The general consensus as to where the Hindu mathematician and astronomner Bhaskara II lived is in favour of the town of Chalisgaon, a medium-sized town on NH 211 in Dist. Jalgaon in Maharashtra. The district-town of Beed, also in Maharashtra, or of Bijapur and Bidar in Karnataka are also named in some studies.
Bhaskara II said the following about himself in the गोलाध्याय of his famous work, the सिद्धान्तशिरोमणि :
आसीत्सह्यकुलाचलाश्रितपुरे त्रैविद्यविद्वज्जने
नानासज्जनधाम्नि विज्जडविडे शाण्डिल्यगोत्रो द्विज:।
श्रौतस्मार्तविचारसारचतुरो नि:शेषविद्यानिधि:
साधूनामवधिर्महेश्वरकृती दैवज्ञचूडामणि:॥
तज्जस्तच्चरणारविन्दयुगलप्राप्
र्मुग्धोद्बोधकरं विदग्धगणकप्रीतिप्रदं प्रस्फुटम्।
एतद्व्यक्तसदुक्तियुक्तिबहुलं हेलावगम्यं विदां
सिद्धान्तग्रथनं कुबुद्धिमथनं चक्रे कविर्भास्कर:॥
In brief, Bhaskara II says that his father, महेश्वर, was a Brahmin of the शाण्डिल्यगोत्र and that he lived in the place called विज्जडविड that was close to the Sahya Mountain (सह्यकुलाचलाश्रित.) Because of the closeness of pronunciation this leads to the immediate surmise that the place of his birth is बीड, बिदर, or विजापूर. The difficulty in accepting any of these surmises is that none of them can really be said to be सह्यकुलाचलाश्रित.
The case in support of Chalisgaon is somewhat stronger. As one nears Chaligaon from Aurangabad on NH 211, a small road branches towards the left. It leads to the small village called Patan पाटण, at the foot of the Satmala सातमाळा Range, a part of the Sahyadri chain. Patan has a dilapilated temple of Bhavani, called Patandevi पाटणदेवी. A long inscription in Sanskrit, found in the ruins of the temple, records the facts that Changadeva, a grandson of Bhaskara, ran a Math in that place where works of Bhaskara like सिद्धान्तशिरोमणि and other works of members of the same family were taught. The inscription also records the lineage of Bhaskara and Changadeva and the grant of charities given to the Math by Soideva and Hemadri of the Nikumbha family, two feudatories of the Yadavas who ruled in Deogiri. The inscription was first noticed by Dr Bhau Daji, who wrote about it in the Journal of the Royal Asiatic Society, New Series, Vol. I. Dr. Keilhorn further edited it in Epigraphia Indica, Vol. I, p. 338. (This book is available in DLI.) A portion of the inscription is as under:
शाण्डिल्यवंशे कविचक्रवर्ती त्रिविक्रमोऽभूत्तनयोऽस्य जात:।
यो भोजराजेन कृताभिधानो विद्यापतिर्भास्करभट्टनामा॥ १७.
तस्माद्गोविन्दसर्वज्ञो जातो गोविन्दसन्निभः।
प्रभाकरः सुतस्तस्मात्प्रभाकर इवापरः॥ १८.
तस्मात्मनोरथो जातः सतां पूर्णमनोरथः।
श्रीमान्महेश्वराचार्यस्ततोऽजनि कवीश्वरः॥ १९.
तत्सूनुः कविवृन्दवन्दितपदः सद्वेदविद्यालता-
कन्दः कंसरिपुप्रसादितपदः सर्वज्ञविद्यासदः।
यच्छिष्यैः सह कोऽपि नो विवदितुं दक्षो विवादी क्वचि-
च्छ्रीमान् भास्करकोविदः समभवत्सत्कीर्तिपुण्यान्वितः॥ २०.
लक्ष्मीधराख्योऽखिलसूरिमुख्यो वेदार्थवित्तार्किकचक्रवर्ती।
क्रतुक्रियाकाण्डविचारसारविशा
सर्वशास्त्रार्थदक्षोऽयमिति मत्वा पुरादतः।
जैत्रपालेन यो नीतः कृतश्च विबुधाग्रणीः॥ २२.
तस्मात्सुतः सिङ्घणचक्रवर्तिर्दैवज्ञवर्यो
श्रीभास्कराचार्यनिबद्धशास्त्
भास्कररचितग्रन्थाः सिद्धान्तशिरोमणिप्रमुखाः।
तद्वंश्यकृताश्चान्ये व्याख्येया मन्मठे नियमात्॥ २४.
The genealogy is thus: कविचक्रवर्ती त्रिविक्रम - विद्यापति भास्करभट्ट - गोविन्द - प्रभाकर - मनोरथ - महेश्वराचार्य कवीश्वर - भास्कर - लक्ष्मीधर - चङ्गदेव. The last two flourished in the courts of the Yadava kings of Deogiri, जैत्रपाल and सिंघण.
(Towards the end of the inscription is a narration in the old Marathi then prevalent of the grants received by the Math for its upkeep.)
The fact that a grandson of Bhaskara II ran a Math or a Pathashalaa so close to Chalisgaon to teach Bhaskara's works has led to the stong belief that Bhaskara II lived in or close to Chalisgaon.
I believe it is possible to narrow this further and say that the village of Patan itself must be the place where Bhaskara II flourished. Its original name could have been विज्जडविड or विज्जडविडपत्तन. Eventually विज्जडविड got dropped out of this long name and only पत्तन or पाटण survived. My evidence for saying so is both literary and physical.
The literary evidence is as under:
a) The Changadeva inscription was discovered in a temple in that village. The village of Patan today appears to be situated in a remote corner but the numerous ruins in it tell us that the place that has seen much better days. I have visited the place and seen several ruins of old temples and other buildings and sevaral ruined मूर्तिs lying in the open. The temple of Bhavani itself, though in partial ruins, is quite large. Apparently the inscription of Changadeva was still in the temple when I visited it, though, according to the Pujari of the temple, it was hidden inside a large collection of other stone fragments by the workers of the ASI to protect it from vandalism and theft. I append below two photos of the temple.
b) Patan पाटण is obviously a corruption of पत्तन, a large town. The very name suggests that in the past it was much bigger than a mere village that it looks now and merited the diecription of a पत्तन. It is obvious that the description of सह्यकुलाचलाश्रित fits it very well, as the cliffs of the Sahyadri-Satamala Range are seen just behind it.
c) The Changadeva inscription refers to it as a पुर, which would be much bigger than a mere village. लक्ष्मीधर, a son of Bhaskara II, lived there and was invited by the King जैत्रपाल यादव from that place. (सर्वशास्त्रार्थदक्षोऽयमिति मत्वा पुरादतः। जैत्रपालेन यो नीतः कृतश्च विबुधाग्रणीः॥ २२.) His son, चंगदेव, lived there as the head of the Math. The whole family of Bhaskara II thus appears to have close ties to this place.
d) Another branch of the family lived not far from this place. About 10 miles north of Chalisgaon is a village called Bahal बहाळ. It has a temple dedicated to Goddess Saraja सारजा. The temple contains an inscription that says that it was built by one Anantadeva, the Chief Astrologer in the court of SighaNa Yadava. The genealogy of Anatadeva is given therein as शाण्डिल्यगोत्री मनोहर - महेश्वर - श्रीपति - गणपति - अनंतदेव. मनोहर and महेश्वर and the gotra शाण्डिल्य being common, it is obvious that श्रीपति here is a brother of Bhaskara II. This inscription, edited by Dr Keilhorn, may be seen at Epigraphia Indica Vol. III p. 110.
There is good physical evidence too that suggests that पाटण at one time must have been a flourishing major town. The enclosed captures from Google Maps show the relative positions of the famous Pitalkhora Caves पितळखोरा and पाटण. Though these caves today appear to be located far in the interior and in the absolute widerness - they are loacted sevaral km towards the west from the town of Kannad on NH 211 north of Aurangabad, - it is also known that all such caves were carved along major trade routes of old India. There is indeed enough evidence that such a trade route existed.
a) A book printed in 1826 at Calcutta, 'Itinerary and directory for Western India: being a collection of routes...' is available in books.google.com. It is written by Capt. John Clunes of the 12th Native Infantry and describes routes of transport of those day, before modern means of transport came into use. Routes LV aqnd LVI in it are 'From DHOOLIA by Mehoonbarra and Gowtulla Ghat to AURAGABAD' and 'From DHOOLIA to AURANGABAD, via Mehoonbarra and Untoor' and mention 'Gowtulla Ghat'. Today the forested area from Pitalkhora to Patan and the surroundings is a protected forest called Gautala Forest. 'Gowtulla Ghat' in Clunes' book would thus be the valley that leads down from Pitalkhora to Patan. I could identify in wikimapia.org 'Mehoonbarra' with the village Mehunbare on the bank of the Girna River at 20.5690 N and 74.94.48 E. Another village mentioned by Clunes, 'Hewurkherah', is 'Hiwarkheda' 20.2924 N and 75.1340 E. He also mentions Kannad in his description.
b) Another book in books.google.com, 'Historical and descriptive sketch of His Highness the Nizam's ...: Volume I' (http://tinyurl.com/d4k8ebb) says the following on p. 384 about the Gautala Ghat.
"The Ghats between His Highness's territory and the British Province of Khandesh are pierced with numerous passes, all of which are more or less used as trade routes. The principle are the Ajanta Pass already mentioned, the Gaotala or Amba Ghat above Kanad, a very old trade route; at the foot of the ghat are the ruins of the ancient city of Patna. The road over this ghat was once so good that it was practicable for carts. It was reconstructed by the Emperor Aurangzeb during his Deccan campaign and was subsequently repaired by Outram when he was Bhil Agent."
Herein it is mentioned that in Aurangzeb's time the Gautala Ghat was passable by carts and that at it foot are the ruins of 'the ancient city of Patna'
c) Yet another book in books.google.com, 'The castes and tribes of H.E.H. the Nizam's dominions: Volume 1' (http://tinyurl.com/bu6sck6) mentines on p.69 how a British Officer by name Outram, appointed at Kannad to bring under control the Bhil Tribes of the Nizam's territory north of Aurangabad, got the Gautala repaired. (The present motorable ghat on NH 211, going down towards Chalisgaon is a later creation. It was made in 1870. See Maharashtra State gazetteers, Volume 4 p. 580. http://tinyurl.com/ct6zfbm. It was named after the same Outram.)
Considering the weight of literary and physical evidence, it looks certain that the village of Patan पाटण is the place where Bhaskara II lived and wrote his famous सिद्धान्तशिरोमणि.
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