Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva: Volume II - Astronomy

624 Pages·2008·43.846 MB·English

by  K. V. Sarma, K. Ramasubramanian, M. D. Srinivas, M. S. Sriram (auth.)

CULTURE AND HISTORY 4
OF MATHEMATICS
Gal}ita-
Yukti-Bh~a
(Rationales in Mathematical Astronomy)
of
Jye~thadeva
CULTURE AND HISTORY OF MATHEMATICS
Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Already Published Volumes
Bruce C. Berndt and Robert A. Rankin (editors): Ramanujan:Essays and Surveys
Richard S. Palais: Seminar on the Atiyah-Singer Index Theorem
Gerard G. Emch, R. Sridharan and M. D. Srinivas (editors): Contributions to the
History ofIndian Mathematics
Gan.i ta-Yukti -Bhas. a
(Rationales in Mathematical Astronomy)
of
Jye~thadeva
Volume II - Astronomy
Malayalam Text Critically Edited
with English Translation
by
K. V. Sarma
With Explanatory Notes in English
by
K. Ramasubramanian
M. D. Srinivas
M. S. Sriram
~o0 HINDUSTAN
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ISBN 978-81-85931-82-1 ISBN 978-93-86279-37-8 (eBook)
DOI 10.1007/978-93-86279-37-8
ISBN-13 978-81-85931-83-8 (2 volume set)
TABLE OF CONTENTS
ENGLISH TRANSLATION 471-617
CHAPTER 8 Computation of Planets ................................. 471
8.1 Planetary motion . . . . . 471
8.2 Celestial Sphere (Bhagola) 472
8.3 Motion of planets: Conception I 472
8.4 Motion of planets: Conception II 474
8.5 The position of Ucca ..... 475
8.6 Ucca, Madhyama and Sphuta 475
8.7 Computation of true Sun . 476
8.8 Computation of the KaT'7}a 481
8.9 Alternative method for finding the Ka'f'7}a 483
8.10 Viparfta-kaT'7}a (Inverse hypotenuse) 484
8.11 Another method for Viparfta-kaT'7}a . 484
8.12 Still another method for Viparfta-kaT'7}a 485
8.13 Manda-sphuta from the Madhyama . . 487
8.14 Sighra-sphuta (True planets): General 488
8.15 True Mercury and Venus. . . . . . . . 493
8.16 Sfghra correction when there is latitude 495
8.17 Calculation of the mean from true Sun and Moon. 500
8.18 Another method for the mean from true Sun and Moon 501
8.19 Calculation of the mean from true planet ........ 502
8.20 Computation of true planets without using Manda-kaT'7}a . 503
vi Contents
CHAPTER 9 Earth and Celestial Spheres .............................. 509
9.1 BhUgola: Earth sphere ........ 509
9.2 Viiyugola: Equatorial celestial sphere. 510
9.3 Bhagola: Zodiacal celestial sphere. . . 511
9.4 Ayana-calana: Motion of the equinoxes. 515
9.5 The manner of Ayana-calana . . . . . . 515
9.6 Changes in placement due to terrestrial latitude . 518
9.7 Zenith and horizon at different locations 518
9.8 Construction of the armillary sphere . . 521
9.9 Distance from a Valita-vrtta to two perpendicular circles 521
9.10 Some Viparfta and Nata-vrtta-s ... 523
9.11 Declination of a planet with latitude 525
9.12 Apakrama-koti ........... . 528
CHAPTER 10 The Fifteen Problems .................................... 533
10.1 The fifteen problems 533
10.2 Problem one. 534
10.3 Problem two 534
10.4 Problem three . 535
10.5 Problem four 535
10.6 Problem five. 536
10.7 Problems six to nine 537
10.8 Problems ten to twelve . 537
10.9 Problems thirteen and fourteen 538
10.10 Problem fifteen .... ..... 538
CHAPTER 11 Gnomonic Shadow 541
11.1 Fixing directions . . . 541
11.2 Latitude (Ak~a) and co-latitude (Lamba) 542
11.3 Time after sunrise or before sunset .... 543
Contents vii
11.4 Unnata-jyii .......... . 544
11.5 Mahii-sariku and Mahiicchiiya . 545
11.6 Drrimar.u!ala . 545
11. 7 Drggolacchiiyii 546
11.8 Chiiyii-lambana. 547
11.9 Earth's radius . 547
11.10 Corrected shadow of the 12-inch gnomon 548
11.11 Viparitacchiiyii : Reverse shadow 549
11.12 Noon-time shadow ....... . 550
11.13 Chayii-bhujii, Arkiigrii and Sarikvagrii 550
11.14 Some allied correlations ..... 551
11.15 Determination of the directions. 552
11.16 Sama-sariku : Great gnomon at the prime vertical 553
11.17 Samacchiiyii ........... . 554
11.18 The Sama-sariku-related triangles 555
11.19 The ten problems ........ . 556
11.20 Problem one: To derive Sariku and Nata 557
11.20.1 Shadow and gnomon at a desired place 557
11.20.2 Corner shadow ............. . 562
11.20.3 Derivation of Nata-jya (Rsine hour angle) 565
11.21 Problem two: Sariku and Apakrama 565
11.21.1 Derivation of the gnomon .. 565
11.21.2 Derivation of the declination 568
11.22 Problem three: Sariku and Asiigrii 568
11.22.1 Derivation of Sariku 568
11.22.2 Derivation of Asiigrii 569
11.23 Problem four: Sariku and Ak§a . 569
11.23.1 Derivation of Sariku (gnomon) 569
11.23.2 Derivation of Ak§a (latitude) 569
11.24 Problem five: Nata and Kriinti ... 570
viii Contents
11.25 Problem six: Nata and Asiigrii 571
11.26 Problem seven: Nata and Ak:9a . 572
11.27 Problem eight: Apakrama and Asiigrii 573
11.28 Problem nine: Kriinti and Ak:9a 574
11.29 Problem ten: Asiigrii and Ak:9a . 574
11.30 I#a-dik-chiiyii: Another method. 574
11.31 Kiila-lagna, Udaya-lagna and Madhya-lagna . 575
11.32 Kiila-lagna corresponding to sunrise 579
11.33 Madhya-lagniinayana . 581
11.34 Drkk:gepa-jyii and K oti . 582
11.35 Parallax in latitude and longitude (Nati and Lambana) 583
11.36 Second correction for the Moon ..... 584
11.37 Chiiyii-lambana: Parallax of the gnomon 587
11.38 Drkka'f'7}a when the Moon has no latitude 589
11.39 Shadow and gnomon when Moon has latitude 589
CHAPTER 12 Eclipse .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 593
12.1 Eclipsed portion at required time 593
12.2 Time for a given extent of eclipse 594
12.3 Computation of Bimbiintara 595
12.4 Orb measure of the planets . 596
12.5 Direction of the eclipses and their commencement 597
12.6 Ayana-valana 598
12.7 Ak:9a-valana . 600
12.8 Combined valana 600
12.9 Graphical chart of the eclipse. 601
12.10 Lunar eclipse ......... . 602
CHAPTER 13 Vyatipiita ................................................ 603
13.1 Vyatipiita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
Contents ix
13.2 Derivation of declination. 603
13.3 Vik$epa . . . . 604
13.4 Vik$epa-calana 606
13.5 Karr},cinayana . 607
13.6 Determination of Vik$epa-calana 608
13.7 Time of Vyatfpiita ... 608
13.8 Derivation of Vyatfpiita 609
CHAPTER 14 Maurf,hya and Visibility Corrections of Planets ........... 611
14.1 Computation of visibility correction 611
14.2 Rising and setting of planets 612
14.3 Planetary visibility . . . . .. 613
CHAPTER 15 Elevation of the Moon's Cusps 614
15.1 The second true hypotenuse of the Sun and the Moon 614
15.2 Distance between the orbs of the Sun and Moon ... 614
EXPLANATORY NOTES 619-856
CHAPTER 8 Computation of Planets ................................. 621
8.1 Planetary motion. . . . 621
8.2 Zodiacal celestial sphere 622
8.3 Motion of planets: Eccentric model. 622
8.4 Motion of planets: Epicyclic model 623
8.5 The position of Ucca ..... 625
8.6 Ucca, Madhyama and Sphuta 625
8.7 Computation of true Sun . 625
8.8 Computation of the Ka'T"1'}a . 628
8.9 Alternative method for the Ka'T"1'}a 633
8.10 Viparfta-ka'T"1'}a: Inverse hypotenuse 635