Brahmagupta, great mathematician of his era was not merely a theorist rather his calculation was based on the observation with the help of instruments or nalikadi yantra which were available at that era. He was a proficient observer who use to observe, calculate and make correction on the basis of his observations. In his treatise Khandakhadyaka he has stressed the need of direct observation.
Brahmasphuta Siddhanta which means corrected treatise of Brahma contains lot of criticism on the work of other mathematicians especially ryabhata. He had many disagreements with Mathematician of that era and most of his chapters of Brahmasphuta Siddhanta deals with dodges in their treatise.
The Indian astronomy was a gift of Aryabhata work. He is said to be the teacher of two distinct astronomy system one is audayika and other one is ardharatrika system. In audayika the astronomical day begins with the sunrise at Lanka and in the ardharatrika it begins with midnight at Lanka. Brahmagupta had made use of astronomical constant of Aryabhata ardharatrika system for his calculation in the first part of Khandakhadyaka, but the methods involved in spherical geometry, calculation of eclipse etc. is just a reproduction of Brahmasphuta Siddhanta. So in short audayika and ardharatrika astronomical constants can be found in Aryabhat ya and may construed from Khandakhadyaka. In connection with this, some authorities had said that Brahmagupta tried to simplify the treatise of ryabhata i.e. Brahmagupta Khadakhadyaka is a simplified version of ryabhata’s ardharatrika and he was also successful in doing this. Brahmagupta in this relationship referred ryabhata in Khandakhadyaka in following ways:
After giving homage to lord Mahadeva, who is the great cause of this World formation, existence and devastation, I shall declare that Khandakhadyaka will yield the same results that of ryabhata treatise.
In Brahmasphuta Siddhanta Brahmagupta takes the astronomical day to begin with the sunrise at Lanka which was same that of ryabhata audayika system, He expresses his surprise as in one treatise ryabhata considers days begin with sunrise at Lanka and in other he considers day begin from midnight at Lanka which results in the difference of one-fourth of the motions. For calculations of days, months, years, yugas and kalpas he had taken first tithi of the month caitra i.e. caitra sukla pratipada and the first day considered is Sunday whereas ryabhata considers Guruvara i.e. Thursday as first day of Kalpa which is wrong according to Brahmagupta.
Brahmagupta differs in the calculation of length of the four yugas from ryabhata. According to Aryabhata all the yugas are of equal length i.e. 1,080,000 years and the caturyuga being of 4,320,000 years, whereas according to Brahmagupta all the four yugas of caturyuga or Mahayuga are not of same length: Kaliyuga is of 432,000 years, Dvapara is of 864,000 years i.e. twice of kali, Treta is of 1,296,000 years, i.e. thrice of kali and Krtayuga is of 1,728,000 years i.e. four times of kali, total comes to be 4,320,000 years but as per both the caturyuga had same length i.e. 4,320,000 years.
Brahmagupta had on one hand critizes Aryabhata works, on the other hand uses his works and expanded them further, which is confirmed by Iranian mathematician and astronomer Alberuni in his work in the early 11th century, Alberuni has defended Aryabhata and addresses Brahmagupta, He says that we shall not argue with Brahmagupta rather only whisper in his ear that why do you begin to calculate the diameter of moon to explain the eclipsing of sun and the diameter of the shadow of the earth to explain its eclipsing the moon after having spoken harsh words for them and why do you compute the above ecliptics on the basis of their theories and not according to them whom you think are proper to agree.
Brahmagupta has criticized Aryabhata in many places in some places without any proper justification, one such point is order of days. Planets motion decreases in following order: Chandra (Moon), Budha i.e. Mercury, Sukra (Venus), Surya (Sun), Kuja (Mars), Guru (Jupiter) and Sani (Saturn). ryabhata in one of his sutra says that starting with the Sun and continuing in increasing order every fourth is the lord of the day which gives the order as follows : Sun, Moon, Mars, Mercury, Venus and Saturn and thus the order of the day on this basis is Ravivara-Sunday, candravara-Monday, Mangalavara-Tuesday, Budhavara-Wednesday, Guruvara-Thursday, Sukravara-Friday and Sanivara-Saturday. In other word Aryabhata has given the same order as given by other Mathematicians of that era then why he was criticized by him was not understood.
Second such point is the Motion of the Earth. According to ryabhata Earth is in motion and the Bhaganas are stationary, where Brahmagupta objected that if Earth is in motion then birds would not be able to return to their nest, the roof and hills
would come down which basically contradicts the observation because of this Brahmagupta is not justified in some places in the criticism of Aryabhata.
Brahmagupta assigns the longitude of the sun apogee to be 80° where as in Uttara-Khandakhadyaka, which is basically a corrected version of Khandakhadyaka, he corrected this longitude to 77° where the Aryabhata value was 78° which is less correct as compare to the modern calculation. The astronomical constant given by Conn. Des Temps, the longitude of Sun opogee in 499 A.D was 77°19′19.44′′ and 76°40 22′′ on the basis of Newcomb’s equation, these two values are almost similar to the value given in Uttara-Khandakhadyaka by Brahmagupta i.e. 77° which means Brahmagupta is more correct than ryabhata.
Brahmagupta noticed that ryabhata had made the Moon’s apogee fast and nodes slow than they actually are and in both the case he made over correction. The length of the anomalistic month given by Brahmagupta = 
days = 27.55454641 and Aryabhata value is 27.554602 but according to Radau the value is 27.5545502, here again Brahmagupta is more accurate. Coming to the length of the sidereal period of the moon apogee the Brahmagupta value is
days = 6792.25396 days and in Khandakhadyaka the value corrected by Brahmagupta is 6794.75083 days. Thus Brahmagupta was successful in attempt to correct the node faster.
Brahmagupta states that the longitude of mars aphelion should be increased by 17° and Jupiter by 10° and here also he was more exact as compare to ryabhata values. The sutra which states this is stated below:
Sutra: 8.1. Uttara Khandakhadyaka: IX-10.
Translation: The apogee of Mars is increased by 17°, Jupiter by 10°. Subtract 74’ with the sughra of Venus, Saturn’s equation of apsis is decreased by one-fifth and the sighra equation of mercury is increased by one sixteenth.
Sutra from Khandakhadyaka i.e. II.6 stated below gives the degrees of longitude of apogees of the planets beginning with Mars which are 11, 22, 16, 8 and 24, where each is multiplied by 10.
Sutra: 8.2. Khandakhadyaka: II-6.
According to above sutra Mars aphelion point which had longitude of 110° and the longitude of Jupiter aphelion was 160° which was corrected by Brahmagupta in the sutra of Uttara Khandakhadyaka -IX.10 which states that the longitude of Mars aphelion point was 127° and the longitude of Jupiter aphelion was 170° in 499 A.D. Whereas ryabhata has given 118° as longitude of Mars aphelion and 180° as longitude of Jupiter’s aphelion.
Newcomb’s rule and Conn. des. Temps rule gives the longitude of Mars aphelion in 499 A.D. as 128°28′12′′ and 128°27′51′′ respectively which states Brahmagupta determination of Mars aphelion is quite satisfactory, again when the longitude of Jupiter is calculated with these two rules it was 170°25′ hence here too Brahmagupta was more accurate.
Figure: 8.1. The apogees of the orbits of planets longitude.
In one of the verses Brahmagupta points out the differences in his and Aryabhata calculations in the pherpheries of the sighra and manda epicycles of
Planets in odd and even quadrants. This differences is illustrated in the table given below:
Table: 8.1. Peripheries of the Epicycles of the Planets.
Aryabhata and Bhaskara have both given the sutra for finding the drkksepajyas of the Sun and the Moon, where drkksepajya is the Rsine of the zenith distance of that point of the ecliptic that is at the shortest distance from the zenith this point is also called central ecliptic point. The rule says: find the product of the Sun/Moon own mahyajya and udayajya then divide it by the radius, square the hence obtained result and subtract this result with the square of madhyajya. The square root of this difference is drkksepajya of the Sun/Moon.
i.e drkksepajya = 
The above rule given by Aryabhata was approximate and has been criticized by Brahmagupta.
Even for finding the Rsine of the agra of the Sun and the Rsine of the Sun prime vertical altitude, the rule which is given in Golapada- Aryabhata ya sutra 30-31was criticized by Brahmagupta.
Sutra: 8.3. Aryabhatta: V-30-31.
Rule says that: find the product of the Sun Rsine of the greatest declination by the Rsine of the Sun true longitude then divide the resultant product with the Rsine of the colatitude, the result thus obtained is the Rsine of the agra of Sun. Now when the agra of the Sun is less than the latitude and Sun is in the northern hemisphere multiply the Sun Rsine of agra with the Rsine of the colatitude the result thus obtained is the Sun’s prime vertical altitude. The error which was created by Aryabhata was that the sun’s agra should be less than the latitude was noticed by Brahmagupta, however Bhaskara corrected this error in his Laghu Bhaskariya according to which Sun’s agra should be not less than latitude rather the Rsine of the Sun northern declination is less than the latitude, this condition is necessary for the prime vertical shadow of the gnomon.
Brahmagupta has also criticized the calculation of determining lambana i.e. the difference of the parallaxes in longitude of Sun and the Moon and the rule for finding avanati or nati i.e. the differences in the parallaxes in latitude of the Sun and the Moon.
Lambana was obtained with the help of the five Rsine values i.e. madhyajya, udayajya, drkksepajya, drgjya and drggatijya.
◆Madhyajya-the Rsine of the zenith distance of meridian ecliptic point=Rsin(∅ +_ declination of meridian ecliptic point ) where ∅ is the place latitude and R is the radius of the celestial sphere.
◆Udayajya-the Rsine of the arc of the horizon intervening between the equator and the ecliptic=
where L is the longitude of the horizon east
where L is the longitude of the horizon east
ecliptic point, ∅ is the place latitude, R is the radius of the celestial sphere and H is the obliquity of the ecliptic.
Drgjya-the Rsine of the zenith distance of the Sun =
where R is the radius of the celestial sphere, L is the longitude of the horizon east ecliptic point and M is the longitude of the sun.
Drggatijya-the Rsine of the altitude of the central ecliptic point=
where R is the radius of the celestial sphere.
where R is the radius of the celestial sphere.
Lambana is the difference between the parallaxes in longitude of the Sun and the
Moon and is given by Aryabhata with the help of following expression:
Lambana = Moon’s lambana – Sun’s lambana.
Where Moon's lambana =
Sun’s lambana = 
Note: Lambana are in terms of minutes of the arc.
Lambana is also expressed in following ways:
Nati or the avanati both terms mean same, is the difference of the parallaxes in the latitude of the Sun and the Moon is given by ryabhata in following ways:
Nati=
minutes. Aryabhata has given this expressions in the following sutra:
minutes. Aryabhata has given this expressions in the following sutra:
Sutra: 8.4. Aryabhatiya: IV-33-34.
Where Brahmagupta has used following expression for lambana and nati which inturn gives accurate values:
Lambana=
ghatis, where R is the radius of celestial sphere, m is the longitudes of the meridian ecliptic and \ is the longitude of the Sun.
ghatis, where R is the radius of celestial sphere, m is the longitudes of the meridian ecliptic and \ is the longitude of the Sun.Nati=
where R is the radius of celestial sphere and ∆ is the difference between the daily motions of the Sun and the Moon.
where R is the radius of celestial sphere and ∆ is the difference between the daily motions of the Sun and the Moon.
Brahmagupta had criticized the ryabhata system of calculation with respect to Lambana i.e. the differences of the parallaxes in longitude, of the Sun and the Moon, drkksena i.e. the Rsine of the zenith distance of the central ecliptic point, srngonnati i.e. elevation of Moon horns and many more points. He opposed ryabhata so intensely that in the end he declares in one of his sutra-Brahmasphuta Siddhanta XI.44 that it is beyond his capability to enumerate all the defects made by ryabhata. Only few of his defects were illustrated here and other defects can be easily traced by intelligent people. Sutra for the same is listed below:
Sutra: 8.5. Brahmasphuta: XI-44.
In one of the other sutra i.e. Brahmasphuta SiddhantaXI.43 Brahmagupta states that ryabhata is neither knowledgeable with the ganita (Mathematics) nor kala i.e. time calculation nor Gola i.e. spherical calculation. Further he adds that it is difficult to enumerate separately the mistakes committed by him in connection with the chapters of the ganitapada, kalakriyapada and golapada.
Sutra: 8.6. Brahmasphuta: XI-43.
ryabhata and Brahmagupta both are scholars of 5th and 6th century respectively have done in-depth works in the fields of Astrology, Astronomy and Math. Brahmagupta while completing his works had lots of disagreement with ryabhata works and has bought it out in his treatise.
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