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This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, Saturn, a superior planet, is on the circumference of the Sheegra Epicycle, where it is met by a radius drawn parallel to the direction of the Sun from the observer.

To the Western scholars, Indian Astronomy is mysterious. Let us see what astro scholars have said about IA.

Dennis Duke, of Florida State University suggests that Indian Astronomy predates Greek Astronomy

“The planetary models of ancient Indian mathematical astronomy are described in several texts.1 These texts invariably give algorithms for computing mean and true longitudes of the planets, but are completely devoid of any material that would inform us of the origin of the models. One way to approach the problem is to compare the predictions of the Indian models with the predictions from other models that do have, at least in part, a known historical background. Since the Indian models compute true longitudes by adding corrections to mean longitudes, the obvious choices for these latter models are those from the Greco-Roman world. In order to investigate if there is any connection between Greek and Indian models, we should therefore focus on the oldest Indian texts that contain fully described, and therefore securely computable, models. We shall see that the mathematical basis of the Indian models is the equant model found in the Almagest, and furthermore, that analysis of the level of development of Indian astronomy contemporary to their planetary schemes strongly suggests, but does not rigorously prove, that the planetary bisected equant model is pre-Ptolemaic” says he.

The earliest Indian Planetary Models are two sets from the writer Aryabhata, both dating from 6th Century AD.

1) The Sunrise System , after the Epoch, which is taken from the sunrise of 18th Feb 3102 ( Arya Paksha ). It appears first in Aryabhatiya

2) The Midnight System, after the Epoch, which is taken from the midnight of 17/18 FEB 3102 ( Ardha Ratri Paksha ). It appears first in Latadeva’s Soorya Siddhanta

The Local Meridien is taken as Lanka, Longitude 76 degrees, Latitude 0 degrees.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given.

In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent.

The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given.

In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent.

The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

This diagram is by courtesy of Jean-Pierre Lacroix and Robert Baywater, www.ancientcartography.net

In the above diagram, both the theories of Manda Kriya and Sheegra Kriya are given.

In the case of a superior planet, a deferent is drawn from an earth based observer. The Center of the Manda Epicyle rotates around the terrestrial observer, travelling around the deferent.

The peripheral end of one radius of this Manda Epicycle determines the center of another epicyle called the Sheegra Epicycle.

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha, the Radius of the Circle is 3438 minutes and Arc is 5400 minutes.

Circumference = 2 Pi R
R = 360/2 Pi
R = 57.3 degrees
R = 57.3 * 60 = 3438 arcminutes
R = 3438 * 60 = 206265 arcseconds

Half Chord of 90 degrees = 90*60 = 5400 arcminutes.

In his astronomical treatise, the Aryabhatiya, he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry.

In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Aryabhata’s Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences of the values of trignometric sines expressed in arcminutes. Because of this, this Table is referred to as the Table of Sine Differences.

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha, the Radius of the Circle is 3438 minutes and Arc is 5400 minutes.

Circumference = 2 Pi R
R = 360/2 Pi
R = 57.3 degrees
R = 57.3 * 60 = 3438 arcminutes
R = 3438 * 60 = 206265 arcseconds

Half Chord of 90 degrees = 90*60 = 5400 arcminutes.

In his astronomical treatise, the Aryabhatiya, he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry.

In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Aryabhata’s Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences of the values of trignometric sines expressed in arcminutes. Because of this, this Table is referred to as the Table of Sine Differences.

Aryabhata, one of the earliest mathematicians and astronomers, ( circa 476-550 CE ) postulated that Vysasardha, the Radius of the Circle is 3438 minutes and Arc is 5400 minutes.

Circumference = 2 Pi R
R = 360/2 Pi
R = 57.3 degrees
R = 57.3 * 60 = 3438 arcminutes
R = 3438 * 60 = 206265 arcseconds

Half Chord of 90 degrees = 90*60 = 5400 arcminutes.

In his astronomical treatise, the Aryabhatiya, he postulated that the Circumference of the Circle is 360*60 = 21600 minutes. All these formulae are useful for the computation of half chords of certain sets of arcs in a circle and became the base of Hindu Trignometry.

In his Sine Tablest, he called 3 degrees 45 minutes divisions by many Sanskrit names, given below.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Aryabhata’s Sine Table is not a set of values of the trignometric sine functions, but rather is a table of the first differences of the values of trignometric sines expressed in arcminutes. Because of this, this Table is referred to as the Table of Sine Differences.

In Hindu Trignometry ( which is derived from Trikonamithi, trikona = triangle and trignon = triangle ), Jya resembles the modern Sine and Koti Jya, the cosine.

But in actuality, Jya is R Sin, that is Radius multiplied by modern sine.

By Jya, Brahmagupta meant 5 degrees of a circle. In Hindu Sine Tables and Tan Tables, the values are given for 5 degrees, 10 degrees, 15 degrees etc so that the Astro Maths students need not bother about using the Indian trignometric and inverse functions. Aryabhata’s sine tables are found to be accurate, when compared to modern sine tables.

In other words, one Zodiacal Constellation, which is 30 degrees is made up of 6 jyas and a total of 72 Jyas constitute the Zodiac.

Koti Jya is R Cos, that is Radius multiplied by modern cosine.

Utkram Jya is the reverse sine, defined as 1- cos x. Since the Reverse sine resembled an arrow, Brahmagupta called it Sara. And since the Arcsine resembled a bow, he called it Chapa.

Bhujajya is radius multiplied by modern sine and bhujachapa is the arcsine. Kotijya is radius multiplied by modern cosine and Kotichapa is arccos. Sparshjya is tan and sparshachapa is arctan.

Aryabhata’s Sine Table was the first ever constructed sine table in the History of Maths.

This is Aryabhata’s Sine Table given for different Kakshyas ( One Kakshya is 3 degrees 45 mins, one eighth of 30 degrees Zodiacal Sign )

Sl. No Angle ( A ) (in degrees, arcminutes) Value in Āryabhaṭa’s numerical notation
(in Devanagari) Value in Āryabhaṭa’s numerical notation (in ISO 15919 transliteration) Value in Arabic numerals Āryabhaṭa’s value of jya (A) Modern value of jya (A)
(3438 × sin (A))

1 03° 45′ मखि makhi 225 225′ 224.8560
2 07° 30′ भखि bhakhi 224 449′ 448.7490
3 11° 15′ फखि phakhi 222 671′ 670.7205
4 15° 00′ धखि dhakhi 219 890′ 889.8199
5 18° 45′ णखि ṇakhi 215 1105′ 1105.1089
6 22° 30′ ञखि ñakhi 210 1315′ 1315.6656
7 26° 15′ ङखि ṅakhi 205 1520′ 1520.5885
8 30° 00′ हस्झ hasjha 199 1719′ 1719.0000
9 33° 45′ स्ककि skaki 191 1910′ 1910.0505
10 37° 30′ किष्ग kiṣga 183 2093′ 2092.9218
11 41° 15′ श्घकि śghaki 174 2267′ 2266.8309
12 45° 00′ किघ्व kighva 164 2431′ 2431.0331
13 48° 45′ घ्लकि ghlaki 154 2585′ 2584.8253
14 52° 30′ किग्र kigra 143 2728′ 2727.5488
15 56° 15′ हक्य hakya 131 2859′ 2858.5925
16 60° 00′ धकि dhaki 119 2978′ 2977.3953
17 63° 45′ किच kica 106 3084′ 3083.4485
18 67° 30′ स्ग sga 93 3177′ 3176.2978
19 71° 15′ झश jhaśa 79 3256′ 3255.5458
20 75° 00′ ङ्व ṅva 65 3321′ 3320.8530
21 78° 45′ क्ल kla 51 3372′ 3371.9398
22 82° 30′ प्त pta 37 3409′ 3408.5874
23 86° 15′ फ pha 22 3431′ 3430.6390
24 90° 00′ छ cha 7 3438′ 3438.0000

Sine Table by courtesy www.wikipedia.org

In Hindu Trignometry ( which is derived from Trikonamithi, trikona = triangle and trignon = triangle ), Jya resembles the modern Sine and Koti Jya, the cosine.

But in actuality, Jya is R Sin, that is Radius multiplied by modern sine.

By Jya, Brahmagupta meant 5 degrees of a circle. In Hindu Sine Tables and Tan Tables, the values are given for 5 degrees, 10 degrees, 15 degrees etc so that the Astro Maths students need not bother about using the Indian trignometric and inverse functions. Aryabhata’s sine tables are found to be accurate, when compared to modern sine tables.

In other words, one Zodiacal Constellation, which is 30 degrees is made up of 6 jyas and a total of 72 Jyas constitute the Zodiac.

Koti Jya is R Cos, that is Radius multiplied by modern cosine.

Utkram Jya is the reverse sine, defined as 1- cos x. Since the Reverse sine resembled an arrow, Brahmagupta called it Sara. And since the Arcsine resembled a bow, he called it Chapa.

Bhujajya is radius multiplied by modern sine and bhujachapa is the arcsine. Kotijya is radius multiplied by modern cosine and Kotichapa is arccos. Sparshjya is tan and sparshachapa is arctan.

Aryabhata’s Sine Table was the first ever constructed sine table in the History of Maths.

This is Aryabhata’s Sine Table given for different Kakshyas ( One Kakshya is 3 degrees 45 mins, one eighth of 30 degrees Zodiacal Sign )

Sl. No Angle ( A ) (in degrees, arcminutes) Value in Āryabhaṭa’s numerical notation
(in Devanagari) Value in Āryabhaṭa’s numerical notation (in ISO 15919 transliteration) Value in Arabic numerals Āryabhaṭa’s value of jya (A) Modern value of jya (A)
(3438 × sin (A))

1 03° 45′ मखि makhi 225 225′ 224.8560
2 07° 30′ भखि bhakhi 224 449′ 448.7490
3 11° 15′ फखि phakhi 222 671′ 670.7205
4 15° 00′ धखि dhakhi 219 890′ 889.8199
5 18° 45′ णखि ṇakhi 215 1105′ 1105.1089
6 22° 30′ ञखि ñakhi 210 1315′ 1315.6656
7 26° 15′ ङखि ṅakhi 205 1520′ 1520.5885
8 30° 00′ हस्झ hasjha 199 1719′ 1719.0000
9 33° 45′ स्ककि skaki 191 1910′ 1910.0505
10 37° 30′ किष्ग kiṣga 183 2093′ 2092.9218
11 41° 15′ श्घकि śghaki 174 2267′ 2266.8309
12 45° 00′ किघ्व kighva 164 2431′ 2431.0331
13 48° 45′ घ्लकि ghlaki 154 2585′ 2584.8253
14 52° 30′ किग्र kigra 143 2728′ 2727.5488
15 56° 15′ हक्य hakya 131 2859′ 2858.5925
16 60° 00′ धकि dhaki 119 2978′ 2977.3953
17 63° 45′ किच kica 106 3084′ 3083.4485
18 67° 30′ स्ग sga 93 3177′ 3176.2978
19 71° 15′ झश jhaśa 79 3256′ 3255.5458
20 75° 00′ ङ्व ṅva 65 3321′ 3320.8530
21 78° 45′ क्ल kla 51 3372′ 3371.9398
22 82° 30′ प्त pta 37 3409′ 3408.5874
23 86° 15′ फ pha 22 3431′ 3430.6390
24 90° 00′ छ cha 7 3438′ 3438.0000

Sine Table by courtesy www.wikipedia.org

In Hindu Trignometry ( which is derived from Trikonamithi, trikona = triangle and trignon = triangle ), Jya resembles the modern Sine and Koti Jya, the cosine.

But in actuality, Jya is R Sin, that is Radius multiplied by modern sine.

By Jya, Brahmagupta meant 5 degrees of a circle. In Hindu Sine Tables and Tan Tables, the values are given for 5 degrees, 10 degrees, 15 degrees etc so that the Astro Maths students need not bother about using the Indian trignometric and inverse functions. Aryabhata’s sine tables are found to be accurate, when compared to modern sine tables.

In other words, one Zodiacal Constellation, which is 30 degrees is made up of 6 jyas and a total of 72 Jyas constitute the Zodiac.

Koti Jya is R Cos, that is Radius multiplied by modern cosine.

Utkram Jya is the reverse sine, defined as 1- cos x. Since the Reverse sine resembled an arrow, Brahmagupta called it Sara. And since the Arcsine resembled a bow, he called it Chapa.

Bhujajya is radius multiplied by modern sine and bhujachapa is the arcsine. Kotijya is radius multiplied by modern cosine and Kotichapa is arccos. Sparshjya is tan and sparshachapa is arctan.

Aryabhata’s Sine Table was the first ever constructed sine table in the History of Maths.

This is Aryabhata’s Sine Table given for different Kakshyas ( One Kakshya is 3 degrees 45 mins, one eighth of 30 degrees Zodiacal Sign )

Sl. No Angle ( A ) (in degrees, arcminutes) Value in Āryabhaṭa’s numerical notation
(in Devanagari) Value in Āryabhaṭa’s numerical notation (in ISO 15919 transliteration) Value in Arabic numerals Āryabhaṭa’s value of jya (A) Modern value of jya (A)
(3438 × sin (A))

1 03° 45′ मखि makhi 225 225′ 224.8560
2 07° 30′ भखि bhakhi 224 449′ 448.7490
3 11° 15′ फखि phakhi 222 671′ 670.7205
4 15° 00′ धखि dhakhi 219 890′ 889.8199
5 18° 45′ णखि ṇakhi 215 1105′ 1105.1089
6 22° 30′ ञखि ñakhi 210 1315′ 1315.6656
7 26° 15′ ङखि ṅakhi 205 1520′ 1520.5885
8 30° 00′ हस्झ hasjha 199 1719′ 1719.0000
9 33° 45′ स्ककि skaki 191 1910′ 1910.0505
10 37° 30′ किष्ग kiṣga 183 2093′ 2092.9218
11 41° 15′ श्घकि śghaki 174 2267′ 2266.8309
12 45° 00′ किघ्व kighva 164 2431′ 2431.0331
13 48° 45′ घ्लकि ghlaki 154 2585′ 2584.8253
14 52° 30′ किग्र kigra 143 2728′ 2727.5488
15 56° 15′ हक्य hakya 131 2859′ 2858.5925
16 60° 00′ धकि dhaki 119 2978′ 2977.3953
17 63° 45′ किच kica 106 3084′ 3083.4485
18 67° 30′ स्ग sga 93 3177′ 3176.2978
19 71° 15′ झश jhaśa 79 3256′ 3255.5458
20 75° 00′ ङ्व ṅva 65 3321′ 3320.8530
21 78° 45′ क्ल kla 51 3372′ 3371.9398
22 82° 30′ प्त pta 37 3409′ 3408.5874
23 86° 15′ फ pha 22 3431′ 3430.6390
24 90° 00′ छ cha 7 3438′ 3438.0000

Sine Table by courtesy www.wikipedia.org