Bhāskara II

Bhāskara II (1114–1185), also known as Bhāskarāchārya (Template:Lit), was an Indian polymath, mathematician, and astronomer.

Quotes

• Let ‘so much as’ (yavattavat) be put for the value of the unknown quantity, and doing with that precisely what is proposed in the instance [i.e. what is proposed in the following], let two equal sides be carefully completed, adding or subtracting, multiplying or dividing, [as the case may require]. Subtract the unknown quantity of one side from that of the other; and the known number of the one from that of the other side. Then divide the remainder of the known quantity by the [coefficient of the] remaining unknown: the quotient is the distinct value of the unknown quantity.
  • ekavarnasamikaranam (equations with one unknown), from theBijaganita, which is part of his monumental work Siddhanta- Shiromani
  • Jha, Achyutananda. The Bijaganita of Sri Bhaskaracharya. Banaras: Kashi Sanskrit Series 148, 1949.
  • Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara. London: John Murray, Albemarle Street, 1817
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• A person has 300 rupees and 6 horses; and another person has 10 horses and 100 rupees debt; and the property of the two is equal; and the price of the horses is the same; what then is the value of each?
  • Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara. London: John Murray, Albemarle Street, 1817
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• On a plane surface describe a circle of any specified radius with a pair of compasses. Mark on its circumference 360 degrees. Draw the east-to-west and north-to-south lines through its center. These lines will divide the circle into quadrants, which should be taken into consideration in the leftwise manner (i.e. anti-clockwise).
  • Bhaskara II (1150) . (Siddhantashiromani Grahaganita ii.19 vasanabhashya):अत्र समायां भमू ौ त्रिज््ययातुल््ययेन कर््क टके न वृत्तं कृ त््ववा भांशै ३६० अङ््कक््यम।् तन््मध््ययेपर््ववापरां याम््ययोत्तरां च रे खां कृ त््ववा प्राच््ययााः सकाशात् सव््यक्रमेण किल पदानि कल््प्ययानिवृत्ते रे खावच््छछिन््ननानि। तेषां क्रमेणायुग््मयुग््मसंज्ञा च।
  • Datta, Bibhutibhushan, and Avadesh Narayan Singh.———. ‘Hindu Trigonometry’ (revised by K.S. Shukla). Indian Journal of History of Science 18 (1) (1983): pp. 39-108. quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• The Rsines of any two arcs of a circle are reciprocally multiplied by their Rcosines; the products are then divided by the radius; the sum of the quotients is equal to the Rsine of the sum of the two arcs, and their difference is the Rsine of the difference of the arcs.
  • Bhaskara II (1150) (Siddhantashiromani Goladhyaya, Jyotpatti 21 and 22):चापयोरिष्टयोर्दोर्ज्ये मिथः कोटिज््यकाहते।त्रिज््ययाभक्ते तयोरै क््ययं तच््चचापैक््यस््य दोर््ज््यका॥चापान््तरस््य जीवा स््ययात्तयोरन््तरसंमिता।अन््यज््ययासाधने सम््यगियं ज््ययाभावनोदिता॥
  • Datta, Bibhutibhushan, and Avadesh Narayan Singh.———. ‘Hindu Trigonometry’ (revised by K.S. Shukla). Indian Journal of History of Science 18 (1) (1983): pp. 39-108. quoted from Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Without the knowledge of upapattis, by merely mastering the calculations (ganita) described here, from the madhyamadhikara (the first chapter of Siddhantashiromani) onwards, of the [motion of the] heavenly bodies, a mathematician will not be respected in the scholarly assemblies; without the upapattis he himself will not be free of doubt….
  • Bhaskaracharya in the introduction to Goladhyaya in the Siddhantashiromani Srinivas, M. D. ———. On the Nature of Mathematics and Scientific Knowledge in Indian Tradition, 2016 (http://iks.iitgn.ac.in/wp-content/uploads/2016/02/On-the- Nature-of-Mathematics-and-Scientific-Knowledge-in-Indian-Tradition-MD- Srinivas-2016.pdf, accessed August 14 2021).
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• The Moon, moving like a cloud in a lower sphere, overtakes the Sun [and obscures its shining disk by its own dark body] hence it arises that the western side of the Sun’s disk is first obscured, and that the eastern side is the last part relieved from the Moon’s dark body: and to some places the Sun is eclipsed and to others is not eclipsed (although he is above the horizon) on account of their different orbits. (1)At the change of the Moon it often so happens that an observer placed at the center of the earth, would find the sun when far from the zenith, obscured by the intervening body of the moon, whilst another observer on the surface of the earth will not at the same time find him to be so obscured, as the moon will appear to him [on the higher elevation] to be depressed from the line of vision extending from his eye to the sun. Hence arises the necessity for the correction of parallax in celestial longitude and parallax in latitude in solar eclipses in consequence of the difference of the distances of the sun and the moon. (2) When the sun and moon are in opposition, the earth’s shadow envelopes the moon in darkness. As the moon is actually enveloped in darkness, and as the earth’s shadow and the moon which enters it, are at the same distance from the earth, there is therefore no call for the correction of the parallax in a lunar eclipse. (3)
• As the moon moving eastward enters the dark shadow of the earth: therefore its eastern side is first of all involved in obscurity, and its western is the last portion of its disc which emerges from darkness as it advances in its course. (4) As the sun is a body of vast size, and the earth insignificantly small in comparison: the shadow made by the sun from the earth is therefore of a conical form terminating in a sharp point. It extends to a distance considerably beyond that of the moon’s orbit. (5)
• The length of the earth’s shadow, and its breadth at the part traversed by the moon, may be easily found by proportion. In the lunar eclipse the earth’s shadow is northwards or southwards of the moon when its latitude is south or north. Hence the latitude of the moon is here to be supposed inverse (i.e. it is to be marked reversely in the projection to find the center of the earth’s shadow from the moon.) (6)
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189
  • In the section Goladhyaya from the Siddhantashiromani, in the chapter Grahanavasana, Bhaskara introduces the phenomenon of eclipses with these verses 1-6 [33].पश्चाद्भागाज््जलदवदधः संस््थथितोऽभ््ययेत््य चन्द्रोभानोर््बबिम््बबं स््फफु रदसितया छादयत््ययात््ममर््त्यया।पश्चात््स््पर्शो हरिदिशि ततो मक्तिरस््ययात एवक््ववापिच््छन््ननः क््वचिदपिहितो नैष कक्षान््तरत््ववात् ॥१॥इदानीीं नतिलम््बनयोः कारणमाह -पर््ववान््ततेऽर्कं नतमद्रुपतिच््छन््नमेव प्रपश््ययेद्भूमध््यस््थथोन तु वसुमतीपृष्ठनिष्ठस््तदानीम।्तद्दृक््ससूत्राद्धिमरुचिरधो लम््बबितोऽर््क ग्रहेऽतःकक्षाभेदादिह खलु नतिर््लम््बनं चोपपन््नम् ॥२॥समकलकाले भभू ा लगतिमृगाङ्के यतस््तयाम््ललानम।्सर्वे पश््यन््तति समं समकक्षत््ववान््न लम््बनावनती ॥३॥पूर््ववाभिमखो गच््छन््ककु च््छछायान््तर््यतः शशी विशति।तेन प्राक् प्रग्रहणं पश्चान््ममोक्षोऽस््य निःसरतः ॥४॥भानोर््बबिम््बपृथुत््ववादपृथुपथिययााः प्रभा हि सूच््यग्रा।दीर््घतया शशिकक्षामतीत््य दरू ं बहिर््ययाता ॥५॥अनुपातात्तद्दैर्व्यं शशिकक्षायां च तद्बिम््बम।्भभू ेन््ददोरन््यदिशि व््यस््ततः क्षेपः शशिग्रहे तस््ममात् ॥६॥
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• If the earth were supported by any material substance or living creature, then that would require a second supporter, and for that second a third would be required. Here we have the absurdity of an interminable series. If the last of the series be supposed to remain firm by its own inherent power, then why may not the same power be supposed to exist in the first, that is in the Earth? For is not the Earth one of the forms of the eight-fold divinity i.e. of Shiva? (Goladhyaya, III.4)
  • In the chapter Bhuvanakosha in the Goladhyaya, Bhaskara proves the absurdity of the supposition that the earth is supported [34]:मर्तोधर््तता चेद्धरित्र्यास््ततोऽन््यस््तस््ययाप््यन््ययोऽस््ययैवमत्रानवस््थथा।अन््त्यये कल््प्यया चेत््स््वशक््तििः किमाद्ये किं नो भमू ेः साष्टमर्तेू श्च मर््ततििः॥४॥

• The property of attraction is inherent in the Earth. By this property the Earth attracts any unsupported heavy thing towards it: The thing appears to be falling [but it is in a state of being drawn to the Earth]. The ethereal expanse being equally outspread all around, where can the Earth fall? (Goladhyaya III.6)
  • in verse 6, he proceeds to elucidate the property by which the Earth attracts all objects to itself:आकृ ष्टशक्तिश्च मही तया यत् खस््थथं गुरु स््ववाभिमखं स््वशक्तया।आकृ ष््यते तत््पततीव भाति समे समन््ततात् क््व पतत््ववियं खे॥६॥

• As the one-hundredth part of the circumference of a circle is (scarcely different from) a plane, and as the Earth is an excessively large body, and a man exceedingly small (in comparison), the whole visible portion of the Earth consequently appears to a man on its surface to be perfectly plane.
  • [Goladhyaya III.13] verse 13 he gives the reasoning why the earth appears like a plane even though it is round,अथ प्रत््यक्षविरोधशङ्का परिहरन््ननाह-समो यतः स््ययात््परिधेः शतांशः पृथ््ववी च पृथ््ववी नितरां तनीयान्।नरश्च तत््पपृष्ठगतस््य कृ त््स्नना समेव तस््य प्रतिभात््यतः सा ॥१३॥
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• From a bunch of lotuses, one third is offered to Lord Shiva, one fifth to Lord Vishnu, one sixth to the sun, one fourth to the goddess. The remaining six are offered to the Guru. Find quickly the number of lotuses in the bunch.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Arjuna became furious in the war and in order to kill Karna, picked up some arrows. With half the arrows, he destroyed all of Karna’s arrows. He killed all of Karna’s horses with four times the square root of the arrows. He destroyed the spear with six arrows. He used one arrow each to destroy the top of the chariot, the flag, and the bow of Karna. Finally he cut off Karna’s head with another arrow. How many arrows did Arjuna discharge?
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• There is no change in infinite (khahara) figure if something is added to or subtracted from it. It is like: there is no change in the infinite Lord Vishnu due to the dissolution or creation of abounding living beings.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• In a triangle or a polygon, it is impossible for one side to be greater than the sum of the other sides. It is daring for anyone to say that such a thing is possible. If an idiot says that there is a quadrilateral of sides 2, 6, 3, 12 or a triangle with sides 3, 6, 9, explain to him that they don’t exist.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• The earth attracts inert bodies in space towards itself. The attracted body appears to fall down on the earth. Since the space is homogeneous, where will the earth fall?
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

Quotes about Bhāskara II

• On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically: as is particularly noticed by Bhaskara himself, towards the close of his algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities.
  • Henry Thomas Colebrooke quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189
  • Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara. London: John Murray, Albemarle Street, 1817.

• Bhaskara […] does not pretend himself to be the inventor, he assumes no character but that of a compiler.
  • from the Bijaganita translated by Edward Strachey,
  • Strachey, Edward. Bija Ganita: or the Algebra of the Hindus, 1813.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Almost any trouble and expense would be compensated by the possession of the three copious treatises on algebra from which Bhaskara declares he extracted his Bijaganita, and which in this part of India are supposed to be entirely lost.
  • from the Bijaganita translated by Edward Strachey,
  • Strachey, Edward. Bija Ganita: or the Algebra of the Hindus, 1813.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189

• Reuben Burrow […] says, he was told by a pundit, that some time ago there were other treatises of algebra.
  • from the Bijaganita translated by Edward Strachey,
  • Strachey, Edward. Bija Ganita: or the Algebra of the Hindus, 1813.
  • quoted in : Bhaskar Kamble, The Imperishable Seed: How Hindu Mathematics Changed the World and why this History was Erased, Garuda Prakashan Private Limited, 2022 ISBN 9798885750189