SOME UNSEEN TEMPLES OF KASHMIR

Chander M. Bhat

Taksaka was one of the major serpents born of the parents, Kasyapa and Kadru. He was the king of 27 races of serpents. It was he who bit and killed the king Pariksit. Enraged by this, the king Janamejaya, son of Pariksit performed the Sarpayaga, aimed at destroying the entire race of the serpents. Taksaka took refuge in Indra, the king of gods. However, when due to the power of the mantra, even Indra along with Taksaka was forcibly pulled towards the sacrificial fire, the young sage Astika appeared on the scene and saved both of them. The sacrifice too was stopped.

Village Zewan [Lat. 34 degrees 2 minutes, Long. 74 degrees 56 minutes] is at a distance of 12 km to the south-east of Srinagar. A link road from Pantha Chowk (ancient Pandu Chak) joins Zewan and is 3 km away from Pantha Chowk. This village was founded by King Kalasha (1089-1011). Takshak Nag was a place of pilgrimage in bygone days and now is visited by the pilgrims who undertake the pilgrimage to Harishore cave up on a hilltop. It is the only holy spring in Kashmir which is distinctively mentioned in the shrine list of the Mahabharata. The stonework for the embankment of this holy spring has been done under Dogra Rule.

The spring measures around 50 feet by 50 feet with a depth of about 4 feet with crystal clear water. Water is oozing in the northeastern corner of the spring. It is said that saffron has originated from this holy spring. Takshaka offered Hakim Waga Bhat, the saffron bulbs as a token of reward for curing his eye ailment. Bilhan the great Sanskrit poet, who flourished in the 11th century, was born at Khunmoh (a village at a distance of 5 km. towards the east of Zewan). 

MARTAND TEMPLE
Martand [Lat. 33 degrees 45 minutes, Long. 75 degrees 16 minutes] is about 52 km south-east of Srinagar and 7 km from Anantnag en-route Pahalgam. It is famous for its temple of Sun built by King Lalitaditya Muktapida (8th century A.D.). Though in ruins, it is considered as one of the beautiful temples of Kashmir. Even in the state of ruin, it appears magnificent.
The ruins of the temple of Martand, or, as it is commonly called, the Pandav Lareh, is situated on the highest part of a Khrewa, where it commences to rise to its junction with the mountains. Occupying undoubtedly the finest position in Kashmir this noble ruin is the most striking in size and situation of all the existing remains of Kashmir grandeur. The temple is not now more than 35 feet in height, but its solid walls and bold outlines towering over the fluted pillars of the surrounding colonnade, give it a most imposing appearance. The temple is of moderate size, 60 feet by 36 feet with two facades in the front.
Brari Aangan, Uma Nagri , Utrasoo
In the Kenopanisad there is the mention of Uma Haimavatisvari (Uma, daughter of Himavan, same as Parvati) teaching Indra about Brahman. She is the personification of Brahmavidya or knowledge of Brahma.
Village Brari Aangan (Uma Nagri) [Lat. 33 degrees 42 minutes, Long. 75 degrees 21 minutes] is a large village in the Kuthar pargana lying east of Achhabal, at the mouth of the Halkan Galli, on the way to village Nowbugh. This village is about 14 km from Achhabal and 19 km from Anantnag.

There are four springs of diameter 12 mts/8 mts and 11 mts. in shrine complex. The springs are having crystal clear water and are located like a triangle in the shape of Omkara known as Brahma Kund, Gauri Kund, Shiva Kund, and Vishnu Kund. The main temple of Goddess Uma has been constructed in the middle of Gauri Kund. On the west side of the shrine, there is a Samadhi of Swami Shivananda.
In the year 1772 AD, Haji Karimdad Khan, (harshest and most tyrannical Afghan Subehdar of Kashmir), deputed a junior executive, Pandit Shiv Ram Kaul, to collect land revenue and other taxes from the Kuthar Pargana. Pandit Shiv Ram Koul had already been initiated by a ranking preceptor in spiritual discipline and on reaching Brari Aangan he intensified his meditative practice and resigned from the job. These facts were reported to the Subehdar who summoned Pandit Shiv Ram Koul to Srinagar. The latter did not go.
The Subehdar himself went to Brari Aangan to check personally and catch hold of the so-called fugitive. Catching a glimpse of Pandit Shiv Ram from a distance he found him flanked by two cats, but when he approached closer he was astounded to find two leopards instead thus Brari Aangan, the courtyard of cats. One of the saints namely Swami Svamyananand of Ganpatyar, Srinagar who was blessed by the Goddess had also contributed a lot for the construction of the shrine at Uma Nagri.
TIRTH RAJ, LOK BHAWAN, DORU, ANANTNAG
Tirth Raj Lok Bhawan [Lat. 32 degrees 17 minutes, Long. 73 degrees 44 minutes] is one of the oldest pilgrimage centers in Kashmir. This ancient shrine was known by the name of Lok-Punya. The word Lok-Punya signifies the Divine greatness of the shrine. With the passage of time the word Punya was replaced by ‘Bhawan’.
Village Lok Bhawan is situated in Doru Tehsil erstwhile Brang Pargana some 10 km from district headquarter Anantnag. A link road from village Larkipora links the village with the rest of the world. This village is situated beneath a small hill. A beautiful spring has its source at the feet of this hill. This place is famous for the ancient pilgrimage, the Lok Bhawan pilgrimage. This pilgrimage, has a shrine, a small holy spring, at an elevation, and big spring, the Lok Bhawan Spring, which receives water from the small holy spring above at the foot of the hillock. The Lok Bhawan spring has plenty of water and abounds in sacred fish. The total land under this shrine is 20 kanal and two springs are spread over an area of 8 kanal and 10 marlas. The big spring is 60 feet long and 54 feet in width and the small spring 10 feet in length and 8 km in width.
It is said the Lok Bhawan was an affluent town in the days gone by. A local ruler named Lok Nath is said to have laid the foundation of this village. Previously it was known as Rudhra Bhawan. Late Shri Anand Koul ‘Bamzai’ in his book ‘History of Kashmir’ has also mentioned that a king named Lok Punya has laid the foundation of village Lok Bhawan. 

 BALA DEVI, BALHAMA, PAMPORE.
‘Bala’ is a Sanskrit word meaning ‘the girl child.’ Worship of Sakti or the Mother Goddess is a very ancient practice in India. Of the several aspects of the Divine Mother Parvati, Lalita or Lalita-Tripurasundari is very important and popular. Bala (also called Bala-Tripurasundari) is an aspect of this Lalita. Sometimes she is described as the daughter of Lalita who helped her to destroy the army of the demon Bhandasura.
As her very name suggests, Bala is pictured as an eternal girl of nine years. Iconographical works describe her as red in colour like a hibiscus flower. She has four arms carrying rosary, noose, goad, 
and book. Bala is specially worshipped, by those desiring psychic powers.
MARTAND TEMPLE
Martand [Lat. 33 degrees 45 minutes, Long. 75 degrees 16 minutes] is about 52 km south-east of Srinagar and 7 km from Anantnag en-route Pahalgam. It is famous for its temple of Sun built by King Lalitaditya Muktapida (8th century A.D.). Though in ruins, it is considered as one of the beautiful temples of Kashmir. Even in the state of ruin, it appears magnificent.
The ruins of the temple of Martand, or, as it is commonly called, the Pandav Lareh, is situated on the highest part of a Khrewa, where it commences to rise to its junction with the mountains. Occupying undoubtedly the finest position in Kashmir this noble ruin is the most striking in size and situation of all the existing remains of Kashmir grandeur. The temple is not now more than 35 feet in height, but its solid walls and bold outlines towering over the fluted pillars of the surrounding colonnade, give it a most imposing appearance. The temple is of moderate size, 60 feet by 36 feet with two facades in the front.
The abode of Bala Devi is at Village Balhama [Lat. 34 degrees 2 minutes Long. 74 degrees 59 minutes] about 4 km north-east of Pampore and 13 km south-east of Srinagar. This village is surrounded by village Wuyan in the east, Khunmoh in the north, and Pampore town in the west. This village is about 3 km on the right side near Pantha Chowk reached by a link road near Pampore Joinery Mill. The temple has been built by Dogra rulers in the year 1941 by constructing a circular brick pillar covering five Devdar trees and thereby joining the pillars by steel fencing. Temple is facing towards the east.

Vivekananda Kendra, Nagdandi, Achhabal
Vivekananda Kendra, Nagdandi [Lat. 33 degrees 39 minutes, Long. 75 degrees 14 minutes] is located in a dense forest in the foothills of Achhabal, 3 km from Achhabal (en-route Sri Uma Devi) reached by a link road, I km from the main road. The temple is dedicated to Sri Ramakrishna Paramahansa.
The ordinary men of this world, sometimes, become so ordinary that extraordinary men have to appear, here to live them up to no ordinary heights, if not extraordinary ones. These tall men who walk this earth may seem to be no different from others at first but soon grow to superhuman proportions revealing their true nature even as Visnu appeared first as Vamana and then as Trivikrama. Sri Ramakrishna belongs to this rare species. His life was religion…in the highest sense…in practice. He lifted up religion from the morass of rituals and dogmas to great mystical heights as effortlessly as Varaha the earth. The earliest study of his life…his doings, and sayings…can not only inspire us but also elevate us to transcendental levels.

This center of Vivekananda Kendra was founded by Swami Ashokanandaji Maharaj (1911-1971) in the year 1947. The shrine complex is spread over more than 107 kanals of land with lush green gardens, Kitchen gardens, fruit trees; a temple dedicated to the holy trinity (Sri Ramakrishna, Sri Sarda Devi, and Swami Vivekananda), a temple dedicated to Swami Ashokanandaji, and several guest houses.
AKHALUK CHAKA PAL
Akhal (Aikshala), is a small village located 3 km to the west of village Murran and 8 km from district headquarter Pulwama en-route village Rohmuh. At this village, we see a “Chaka-Pal” (Chakrautapala), a huge boulder with a Siva linga and Mahavera engraved on it.
It is believed that the Pandavas while visiting the pilgrimage centers of north India have visited this place. They have stayed here for a couple of days. It is said that Pandavas tired to lift this boulder and for this purpose, they tied the boulder with iron chains to have a perfect grip. The scars of these chains are still visible on the boulder.
Lord Shiva is engraved on the left side of the boulder and Mahaveera on the front side. It is a pilgrimage center even though there was no temple or Dharamshala at this spot.
Pandits of the neighboring villages i,e. Murran, Frastipora, Diear, Haarpur, Moonghama, Sirnoo, Putrigam and Rohmuh used to visit this place for worship and offer vermilion (सिंदूर), a paste made by mixing the सिंदूर with ghee which was being pasted on the engraved statue of Mahaveera. A big stream is flowing nearby making the place more attractive.
Village Akhal is in the border of the Pulwama, Badgam and Shopian districts. Murran 2 km, Kangan 3 km, Monghama 4 km, Mitrigam 1 km, Rohmuh 2 km are the nearby Villages to Akhal. 

 Wagh Bhat of Village Waghhama
by Chander M. Bhat
Those who believe in the austerity of thought and those who believe that specialties are not uncommon, leave an impact to such a personality, for whom there are no scriptures and any written history to make him immortal. Wagh Bhat belongs to this tiny village some 5 km from Bijbehara. There was no Kashmiri Pandit family in this village at the time mass exodus. Only what remains of his memories is that oral stories flow from generation to generation.
It is said that Adi Guru Shankaracharya, while on his visit to Kashmir, stayed at Vijeshwara Temple at Bijbehara. He was enquiring about blissful and learned people with whom he could exchange his ideology. Apparently little known Wagh Bhat came to his knowledge by some local resident as being the spiritual personality with magical powers.
He left in search of him and landed at Waghama. During his quest he found him ploughing his field and was advised to go to his home at the village. Adi Guru proceeded to his home. What he saw in the corridor [Voouz in Kashmiri], felt like a hammer on his pious heart. The wife of the spiritual Wagh Bhat was cleaning a small fist locally known as ‘Gurran’ in the corridor. Adi Guru was helpless. The housewife asked him to come in and advised him to sit in the room adjacent to the corridor called ‘Wuot’ in Kashmiri. She continued cleaning the fish and Adi Guru did not like it because of being a vegetarian.
He waited for Wagh Bhat for an hour or so. As he felt restless, he asked the housewife about the reason for his late coming. On knowing, that Adi Guru was intentional to have a dialogue with Wagh Bhat. She wanted to break the illusion of Adi Guru about Shiva being the supreme with only an unimportant secondary place to Shakti.
She took some water from the pot in which she was cleaning then and threw it on the broom and urn, containing cleaned combine of water and red clay for purification of rooms and corridor [locally known as Levan Dul]. Surprisingly for Adi Guru, was that he experienced the flow of Shalukas from both these broom and urn kept in a corner of the corridor. He was moved and accepted that Shakti is equally powerful and is part and parcel of Shiva. He had a mission that got completed even before meeting Wagh Bhat.
Hats down for those Kashmiris who have had at their command such spirituality that even Adi Guru Shankaracharya was outsmarted .

KSHIR BHAVANI, TIKKAR
TIKKER [Lat. 34 degrees 32 minutes, Long. 74 degrees 18 minutes] a Sedd Peth, is believed that Mata Raginya had stayed at this place at the time of her arrival in Kashmir, when Hanuman along with 360 Nagas (snake-chieftains) brought Devi to Kashmir from Lanka. The nagas too were given appropriate places to live.
Maha Raginya shrine at village Tikkar is situated on Kupwara-Chowkibal road; just 3 km from the district headquarter Kupwara. The temple is a small structure of marble constructed inside a spring, as at Kshir Bhawani, Tullamulla. Swami Lal Ji Maharaj, Swami Nand Lal Ji and Swami Kral Bab used to perform their Sadhana at the shrine. The shrine has some old houses constructed from mud and wood on a mound which were used by these great saints.

The shrine also has a Sharda temple with a Shila in the form of Paad, many Dharamshalas and a large landed property.

BHADRAKALI SHRINE, WADIPORA, HANDWARA.
BHADRAKALI SHRINE also known as Bhadraheuur is an ancient shrine situated at the extreme end of village Wadipora some 8 km from Handwara town in Kupwara District.
The shrine is situated on a small hillock and is covered with thick forest. One has to climb 350 steps to reach the shrine. The importance of this shrine is described in Bhadrakalipradurbhave. The holy shrine has an Icon in the form of ancient deodar (cedar) tree, considered as the manifestation of Shakti and believed to be more than 1000 years old, around which the temple has been constructed 

NAGBAL, TRAL:
Tral [Lat. 33 degrees 56 minutes, Long. 75 degrees 10 minutes] is situated on the sloping plateau at the foot of the mountain [Vastur Van] near the east side of the Wullar pargana. It lies about 9 km east of Awantipora.
There are 12 springs in Tral town and among these 12 springs, that of the Diva Nag is the most famous. This spring lies on the east side of the town and is shaded by magnificent Chinar and other trees. The water which is very cool and clear rises into a pool about 60 feet square and about 6 feet deep. The waters of this spring are considered sacred by Kashmiri Pandits. The usual encamping ground is on the grassy plain by this spring.
Another spring known as Davabal spring is worthy of notice. In lower Tral there are two springs called Mertser Pukkur, also Kara Nag, Brim Sar, and Konchibal. The remaining three springs are situated in the middle of the town. Fatehgarh temple is situated at Sheergaon, Fatehgarh [Lat. 34 degrees 30 minutes, Long. 74 degrees 35 minutes] 2 km to the South of village Sheeri and 14 km from Baramulla. This temple stands in the middle of the main village. This temple is under the management of the Archaeological Survey of India.
The compound of the temple is 47 feet square. The temple is made of black stones, some of which measure 12 feet in length and 4 feet in height. The interior area of the main temple is 29 feet square. The greater part of cella was occupied by a massive platform on which was placed in the colossal Shivlinga. Two fragments of a water spout are also lying in the sanctum. A krtrima Shivlinga, made of brownstone was found during excavation with carving on two sides representing the Bhairava roop of Siva. The carving on one side is that of Shiva with three heads and on the other, a figure representing his consort Parvati is actually a ‘Rudralinga’.PARIHASPUR
PARIHASPUR [Lat. 34 degrees 08 minutes Long. 74 degrees 38 minutes] is situated over an elevation some 2 km from Narbal and 22 km from Srinagar en-route Baramulla. Wullar Lake forms the northern portion of these magnificent remains of yesteryears.
Lalitadiya Muktapida (695–731), built a number of towns with great temples during his regime. Kalhana writes “There is no town or village, no river or lake, no island where the king did not build a sacred foundation”. It was he who built the famous and elegant temple at Parihaspur. This temple was known as Parihaskesava. Parihaspur was the capital city of the great ruler. It is believed that Laltaditya built four Vaisnava temples namely Parihaskesava, Mahavarha, Govardhanadhara, and Muktadesva. The whole area consisting of about 50 kanals of land is under the management of the Archeological Survey of India. Rajtarangani further discloses “that the emperor constructed a separate stone shrine by the side of Shiva at Parihaspura and installed two images of Lord Rama in it which had been unearthed near the village of Suravardhamana 

BRARI MEUUJ, VILLAGE MURRAN, PULWAMA
Village Murran [Lat. 33 degrees 52 minutes, Long 74 degrees 54 minutes] is located about three km to the west of Pulwama town and two km. to the east of village Mitrigam… the birthplace of famous patriotic poet Pirzada Ghulam Ahmad ‘Mehjoor’. The natural scenery of the village is very charming. This village, surrounded on all sides by green pastures, untrimmed meadows, trees, shrubs and paddy fields, breathes a typical rural atmosphere.
Brari Meuuj, an ancient shrine is situated in North West of the village. This shrine came into existence when the Bhat dynasty migrated from village Sumbal.

“SUMBAL”…a beautiful village in north Kashmir was the birthplace of great grand ancestors of Bhat’s of village Murran. Village Sumbal was often under floods during the rainy season. Pandit Bhawani Bhat (1765-1835), a pious and orthodox person had a wonderful dream on Magh Ashtami night. Divine effulgence filled his muddy room. The Divine Mother looked affectionately at Pandit Bhawani Bhat, beckoned him, and said, “Your extraordinary devotion has made me very happy. The time has come to shift you from this place. Early the next morning, a cat will appear in your compound. Pack up your belongings and follow the cat and settle down where the cat will give you an indication. That will be your next place of settlement"... Divine Mother disappeared after reciting the verdict to Pandit Bhawani Bhat.
Early the next morning a Divine Cat appeared in the compound of Pandit Bhawani Bhat and by that time he had already packed up his little belongings and followed the Divine Cat till both of them reached a village. The Divine Cat gave an indication to him by waving the tip of its tail and disappeared. When the Divine Cat disappeared a small spring emerged on the spot, which was afterward expanded and a beautiful temple was built exactly on the spot where the Divine Cat made Pandit Bhawani Bhat settle. The temple was later called Brari Meuuj (Cat Mother). The construction of the temple started in 1847 and was completed in about four months at the expense of eighty-six rupees. The icon of the Divine Mother was installed in 1849 under the supervision of Shri Loket Bhat (1801-1876).
In the dark nights, under the canopy of the Chinars, the sacred place presents an atmosphere where one becomes rapt in his meditation and in the moonlit nights, a devotee experiences something mystic all over.
PATTAN TEMPLES:
PATTAN [Lat. 34 degrees 10 minutes, Long. 74 degree 36 minutes] is situated on Srinagar Baramulla road at a distance of 28 km on North West of Srinagar and 29 km southeast of Baramulla. When the valley of Kashmir was partitioned into parganas in the time of the emperor Akbar, the village of Pattan was forgotten. On discovering the mistake, he ordered to be constituted the 34th pargana by itself, to be called the Pattan pargana. The ancient name of this place was Sankarpura.

This place is known for three ancient temples. Two of these ancient temples, presently in ruins are located on the right side of the road and are some 400 meters away from each other. Accordingly to Rajatarangini, King Samkaravarman built one temple known as Shankar Gauresha and his wife Queen Sugandha built another temple known as Sagandhesvara. Both of the temples are dedicated to Lord Shiva.
Each temple has a Cella, porches and a square chamber for lingam. The design of construction reveals a similarity to Mattan Temples. A short distance away from these two temples are some remains of another ancient temple, standing inside a spring, known as Ratnavardhanesa, believed to be built by Ratna Vardhan, King Samkaravarman’s minister. According to Pandit R.C.Kak, a boali (spring), whose waters are confined in three regular reservoirs, which are connected with each other, were excavated some time back.
LINGODBHAVAMURTI SHIV LINGA, SHEERI, BARAMULLA
Village Sheeri [Lat. 34 degrees 9 minutes and Long. 74 degrees and 79 minutes] is located at a distance of 12 km from Baramulla en-route Uri and is reached by a link road. The village has two adjoining areas namely "Sheeri Bala" and "Sheeri Payeen".
Eleven feet Acala Shiv Linga is located on an elevated platform of land, 1 km to the south of the village in an open area surrounded by paddy fields. The image of Siva is carved on three sides at ‘Brahmabhaga’ of this ‘Lingodbhavamurti’ linga.

1. Acala: Immovable Linga.
2. Brahmabhaga: The lowest part of the Linga.
3. Lingodbhavamurti: The image of Siva carved on the middle
portion of the Linga.

PRATAPSWAMIN TEMPLE, TAPPER
Tapper Waripora is located 23 km towards east from Baramulla and 8 km to the west of Pattan. Ancient ruins of Pratapswamin temple with four subsidiary shrines are situated in this village. Tapper Waripora is surrounded by Kunzer towards the South, Narbal towards East and Singhpora towards west.
The temple is in the middle of a courtyard enclosed by a peristyle. It is assignable to the eleventh century A.D. and consists of a Garbhagriha and a Mandapa. It has subsidiary shrines in the courtyard. Only the plinth is preserved.

NAGBAL, ANANTNAG: Anantnag is one of the 10 districts of the Kashmir Valley situated in its south and southwestern direction. Geographically the district lies between 33 degrees 20 minutes to 34 degrees15 minutes north latitude and 74 degrees 30 minutes to 70 degrees 35 minutes east longitude, situated at a distance of 65 km of the south-east of Srinagar.
Nagbal, located in the Eastern part of Anantnag, is a religious centre. The shrine has been developed into a complex. The entire complex is known popularly as Nagbal and is famous Hindu religious centre in Anantnag District. The holy spring which originates from here and the formation of which is attributed to Vishnu or Narayana is said to be a Vedic pilgrimage centre. The spring rises beautifully from the foot of a small hill-lock and is dedicated to the worship of Ananta or Vishno. The shrine has a temple in the left corner, a tank, smaller than the holy spring at a lower elevation, with a Shiv Lingam in its centre and a garden with a number of Chinar trees dotting and surrounding the lower tank. Both the holy springs and the tank abound in fish, which are considered sacred. The holy spring attributed to Ananta (Lord Vishnu) and is believed to be a Vedic pilgrimage. In ancient times, it was called the Inder Nag. The small temple dedicated to Ananta and standing in the left corner was built during the reign of Maharaja Ranbir Singh. The second temple was built during the reign of Maharaja Pratap Singh and dedicated to Lord Shiva. Besides the holy spring and the two temples, there is also sulphur spring and a Gurdwara within the complex. The complex is barely 200 yards from Devibal Anantnag. The annual festival is held on beuuderpeth gat’iu pachh tsodah.
NIL NAG [Lat: 33 degrees 51 minutes, Long: 74 degrees 44 minutes] is an oval sheet of water, about 150 meters long and 30 meters wide, lying in a deep hollow on the slopes of the hills, on the south side of Srinagar, about 7 km west of Tsar Sharief. The water is derived from the springs, and the place is considered very holy the Kashmiri Pandits.
Abu’l Fazl (14 January 1551 – 12 August 1602) in his mention of this lake states it was “held sacred, and many fanatics consume themselves with fire on its border. They likewise try their fortunes by it in the following manner:

A walnut divided into four parts is thrown into the spring; if an odd number floats, it is accounted a good omen, and an even number is deemed unlucky. They also throw milk into it, sinking of milk indicates a good omen, but if it floats, the omen is bad. In ancient times there was, in this spring, a book entitled “Nilmat,” containing a particular description of Kashmir, with a history of this place of worship. It is asserted that at the bottom of the spring there is a large inhabited city and that a Pandit and went and remained there two or three days, and on his return gave a wonderful description of it.”

MANASBAL LAKE [Lat: 34 degrees, 15 minutes, Long: 74 degrees, 44 minutes]
Manasbal is situated in the North of Srinagar and is about 30 km from Srinagar on Srinagar Bandipur road in the same direction as the Wular Lake; it lies on the north side of Vitasta, with which it is connected by a canal which opens into the river about 2 km below Sumbal. The canal is about 2 km long from its mouth to its junction with the lake; it is about 18 yards wide and varies in depth according to the height of the river. Manasbal Lake is oblong in shape. Its length is 4 km, and breadth is about one km. It is the deepest of all the lakes in Kashmir.

Manasbal is famous not only for Manasbal Lake but also for a small ancient Shiva Temple situated on the North East side of the lake.
The temple is 6 feet square and has a two pyramidal roof and the main door. The temple is the mode of large limestone slabs. At present, the lower portion of the temple is submerged in water.
GOUSEIN NAAR, VILLAGE LADHU
Ladhu (Lat. 34 degrees, Long: 75 degrees and 2 minutes), a village in the Bihu pargana, situated on the table-land at the foot of the mountains (Wastarwan), about 9 km each of Pampore. It may also be reached from the village of Latapur, on the right bank of the Vitasta, by a metallic road over the table-land; following the base of the mountain, and the distance is about 5 km.
GOUSEIN NAAR is in the lap of Wastarwan one km from Village Ladhu. Jeewan Sahib passed the last days of his life in a house at Gousein Naar (see pic.). Jeewan Sahib, an outstanding saint of the 18th century was born in Motiyar Mohalla of Rainawari, Srinagar. It is said that this great saint shifted from Rainawari to Gousein Naar in 1779 and practiced sadhana in this village. All-time Dooni was on during his lifetime. The then Maharaja of Kashmir allotted him a jaggir of 80 kanals of land at Gousein Naar, Ladhu. Ten Kashmiri Pandit families were residing in this mohalla of the village, who were brought there from Srinagar by the then Maharaja of Kashmir to look after the land gifted to Jeewan Sahib.

This sacred shrine is being maintained by Shri Ravi Bhat, a resident of village Ladhu. SALUTE HIM.

SWAMI MERZA KAK ASHRAM, HANGULGUND, KOKERNAG
Hangulgund (Lat: 33 degree 36’minutes. Long: 75 degree 20 minutes) is a village in Bring pargana, situated on the path from village Sof towards Verinag.
Swami Merza Kak Ashram is located on the bank of Kokarnag Nala at Village Hangulgund, 25 km from Anantnag en-route Kokernag (known for its gardens, pristine fresh water springs and rainbow trout farms). The shrine has besides a Samadhi Sathaal, constructed by the followers of Swami Merza Kak after his Nirvana, a temple, a Dharamshala, number of Chinar and other trees and some landed property. The Samadhi, on which a tree has come up, is enclosed on all the four sides. The devotees pay obeisance to the great saint at his Samadhi.

In his honour, his followers have created a trust and built another Ashram, named Swami Merza Kak Peeth Ashram, at Nagrota, Jammu and perform Hawan on Jyestha Krishna Pakh Dwitiya at the shrine every year.

NARANTHAL is located on old Muzzarrarabad road about 4 km below Baramulla on the right bank of Vitasta in an isolated place. Situated at a short distance from the village is a small shrine which is said to have stood in a tank, though today it is on dry ground and no traces of a tank are visible. There is a spring nearby. Only the superstructure of the temple is above ground. It is built of a plain block of slate which is now very much the worse for wear. There is only one arched entrance on the east side. Traces of stone floor are visible inside.
The topmost stone of the roof has a circular mortice in the center which was originally intended to hold the finial which crowned the apex of the pyramid.

GANGNOOR, BARAMULLA
GANGNOOR also called Guptganga is an ancient Holy spring at Mohalla Rajghat, near Koti Tirtha, Baramulla. The spring is 6 feet wide and 10 feet long. The spring receives water from seven springs of Gosain Teng existing above at an elevation on the small hillock. The shrine has about 8 kanal of land attached to it. Devotees from around the area used to have a bath in the spring and proceed to Koti Tirtha for offering their puja/prayers.
Devotees congregate at the shrine on Kartik Purnima (15th day of Moonlit fortnight of Kartik i.e. October-November).

GURDUWARA PARAMPILLA, URI (BARAMULLA)
Gurudwara Sri Chevin Patshahi Sahib, Parampilla is situated in village Parampilla, 24 km from Baramulla en-route Uri, on the right bank of Vitasta. Sri Guru Hargobind Singh Ji has visited this place along with Jahangir. Guru Sahib stayed here for some time and further proceeded to Muzaffrabad.
Guru Ji proceeded to Uri via Village Khatnayar, Peernian and stayed near Sultan Dhaki. A plate of stone is still preserved in Gurdwara, on which five Muslim faqirs (viz: peer Gulsher, per Bhur Sultan, Peer Rangi Iman, Peer Noor Nihal, and Peer Abdul Gaffoor) had religious discourses with Guru Ji. In the end, Guru Ji proceeded towards 'Dolanga' Village. The footprints of Guru Ji's horse are still preserved near 'Dolanga' site.

In 1936-37 A.D., a small Gurdwara was constructed by the Devotees of Slamabad Dardkot, etc. Sardar Narian Singh of Slamabad, Sardar Gurmukh Singh, Sardar Damodar Singh engineer, and other Gursikhs helped in the construction. Sahajdhari Sikhs, Brahmans, and Muslim Sayeeds visited regularly to Gurdwara for blessings, Gopal Singh Granthi of Chandanwari traveled 20 km daily to perform duties of Granthi regularly. Baisakhi and Guru Har Gobind Sahib's birthday are celebrated with great enthusiasm. A small bridge constructed on the river Jehlum in 1970-71, connects the Gurdwara with the main road.
Gurdawara Nangali Sahib, Poonch
Gurudwara Nangali Sahib is situated in the lap of a picturesque hill and on the bank of Drungali Nallah, about 4 km from Poonch town. It is one of the oldest shrines of the Sikhs in Northern India.
The Gurdwara was established by Thakur Bhai Mela Singh Ji (Fourth successor of Sant Bhai Feru Singh Ji) in the 1803 AD. Maharaja Ranjit Singh visited Gurudwara Nangali Sahib in 1814 and was very much impressed by it. He attached an estate with the Gurudwara sahib. He again attached four villages with the Gurudwara Sahib in the year 1823.

In 1947, the original building was completely burnt by Kabalis and with the efforts of the local Sangat, the Gurudwara was reconstructed by Mahant Bachitar Singh Ji.

KHARDUNGLA TOP [LEH] (Height: 18,380 feet).
The pass on the Ladakh Range is north of Leh and is the gateway to the Shyok and Nubra valleys. The Siachen Glacier lays partway up the latter valley. Built-in 1976, it was opened to public motor vehicles in 1988. 40 km from Leh, Khardungla is the highest motorable all-weather road in the world.
Khardong La is historically important as it lies on the major caravan route from Leh to Kashgar in Central Asia. About 10,000 horses and camels used to take the route annually, and a small population of Bactrian camels can still be seen at Hunder, in the area north of the pass.

During harsh winters temperature at Khardungla goes down to Minus 40 degrees Celsius.

SHARDA TEMPLE, VILLAGE GUCHI, KUPWARA.
Sharda Temple, also known as Shardibal, is dedicated to Mother Sharda. The shrine is located on the main Handwara-Guchi-Kupwara road at village Guchi, just 6 km from Kupwara. The shrine has besides the Sharda temple, a huge chunk of land attached to it.
The Goddess of the Divine Lute (Veena). She represents original knowledge transcending all limitations. Sharda is an epithet of Saraswati; the Goddess of learning. She grants boons. Her effulgence is the means of obtaining final emancipation…Bhavani Nama Sahasra Stutih शारदा वरदा देवी मोक्षधारत्री सरस्वती (Sharda Varada Devi Mookshdhartri Saraswati) is a hymn meaning Sharda grants the boon of emancipation through wisdom.

NANDKISHORE, village Villgam. This village is about 8 km from Kupwara.
In 1948, a pious man Pt. Raghav Ram Bhat father of Swami Anandji Maharaj migrated to this village from Vaskura, Sumbal in search of his livelihood. Later, after his settlement in the village Villgam he got married to Pt. Savram’s daughter from Sumbal village. This pious lady was a daily visitor to the shrine of Nandkishore of Sumbal.
After her marriage, this pious lady went daily (by foot) to Sumbal to pay obeisance to Nandkishor. One night Nandkishore came in a dream and directed Pandit Raghav Ram to go to Villgam to prevent her wife from coming to Sumbal. The next day Nandkishore appeared under a willow tree in the form of a black boulder in Villgam near a stream. A Yagna was performed on Chetra Purnima and this black boulder along with a Shivlingam was installed near the willow. Later Swami Anandji Maharaj constructed a temple at this spot and after few days a spring oozed from there to the surprise of everyone.
Ragyna Shrine at village Kachwa Muqam, Wagoora, Baramulla, is situated on a slope with spring and ten poplar trees. Pandits of this village used to worship and offer Kshir (thickened milk) in the spring. Shri Abdul Rashid Lone a retired Lecturer who is presently residing near the shrine disclosed that he has seen a white cat with earrings many a time, going towards the spring in wee hours. He further disclosed that a lion visits this shrine every Thursday. Shri Abdul Rashid Lone further disclosed that some persons vandalized this shrine and in a month they were reduced from riches to rags.

Kachwa Muqam (Lat. 34-10 and Long. 74-28) is a village in the Kruhin Pargana, situated on the right bank of the Ningili stream, about 19 km from Baramulla. Ningili stream is formed by the glacier waters from Afarwat Mountain and waters from Alpather spring. The stream after passing through Gulmarg and adjoining villages join Wular Lake. This village is situated on the slope of the Karewa (Wudar). 15 well off Kashmiri Pandit families were residing in this village.

Brahmagupta

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Brahmagupta
Born	c. 598 CE
Died	c. 668 CE
Known for	Zero Modern number system Brahmagupta's theorem Brahmagupta's identity Brahmagupta's problem Brahmagupta-Fibonacci identity Brahmagupta's interpolation formula Brahmagupta's formula
Scientific career
Fields	astronomy, mathematics
Institutions	Bhillamāla, Gurjaradesa[1] Ujjaini, Avanti[2]

Brahmagupta (born c. 598 CE, died c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.
Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were in elliptic verse^{[clarification needed]} in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.^[3]

Contents

• 1 Life and career
• 2 Controversy
• 3 Reception
• 4 Mathematics
  • 4.1 Algebra
  • 4.2 Arithmetic
    • 4.2.1 Series
    • 4.2.2 Zero
  • 4.3 Diophantine analysis
    • 4.3.1 Pythagorean triplets
    • 4.3.2 Pell's equation
  • 4.4 Geometry
    • 4.4.1 Brahmagupta's formula
    • 4.4.2 Triangles
    • 4.4.3 Brahmagupta's theorem
    • 4.4.4 Pi
    • 4.4.5 Measurements and constructions
  • 4.5 Trigonometry
    • 4.5.1 Sine table
    • 4.5.2 Interpolation formula
• 5 Astronomy
• 6 See also
• 7 Citations and footnotes
• 8 References
• 9 Further reading
• 10 External links

Life and career

Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala (modern Bhinmal) during the reign of the Chavda dynasty ruler, Vyagrahamukha. He was the son of Jishnugupta and was a Shaivite by religion.^[4] Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.^[5]
Bhillamala, called pi-lo-mo-lo by Xuanzang, was the capital of the Gurjaradesa, the second largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional siddhanthas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.^[5]
In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he incorporated a great deal of originality to his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.^[5][6][7]
Later, Brahmagupta moved to Ujjain, which was also a major centre for astronomy. At the age of 67, he composed his next well known work Khanda-khādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.^[2]
Brahmagupta lived beyond 665 CE. He is presumed to have died in Ujjain.

Controversy

Brahmagupta directed a great deal of criticism towards the work of rival astronomers, and his Brahmasphutasiddhanta displays one of the earliest schisms among Indian mathematicians. The division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmagupta's case, the disagreements stemmed largely from the choice of astronomical parameters and theories.^[8] Critiques of rival theories appear throughout the first ten astronomical chapters and the eleventh chapter is entirely devoted to criticism of these theories, although no criticisms appear in the twelfth and eighteenth chapters.^[8]

Reception

The historian of science George Sarton called him "one of the greatest scientists of his race and the greatest of his time."^[2] Brahmagupta's mathematical advances were carried on further by Bhāskara II, a lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle of mathematicians). Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka.^[9] Further commentaries continued to be written into the 12th century.^[2]
A few decades after the death of Brahmagupta, Sindh came under the Arab Caliphate in 712 CE. Expeditions were sent into Gurjaradesa ("Al-Baylaman in Jurz", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated but Ujjain repulsed the attacks. The court of Caliph Al-Mansur (754–775) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) astronomical texts, including those of Brahmagupta. Brahmagupta's texts were translated into Arabic by Muhammad al-Fazari, an astronomer in Al-Mansur's court under the names Sindhind and Arakhand. An immediate outcome was the spread of the decimal number system used in the texts. The mathematician Al-Khwarizmi (800–850 CE) wrote a text called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Indian Arithmetic), which was translated into Latin in the 13th century as Algorithmi de numero indorum. Through these texts, the decimal number system and Brahmagupta's algorithms for arithmetic have spread throughout the world. Al-Khwarizmi also wrote his own version of Sindhind, drawing on Al-Fazari's version and incorporating Ptolemaic elements. Indian astronomic material circulated widely for centuries, even passing into medieval Latin texts.^[10][11][12]

Mathematics

Algebra

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,

The difference between rupas, when inverted and divided by the difference of the [coefficients] of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted.^[13]

which is a solution for the equation bx + c = dx + e equivalent to x = e − c/b − d, where rupes refers to the constants c and e. He further gave two equivalent solutions to the general quadratic equation

18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].
18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.^[13]

which are, respectively, solutions for the equation ax² + bx = c equivalent to,

${\displaystyle x={\frac {\pm {\sqrt {4ac+b^{2}}}-b}{2a}}}$

and

${\displaystyle x={\frac {{\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}.}$

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns.

18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].^[13]

Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.^[14] The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.^[14]

Arithmetic

The four fundamental operations (addition, subtraction, multiplication, and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus "The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier". ^{[15][page needed]} Indian arithmetic was known in Medieval Europe as "Modus Indorum" meaning method of the Indians. In Brahmasphutasiddhanta, multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation, Brahmagupta details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions: a/c + b/c; a/c × b/d; a/1 + b/d; a/c + b/d × a/c = a(d + b)/cd; and a/c − b/d × a/c = a(d − b)/cd.^[16]

Series

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed].^[17]

Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.^[18]
He gives the sum of the squares of the first n natural numbers as n(n + 1)(2n + 1)/6 and the sum of the cubes of the first n natural numbers as (n(n + 1)/2)².

Zero

Brahmagupta's Brahmasphuṭasiddhanta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers.^[19] The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,

18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero.

[...]

18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.^[13]

He goes on to describe multiplication,

18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.^[13]

But his description of division by zero differs from our modern understanding:

18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.
18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.^[13]

Here Brahmagupta states that 0/0 = 0 and as for the question of a/0 where a ≠ 0 he did not commit himself.^[20] His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

Diophantine analysis

Pythagorean triplets

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta provides a formula useful for generating Pythagorean triples:

12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.^[21]

Or, in other words, if d = mx/x + 2, then a traveller who "leaps" vertically upwards a distance d from the top of a mountain of height m, and then travels in a straight line to a city at a horizontal distance mx from the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city.^[21] Stated geometrically, this says that if a right-angled triangle has a base of length a = mx and altitude of length b = m + d, then the length, c, of its hypotenuse is given by c = m(1 + x) − d. And, indeed, elementary algebraic manipulation shows that a² + b² = c² whenever d has the value stated. Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can therefore be obtained from a, b and c by multiplying each of them by the least common multiple of their denominators.

Pell's equation

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as Nx² + 1 = y² (called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces.^[22]

The nature of squares:
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.^[13]

The key to his solution was the identity,^[23]

${\displaystyle (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_{1}y_{2}+x_{2}y_{1})^{2}}$

which is a generalisation of an identity that was discovered by Diophantus,

${\displaystyle (x_{1}^{2}-y_{1}^{2})(x_{2}^{2}-y_{2}^{2})=(x_{1}x_{2}+y_{1}y_{2})^{2}-(x_{1}y_{2}+x_{2}y_{1})^{2}.}$

Using his identity and the fact that if (x₁, y₁) and (x₂, y₂) are solutions to the equations x² − Ny² = k₁ and x² − Ny² = k₂, respectively, then (x₁x₂ + Ny₁y₂, x₁y₂ + x₂y₁) is a solution to x² − Ny² = k₁k₂, he was able to find integral solutions to Pell's equation through a series of equations of the form x² − Ny² = k_i. Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that if x² − Ny² = k has an integer solution for k = ±1, ±2, or ±4, then x² − Ny² = 1 has a solution. The solution of the general Pell's equation would have to wait for Bhaskara II in c. 1150 CE.^[23]

Geometry

Brahmagupta's formula

Diagram for reference

Main article: Brahmagupta's formula

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,

12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.^[17]

So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is p + r/2 · q + s/2 while, letting t = p + q + r + s/2, the exact area is

√(t − p)(t − q)(t − r)(t − s).

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case.^[24] Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

Triangles

Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:

12.22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment.^[17]

Thus the lengths of the two segments are 1/2(b ± c² − a²/b).
He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:

${\displaystyle a={\frac {1}{2}}\left({\frac {u^{2}}{v}}+v\right),\ \ b={\frac {1}{2}}\left({\frac {u^{2}}{w}}+w\right),\ \ c={\frac {1}{2}}\left({\frac {u^{2}}{v}}-v+{\frac {u^{2}}{w}}-w\right)}$

for some rational numbers u, v, and w.^[25]

Brahmagupta's theorem

Main article: Brahmagupta theorem

Brahmagupta's theorem states that AF = FD.

Brahmagupta continues,

12.23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes].^[17]

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is √pr + qs.
He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to Brahmagupta's famous theorem,

12.30–31. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular].^[17]

Pi

In verse 40, he gives values of π,

12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.^[17]

So Brahmagupta uses 3 as a "practical" value of π, and ${\displaystyle {\sqrt {10}}\approx 3.1622\ldots }$ as an "accurate" value of π. The error in this "accurate" value is less than 1%.

Measurements and constructions

In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides. He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral.
After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas (or empty spaces dug out of solids). He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a frustum of a pyramid, he gives the "pragmatic" value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the "superficial" volume as the depth times their mean area.^[26]

Trigonometry

Sine table

In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary True Longitudes, Brahmagupta presents a sine table:

2.2–5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns [...]^[27]

Here Brahmagupta uses names of objects to represent the digits of place-value numerals, as was common with numerical data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents the number of sides of the tradition die or 6, and so on. This information can be translated into the list of sines, 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270.^[28]

Interpolation formula

Main article: Brahmagupta's interpolation formula

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated.^[29] The formula gives an estimate for the value of a function f at a value a + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when its value is already known at a − h, a and a + h.
The formula for the estimate is:

${\displaystyle f(a+xh)\approx f(a)+x\left({\frac {\Delta f(a)+\Delta f(a-h)}{2}}\right)+{\frac {x^{2}\Delta ^{2}f(a-h)}{2!}}.}$

where Δ is the first-order forward-difference operator, i.e.

${\displaystyle \Delta f(a)\ {\stackrel {\mathrm {def} }{=}}\ f(a+h)-f(a).}$

Astronomy

Some of the important contributions made by Brahmagupta in astronomy are his methods for calculating the position of heavenly bodies over time (ephemerides), their rising and setting, conjunctions, and the calculation of solar and lunar eclipses.^[30]
In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun.^{[clarification needed]} He does this by explaining the illumination of the Moon by the Sun.^[31]

1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the longitude of the moon? The near half would always be bright.

2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.

3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...]^[32]

He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies.^[31]
Further work exploring the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are discussed in his treatise Khandakhadyaka.

अश्वमेध यज्ञ। Ashwamedh Yagya, अश्वमेध यज्ञ क्या है? Ashwamedh yagya in ...