by navigators, not listed with coordinates but as practical tools, are also included. The origin of the names are described—some were indigenous, and some were borrowed from the Arabs and later from the Europeans. In this preliminary study we provide an overview of the positions of the stars.
Here the author emphasizes the importance of actual observations of the celestial objects through... more Here the author emphasizes the importance of actual observations of the celestial objects through the measurement of their shadows etc. in determining their longitudes and latitudes. In the case of the Sun, the measurements associated with the shadow at noon and other times pose no difficulty, in principle at least, as these are done during the day. Observations pertaining to the Moon’s shadow are also possible during the night. But what about the planets and stars?
The ascensional difference (cara) and the prāṇakalāntara are to be applied to the [longitude of t... more The ascensional difference (cara) and the prāṇakalāntara are to be applied to the [longitude of the] end of the desired zodiacal sign (rāśyanta) which is corrected for the movement of equinox (ayanacalana). The result thus obtained is stated to be the kālalagna corresponding to the end of the desired zodiacal sign (rāśyanta).
It is well known that ancient Indian calendar dwelled on the 27 nakśatra system for fixing the po... more It is well known that ancient Indian calendar dwelled on the 27 nakśatra system for fixing the positions of the sun, moon and the planets. Several attempts to identify these 27 stars in the sky have yielded very precise results for stars bright enough not to be misidentified, which is not so for the fainter ones. The basis for identification is the coordinate system available in the texts. Here, we try to understand the ambiguity and offer a possible solution by using the measured coordinates, which have not been utilized for this purpose so far. This also provides clues on the techniques used for measuring the coordinates.
In the vākya system of astronomy prevalent in south India, the true longitudes of the Sun, the Mo... more In the vākya system of astronomy prevalent in south India, the true longitudes of the Sun, the Moon, the planets, and associated quantities can be directly found using vākyas or mnemonics. The set of vākyas for a specific physical variable presented at regular intervals is essentially a numerical table. The text Karaapaddhati of the Kerala astronomer Putumana Somayāji (ca. 1732 AD) describes methods to obtain the set of vākyas, based on the general principles of Indian astronomy. In particular, it presents the rationale for obtaining the various vākyas pertaining to the Sun, namely ‘māsavākyas’, ‘sakrāntivākyas’, ‘nakatrasakramaavākyas’, and ‘yogyādivākyas’. In this article, we explain the procedures outlined in Karaapaddhati to obtain the sets of vākyas pertaining to the Sun.
The product of 200 (jnānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be a... more The product of 200 (jnānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be added to or subtracted from the product of the [śakābda]-hāras [of the planets], beginning with manda (85), and their respective revolution numbers. These are guṇas [of the planets]. The [śakābda]-hāras, beginning with manda, multiplied by the the number of civil days (bhūdina) in a mahāyuga are the hāras. These are the mahāguṇas and mahāhāras, whose apavartana is to be done (they have to be factored by their GCD).
Hindustan Book Agency, 2017
Sources and Studies in the History of Mathematics and Physical Sciences
May the guru, an embodiment of intelligence and bliss, keep ever rising in the space of my heart,... more May the guru, an embodiment of intelligence and bliss, keep ever rising in the space of my heart, like the Sun dispelling the darkness of ignorance.
Sources and Studies in the History of Mathematics and Physical Sciences
The longitudes (sphuṭas) and latitudes (vikṣepas) of the planets have been instructed in various ... more The longitudes (sphuṭas) and latitudes (vikṣepas) of the planets have been instructed in various ways by different scholars. Therefore, they have to be established after examination of their shadows etc. (chāyādi), as observed by the instruments.
Sources and Studies in the History of Mathematics and Physical Sciences
mandādihāraguṇitā bhagaṇā yutonā jñānīndrasaṃguṇadhanādiguṇairguṇāḥ syuḥ | mandādihārahatabhūdiva... more mandādihāraguṇitā bhagaṇā yutonā jñānīndrasaṃguṇadhanādiguṇairguṇāḥ syuḥ | mandādihārahatabhūdivasāśca hārāḥ proktā mahāguṇaharāsta ime'pavartyāḥ || 1 || The product of 200 (jñānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be added to or subtracted from the product of the [śakābda]-hāras [of the planets], beginning with manda (85), and their respective revolution numbers. These are guṇas [of the planets]. The [śakābda]-hāras, beginning with manda, multiplied by the the number of civil days (bhūdina) in a mahāyuga are the hāras. These are the mahāguṇas and mahāhāras, whose apavartana is to be done (they have to be factored by their GCD).
The above verse defines the mandakendrahārakas of the planets. It may be recalled that the mandak... more The above verse defines the mandakendrahārakas of the planets. It may be recalled that the mandakendra is the difference between the planet and its mandocca (apogee). The mandakendra-guṇakāras and the mandakendrahārakas determine the successive approximations to the rate of motion of the mandakendra.
Sources and Studies in the History of Mathematics and Physical Sciences
May the diameter be multiplied by four, kept separately at several places, and divided by the odd... more May the diameter be multiplied by four, kept separately at several places, and divided by the odd numbers 3, 5, 7 etc. [The results] may be sequentially applied negatively and positively to the diameter multiplied by four. Then we obtain a very accurate [value of the] circumference.
In continuation of the compilation of all bright stars in various texts based on the listed coord... more In continuation of the compilation of all bright stars in various texts based on the listed coordinates, we present here the coordinates of the stars in the region of Gemini. Using the 27 nakṣatras on the ecliptic, the coordinates were matched for the epochs of the catalogues to resolve possible ambiguity in the identification of faint stars. In an attempt to cover the stars beyond the zodiac, conversions of coordinates are very useful. Those lists which hint at the actual observational data are chosen for the study. Although the region specified is Gemini, it covers all Declinations within the Right Ascension 4 hours to 6 hours.
Sources and Studies in the History of Mathematics and Physical Sciences
Let the heap of days (dharādinaugha), and the difference in the revolutions of the Moon and its a... more Let the heap of days (dharādinaugha), and the difference in the revolutions of the Moon and its apogee be mutually divided. Then, from these results, the guṇakāras and hārakas associated with the Moon’s anomaly (candra-kendra) are to be obtained as stated earlier.
In his Tantrasaṅgraha, Nilakan. t.ha Somayāji has given a method for determining the arc (cāpa) c... more In his Tantrasaṅgraha, Nilakan. t.ha Somayāji has given a method for determining the arc (cāpa) corresponding to a given Rsine (bhujā), when both are small, using an iterative procedure. Nilakan. t.ha also gives a method for finding the arc length for small Rsines, when the difference between the arc and the Rsine (bhujācāpāntara) is equal to an integral number of seconds of arc. These are described in greater detail in Laghuvivr . ti and Yuktidipikā— commentaries on Tantrasaṅgraha—and also in Karan. apaddhati of Putumana Somayāji. In this paper, we discuss these methods of finding the (small) arc, given the Rsine.
Journal of Astrophysics and Astronomy
Based on a thorough examination of the position of the planets at the time of eclipses, planetary... more Based on a thorough examination of the position of the planets at the time of eclipses, planetary conjunctions, and so on, the revolution numbers etc. [of the planets] in a kalpa have to be conceived of for achieving concordance with observations.
The Growth and Development of Astronomy and Astrophysics in India and the Asia-Pacific Region
Based on their listed coordinates, we have compiled a catalogue of more than 100 bright stars inc... more Based on their listed coordinates, we have compiled a catalogue of more than 100 bright stars included in various texts from the Surya Siddhanta to the Siddhanta Darpana by Chandrashekhara Samanta in the nineteenth century. Using the 27 nakshatras on the ecliptic, which fix the position of the Solar System bodies, the coordinates were matched for the epochs of the catalogues. This resolved some ambiguity in respect of the identification of faint stars and provided a means to extend the method to other stars outside the zodiac. We have specifically chosen those lists that are characterized by observations, which are highlighted in the discussion. Our study reveals that a scale similar to the magnitude scale of brightness (currently in use) was in vogue in ancient Indian astronomy. Stars used by navigators, not listed with coordinates but as practical tools, are also included. The origin of the names are described—some were indigenous, and some were borrowed from the Arabs and later from the Europeans. In this preliminary study we provide an overview of the positions of the stars.
Here the author emphasizes the importance of actual observations of the celestial objects through... more Here the author emphasizes the importance of actual observations of the celestial objects through the measurement of their shadows etc. in determining their longitudes and latitudes. In the case of the Sun, the measurements associated with the shadow at noon and other times pose no difficulty, in principle at least, as these are done during the day. Observations pertaining to the Moon’s shadow are also possible during the night. But what about the planets and stars?
The ascensional difference (cara) and the prāṇakalāntara are to be applied to the [longitude of t... more The ascensional difference (cara) and the prāṇakalāntara are to be applied to the [longitude of the] end of the desired zodiacal sign (rāśyanta) which is corrected for the movement of equinox (ayanacalana). The result thus obtained is stated to be the kālalagna corresponding to the end of the desired zodiacal sign (rāśyanta).
It is well known that ancient Indian calendar dwelled on the 27 nakśatra system for fixing the po... more It is well known that ancient Indian calendar dwelled on the 27 nakśatra system for fixing the positions of the sun, moon and the planets. Several attempts to identify these 27 stars in the sky have yielded very precise results for stars bright enough not to be misidentified, which is not so for the fainter ones. The basis for identification is the coordinate system available in the texts. Here, we try to understand the ambiguity and offer a possible solution by using the measured coordinates, which have not been utilized for this purpose so far. This also provides clues on the techniques used for measuring the coordinates.
In the vākya system of astronomy prevalent in south India, the true longitudes of the Sun, the Mo... more In the vākya system of astronomy prevalent in south India, the true longitudes of the Sun, the Moon, the planets, and associated quantities can be directly found using vākyas or mnemonics. The set of vākyas for a specific physical variable presented at regular intervals is essentially a numerical table. The text Karaapaddhati of the Kerala astronomer Putumana Somayāji (ca. 1732 AD) describes methods to obtain the set of vākyas, based on the general principles of Indian astronomy. In particular, it presents the rationale for obtaining the various vākyas pertaining to the Sun, namely ‘māsavākyas’, ‘sakrāntivākyas’, ‘nakatrasakramaavākyas’, and ‘yogyādivākyas’. In this article, we explain the procedures outlined in Karaapaddhati to obtain the sets of vākyas pertaining to the Sun.
The product of 200 (jnānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be a... more The product of 200 (jnānīndra) and the [śakābda]-guṇakāras, beginning with dhana (9), has to be added to or subtracted from the product of the [śakābda]-hāras [of the planets], beginning with manda (85), and their respective revolution numbers. These are guṇas [of the planets]. The [śakābda]-hāras, beginning with manda, multiplied by the the number of civil days (bhūdina) in a mahāyuga are the hāras. These are the mahāguṇas and mahāhāras, whose apavartana is to be done (they have to be factored by their GCD).