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The Siddh!anta-sundara of Jn!anar!aja:An English Translationwith Commentaryby Toke Lindegaard Knudsen

BALTIMORE: JOHNS HOPKINS UNIVERSITY PRESS, 2014, XVII+351 PP., US

$89.95, ISBN: 978-1-4214-1442-3

REVIEWED BY JAYANT SHAH

TThe tradition of geometric astronomy in India, fromAryabhata (c. 500 CE) to Nı lakan

_t_ha (c. 1500 CE), is

based on the pre-Ptolemaic Greek astronomy thatwas introduced in India sometime early in the firstmillennium. Thus the basic framework of the Indianastronomy was single epicycle models for the Sun andthe Moon and models with two epicycles for the planets.Menelaus’s spherical trigonometry was unknown. Indianastronomers derived the necessary formulae for handlingspherical geometry from a set of basic plane right triangles.With the exception of the introduction of lunar evection byManjula in the 10th century, this framework remainedunchanged throughout the development of Indian as-tronomy. Although Greek astronomy arrived in India,Greek tradition did not. Indian astronomers followed thetradition of P!an

_ini, Pingala, and pur!an

_as as they sought to

refine and adapt the Greek import. There is an increasingdegree of sophistication culminating in M!adhava’s powerseries and N ı-lakan

_t_ha’s implicitly heliocentric models

for the inferior planets. At the same time, there areapproximations and tables for everyday religious andastrological calculations. Even though many importantIndian astronomical manuscripts have been translated,our understanding of the Indian tradition remains incom-plete. Many manuscripts still remain to be explored andtranslated. The translation of Jn!anar!aja’s Siddh!antasun-dara by Toke Lindegaard Knudsen is a welcome addition.It is to be hoped that as more manuscripts are studied, amore complete picture will emerge.

Medieval Indian astronomical literature falls into threedistinct genres: Comprehensive treatises (siddh!antas andtantras), Manuals (karan

_as) containing simplified algo-

rithms, and Tables (kos_t_hakas) ready for use by astrologers

and calendar makers. As its name indicates, Siddh!anta-sundara belongs to the first category. These texts expoundformulae and algorithms for solving a typical sequence ofproblems in astronomy. Some are more wide-ranging thanothers, covering topics in arithmetic, algebra, and evenprosody, often with no apparent application. As usual thesetexts are written, as are a lot of Indian technical texts, interse Sanskrit verses. Brevity and rules of prosody imposedconstraints. Writers not only used several synonymous

terms to indicate a particular concept, but also sometimesused the same term to mean different concepts dependingon its context. The result is to render the task of translationa major undertaking. Teasing out the true meaning from acorrupted text is exceedingly difficult.

Siddh!antasundara is one of the two major siddh!antascomposed after the monumental siddh!antasiroman

_i of

Bh!askara II (c. 1150 CE), the other being Nı lakan_t_ha’s

Tantrasangraha composed at about the same time asSiddh!antasundara (c. 1500 CE). The two siddh!antas offer astudy in contrast. While Jn!anar!aja is preoccupied withreconciling the ancient Indian cosmology of vedas andpur!an

_as with Hellenistic astronomy, his contemporary 900

miles to the south, Nı lakan_t_ha, is introducing mathematical

innovations and refining planetary models. Consequently,Jn!anar!aja divides his treatise into two parts. He devotes thefirst part to the cosmology of pur!an

_as and tries to reconcile

it with the Hellenistic astronomy of siddh!antas. The secondpart follows a typical sequence of traditional topics in In-dian mathematical astronomy with frequent digression topur!an

_as.

Siddh!antasundara is not easy to follow. Topics do notalways follow a logical progression. Chapter contents are notunified. Jn!anar!aja routinely uses technical terms in his pre-liminary discussions before these terms are defined in detailin a later chapter. Knudsen’s introductory chapter andcommentary provide a helpful guide. He has done an ad-mirable job of providing cross-references to navigatethrough Jn!anar!aja‘s exposition. Despite his best efforts, themathematical content of some of the verses remains obscure.

The book is organized as follows:

Introduction

An introduction to Indian astronomy, a brief description ofthe cosmology of pur!an

_as, an overview of Siddh!antasun-

dara, and a discussion of Jn!anar!aja’s place and time.

Part I: General Description

1. Lexicon of the Worlds: Jn!anar!aja starts by pointing outthat jyotihs!astra (science of luminaries) is one of the sixlimbs of vedas (ved!angas). It includes three divisions:astronomy, astrology, and omens. Three different ac-counts of creation are recounted. Earth’s sphericity isreconciled with the pur!an

_a’s geography of seven ring-

shaped oceans. This is followed by terrestrial geographyand cosmology.

2. Rationale of Planetary Motion: Planetary motions arequalitatively described. The reader is assumed to begenerally familiar with technical terms such as latitudes,declination, nodes, apogee, epicycles, and so on. Differ-ent measures of year, month, and day are provided.

3. Method of Projections: The theoretical framework ofplanetary models is explained in terms of the eccentriccircle as well as the epicycles.

4. Description of the Great Circles: Great circles defined bythe three coordinate systems, ecliptic, equatorial, and

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DOI 10.1007/s00283-015-9555-8

Author's personal copy

horizontal, are described. These are fundamental to theIndian method of astronomical calculations.

5. Astronomical Instruments: Jn!anar!aja lists seven differentinstruments, but describes in detail only one of them,namely, the sine-quadrant. In essence, it is a device toread off values of sine graphically. It also has sightingvanes. A number of applications are provided, butdescriptions are not entirely clear.

6. Description of the Seasons: This is a nontechnicaldescription of the six seasons in a year. The poeticdescription more appropriately belongs to the traditionof the poet and playwright K!alid!asa (4th or 5th centuryCE) rather than astronomy. Jn!anar!aja uses the techniqueknown as sles

_a so that each verse has two narratives.

Either it can be read as some story from Indianmythology or it can be seen as the description of aseason.

Part II: Mathematical Astronomy

1. Mean Motions: The traditional great periods are listed.A kalpa consists of 4,320,000,000 sidereal years, di-vided into smaller periods, mah!ayugas and yugas.Planets, nodes, and apogees make a whole number ofrevolutions around the earth in a kalpa. All the planets,nodes, and apogees are postulated to be at the zero-degree longitude when the planetary motion begins. Aprocedure for calculating the number of elapsed yearsup to the start of the saka era (78 CE) is presented. Fromthese data, one can calculate the mean longitudes atany given time. Jn!anar!aja now proceeds to describe anapproximate method for checking the theoretical meanmotion by means of observations. Next a more practi-cal method is provided for calculating the meanlongitudes from their values at epoch saka 1425. Fi-nally, a formula for longitudinal correction for placesnot on the prime meridian is provided.

2. True Motions: Mean motions are corrected by takinginto account the geometry of the actual orbits based oneccentric and epicyclic models. The chapter beginswith a table of sines. As usual in Indian astronomy,epicycles have variable radii. Jn!anar!aja shows amethod for computing square roots, which is the sameas the method given by Heron of Alexandria (c. 1stcentury CE) and apparently recorded here for the firsttime in an Indian treatise. Jn!anar!aja follows the Indiantradition of using a multistep procedure for combiningthe contributions of the two epicycles in the case of thestar-planets. Jn!anar!aja’s omission of lunar evection isperhaps indicative of his relative priorities. A formulafor the ascensional difference is provided for use incomputing the lengths of days and nights. The chapterends with a procedure for calculating the ascendant.

3. Three Questions: This is the traditional title of a chapterdiscussing locally observed astronomy and refers todirection, place, and time. A procedure for determiningthe cardinal direction by means of a gnomon isdescribed. Eight plane right triangles involving various

coordinate circles are introduced and form a basis forderiving the formulae for local astronomical phe-nomena. The mathematical meaning of a lot of versesremains unclear.

4. Occurrences of Eclipses: Methods for computingquantities related to eclipses are presented. Much ofthe material in this chapter is repeated in the individualchapters on lunar and solar eclipses that follow.

5. Lunar Eclipses: Apparent diameters of the Moon, theSun, and the earth’s shadow at the distance of theMoon are computed. From the lunar latitude at the timeof conjunction, the magnitude of the eclipse isobtained. A procedure for calculating the angle be-tween the ecliptic and the east-west line on the Moon’sdisk is given.

6. Solar Eclipses: Solar eclipses are more complicated topredict because of the lunar parallax. Approximateformulae of Jn!anar!aja for computing the zenith dis-tance of the nonagesimal are the same as inSuryasiddh!anta. After computing the longitudinalparallax, the time of apparent conjunction is obtainedby an iterative procedure. From the lunar latitudecorrected for the latitudinal parallax, the magnitude ofthe eclipse is obtained. At the end of the chapter, therationale for the formulae is briefly explained.

7. Rising and Setting of the Planets: Limits on elongationof a planet to be visible are given and then correctedfor the true latitude of the planet.

8. Shadows of Stars: Shadows of planets are to be foundjust as in the case of the Sun. Polar longitudes andlatitudes of Naks

_atras (constellations) and positions of

the seven stars of Ursa Major are given.9. Elevation of Lunar Horns: The objective is to determine

the shape and orientation of the lunar crescent withrespect to the cardinal directions. The meaning of someof the verses is unclear.

10. Conjunctions of Planets: This brief chapter calculates thetimeof conjunction in a straightforwardmanner from thetrue positions and true velocities of the planets.

11. p!atas: These are astrologically inauspicious events thatdepend on the relative positions of the Sun and theMoon.

Siddh!antasundara is of interest perhaps mostly to experts,but Jn!anar!aja’s efforts to defend the pur!an

_a’s cosmology is

of general interest. The book does not include the Sanskrittext, but the author plans to publish a critically editedSanskrit text in the future. Siddh!antasundara provides animportant link in the multifaceted development of the In-dian astronomy after Bh!askara II. As the author points out,the next step is a detailed analysis of its sources and itsinfluence on the subsequent development of astronomicalthought in India.

Mathematics Department

Northeastern University

Boston, MA 02115

USA

e-mail: j.shah@neu.edu

THE MATHEMATICAL INTELLIGENCER

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