लघुभास्करीयम्

by भास्कराचार्यः

लघुभास्करीयम् भास्कराचार्यः

PART 11 With Explanatory and Critical Notes Published by and Comments, etc. श्रीभास्कराचार्यविरचितम् लघुभास्करीयम् लखनऊ-विश्वविद्यालयस्य गणिताध्यापकेन एम० ए०, डी० लिट्० इत्युपाधिारिणा श्री कृपाशंकर शुक्लेन आङग्लानुवाद - व्याख्या-टिप्पण्यादिभिः सहितं सम्पादितम् लखनऊ विश्वविद्यालयस्य गणित-ज्योतिष-विभागेन सं० २०२० वि० Price: Rs. 60.00 $ 10.00 . Chandra, Star Press, Lucknow ( Phone 24768) PREFACE The object of the "Hindu Astronomical and Mathematical Texts Series" is to bring out authoritative and critical editions of important unpublished works dealing with ancient Hindu astronomy and mathematics. The present edition of Bhaskara I's Laghu-Bhaskarlya is No. 4 of this series. v • The idea of bringing out the above series is due to Dr. A. N. Singh, late Professor of Mathematics, Lucknow University, who organised a scheme of research in the history of Hindu mathematics and astronomy. in the Depart- ment of Mathematics, Lucknow University, with the object of collecting, studying, and editing important works on Hindu mathematics and astronomy. Under his able supervision remarkable progress was made in this direction and a number of manuscripts were acquired, studied and edited. The work is being continued since his death in 1954 by our colleague, Dr. Kripa Shankar Shukla, Reader in Mathematics, Lucknow University, who has already been actively engaged in this work since 1941. The scheme of research in the history of Hindu mathematics and astronomy referred to above has been financed by the Government of Uttar Pradesh, through the help of Dr. Sampurnanand, its then Education Minister* for which we offer our sincere thanks to them. We are particularly grateful to Dr. Sampurnanand for taking keen and abiding interest in the progress of our research and encouraging us from time to time. The present publication has been made out of a grant of Rs. 3,000/- kindly sanctioned by the Government of Uttar Pradesh, for which we onc% again express our sincere thanks to them. Our thanks are especially due to Acharya Jugat Kishore, our present Minister of Education, for sanctioning the above grant. R. Ballabh CONTENTS Page Introduction , i Sanskrit Text fgrftq^SSJTm: X English Translation Chapter I. MEAN LONGITUDES OF THE PLANETS 1 Homage to the Sun vs. 1— Homage to Aryabhata I 2 — Appre- ciation of Aryabhata I and his work 3— Calculation of the Ahargana 4- 8— Revolution -numbers of planets 9-14— Calcu- lation of mean longitudes of planets 15-17— Positions of apogees of planets 18— Epicycles of planets 19-22-,Hindu prime meridian 23— A rule for the distance from the prime meridian and its criticism 24-27— Criticism of another rule 28— Longitude in time 29— Criterion for knowing whether the local place is to the east or to the west of the prime meridian 30— The longitude correction and its application 31 — Another rule for distance from the prime meridian 32— Alternative rule for the longitude correction 33- Justi- fication of the longitude correction 34 — Demonstration of the same 35 — Consequences of improper application of the longitude correction 36 — Comparison of corrections for longitude and parallax (lambana) 37. CONTENTS Page Chapter II. TRUE LONGITUDES OF THE PLANETS 16 Sun's mean anomaly and its Rsine l-3(i)-Sun's equation of the centre (bkujnphala) 3(ii)-4(i)-Sun's correction for the equation of time due to the eccentricity of the ecliptic 4(ii)-5— True distances of the Sun and Moon 6-7— True daily motion of the Sun and Moon, by various methods 8-15— Sun's declination, earthsine, and ascensional differ- ence 16-18— Correction for the Sun's ascensional difference 19-20— Lengths of day and night 21 -Corrections for the Moon 22-24— Kaksatra 25-26()—Tithi 26(ii)-27— Karana 28— The three kinds of VyatipZta 29- Corrections for the planets, Mars, etc. 30-39— Criterion for knowing whether a planet is stationary 40— Retrograde and direct motions 41. Chapter III. DIRECTION, PLACE, AND TIME FROM SHADOW 41 Determination of directions 1— Latitude and colatitude of the local place 2-3— Ascensional differences of the signs 4— Times of rising of the signs at the equator 5— Times of rising of the signs at the local place 6— Sun's zenith distance and the shadow of the gnomon for the given time 7-11 Time corresponding to the given shadow in the forenoon or afternoon 12-15— Sun's sahkvagra 16— Longitude of the rising point of the ecliptic for the given time and vice versa 17-20— Sun's agra 21— Sun's altitude and longitude when the Sun is on the prime vertical 22-25— Reduction of Rsine to the corresponding arc 26- Midday shadow from the Sun's declination and the latitude of the place 27-28— Sun's declination and longitude, and the latitude of the local place from the midday shadow of the gnomon 29-35. Chapter IV. THE LUNAR ECLIPSE Longitudes of the Sun and Moonwhen they arein opposition or conjunction in longitude 1— Distances of the Sun and Moon 2-3— Diameters of the Sun, Moon and Earth 4-5— Length of the Earth's shadow 6— Diameter of the Earth's shadow where the Moon crosses it 7— Moon's latitude 8— Measure of the Moon's diameter unobscured by the shadow 9— Sparsa and mokfa stkityardkas {i.e., durations of eclipse before and 58 CONTENTS after the time of opposition) 10-12— Times of first and last contacts 13—Sparsa and moksa vimardardhas {i.e., durations of totality before and after the time of opposition) 14 Projec- tion of an eclipse: Aksavalana 15-16— Ayanavalana 17— Resultant valana 18— Corrected valana 19-20— Valana for the middle of an eclipse 21 — Conversion of minutes of arc into ahgulas 22— Construction of the figure of an eclipse 23-30— Construction of the phase of an eclipse for the given time 31-32. Chapter V. THE SOLAR ECLIPSE Longitude of the meridian-ecliptic point 24(i)— Drkksepa for the time of geocentric conjunction in longitude of the Sun and Moon 1, 4(n)-7 (i)-Drggatijya for the same time 7(H)- B{)-Lambana for the time of apparent conjunction of the Sun and Moon 8-W-Mti for the same time 1 1 -Moon's true latitude for the same time ~Sparsa and mohfa sthit- yardhas (i.e., durations of eclipse before and after the time of apparent conjunction) 13-14-Condition for the impossi- bility of a solar eclipse 15. Chapter VI. VISIBILITY, PHASES, AND RISING AND SETTING OF THE MOON Visibility corrections 1-4-Minimum distance of the Moon from the Sun, in degrees of time, at which she becomes Tu w Msasures of illuminated and unilluminated parts of th e Moon 6-7-Elevation of the lunar horns : Moon's nnkvagra for sunset 8-Moon's true declination and agra v-W—JSahu (t. e ., base of the elevation triangle) ll-12(i)— Construction of the figure exhibiting the elevation of the uLhThtTf , I2(iiM8 ~ Durati - of Moon-, visibility in the light half of the month 19-Time of Moonrise on the full moon day 20-21-Shadow of the gnomon due to the Moon 23-25 m ° 0nrise iQ thc dark half -of the month Chapter VII. VISIBILITY AND CONJUNCITON OF THE PLANETS Minimum distances of the planets from the Sun, in degrees of time, at which they become visible 1-2-Degrees of time between the Sun and a planet 3-Time and common longuudc of two planets when they are in conjunction in CONTENTS Page longitude 4-5— Latitudes of those planets for that time 6-9(i)— Distance between two planets when they are in conjunction in longitude 9-10. CHAPTER VIII. CONJUNCTION OF A PLANET AND A STAR 95 Longitudes of the junction-stars of the twenty-seven nak$atras i-4--Conjunction of a planet with a star 5— Celestial latitudes of the junction-stars of the twenty-seven nak§atras 6 * 9 — Definition of absolute conjunction of the Moon with a star 10— Celestial latitudes of the Moon when she occults some of the prominent stars of the zodiac 11-16 — Two astro- nomical problems on indeterminate equations 17-18— Object, scope, and authorship of the book 19. Appendices 1. Theory of the pulveriser as applied to problems in astro- nomy by Bhatta Govinda 103 2. Passages from the Laghu-Bhaskariya quoted or adopted in later works 115 Glossary 120 Short : Non-aspirant : Classiffed: Compound: अ आ ठ ऽ चच् ट् त् इ ई = छु ट् }; थ् }; उ ऊ ज् ड् द् ऋ ए ८ छ् द् ध्रु b ऐ ८ व् ण् न् ओ 0 औौ 200 6 B. G. } LB Pr S 14B Si77 TS 7. 4a INTRODUCTION This Part contains a critically edited text of the Laghu- BfiSskarlya ("the smaller work of Bhaskara I") and its English translation with notes and comments where necessary. Sanskrit Text. In editing the text I have made use of the following four manuscripts in the collection of the late Dr. A. N. Singh : MS. A— Containing the text only ; MS. B— Containing the text together with the commen- tary of 6ankaranarayana (869 A. D,); MS. C — Containing the text together with the com- mentary of Udayadivakara (1073 A. D.) ; MS. D — Containing the text together with the com- mentary of Paramesvara (1408 A.D.). The manuscripts consulted by me are generally in good condition but at places there are imperfections and omissions. In none of them are the verses numbered. MSS. A, B, and C are complete whereas MS. D breaks off at the end of the seventh chapter. B. D. Apte acquired a complete copy of MS. D which he has published in the Anandasrama Sanskrit Series. I have called his edition P. The following is a chapterwise analysis of the extents of the manuscripts consulted by me : ii INTRODUCTION Number of verses MS. A MS.B MS. G MS. D P Common to all • I 38 37 37 37 37 37 II 1 41 40 41 41 41 40 III 35 35 35 35 35 35 IV 32 32 32 32 32 32 V 15 15 15 15 15 15 VI 25 25 25 25 25 25 VII 10 10 12 12 10 VIII 19 19 1 19 19 19 19 Total number of verses common to all manuscripts = The above table shows that (1) MS. A contains an additional verse in Chapter I, (2) MSS. A, C, D and P contain an additional verse in Chapter II, and (3) MS. D and P contain two additional verses in Chapter VII. Of these additional verses, the one belonging to Chapter II possibly belonged to the original text of the Laghu-BK&skanya. The other additional verses are interpolatory as the following discussion will show. Discussion of Additional Verses (1) Additional Verse in Chapter I. This verse occurs in MS A between verses 17 and 18, and runs as follows:- वाग्भावोनाच्छकाब्दाद् धनशतलयहान्मन्दवैलक्ष्यरागै: प्राप्ताभिलिप्तिकाभिर्विरहिततनवश्चन्द्रतत्तुंगपाताः । शोभानीरूढ़संविद्गणकनरहतान्मागराप्ताः कुजाद्याः संयुक्ता ज्ञारसौराः सुरगुरुभृगुजौ वजितौ भानुवज्र्यम्। 11 [Translation. The mean longitudes of the Moon, its ap0ge and ascending mode should be (respectively) diminished by the minutes of arc which are obtained by diminishing the (elapsed) years of the Saka era by 444, (then severally) multiplying (that difference) by 9, 65, and 13 and dividing them by 85, 134, and 32 (respectively). (Severally) multiplying (the same diffe rence) by 45, 420, 47, 153, and 20 (respectively) and dividing (all of them) by 235 are obtained (the corrections in minutes of arc) for Mars, etc. (The corrections) for (the sigr00 of) Mercury, Mars and Saturn should be added (to their mean longitudes); (those) for Jupiter and (the stglr0८a of) Venus should be subtracted (from their mean longitudes). The Sun is to be excluded (from this correction) .]] This verse states the so-called sakabda stated in the tabular form, is a follows: correction, which, 17 Mars Jupiter Saturn Planet Pp. 53 .ि of Mercury Correction per annum Ni। -985 minutes or –6'21" 65/134 minutes or -29"6" -13/32 minutes or -24"2"' +45/235 minutes or +11"29” +420/235 minutes or +147-14” 47/235 minutes or -12" -153|235 minutes or -39"4" +20|235 minutes or +-5"6" The above verse has already been proved to be interpolatory and not belonging to the original. The reasons for that con clusion may be summarised here as follows : (i) The correction stated in the above verse does not occur in the author's bigger work, the Mala-Bhaskarya, mor in his commentary or the Aryablba yu. (ii) The system of numeral notation used for forming number-chromograms in the above verse is alhabeti० (kata29d system) whereas at other places in the Logbu-Bhaskarya the auth0r has used the word mumeral system. In the other works of Bhaskara 1, too, the latter system is used. (iii) The language and style of this verse are not in con formity with the rest of the Laghu-Bhaskaryou.

• See Part , Chapter II, 2:31.
• For the alphabetic and word-numeral systems of notation, the reader

is referred to B. Datta and A. N. Singh, History of Hindu Mathematics, Part I,pp.53 f.], (2) Additional Verse in Chapter II. The verse in question is तिथ्यर्धहारलब्धानि करणानि बवादितः । विरूपाणि सिते पक्षे सरूपाण्यसिते विदुः । [t occurs in MSS. A, C, ID, and P and also in the Mahā Bhaskarya. In MS. B, too, it is found to occur ; but from the opening remarks of the commentator Saikaranarayama it appears that he does not take it as forming part of the Logbu-Bhaskary0. “How is the kara10 to be known ? Bhaskara I) has stated (the method for This very darya (16., determining) it in the तिथ्यर्धहारलब्धानि करणानि बवादितः । विरूपाणि सिते पक्षे सरूपाण्यसिते विद्ः ।।' The above verse may not have occurred in the Loghu Bhaskaryot as Sarikaranarayapa seems to believe, but as the verse is a composition of Bhaskara I and occurs in most of the manus cripts of the Loghu-Bhaskarju , and is relevant to the context, I have included it in the edited text. In my opinion the text would be incomplete without this verse. For, when the text gives rules for the titlhi, 10ksatra and yutt}ळta, there is no reason why there should be no rule for the karapa which is an equally important element of the Hindu Calendar (Paffciga) (3) Additional Verses in Chapter VII. The following two verses are found to occur in MS. D in the seventh chapter between verses 9 and 10 of our edited text. In P they are included in the text and are numbered as 10 and 11. अत्यष्टिविश्वरुद्राङ कतिथ्याप्ता बाणसागराः । बिम्बानि भूसुताद्यासशीघ्रकर्णान्तरैः पुनः । हत्वा पृथक्शीघ्रकर्णव्यासयोगेन भाजितम् । कर्णे हीनेऽधिके स्वर्ण कुर्याद् बिम्बे स्फुटं भवेत् । [Translation. 45 severally divided by 17, 13, 11, 9, and 15 are the mean diameters (in minutes of arc) of the planets beginning with Marऽ (*.e., of Mars, Mercury.Jupiter, Venus, and Saturn). Each diameter should be multiplied by the diffe rence between the ॐghra-kar70 and the radius and then divided by the sum of the ॐgh70-5ar70 and the radius ; and whatever is obtaimed should. be added to or subtracted from the mean diameter, according as the ॐgra-kar10 is less or greater (than the radius '. Thus are obtained the true diameters ( in minutes of arc).]] These two verses do not belong to the text because their contents are not in conformity with the teachings of Bhaskara I The first of the two verses gives the mean diameters of the planets which are different from those given in the Mahā Bhaskary0 both in absolute and relative magnitudes as is clear from the following table : Planet Mercury Jupiter Venus | 32|25=128 32/15=2-13 32/10=3:2 32|5=6*4 32/30=1'06

• The word pasa here means “radius'

45|17=264 45|13=3"46 45|11=4'09 45/9=5 45/15=3 INTRODUCTION vii The second verse gives the following formula for the true diameter of a planet: True diameter = mean diameter mean diameter X (slghra-karna— radius^ slghra-k&rna+ radius mean diameter X radius $ (slghra-karna + radius) The corresponding formula given by Bhaskara I in the MahH- Bhaskarlya is 1 mean diameter x radius true diameter = — • (stghra-karna x manda-karna) /radius The two formulae are fundamentally different, because the former is based on the assumption that the true distance of a planet is equal to 2 £ (Stghra-karna + radius), and the latter is based on the assumption that the distance of a planet is equal to (slghra-karna x manda-karna) /radius, The verses in question occur in MS. D, which contains the commentary of Paramesvara, but have not been commented upon by the commentator. Evidently they are quotations cited by the commentator. Discarding the additional verses of Chapters I and VII and counting that of Chapter II, the text edited by me comprises 214 verses. The size of the Laghu-Bti&skarlya is thus approxi- mately half that of the MahS-BhUskariya which contains 403^ verses. 1 Cf. MBk, vi. 2 This is in agreement with what Burgess interprets to be the meaning of SJiSi, vii. 13-14. Cf. E. Burgess, Translation of the Sury a-Siddhanta, Calcutta (1935), p. 195. The mean diameters of the planets given in xhtSurya-Siddhanta, however, do not agree with those stated in the first verse above. viii INTRODUCTION Reading-differences, In the determination of correct readings I have adopted the same principle as followed by me in the Maha- BhSskartya. English Translation. The English translation supplied by me is as far as possible literal. Where necessary additional explanatory matter is enclosed within brackets. The translation is preceded by a brief N gist of the passage translated and is followed where necessary by short notes and comments. To avoid repetition passages having parallels int^e Mahi-BhSskanya have not been commented upon in detail. Parallel passages in the Maha-Bhaskartya have been indicated in the foot-notes and the reader should refer to them for details. Technical terms are explained in the (glossary given at the end of the book and the reader can conveniently refer to it when necessary. In the end of this Part, I have added two appendices con- taining 1. Theory of the pulveriser as applied to problems in astronomy by Bhatta Govinda. 2. Passages from the Laghu-BhZskartya quoted or adopted in later works. , Contents of the Lagku-BhSskariya. The Laghu-BMskanya^as its name implies, is the smaller work on astronomy by the author. From the closing stanza of this work, it is clear that the author wrote this work for the benefit of young students with immature mind by condensing and simplifying the contents of his bigger work, called MahS-Bhdskdriya or Kwrma-nibandha: "For acquiring a knowledge of the true motion of the planets by those who are afraid of reading voluminous works, the Karma-nibandha has been briefly told by Bhaskara." The Laghu-Bhdskariya is divided into eight chapters. The first chapter contains 37 verses and deals with the calculation mean longitudes of the planets. INTRODUCTION 1X Verse 1 pays homage to the Sun and Verse 2 to Myabhata I. vZ 3 ' an appreciation of Aryabhata I and h, S work. Ya -Ja method for determining the number of days yCarS ' u n »™ rule for calculating the mean Verses 15-17 give the general rule loneitudes of the planets, etc Vers 18-22 state the positions of the apogees of the p.anets and ,he dimensions of the epicycles of the planets. Verse 23 specifies the position of the Hindu pome mendta. Verses 24-29 are devoted to the determination of the long.tude of a place. Verse 30 gives the criterion for knowing whether the local place is to the east or to the west of the prime meridian. Verses 31-36 relate to the longitude correction to the mean longi- tudes of the planets and its justification and importance. Verse 37 differentiates between the longitude-correction and the fcwnfowfl-correction. A detailed treatment of the longitude-correction is a remarkable feature of this chapter. As many as fourteen verses are devoted to this topic only. The second chapter contains 41 verses and is devoted to the calculation of the true longitudes of the planets. Verses 1-20 relate to the determination of the Sun's true longi- tude. Of these, verses 1-4 deal with the Sun's equation of the centre, and the Sun's correction for the equation of time due to the eccentricity of the ecliptic; verse 5 gives approximate formulae for the latter correction m the case X INTRODUCTION of the Sun and the Moon ; verses 6-7 give a rule for finding the true distances of the Sun and the Moon; verses 8-15(ii) relate to the calculation of true daily motion (in longitude) for the Sun and the Moon; verse 16 gives a rule for finding the Sun's declination from the Sun's longitude; verses 17-18 give a rule for finding the Sun's ascensicnal differ- ence; and verses 19-20 relate to the Sun's correction for the Sun's ascensional difference {i.e , for the difference of times of sunrise at the local place and at the place where the local meridian intersects the equator). Verse 21 gives a rule for finding the lengths of day and night when the Sun is in the northern or southern hemisphere. Verses 22-24 deal with the corrections for the Moon. Verses 25-28 relate to the calculation of naksatra, titki, and kamna, which form three important elements of the Hindu Calen- dar. Verse 29 relates to the classification of the phenomena called vyaiipata. The remaining chapter deals with the planets, Mars, etc. Verse 30 gives general instructions relating to the planets. Verses 31-32 relate to the conection to be applied to the tabu- lated epicycles of the planets. Verses 33-37 (i) give the method for finding the true longitude in the case of Mars, Jupiter and Saturn. Verses 37(ii)-39 give the corresponding method for Mercury and Venus. Verse 40 gives the criterion for knowing whether a planet is stationary. Verse 41 states the method for finding the true daily motion of a planet, direct or retrograde. The third chapter comprises 35 verses and deals with the determination of directions, time, and place, with the help of the shadow of the gnomon. INTRODUCTION xi Verses 1-2 (i) give the method for finding the directions— east, west, north, and south. Verses 2(ii)-3 give a rule for finding the local latitude from the equinoctial midday shadow. Verses 4-6 relate to the times of rising of the signs at the equator and at the local place. Verses 7-11 and 12-15 give rules for finding the Sun's altitude and zenith distance with the help of the time elapsed since sunrise (in the forenoon) or to elapse before sunset (in the afternoon), and vice versa. Verse 16 relates to the determination of the sahkvagra (i.e., the distance of the Sun's projection on the plane of the celestial horizon, from the Sun's rising-setting line). Verses 17-19 give a method for finding the longitude of the rising point of the ecliptic with the help of the Sun's instan- taneous longitude and the time elapsed since sunrise. Verse 20 gives a rule for finding the time elapsed since sunrise with the help of the instantaneous longitudes of the Sun and the rising point of the ecliptic. Verse 21 relates to the determination of the Rsine (= Radius X sine) of the Sun's agrt [i.e., the distance between the east- west line and the Sun's rising-setting line). Verses 22-23 relate to the calculation of the Sun's prime vertical altitude and the derivation of the shadow of the gnomon therefrom. Verses 24-25 give a rule for finding the Sun's longitude with the help of the prime vertical shadow of the gnomon. Verse 26 gives a rule for finding the arc corresponding to a given Rsine. [The converse of this was already given in ii. 2(ii)-3(i).] Verses 27-28 give a rule for finding the Sun's altitude and zenith- distance, and the midday shadow of the gnomon with the help of the Sun's declination and the local latitude. jj INTRODUCTION Verses 29-33 relate to the determination of the Sun's longitude from the midday shadow of the gnomon. Verse 34 gives a rule for finding the Sun's declination with the help of the local latitude and the Sun's meridian zemth distance. , t . , . , Verse, 35 relates to the determination of the local latttude with the help of the Sun's declination and the midday shadow of the gnomon. The fourth chapter contains 32 verses and is devoted to the calculation of a lunar eclipse and also to the graphical represen- tation of an eclipse. Verse 1 gives an approximate rule for finding the longitudes of the Sun and the Moon for the time of geocentric opposition or • conjunction of the Sun and Moon. Verse 2 states the mean distances of the Sun and the Moon, and verse 3 gives a rule for finding their true distances. Verse 4 states the measures of the diameters of the Sun, Moon, and the Earth. Verse 5 gives a rule for finding the angular diameters of the Sun and the Moon. Verses 6-7 relate to the determination of the diameter of the Earth's shadow where the Moon crosses it. Verse 8 gives a rule for finding the Moon's latitude for the time of opposition of the Sun and Moon. Verse 9 gives a rule for finding the measure of the Moon's diameter unobscured by the shadow. Verses 10-13 relate to the determination of the durations of eclipse before and after the time of opposition of the Sun and Moon and of the times of the first and last contacts. Verse 14 gives a rule for finding the durations of totality before and after the time of opposition of the Sun and Moon. Verses 15-21 relate to the determination of the so called valana, which is required in the construction of the figure of an eclipse. INTRODUCTION xiii Verse 22 relates to the conversion of minutes of arc into ahgulas. Verses 23-30 relate to the construction of the figure of an eclipse. Verses 31-32 relate to the construction of the phase of an eclipse for the given time. The fifth chapter consists of 15 verses and deals with the calculation of a solar eclipse. Verse 1 gives the definition of the so called "local divisor" to be used later. Verses 2-8 (i) relate to the determination of the drkksepa-jylk and dfggati-jy*. Verses 8(ii)-10 and 1 1 relate to the determination of the lambam- ghafh (i.e., the difference between the parallaxas in longitude of the Sun and Moon, in terms of ghafis) and the nati (i.e., the difference between the parallaxes in latitude of the Sun and Moon) for the time of apparent conjunction of the Sun and Moon with the help of drkksepa-jyH and drggati-jyi. Verse 12 relates to the determination of the Moon's true latitude {i.e., Moon's latitude corrected for parallax) for the same time. Verses 13-14 give a rule for finding the durations of a solar eclipse before and after the time of apparent conjunction of the Sun and Moon. Verse 15 gives the condition for the impossibility of a solar eclipse. The sixth chapter contains 25 verses and deals with the visibility of the Moon, the phases of the Moon including the elevation of the Moon's horns, and the rising and setting of the Moon. Verses 1-4 deal with the visibility corrections (aksa-drkkctrma and ayana-drkkarma) . Verse 5 gives the minimum distance of the Moon from the Sun at which she becomes visible. x - v i INTRODUCTION Verses 6-7 give a rule for finding the measure of the Moon's illuminated part in the light half of the month and the measure of the Moon's unilluminated part m the dark halt of the month. Verses 8-12(i) relate to the calculation of the base of the eleva- tion triangle. Verses 12(ii)-18 relate to the construction of the figure exhibiting tL elevation of the lunar horns in the first and second quar- ters of the month. Verse 19 Rives a rule for finding the duration of visibility of the Moon in the light half of the month. Verses 20-21 relate to the time of rising of the Moon on the full moon day. Verse 22 relates to the determination of the shadow of the gnomon due to the Moon. Verses 23-25 gives a rule for finding the time of moonrise in the dark half of the month. The seventh chapter comprises 10 verses and deals with the visibility and conjunction of the planets. Verses 1-2 give the minimum distances of the planets from the Sun. in degrees of time, at which they become vsrble. Verse 3 gives the method for finding the degrees of time between the Sun and a planet. Verses 4-5 give a rule for finding the time and common longi- Le of two neighbouring planets when they are m conjunc- tion in longitude. Verses 6-9(i) give the method for finding the latitudes of the planets. Verses 9-10 relate to the determination of the distance between two planets which are in conjunction in longitude. INTRODUCTION XV The eighth chapter is composed of 19 verses and deals with the conjunction of a planet with a star. Verses 1-4 state the longitudes of the junction-stars of the twenty- seven zodiacal asterisms. Verse 5 defines the conjunction of a star with a planet. Verses 6-9 state the latitudes of the junction-stars of the twenty- seven zodiacal asterisms. Verse 10 relates to the conjunction of the Moon with a star. Verses 11-16 give the latitudes of the Moon when she occults some of the prominent stars of the zodiac. Verses 1 7-18 give two astronomical problems on indeterminate equations. Verse 19 states the object, scope and authorship of the book. A comparative study of the contents of the Mahi-Bhdskartya

• and the Laghu-Bh&skariya confirms the author's claim that the

latter work is an abridgement of the former. The Laghu-Bhas- karlya is, truly speaking, a well-planned summary of the MaK&- Bhllskarlya, in which the unnecessary or irrelevant rules have been omitted, the defective or erroneous rules ha^e been rectified or replaced, and some new rules which were considered important for the beginner have been added. The following table provides a comparative analysis of the rules occurring in the two works. It will show at a glance which of the rules of the MahZ-BhZskarlya occur in the Laghu-Bhaskariya in abridged or modified form, or have been omitted in the Laghu- Bh&skarlya, or which of the rules of the Laghu-Bficlskarlyd have no counterpart in the Maha-Bttiskariya. ] Comparative Analysis of the rules of the Laghu-Bhaskarya 1. 4-8 1. 9-14 1. 15-17 1. 23 1. 24 1. 25-26 i. 18, 19-21, 22 || wi. 11, 12 ), 13-16 1. 27 1. 28 1.29 1. 32 1. 33 1.34 1. 35 1.36 1.37 1. 4-6: wi. 6-7 wi. 1-5, 8 1. 8. 40 i. 1-2 i. 10(iii) i. 5 i. 7 ii. 10(i) i. 10(ii) t. 7 1. 9 1. 10 1. 11-12 i. 1-2() i. 2(ii)-3(i) i. 3(ii)-4(i) i. 4(ii) i. 6-7 i. 9-10 i. 11-13 i. 14-15() i. 15(i) 1i. 16 ii. 17-18 i. 19-20 i. 2] i. 25-26(i) i. 13-19 1. 20 1. 21-39 i. 41-52 i. 8 iv. 1; 8(i) || iv. 3-4(i) || iv. 6 iv. 9-12 iv. 13 iv. 14 iv. 15-17 iv. 18 iां. 6(i) i. 6(ii)-7 iv.26-27 || iv. 34 and the INTRODUCTION xvii Laghu-Bhaskariya Jl/fsihH RhUtlr/lTiv/] Loghu-Bhiskarlya Maha-Bhaskariya ii. 26(ii)-27 iv. 31-32 iii. 4 iii. 8 ii. 28 iv. 33 iii. 5 iii. 10 (i) ii 29 iv. 35 iii. 6 iii. lO(ii) ii. 30 iv. 37 iii. 7-10 iii. 18-20 ii. 31-32 iv. 38-39 iii. 11 (i) iii. 25 it. 33-37 (i) iv. 40-43 iii. 11 (ii) iii. 26 ii. 37(ii)-39 iv. 44 Hi. 12-15 iii. 27-28(1) ii. 40 — iii. 16 iii. 54 ii. 41 — iii. 17-19 iii. 30-32 — iv. 2 iii. 20 iii. 34-36 — iv. 4(ii)-5 iii. 21 iii. 37 — iv. 19-20 . iii. 22-23 iii. 37-38 — iv. 21-23 iii. 24-25 iii. 41 — iv. 24 iii. 26 viii. 6 iv. 25 iii. 27-28 ui. 1 1 — iv. 36 iii. 29-33 iii. 13-16 — . iv. 45-46 iii. 34 — — iv. 47 iii. 35 iii. 17 — iv. 48-54 — iii. 3 — iv. 55 — j iii. 9 iv. 56-57 iii. 12 iv. 58-63 iii. 21-24 iii. 1 iii. 1-2 iii. 29 iii. 2-3 iii. 4-5(i-iii) iii. 33 xviii INTRODUCTION Laghu-Bhaskarlya iv. 1 iv. 2 iv. 3 iv. 4 iv. 5 iv. 6 iv. 7 iv. 8 iv. 9 iv. 10-12 iv. 13 iv. 14 iv. 15-16 iv. 17 Maha-Bhaskarlya Laghu-Bhaskartya iii. 39 iii. 40 iii. 42-45 iii. 46-51 iii. 52 iii. 53 Hi. 55 iii. 56-60(i) iii. 60(ii)-61 iii. 62 iv. 64 v. 2 v. 3 v. 4 v. 5 v. 71 s 72(i) v. 72(n)-73 v. 30-31 (i) v. 74-76(i) v. 35 v. 76(H) v. 42-44 v. 45 iv. 18 iv. 19 iv. 20 iv. 21 iv. 22 iv. 23-30 iv. 31-32 Maha-Bhaskariya v. 2-4(i) v. l,4(n)-7(i) v. 8(ii)-10 v. 11 v. 12 v. 13-14 v. 15 vi. 1-2 vi. 3-4 vi. 5 vi. 6-7 v. 46-47 (i) v. 47(H) v. 54(i),77 v. 53(H) v. 48-58,61 v. 62-65 v. 6-7 v. 32 v. 40 v. 41 v. 59-60 v. 66-67,68-70 v. 8-11 v. 12-23 v. 24-27 v. 28-29 v. 31(H) v. 34-39 v. 33 vi. l-2(i) vi. 2(ii)-3 vi. 4(ii)-5(i) vi. 5(ii)-7 wi. 8-12(i) vi. 12(ii)-17 w. 19 wi. 20-21 w. 22 wi. 23-25 wi. 6-10 wi. 8-12 | wi. 13-17 wi. 27 wi. 22 wi. 4 , 46(i) wi. 46(ii)-47 wi. 49-51 wi. 20-21 wi. 48,52-55; wii. 9-10 || wi. 10 wi. 23-26 wi. 1-4 wii. 11-16 wi. 32-38 wi. 56-60 wi. 20-35 i. 63-66(i) i. 70(ii) wi. 6(ii)-70(] i. 71 (i) i. 7] (ii)-75(i) The arrangement of the contents of the Laghu-Bhaskary0 is more systematic and logical than that of the Maha-Bhaskarya and is, at the same time, in keeping with the general practice fol lowed by the other Hindu astronomers. P0}ularity g the Loghu-Bhaskarya. In Part 1, I have shown that both the Maha-Bhaskarya and the Laghu-Bhaskar ya were Popular works, having been studied in south India up to the end of the fifteenth century A. D., the former due to its being an authoritative work on Aryabhata I's system of astronomy and the latter being an excellent text-book for beginners in astro Evidence of popularity of the Laghu-Bhaskaya is furnished by the numerous quotations from this work that are found to INTRODUCTION occur in the annotative works of Suryadeva (b. 1191 A.D.), Yallaya (1480 A.D.), NUakantha (1500 A.D.), Raghunatha Rnja (1597 A.D.), Govinda Somayajt and Visnu 6arma, and in the Prayoga-racariS, an anonymous commentary on the MaKi- BHiskarlya. Quotations from the Laghu-BhZskarlya are found to occur not only in astronomical and astrological works but also in works on other subjects. For example, one quotation occurs in Karavinda Svaml's commentary on the Ap&stamba-sulba-butra. Some passages from the Laghu-BhZskariya have also been adopted verbatim or with slight verbal alterations in the Tantra-sahgraha of Nllakantha (1500 A. D.). A list of passages quoted or adopted in later works is given in Appendix 2 at the end of this book. 1 Another evidence of the popularity of the Laghu-Bhiskarlya is the occurrence of commentaries on this work, written in Sanskrit as well as in provincial vernaculars, such as Malayalam and Tamil. Amongst the notable commentators may be men- tioned the names of &ankaranarayana (869 A. D.), Udaya- divakara (1073 A.- D.) and Paramesvara (1408 A.D.). Authorship. The author of the Lagku-BhSskartya bears the name Bhaskara as is evident from the closing stanzas of his works, the MakS-Bfi&skarlya and the Laghu-BhSskartya. But he is a different person from his namesake of the twelfth century A. D., the celebrated author of the Siddhlinta~kiromani> IMHvaii t and Bijaganita, etc. He lived in the seventh century of the Christian era and was a contemporary of Brahmagupta (628 A. D.). To distinguish between the two Bhaskaras, I have called the author of the Mah&'bhftskariya and the Laghu-Bhaskarlya by the name Bhaskara I and the author of the Siddhinta-siromani by the name Bhaskara II. 1 Sec pp. 115-119. INTRODUCTION xxi In addition to the two works mentioned above, Bhaskara I wrote one more work on astronomy, viz. a commentary on the Aryabhatiya. In Part I, I have shown that the three works of Bhaskara I were written in the following chronological order: ( 1 ) The MaM-BUskarxya (2) Commentary on the Aryabhatiya (3) The Laghu-Bh'Sskartya At two places in the commentary on the Iryabhaflya, Bhas- kara I has mentioned the time elapsed since the beginning of the current Kalpa (Aeon). Thus in his commentary on the eighth gttikZ-sutra (1, i. 9), he writes: "Since the beginning of the (current) Kalpa (Aeon) the number of years elapsed is this: zero, three, seven, three, twelve, six, eight, nine, one (proceeding from right to left) years. The same (years) in figures are 19861 23730." 1 Under the same gltikH-sutra, he again writes: "The time elapsed, in terms of years, since the com- mencement of the (current) Kalpa is zero, three, seven, three, twelve, six, eight, nine, one (years). The same (years written in figures) are 1986123730." 2 Now the number of years elapsed since the beginning of the current Kalpa at the commencement of Kaliyuga 3 = 6 manus +27% yugas ~6 x 72 yugash 21 yugas = (432 + 11 1/4) x 4320000 years = (1866240000 + 119880000) years = 1986120000 years. 1 ^FTT^5^T^^T Srs^TfofefrntrT: «|i4fc<MI*<WNlTON*: I 1986123730. a w 1986123730. 1 See a. i. 5. xxii INTRODUCTION Therefore the number of years elapsed since the beginning of Kaliyuga at the time of writing the commentary -1986123730-1986120000 years =3730 years. The year when 3730 years of Kaliyuga had elapsed was the year 629 of the Christian era. Bhaskara I's commentary on the Aryabhafiya was, therefore, written in 629 A. D., i.e., exactly one year after Brahmagupta wrote his Br^kma-sphufa-siddkSnta. The MaKi'Bk&sharlya was written earlier and the Laghu-BKiskariya later than this date. The place of birth and activity of Bhaskara I is not definitely known. On the basis of circumstantial evidence supplied by his works I have shown in Part I that he had associations with the countries of Asmaka and Surastra. His commentary on the Aryabhafiya was probably written in the city of Valabhl in Surastra. It may be that Bhaskara I was born and educated in Asmaka and migrated to Valabhl where he wrote his commen- tary on the Aryabhafiya, or that he was a native of Valabhl and got his education in the Asmaka country. (For details, see Parti). Bhaskara I has a special predilection for calling Aryabhata I by the name Asmaka, his Aryabhafiya by the name A'smaka-tantra or Asmakha, and his followers by the epithet AsmaktyZh. Preference for these unusual names to the usual ones seems to suggest that either Bhaskara I belonged to the Asmaka country or that there was a school of astronomy in that country whose exponents where "followers of Aryabhata" and to which Bhaskara I himself belonged. As Datta has observed, 1 Bhaskara I was undoubtedly the most competent exponent of Aryabhata I's school of astronomy (the Asmaka school). (For details, see Part I). 1 B. Datta, "The Two Bhaskaras," Indian Historical Quarterly, Vol. VI, 1930, pp. 727-736. INTRODUCTION xxiii The Asmaka country (or Asmaka Janapada) is mentioned in both Hindu and Buddhist literatures, where it means either (i) a country in the north-west of India, or (ii) a country lying between the rivers Narmada and Godavar*. The Asmaka of Bhaskara I was evidently the latter one. As regards the personal history of Bhaskara I, it appears from his works that he was a Brahmana, a worshipper of God &va. He seems to have been a teacher by profession, in which capacity he earned a great name and fame. Later writers have shown their respect to him by addressing him by the epithet guru. Thus Sahkaranarayana, in the beginning of his commentary on the Laghu-Bftiiskariya, says : "Having paid homage by lowering my head to Acarya Aryabhata, Varahamihira, hnmadguru Bhaskara, Govinda, and Haridatta, one after the other, I give out ... ***■ So also says Udayadivakara : "Having bowed to Muiari, the Lord of the entire world, and also having paid respectful homage to Acarya Aryar bhata, I write an extensive exposition of the smaller work on astronomy composed by guru Bhaskara." 2 The professional ability of Bhaskara I is clearly evinced by his works which were studied in India up to the end of the fifteenth century A. D. (or even after) and on which a number of commentaries were written. His commentary on the Arya- bhailya, in particular, has been recognized as a work of great scholarship, and he has been called sarvajna bKisyakHra ("all- knowing commentator*' ) . xxiv INTRODUCTION Though essentially an astronomer and mathematician, Bhas- kara I, in his commentary on the Iryabhatlya> displays a thorough knowledge of Sanskrit grammer and Vedic literature m general, and seems to be well-read in other branches of Sanskrit learning also. As an astronomer Bhaskara I was a follower of Aryabhata I and, as already mentioned, belonged to the Asmaka school of astronomy. His works put before us a complete and clear picture of the teachings of Acarya Aryabhata I and throw fresh light on the development of astronomy during the sixth and seventh centuries A. D. His works are thus of special signi- ficance to historians of Hindu mathematics and astronomy, who are now in a position to have 1 a clear glimpse of the astronomical conditions prevailing in the sixth and seventh centuries A. D. in the Asmaka country which was the main seat of Aryabhata I's system of astronomy. In the absence of the works of Bhaskara I, many a passage in the Iryahhafiya of Aryabhata I would have remained obscure to us. In conclusion I take the opportunity to express my sincere thanks to Dr. Ram Ballabh, Professor of Mathematics, Lucknow University, for taking keen interest in my work and offering helpful suggestions and advice from time to time, and for affording all facilities in my researches. I must also express my thanks to my Research Assistant, Sri Markandeya Misra, Jyotisacharya, for the assistance rendered by him to me. My thanks are also due to Sri R. Chandra of Star Press, Lucknow, for their unfailing courtesy and fare in the printing of this book. K. S. Shukla लघुभास्करीयम् प्रथमोऽध्यायः भास्कराय नमस्तस्मै' स्फुटेयं ज्योतिषां गतिः । प्रक्रियान्तरभेदेऽपि यस्य गत्याऽनुमीयते ।। १ ।। काले महति देशे वा स्फुटार्थ यस्य दर्शनम् । जयत्यार्यभटः सोऽब्धिप्रान्तप्रोल्लङ्घिसद्यशाः' ।। २ ।। नालमार्यभटादन्ये' ज्योतिषां गतिवित्तये । तत्र' भ्रमन्ति तेऽज्ञानबहलध्वान्तसागरे ।। ३ ।। नवाद्रयकाग्निसंयुक्ताः* शकाब्दा द्वादशाहताः । चैत्रादिमाससंयुक्ताः पृथग्गुण्या' युगाधिकैः ॥ ४ ॥ ते च षट्त्रिकरामाग्निनवभूतेन्दवो युगे । भागहारोऽब्धिवस्वेकशराः स्युरयुताहताः ॥ ५ ॥ अधिमासान्पृथक्स्थेषु प्रक्षिप्य त्रिशताहते । युक्त्वा' दिनानि यातानि प्रतिराश्य युगावमैः ।। ६ ।। सङ्गुणय्या'"'म्बराष्टेषुद्वयष्टशून्यशराश्विभिः । छेदः खाष्टवियद्व्योमखखाग्निखरसेन्दव ' ।॥ ७ ।।

• नमस्तुभ्यं B, P . * प्रक्रियातदभेदेऽपि A. 3 स्फुटार्था A. ४ सोऽब्धिप्रान्तप्रोल्ला

घिसद्यशाः A. Udaya Divakara refers to the readingए वार्धि in place of सोऽब्धि. * अलमार्यभटादन्ये B. ६ यत्र A, C. ७ ते ज्ञानबहुलभ्रान्तिसागरे A.

• नवाद्येकाग्निसायुक्ता A; नवाद्रीन्द्वग्निसंयुक्ता: C. * पृथ ************ D. *** युङ क्त्वा C.

११ सगुणस्या A. १ *३ काष्ठविय० A. The second line of this verse is missing from D. [लघुभास्करीये लब्धान्यवमरात्राणि तेषु शुद्धेष्वहर्गणः। वारः सप्तहृते शेषे शुकादिर्भास्करोदयात् ।। ८ ।। दस्राग्निसागरा भानोरयुतघ्नाः' निशाकृतः । अङ्गपुष्कररामाग्निशरशैलाद्रिसायकाः ।। ६ ।। कौजा वेदाश्विवस्वङ्गनवदस्रयमा' गुरो सागराश्वियमाम्भोधिरसरामा:" प्रकीर्तिताः ।। १० ।। शनेरपि च वेदाङ्गभूतषट्कसुराधिपाः । १११ सावित्रा' राजपुत्रस्य भगण भार्गवस्य च ॥ ॥ इन्दूच्चस्य नवैकाश्विवसुप्रकृतिसागराः । बौधाः खाश्विखसप्ताग्निरन्ध्रशैलनिशाकराः ।। १२ ।। भार्गवस्याष्टवस्वग्नियमदस्राम्बराद्रयः । मध्यमो भास्कर ** शीघ्रः*** शेषाणां पातपर्ययाः * ।। १३ ।। अङ्गाश्वियमदस्राग्नियमलाः भूदिनानि तु। व्योमशून्यशराद्रीन्दुरन्ध्राद्रयद्रिशरेन्दव *3 ।। १४ ।। पर्ययाहर्गणाभ्यासो' ह्रियते” भूदिनैस्ततः । लभ्यन्ते पर्ययाः शेषाद्राशिभागकलादयः ।। १५ ।। भास्करैस्त्रिशता षष्ट्या सङ्गुणय्य पृथक् पृथक् । तेनैव भागहारेण लभ्यन्तेऽकॉदयावधे * ।। १६ ।। विलिप्तान्ता ग्रहा मध्याः शश्युच्चे' राशयस्त्रयः । १७ ।। क्षिप्यन्ते षट् तमोमूतौ*चक्रात् स च विशोध्यते ** ।। शतमष्टादशोपेतं द्विशती दशसंयुता । चक्रार्धभागा* नवतिः षट्त्रिदस्राः कुजादितः*' ।। १८ ।। 3 तौजा A १ दानोर" A. * अङ्गपुष्कररामाग्निशरखैः प्राद्रिसायकाः A. ४खेदाश्वि० . ५ . ६ सवित्रा D. ८ ०रस्त्रशैल° A. ७ भास्करस्य A A सागरोऽश्वि° A ९ १०ग्निनवदस्राम्बरादय: B. १० भास्करो मध्यमः (C. ११ शीघ्रः is missing from D. १ २ पातपयया D. . १५ क्रियते (C. १६ °न्तेतोदया" A. १3 ० रखाद्यद्रि० A १४ पर्याया० P २०० चक्रेऽर्धभाग १८ ID) .

• ७ शश्य“ तमोमुक्तेः A . १ *५ विशोधयेत A.

A. ३ १ ०दस्रकुजाभित: B. प्रथमोऽध्यायः] मन्दाः सुराधिपाः सप्त शैला' जलधयो नव । अष्टादश च पञ्चाष्टौ द्वौ च युग्मे त्रयोदश ।। १६ ।। पञ्चाशत् त्रिकसंयुक्तास्त्रिशद्रपेण संयुताः । षोडशैकोनषष्टिश्च शीघ्रा नव च कीर्तिताः ।। २० ।। द्वाभ्यां द्वाभ्यामथैकेन द्वाभ्यामेकेन वजिताः । त एव स्युः क्रमाद्युग्मे दृष्टाः परिधयो निजाः ।। २ भास्करस्यापि मन्दांशाः' सप्ततिर्वसुसंयुताः" । परिधिश्च त्रिकस्तस्य ' सप्त चामृततेजसः ।। २२ ।। लङ्कावात्स्यपुरावन्तीस्थानेश्वरसुरालयान् । अवगाह्य स्थिता रेखा देशान्तरविधायिनी ।। २३ ।। लम्बकेनाहतं भूमेर्नवरन्ध्राश्विवह्नयः* । व्यासार्धापहृतं वृत्तं' स्वदेशे तत्प्रकीत्र्यते ।। २४ ।। समरेखास्वदेशाक्षविश्लेषान्तरसङ्गुणम्' ' । वृत्तं स्वदेशजो* भूमेबहुश्चक्रांशकोट्टतम् '3 ।। २५ ।। कर्ण:*४ स्वदेशतस्तिर्यक् समरेखावधेः' स्थितः*७. । तद्बाहुवर्गविश्लेषमूलं देशान्तरं स्मृतम्' ।। २६ ।। इत्याहुः केचिदाचार्या* नैवमित्यपरे जगुः* । स्थूलत्वात्कर्णसङ्ख्याया वक्रत्वात्परिधेर्भुवः ।। २७ ।। मध्यच्छायादिनाधत्थतिग्मरश्म्योर्यदन्तरमृ*** । न तत्पलस्य* तुल्यत्वात्समपूर्वापराशयं * ।। २८ ।।

• शैला is missing from C. * पञ्चाशत्रिकसंयुक्ता B. ॐ शीघ्र B, C.
• मन्दांशा B; मन्दांश: C . ५ १०युता B, C. ६ त्रिको यस्य A. ७ लङ्कामात्स्य

पुंरावन्तिस्थानेश्वरसुरालयात् A; ०स्थानैश्चर° B; "वन्ति• C, ID); °मात्स्यपुरावन्ति" P. ८ “हता A. * १० खाद्रयः A. १• १० हृता वृत्ताः A . **०खासदेशाक्षविश्लेषान्तर सङगुणा A; *न्तसङगु° ID) . १२ वृत्तास्वदेशजा A; स्वदेशज B, C, ID . १3 °द्धतः 4A, C, D, P. १ ४ तत्र C; कर्ण B, 10 . १५ ०तियक् D. १६ ०वधौ A.

• स्थिता D. १८१०न्तरा स्मता A. १ १ १० दाचार्यो B. ३० विदुः A. ३१ “तिग्मराश्योर्य० A;
• नाधर्मोत्थं तिग्म० D. २२ तत्फल० A, B, C, ID, P. २3 ° परांशयोः A .

. गणितप्रक्रियावाप्तप्रत्यक्षीकृतकालयोः । विश्लेषो यो ग्रहणयोः कालो देशान्तरस्य सः ।। २८ ।। अतीत्य गणितानीतं४ यदा स्यातामुपप्लुती' । पूर्वेण समरेखाया द्रष्टा स्यात् पश्चिमेऽन्यथा ।। ३० ।। देशान्तरघटीक्षुण्णा मध्या भुक्तिर्छुचारिणाम् । षष्ट्या भक्तमृणं प्राच्यां* रेखायाः पश्चिमे धनम् ।। ३१ ।। स्वदेशभूमिवृत्तेन' हत्वा देशान्तरा घटीः ** । षष्ट्या विभज्य लभ्यन्ते योजनानि स्वदेशतः ।। ३२ ।। योजनैर्मध्यमां भुक्ति हत्वा तद्देशजैः सदा' । स्वभूवृत्तेन यल्लब्धं** शोध्यं क्षेप्यं स्वमध्यमे' ।। ३३ ।। देशान्तरघटीभोगप्रक्षेपापचयो विधिः । ऊनाधिकतिथेहेंतुस्तेन दृष्टं न हीयते ।। ३४ ।। मोक्ष्यमाणे तु शीतांशौ* * नाडिकायामिहास्तगे' । । मुक्त्वाऽस्तं* पश्चिमे यातः ** प्राच्या'* प्राहुस्तदा ग्रहः' ' ।। ३५ ।। विपरीतधनर्णत्वे यथा दृष्टा* तिथिर्न सा । अन्यथा प्रक्रियाप्राप्तिर्गत्यन्यत्वं* ग्रहस्य च * ॥ ३६ ।। धनणें स्तस्तिथेस्तस्य3 कालस्येन्द्वर्कयोस्ततः । लम्बनस्यैव* नात्र स्याद्युक्तिर्देशान्तरस्य सा ।। ३७ ।। इति लघुभास्करीये प्रथमोऽध्यायः । [लघुभास्करीये

• *प्रक्रियाप्राप्त° P. * विश्लेषो ग्रहणं योर्य: B. ॐ काले B. ४ तं is missing

from A. " °प्लुतिः A. ६ भ्रष्टा स्यात्प° A; द्रष्टास्याः प* C. ७ °भुक्तिविचा" B; मध्यभुक्ति° P. ८ भक्तमृणप्राच्या A. * सन्देशभुविवृत्तेन A . १* घटि: A; हत्वा देशान्तरा घटी: is missing from B. **१ हत्वा तद्देशजैः सदा is missing from B. १२ भूवृत्तेन तु य° A; स्वभूवृत्तेन यत् is missing from B. *3 च मध्यमे B.

• ४ शितांशौ ]P. १५ नाडिकायामथास्तके A. १६ मुक्तास्तं B . १७ पश्चिमे यातं A;

१०मे याताः' B; "भायाताः (C. १०८ प्राच्या: B, C . *** प्रागुदयाद्ग्रहम् A; प्राहुस्तथा ग्रहाः . * दृष्ट्वा A; दृष्टि: C. २१ प्रक्रियावाप्तिर्गद्यवत्व A. २२ तु 10 . ३३ घनणेऽस्तास्थितेतस्य A; धनर्णस्ते तिथेस्तस्या B; ०स्तिथेस्तद्वत्त (C. २४०नस्येव A, B . द्वितीयोऽध्यायः मध्यमं पद्मिनीबन्धोः केन्द्रमुच्चेन वर्जितम्' । पदं* राशित्रयं तत्र भुजाकोटी गतागते ॥ १ ॥ ओजे युग्मे क्रमाज्ज्ञेये कोटिबाहू४ इति स्थितिः । लिप्तीकृत्य' धनुर्भागैर्जीवाः कल्प्या' भुजेतराः ॥ २ ॥ वर्तमानाहतं शेषं धनुषाप्तं विनिक्षिपेत् । ते परिध्याहतेऽशीत्या लब्धे कोटिभुजाफले ।। ३ ।।*** भुजाफलं* धनर्ण* स्यात् केन्द्रे जूकक्रियादिके * । भुजाफलहते भोगे' * चक्रलिप्ताप्तमेव' च ।। ४ ।। भजाफलस्य षड्भागस्तिग्मांशोर्वा विलिप्तिका * । त्रिरभ्यस्ता' ७ द्वयशीत्याप्ता लिप्तिकाद्या *** निशाकृतः ।। ५ ।। कोटिसाधनयुक्तोनं व्यासार्ध मृगककिंतः***. । तद्बाहुवर्गसंयोगमूलं कर्ण:* फलाहत* ।। ६ ।। व्यासार्धाप्तफलावृत्या कर्णः कार्योऽविशेषितः * । शीतांशोरप्ययं ज्ञेयो विधिः कणविशेषणे *3 ।। ७ ।। व्यासार्धसङ्गुणा भुक्तिर्मध्या* कर्णेन लभ्यते । स्फुटभुक्तिः सहस्रांशोः शीतांशोरप्ययं विधिः ।। ८ ।। अन्त्यमौर्वीहतां भुक्ति मध्यमां धनुषा हरेत्" । लब्धं स्ववृत्तसंक्षुण्णं* छित्वाऽशीत्या* विशोधयेत् ॥ ६ ॥

• वजिता A. २ पदा A. 3 भुजाकोटि A, D; भुजकोटी B. ४ कोटिवाह्य B.
• ज्ञप्तिकृ* A. ६ धनुर्भागे जीवः:P. ७ कल्य A; कथ्या 10 . ८ ०तरा A, C, D.
• वर्तमानाहता शेषा धनुषाप्ता A . १० This verse is missing from B, but

it has been commented upon by Sarikaramārāyama and occurs in all other Ms . ११ भुजावाप्तं B, C. १३ धनं हि C . १3 °यादिंगे A.

• ४ भागे A. १५ वक्रिलिप्ता" A. १६ १०वपि लिप्तिकाः A. १७ त्रिरभ्यस्य A, B;

त्रिकाभ्यस्ता (C. १८ काद्यां P. १५०कर्तितम् D. २० कर्म A. २१ फलाहतः is missing from D. २२ कर्णः कार्यो विशेषतः A; कर्णकोट्योर्विशेषज: B. २.3 कर्णाविशेषणै: A, B; कर्णविशेषणे D; कणोंविशेषजः P. २४भुक्तिर्मध्य: P. २५ धनुराहरेत् B. २६ लब्धा स्ववृत्तसंक्षुण्णा . 4. २७ हृत्वा° A; हत्वां D [लघुभास्करीये मकरादिस्थिते केन्द्रे कर्कटादौ तु योजयेत्' । मध्यभुक्तौ सहस्रांशोः स्फुटभुक्तिरुदाहृता ।॥ १० ॥ उत्क्रमक्रमतो ग्राह्याः पदयोरोजयुग्मयो वर्तमानगुणादिन्दोः केन्द्रभुक्तेः कलावशात् ॥ ११ ॥ आद्यन्तधनुषोज्ञेयं फलं त्रैराशिकक्रमात् । गतगन्तव्यधनुषी केन्द्रभुक्तेर्विशोधयेत् ।। १२ ।। इत्यमाप्तगुण हत्वा वृत्तनाशातसहृतम् । प्राग्वत् क्षयोदयाविन्दोर्मध्ये भोगे स्फुटो मतः ।। १३ ।। अभिन्नरूपता भक्तेश्चापभागविचारिण रवेरिन्दोश्च जीवानामूनभावाद्यसंभवात् ।। १४ ।। एवमालोच्यमानेयं जीवाभुक्तिविशीर्यते । कर्णभुक्तिस्फुटाह्वोर्वा विश्लेषस्फुटयोर्द्धयो ' ।॥ १५ ॥ सप्तरन्ध्राग्निरूपाणि परमापक्रमो गुणः । तत्स्फुटार्कभुजाभ्यासस्त्रिज्ययेष्टापम ' हृतः' ।॥ १६ ॥ तद्वर्गव्यासकृत्योर्यद्विश्लेषस्य' * पदं' ॐ भवेत् । स्वाहोरात्रार्धविष्कम्भः** पलज्येष्टापमाहता " ।। १७ ।। क्षितिज्या लम्बकेनाप्ता व्यासाधेनाहता हृता । स्वाहोरात्रेण यल्लब्धं *६ चरजीवार्धमिष्यते *७ ।। १८ ।। तच्चापलिप्तिकाः प्राणाः स्फुटभुक्त्या समाहताः** । खखषडघनभागेन लभ्यन्ते लिप्तिकादय ।। १८ ।।' १ चयो भवेत् B . * "धनुषो ज्ञेया A; आद्यर्घ घनुषो" C. 3 ०राशिकक्र* , (0, D. ४ °र्मध्यभोगः स्फु* A; "र्मध्यभोगे" C, ID) * अभिन्नरूपतो भक्तेश्चापभोग० P. ६ ०भावादिसं० A . ७ °मानायां A . ८ °क्तिर्विशिष्यते A. * कर्णभुक्तिः स्फुटानां वा विश्लेषं A; कर्णभुक्तिस्फुटाहो* C; कर्णभुक्तिस्फुटाहौव*ि D; कर्णभुक्तिस्फुटाह्वोर्वा विश्लेषस्फु० P. १००भ्यासात्त्रिज्यापक्रमो A; ‘ष्टावमो (C, P; ०भ्यासं त्रिज्ययेष्टावमो ID.

• हृतम् P . ***कृत्योस्तु विश्ले" P. १3 पदा A. १४ °ष्कम्भं A. १५ फलज्येष्ठाव

माहता A ; पलज्येष्टावमाहता C; *ज्येष्ठाप* D; ज्येष्ठावमा° P. *६ यल्लब्धा A. १७ चरं जीवा० P. १८ तस्यावलिप्तिकाः प्राणाः स्फुटभुक्तिसमाहता : A. * ** This verse is missing from B. द्वितीयोऽध्यायः] उदग्गोलोदये शोध्या देया याम्ये विवस्वति । व्यत्ययोऽस्तस्थिते काय न मध्याह्नार्धरात्रयोः ।। २० ।। उदग्गोले द्विरभ्यस्तैश्चीयतेऽहश्चरासुभिः । निशाऽपचीयते तत्र गोले याम्ये विपर्ययः ।। २१ ।। भास्वद्भुजाफलाभ्यस्ता' मध्या भुक्तिर्निशाकृतः । रविवच्चक्रलिप्ताप्तमिन्दुमध्ये धनक्षयौ ।। २२ ।। चरप्राणै रवेर्हत्वा' स्फुटभुक्ति निशाकृतः । अहोरात्रासुभिश्छित्वा '* यत् फलं लिप्तिकादि तत्' ।। २३ ।। धनक्षयौ स्फुटे चन्द्रे भास्करस्य वशात्सदा । आदित्यकर्मणा तुल्यं शेषमिन्दोविधीयते ॥ २४ ॥ लिप्तीकृतो निशानाथः शतैर्भाज्योऽष्टभिः फलम् । अश्विन्यादीनि भानि स्युः षष्ट्या हत्वा गतागतम् ' * ॥ २५ ॥ गतगन्तव्यनाड्यस्ताः* स्फुटभुक्त्योदयावधैः । अर्कहीनो निशानाथो लिप्तीकृत्य विभज्यते ।। २६ ।। शून्याश्विपर्वतैर्लब्धास्तिथयो** या गताः क्रमात् । भुक्त्यन्तरेण लभ्यन्ते' षष्ट्या हत्वा'* गतागतम्' ।। २७ ।। तिथ्यर्धहारलब्धानि करणानि बवादित:** । विरूपाणि सिते पक्षे ** सरूपाण्यसिते* विदुः ।। २८ ।। सूर्येन्दुयोगे* चक्राधे व्यतीपातोऽथ वैधृतः । चक्र च ** मैत्रपर्यन्ते विज्ञेयः सार्पमस्तक :* ।। २८ ।।

• व्यत्ययोस्तमिते A; व्यत्ययोऽन्तःस्थिते ID, P. २ ०भ्यस्ते हीयतेऽह' A; ०गोलैरि• B.

ॐ निशापिचीयते B; निशावचीयते D. ४ याम्ये गोले विपर्ययात् A; गोले याम्यविपर्ययः D, P. " "भ्यस्तो A. ६ मध्याभक्ति० (C . ७ मिन्दोर्मध्ये C. ८ रवेहृत्वा A. ९ स्फुटा भुक्तिर्नि" A; स्फुटभुक्तिनि• B. १००सुभिर्भक्त्या A; *भिहत्वा B; “त्रादिभिर्भक्ता (C. ११ “कादिका A. १२ गतागते A ; गतागतम् B. १ ३ गतगन्तव्यना• *******A; "नाड्यः स्युः B. १४०र्लब्धास्थितयो A. १ ६ गत्यन्तरेण नास्य स्युः A; गत्यन्तरेण ल" B. '* गत्वा B. १७ गतागते A, B. १८ बाधितः A. १ ५ पक्वे B. २० स्वरूपा० D. २१ अर्केन्दु D.

• २२ .

चक्रेश A २3 सर्वम° A. [ लघुभास्करीये केन्द्रकोटिभुजामौवीं तत्फलर्णघनादयः । भास्करादवबाद्धव्या ग्रहाणां मन्दशीघ्रयोः ।। ३० ।। क्रमोत्क्रमभवां* जीवां पदयोरोजयुग्मयोः । वृत्तान्तरेण संक्षुण्णां हरेद् व्यासदलेन ताम्' ।। ३१ ।। लब्धमूने क्षिपेद् वृत्ते शोध्यमभ्यधिके फलम् । स्फुटवृत्तमन्यथा स्यान्मण्डूकप्लुतिवद्गतिः ।। ३२ ।। मन्दोच्चफलचापार्ध प्राग्वन्मध्ये धनक्षयौ । कृत्वा शीघ्रोच्चतः शोध्यं शीघ्रकेन्द्रं तदुच्यते ।। ३३ ।। तस्माद् बाहुफलं हत्वा व्यासाधेन विभज्यते । कर्णेनाप्तस्य चापार्घ धनणे' मेषतौलितः ।। ३४ ।। शोधयित्वा ततो मन्दं बाहोः कृत्स्नं फलं ततः'। काष्ठितं' मध्यमे कुर्यात् स्फुटमध्यः स उच्यते ।। ३५ ।। शोधयित्वा' * तु तं'ॐ शीघ्राच्छीघ्रन्यायागतं फलम् । चापितं* सकलं कुर्यात् स्फुटमध्ये स्फुटो भवेत् ।। ३६ ।। कुजार्कसुतसूरीणामेवं** कर्म विधीयते । बुधभार्गवयोश्चाथ** प्रक्रिया परिकीत्र्यते ।। ३७ ।। प्रागेव चलकेन्द्रस्य फलचापदल' स्फुटम्'* । व्यस्तं*** स्वकीयमन्दोच्चे धनणें* परिकल्पयेत् ।। ३८ ।। तेन मन्देन यल्लब्धं सकलं तत् स्वमध्यमे । स्फुटमध्यश्चलोच्चेन* संस्कृतः स स्फुटो ग्रहः ।। ३८ ।। वर्तमानो ग्रहस्तुल्यः श्वस्तनेन यदा भवेत् । वक्रारम्भस्तदा तस्य निवृत्तिर्वाऽथ कीर्तिता ।। ४० ।। १ केन्द्रकोटि० P. २ त्क्रमभवा A, P. 3 जीवा P. ४ १० क्षुण्णा P. "ाः A, P. ६ मूनो A. ७ °मप्यधि० P. ८ "वृत्तमथान्ये च मण्डूक° A. * धनर्ण D. १० फलं तु तत् A; कृत्स्नफलंत० P. *** काष्ठिका A. १ २ पातयित्वा A. १3 कृतं B, P. १४ यापितं A . १५ कुजाकिंशुक्रसूरीणां एवं B. १६ "श्चापि A, D. १७ फलं चाप० D, P. *** स्मृतम् D)

• * व्यक्तं A, B; यस्तं P . ** स्वकीये मन्दोच्चधनणें D, P. २१ स्फुटमध्यंचलो*

C, D. द्वितीयोऽध्यायः] श्वस्तनेऽद्यतनाच्छुद्धे वक्रभोगः प्रकीर्तितः । विपरीतविशेषोत्थश्चारभोगस्तयोः* स्फुटः ।। ४१ ।। इति लघुभास्करीये द्वितीयोऽध्यायः । ।

• २०षांनश्चारभोगास्तयो.

प्रकीत्यंते A. : A; विपरीते विशै० p ततीयोऽध्यायः इष्टमण्डलमध्यस्थशङ्कुच्छायाग्रवृत्तयोः' । योगाभ्यां कृतमत्स्येन ज्ञेये याम्योत्तरे दिशौ ॥ १ ॥ समायां कौ दिशां* मध्ये शङ्कोज्ञतार्जवस्थितेः । विषुवद्दिनमध्याह्नच्छायाया वर्गसंयुतात् ॥ २ ॥ शङ्कुवगद्ििध यन्मूलं तेन त्रिज्या विभज्यते । शङ्कुच्छायासमभ्यस्ता लम्बकाक्षगुणौ" फले. ।। ३ ।। राश्यन्तापक्रमैः कार्या: पूर्ववत्तच्चरासवः । पूर्वशुद्धाः क्रमात्ते स्युर्मेषगोवल्लकीभृताम् ' ।। ४ ।। शून्याद्रिरसरूपाणि भूतरन्ध्रमुनीन्दवः पञ्चाग्निरन्ध्रशशिनो मेषादीनां निरक्षजाः ।। ५ ।। चरप्राणाः क्रमाच्छोध्या दीयन्ते व्युत्क्रमेण' ते । स्वदेशभोदया** मेषाद् व्यत्ययेन तुलादितः ।। ६ ।। गतगन्तव्यघटिका दिनपूर्वापरार्धजाः षष्ट्याऽभ्यस्ताः पुनः षड्भिः प्राणास्तेभ्यश्चरासवः ।। ७ ।। उदग्गोले विशोध्यन्ते* क्षिप्यन्ते दक्षिणे तु ते* * । तेषां जीवा समभ्यस्ता'3 स्वाहोरात्रदलेन** सा ** ॥ ८ ॥ व्यासार्धाप्तफले' कुर्याद् भूज्यां तस्य विपर्ययात् । लम्बकेन पुनर्हत्वा त्रिज्या शङ्कुराप्यते ।। ६ ।। तद्वर्गव्यासकृत्योर्यद्’७ विश्लेषान्तरजं* पदम् ।

छाया सा'* द्वादशाभ्यस्ता* शङ्कुभक्ता प्रभा स्फुटा ।। १० ।। १ इषुमण्डल० A; ०मध्यस्थ:० P. २ समायान्तौ दिशोः A ; ० दिशौ D. 3 मध्य A. ४ शङ्कोजीतार्जवस्तथा A; शङ्कोजतिाज्जव० P . ५ “गुणे A, B; "गुणो P . फलम् ६ P. ७ :"गोवलतद्युताः A; ०षकोवल्लकीभूता D. * भूतरन्ध्र मु’ P

• व्यत्ययेन A. १० ० शजोयया A; ०शभोदयो D) १ १ ०गोलेऽपिशा० A . *** तु तत्

A; कृते D. * *3 समाम्यस्ता B. १४ साहोरात्र" P . १५ सः A. * १६ ०साधेन फले P. १७ ‘कृत्योस्तु P. १८ ०न्तरजा A. १९ छायाया A; च्छायासा B. २० द्यादशा" B . तृतीयोऽध्यायः] इष्टासुभ्यश्चराशुद्धौ' . व्यत्ययः शेषजीवया । शर्वर्या शङ्कुरर्कस्य कार्यो व्यस्तेन कर्मणा ।। ११ ।। शङ्कुच्छायाकृतियुतेमूलच्छेदेन ' संहरेत् । त्रिमौवीं ७ शङ्कुनाऽभ्यस्तां* शङ्कुस्तद्व्यत्ययाद् घटीः ॥ १२ ॥ व्यासार्धसङ्गुणः शङ्कुलैम्बकेन समुदृतः । लब्धे क्षयोदया' भानौ क्षितिज्या* सौम्यदक्षिणे ।। १३ ।। व्यासार्धनिहते भूयः': स्वाहोरात्रार्धभाजिते । लब्धचापे** चरप्राणा देयाः शोध्याश्च गोलयोः ।। १४ ।। सौम्यदक्षिणयोः षड्भिः षष्ट्या भूयश्च नाडिकाः । गतगन्तव्यजा ज्ञेया दिनपूर्वापरार्धजा ॥ १५॥ अक्षजीवाहतः शङ्कुर्लम्बकेन समुद्धृतः* । अस्तोदयाग्ररेखायाः* शङ्क्वग्रं नित्यदक्षिणम् ॥ १६ ॥ स्वदेशोदयसंक्षुण्णं राशिशेषं विवस्वतः । राशिलिप्ताहृतं ***- लब्घमिष्टासुभ्यो विशोधयेत् ।। १७ । राशिशेषं रवौ क्षिप्त्वा शेषासुभ्योऽपि यावताम् । प्राणा विशुद्धोस्तावन्तो दातव्यां राशयः क्रमात् ।। १८ ।। त्रिंशदादिगुणे* शेषे वर्तमानोदयोदृते* । लब्धांशलिप्तिकायुक्तं ' विनिर्दिशेत् ॥ १८ ॥ प्राग्विलग्नं प्राग्विलग्नगतान्प्राणान्संपिण्डय** व्युत्क्रमाद्रवेः । अभुक्ताशावधः काल * कल्प्यते * कालकाङ्क्षणा*॥ २० ॥ ] १ ११

• *श्चराशुद्धा A. * "त्याच्छेष* A, B. ॐ व्यत्यस्त A. ४ o मूलं'छेदेन A ;
• कृति युक्तेमूल° D; ०च्छायागतियुते त्रिमौर्वी ऽभ्यस्तां is

मूलच्छेदं न P. ५ A. ६ Imissing from D. ७ ०घटीम B; ०घटी (C, D, P . ८ *दृते A. **दयौ B, C; लब्धेऽक्षयोदयो ०दये १००ज्यां भूयात् A. १ लब्धं चापे P. D; P. A. ११ *

• 3 विभाजितः p . १४०रेखायां A. १५ राशिलिप्ताच्छतं P. १६ यावता A, C.

१७ त्रिशतादि०:A; विंशदादिगुणे D. १८०दयाहृते B. A, C, D; वर्तमानोदयादते १ * प्राविलग्नं B. २००लग्नागतान्प्राणान् संविन्द्याद A . *०१ कालात् A. * कल्यते P. काङक्षिण: A १२ [लघुभास्करीये क्षुण्णां* परमया क्रान्त्या भुजज्यामुष्णदीधितेः । लम्बकेन विभज्याप्तामकग्रां तां3 प्रचक्षते ॥ २१ ॥ पलज्योनामुदक्क्रान्तिं* विष्कम्भार्धहतां हरेत् । समपूर्वापरः शङ्कुर्लब्धोऽर्कस्य" पलज्यया ।। २२ ।। शङ्कुवर्गविहीनाया' विष्कम्भार्धकृतेः पदम् । द्वादशाभिहतं* भक्तं* शङ्कुना लभ्यते प्रभा ॥ २३ ॥ छायाविधानसम्प्राप्तः शङ्कुः क्षुण्णः * पलज्यया* । क्रान्त्या परमया भक्तो* लब्धजीवाकलाधनुः * ।। २४ ।। तिग्मांशुर्मण्डलार्धाच्च'* परिशुद्धो" विधीयते* । सममण्डलदिङ्मार्गशङ्कुच्छायाप्रसाधितः ।। २५ ।। पिण्डतः'७ प्रविशुद्धानां ज्यानां सङ्ख्या * समाहता* । तिथिवर्गेण शेषं च स्वान्त्यज्याप्तयुतं* धनुः ।। २६ ।। पलापक्रान्तिचापानां* योगविश्लेषजो गुणः छाया याम्योत्तरे भानौ नभसो मध्यसंस्थिते * ।। २७ ।। तच्छायावर्गहीनस्य* त्रिज्यावर्गस्य* यत्पदम् । शङ्कुद्धदशसङ्ख्यस्य** छाया ज्ञेयाऽनुपाततः ॥ २८ ॥ शङ्कुवर्गेण युक्ताया मध्यच्छायाकृतेः पदम् । छेदस्त्रिराशिजीवायाश्छायाघ्नायाः** फलं नतिः* ॥ २८ ॥ नतभागाः* पलान्यूना:* पलाच्छोध्या* रवेरुदक् । दक्षिणेन यदा छाया योगः क्रान्तेर्धनुस्तदा ॥ ३० ॥ 38१ १ क्षुण्णं A, P . * चरमया D. 3 ०ज्याप्तमकर्काग्रान्ता A; ०मकग्रेिति (C. ४पलज्यो नमुदक्रान्ति A; फलज्योनामदक्रान्ति P. *समपूर्वापराशङकु० A. ६ फलज्यया P. ७ शङ कु वर्गविहीनयां P. * द्वादशात्त्रिहतं P. * लब्धं A. ***सम्प्राप्तं शङ कुं क्षुण्णं A; च्छाया० B, D. *** फलज्यया P. * * भक्ता A. १3 लब्धं जीव० 0; लब्धजीवकला० P . १४ तिग्मांशोर्मण्डलाघर्वाञ्च A; तिग्मांशुमण्डलार्धाच्च B, C, D. *** परिशुद्धा A. १ ६ मिधीयते A; ऽभिधीयते B, C, D. १७ पिण्डितः B. १८ सख्या A; यत्ता B, P . १ * समाहतः A. ** स्वान्त्यज्याप्तहतं P. * १०न्तिभागानां A, B, C. *** °संस्थितौ B. *3 तद्वर्गहीनसङख्यस्य D. ३४ स्य is missing from B . *** शङकोद्वf° P. ६ छेदास्त्रिराशिजीवायाश्चानेयाः A; च्छेदस्त्रिराशिजीवायां च्छायायां B; च्छेद° D. २७ फलतोन्नति 4. १८ नतभाग D. *** फलन्यूनाः A; पलान्यूनाः D. 3० फलाच्छोध्या A, B. *** खावुदक (C, D. पर्वनाडयो* रवौ देयास्ताः सलिप्ता' निशाकरे । एवं प्रतिपदः शीघ्याः समलिप्तादिदृक्षुणा ।॥ १ ॥ पञ्चवस्विषुरन्ध्रषुसागरास्तिग्मतेजसः । कर्णः पर्वतशैलाग्निवेदरामा निशाकृतः ।॥ २.॥ अविशेषकलाकर्णताडितौ त्रिज्यया हृतौ । स्फुटयोजनकणौ तौ तयोरेव यथाक्रमम् ।। ३ ।। पङ्क्तिसागरवेदाख्यो रवेस्तिथिशिखीन्दुजः । व्यासो वसुन्धरायाश्च व्योमभूतदिशः स्मृतः ।। ४ ।। ोजनव्याससंक्षुण्णं विष्कम्भार्ध विभाजयेत् । स्फुटयोजनकर्णाभ्यां लिप्ताव्यासौ** स्फुटौ तयोः' ५ ।। ५ ।। कर्णः** क्षुण्णः सहस्रांशोर्मेदिनीव्यासयोजनैः । मेदिन्यर्कविशेषेण * भूच्छायादैध्र्यमाप्यते ।। ६ ।। चन्द्रकर्णविहीनेऽस्मिन् भूमिव्यासेन ताडिते । छायादैघ्र्यहृते व्यासश्चन्द्रवत्तमसः कलाः ।। ७ ।। पातोनसमलिप्तेन्दोजीवा खत्रिघनाहता** । कर्णेन*** हृियते लब्धो विक्षेपः सौम्यदक्षिणः ॥ ८ ॥ इन्दुहीनतमोव्यासदललिप्ताविवर्जिताः । विक्षेपस्य ' न गृह्यन्ते तमसा शशलक्ष्मणः' * ।। ६ ।। विक्षेपवर्गहीनायाः सम्पर्कार्घकृतेः* ** पदम् । गत्यन्तरहृतं हत्वा षष्टया'* स्थित्यर्धनाडिकाः ॥ १० ॥

• पर्वनाड्या A. . ३ देयास्समलिप्ता A. ॐ "लिप्तौदि* D. ४ पञ्चवस्विष्ट

रन्ध्रष्ट साग" A. * कर्ण B. ६ °ताडिताः. A . ७ हृतः. A. * "न्दुजा: A; ०शिखी षुजः 0; [तिथिशिखीन्दुजः = तिथिशिखि (without case-ending) + इन्दुजः ]

स्मृताः A. *** लिप्तव्या° A. *** तयोः स्फुटौ A, C, P; स्फुटा तयो: D .

१ २ कर्ण A. १३ मेदिन्यार्क० A. १४वत्त्रिघना० A. १५ In place ofकर्णेन the commentator Parame5vara refers to the reading व्यासेन०१९ क्षेपलिप्ता A. १८ सवर्गार्घ० A. *** षष्ट्या हृत्वा A. १० ०लक्षणः B. चतुर्थीऽध्यायः] पर्वमध्याद् A; स्फुटभुक्तिहता' नाड्यः षष्ट्या नित्यं समुद्धृता लब्धलिप्ताः क्षयश्चन्द्रे क्षेपश्च स्पर्शमोक्षयोः ११ ।। विक्षेपश्चन्द्रतस्तस्मान्नाडिका लिप्तिकाः शशी । । १२ ।। स्थित्यर्धेनाविशिष्टेन" हीनयुक्ता तिथिः स्फुटा° । स्पर्शमोक्षौ तु तौ* स्यातां पर्वमध्यं* ग्रहस्य' तत्' ॥१३॥ ग्राह्यग्राहकविश्लेषदलविक्षेपवर्गयोः विश्लेषस्य* ३ पदं प्राग्वद् विमदर्धस्य नाडिकाः ॥ १४ ॥ ११ 3 विषुवज्या हता** भाज्या*** त्रिमौव्य लब्धदिक्क्रमः १५ ।। प्राक्कपाले तु बिम्बस्य पूर्वपश्चिमभागय वलनं पश्चिमेऽन्यथा ॥ १६ ॥ तत्कालेन्द्वर्कयोः कोटयोरुत्क्रमज्यापमो'* गुणः । अयनाद्विम्बपूर्वाधे पश्चाधे व्यत्ययेन दिक् ॥ १७ ।। योगस्तद्धनुषोः साम्ये दिशोभेदे' * विपर्ययः । सम्पकर्धहता तज्ज्या* त्रिज्याप्तं वलनं हि तत् ।। १८ ।। एकदिक्कं* क्षिपेत् क्षेपे विदिक्कं* तद्विशोधयेत् वलनं तत् स्फुटं ज्ञेयं सूर्याचन्द्रमसोग्रेहे ।। १ ।। सम्पकधििधकं** तद्धि सङख्यया यत्र लभ्यते सम्पकत् िसकलाद्धित्वा वलनं तन्त्र शिष्यते२६ ॥ २० ॥ ११ ८ ० स्फुटभुक्त्या हता P. २ ० येन A; आवृत्तिकर्मणानेन D; आवत्या ॐ लिप्तिका B,C. ४ णा कर्मणानेन P *त्यधनाचशि०. B धन विशिष्टेन P: ६ हीना युक्ता B, C: ७ स्थितिस्फुट: A; तिथि स्फुटा (C. तत: A, B मध्य :B. १ • गतस्य (C. १ १ च A. *** विक्षेपस्य A, B. १3 4. १४ ि वक्षपज्याहता 4. १५ भज्या A; हाज्या B कोष्ठस्य D दिक्क्रमात् A. १७ ० दक्षिण ज्यावमो A; *योव्र्युत्क्रमज्यावमो (C दिशोभेदे A, C, Pः ३० त्रिज्या एकदिक्स्थां A; एसद्दिक्कं B २२ विदि सम्पर्काकर्काधिकं C. ३४ लक्ष्यते A.C. * २५

• निर्दिशेत् A, B, C, D सकलं हित्वा A

७

२१. २४ [लघुभास्करीये असंयुक्तमविश्लिष्टं स्पर्शवत् केवलं स्फुटम् । विक्षिप्त्या ' ग्रहमध्यस्य तस्य स्याद् व्यस्तदिवक्रमः ।। २१ ।। भास्करेन्दुतमोव्यासविक्षेपवलनोद्भवा अङ्गुलान्यर्धिता' लिप्तास्ता एव हरिजस्थिते ।। २२ ।। ग्राह्याङ्गुलार्धविस्तृत्या वृत्तं सूत्रेण लिख्यते । ग्राह्यग्राहकसम्पर्कदलसङ्ख्येन चापरम् ॥ २३ ॥ पूर्वापरायतं सूत्रं* तन्मत्स्यात्' सौम्यदक्षिणम्' । कृत्वा यथादिशं केन्द्राद्वलनं नीयते स्फुटम् ॥ २४ ॥ विन्यस्तमत्स्यमध्येन सूत्रं पूर्वापरे दिशौ । नीत्वा तु बाह्यवृत्तान्तं ततः केन्द्रं समानयेत् ॥ २५ ॥ ग्राह्यमण्डलतद्योगो* व्यक्तं यत्रोपलक्ष्यते । प्रग्रासग्रहमोक्षौ' 3 स्तस्तत्र' ४ देशे** निशाकृतः ।। २६ ।। तुल्यदिग्वलनक्षिप्त्योर्वलनं* बारुणीं नयेत्' । अन्यथैन्द्रीं'* रवेव्यैस्तं सूत्रं तन्मत्स्यतो'* बहिः* ॥ २७ ॥ विक्षेपस्य वशात् केन्द्रमानयेत्** तत् यथादिशम् । विक्षेपं केन्द्रतो नीत्वा विन्दं तत्र प्रकल्पयेत् ॥ २८ ॥ ग्राहकाङ्गुलविष्कम्भदलसङ्ख्येन:* खण्डयेत् । ग्राह्यबिम्बं तथा मध्ये* ग्राहकस्यावतिष्ठते* ।। २८ ।। प्रग्रासमध्यमोक्षाणां बिन्दूनां* मस्तकानुगम् । मत्स्यद्वयोत्थवृत्तं यद् वत्र्म स्यात् ग्राहकस्य तत्* ॥ ३० ॥

स्थित्यर्धेनेष्टहीनेन हत्वा गत्यन्तरं हरेत् । षष्टया लब्धकृति युक्त्वा विक्षेपस्य कृतेः पदम् ॥ ३१ ॥

• "विश्लिष्ट 10. : * विक्षिप्य P. 3 ग्रहमध्यः स्याद् A. * व्यस्तस्तस्यास्तु दिक्क्रमः

A, C; तस्य स्याद्धस्तदिक्क्रमः P. **वलनोद्भवात् A. * *न्यर्धितं A. ७ लिप्तास्थायेव हरिति' स्थिते A; "स्थितेः D, P. ८ तत्र P. * तन्मध्ये A; तन्मध्यात् B, C, D. १० दक्षिणोत्तरम् A, D. *** विन्यस्तमध्यमध्येन C; विन्यस्यमत्स्य" P. *** ०तद्योगे (C. १ 3 प्रग्राहग्रह " A; प्रग्रासाद् ग्रह * P. *४ स्तांतत्र A. *** शे is missing from B. १६ ०लनाक्षि० A; १० नक्षित्योवर्लनं P. १७ नये is missing from B .

• अन्य is missing from B. * ** तस्मा B. *** बहिः is missing from B
• केन्द्रानानयेत् A. * *काङ गुलिवि" A . २.3 मत्स्ये ID. ४ “कस्तावदिष्यते A.
• बिन्दूनं A" ३६ वत्र्म तद् ग्राहकस्य तुA, D. चतुर्थोऽध्यायः]

तन्नयेत् केन्द्रतो वत्र्म' यत्र सम्यक् तयोर्युतिः । तत्रेष्टकालजो ग्रासो ग्राहकार्धेन लिख्यते ।। ३२ ।। इति लघुभास्करीये चतुर्थोऽध्यायः । १ तन्न चेत्केन्द्रतो वत्र्मा . ।

१७

पञ्चमोध्यायः

लम्बकाभिहता' त्रिज्या परमक्रान्तिसंहृता।
लब्धं स्वदेशसम्भूतो* व्यवच्छेदः प्रकीर्तितः ॥ १ ॥
लङ्कोदयानुपाताप्तानवगम्य रवेरसून् ।
तिथिमध्यान्तरासुभ्यो हित्वा शोध्यं गतं" ततः ।। २ ।।
शेषेऽपि* यावतां सन्ति व्युत्क्रमात् तावतस्त्यजेत् ।
भागा' लिप्ताश्च पूर्वाह्न मध्यलग्नमुदाहृतम् ।॥ ३ ॥
अपराह्न चयः कार्यो गन्तव्यादेर्विवस्वतः ।
पातहीनात्ततः कल्प्यो* विक्षेपः सौम्यदक्षिणः ।। ४ ।।
मध्यलग्नापमक्षेपपलज्याधनुषां* युतिः ।
तुल्यदिक्त्वे विदिक्कानां* विश्लेषश्शेषदिग्वशात्' ।। ५ ।।
मध्यजीवा तया क्षुण्णां* प्राग्विलग्नभुजां' हरेत् ।
व्यवच्छेदेन यल्लब्धं वर्गीकृत्य विशोधयेत् ॥ ६ ॥
मध्यज्यावर्गतः शेषो वर्गे दृक्षेपसंभवः ।
तत्कालशङ्कुवर्गेण युक्त्वा तं प्रविशोधयेत् ।। ७ ।।
विष्कम्भार्धकृतेर्मलं रूपरन्ध्रनिशाकरैः ।
हृत्वा लब्धस्य भूयोंऽशो*४ विज्ञेयो योऽर्धपञ्चमैः ।। ८ ।।
लम्बनाख्यो' भवेत्कालो नाडिकाद्यो* रवेग्रहे ।
पर्वणः शोध्यते प्राह्म१७ दीयते मध्यतोऽपरे *८ ॥ ८ ॥
एवं कृतेन भूयोऽपि पर्वणा * * कर्म कल्प्यते ३० ॥
कालस्य लम्बनाख्यस्य निश्चलत्वं दिदृक्षुणा *' ।। १० ।।

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* "हत A; लम्बकेन हता (C. ३ ०न्तिताडिता A; °संहता P. 3 लब्ध: D, P.४ स्वदेशजो भूमेः P. * नतं (C. ६ शेषोऽपि A . ७ भाग B, C, D. * कार्यो A:* लग्नावमक्षेपवलज्या° A, C; ०क्षेपफलज्या° D; •लग्नावमक्षेप* P. १० दिक्चविदिक्स्थानां A; ०दिक्के विदिक्कानां B . ११ विश्लेषश्लेषदिग्व० P. १ २. मध्यजीवायतक्ष० P. १ ३ प्राग्वल्लग्न० A. १४ भूतांशो A. *** लम्बनाडयो A. १ *६ नाडिकाभ्योA. *७ प्रालो A. १०८ मध्यतः परे A. १९ पर्वणः A . २० कथ्यते D. २१ निश्लत्वं दि* B; निश्चलत्व दि* P. पञ्चमोऽध्यायः ] दृक्क्षेपज्यामविश्लिष्टां गत्यन्तरहतां हरेत् खस्वरेष्वेकभूताख्यैर्लब्धास्ता* लिप्तिकादयः ।। ११ ॥ तत्कालशशिविक्षेपसंयुक्तास्तुल्यदिग्गताः । भिन्नदिक्का विशेष्यन्ते रवेरवनतिः स्फुटा ॥ १२ ॥ अर्केन्दुबिम्बसम्पर्कदलादवनतेः स्फुटात् । स्थित्यर्धनाडिका साध्या प्राग्वद् वलनकर्म च ॥ १३ ॥ प्रग्रासमोक्षयोरेवं लम्बनावनती सकृत् । लम्बनान्तरसंयुक्ते' स्थित्यर्थे निर्दिशेत् स्फुटे ॥ १४ ॥ सम्पकीर्घकलातुल्यकलासङ्ख्यानतौ* शशी । न रुणद्धि** रवेबिम्बं*** ध्वान्तविध्वंसदीधितेः* * ॥ १५ ॥ इति लघुभास्करीये पञ्चमोऽध्यायः । । १९ १ विक्षेपजेामवि० A; विक्षेपज्यामवि० B; दिक्क्षेपज्या° ३ खस्वराद्भयेक" P.

• विशोध्यन्ते A, C. D; विशिष्यन्ते P . ४ ०दवनतै: A; ०दलावनतित: B; १०सम्पर्का

६लेनापनते D; "दलेनानवते: P. * स्फुटाः B. ६ प्राग्व“लन° B; तथा वलन" D.

• पृथक् A . ८ "संयुक्त A; र is missing from D. * *नते A; °संख्यनतौ D.P.
• रुणद्धि हि A. ११ रवेबिम्बादु P. १* ध्वान्तविच्छवासदीधितेः A. षष्ठोऽध्यायः

विक्षेपज्यां क्षपाभर्तुरक्षज्याक्षुण्णविग्रहाम्' । लम्बकेन हरेल्लब्धं विशोध्यं तत्स्फुटेन्दुतः ॥ १ ॥ उदये सौम्यविक्षेपे देयमस्तमये सदा४ । व्यस्तं तद्याम्यविक्षेपे कार्य" स्यादुदयास्तयो: ।। २ ।। त्रिराश्यूनोत्क्रमक्षुण्णां तत्कालक्षिप्तिमाहताम् । क्रान्त्या परमया भूयो हरेद् व्यासदलस्य ताम् ।। ३ ।। कृत्या लब्धकलाः शोध्या विक्षेपायनयोर्दिशो: । तुल्ययोव्र्यत्यये' क्षेप्यं ** शीतांशोस्तत्फलं** सदा ।। ४ ।। एवं कर्मक्रमात् सिद्धो दृश्यतेऽन्तरितः शशी । भागैद्वादशभिः सूर्याद् व्यश्रे' * नभसि निर्मले ।। ५ ।। अन्तरांशोत्क्रमां जीवां** स्फुटेन्दुव्यासताडिताम् । षण्णगाष्टरसैहृत्वा* * सितमानं " पदाधिके ॥ ६ ॥ क्रमज्यामधिकोत्पन्नां त्रिज्या योज्य तत् सितम् । आनयेदसितेऽप्येवमुत्क्रमक्रमतोऽसितम् ।। ७ ।। अन्तरालासुभिः कार्यश्चन्द्रभूज्याचरासुभिः ** । शङ्कुः शङ्क्वग्रमप्यस्मात् साध्यते नित्यदक्षिणम् ।। ८ ।। क्षेपक्रान्तिधनुषोभिन्नतुल्यस्वदिग्वशा ' । विश्लेषयोगजा जीवा सेन्दोः क्रान्तिस्ततः स्फुटा '* ॥ ६ ॥ १ क्षेपभक्तामक्षज्यांक्षुण्ण० A; विग्रहम् (C. २ उभये A: उदय D . 3 उदयास्त मये D. ४ यदा B. ५ कार्य: A. ६ स्यादुभयास्तयो: B ७०क्षिप्तमा० B. ८०पान नयो० A; ०पायनयार्दिशो: D . * तुल्ययो व्यत्ययो D. १० रक्षव्यं D; क्षेप्य P. ११ शीतांशौतत्फ* A, B. १२ तुर्याद्व्यभ्र' A; सूर्याद्व्यभ्ये B. ** अनुरांशोत्क्रमा जीवा A; ०त्क्रमाज्जीवां D, P . *४ षण्णवाष्ट* A; षण्णागाष्टनुसैर्हत्वा D; षण्नगाष्ट० P १५ सितमाना A; सितं मानं P १०८ . १६ कार्याश्च" A. १७ °तुल्यस्यदि* B. क्रान्तिः स्फुटामता A, C, D, P. षष्ठोऽध्यायः] २१ स्वाहोरात्रादयः साध्या व्यासार्धाभिहतां हरेत् । लम्बकेन शशिक्रान्तिमिन्द्वग्रं तत्र' लभ्यते ॥ १० ॥

शङ्क्वग्रतुल्यदिक्त्वे * स्याद्युक्तं" विश्लिष्टमन्यथा अकग्रा तद्विशेषः स्यात्तुल्यदिक्त्वेऽन्यथां* युतिः ॥ ११ ॥ एवं सिद्धो भवेद् बाहुरकर्कात् सम्यक्प्रसार्यते । कोटिसूत्रं तदग्रोत्थमत्स्यपुच्छास्यनिःसृतम् ॥ १२ ॥ चन्द्रशङ्कुमिता' कोटिः* पूर्वतो* नीयते स्फुटम्'* । तद्भुजामस्तकासक्तं कर्णसूत्रं विनिर्गतम् ॥ १३ ॥ कर्णकोटयग्रसम्पातकेन्द्रेणालिख्यते शशी । कर्णानुसारतस्तस्य * 3 सितमन्तः प्रवेश्यते ।। १४ ।। कर्णः* पूर्वापरे तन्मत्स्याद्दक्षिणोत्तरे काष्ठ दक्षिणोत्तरयोर्विन्दू तृतीयः सितमानज * ।। १५ ।। त्रिशर्कराविधानोत्थमत्स्यद्वयविनिःसृतम् । विन्दुत्रयशिरोग्राहिवत्र्मवृत्तं** समालिखेत् ।। १६ ।। वत्तान्तरसितोद्भासिश्रङगोन्नत्या१८ प्रदश्यते* * । ज्योत्स्नाप्रसरनिर्धतध्वान्तरराशिनिशाकर * ॥ १७ ॥ प्राक्कपाले* शशाङ्कस्य लग्नेन्द्वग्रादिभिः* स्फुट * । साध्यो बाहुरनादिष्टमपराभिमुखं स्मृतम्* ।। १८ ।। मण्डलार्धयुतार्केन्दुविवरोत्पन्ननाडिका * । कृताविशेषकर्माणो दृश्यकाल :* सिते स्फुट :* ।। १८ ।।

• व्या is missing from D; "सार्धान्निहतां P. * * मिन्द्रग्रा B. 3 चात्र A .

४शङ क्वग्रतुल्या दिक्त्वत्र A; "तुल्यदिक्के B. * युक्त A. * विशिष्ट* D. ७ °दिक्के ऽन्यथा B; तद्विशेष्या तुल्य- D. ८ १० ग्रोत्थंम* B. P ; "पुच्छस्यनिस्सृतः A. * चन्द्रशङ्कु मत: A. १० कोटि B. ११ पर्वतो B . १२ स्फुटा B. १३ °तस्तस्या A. *४ कर्णात् A; कर्ण B; कर्ण P. १५ तन्मध्याद्दक्षि* A; तन्मत्स्यात्सौम्यदक्षिणे (C. १६ योबिन्दु स्तृतीयास्सितमानजाः A. १७ *"पद्मवृत्तं B. *** *न्तरे सितोद्भानि ग्रहोन्नत्या A. १ * प्रद श्र्यतेि (C. २० *प्रवाहनि° A; "प्रकरनि° B; "प्रसारनि" (C. ३१ प्रोक्तपाले A. २२ लग्ने न्द्वकर्कादि* B; लग्नेन्द्वग्रादिति P. * स्फुटम् A, B, D; स्फुटा P . २४ स्फुटम् A २५ oनालिकाः B. २६ "कालाः A, B; दृश्यंकाले P. ३७ स्फुटम् A, P . अर्कोन्दुसमलिप्तात्वे' पौर्णमास्यां समोदयः । प्रागेवाभ्युदितो हीनः पश्चादभ्यधिको रवेः ॥ २० ॥ ऊनाधिककलाक्षुण्णास्तद्ग्रहेष्टासवो' हृताः । राशिलिप्तासमूहेन लब्धः कालो विशेषितः' ।। २१ ।। उदयेन्द्वन्तरप्राणैरस्तचन्द्रान्तरैरपि । स्वाहोरात्रादिभिश्चान्दैः शङ्कुदृग्ज्ये * ततः प्रभा* ॥ २२ ॥ दिनान्तोदयलग्नस्य गन्तव्या' लिप्तिकाहता ' स्वभोदयासुभिर्लब्धाः* प्राणराशिकलाहृताः'* ॥ २३ ॥ सम्पिण्डय** शशिनो* यावद्भुक्तलिप्तावधेरिति" । स्फुटभोगानुपाताप्तमिन्दोः क्षिप्त्वाऽविशेषयेत् ' ।। २४ ।। अविशिष्टेन कालेन शर्वर्या दृश्यतेऽसिते*७ । विध्वस्तध्वान्तसंघातधामराशिनिशाकर:* ॥ २५ ॥ इति लघुभास्करीये षष्ठोऽध्यायः । लघुभाकरीये

• "लिप्तत्वे B. ३ रविः A. C.

ॐ ऊनाधिक“लाक्षु" B; "स्तत्कालेष्टॉसंव ४ कार्यो b. ५ विशेषतः 10 . ९ शङ कुं दृश्येत D. ७ तत् D . ८ प्रभा: P.

• शङ्कोर्वा A. '*** *हता A. *** लब्धा C. १ ३ प्राणाराशि० P. १ उ संपिण्ड्याः P .

१४ म्पिण्ड्यशा is missing from D . १५-भुक्तिलि• D. १ ** ०पाताप्तामिन्दोः कृत्वा वि" A; ‘मिन्दौ क्षि* C, P. *७ ते शशी A; “ते सिते B, C, D, P. १८ घात**** मराशि A. सप्तमोऽध्यायः कृतदर्शनसंस्कारो भार्गवोऽकन्तरस्थितैः । अंशकैर्नवभिस्तेभ्यो* द्वयधिकैर्द्धयधिकैः क्रमात् ।। १ ।। दृश्यन्ते सूरिवित्सौरिमाहेया' निर्मलेऽम्बरे' । कालभागा दिगभ्यस्ता विज्ञेयास्ता विनाडिकाः ।। २ ।। राशेस्तस्यैव पूर्वस्यां७ सप्तमस्यापरोदये । स्वदेशभोदयै:८ कालं ज्ञात्वा दर्शनमादिशेत् ॥ ३ ॥ इष्टग्रहान्तरं भाज्यं प्रतिलोमानुलोमगम्' । भुक्तियोगविशेषेण दिनादिस्तत्र लभ्यते ॥ ४ ॥ स्फुटभुक्त्यानुपाताप्तफलेनासन्नयोगिनाम्' । ग्रहाणां शुद्धिकल्पाभ्यां* कुर्यात् समकलावुभौ ।। ५ ।। पातभागास्ततः शोध्याः शीघ्रोच्चात सितसौम्ययोः कृतद्वयष्टर्तुककुभो दिग्गुणास्ते कुजादितः * 1॥ ६ ।॥ नवार्कत्र्वर्करवयो** दशघ्नाः क्षिप्तिलिप्तिका:*४ । पातांशोनभुजामौवसङ्गुणाः** सौम्यदक्षिणाः ॥ ७ ॥ विष्कम्भार्धहृतो घातो'* मन्दशीघ्रोच्चकर्णयोः । भूताराग्रहविवरं भागहारः ** प्रकीर्तितः* ८ ।। ८ ।। विक्षेपलिप्तिका लब्धास्ताभिरन्तरमिष्टयो:** । एकदिक्त्वे* ि वशिष्टाभिर्यक्ताभिभिन्नदिक्कयो :* ॥ ६ ॥

• स्थितः A. • २ ०भिदृश्यो A. ॐ द्वयधिकैः is missing from D. * सूरि

वत्सौरि A,B,D. * निर्मलाम्बरे A, D. ६ १०यास्ते C. ७ पूर्वस्याः A . ८ सन्देश भोदयै: A. * ०नुलोमकम् C . *** "क्त्यासपाताप्ताफले० A . ११ * कल्प्याम्यां P .

• * कुजादिकाः B. १ 3 नवार्कचर्क० A; नवार्कत्वरवयो B. *४क्षेपलिप्तिकाः A; क्षिति

लिप्तिकाः B.P. १५ * "मौर्वीसौगुणा B. *** घातो: B. १७ “हाराः A. *** *र्तिताः A.

• लब्धास्माभि० (C; लब्धास्ताराभि* D. - *** एकदिक्के B. १ ०भिन्नदिक्स्थयोः A. २४

चतुर्भागाङ्गुला" लिप्ता ग्रहयोरन्तरं स्फुटम् । वर्णरश्मिप्रभायोगादूह्यमन्यत् स्वया धिया ।। १० ।। इति लघुभास्करीये सप्तमोऽध्यायः । ।

• *. 3 “

गाङगुले D. * लिप्तं A दूह्य' वान्यत् B [लघुभास्करीये अष्टमोऽध्यायः अष्टावष्टादश दिशो मनवोऽक*ि द्वयोर्घनः । द्वाविंशतिश्च विश्वे च नव शक्रास्त्रयोदश ।। १ ।। विश्वे विंशतिरेकोना४ द्वादशाक दिनानि च' । दिशो रसाश्च विश्वे च विश्वे सूर्या धृतिस्तथा । ॥ २ ॥ रुद्राः सूर्यास्त्रिसप्ताथ शैलेन्दुतिथयस्तथा । पूर्वपूर्वयुता' ज्ञेया योगभागा यथोदिताः ।। ३ ।। आप्यवैष्णवमूलानां* पित्र्यवासवयोरपि* । त्रिशल्लिप्ता: संयाम्यानां क्षेप्या वैश्वस्य १० शेषत: ।। ४ ।। योगभागसमः सर्वः संयुक्तो** लक्ष्यते ग्रहः । अधिकोनकलाकालविज्ञानं चानुपातत ।। ५ ।। उदग्दिशोऽर्कभूतानि** याम्ये पञ्च दिशो* भवाः** । उदग्रसास्तथा व्योम दक्षिणे मुनयोऽम्बरम् ।। ६ ।। उदगकर्कास्तथा विश्वे दक्षिणे मुनयोऽश्विनौ । सौम्ये रसकृतिः सैका याम्ये साधस्तथाग्नयः*** ।। ७ ।। अब्धयो वसवः सार्धाः सप्तशैलास्ततः परम् । उदक् त्रिंशत् कृतिः षण्णां याम्ये लिप्तोस्त्रिषटकका ।। ८ ।। उदगकश्च विश्वै च द्विरभ्यस्ता नभस्तथा । विक्षेपांशाः क्रमाद् दृष्टाः पण्डितैर्वाजिभादितः ।। ६ ।। यावत्या' यद्दिशाक्षिप्त्या*७ यावांस्तारासमागमः१८ ।। तावत्या' तद्दिशाक्षिप्त्या* तावानिन्दुः समो** भवेत् ॥१०॥

• Missing from D. * दैिशो: A; दिशा P. 3 मनवोक्ता A. ४विंशतिर

कॉना B. ५०शाकर्कास्त्रिपञ्चकाः A, C. * ०तिथयः क्रमात् A, G. ७ पूर्वभागयुता A.

• *वैणवमू० B . * विश्ववासव० A. १ • विश्वस्य A. ११ सम्यक्तौ A. १ * उदग्दि

गर्कभूतानि A, B. १३ याम्येऽपदिशो P. १४ द्भवाः B, C. १५ सार्ध तथाग्नयः A;

• थानय: B. १६ यावन्त्या A. १ ७ यद्दिशाप्त्या A. १०८ १०समागमे B. १९ * तावन्त्या A.
• तद्दिशाक्षिप्या A . * *निन्दुसमो A . [लघुभास्करीये

अष्टिर्दशगुणा लिप्ता विक्षेपस्य यदोत्तरे' । निरुणद्धि तदा* व्यक्तं कृत्तिकातारकां शशी ।। ११ ।। उत्तरां परमां क्षिप्ति गत्वा शिशिरदीधितिः । आवृणोति स्वबिम्बेन मघामध्यस्थतारकाम्' ।। १२ ।। आरोहति शशी षष्टया प्राजेशशकटं " स्फुटम् । अष्टिवर्गेण याम्यायां योगतारा विलिख्यते ॥१३॥ याम्यगं पञ्चहीनेन शतेन त्वाष्ट्रतारकम् । मैत्रं शतेन साधेन द्विशत्या' शक्रतारकाम्' ।। १४ ।। सप्ताशीत्या शशी हन्ति तारां सौम्यविशाखयोः* । याम्यगो दक्षिणाशास्थो" ३ व्यक्तं* शतभिषग्जिनैः*४ ॥ १५॥ पुष्यं पौष्णं च पातस्थो निरुणद्धि निशाकरः । यष्टियुक्तकलाक्षिप्त्या " भेदः स्याद् ग्रहधिष्ण्ययोः ।। १६ ।। शेषौ मण्डलजौ१ ६ यमक्षितिजयोः संयूक्तविश्लेषिता वन्यान्याहृतावग्रहा च पददौ' ७ रूपेण संयोजितौ । एवं** साधु विचिन्त्य वर्गविधिना द्वित्रिक्रमाद्वत्सरं संगण्या* द्युगणार्कजक्षितिसुताः* कालेन कालोद्भवाः॥ १७ ॥ लिप्ताशेषः कुजस्य द्विकधनगुणितो* मूलदो रूपयुक्तः सप्ताभ्यस्तः सरूपः पुनरपि पददो**वर्गरराशिः स एव । इत्थं शेषं विचिन्त्य क्षितिजदिनगणौ २४ वेत्ति यो वर्षपूगैः स स्यादम्भोधिकाञ्च्यां गणितपटधियामग्रगामी धरायाम् ।। -॥ १ यदोत्तरम् . ३ यदा A; तथा B. A ॐ व्यक्ति A . C. ४ मखामध्य° ५ प्तांजेशशकटं B. . ७ याम्यग: A; याम्याः B. ८ त्वाष्टितार° A; ६ अष्टवर्गेण A, P “तारकम् . P * . * शक्रदैवतम्A; शाक्रसुप्रभाम् B; शक्रदेवताः । . विंशत्या A, C, P ११ तारकां तु विशाख० P. १२ °णाशास्थौ A . १३ व्यक्तां A. १४ शाखाभिषज्जनै: A. १६ मण्डलजो १५ दृष्टियुक्त्याकला० A; दृष्टियुक्त• C, P. A. १७ पटदौ 13

• ८ इत्थं A, C. १* "माद्वासरैः P. * संगुण्या B, P. * द्विगुणा" B; द्विगुणार्कजक्षितिसुतं

P. २३ द्विघन• B. ३3 पुनरखिलपदो B. २४ ०नगणो B; •नगणान् (C . अष्टमोऽध्यायः] विस्तारग्रन्थभीरूणां ग्रहसद्वत्र्मवित्तये । निबन्धः कर्मणां प्रोक्तो भास्करेण समासतः ।। १६ ।।' इति लघुभास्करीये अष्टमोऽध्यायः ।

• This verse is missing from (C.

। २७ अंश 1. २२; i. १९; wi. ६; wi. ७ अंशक vii. १. अक्ष 1. २५; 11. ३४; 1V. १६ अक्षगुण 111. ३ अक्षजीवा i. १६ अक्षज्या wi. १ अक्षस्य वलनम्, iv. १६ अगत i. १, २५,.२७ अग्नि 1.४, ५, ७, ९, १२, १३, १४ ; 11. १६; 11. ५; iv. २; wili. ७ अग्र i. १; wi. १०, १२, १४ अङ्ग . ९-११, १४ अङगुल iv. २२, २३, २९; wi. १० अद्यतन 11. ४१ अद्रि 1. ४, ९, १३, १४; i. ५ अधिमास 1. ६ अनुपात i. २८; v. २’ v1 २४ ! ५; wi. ५ लघुभास्करीये प्रयुक्तपारिभाषिक शब्दानाम्. अनुक्रमणिका अन्तराल iv: १५; wi.८ अपुत्रक्रसi॥ १६, i.४८ अपक्रान्तिचापधांis २७८ः अषम-i. १६, १७; iv. १७; v: ५” बखमोगुण अपर iv. २४; v. ९; wi. ३ अपरा iv. २५; vi. १५, १८ अपराह्न V. ४ अब्धि i. ५; wi. ८ अभुक्तांश i. २० अभ्यासःi. १५; i. १६ अमृततेजस्'i. २२ अम्बर iः ७, १३;ः viii. ६ . अम्भोधि 1. १० अयन 1V. १७; v1. ४ अयुत 1. ५, ९ अर्क 1. ३७ ; 11. १६, २६; 11. ११, २१, २२; iv. ६, १७; v.१३; vi.११, १२, १९, २०; wi. १, ७; wi. १, २, ६, ७, ९ अर्कज wiii. १७ अकेसुत 11. ३७ अर्काग्रा iii. २१; wi. ११ अकॉदय 1. १६ अर्धरात्र i. २० • अक्नति v. १२-१४ अविशिष्ट iv. १३; vi. २५ wi. १९ अविशेषकलाकर्ण v. ३ 11. ७ अश्विi.७ , १०, १२, १४, २४; i. २७ अनुक्रमणिका] अष्टि wiii. ११ असित 11. २८; vi ७ असु . ११, १७, १८; 1v. १५; v. २; 111 अस्त 1. ३५; 1२०; wi. २, २२ i. अस्तमय vi. २ अस्तोदयाग्ररेखा iii. १६ अहन् 11. १५ अहमण 1. ८, १५ अहोरात्र 11. १८; wi. १०, २२ अहोरात्रदल . ८ 11 अहोरात्रासु i. २३ अहोरात्रार्ध . १४ 11 अहोरात्रार्ध-विष्कम्भ i. १७ आदित्य i. २४ आप्य viii. ४ आशा 1. २८ इन्दु 1. ५, ७,१४; i. ११, १३, १४, २२ , २४, २९; i. ५; iv. ४, ८, ९, १७, २२; v. १३; vi . १, ६, ९, १०, १९, २०, २२, २४; wi. ३ , इन्दूच्च 1. १२ इन्द्वग्र vi. १०, १८ इषु i. ७; iv. २; v. ११ इष्ट i. १६, १७; i. १, ११, १७ iv. ३१, ३२; vi. २१; vi. ४, ९ इष्टकाल jv. ३२ इष्टग्रह wi.४ इण्टासु i. ११, १७; wi. २१ उच्च i. १ उत्क्रम 1i. ११; wi. ३, ७ उत्क्रमजीवा wi. ६ उत्क्रमज्या iv. १५, १ उत्क्रमज्याफल iv. १५ उत्क्रमभवा जीवा i. ३१ उत्तर i. १; wi. १५; wi. ११, १२ उदक् i. २२, ३०, ३५; iv. १६; wii . ६-९ उदग्गोल i. २०, २१; i. ८, ३३ उदय 1. ८; i. १३, २०, २६; i. १३ v1. २, २०, २२; wi. ३ उपप्लुति i. ३० उष्णदीधिति i. २१ ऋतु v1. ६, ७ ऐन्द्रीiv. २७ ओज i.२, ११, ३१ 11 करण 1i. २८ २९ कर्ण 1. २६, २७; i. ६-८, ३४; iv. २, ६, ८; wi. १४, १५ कर्णभुक्ति i. १५ कर्णसूत्र vi. १३ कलाi. १५; i. ११; i. २४; iv. ७; v. १५; vi.४, २१; wi. ५, १६ कल्प wi. ५ कार्मुक i. ३४ काल i. २, २९, ३७; . २०; iv. १७; 11 v. ७, ९, १०, १२; wi. १९, २१, २५; wi. ३; wi. ५, १७ कालभाग viां. २ काष्ठा wi. १५ कु 11. २ कुज i. १०, १८; i. ३७; wi. ६; wi. कृति i. १७; i. १०, १२, २३, २९ ; iv. १०, ३१; v. ८; vi. ४ कृत्तिका wi. ११ केन्द्र . १, ४, १०, ३०; , 1iiv. २४, २५ २८, ३२; wi. १४ केन्द्रभुक्ति . ११, १२ 11 कोटि i. १, २, ३०; iv. १७; wi. १३, १४ कोटिफल i. ३ 11. ६ कोटिसूत्र vi. १२ ऋक्रम 11. ११; W1. ७ ऋक्रमज्या wi. ७ क्रमभवा जीवा i. ३१ क्रान्ति ३४ः i. २१, २२, २४, ३०, ३२, w. १; wi. ३, ९, १ क्रिय 11. ४ क्षपाभतृ wi. १ क्षय i. १३, २२, २४, ३३; i. १३ ; क्षितिज wiii. १७, १८ क्षितिज्या i. १८; i. १३ क्षितिसुतं wi. १७ क्षिप्ति iv. २७; wi. ३; vi. १०, १२, क्षिप्तिलिप्तिकाः'wi. ७ क्षेप v. १९; v. ५ ख 1. ७, १२; i. १९; iv. ८; v. ११ गणित 1. २९, ३०; vi. १८ गणित प्रक्रिया i. २९ गत i. १, १२, २५-२७; i. ७, १५, गति i. १, ३, ३६; i. ३२ गन्तव्य i. १२, २६; i. ७, १५; v. ४; v1. २३ गुण i. ११, १३, १६; i. १९, २७; iv. १७; wi. ६; wi. ११ गुरु 1. १ गो 1i. ४ गोल . २१; i. ८, १४ 1i ग्रह i. १७, ३५, ३६; i. ३०, ३९, ४०; iv. १३, १९, २६; v. ९; vi. २१; wi. ५, १०; wi. ५, १६ ग्रहण 1. २९ ग्रहमध्य 1v. २१ ग्रहसद्वत्र्म wi. १९ ग्रास jv. ३२ ग्राहक iv. १४, २३, २९, ३०, ३२ iv. ३२ ग्राह्य iv. १४, २३, २९ ग्राह्मबिम्ब iv. २९ ग्राह्ममण्डल iv. २६ घटिका 1i. ७ घटी i. १२ घन i. १९; wiii. १८ [लघुभास्करीये ' चक्र 1. १७; i. २९ चक्रलिप्ता i. ४, २२ चक्रार्ध i. १८; i. २९; i. ३२-३३ चक्रांशक 1. २५ चन्द्र i. २४; iv. ७, ११, १२; wi. ८, १३, २२ चन्द्रमस् v. १९ चन्द्रशङ्कु v1. १३ चर i. ११ 11. १८ अनुक्रमणिका] चरप्राण i. २३; i. ६, १४ चरासु i. २१; i. ४, ७; wi. ८ 11. ३८ चलकेन्द्रफल 11. ३८ चलोच्च i. ३९ चाप 11. १९, ३३, ३४, ३८ ; i. १४, २७, ३२, ३३ चापभाग 11. १४ चापित i. ३६ 11. ४१ छाया i. १, ३, १०, १२, २५, २७-३०, ३४, ३५; 1v. ६ 17. ७ छायाविधान iii. २४ छेद 1. ७; i. १२, २९ जलधि i. १९ जिन wiii. १५ जीवा i. २, १४; i. ८, ११, २४, ३२; ीवाभुक्ति i. १५ ज्या 1i. २६; iv. १८ ज्योतिस् 1. १, ३ तमोमूर्ति i. १७ तमोव्यास iv. २२ तारका wi. १२, १४ तिग्मरश्मि 1. २८ तिग्मांशु i. ५; i. २५ तिथि i. ३४, ३६, ३७; i. २७; iv. ४, १३, १५; v. २; viii. ३ तिथिवर्ग i. २६ तिथ्यर्धहार i. २८ तिर्यक 1. २६ तुला i. ६ ३१ तुल्यत्व 1. २८ तुल्यदिक् iv. २७; v. १२; wi. ९ तुल्यदिक्त्व v. ५; wi. ११ त्रिज्या i. १६; i. ३, ९, २८, ३२; iv. ३,१८; v. १; wi. ७ त्रिमौर्वी i. १२; iv. १५ त्रिराशि vi. ३ त्रिराशिजीवा i. २९ त्रिशर्कराविधान wi. १६ त्रैराशिक i. १२ त्वाष्ट्र viii. १४ दक्षिण 1ii . ८, १३, १५, १६, ३०, ३१, ३३; iv. ८, १६, २४;"v. ४; wi. ८, १५; wi. ७; vii. ६, ७ दक्षिणाशा wi. १५ दर्शन-सस्कार v1. १ दस्र i. ९, १०, १३, १४, १८ दिक् i. १, २; iv. १५, १७, २१; v. ५, १२; wi. ९; wi. २, ६; wi. दिक्क v. १२; wi. ९ दिन vi. ४; wi. २ दिनगण viii. १८ दिनपूर्वापरार्ध i. ७, १५ दिनान्तोदयलग्न wi. २३ दिनार्ध i. २८ दिशु i. १, २; iv. ४, १८, २४, २५ , २८; wi. ४; wi1. १, २ दिशा viii. १० दृक्क्षप v. ७ दृक्क्षेपज्या v. ११ दश्यकाल wi. १९ देशान्तर i. २३, २६, २९, ३१, ३२, ३४ ; ३७ देशान्तर-घटी i. ३१, ३२, ३४ द्युगण viii. १७ द्युचारिन् vi. ३१ द्रष्टा 1. ३० घन 1. ३१, ३६, ३७; i. ४, २२, २४ , ३०, ३३, ३४, ३८ धनुः i. १२; i. २४, २६, ३०, ३४ धनुर्भाग i. २ धनुस् i. ३, ८; iv. १९; v. ५; wi. ९ धिष्ण्य vi. १६ घृति viii. २ नग v1. ६ नतभाग i. ३०, ३१, ३५ नति i. २९; v. १५ 11. २७ -नाडिका i. ३५; i. १५; iv. १०, १२, १४; v. ९, १३; vi. १९ नाडी i. २६; iv. १, ११ निरक्षजा असवःi. ५ निशा i. २१ निशाकर 1. १२; iv. १; v. ८; wi. १७ , २५; v1. १६ निशाकृत् i. ९; i. ५, २२, २३; iv. २, निशानाथ i. २५, २६ पद i. १, ११, १७, ३१; i. १०, २३, २८, २९; iv. १०, १४, ३१; wi. ६; पद्मिनीबन्धु i. १ परमक्रान्ति 11. २१, २४, ३२; v. १: परमक्षिप्ति viii. १२ परमापक्रम 11. १६ परमापक्रमो गुणः i. १६ परिधि i.२१, २२, २७; i. ३ पर्यय 1. १३, १५ पर्व iv. १; w7. ९, १० पर्वत 11० २७; iv. २. पर्वमध्य iv. १३ पर्वनाडी iv. १ पल 1. २८; i. २७, ३०, ३१, ३५ पलज्या i. १७; i. २२, २४; v. ५ पश्चार्ध iv. १७ पश्चिम i. ३०, ३१, ३५; iv. १६ पात 1. १३; iv. ८; v. ४; vi. ६, ७: पातभाग vi. ६ पित्र्य viii. ४ पुष्कर 1. ९ [लघुभास्करीयै पूर्वाह्न v. ३ पूर्व . ३०; 1V. १६, १७, २५; v1. १३, पौष्ण vi. १६ प्रकृति 1. १२ प्रक्षप 1. ३४ अनुक्रमणिका] प्रक्रिया 1. १, २९, ३६; i. ३७ प्रग्रास iv. २६, ३०; v. १४ प्रतिपद् iv. १ प्रभा i. १०, २३; wi. २२ प्राक्कपाल iv. १६; wi. १८ प्राग्विलग्न iii. १९, २०; v. ६ प्राची 1. ३१, ३५ प्राजेशशकट wi. १३ प्राण 11. १९; i. ७, १८, २०; wi. २२, २३ प्राह्य v. ९ फल 11. ६, ७, १२, २३, २५, ३०, ३२ , ३६, ३८ ; 11. ३, ९; vi. ४: बव 11. २८ बाहु 1. २५, २६; i. २, ६; wi. १२, १८ बाहुपफल 11. ३४ बिन्दु iv. २८, ३०; v1. १५, १६ बिम्ब iv. १६, १७, २९; v. १३, १५; बुध 1. १२; 11. ३७ भ i. ६; wi. ३ भगण 1. ११ भव wi. ६ भाग 1. १५, १८; i. ५, १४, १९ ; iv. १६; v. ३; vi. ५; vi. ६ भागहार 1. ५, १६; wi. ८ भाज्य 11. २५; wi. ४ भानु i. ९; i. १३, २७, ३५ भार्गव 1. ११, १३; i. ३७; vii. १ भास्कर 1. १, ८, १३, १६, २२; i. २४ , 1w. २२ भास्वत् iां. २२ भिन्नदिक wi. ९ भुक्त wi. २४ भुक्ति i. ३१, ३३; i. ८, ९, ११, १४ २२, २७ भुक्तियोग vi. ४ wi. ४ 11. २७ भुजज्या i.२१ भुजा 1. १, २, १६, ३०; v. ६; vi. १३; wi. ७ भुजाफल i. ३, ४, ५, २२ भुजामौर्वी wi. ७ भू 1. २७, ३३; iv. ६ भूज्या i. ९; wi. ८ भूत 1. ५, ११; i. ५; iv. ४; v. ११ : 11. भूताराग्रहविवर vi. ८ भूदिन i. १४, १५ भूमि i. २४, २५, ३२; iv. ७ भूमिव्यास iv. ७ भूमेः वृत्तम् i. २४, २५ ३३ भेद wi. १६ भोग 1. ३४; i. ४, १३ मकर 1i. १० मघा wi. १२ मघामध्यस्थतारकम् viii. १२ मण्डल i. १; wi. १७ मण्डलमध्य 11. १ 11. २५ मत्स्य i. १; iv. २४, २५, ३०; wi. १२, १५, १६ मध्य 1. १७, ३१; i. ८, १३, २२, ३३; ३४ 1i. १; iv. १५, ३०; v. २, ९ मध्यच्छाया 1. २८ ; i. २९ मध्यजीवा v. ६ मध्यभुक्ति 11. १ मध्यम 1. १३, ३३; i. १, ९, ३५; ३९ भध्यमा भुक्तिः . ३३; i. ९ मध्यलग्न V. ३, ५ मध्या भुक्तिः i. ३१; i. ८, २२ मध्याह्न 11. २० मध्याह्नच्छाया 11. २ मनु v1. १ मन्द 1. १९, २२; i. ३९ 11. ३८ मन्दोच्चफल 11. ३३ मन्दांश 1. २२ मास 1. ४ 11. ८ माहेय vi. २ मुनि i. ५; viii. ६, ७ मूल 1. २६; 1i. ६; 11. ३, १२; v. मृग 11. ६ मेदिनी iv. ६ मेष i. ३४; i. ४-६ मैत्र i. २९; wiii. १४ मोक्ष iv. ११, १३, २६, ३०; v. १४ मौर्वी i. ३०; wi. ७ यम 1. १०, १३, १४; viii. १७ यमल 1. १४ यात 1. ६ याम्य 1i. २०, २१; 11. १, २७; vi. २; wi. ४, ६, ७, ८, १३-१५ याम्योत्तर iii. २७, ३४ युगाधिक 1. ४ युगावम 1. ६ युग्म 1. १९, २१; i. २, ११, ३१ युति iv. ३२ योग i. १, २७, ३०, ३४, ३५; iv. १८; [लघुभास्करीये योगतारा viii. १३ योगभाग W111. ३, ५ योजन 1. ३२, ३३ योजनकर्ण iv. ३, ५ योजनव्यास v. ५ रन्ध्र 1. १२, १४, २४; i. १६; i. ५; 1V. २; V. ८ रवि i. १४, २२, २३; i. १८, २०, ३०, ३२; iv. १, ४, २७; v. २, 8, १२, १५; V1. २०; vii. ७ रस 11 ७, १०; 11. ५; wi. ६; V111 . २, ६ राजपुत्र 1. ११ राम 1. ५, ९, १०; iv. २ राशि i. १५, १७; i. १; i. ४, १८ ; राशिकला wi. २३ राशित्रय i. १ राशिशेष i. १७, १८ रूप 11. १६, २८; i. ५; v. ८; . 7111 १७, १ रेखा 1. २३, ३१ लङ का 1. २३ लङ कोदय v. २ लम्बक 1. २४; i. १८; i. ९, १३, १६, २१; v. १; wi. १, १ अनुक्रमणिका] 11. ३ लम्बन 1. ३७; v. ९, १०, १४ लिप्ता iv. १, ५, ९, ११, २२; v. ३; vi. २४; wi. १०; viii. ४, ८, लिप्ताशेष wi. १८ लिप्तिका i. ५, १९, २३; i. १९; iv. १२; v. ११; vi. २३ वक्रत्व 1. २७ वक्रभोग i. ४१ वत्सर wiii. १७ वर्ग , 1. २६; 11. ६, १७; 11. २, ३, १० २८, २९;iv. १०, १४; v. ७; wi. १३, १७ वर्गविधि wiii. १७ वर्गराशि viii. १८ वर्तमान i. ३ 11. ११ वतमानग्रह 1i. ४० 11. १९ वत्र्म v. ३०, ३२ वत्र्मवृत्त v1. १६ वर्ष wi. १८ वषपूग V1. १८ वलन iv. १६, १८-२०, २२, २४, २७ वल्लकाभृत् 1i. ४ वसु i. ५, १०, १२, १३; iv. २ वह्नि 1. २४ वार 1. ८ वारुणी iv. २७ वासव viii. ४ विक्षिप्ति iv. २१ विक्षेप iv. ८-१०, , , , २८, १२१४२२ ३१; v. ४, १२; wi. २, ४, ९; v11. ११ 71. १ विक्षेपलिप्तिका wi. ९ विक्षेपांश viii. ९ वित् vi. २ विदिक्क iv. १९; v. ५ विधि i. ३४; i. ७, ८; i. ३३ विनाडिका vi. २ विमर्दार्ध iv. १४ वियत् 1. ७ विलिप्ता 1. १७ विलिप्तिका i. ५ विवर vi. १९ विवस्वत् 11. २०; 11. १७, ३१; v. ४ विशाखा wiii. १५ विश्लेष i. २५, २६, २९; i. १५, १७; 11. १०, २७, ३४, ३५; iv. १४; v. ५; v1. & विश्व vi. १, २, ७, ९ ३५ विषुवद्दिन ji. २ विषुवद्दिनमध्याह्नच्छाया विष्कम्भार्ध i. २२, २३; iv. ५; v. विस्तृति iv. २३ वृत्त 1. २४, २५, ३२; i. ९, १३, ३१, ३२; 111. १; iv. २३, ३०; wi. १७ वेद 1. १०, ११; iv. २,४ वैश्व viii. ४ वैष्णव viii. ४ व्यतीपात 1i. २९ व्यवच्छेद v. १, ६ व्यास i. १७; i. १०; iv. ४, ६, ७; vi. व्यासदल 1i. ३१; iv. ९; vi. ३ व्यासयोजन iv. ६ व्यासार्ध 1. २४; i. ६, ७, ८, १८, ३४; iii. ९, १३, १४; vi. १० व्योम 1. ७, १४; iv.४; vi. ६ शकाब्द 1. ४ शक्र v1. १ शक्रतारकम् vi. १४ शङ कु i. १-३, ९-१३, १६, २२-२५, २८, २९.१ ३४; v. ७; v1 . ८, २२ शङ क्वग्र 1i. १६; vi. ८, ११ शतभिषक viii. १५ शनि 1. ११ शर 1. ५, ७, ९, १४ शशि i. ५; iv. १२; v. १२, १५; wi. ५, १०, १४, २४; wi. ११, १३, १५ शश्युच्च 1. १७ शिशिरदीधिति vi. १२ शीघ्र i. १३, २०; i. ३०, ३ शौघ्रकेन्द्र i. ३३ शीघ्रन्यायागतं फलम् i. ३६ शीघ्रोच्च i. ३३; wi. ६ शीघ्रोच्चकर्ण vi. ८ शीतांशु i. ३५; i. ७, ८; wi. ४ शुक्र 1. ८ शून्य 1. ७, १४; i. २७; i. ५ श्रङ्गोन्नति wi. १७ [लघुभास्करीये ३शेष 1. ८, १५; 11. ३, २४; 11. ११ . १७-१९, २६, ३१; v. ३, ७; viii. १७, १८ शैल 1. ९, १२, १९; v. २ 1. २९ संयुक्त vi. ५ सस्कृत 1. ३९ सकृत् v. १४ सङख्या i. २७; 11. २६; 1v. २९; v.१५ समकल . ५ wii समपूर्वापर i. २२ समपूर्वापरः शङ कुः i. २२ सममण्डल 11. २५ समरेखा 1. २५, २६ समलिप्तेन्दु 17. ८ सम्पक 1V. २०, २३ सम्पर्कदल iv. २३; v. १३ सम्पर्कार्घ 1V. १०, १८, २०; v. १५ 11. ८, १०; 1V. ६ सागर 1. ९, १०, १२; iv. २, ४ सायक 1. & सार्पमस्तक सावित्र 1. ११ सित i. २८; wi. ७, १४, १७ १९:

सितपक्ष i. २८ सितमान wi. ६, १५ सुराधिप i. ११, १९ सूरि i. ३७; vii. २ सूर्य i. २९; iv. १९; wi. ५; viii. अनुक्रमणिका] सौम्य i. १३, १५; iv. ८, २४; v. ४; wi. २; wi. ६, ७; . ७ wiii सौरि vi, २ स्थित्यर्ध iv. १०, १२, १३, ३१; v. १३, स्थित्यर्धनाडिका iv. १० स्थूल 1. २७ स्पर्श v१११३, २१ * . , स्फुट 1. १, २; i. १३, १५, १६, २४ , ३६, ३८, ३९, ४१; i. १०; iv. ३, ५, १३, १९, २१, २४; v. १२, १३, १४; vi. १, ६, ९, १३, १८, १९; wi. १० स्फुटग्रह i. ३९ स्फुटभुक्ति i. ८, १०, १९, २३, २६; iv स्फुटभोग vi. २४ स्फुटमध्य . ३५, ३६, ३९ 11 स्फुटयोजनकर्ण iv.३, ५ 11. ३२ स्वदेशभूमिवृत्त 1. ३२ स्वदेशभोदय i. ६; vii. ३ स्वदेशाक्ष 1. २५ स्वदेशोदय i. १७ स्वभूवृत्त 1. ३३ स्वर v. ११ हरिज iv. २२ लघुभास्करीये प्रयुक्त छन्दसाम् अनुक्रमणिका ३७ अनुष्टुभ् (श्लोक) i. १-३७; i. १-४१; i. १-३५; iv. १-३२ ; v. १-१५; wi. १-२५; wi. १-१०; wi. १-१६, १९ शार्दूलविक्रीडित wi. १७ स्रग्धरा viii. १८ English Translation OF THE LAGHU-BHASKARIYA CHAPTER I MEAN LONGITUDES OF THE PLANETS Homage to the Sun : 1. I bow to the Sun— to Him with the help of whose mo- tion this true motion of the heavenly bodies is inferred even though the methods (adopted for the purpose by different writers) be different. Homage to Aryabhata I : 2. Victorious is Aryafchata whose excellent fame has crossed the bounds of the (Indian) oceans and whose (treatise on astronomical) science leads to accurate results in far off places (even) after the lapse of so much time. Appreciation of Aryabhata I and his work : 3. None except Aryabhata has been able to know the motion of the heavenly bodies : there the others (merely) move in the ocean of utter darkness of ignorance {ajnanabahaladhvanta- sagara) . Sankaranarayana reads alafn in place of nalam and so he interprets the passage as follows : "Those who endeavour to determine the motion of the planets with the help of other astronomical works than the Aryabhatlya move in vain in the ocean of utter darkness of ignorance". The compound word ajnanabahaladhvatitasagaramzy also be interpreted to mean "the ocean of the darkness of utter ignorance". in th? T ,0giZing * r yabhata T and his work on mathematics and astronomy n the above stanzas the author has indicated the system of astronomy that he 1S going to follow in the present work. A rule for calculating the ahargana : 4-8. Add 3179 to the (number of elapsed) years of the fcaka era. (then) multiply (the resulting sum) by 12, and (then) add the (number of lunar) months (expired) since the com2 MEAN LONGITUDES OF THE PLANETS [CH. I mencement of Caitra. Set down (the result thus obtained) at (two) separate places; multiply (one) by (the number of) in- tercalary months in a juga, which are 15,93,336 in zyuga ; and divide (the product) by 5184 into 10,000 (i.e., by 5, 18,40,000). Add the (resulting complete) intercalary months to the result placed at the other place. Then multiply (that sum) by 30 and (to the product) add the (lunar) days (i.e., tithis) expired (of the cur- rent month). Set down (the result thus obtained) in two places; multiply (one) by the (number of ) omitted lunar days in ayuga, i.e., by 2,50,82,580, and divide by 1,60,30,00,080. The result- ing (complete) omitted lunar days when subtracted from the result put at the other place give the (required) ahargana. The remainder obtained on dividing (the ahargana) by 7 gives the day beginning with Friday at sunrise (at Lanka). 1 • The above rule tells us how to calculate the ahargana, i.e., the number of mean civil days elapsed at mean suniise at Lahk^. 2 on a given lunar day (titki), since the beginning ofKaliyuga. The beginning ofKaliyuga, which is taken as the starting point of the reckoning of ahargana in the above rule, occurred on Friday, February 18, B.C. 3102, at mean sunrise at Lanka, when the Sun, Moon, and the planets are supposed to have been in conjunction at the first point of the naksatra Asvini (which is a fixed point situated near the star ^-Piscium). The duration of Kaliyuga, according to Aryabhata I, is 10,80,000 solar years. Four times this (i.e., 43,20,000 solar years) is the dura- tion of a bigger period called pjyuga (or maha-yuga). The following table gives the number of lunar months, solar months, intercalary months (i.e., lunar months minus solar months), lunar days, civil days, and omitted lunar days (i.e., lunar days minus civil days) in ayuga : 1 Cf. MBh, i. 4-6; vii. 6-7. 2 Lanka is a hypothetical place on the equator where the meridian of Ujjain (long 75°52' E from Greenwich) intersects it. Months and Days in a Tuga Lunar months Solar months Intercalary months Lunar days Civil days Omitted lunar days 1,60,30,00,080 1,57,79,17,500 2,50,82,580 5,34,33,336 5,18,40,000 15,93,336 VSS. 4-8] AHARGANA 3 A lunar month is reckoned in Hindu astronomy from one conjunction of the Sun and Moon to the next. The first month of the year is called Caitra. A solar month is reckoned from the Sun's one transit into a sign to the next. A civil day is reckoned from one sunrise to the next. The Saka era referred to in the above rule started exactly 3179 solar years after the beginning of Kaliyuga. The following example will illustrate the above rule : Example. Calculate the ahargana for January 1, 1963 A D. From the Hindu Calendar we find that January 1, 1963 A.D., falls on Tuesday, the 6th lunar day (tithi), in the light half of the 10th month (Pausa), in the Saka year 1884 (elapsed). We therefore proceed as follows. Calculation : Adding 3179 to 1884, we get 5063. ... ... _ (J j Multiplying this by 12 and adding 9 (i.e., the number of lunar months elapsed since the beginning of Caitra), we get 60,765. ... (2) . ,«^nnn Plying ^ 15 ' 93 ' 336 " d dividin S the P^duct by 5 18,40,000, we get 1867 as the quotient. (The remainder is discard- • ed, as it is not needed).

• * *" * • . . (3)

Adding this number (i.e., 1857) to the previous one (i.e., 60 7651 we get 62,632: «>' u *v» .. (4) Multiplying this by 30 and adding 5 (i.e., the number of lunar days elapsed smce the beginning of the current month) to the pro- duct, we get 18,78,965. (5) 1,60,30,00,080, we get 29,400 as the quotient. (The remainder is dis- carded, as it is not needed.) / ** " ••• ... (6) iq ^n^r^ 5 th!S nUmber (i ' e -> 29 ' 40 °) from thc P™™™ one (i.e., 18,78,965) we get 18,49,565. ... _ (7) This is the required ahargana. Verification : Dividing this ahargana by seven, we get 4 as the remainder. This shows that January 1, 1963 A.D., falls on the 5th day counted with Friday, i e on 1 uesday, which is correct. Explanation : Kalifuga* 1 * ^ 8 ' VeS nUmber of solar y ears elapsed since the beginning of 4 MEAN LONGITUDES OF THE PLANETS [CH. I Result (2) gives the number of mean solar months elapsed up to the beginning of the 10th mean solar month of the current year. Result (3) gives the number of complete mean intercalary months corres- ponding to (2). Result (4) gives the number of mean lunar months elapsed up to the be- ginning of the 10th mean lunar month of the current year. Result (5) gives the number of mean lunar months up to the beginning of the 6th mean lunar day of the 10th mean lunar month of the current year. Result (6) gives the number of complete mean omitted lunar days corres- ponding to (5). Result (7) gives the number of mean civil days up to mean sunrise (at Lankaj on the 6th mean lunar day of the 10th mean lunar month of the current year. Verification shows that this is equal to the number of mean civil days up to mean sunrise (at Lanka) on the 6th lunar day of the 10th lunar month of the current year. Also see my notes on MBk, i. 4-6. The mean lunar day may, sometimes, differ from a true lunar day by one, so that the ahargana obtained by the above rule may sometimes be in excess or defect by one. To test whether the ahargatja is correct, it should be divided by seven and the remainder counted with Friday. If this leads to the day of calculation, the ahargana is correct; if that leads to the preceding day, the ahargana is in defect ; and if that leads to the succeeding day, the ahargana is in excess. When the ahargana is found to be in defect, it should be increased by one; when it is found to be in excess, it should be diminished by one. Similarly, when a true intercalary month has recently occurred prior to the given lunar month or is about to occur thereafter, the true lunar month may differ from the mean lunar month by one. When a true inter- calary month has occurred prior to the given month and the intercalary frac- tion (which is discarded) amounts to one month approximately, then the quotient denoting the complete intercalary months is increased by one. When a true intercalary month occurs shortly after the given month and the inter- calary fraction is small enough, the quotient denoting the complete inter- calary months is diminished by one. VSS. 15-1 7(i)] CALCULATION OF THE MEAN LONGITUDE 5 Revolution-numbers of the planets, etc., 1 in a period of 43,20,000 solar years (called a yuga): 9-14. (In a. yuga) the revolution-number of the Sun has been stated to be ten thousand times 432 (i.e., 43,20,000); of the Moon, 5,77,53,336; of Mars, 22,96,824; of Jupiter, 3,64,224; of Saturn, 1,46,564; of Mercury and Venus, the same as that of the Sun; of the Moon's apogee {mandocca), 4,88,219; of {the sigh- rocca of) Mercury; 1,79,37,020; and of (the sighrocca of ) Venus, 70,22,388. The mean Sun is the sighrocca of the remaining pla- nets. The revolution-number of the Moon's ascending node {pata) is 2,32,226; and the number of civil days (in a vuea) 1,57,79,17,500. 2 K S ] The fighroccas of Mercury and Venus are imaginary bodies which are supposed to revolve around the Earth with heliocentric mean angular veloci- ties of Mercury and Venus respectively, their directions from the Earth always remaining the same as those of Mercury and Venus from the Sun. It will thus be seen that the revolutions of Mars, the sighrocca of Mercury', Jupiter, the fighrocca of Venus, and Saturn, given above, are equal to the revo- lutions of Mars, Mercury, Jupiter, Venus and Saturn respectively around the Sun. Tne mean longitudes of Mars, the fighrocca of Mercury, Jupiter, the sighrocca of Venus, and Saturn, which are obtained by the rule given below are therefore, equivalent to the heliocentric mean longitudes of Mars, Mer- cury, Jupiter, Venus, and Saturn respectively. A rule for calculating the mean longitudes of the planets for mean sunrise at Lanka : 15-17(i). Divide the product of the revolution-number ot a planet and the ahargana by the (number of) civil days (in a yuga); thus are obtained the (number of) revolutions (per- formed by that planet). From the (successive) remainders multi- plied respectively by 12, 30, and 60 and divided by the same divisor (viz. the number of civil days in a yuga) are obtained the signs, degrees, and minutes, etc. (of the mean longitude of that pl anet) for (mean) sunrise (at Lanka). 3 the EaTth. 1 h ' nUmbCr ° f rcvolutions that the P ,ancls > etc , make around a Gf. MBh, vii. 1-5, 8. 1 Cf. MBh, i. 8. ° MEAN LONGITUDES OF THE PLANETS [CH. 1 (In this way should be obtained) the mean longitudes of the planets up to seconds of arc. The point from which the longitudes are measured in Hindu astronomy is the first point of the naksatra AsvinI where the Sun, Moon T and the planets are supdosed to have been situated at the beginning of Kaliyuga, the epoch of reckoning the ahargana. The naksatra AsvinI is a fixed point on the ecliptic near the star ^-Piscium. Correction to be applied to the mean longitudes of the Moon's apogee and the Moon's ascending node obtained by the previous rule : 17(ii-iv). To the (mean) longitude of the Moon's apogee (obtained by the above rule) add three signs and to that of the Moon's ascending node add six signs, and subtract (the latter result) from a circle (i.e., from 360 ). 1 These corrections are made to the longitudes of the Moon's apogee and ascending node, because in the beginning of Kaliyuga their longitudes were 3 s 0° 0' 0* and 6 s 0° 0' 0" respectively and not s 0° 0' O'as those of the Sun, Moon and the planets. The longitude of the Moon's ascending node is subtracted from 360° because the motion of the Moon's ascending node is retrograde. Positions of the apogees of the planets : 18. (The longitudes of the apogees of the planets) beginn- ing with Mars are 100 plus 18, 200 plus 10, half a circle (i.e., 180), 90 and 236 degrees respectively. 2 Dimensions of the epicycles of the planets : 19-21. The manda epicycles (of the planets beginning with Mars) are 14, 7, 7, 4, and 9 (in the beginnings of odd quad- rants) and 18, 5, 8, 2, and 13 in the (beginnings of) even quad- rants. 50 plus 3, 30 plus 1, 16, 59, and 9 have been stated to be the slghra epicycles (of the same planets) (in the beginn- ings of odd quadrants) and the same diminished respectively by 2, 2, 1, 2, and 1 are their own {slghra) epicycles in the (beginn- ings of) even quadrants. 3 1 Cf. MBh, i. 40. 1 Gf. MBh, vii. 13. •Cf. MBh, vii. 13-I6(i). VS * 23] THE HINDU PRIME MERIDIAN *j ^ The Hindu astronomers generally state the dimensions of the mania and sighra epicycles of a pia net in terms of degrees and minutes, where a degree stands for the 360th part of the planet's mean orbit and a minute for the 60th part of a degree. The author of the present work, following Xryabhata I, has stated here the dimensions of the mmda and Bghra epicycles of the planets in terms of degrees, after dividing them by 4J. This division has been evi- dently made to simplify calculation. These epicycles will be required in tfee next chapter in finding the true longitudes of the planets. 1 Position of the Sun's apogee and the epicycles of the Sun and the Moon : 22. (The longitude of) the Sun's apogee, in degrees, is 70 plus 8; his epicycle is 3, and that of the Moon 7. 2 The previous remark applies to these epicycles also. Position of the Hindu prime meridian : 23. The line which passes through Lanka, Vatsyapura, Avanti, Sthanesvara, and "the abode of the gods'* is the prime meridian. 3 Lanka in Hindu astronomy denotes the place where the meridian of Ujjain (latitude 23°11'N ) longitude 75°52'E from i Greenwich) intersects the equator. It is one of the four hypothetical cities on the equator, called Lanka, Romaka, Siddhapura and Yamakoti (or Yavakoti). Lanka is des- cribed in the Surya-siddhlnta as a great city (mahapuri) situated on an island (dvipa) to the south of Bharata-varsa (India). The island of Ceylon, which bears the name Lanka, however, is not the astronomical Lanka, as the former is about six degrees to the north of the equator. Vatsyapura is the same place as the Vatsagulma of the Maha-Bhaskariya* It may be identified with the town of Basim or Wasim (pronounced as Basim or Vasim), situated at a distance of 52 miles from the city of Akola 1 See infra, ii. 9-10, 1 1-13, etc. 2 Gf. MBh, vii. 12(i), 16. 3 Gf. MBh, ii. 1-2.

• xii. 37, 39.
• ii. 1-2. 8 MEAN LONGITUDES OF THE PLANETS [CH. I

in the state of Madhya Bharat. 1 Basim originally was the seat of hermitage of Sage Vatsa and was called Vatsa-gulma. 2 Later on it became a sacred place and grew into a town called Vatsyapuca after the name of the sage. Its present name Basim or Wasim is evidently a corrupt form of Vatsam (or Vatsapuram). Basim is now a seat of Hindu religion, famous for its sacred tank called Padma-tirtha. The above identification of Vatsyapura or Vatsagulma with Basim seems to be more plausible than our previous identification with KausambI (modern Kosam), the ancient capital of the Vatsa country, for two reasons : (1) Basim (longitude 77°irE from Greenwich) is nearer from the Hindu prime meridian (longitude 75°52'E from Greenwich) than KausambI (longitude 81°24'E from Greenwich). (2) Basim fits in the order in which the places lying on the Hindu prime meridian have been stated in the Maha-Bhaskariya. For, Basim lies to the north of "the White Mountain" and to the south of AvantT (modern Ujjain), as it should be. KausambI does not fit in that order. It may, however, be pointed out that the commentator Udaya Divakara seems to identify Vatsyapura with KausambI, for he writes: "The town called Vatsyapura is well known in the Vatsa country." And we know that KausambI was the capital of the Vatsa country in ancient times. AvantI is modern Ujjain in Madhya Bharat. Sthanesvara (or Sthanvlsvara) is a sacred place in Kuruksetra, famous for its sacred tank and temple of God Siva (called Sthanu Siva). 8 It is situa- ted at a distance of about two furlongs from the city of Thanesar in East Panjab. 4 "The abode of the gods" is Mem, the north pole. Circumference of the local circle of latitude : 24. 3299 (yojanas), (the circumference) of the Earth, mul- tiplied by the Rsine 5 of the colatitude (of the local place), and 1 See V. V. Mirashi, "Historical data in Dandin's Dasa-kumara-carita", Nagpur University Journal, Number 11, December 1945, lines 6-7. a See Kalyana* Tlrthanka, p. 229. Also see W. W. Hunter, The Imperial Gazetteer of India, Volume II, London (1885), p. 188. • "sthanesvarafn ievayatanam, tadapi kuruk$etre." Udaya Divakara. 4 See Kalyana, Tlrthanka, p. 80.

• Rsine means "radius X sine". vss. 25-28]

DISTANCE FROM THE PRIME MERIDIAN 9 divided by the radius (i.e., 3438') is known as the (Earth's) cir- cumference at the local place. 1 The Earth's circumference at the local place means "the circumference of the local circle of latitude", A rule for finding the distance of the local place from the prime meridian : 25-26. The circumference of the Earth multiplied by the differe*;e between the latitudes of (a place on) the prime meri- dian and the local place and divided by the number of degrees in a circle (i.e., by 360) gives the bahu (i.e., the base of the longitude triangle) due to the local place. The oblique distance from that local place to (the place on) the prime meridian is the hypotenuse (of the triangle). The square root of the difference between the squares of that (hypotenuse) and the bahu is said to be the longitude (in yojanas of that place). 2 The longitude in yojanas of a place means the distance of the place from the prime meridian in terms of yojanas measured along the local circle of latitude. In the Maha-Bhaskariya, the above bahu has been called the koti (i.e., the upright of the longitude triangle). For details, see my notes on MBh, ii.3-4. Criticism of the above rule : 27. Some learned scholars say like this; others say that it is not so, because of (i) the grossness of the hypotenuse and (ii) the sphericity of the Earth. 3 Srlpati (1039 A.D.), too, has criticised the above rule for the same reasons. Criticism of another rule : 28. (It has been said that) the difference between (the longitude of) the Sun derived from the midday shadow (of the g nomon at the local place) 4 and that calculated for the middle 1 Cf. MBh, ii. 10(iii).

• Gf. MBh, ii. 3-4.

• Gf. MBh, ii. 5. 4 See infra, iii. 29-33. 10 MEAN LONGITUDES OF THE PLANETS of the day (without the application of the longitude correction) (gives the longitude correction for the Sun). But that is not so, as to the east and west of a place on the prime meridian (i.e., on the same parallel of latitude) the latitude (and therefore the shadow of the gnomon) remains the same. 1 This rule has also been criticised by Srlpati, who says : "Whatever is obtained here as the difference between the longitudes of the Sun derived from the midday shadow (of the gnomon) and that obtained by calculation (for midday, without the application of the longitud&correc- tion) when multiplied by the (local) circumference of the Earth and divided by the (Sun's daily) motion gives the yojanas of the longitude (i.e., the distance xxyojanas of the local place from the prime meridian). This is gross on account of the small change in the Sun's declination." 2 The reader will also note that the longitude derived from the midday shadow will be tropical, whereas the other is not. A rule for the longitude in time : 29. The difference between the computed and observed times of an eclipse is the longitude in terms of time. 3 The computed time is the local time for the place lying at the intersection of the prime meridian and the local circle of latitude, while the observed time is the local time for the local place. The difference between the two is obvi- ously the longitude in time for the local place. It may be pointed out that in Hindu astronomy time is measured from sunrise. Criterion for knowing whether the local place is to the east or to the west of the prime meridian : 30. If the (lunar and solar) eclipses occur after the calcula- ted time, then the observer is to the east of the prime meridian ; otherwise, to the west. 4 The calculated time is the local time for the place lying at the intersec- ti on of the prime merid ian and the local circle of latitude. 1 Cf. MBh. ii. 6. 2 SiSe, ii. 103. » Cf. MBh. ii. 7.

• Cf. MBh, ii. 9. vss. 31-33]

THE LONGTIUDE CORRECTION 11 The longitude correction and its application: 31. The mean daily motion of the planet multiplied by the longitude (of the place) in terms of ghatls and divided by 60 should be subtracted (from the mean longitude of the planet for mean sunrise at Lanka) (if the place is) to the east of the prime meridian and added (to it) if it is to the west. 1 Sankaranarayana gives the following table for the mean daily motion of the planets : Planet Moon daily motion correct to seconds of arc Mean Sun Moon Moon's apogee Moon's ascending node Mars Sighrocca of Mercury Jupiter Sighrocca of Venus Saturn 59' 8* 13 3 10' 35* 6' 41" 3' 11" 31' 26" 4° 5' 32* 4' 59" 1° 36' 8* 2' Another rule for finding the distance of the local place from the prime meridian : 32. The yojanas (of the distance of the prime meridian)from the local place are obtained on multiplying the longitude in ghatts by the local circumference of the Earth and dividing (the product) by 60. 2 An alternative method for the longitude correction : 33. Whatever is obtained on multiplying the mean daily motion (of the planet) by the yojanas (of the distance from the prime meridian) for that place and dividing by its own (local) earth-circumference is to be subtracted from or added to the mean longitude of the planet (for mean sunrise at Lanka) 1 Gf. MBh, ii. 10(i). 2 Cf. MBh. ii. 10(ii). 12 , MEAN LONGITUDES OF THE PLANETS [CH. I (according as the local place is to the east or to the west of the prime meridian). Application of the longitude correction to the mean longitude of a planet for mean sunrise at Lanka gives the longitude of the planet for mean sunrise at the place where the local meridian intersects the equator. The place where the local meridian intersects the equator is called svaniraksa (i.e., "local equatorial place"), Justification of the longitude correction : 34. The method of adding or subtracting motion corres- ponding to the longitude (of the local place) in ghatis (taught above) is the cause of the decreased or increased tithi 1 (i.e., the local time of observation of the eclipse); seeing is unaffected by that correction. • Sankaranarayarta comments on this verse as follows : "The rule which has been stated here in accordance with which the correction obtained from the longitude, in ghatis or yojanas, is to be subtracted from or added to the mean longitude of (the Sun and) the Moon according to the direction of the local place (east or west of the prime meridian) is the cause of the decrease or increase of the tithi (i.e., of the local time of observation of an eclipse). Therefore the previous remark that by those who are situated to the east of the prime meridian an eclipse is seen after the time calculated for its occu- rence (on the meridian of Lanka), and by those who are situated to the west of the prime meridian it is seen in advance (of the calculated time) remains unaffected. How ? For the time of occurrence of an eclipse as obtained by subtracting the longitude of the Sun from that of the Moon without making allowance for the longitude correction is certainly less or greater than the local time which is obtained by properly subtracting or adding the longitude correction. Therefore at new moon or full moon an eclipse is naturally seen on the two sides (of the prime meridian) after or before the time calculated for its occurrence (on the meridian of Lanka), Otherwise (i.e., if the longi- tude-correction be not made), the difference between the times of observation of an eclipse by those situated to the east and to the west (of the prime meridian) would not be explained." Paramesvara observes : "That is the reason for the calcula ted time (of occurrence of an eclipse on the meridian of Lanka) being less or greater than 1 See Glossary. vs. 35] the local time ofobservation. But the calculated time (of occurrence of an eclipse) obtained after the correction for longitude has been applied does not differ from that (i.e., the local time of observation)." Similar remarks have also been made by Udaya Divakara. 13 Demonstration of the justification of the longitude correction : 35. When at a certain nalk1 (of local time) the Moon is at the point of emersion and is (at the same time) at the point of setting here (i.e., at a place on the prime meridian), then (at the same local time) (people residing) in the west (of that place) say: “The Moon sets after its separation from the shadow” and (those living) in the east (of that place) say : *The Moon sets with the eclipse'.* Udaya Divakara comments: “For people residing on the same parallel of latitude the lengths of day and might being equal, the Moon is seem to set as many gha s after sunrise (or sunset) in the countries which lie to the east or west of a place on the prime meridian as in the place on the prime meridian. But in the west the Moon as separated from the shadow is seen. The Moon at the point of separation as observed on the prime meridian is seen there earlier. So the (corresponding) till (i.e., the local time of observation of the separating Moon) is smaller. Therefore, in order to get that (titl) the Imotions corresponding to the intervening time should be added to the longi tudes of the Sun and the Moon. Similarly, in the east the eclipsed Moon is observed ; the separating Moon is seen later, and so the corresponding till is greater than the other. So here also the motions (corresponding to the in tervening time) are rightly subtracted The word radika in the opinion of Udaya Divakara denotes the mats of the time in the might when the Moon separating from the shadow at mojnset is seen on the prime meridian. east According to the commentator Parame5vara the word nadia is used in the sense of time in general (moment, etc.). So according to him the verse would be translated as follows : 1.6e In his comm. on 4, 1. 2, *A lumar eclipse which occurs here when one gho has elapsed.in the might is said to occur at the end of the day by those who live at a distance of one gthat (of longitude) towards the west and to occur later (in the might) by those who reside towards the 14 MEAN LONGITUDES OF THE PLANETS [CH. I "When the Moon is (just) setting at the end of an eclipse here (at a place on the prime meridian), then for those situated to the west the Moon sets after the eclipse is over whilst for those situated to the east it has set with the eclipse." Or, literally as follows : "When the Moon is at the point of separation (from the shadow at the end of a lunar eclipse) and is (at the same time) at the point of set- ting here (at a place on the prime meridian), then in the west (of that place) (people say) : "The Moon has set after its separation (from the shadow)", and those in the east say : "There is eclipse." Consequences of improper application of the longitude cor- rection : 36. When improper (viparlta) application of the positive- negative (longitude) correction (to the longitudes of the Sun and the Moon) is made the resulting tithi is not the correct one. (Also) the results derived from (the correct) procedure become otherwise and the motion of the planet also becomes different. Sankaranarayana interprets this stanza thus: "(By the word mpartta is meant the case) when at the place where the correction for the longitude has been stated to be negative (positive) it is applied contrarily, i.e., positively (negatively). Or, the word viparlta may mean that the correction for the longitude is not made at all. The titki obtained in both these cases is not considered to be correct for the purposes of religious sacrifices, etc. Without making allowance for the longitude correction the planetary motion is also incorrect." Paramsevara says : "When the correction for the longitude is applied contrarily to that stated, the tithi obtained is incorrect for the purposes of religious sacrifices, etc. The calculated time of occurrence of an eclipse is also different from the time of actual observation. The positions of the pla- nets are also wrong." The first half of the verse may also be translated as follows : "When improper application of the positive-negative (longitude) correction (to the longitudes of the Sun and the Moon) is made, the (computed) tithi (i.e., time of an eclipse) does not tally with that of observation." VS. 37] COMPARISON WITH LAMBANA CORRECTION 15 This translation is in agreement with the interpretations of Udaya Diva- kara and Paramesvara. Comparison of the longitude correction with the lambana correction : 37. The lambana correction is (also) additive or subtrac- tive to the tithi and to the longitudes of the Sun and the Moon for that time, but the law of this (positive-negative) correc- tion in the case of the longitude is different from that of the lambana. The term lambana means the difference between the parallaxes in longi- tude of the Sun and the Moon. For the lambana correction, see infra, chap- ter V, stanzas 8-10. The tithi in the above passage stands for the time of conjunction in longitude of the Sun and the Moon (called parva-tithi, or simply parva). Udaya Divakara reads tadvat in place of tasya and interprets the verse as follows : "Since the lambana correction is applied positively (or negatively) to the longitudes of the Sun and the Moon and exactly in the same way it is posi- tively (or negatively) applied to the time of the tithi, therefore the process of the longitude correction is not like that of the lambana correction." He continues : "This is what has been said : In the case of the lambana correction, when the corresponding motions are added to the longitudes of the Sun and the Moon, then the parva (i.e., the time of conjunction of the Sun and Moon) is also increased by that time. When the motions corresponding to the lambana are subtracted from the longitudes (of the Sun and the Moon) then the parva is also diminished by that time. Here (in the case of the longitude correc- tion) it is just the reverse. For, when the longitudes of the Sun and the Moon are increased, the parva is diminished ; and when the longitudes of the Sun and the Moon are diminished, the parva is increased. The lambana and the longitude corrections being thus of unlike natures, the longitude correc- tion is incomparable with the other." CHAPTER II TRUE LONGITUDES OF THE PLANETS Definitions of the Sun's mean anomaly and the corresponding bhuja and kofi : l-2(i). The mean longitude of the Sun diminished by the longitude of the (Sun's) apogee is (called) the (Sun's mean) anomaly. There (in that anomaly) three signs form a quadrant. In the odd quadrant, the arc traversed and the arc to be traver- sed are known as bhuja (or bahu) and kofi (respectively); in the even quadrant, (they are known as) kofi and bhuja (or bahu) respectively. This is the position. 1 That is, Sun mean anomaly = Sun's mean longitude — longitude of Sun's . , . „ apogee. And if the Sun's anomaly be degrees, then bhuja=f) 1 180° -01 0—180°) 360°— fll koti=90"- 6 f > 0-90°/ » 27O°-0/> or 0-270°} ' according as O<0< 90°, 9O°<0<18O°, 18O°<0<27O° or 270° < < 360°. A rule for calculating the Rsines of the bhuja and kofi : 2(ii)-3(i). After converting the bhuja and the other (i.e., the koti) into minutes of arc and dividing by 225, (in each case) take (the sum of ) as many Rsine-differences as the quotient. Then multiply the remainder (in each case) by the current (i.e., next) Rsine-difference and divide by 225 and add the result (to the corresponding sum of the Rsine-difTerences obtained above). (The sums thus obtained are the Rsines of the bhuja and the koti) . 2 1 Gf. MBh, iv. 1, 8(iJ. 2 Gf. MBh, iv. 3-4(i). -- vs. 2(ii)-3(i)] - -- ------

---

- This rule tells us how to 6nd the Rsine (*'radius x sime") of the big or bot (or of any given arc or angle) with the help of the following table of Rsine-differences given by ryabhata I:

225

---

--

224 222 219 - --- -- ---

---

215 210 10

---

- -

11] 12 RsINEs or

---

- - = = - -- 205 199 183

---

174 164 Suppose that bjळ =24. will be obtained as follows: --- --- - Bhuja AND 13 15

---

16

---

17 | - = | ------------ = 154 = 143 131 - 119 -- 106 ४ort 93 || 19 - --- -- -- ------- --- --- - 20 | 22 21 -

---

---,

23 24 -- = --- = = 79 51] 37

17 22

---

---

---

--------

Then, according to the above rule, Rsine 24* Converting the blugja into minutes (of arc), we get 140. Dividing this by 225, we get 6 as the guotient and 90' as the remainder. So taking the sum of the first six Rsine-differences, we get 225'+224'+-222'+219'+215'+210'=1315. Now multiplying the remainder, i.e., 90', by the next (i.e., 7th) Rsine difference (i.e., 205) and dividing by 225, we get 90 * x205 225 Adding this to the previous sum of six Rsine-differences, we get 1315+ 82'=1597'. This is the reguired value of Rsin 24 ". IS TRUE LONGITUDES OF THE PLANETS [ CH. II Calculation of the bhujaphala and the kotiphala : 3(ii). They (i.e., the Rsines of the bhuja and the koti) multiplied by the (planet's tabulated) epicycle should be divided by 80 : the results are (known as) bhujaphala and kotiphala. 1 That is, bhujaphala = Rsin $huja) x tabulated epicycle 80 kotibhala = Rs,n f* * 1 ") x tabulated epicycle

• • ~~ 80~ *

In the case of the Sun and the Moon, the bhujaphala corresponds to the equation of the centre of modern astronomy. For details, the reader is referred to my notes on MBh, iv. 6. Application of the bhujaphala correction: 4(i). The bhujaphala is additive or subtractive according as the (mean) anomaly is in the half-orbit commencing with the sign Libra or in that commencing with the sign Aries. 8 In other words, the bhujaphala is additive or subtractive according as the mean anomaly is greater than 180° or less than 180°. The bhujaphala correction is applied to the Sun's mean longitude as cor- rected for the longitude correction. This correction having been applied we obtain the Sun's true longitude for mean sunrise at the 'local equatorial place' (i.e., at the place where the local meridian intersects the equator). Calculation and application of the bhujantara (or bhujaviva? a) correction: 4(ii). So also is applied (the bhujantara correction) which is obtained by multiplying the (mean daily) motion of the planet by the (Sun's) bhujaphala and dividing by the numberof minutes of arc in a circle (i.e., 21600). 3 1 Gf. MBh, iv. 4, 8(ii).

• Cf. MBh, iv. 6.

» Gf. MBh, iv. 7. vs. 6-7: ] That is, This correction is subtracted from or added to the Sun's true longtude for mean sumrise at the local equatorial place, according as the Sun's b ja 24ala is subtractive or additive. Thus we obtain the Sun's true longitude for true sunrise at the local cपृuatorial place. Sun's blujahala x planet's mean daily motion 21600 The blgjantara correction is, thus, the correction for the euation of time due to the Sun's cपृuation of the centre (i.e., due to the eccentricity of the ecliptic). and Approximate formulae for the blujantara corrections for the Sui and the M001 : 5. One-sixth of the (Sun's) bhujahala is, in seconds of arc, (the bltgantara correction) for the Sun; that for the Moon is obtained in minutes of arc etc. by multiplying (the Sun's blugja }/hala) by 3 and dividing by 82. That is blujantara correction for the Sun = bhujantara correction for the Moon = 19 These formulae can be easily derived similar formulae see 62 seconds; from the previous rule. minutes. For other A rule for finding the true distances of the Sun and the Moon in minutes (called mandalkar70): 6-7. Increase or diminish the radius by the (Sun's) oft }lhala (according as the mean Sun is) in the half-orbit commenc ing with the anomalistic sign Capricorn or in that commending with Cancer. The s५uare root of the sum of the squares of that and the (Sun's) balhuploala is the (first approximation to the Sun's) distance. (Severally) multiply that by the (Sun's) bahu42bala and kot}}hala and divide (each product) by the radius: (the 20 TRUE LONGITUDES OF THE PLANETS [ CH. II results are again the Sun's bahuphala and kotiphala). (Making use of them calculate the Sun's distance afresh: thus is obtained the second approximation to the Sun's distance). (Repeat this process again and again and thus) by the method of successive approximations obtain the nearest approximation to the Sun's (true) distance. For the Moon, too, this is to be regarded as the method for finding the nearest approximation to the true distance. 1 The distance obtained by the above method is in terms of minutes and is called mandakarija. As it is based on the method of successive approxima- tions, it is also known as asakrtkalakarna or avisesakarna. For the rationale, see my notes on MBh, iv. 9-12. A rule for finding the true daily motion (called karnabhukti) of the Sun and the Moon: 8. Multiply the mean daily motion (of the Sun) by the radius and divide (the product) by the (Sun's true) distance (in minutes): the result is the Sun's true daily motion (known as karnabhukti or karnasphufabhukti). For the Moon, too, this is the method. 2 That is, e > » j m /i ,, , ., Sun's mean daily motion X R Sun's true daily motion (karnabhukti) =- — ; - Sun s true distance in minutes » w > * ... . ' .. , A .. Moon's mean daily motion X R Moon s true daily motion (karnabhukti) = ' Moon's true distance in minutes where R is the standard radius (=3438'). The true daily motion obtained by the above formulae was called karna- bhukti (meaning, "motion derived from the distance") because it was obtained by proportion from the true distance of the Sun or Moon. A rule for the determination of the Sun's true daily motion (called jiuabhukti): 9-10. Divide by 225 the (Sun's) mean daily motion as multiplied by the current Rsine-difFerence. Multiplying the result

• 'Cf.'Af£A, iv. 9-12.
• Cf. MBh, iv. 13. vss. 9- 10 ] Jivabhukti of the sun 21

(thus obtained) by its (tabulated) epicycle and dividing by 80, subtract that from the Sun's mean daily motion if the (Sun's) anomaly is in the half-orbit commencing with Capricorn and add that to the same if (the Sun's anomaly is) in the half-orbit com- mencing with Cancer. (The sum or difference thus obtained) is known as the (Sun's) true daily motion. 1 Let M and M' be the mean longitudes and S and S' the true longitudes of the Sun at sunrise yesterday and today respectively. Also let Q and 0' be the corresponding values of the bhuja (due to the Sun's mean anomaly). Then, we have where r t is the Sun's tabulated epicycle, — or -f- sign being taken according as the Sun's mean anomaly is less than or greater than 180°. Therefore S'-S= (AT- J*} T (Rsi " . . 80 (Rsin Rsin $) x r, = m f 80 where m denotes the Sun's mean daily motion, •— or 4- signs being taken according as the Sun is in the first and fourth or in the second and third anomalistic quadrants. Neglecting the motion of the Sun's apogee and assuming that the Rsines vary uniformly, we have Rsin &' - Rsin = (current R sine-di fference) X m Therefore C c « -r (current Rsine-difference) X m X r t Hence the above rule.. Since the Sun's true daily motion has been obtained here with the help of Rsines ( jha ), therefore it is generally called jhabhukti. 1 Cf. MBh, iv. 14. 2?

• TRUE LONGITUDES OF THE PLANETS [ CH. n

A rule for finding the Moon's true daily motion (known as jhabkukti) : 11-13. From the (mean daily) motion of the (Moon's) mean anomaly subtract the preceding or succeeding arc (of the current element of the arc, i.e., the elementary arc 1 containing the Moon) (according as the Moon is in the odd or even anom- alistic quadrant). (Then) take (the tabulated Rsine-differences) on the basis of the (residue in) minutes of the (mean daily) mo- tion of the Moon's mean anomaly, starting from the current Rsine-difference reversely and directly in the odd and even anom- alistic quadrants respectively. The results (i.e., the Rsine-dif- ferences) corresponding to the fractions of the first and last elementary arcs should be determined by proportion (and added to the sum of the previous Rsine-differences). The Rsine-dif- ference (corresponding to the daily motion of the Moon's mean anomaly) thus obtained multiplied by the (Moon's tabulated) epicycle and divided by 80 should be subtracted from or added to the Moon's mean daily motion as before (in the case of the Sun, i.e , according as the Moon's anomaly is in the half-orbit commencing with the sign Capricorn or in that commencing with the sign Cancer). This is known as (the Moon's) true (daily motion). 2 The commentator Paramesvara explains the above method as follows: "From the mean (longitude) of the Moon subtracting its apogee, (then) obtaining the (corresponding) bkuja, (then) reducing that to minutes of arc, (then) dividing that by 225, (then) setting down separately the preceding portion of the current elementary arc as also the succeeding one, (then), the (anomalistic) quadrant being odd, having multiplied the preceding portion of the current element of the arc by the current Rsine-difference and divided by 225 and taken (down) the resulting Rsinc-difTerence, subtract the preced- ing portion of the current elementary arc from the (daily) motion of the Moon's mean anomaly. Then, having divided that remainder by 225, add to the Rsine-difference obtained before as many(tabulated)Rsine-difTerences, 1 The twenty-four divisions of a quadrant, each equal to 225' t the Rsine-differences of which have been tabulated by Aryabhata I, are called "elements of arc", or "elementary arcs". 2 Cf. MBh, iv. 15-17. vss. 14-1 5 (i)] DEFECTS OF THE jfivdbhukti 23 in the inverse order, from thecurrent Rsine-difference as the quotient-number. Then having multiplied the remainder, in minutes of arc, obtained (above) by dividing the (daily) motion of the (Moon's) mean anomaly by 225, by the next Rsine-difference, in the inverse order, and divided by 225, add the resulting Rsine-difference, too, to the Rsine-difference obtained before (by addition). This is (the process) in the odd anomalistic quadrant. In the even (anomalistic) quadrant, on the other hand, having multiplied the succee- ding portion of the current elementary arc by the current Rsine-difference and divided that by 225 and (then) having taken the resulting Rsine-differ- ence, subtract from the (daily) motion of the (Moon's) mean anomaly the succeeding portion of the current elementary arc. Also, then, take, in the direct order, the Rsine-differences resulting from the remaining motion of the (Moon's) mean anomaly. Thus are to be taken the Rsine-differences in the inverse and direct order. If here (i.e., in the above process) the Rsine- differences to be taken in the inverse order come to an end (due to the end of the odd quadrant falling within the arc corresponding to the motion of the Moon's anomaly), then for the remaining arc take the Rsine-differences in the direct order. When the Rsine-differences to be taken in the direct order come to an end, then take the Rsine-differences in the inverse order. There (i.e., in such cases) for the Rsine-differences taken in the direct and inverse order motion-correction is obtained separately. Having mutiplied the Rsine- difference (corresponding to the daily motion of the Moon's mean anomaly), thus obtained, by her (tabulated) epicycle viz. 7 and divided (that) by 80, the result should, as before, be subtracted from or added to the (Moon's) mean (daily) motion (according as the Moon is) in the half-orbit beginning with the (anomalistic) sign Capricorn or in that beginning with Cancer. Where, however, there are (two) corrections derived from the Rsine-differ- ences taken in the direct order as well as in the inverse order, there the two corrections are applied to the mean (daily) motion (of the Moon) in accor- dance with their (anomalistic) quadrants. That is the true (daily )motion (of the Moon)." The rationale of the above rule is exactly similar to that of the previous one. The difference is that the motion of the Moon'i apogee is also taken into account in this case. Defects of the jivabhukti : 14-15 (i). (According to the rules stated above) whilst the Sun or the Moon moves in the (same) element of arc 1 , there is 1 Vide supra, p. 22 footnote [}) 2 * TRUE LONGITUDES OF THE PLANETS [CH. II no change in the rate of motion because (the current Rsine- difference being fixed throughout that element) the Rsine- difference does not decrease or increase : when viewed in this way, this jhabhukti is defective. Rule 9-10 shows that so long as the Sun remains in the same element- ary arc (measuring 225') the Sun's jlvabhukti does not vary. Since the Sun remains in the same element for three consecutive days, its jhabhukti remains the same for three consecutive days. This is defective, because the rate of motion varies from instant to instant. 1 Similarly, so long as the Moon remains in the same elementary arc, its velocity remains the same because throughout that element the Rsine-diffe- rence is constant. Thus, in the case of the Moon, the instantaneous daily motion obtained with the help of the Moon's current Rsine-difference is defective. Author's opinion regarding the true daily motion : 15 (iij. The karnabhukti* or the difference between the true (longitudes) for two consecutive days is the true (daily) motion. The commentator Paramesvara thinks that the karnabhukti is the instan- taneous daily motion. . The comparative merits and demerits of the jhabhukti and the karnabhukti have been examined in detail by Mlakantha in his commentary on i,ii.22-25. A rule for finding the Sun's declination with the help of the Sun's tropical (sayana) longitude : 16. 1397 is (in minutes of arc) the Rsine of the (Sun's) greatest declination. The product of that and the Rsine of the bhuja due to the Sun's true (tropical) longitude divided by the radius is the Rsine of (the Sun's) desired declination. 3 1 Instantaneous change of velocity was recognised by the Hindu astro- nomer Manjula (932) who, on the basis of the idea of the "infinitesimal increment", gave a rule for the instantaneous velocity of a planet. 9 See stanza 8 above. » Cf. MBh, iii. 6(i). wss. 17-18 ] That is, where 8 is the Sun's declimation and x the Sun's longitude. In Fig. 1, let S denote the position of the Sun on the celestial sphere, SI. the perpendicular from S on the plane of the celestial c१uator, and SM the perpendicular from S on the line joining the first point of Aries and the first point of Libra. Then in the plane triangle SLM, we have SL = Rsin 8, SM = Rऽin (blugia x), Therefore ८ SLM = 90० talking Rsांn (bhuja ) ) x 1397 Rsin 6 = '. 1397 SL|SM = , Rsin { // Rsin 90

• C. MBh, i. 6(ii)-7

Rsin(bhujax)x1397 25 Fig. 1 A rule for finding the earthsine and the for the Sun: ascensional difference 17-18. Whatever be the square between the squares of that(i.e ., of the Rsine of the Sun's decli root of the difference: mation) and of the radius is the (Sun's) day-radius. The Rsine of the latitude multiplied by the Rsine of the (Sun's) and divided by declination the Rsine of the colatitude is (known as) the (Sun's) carthsine.This is multiplied by the radius and divided by the (Sun's) day-radius: whatever is obtained Rsine of the (Sun's) ascensional difference. " is called the The Sun's day-radius is the radius of the Sun's diurmal circle, along which the Sun moves in its diurnal motion. It is eggual to the Rsine of the Sun's codeclination. Hence day-radius = where 8 is the Sun's declimation. 26 [ CE. I The Sun's earthsine is the distance between (1) the Sun's rising-setting line and (2) the line joining the points of intersection of the Sun's diurnal circle and the six o'clock circle. In Fig.2, let K be the point of intersection of the Sun's diurnal circle and the six o'clock circle, KB the perpendicular from K on the Sun's rising setting line, and KA the perpendicular from K on the east-west line. Then in the triangle KAB, we have KA=Rsin 8, KB=Sun's earthsine, and ZKAB= }. Therefore we have Rऽin 8 Rsin (90-) The Sun'ascensional is of s difference the arc the celestial equator 1ying between (1) the hour circle of the Sun's rising point on the eastern horizon and (2) the six o'clock circle. It can be seen from the celestial sphere that Rsirm (Sun's ascensional difference) Sum's earthsine Therefore Fig. 2 Rsin (Sun's ascensional difference ) = Correction for the Sun's ascensional difference (car0-talkskra) : 19-20. The minutes ofarcin the ar०ofthat (Sun's ascensional difference) are known as brama (or asu). On multiplying them by the true daily dividing by (Sun's) motion and 21600 are obtain ed the minutes, etc., (of the Sun's motion corresponding to its ascensional difference). (In order to obtain the Sun's true longitude) at sumrise (for the local place) these (minutes, etc.) should be subtracted (from the Sun's true longitude at sumrise vss. 19-20 ] Cara correction 21 for the local equatorial place) provided the Sun is in the northern hemisphere (i.e., to the north of the equator) and added if the Sun is in the southern (hemisphere). In the case of sunset, (the law of correction is) the reverse. In the case of midday or midnight, this (correction) should not be performed. 1 The asu is a unit of sidereal time equivalent to 1/21600 of a Sidereal day. The Sun's ascensional difference measured in asus denotes the time-interval, in asus, between the Sun's rising or setting at the local and local equatorial places. The above correction for the Sun's ascensional difference, therefore, makes allowance for the difference between the times of Sun's rising or setting at the local and local equatorial places. The general formula for the above correction is; Correction for the Sun's ascensional difference {cara correction) _ Sun's asc. diff. in asu sX planet's true daily motion 21600 minutes of arc. When the Sun is in the northern hemisphere, sunrise at the local place occurs earlier than at the local equatorial place, and sunset at the local place occurs later than at the local equatorial place. When the Sun is in the southern hemisphere, it is just the contrary. Hence the law of addition and subtraction of the correction. Since midday or midnight occurs simultaneously at the local and local equatorial places, therefore there is no need of such a correction at that time. When the above correction has been applied to the Sun's true longitude for true sunrise at the local equatorial place, we get the Sun's true longitude for true sunrise at the local place. This is called the Sun's true longitude. We thus see that, in the case of the Sun, to obtain the true longitude for true sunrise at the local place we have to apply to the mean longitude for mean sunrise at Lanka the following four corrections in their respective order: (1) the longitude correction, (2) the bhujaphala correction (i.e., the equation of the centre), (3) the bhujantara correction (i.e., the correction for the equation of ^^^^ time due to the Sun's equation of the centre), 1 Gf. MBh, iv. 26-27(i). 28 (4) the ८ara correction (i.e., the correction sional difference) In the case of the Moon, the same four following order: (1) the longitude correction, (2) the bltgantara correction, (3) the blujahala correction, (4) the ८ara correction. due to the Sun's ascen corrections are applied in the The correction for the euation of time due to the obliguity of the ecliptic has been neglected by the author of the present work, like all other early Hindu astronomers. This correction occurs for the first time in the works of Sripati (1039) and Bhaskara II (1150). Bhaskara II called it udayantara-satiskara and prescribed it for all planets. Lengths of day and might 21. (When the Sun is)in the northern hemisphere, the day increases and the might decreases by twice the 0.5us of the (Sun's) ascensional difference. (When the Sun is) in the southern hemisphere, the contrary is the case. What is meant is that when the Sun is the northern hemisphere length of day=30 ghaऽ+ twice the Sun's asc. diff. in usus, and length of might=30 ghatऽ- twice the Sun's ase. diff. 'in asus, and when the Sun is in the southern hemisphere length of day=30 gha! and length of might=30 gha ऽ+twice the Sun's asc. diff. in asu७ The truth of this can be easily seen from the celestial sphere The blugjantara correction for the Moon 22. The mean daily motion of the Moon multiplied by the Sun's blugjalala and divided by 21600 should be added to or subtracted from the mean longitude of the Moon (corrected for the longitude correction) as in the case of the Sun (i.e., ac cording as the Sun's mean anomaly is in the half-orbit commen cing with Libra or in that commencing with Aries).* vs. 23-24 ] (Other corrections for the Moon : 23-24. The result in minutes of arc, etc., which is obtained on multiplying the true daily motion of the Moon by the usus of the Sun's ascensional difference and dividing (that product) by the number of a15us in a day and might (i.e., by 21600) should always be added to or subtracted from the true longitude of the Moon (for true sumrise at the local equatorial place) according to (the position of) the Sun. The remaining (bltgjahal८) correction for the Moon is applied (to the Moon's longitude corrected for the longitude and blujantara corrections) in the same manner The correction stated in the first part of the above passage is the Moon's ८ara-5aौskara, i.e., correction to the Moon's longitude due to the Sun's ascen The blugjabial correction for the Moon, which is the caाra correction, is given by the formula: blujahala correction for the Moon = Rsin(bgja due to Moon's mean anormaly)x Moon': tabulated epicyle 80 or + sign being taken according as the Moon's mean anomaly is less or greater than 180° From the above, we see that in the case of the Moon, the order corrections to be applied is, as stated before, as follows: (1) the longitude correction, (3) the blujaphala correction centre) to be applied before (2) the bigjantara correction (i.e., correction due to the Sun's cपृua tion of the centre),

• wid८ supra, stanzas 19-20.

(i.e., the 29 Moon's equation (4) the ८ara correction (i.e., correction due to the Sun's difference ). of the of the ascensional

• The commentator Parame5vara suggests, as a , alternative, the appli 30

The next five starm2as relate to the application of the true longitudes of the Sun and the Moon to the computation of three of the elements of the Hindu (Calendar (Paffcatiga) , vi८, 100ks.ation, tillhi and kar010 and to the deter mination of the phenomena of yat;/2ata. It must be noted that the calcula 25-26(i). (The true longitude of) the Moon reduced to Iminutes of arc should be divided by 800: the quotient (thus obtained) denotes the (number of ) 70ks.atra5 Asvini, etc., (pass ed over by the Moon). The traversed and the untraversed por tions (of the current 70 0tra) should be divided by the true daily motion (of the Moon in minutes of arc) after having Imultiplied them by 60: thus are obtained the 12d:ऽ elapsed and to elapse at sunrise. Beginning with the first point * of the maksatra A5vinl, the ecliptic is divided into 27 parts , each of 800 minutes of arc. These parts are called maksatras and are given the same names as the 20diacal asterisms, i. c: 2. Bharati 4. Rohini 5. . Mgakira

10. 11. 12. 13 Magha Purva Phalguni 18. Uttara Phalguni 19. Hasta 21 . Jyestha Mula Uttarasadha 22. 27. Sravarma drapada 8 The above rule enables us to know the maksatra in which the Moon lies at and the clapsed she entered sumrise time since that makotra as also the time to clapse before she enters the next malatra. drapada Revati The first point of the magatra A5vinl is the fixed point from longitudes of the which the planets are measured in Hindu astronomy This point coincides with the junction star of the maksatra Revati, i.e., with -Piscium vs. 28 ] Calculation of the titlt : 31 26(ii)-27. Having reduced (the longitude of) the Moon Iminus (the longitude of ) the Sun to minutes of arc, divide it by 720: the quotient is the (number of ) tilhas clapsed (since Imew moon). On multiplying (the portions of the current till, elapsed and to be clapsed severally) by 60 and dividing by the difference between the (true) daily motions (of the Sun and Moon ) are obtained (the ghapts) clapsed and to be clapsed (of the current titlhi).1 A 1umar month, which is defined in Hindu astronomy as the period from one new moon to the next, is divided into 30 parts. called titlis (or lumar days). The first titl begins just after new moon (when the Sur and Moon have the same longitude) and continues till the Moon is 12" (or 720') in ad vance of the Sun; the second tith then begins and continues till the Moon is 24* in advance of the Surn; the third titlt' then begins and continues till the Moon is 36" in advance of the Sun; and so on. The fifteenth titlhi is called Purplina or Purpimals (“the full moon tithi'), and the thirtieth titlt is called Amāvasya or Amavasya (“the tithi in which the Sun and Moon are in conjuc tion', i.e., “the new moon titlhi ) The first feen tilis fall in the light half of the lunar month and the remaining fifteen titlhis in the dark half of the lumar month. The tilis falling in either of the two halves are numbered 1, 2, 3, the thirtieth titl:# being, however, numbered 30. The rule given above gives the number of tilliऽ clapsed since new moon, and the time clapsed at sumrise since the beginning of the current titlt as also the time to clapse at sumrise before the commencement of the next tith 28. The kara11as (elapsed) are obtained by talking “half the measure of the titlhi (i.e., 360 minutes)” for the divisor, and are counted with Bava. But the number of karands elapsed in the light half of the month should be diminished by 076, whereas those clapsed in the dark half of the month should be increased 32 by 01८. This is what has been stated." A lunar month is also divided into sixty parts called karapas. These sixty karaas are divided into eight cycles of seven movable karar:05, bearing the names Bava, Balava, Kaulava, Taitila, Gara, Vapija, and Visti respect ively, and four immovable karanas, bearing the names Sakumi, Catuspada, Naga, and Kinstughma respectively The first round of the movable karanas begins with the second half of the first titlt in the light half of the month, and the eighth round ends with the light half of the month, the second kararror is Bava, the third karana is Balava, the fourth kara८ is Kaulava, and so on; and in the dark half of the month, the first karapa is Balava, the second kar८a is Kaulava, and The four immovable karapas occur in succession after the eighth round of the cycle of the seven movable karap05. The following table will clarify the with respect to the tilis. occurrence of the various karanas 1 That is to say: If it is the light half of the month, divide the true longitude the Moon as diminished by that of the Sun, reduced to minutes, of by 360. The uotient diminished by one should be divided by seven and the remainder counted with Bava. This gives the karand clapsed before If it is the dark half of the month;subtract the longitude of the Sun from that of the Moon, and diminish that difference by six signs. Reduce it to minutes and divide by 360. The quotient increased by one should be divided by seven and the remainder counted with Bava. This gives the karana clapsed before sunrise.

• The karapa Vist is also

.. 'The time elapsed at sunrise since the beginning of the current kara१० should be determined from the remainder obtained after division by 360 as in the case of the titl. nown as Bhadra and is considered inauspicious. vs. 29 ] }; 1. Pratipada 2. Dvitya 6. Sasth 7. Saptaml 13. Tray०da8] {{ ८८८ Kaulava Kaulava Taitila {# {{ # {# a " Balava Kaulava {a { # . Dark 1. Pratipada 7. Saptam 13. Trayoda5 14. Caturda5 Half {#

Vaja Blava Kaullava {{#. { # Taitila Taittila {#. 33 29. When the sum of the (true) longitudes of the Sun and the Moon amounts tohalf a circle (i.e., 180°), the phenomenon is called (lat०) wat ata; when that (sum) amounts to a circle (i.e., 360°), the phenomenon is called uidlhta (90t ata); and when that (sum) extends to the end of the maks८tra Anurādha (i.e., when the sum amounts to 7 signs, 16 degrees, and 40 minutes), the The term yat2ata (or, what is generally known as ata or malhayata) literally means "a very great portentious calamity astronomical phenomenon which is considered to be extremely imauspicious “The time intervening between the moments of the beginning and end is to be looked upon as exceedingly terrible, having the likeness of the consuming fre, for bidden for all work. While any part of the discs (of the Sun and the Moon so long is there a continuance of this aspect causing the destruction of all works he continues, “from a (pre ious) knowledge of the time of its occurrence, very great advantage is obtained, by means of bathing, giving, prayer, ancestral offerings, vows oblations, and other । * The phenomenon of yot ala is said to have a universal effect. According to our author *when the phenomenon of pyatala occurs, even on cutting the branches of a milk-tree ( ralaru), there is absence of millk”. The text describes the three varieties of the pat2ata, [2:40, paidlta and 20amustaka, giving simply the regions of their occurrence. It does not go into the details of their calculation. The subject, however, is so important for the astrologer that works on Hindu astronomy generally include a chapter giving a detailed discussion of this subject.* In modern Hindu Calendars (called Paffcatiga) are gven the titi, karapa, maksatra and yoga current at sumrise for every day ofthe year and also the times when they end and the next ones begin. The yogu has not been reated by Bhaskara I, but it forms one of the five important clement of the Hindu (Calendar. Like the makऽatras, the number of yogas is also twenty-seven. The method of finding the number of yogas passed 'over and the time clapsed at sunrise since the commencement of the current yoga is similar to that prescribed for the maksatra. The differ ence is that in the case of the yoga calculation is made with the sum of the longitudes of the Sun and the Moon , whereas in the case of the nal०rtra calculation is made with the help of the longitude of the Moon only. The PS, i. 20-22; $D7, 1, xi. 1; B७S, xiv.33-34; MS, xi. 1; ऽऽ०, wi. 1-2; ऽ6: , xi. 8; 7ऽ, wi. 1.2

• Cf. Burges B., Siya-iddhanta (English Translation), Calcutta (1935)
• See e.g. Sist, xi; BS2७, ४iv; ऽऽ०, wi;Si8, 1; xi. TS, wi. vs. 31-32 ]

first yoga (called Viskambha) begins when the sum of the longitudes of the Sun and the Moon is zero , and continues til that sum amounts to 13-20 '; the second yog८ (called Prti) then begins and continues till the sum of the longitudes of the Sun and the Moon amounts to 26°40'; and so on." It is noteworthy that the sapamastaka -yuathala occurs when the seventeenth yoga, bearing the name yat2ata, ends. (See footnote 1) 35 General instruction regarding the planets : 30. In the muda and 5:gra(operations) of the planets(Mars etc.) the kendra (anormally ), (their) kor and blugja, (their) Rsines, the corresponding bhalaऽ (corrections) and their addition and subtraction, should be understood as in the case of the Sum.2 There is one exception, viz. that the gौrakendra is defined as 3ghrak८rdro=longitude of planet's 1gro८८८ - longitude of planet , and not as mandakendra, which is defined as mandakermdra=longitude of planet - longitude of planet's mardocra . A rule for finding the corrected epicycle in the case of the planets Mars, etc 31-32. One should divide by the radius the Rsine or the Rversed-sine(ofthe part ofthe karadra lying in the current quadrant*) as multiplied by the differencebetween the epicycles (for the begin Imings of the odd and even guadrants) according as the (current) quadrant is odd or even. If the epicycle (in the beginning of the current quadrant) is smaller , add the (above) result to it;

• The names of the twenty-seven yogas are:-(1) Vislambha, (2) Prti,

(3) Ayusman, (4) Saubhāgya, (5) Sobhana , (6) Atigapa, (7) Sulkarma, (8) Dhti, (9) Sla, (10) Gapda, (11) Vrddhi, (12) Dhruva, (13) Vyaghata, (14) Harsapa, (15) Vajra Parigha, (20) Siva, (21) Siddha, (22) Sādhya, (23) Subha, (24) Sukla, (25) Brahma, (26) Indra, and (27) Vaidhra. There is another system of twenty-cight yogas, beginning with Ananda In some Hindu Calendars yogar ofthis system are also given for every day of the timonth. But these yogas are of astrological interest only in

• The k८ndra is said to be in the first quadrant when it is less than 909)

the second quadrant when it is between 90° and 180° , and so on. 36 if the epicycle (in the beginning of the current quadrant) is greater, subtract the (above) result from it. Then is obtained the corrected epicycle. If this correction is not made, the motion (of the planet) would be like the jumping of a frog . From chapter I, starm2as 19-21, we know that the planets Mars, Mercury, Jupiter, Venus, and Saturn have two types of epicycles, manda and 31gra, which vary in size from place to place. Their values for the beginnings of odd and even quadrants were tabulated in those starm2as. The above rule gives their values for any other place of the orbit Let ८ and 3 be the values of the epicycles (numda or 3ghra) of a planet for the beginnings of odd and even quadrants respectively. Then (i) if the planet be in the first anomalistic १uadrant, say at P, and its amounaly be 8, (P-८)x Rsin 0 epicycle at P=८ + when ८. < 3, 3 + ८ - 3) x Rsin 6 and (ii) if the planet be in the second anomalistic guadrant, say at 2, and its anomaly be 90°++; ( 8 epicycle at 9=Bि when ८ < P, when ०८ >P, d-B ) x Rversin whern d. > B. Similarly in the third and fourth anomalistic quadrants. A rule relating to the calculation of the true (geocentric) long: tudes of the superior planets, Mars,Jupiter, and Saturn 33-37(i). Having added half the bahu4lhala due to the mand०८ (apbgce) t0 or subtracted that from the mean longi tude of the planet as before, the result should be subtracted from (the longitude of) the ॐgthr0c0 : that (difference) is called the ॐgrak०7dra. From that obtain the balu4hala : (and) having multiplied that by the radius, divide (the product) by the (3ghra)karta. Half the arc corresponding to the result obtained should be added or subtracted according as the 5ःghrakendra is in the half-orbit beginning with Aries or in that beginning with vs. 33-37(i)] Libra. Then after subtracting (the longitude of) the mamdb८a from that (result), the entire bahuplala (derived from that differ ence), reduced to arc, should be applied (as correction, positive or negative) to the mean longitude of the planet (according as the 7207idak८rada is in the half-orbit beginning with the sign Libra or in that beginning with the sign Aries) this (result) is known as the true-mean longitude (of the planet) (Then) after subtracting the resulting quantity (viz. the true mean longitude of the planet) from the ॐghr0ca, the entire cor rection obtained by the sgrocca process, reduced to arc, should be applied (as correction, positive or negative,) to the true-mean longitude of the planet (according as the $ghrakendra is in the half-orbit beginning with Aries or in that beginning with Libra): thus is obtained the true longitude of the planet.! 37 This procedure is adopted in the case of Mars,Jupiter, and Saturrm Thus, in order to obtain the true longitude of Mars,Jupiter, or Saturn, one should proceed as follows First calculate the mean longitude of the planet (as corrected for the longitude, bhajantara, and ८ara corrections). Then subtract therefrom the longitude of the planet's mando८a (apoge): this gives the mardalk८rudra. Cal culate the corresponding bgja, and therefrom the blujahala by the appli cation of the formula: Rsin (bhuja)xcorrected manda epicycle 80 Subtract half of it from or add that to the mean longitude of the planet, according as the mandakendra is less or greater than 180° . Subtract the result ing longitude from the planet's 3gloc८८: this gives the 3ghrakendra. Calculate the corresponding bhuja, and therefrom the bhujahala by the application of the formula: (1) (2)

• See Parame5vara's commentary . According to the commentator

Sarikaranārāyapa, one should take here the mean longitude as corrected for the longitude and the blujantara corrections, and should apply the ८ra correction when all the other corrections have been performed. 38 Multiply this blujahala by the 3ghrakarr०, which is equal to radius (i.e., 3438') and divide by the (3) + or – sign being taken according as the 3gbrakendra is in the frst and fourth or second and third quadrants. Thern find the corresponding arc. Add half of it to or subtract that from the mean longitude of the planet, already corrected for half the bhaja than 180° From the result thus obtained subtract the longitude of the planet's maा। doca (apoge) : this gives the mandakerudra. Find the corresponding bluja, and therefrom calculate the bhujळ2hala by applying the formula (1) above Subtract this blujahala from or add that to the mean longitude of the planet (as corrected for the longitude, bhujantara and caru corrections), according as the mandak८rudra is less or greater than 180° : this gives the true-mean longi tude of the planet. Subtract this true-mean longitude from the longitude of the planet's 3ghroc८० : this gives the sigraटाdra. Find the corresponding bluja, and therefrom, by the application of formula (2) above, calculate the tained affresh by formula (3). Then find the corresponding arc, and add that to or subtract that from the true-mean longitude of the planet, according as the 51ghralk८nda is less than or greater than 180". The result thus obtained i the true longitude of the planet for true sunrise at the local place. For the Hindu epicyclic theory on which the above procedure is based, see my notes on MBl, iv. 40-44 A rule relating to the determination of the true (geocentric) longitudes of the inferior planets, Mercury and Venus 37(ii)-39. The method used in the case of Mercury and Venus is being described now First add or subtract half the arc corresponding to the ॐgh72}}|hala in the reverse order (i.e., according as the 3ghral८ndra is in the half-orbit beginning with Libra or in that beginning with Aries) to or from its own marud0८. Whatever correction is (then) derived from that (corrected) 7107do८a should, as a whole, be applied as correction to the mean longitude of the In the case of Mars,Jupiter and Saturn, the true-mean longitude is roughly the true heliocentric longitude and the true longitude, the true geo centric longitude. ws. 37(ii)39] planet : them is obtained the true-mean longitude (of the planet). That corrected for (the correction due to) the ॐghro८a gives the true longitude (of the planet). That is, first calculate the mean longitude of the planet (as corrected for the longitude, bhugjantara and ८ara corrections). Then subtract it from the longitude of the planet's 3gro८ : this gives the 3gra-kendra. Calculate the corresponding bugja, and therefrom the blujahala by the application of the formula : Rsin (buja) x corrected 31gra epicycle + or - sign being taken according as fourth or second and third guadrants. Multiply that by the radius and divide the product by thesigrakara, which is e१ual to ८ 80 the = 3grakeruda is in the 39 first Rsin (bugja) x corrected manda epicycle 80 (1) Thern fnd the corresponding arc. Add halfof that are to 0r subtract that from the mean longitude of the planet's mardoc८ (apoge), according as the grakendra is greater or less than 180°. Thus is obtained the corrected longitude of the planet's mand००८ (apoge) Now subtract the corrected longitude of the planet's randocra from the mean longitude of the planet: this gives the marudakerda. Calculate the corre5 ponding bhugja, and therefrom the blujalhala by the application of the (2) true and Subtract it from or add it to the mean longitude of the planet, according as the manda-terndra is less or greater than 180": this gives the true-mean long tude of the planet. Subtract this true-mean longitude from the longitude of the planet's 3ghrocca: this gives the 3grakendra. Calculate the correspond ing bhagja, and therefrom the blujahala by the application of the formula (1) 1 C. MBh, iv. 44. We have pointed out before (uide supra, i-9-14) that the mean longitude of the sigro८, in the case of Mercury and Venus, is the heliocentric mean longitude of the planet . The heliocentric longitude may be obtained by applying the planet's mandalhala correction to that. The true longitude obtained above is the true geocentric longitude. 40 above. Multiply that by the radius and divide the product by the 5.ghra aाःa which is obtained by formula (2) above. Then find the correspond ing arc, and add that to or subtract that from the true-mean longitude of the planet, according as the girakendra is less or greater than 180". The result thus obtained is the true longitude of the planet for true sumrise at the local place. Criterion for knowing whether a planet is stationary : 40. When (the longitude of ) a planet for today is equal to that for tomorrow, then is said to be the commencement or conclusion of the retrograde motion of that planet. A rule for finding the amounts of the retrograde and direct motions of a planet : 41. (Whatever is obtained on) subtracting the longitude (of a planet) for tomorrow from the longitude for today (when it is possible), is called the retrograde motion (of the planet for the day); and whatever results on performing the subtract ion reversely gives the direct motion (of the planet for the day). The commentator -Sarikaranarayapa interprets the text as follows:

• When the longitude of a planet calculated for tomorrow is less than the

longitude for today, the motion (ofthe planet) is said to be retrograde; when it is the contrary, the motion is direct.” DIRECTION, PLACE AND TIME FROM SHADOW Determination of the directions with the help of the shadow of a 1. The north and south directions should be determined by means of the fish-figure constructed with the two points where the end of the shadow of the gnommon, situated at the centre of an arbitrary circle (drawn on the ground) meets that circle (in the foremoon and in the aftermoon). Let ENWS (See Fig.3) be the circle drawn on the ground, and CO) its centre where the grnommon is situated. Let W be the point where the shadow of the grnommon enters into the circle in the foremoon arid B the point where the shadow passes out of the circle in the aftermoon. Then W and B are Fig. 3 the points where the end of the shadow of the grnommon meets the circle in the foremoon and aftermoon respectively. Join EW . The line EW, is directed cast to west. With B, as centre and with a radius greater than E,W, draw an arc of a circle, and with W1 as centre and with the same radius draw another arc cutting the former at the points N, and S. Join N, and S. The line NS, is directed north to south. Let the line N,S. meet the circle in the points N and S and the line through O drawn paralle to E.W, in B and W. Then B, W, N and S are respectively the east, west, morth and south directions with respect to the point (O. 42 The figure NBSWN, which looks like a fish has been called a “iिsh figure Also see my notes on MBh, i. 1, 2. ': A rule for finding the latitude and colatitude of a place from the eguin0ctial midday shadow of the grnommon : 2-3. By whatever results as the suare root of the sum of (1) the sपृuare of the eguin0ctial midday shadow of the gnommon, which is erected on level ground at the intersection of the direc tion-lines and wh0se perpendicularity has been tested, and (ii) the square of the gnormon, divide the radius (severally) multipli ed by the gnommon and the (eguinoctial midday) shadow : the results are the Rsines of the colatitude and the latitude. That is, if R be the radius of the celestial sphere, g the length.of the. gion0n, and s the length of the equinoctial midday shadow, } the latitude of the place, and C(=90- ) the colatitude, then s11

• *

= [ cr. Iा x R Fig. 4 Fig. 4 is the celestial sphere for a place in latitude . SENW is the hori टon, S, E, N and W being the south, east, north and west points; 2८ is the i.e., at the centre of the circle drawn on the ground. C. MBh, i. 4-50(i-iii). See stam (1) 1. (2) vs. 4 ] ASCENSIONAL DIFFERENCES OF SIGNS 43 zenith. RETW the equator, and P the north pole. R is the point where tht equator intersects the local meridian. Then the arc ZR defines the latitude of the place. OY is, the gnomon erected at the local place O perpendicular to the plane of the horizon. Let RD be perpendicular to the plane of the horizon and YX parallel to RO. Then we have two right-angled triangles YOX and RDO, right angled at O and D respectively. These triangles are similar and their corresponding sides are as follows: base upright hypotennse DO(=Rsin^) RD^(RsinC) RO(=R). OX(=s) YO(=g) YX^Jg^Tj Comparing these triangles, we have (1) and (2). A rule for finding the ascensional differences of the tropical signs Aries, Taurus, and Gemini* : 4. From the declinations of the last points of the (first three) signs should be obtained, as before, 1 their ascensional differences in terms of asus. When (each of them is) diminish- ed by the preceding (ascensional difference, if any,) they become (the asus of ascensional difference) for Aries, Taurus, and Gemini respectively. That is, if x,y, and z be the ascensional differences of the last points of the signs Aries, Taurus, and Gemini respectively, then the ascensional differ- ences of the signs Aries, Taurus, and Gemini are #, y— x, z-y respectively. We have already seen that the ascensional difference of the Sun is the difference between the times of rising of the Sun on the local and equatorial horizons. The ascensional difference of the sign Aries is the difference between the times of its rising above the local and equatorial horizons. Since the first point of Aries rises simultaneously at both the horizons, therefore the ascensional difference of Aries is equal to the ascensional difference of the last point of Aries (for which the celestial longitude X is equal to 30 d ). Simi- larly, the ascensional difference of Aries and Taurus (taken together j is equal to the ascensional difference of the last point of Taurus (for which X=60°). 1 Vide supra, Chapter II, stanza 18. The ascensional difference of Taurus is equal to the ascensional differ ence of Aries and Taurus minus the ascensional difference of Aries. That is to say, it is equal to the ascensional difference of the last point of Taurus minus the ascensional difference of the last point of Aries. The ascensional difference of Gemini, similarly, is egual to the ascen ional difference of the last point of Germini minus the ascensional difference of the last point of Taurus. The times of rising of Aries, Taurus, and Gemini at the equator: 5. 1670, 1795 and 1935 are (in 1ऽus) the times of rising of (the first three tropical signs) Aries etc., at Larika." It can be easily seen from the celestial sphere that the time of rising of Aries at the equator is cपृual to the right ascension of the last point of Aries, and the time of rising of Aries and Taurus (taken together), egual to the right ascension of the last point of Taurus. Thus the time of rising of Taurus is eual to the right ascension of the last point of Taurus minus the right ascension of the last point of Aries. Similarly, the time of rising of Gemini at the equator is cपual to the right ascension of the last point of Gemini minus the right ascension of the last point of Taurus where ८ is the right ascension of a point on longitude and 8 its declimation. 0 0 Therefore Putting 11 2 3141 x Rsin). Rcos 8 007 Hence the times of rising of the =30", 60° and 90" successively in (3), we Aries, Taurus ecliptic, . its (1) tropical (2) (3) casily get ८ =1670 and Gemini at the equator 1670 aऽus, (3465-1670) aऽus, and (540-3465) 454 i.e., 1670 ८sus, 1795 usus, and 1935 asur respectively. A rule for the determination of the times of rising of the tropical signs at the local place 6. (From the above times of rising of Aries, Taurus, and Gemini at Latikā should be subtracted the asus of their (own) ascensional differences, in order, a11d (then) (to the same times of rising of Aries, Taurus, and Gemini at Larika) they should be added in the reverse order : the results (in order) are the times (in 0545) of rising at the local place pf the tropical signs begin ming with Aries, and (the same results) in the reverse order (are for th0se) beginning with Libra.! Sign If a, b, ८ denote the ascensional differences of Aries, Taurus and Gemini respectively, then the times of rising of the signs at the local place are as given in the following table 1. Aries 2. Taurus 3. (Gemini 4. Cancer 6. Virgo Time of rising at the local place 1

• C. 4Bh, i. 10(ii)

1 45 1935-८ 1935-+८ 1795+b 1670+0 Sign 12. Pisces 45 10. Capriorm From what has been said above, we have Time of rising of a sign at the local place = time of rising of the sign at the e१uator {(ascensional difference of the last point of the sign) -(ascensional difference of the first point of the sign)} 9. Sagitarius 8. Scorpo 46 Hence the above rule. The following table gives the times of rising in ast4s of at Lucknow Sign 2. Taurus 3. (Gemini 4. Cance DIRECTION, PLACE AND TIME 6. Virgo Time of Rising in asus

1670-354= 1316 1795-290=1505 1935-119=1816 1935-+119=2054 1795-+290=2085 1670-+-354-2024 C. MBh, i. 18-20. the tropical signs Sign) 12. Pisces A rule for finding the Rsine of the Sun's zernith distance and the length of the shadow of the grnommon from the time elapsed since sunrise in the foremoon or to elapse before sunset in the 10. Capricorn 9. Sagitarius 8. Scorpio 7. Libra 7-10. The ghats clapsed (since sumrise) and to be elapsed (before sunset), in the first half and the other half of the day (respectively), should be multiplied by 60 and again by 6: then they (i.e., those ghats) are reduced t0 05us. (When the Sun is) in the northern hemisphere, the usus of the Sun's ascensional difference should be subtracted from them and (when the Sun is) in the southern hemisphere, they should be added to then (Then) calculate the Rsine (of the resulting difference or sum) and multiply that by the day-radius. Then dividing that (pro duct) by the radius, operate (on the quotient) with the earth sine contrarily to that above (i.e., add or subtract the earth sine according as the Sun is in the northern or southern hemis phere). Multiply that (sum or difference) by the Rsine of the colatitude and divide by the radius : the result is the Rsine of the Sun's altitude. The square root of the difference between the souares of that and of the radius is the Rsine of the Sun's टenith distance. That multiplied by twelve and divided by the Rsine of the Sun's altitude is the true shadow (of the grnommon). vs. 7-10, ] The ghatऽ clapsed since sumrise in the foremoon or to clapse before sunset in the aftermoon, being multiplied by 60 and again by 6, give the minutes of arc in the arc of the celestial'euator 1ying between the hour circles passing through the Sun at that time and through the Sun's positioth on the horizon t sumrise or sunset. When these asus are diminished or increased by the arsus of the Sun's ascensional difference (according as the.Sum is in the north ern or southern themisphere), the usus of the difference or sum give the minutes of arc in the arc : of the celestial cपृuator lying between the or sum multiplied by the day-radius and. divided by the radius gives the distance of the Sun from the line joining the points of intersection of the six o'clock circle and the Sun's diurrnal circle. This increased or diminished by the earthsine (according as . the Sun is in the northeri or southern hemis phere) gives the distance of the Sun from the Sun's rising - setting line (i.e., the line joining the points of the horizon where the Sun rises and sets) S3 In Fig.5 let S denote the position of the Sun on the celestial sphere, SA the perpendicular from S on the plane of the celestial horizon, and SB the perpendicular from S on the rising-setting line. Then in the plane triangle SAB, we have 47 ८ SBA=90० Rऽांn (given time in 05us - + asc. diff.) x day radius

---

+ earthsilue,

and Z८ SAB=90, where a is the Sun's altitude, and # the latitude of the place Therefore, we have Fig. 5 Als०since the Sun's 2ernith distance ८ is the complement of ८, therefore Rsin 2 = Rcos ८ = */फलकम्:R* - (Rsin ८)* 48 Now the triangle of shadow for that time in which base=shadow of the grmommon, upright=length of the mommon, and hypotenuse=hypotenuse of shadow is similar to the triangle of the great shadow in which base upright = Rsin a , and hypotenuse = R Hence comparing the bases and uprights of the two triangles, we have Rsin ८ x length of the grnommon shadow of the gnommon= taking length ofgnommon=12 argul05 Ifinstead of the time clapsed since sumrise or before sunset, the

t०८lapse:

longitude of the Sun and of the rising point of the cliptic be known, then the time clapsed since sunrise or t०८lapse before sunset may be determined by the following formula : time clapsed since sumrise =time or rising at the local place of the arc of the ecliptic 1ying between the Sun and the rising point of the ecliptic; time to elapse before sunset =duration of the day-time clapsed since sunrise (: twice the Sun's ascensional difference 30ghatऽ+ - time clapsed since sunrise, +or – sign being taken according as the Sun is in the southern hemisphere northern or Two particular cases of the above rule, viz. (i) when the Sun's ascensional difference is greater than the given time, and (ii) when the Sun is below the horizon : 11. When the Sun's ascensional difference cannot be sub tracted from the given (time in) asus, reverse the subtraction (..subtract the latter the ) and with the e, from formerRsine of the remainder (proceed as above). 1. In the might the Rsine of vs. 12-15 ] the Sun's altitude should be obtained contrarily (i.e., by revers ing the laws of addition and subtraction). The first part of the rule relates to the case when the Sun is in the nor thern hemisphere and lies between the local and equatorial horizons, i.e., shortly after sumrise or before sunset. 49 The second part of the rule indicates the method to be used for finding the Sun's altitude in the night. The details of the method are given by the

• (When the Sun is) in the northern hemisphere, having calculated the

Rsine of the given mocturnal aऽus (i.e., those clapsed since sunset in the first half of the might or those to clapse before sumrise in the second half of the might) as increased by the (Sun's) ascensional difference, (thern) multiplying (that) by the day-radius and dividing by the radius, then from the (resulting) guotient subtracting the earthsine, and (finally) multiplying the remainder by the Rsine of the colatitude and dividing by the radius is obtained the Rsine of the Sun's altitude. (When the Sun is) in the southern hemisphere, the (Sun's) ascensional difference and the earthsine are (respectively) subtractive

The Rsine of the Sun's altitude in the might is required (i) in the calcu lation of the elevation of the lunar horms, and (2) in the calculation of the solar eclipse. A rule for calculating the time elapsed since sumrise in the fore moon or to clapse before sunset in the aftermoon with the help of the shadow of the gnommon : 12-15. By the divisor, which is the suare root of the sum of the sपuares of the grnommon and its shadow, should be divided the radius multiplied by the gnormon: (the result is) the Rsine of the Sun's altitude. From that are obtained the ghas (of the time elapsed since sunrise in the foremoon or to clapse before sunset in the aftermoon) (by proceeding) conversely (to Rule 7-10) (in the following manner): C. MBh, i. 26. The Rsine of the Sun's altitude should be multiplied by the radius and divided by the Rsine of the colatitude. In the 50 (resulting) quotient should be subtracted or added the earthsine according as the Sun is in the northern or southern hemisphere. Then having multiplied that (result) by the radius and divided by the day-radius, to the arc of the (resulting) quotient add or subtract from the same arc the 05us of the (Sun's) ascensional difference (according as the Sun is) in the northern or southern hemisphere. (Dividing the resulting 25us) by 6 and again by 60 should be determined the ghas elapsed (since sumrise) and to elapse (before sunset) in the first half and the second half of the day (respectively). This rule is the converse of that given in starm2as 7-10 above. A rule for the calculation of the Sun's saikuagra : 16. The Rsine of the Sun's altitude multiplied by the Rsine of the latitude and divided by the Rsine of the colatitude is the (Sun's) 5aikuagra, which is always to the south of the rising-setting line. The Sun's 5aikuagra denotes the distance of the Sun's projection from the (sum's) rising-setting line . In Fig. 6,S denotes the Sun, A the foot of the perpendicular dropped from the Sun on the plane of the celestial horizon, SB the perpendicular from S on the rising-setting line, and AB the perpendicmlar from A on the rising setting line. So AB is evidently the 5aiku4grou. Since SA = Rsin ८, AB = saikण्0gra, //SBA = 90-4, //८ASB = }, therefore, we have giving AB 1 Cf. MBh, i.27-28(). SA Rcos Fig. 6 vs. 17-19 ] 51 It can be easily seen from the celestial sphere that, whatever be the position of the Sun, the aikuagra will always lie to the south of the rising setting line A rule for the determination of the tropical longitude of the rising point of the ecliptic at any given time with the help of the time elapsed since sumrise and the corresponding tropical longi 17-19. The residue (i.e., the untraversed portion) of the Sun's (tropical) sign (in minutes of arc) should be multiplied by the time of its rising at the local place and divided by the num ber of minutes in a sign (i.e., 1800): the result should be sub tracted from the given (time elapsed since sumrise, in) 0545 (Then)having added the residue of the (Sun's) sign to (the tropi cal longitude of) the Sun, one should (further) add successively the (subsequent) signs whose times of rising, in 25us, at the local place can be subtracted from the remaining (time in) 0545. That (further) increased by the de8rees and minutes obtained on multiplying the remainder (in 0545) by 30, etc., and dividing by the time of rising at the local place of the (tropical) sign occupied by the rising point of the ecliptic should be declared as the (tropical) longitude of the rising point of the ecliptic. We shall illustrate this rule by an example. Suppose that 14 ghags after sunriseat Lucknow (lat.26°55' E, long. 80°45'B.) the tropicallongitude of the Sun is 3 4°20'. Then to find the longitude of the rising point of the cliptic we shall proceed as follows: The Sun lies in the 4* sign, the untraversed portion of that sign being 25० 10' (=1510'). The time of rising of this sign at Lucknow is 20054 15us (See the table of times of risings of signs at Lucknow on page 46). There fore, we multiply 1510' by 2054 and divide the product by 1800; thus we get 1510x2054 172 05us approx 1800 This is the time of rising of the untraversed portion of the Sun's sign Subtracting this from 14 gats, i.e., 5040 05us, we get 3317 25us. Sub tracting from this the 25us of rising of the 5b sign, i.e., 2085 25u, we get 232 ८45us. The atus of rising of the 62 sign cannot be subtracted from it, so we 52 multiply this by 1800 and divide the product by the asus of rising of the 6* sign, i.e , by 2024. Thus we get 232x1800 =202' or 3"22' approx . 2024 Thus we see that in 14 gla!ऽ 25"10" of the 4" sign, the whole of the 5७ sign and 3°22'of the sixth sign have risen above the horizon of Lucknow. Adding the७e to the Sun's longitude, we get 53°22' as the tropical longitude of the rising point of the ecliptic. A rule for obtaining the time elapsed since sunrise with the help of the tropical longitude of the rising point of the ecliptic and the tropical longitude of the Sun : 20. One who desires to know the time (elapsed after sum rise) obtains that time on adding together, in the reverse order the times of rising at the local place of the signs (and parts thereof, if any) traversed by the horizon-ecliptic point up to the untraversed portion of the Sun's (tropical) sign.! This rule is the converse of the preceding one . A rule for calculating (the Rsine of) the Sun's graः 21. The result obtained on dividing the Rsine of the bhaga of the Sun's (tropical) longitude as multiplied by the Rsine of the Sun's greatest declination, by the Rsine of the c6latitude is known as (the Rsine of ) the Sun's agra.* That is, where x is the Sun's (tropical) longitude, e the Sun's greatest declimation (i.e., the obliguity of the ecliptic), and # the latitude of the local place [For the rationale of this formula, see under stanzas 22-23 below.]] () The term agra, in Hindu astronomy, has been used in two senses : The are of the celestial horizon 1ying between the point where the heavenly body concerned rises. cast point and the vs. 22-23 ) the east (ii) The Rsine of that are, which is equal to the distance between west line and the rising-setting line of the heavenly body concerned. To avoid this ambiguity, we have translated the term agra by *(the arc (the Rsine of) aga', according as it is used in the former or 1atter sense. hcrnever the qualifying phrases have not been used the mean ing should be understood from the context A rule for calculating the prime vertical altitude of the Sun and the corresponding shadow of the grnommon: 22-23. The Rsine of the Sun's northern declimation, when less than the Rsine of the latitude, multiplied by the radius should be divided by the Rsine of the latitude : the result is the Rsine of the altitude of the Sun when it is on the prime vertical The sguare root of the square of the radius diminished by that of the Rsine of the Sun's altitude (obtained above) when multi plied by twelve and divided by the ( same) Rsine of the Sun's altitude gives the shadow (of the gnommon corresponding to the Sun on the prime vertical). 0 ) That is, when the Sun is on the prime vertical, () (ii) Rsin a = where a is the Sun's the place Rsin 8 Shadow of the gnommon 12 x R altitude, ४ the Sun's declination, and * the latitude of In Fig. 6 on page 50, let S denote the position of the Sun when it is on the prime vertical, SA and SB the perpendiculars from S on the east-west and rising-setting lines respectively, and C the point where SB meets the |ime joining the points of intersection of the Sun's diurnal circle and the six o'clock circle. Since SB 1ies in the plane of the diurmal circle and AC in the plane of the six o'clock circle, and the two circles are at right angles, therefore AC and SB are at right angles 54 Therefore, Therefore But, from i. 16, In the triangle ABC, we have where 8 is the Sun's declination, and ; the 1atitude of the place. AC = Rsin ० , Therefore and ZACB = 90°, AB = Rsin{Sun's agra), Rsin (Sun's agra) = Rcos where x is the Sun's tropical longitude. [See rule 21 above.1 SA = Rsim a , AB Now in the triangle SAB, right-arngled at A, we have and 4८ SBA = 90"-4. Rऽin 8 C0S Rsin x R x -, using (1), (1) (2), where ८ is the Sun's altitude. This formula can also be derived directly by considering th८ triangle SAC. The formula for the shadow of the grommon easily follows from the shadow triangle. The condition that “the Rsine of the Sun's northern declimation should the R७ine of the latitude" is necessary for the of be less than existence vs.27-28 ] 55 the prime vertical shadow of the gnormon. When this condition is not satis fed, the Sun in the northern hemisphere would not cross the prime vertical and likewise the prime vertica1 shadow of the grommon would not exist When the Sun is in the southern hemisphere, the Sun does not cross the prime vertical above the horizon and so the prime vertical shadow of the grnommon does not exist A rule for the determination of the Sun's tropical longitude from the Sun's prime vertical altitude 24-25. The Rsine of the Sun's altitude (when the Sun is on the prime vertical), determined from the method of the shadow, should be multiplied by the Rsine of the latitude and divided by the Rsine of the (Sun's) greatest declination: the resulting Rsine, in minutes of arc, reduced to arc or that subtracted from half a circle (i.e., 180°) is known as the (tropical) longitude of the Sun determined from the shadow of the grommon when the Sun is on the prime vertical (according as the Sun is in the first or second quadrant, i.e. according as the prime vertical shadow or midday shadow of the grnommon is decreasing or increasing day to day).* This rule follows from the previous one combined with rule i. 16 A rule for finding the arc corresponding to a given Rsine : 26. The number of the tabulated Rsine-differences which can be subtracted from the given Rsine, as also the remainder (of that subtraction, if any) 'divided by the current (i.e., next) Rsine-difference, should be (severally) multiplied by 225: their sum is the (required) arc.* This rule is the converse of that given in i. 2(ii)-3(i) above. It has been stated here because it is required in the preceding rule for calculating the are corresponding to the Rsine of the Sun's longitude. A rule for finding the midday shadow of the grnommon with the help of the Sun's declination and the latitude of the place: 27-28. In case the Sun is situated on the meridian (1it. in 56 the middle of the sky), the Rsine of the sum or difference of the arcs of the latitude and the (Sun's).declimation according as the Sun is in the southern or morthern hemisphere, is the (great) shadow. Whatever is the sguare root of the number which is obtained on subtracting the square of that fro1 the square of the radius is the (great) grnommon. The shadow of the gnommon of twelve (aigulds) should be determined by proportion. The great shadow is the Rsine of the Sun's zernith distance, and the great gnommon is the Rsine of the Sun's altitude . When the great shadow and the great grnormon are known, the shadow of the gnommon of twelve aigulas is obtained by the formula: great shadow x 12 shadow of the gnommon= great A rule for the determination of the Sun's longitude from the midday shadow of the gnommon 29-33. The sguare root of the sum of the squares of the grommon and its midday shadow is the divisor of the product of the (midday) shadow. and the radius: the resulting quotient is the Rsine of the (Sun's) meridian Zermith distance. (When the midday shadow falls) towards the north, the Sun's meridian 2emith distance, if less than the latitude, should be subtracted from the latitude; when the (midday) shadow falls towards the south, take their sum: the result (in b0th cases) is the (Sun's northern) declimation. * In the contrary case (i.., when the midday shadow falls towards the north but the Sun's meridian zernith distance is greater than the latitude), the latitude should be subtracted from the(Sun's) meridian 2emith distance the (resulting) remainder is the Sun's southern declination.* The Rsine of that (i.e., the Sun's declimation, north or south) should be multiplied by the radius and divided by the (Sun's) greatest declination: the arc corresponding to the quotient or that vs. 35 ] 57 subtracted from half a circle (i.e., 180°) is known as (the tropical longitude of) the Sun (according as the Sun is in the first पृuadrant beginning with the tropical sign Aries or in the second guadrant beginning with the tropical sign Cancer, i.e., according as the midday shadow, if falling towards the north, is decreasing or increasing day to day, and, if falling towards the south, is in creasing or decreasing day to day). This method is for (the Sun in) the northern hemisphere. Now we describe the method for (the Sun in ) the southern hemisphere. (There) the arc (obtained above) should be added to half a circle or subtracted from 12 (signs) (i.e , from 360°) (according as the Sun is in the third quadrant beginning with the tropical sign Libra or in the fourth quadrant beginning with the tropical sign Copriorm, i.e according as the midday shadow falling towards the north is increasing or decreasing day to day). A consolidated rule for finding the Sun's declimation with the help of the Sun's meridian Zenith distance and the latitude 34. The sum or difference of the meridian 2ermith distance and the latitude, according as the (midday) shadow of the gno m0rn falls towards the south or towards the north, is known as declimation. A rule for finding the local latitude with the help of the meridian 2emith distance and declination of the Sun and the direction of the midday shadow of the gmon01) 35. When the Sun is in the northern hemisphere, the (meridian) 2emith distance and the declination of the Sun should be added together (if the midday shadow of the grnommon falls towards the north). In the contrary case (viz. when the Sun is in the southern hemisphere), or when the (midday) shadow falls in the contrary direction (i.e., towards the south), one should take their difference. The result (in each case) is the latitude. " C. MBh, i. 17 A rule for the determination of the longitudes of the Sun and the Moon when they are in opposition or conjunction in longitude 1. One who wants to obtain (the longitudes of the Sun and the Moon when there is) euality in minutes of arc! should add as many minutes of arc as there are 20ruard ts, to the Sun's longitude (at sumrise) and the same together with the minutes of arc (of the difference between the longitude of the Sun as increased by 6 signs, and the longitude of the Moon in the case of opposition, or of the difference between the longitudes of the Sun and the Moon in the case of conjunction) to the Moon's longitude (when opposition or conjunction of the Sun and Moon is to occur); similarly, (when opposition or conjunc tion of the Sun and Moon has occurred) one should subtract the 2ratjud-tatऽ (etc. from the longitudes of the Sun and the Moon) In other words, if S and 4 denote the longitudes of the Sun and the Moon at sumrise on the full moon day, then Sun's longitude at the time of opposition of the Sun and Moon and Moon's longitude at the time of opposition of the Sun and Moon +(S+6 signs-4); and ifS' and M'denote the longitudes of the Sun and the Moon at sumrise on the new m00n day, then Sun's longitude at the time of conjunction of the Sun and Moon

• When the Sun and Moon are in opposition, their longitudes differ

by six signs; when they are in conjunction, their longitudes are the same. The minutes, however, are the same. The equality in minutes of arc refers here to the time ofopposition or conjunction. VSS. 2-3 ] DISTANCES OF THE SUN AND MOON 59 and Moon's longitude at the time of conjunction of the Sun and Moon =M'+patvanadh treated as minutes of arc + (S r -M'). If Si and Mj denote the longitudes of the Sun and the Moon at sunrise on the day following full moon, then Sun's longitude at the time of opposition of the Sun and Moon =Si" pratipad-riadls treated as minutes of arc, and Moon's longitude at the time of opposition of the Sun and Moon ^M^pratipad-riedis treated as minutes of arc -[Mi — (^+6 signs}]; and if 'SJ and M/ denote the longitudes of the Sun and the Moon at sunrise on the day following new moon, then Sun's longitude at the time of conjunction of the Sun and Moon —5^' ^pratipad-riadls treated as minutes of arc, and Moon's longitude at the time of conjunction of the Sun and Moon s= M t ' —pratipad-riadls treated as minutes of arc By parvariadls is meant "the riddis of the full moon or new moon tithi (called parva) which are to elapse at sunrise on that day". Similarly, by pratipad-riadls is meant "the riadls of the next tithi (called pratipad or pratt- pada) which elapse at sunrise on that day." The above rule gives only an approximate result because it is based on the assumption that the Sun travels at the rate of one minute of arc per riadl, but for practical purposes it is good enough. Mean distances of the Sun and the Moon in terms oiyojanas : 2, 459585 is (myojanas) the (mean) distance of the Sun and 34377 that of the Moon. 1 A rule for finding the true distances of the Sun and the Moon in terms oiyojanas : 3. These (above-mentioned mean distances of the Sun and the Moon) multiplied by their true distances in minutes

• The same distances are given in MBh, v. 2; and SiDVr, I» iV. 4. []

obtained by the method of successive approximations and divided by the radius (i.e., by 3438) give their true distances in That is, Sun's true distance in ygja705 and Moon's true distance in y9jaras Sun's meam distance in y9a71015xSun's true distance in minutes A rule for finding the and Moon's meam distance in y9jarl45 x Moon's 20C. Diameters of the Sun, the Moon and the Earth : 4. The diameter of the Sun is 410 (19jaraऽ); of Moon, 315 (wgjaras); and of the Earth, 1050 (19ja745). * 343ॐ' That is, angular Moon's diameter in true distance in diameters of the Sun 5. Multiply the radius ( i.e., 3438') (separately) by their diameters in y9jar105 and divide by their true distances in 19jaraऽ: then are obtained their true (i.e., angular) diameters in minutes Sun's diameter in y9jaras x 3438' Sun's true distance in 199jaras minutes of arc: minutes Moon's diameter in yojards x 3438 Moon's true distance in y9jaras and the the

• See supra, i.7.
• The ame rule is given in MBh, v.3; 5:D7, 1, iv. 5(); S:5, v. 4 (ii);

SiSi, 1, v. 5(); and 7S, iv. 10(ii)-11 .

• The same walues are given in MBi, v. 4; SiDW, I, iv. 6 (); and 7७,

iv. 10(i).

• Cf. AMBh, wः 5. vs. 7 ]

4 rule for the determination of the length of the Earth's shadow: 6. Multiply the Sun's (true) distance (in y9jaras) by the diameter of the Earth in 19janus and divide by the difference between (the diameters of) the Sun and the Earth. Then is obtained (in.99jaras) the length of the Earth's shadow. That is, length of the Earth's shadow Sun's true distance in 19ja70s x Earth's diameter Sun's diameter - Earth's diameter 61 By “the length of the Earth's shadow” is meant “the distance of the ver A rule for the determination of the diameter of the Earth's shadow at the point where the Moon crosses it, in terms of 7. This (length of the Earth's shadow) diminished by the (true) distance of the Moon and multiplied by the diameter of the Earth and (then) divided by the length of the Earth's shadow gives (in 19jamas) the diameter of the Earth's shadow (at the point where the Moon crosses it). This should be reduced to minutes of arc like (the diameter of) the Moon. That is Diameter of the Earth's shadow (length of Earth's shadow - Moon's true distance) Earth's diameter length of Earth's shadow By “the diameter of the Earth's shadow' we mean “the diameter of the section of the Earth's shadow come where the Moon crosses it at the time of For the Hindu method of deriving the formulae of stanzas 6 and 7, see my notes on MBh, v. 71-73 62 A rule for finding the Moon's latitude : 8. Multiply the Rsine of the difference between the long tudes of the Moon, when in opposition with the Sun, and its as cending mode by 270 and divide (the product) by the true dis tance of the Moon, in minutes : the result is the Moon's (true) latitude, north or south.1 That is, where M, 68 denote the longitudes of the Moon and Moon's ascending node. This formula is evidently wrong normers. The correct formula is ) x 270' and has been discarded by later astro Rsin (M-68) x 270' approx. A rule for finding the measure of the Moon's diameter unobs cured by the shadow 9. Diminishing the (minutes of arc of the) Moon's latitude (obtained above) by half of the minutes of arc resulting on diminishing the diameter of the shadow by that of the Moon are obtained those of (the diameter of) the Moon which remain umobscured by the shadow.* It is easy to see that the obscured part of the Moon's diameter (at the time ofopposition in the case of a partial lunar eclipse) = } (diameter of shadow + Moon's diameter) - Moon's latitude, and hence the unobscured part of the Moon's diameter at that time e= Moon's latitude - (diameter of the shadow - Moon's diameter) Bhaskara I does not make any distinction between the time of opposition and the time of the middle of the clips. Hence the above rule.

• C. MBh, v. 30-31(i). This rule occurs also in 7S, iv. 17(ii)-180()
• This rule occurs also in 2, iv.43; 5#D7, 1, v. 13; S:5, v. 11; /ऽ,

w.7; TS, iv. 19(ii)-20(i). Also see SiS, iv. 10; BSS, iv. 7; SS, 1, v. 11. vss. 10-12 ] STHITYARDHAS 63 A rule relating to the calculation of the sparsa- and moksa- stkityardhas : 10-12. Diminish the square of half the sum of the diameters of the Moon and the shadow (samparkardha) by the square of the (Moon's) latitude (for the time of opposition of the Sun and Moon) and then take the square root (of that). That divided by the difference between the (true) daily motions (of the Sun and Moon) and multiplied by 60 gives, in nadis, the (first ap- proximation to the sparsa- or moksa-) sthityardha. (Then) multiply those nadls by the true daily motion (of the Moon) and always 1 divide by 60. The resulting minutes should then be severally subtracted from and added to the longitude of the Moon (calculated for the time of opposition) to get the longi- tudes of the Moon for the times of the first and last contacts. From the Moon's longitude (for the first contact as also for the last contact) calculate the Moon's latitude; and from that successively determine the (corresponding sthityardha in terms of) riadis, the corresponding minutes of arc (of the Moon's motion), and the longitude of the Moon (for the first contact as also for the last contact). Repeating this process again and again, find the nearest approximations to the (spar'sa~ and moksa-) sthityardhas? The t#rm samparkardha means "half the sum of (the diameters of) the eclipsed and eclipsing bodies". In the case of a lunar eclipse, it denotes the sum of the diameters of the Moon and the shadow. The term sthityardha means "half the duration (of the eclipse)" and denotes, in the case of a lunar eclipse, the time-interval between the. first contact and opposition or bet- ween opposition and the last contact. The interval between the 6rst con- tact and opposition is called the sparsa-sthityardha (or sparsika sthityardha) and that between opposition and the last contact is called the moksa' sthityardha (or mauksika sthityardha). The above three verses say how to find the sparsa- and moksa- Sthityardhas. The method used is the method of successive approximations and may be ex- plained as follows : 1 i.e., in every approximation. 1 Cf. MBh, v. 74-76(i). See Fig. 7. AB is the ecliptio; S is the centre of the shadow for the time ofopposition, the circle around S being the circumference of the shadow CD is the Moon's orbit relative to the shadow centred at S, and Mis the posi tion of the Moon at the time ofopposition (CD, is drawn through M para Fig.7 the parsa-stiyardha could be obtained at once by considering the triangle M'L'S, right-arngled at L. But the Moon's latitude for the time of the first contact (viz. M'I') itself depends on the knowledge of the spor50-ऽthiyardha. Hence we use the method of successive approximations. To begin with we neglect the variation of the Moon's latitude and take MS as the Moon's latitude throughout the eclipse. Thus we take M, to be the position of the Moon for the time of the first contact. where Let M,[1 be the perpendicular to the ecliptic. Then from the triangle 7 and MS=half the sum of the diameters of the Moon and the shadow (1) . gives LS, i.e., the distance along the ecliptic to be traversed by the Moon with respect to the shadow during the baा50-ऽthijardlha. Thus if mr denote the daily motion of the Moon with respect to the shadow, then ia (1) . 1 Neither the ecliptic mor the Moon's orbit is a straight ime but their arcs which we are considering are so small that they may be regarded as such without much error. VS. 13] TIMES OF FIRST AND LAST CONTACTS 65 This is the first approximation to the sparsa sthityardha. Let us denote it by t r Now we calculate the displacement of the Moon for the sparsa-sthityardha t v then diminish the Moon's longitude (calculated for the time of opposition) by that displacement, and then with the help of the resulting longitude calculate the Moon's latitude. Treating this as the Moon's latitude for the time of the first contact, we calculate, as before, the sparsa-sthityardha again. This is the second approximation to the sparsa-sthityardha. Let us denote it by t 2 . Repeating the above process, we calculate the successive approximations t,, t 4 , t 5 to the sparsa-sthityardha. It can be easily seen that ^ - . . . 60xL'S tx <t, <t 3 <... <t n <... < — ~ . Therefore, the sequence of the successive approximations to the sparsa- sthityardha is convergent. The convergence is also rapid, so that the third or fourth approximation generally gives a fairly good approximation to the sparsa sthityardha. The method for finding the moksa-sthityardka is similar. The only differ- ence is that in the second and the next successive approximations calculation is made of the Moon's latitude for the time of the last contact instead of that for the first contact. A rule relating to the determination of the times of the first and the last contacts : 13. Diminish and increase the true time of opposition by the {sparsa^ and moksa-) sthityardhas, obtained by the method of successive approximations, (respectively): then are obtained the times of the first and the last contacts. The time of the middle of the eclipse is the same as that (of opposition of the Sun and the Moon). 1 This is how the exact times of the beginning and end of a lunar eclipse are determined. In practice, however, the exact beginning and end of an ' eclipse are not perceived with the unaided eye. A lunar eclipse is seen to begin after a portion of the Moon's disc is already obscured by the shadow.

• Cf. MBh,v. 35. 66

THE LUNAR ECLIPSE [CH. IV Sarikaranarayapa tells us how to find the times when a lunar eclipse is actually seem to begin and end. He says: “At the beginning, having diminished the sixteenth part of the Moon's diameter from half the sum of the diameters of the Moon and the shadow (then) having squared it and subtracted from it the s9uare of the Moon'slati tude, one should obtain half the (apparent)duration of the lunar e:lipse by the method of successive approximations. Or, one should multiply the sixtee nth portion of that (semi-duration) in minutes by 60 and divide by the difference between the daily motions of the Sun and the Moon, and then reduce that to ugbats etc. Having thus ascertained the corresponding time (in igha s etc.), the apparent instant of the first contact should be declared by adding that to the instant of the first contact. After that, in order to determine the instant of the last contact, the mosa-stliyardha obtained by the method of successive approximations should be added to the instant of opposition and the result taken, as before, as the instant of the last contact. There also the (apparent) time should be announced affer diminishing it by one-sixteenth (of the time corresponding to the m0 0-sthijyardha). Then adding the two ऽthijyardlhas (i.e., the spur3a - and moksa-ऽthijuardlhas), the sum) should be declared, in ghas etc., to be the duration of the eclipse. of In support of his statement , Sarikaranārāyapa* quotes the following verse carya Buatta Govinda bharati taderndagrahar:०मौः na bharatyale'rdla50ाpark८.* 1.८., When half the sum of the diameters of the Moon and the shadow dimi mished by the sixteenth portion of the Moon's diameter is greater than the Moon's latitude (for the time ofopposition), then does a lunar eclipse occur (i.e., is observed). When half the sum.of the diameters of the Moon and the shadow (diminished by the sixteenth part of the Moon's diameter) is smaller a lumar eclipse does not occur (i.e., is not observed The statement that the time of the middle of the eclipse is the same as that of opposition of the Sun and Moon is only approximately true. Am accurate expression for the difference between the two instants was first given by Game5a Daivaja (1520) From Saikaranārāyapa's comm. on the verse under consideration. See his comm. on LB, v. 9 शशिदेहाष्ट्यंशोनं सम्पर्कदलं यदा नतेरधिकम भवति तदेन्दुग्रहृपं च भवत्यल्पेऽर्धसम्पर्के ।। vs. 14] VIMARDARDHAS 67 A rule for finding the sparse- and moksa- vimardardhas: 14. The square root of the difference between the squares of the Moon's latitude and half the difference between (the diameters of) the eclipsed and eclipsing bodies leads, as before, to the determination of the (nearest approximation in) riadis of the (spar'sa-vimardardha as also of themoksa~) vimardardha. ¹ The term vimardardha means "half the duration of totality (of an eclipse)"* and denotes, in the case of a lunar eclipse, the interval between the times of immersion (of the Moon into the shadow) and opposition (of the Sun and Moon), or between the times of opposition (of the Sun and Moon) and emersion (of the Moon out of the shadow). The interval between the times of immersion and opposition is called the spar'sa-vimardardha; and the interval between the times of opposition and emersion is called the moksa-vimardardha. The method for finding the sparsa- and moksa- vimardardhas, given above, is similar to that for rinding the sparsa- and moksa- sthityardhas, stated in stanzas 10-12 above. The difference is that in place of the sum of the semi-diameters of the Moon and the shadow use is made in the present case of their differ- ence. The remainder of this chapter deals with the graphical representation of an eclipse. This requires the knowledge of valana, i.e., the deflection of the ecliptic from the prime vertical on the horizon of the eclipsed body (i.e., on the great circle having the eclipsed body at either of its poles). For the con- venience of calculation, this valana is broken up into two components called' the aksa-valana and the ayana-valana. The former is the deflection of the equa- tor from the prime vertical on the horizon of the eclipsed body, whereas the latter is the deflection of the ecliptic from the equator on the horizon of the eclipsed body. Thus if A, B, G be the points where the prime vertical, the equator, and the ecliptic intersect the horizon of the eclipsed body towards the east of the eclipsed body, then the arc AB denotes the aksa-valana, the arc BG denotes the ayana-valana, and the arc AC denotes the valana. « Cf. MBh, v. 76(H). 68 A rule relating to the determination of the magnitude and direction of the alka-ular0 : 15-16. Multiply the Rsine of the (local) latitude by the Rversed-sine of the 15us between the times of (the beginning , piddle, or end of) the eclipse and the middle of the might or day1, and divide by the radius (.८.,3438') : (the result is the Rsine of the nk50-20alu710). The direction of the result (i.e., nk90 ulamu) is (determined) in the following manner : (If the eclipsed body, at the time of the first or last contact is) in the eastern half of the celestial sphere, the directions of the aks0-0alamu for the eastern and western halves of the disc (of the eclipsed body) (.e., of the sharsa-and m20k90-0alam41 in the case of the Moon and nice uerऽa in the case of the Sum) are 10rth and south (respectively); (if the eclipsed body is) in the western half of the celestial sphere, (they are to be taken) reversely.* “The usus between the times of (the beginning, middle, or end of) the eclipse and the middle of the might or day are the ८sus of the hour angle* of the eclipsed body for that time. Thus the rule given in the text is egua valent to the following formula :

3438 3 We where H is the hour angle of the eclipsed body, and # the latitude of the local place. This formula, as pointed out by me in the Maha-Bhaskary0 is inaccu rate. For details see my motes on MBi, v. 42-44 . s Night when the eclipse is lunar and day when the eclipse is solar . Measured east or west of the local meridian vs. 17] A rule relating to the determination of the magnitude and direc tion of the (90710-0alam0 17. The Rsine of the declimation calculated from the Rversed sine of the kott of the tropical (sy0 ) longitude of the Sun or Moon1 for that time (i.e., for the beginning, middle, or end of the eclipse) (is the Rsine of the 4y270-0ala70). In the eastern half of the disc (of the Sun or Moon), the direction (of the guru0 ula710) is the same as that of the para0* (of the Sun or Moon) In the western half, the direction is contrary to that of the gyna. If ). be the tropical longitude of the eclipsed body, then its k0t is 2 -) , 2.-90, 270"-५, or ).-270", according as the clipsed body is in st second, third, or fourth quadrant 69 If denote the 501 of the tropical longitude of the eclipsed body, then, according to the above rule where e is the obliguity of the ecliptic. Rsin (gyan0-0al474) == This formula is equivalent to that given by the author in the Malta-Bhas karya, where I has been replaced by the bhaja of .+90-.* For the bluja of .-+-90° is equal to 90°-).,2-909, 270°-*, or ).-270", according as the eclip sed body is in the first, second, third, or fourth quadrant. Correct. As pointed out by me in the Mala-Bhaskarya, the above formula is in 1 The Sun is taken when the eclipse is solar, and the Moon is taken when the eclipse is lumar.

• 4jyana means “the northerly or southerly course (of a planet)”. The

course (yana) is north or south according as the planet lies in the half orbit beginning with the tropical sign Carpricorn or in that beginning with the tr० pical sign Cancer

• In my note to MBh, v. 45, Rversi0 (.+90°) stands as usual for Rversin

{tlja (.+-90")} w. 45, notc. 70 THE LUNAR ECLIPSE [CH. IV A rule relating to the determination of the resultant valana cor- responding to the circle having half the sum of the diameters of the eclipsed and eclipsing bodies for its radius : 18. Take the sum of their arcs (i.e., of the aksa-valana and ayana-valana) when they are of like (directions) and the diffe- rence when they are of unlike directions. Multiply the Rsine of that (sum or difference) by the sum of the semi-diameters of the eclipsed and eclipsing bodies and divide by the radius: this result is the valana¹ . The valana obtained by this rule is the Rsine of the valana corresponding the circle of radius equal to the sum of the semi-diameters of the eclipsed and eclipsing bodies. A rule relating to the determination of the corrected valana {sphuta- valana): 19-20. If the valana (obtained above) is of the same direc- tion (as the Moon's latitude) add it to the Moon's latitude; if it is of the contrary direction, subtract it (from the Moon's latitude). The (sum or difference thus obtained) is known as the corrected valana {sphufa-valana) in the case of solar and lunar eclipses 2 . In case that (corrected valana) is found to be greater than the sum of the semi-diameters of the eclipsed and eclipsing bodies, it should be subtracted from the entire sum of the semi-diameters of the eclipsed and eclipsing bodies and the remainder (thus ob- tained) should be taken as the (corrected) valana. The corrected valana is supposed to give the distance of the centre of the eclipsing body from the east-west line drawn through the centre of the ec- lipsed body in the projected figure. As pointed out by me in the Maha-Bhaskarlya, the addition or subtraction of the valana and the Moon's latitude is not proper. Both the quantities i Cf. MBh,v. 46-47 (i).

• Gf. MBh> v. 47. VSS. 23-30] GRAPHICAL REPRESENTATION 71

should be kept separately and laid off one after the other in the projected figure. A rule relating to the valana for the middle of the eclipse : 21. The (resultant) valana for the middle of the eclipse ob- tained in the same way as for the first contact without any further addition or subtraction of the Moon's latitude is the corrected {valana for the middle of the eclipse). The direction of that (Moon's latitude) is to be taken reversely (in the projection of a lunar eclipse). 1 What is meant is that the valana for the middle of the eclipse (which is calculated according to the rule stated in stanza 18) should not be combined with the Moon's latitude for that time (although such a rule is given in stanzas 19-20). The two quantities should be kept separately and laid off one after the other in the projected figure in the manner prescribed in stanzas 23-30 below. The latter part of the stanza says that in drawing the figure of a lunar eclipse, the direction of the Moon's latitude is reversed. That is, when it is north, it is taken as south; and when it is south, it is taken as north. The rea- son is that in the case of a lunar eclipse, we find the position of the shadow with reference to the Moon; and when the Moon is north of the ecliptic (i.e., when the Moon's latitude is north), the shadow is to the south, and vice versa. A rule for converting minutes of arc into ahgulas : 22. The minutes of arc of the diameters of the Sun, Moon, and the shadow and those of the (Moon's) latitude and the (cor- rected) valana when divided by two are reduced to ahgulas, (But when the Sun and Moon are) on the horizon, they (i.e., minutes of arc) are the same (as ahgulas)? A rule relating to the construction of the figure of an eclipse : 23-30. Draw a circle with a thread equal in length to half the ahgulas of the diameter of the eclipsed body (as radius) and another (concentric circle) with a thread equal in length to half the sum of the diameters k the eclipsed and eclipsing bodies. 1 Cf. MBh, v. 54, 77. 2 Cf. MBh, v. 53(ii). 72 THE LUNAR ECLIPSE [CH IV (Then) having drawn (through the common centre) the east-west line and with the help of a fish-figure the north-south line, lay off from the centre (of the circle) the corrected valana (for the first or last contact) according to its direction. About that point draw a fish-figure (in the east-west direc- tion). (Then) pass a thread through the middle of that fish- figure and produce it towards the east or west (as the case may be) to meet the outer circle and from there carry it to the centre. The point where the junction of the circle of the eclipsed body and that (thread) is clearly seen (in the figure) is the place where the Moon is eclipsed or is separated (from the shadow). When the valana and the Moon's latitude (for the middle of the eclipse) are. alike in direction, the valana should be laid off towards the west (from the centre); otherwise, towards the east. In the case (of the eclipse) of the Sun, it should be done reverse- ly. (Then) through the fish-figure drawn (along the north-south direction) about that point, pass a thread and extend it beyond the fish-figure (towards the north or south), according to (the direction of) the Moon's latitude to meet the outer circle, and from there carry the thread to the centre. Then from the centre along that thread lay off the Moon's latitude in the proper direction and put there a point. (With that point as centre and) with the ahgulas of the semi- diameter of the eclipsing body (as radius), draw a circle cutting the disc of the eclipsed body. The portion of the eclipsed body thus cut off lies submerged in the eclipsing body. 1 The circle which is drawn through the points (i.e., the cen- tres of the eclipsing 'body) corresponding to the beginning, mid- dle, and end of the eclipse, with the help of two fish-figures, is the path of the eclipsing body. 2 1 » Cf. MBh, v. 48-57. • Cf. MBh, v. 61. VSS. 31-32] GRAPHICAL REPRESENTATION 73 Construction of the phase of the eclipse for the given time: 3 1 -32. Multiply the difference between the (true) daily mo- tions (of the Sun and Moon) by the sthityardha minus the given time and divide that (product) by 60. Then adding the square of that to the square of the Moon's latitude (for the given time), take the square root (of that sum). (The square root thus ob- tained is the distance between the centres of the eclipsed and eclipsing bodies at the given time). Lay that off from the centre so as to meet the path of (the centre of) the eclipsing body. With the meeting point as centre and half the diameter of the eclipsing body as radius, draw the eclipsed portion for the given time. 1 1 Cf. MBh, v. 62-65. CHAPTER V THE SOLAR ECLIPSE Definition of the local divisor : 1. Multiply the radius by the Rsine of the colautude and divide by the Rsine of the (Sun's) greatest declmaton . the result is called the local divisor. The divisor defined here will be used in stanza 6 below. It is called local, because it depends on the latitude ot the local place. A rule relating to the determination of the tropical longitude ct the meridian ecliptic point for the time of geocentric conjunc tion of the Sun and Moon : , 2-4(i). Having calculated the asus (of the right ascension) of the traversed portion of the Sun's sign, by proportion with the right ascension of the Sun's sign,* and (then) having subtracted them from the asus between the times of geocentric conjunction of the Sun and Moon and midday, subtract the traversed por- tion of the Sun's sign from the Sun's longitude. From the re- mainder also subtract, in the reverse order, as many signs as have their right ascensions included (in the ^"^f^^" also) the degrees and minutes (of the fraction) of a g?> ^ The result (thus obtained) is known as the (tropical) longitude of the meridian ecliptic point in the forenoon. (When the geocentric conjunction of^ the Sun and Moon ocean) in the afternoon, addition should be made of the untra- versed portion of the Sun's sign, etc. 2 As regards the determination of the asus between the times of geocentnc conjunction of the Sun and Moon, and midday, the commentato. : nUyana says : "On the desired day whatever be the tun. * junction of the Sun and Moon, convert that >nto asus and also reduce to T^hTIscension of the Sun's sign" is the same as "the time of rising of the Sun's sign at Lanka."

• Cf. MBh, v. 8-11. VSS. 2-4- (i)] MERIDIA-ECLIPTIC POINT 75

the true semi-duration of the day. If the geocentric conjunction of the Sun and Moon occurs in the forenoon, subtract the time of geocentric conjunction (in asus) from the true semi-duration of the day (in asus); and if the geocentric conjunction occurs in the afternoon, then from the time of geocentric, conjunc- tion (in asus) subtract the (true) semi-duration of the "day (in asus) : in both the cases the remainder denotes the asus between the times of geocentric con- junction and midday." Sankaranarayana has given the full method for finding the tropical (sayana) longitude of the meridian ecliptic point for the time of geocentric con- junction of the Sun and Moon when the geocentric conjunction occurs in the afternoon. He writes : "When the geocentric conjunction of the Sun and Moon occurs in the afternoon, then the semi-duration of the day is subtracted from the time of geocentric conjunction and thus is obtained the difference in asus between the times of geocentric conjunction and midday; the result is set down at some place; from these asus of the difference between the times of geocentric conjunction and midday are then subtracted the asus which are obtained by proportion from the untraversed portion in minutes of arc of the sign occupied by the Sun or Moon at the time of geocentric conjunction and the right ascension of the sign (i.e., the asus of the right ascension of the un- traversed portion of the Sun's sign); the untraversed portion of the Sun's sign is then added to the Sun's tropical (sayana) longitude for the time of geocen- tric conjunction; from the remaining asus are then subtracted in serial order the right ascensions of as many signs as possible and these signs are added to the Sun's longitude; finally, adding the degrees, minutes, etc., obtained on multiplying the remaining asus by 30 and dividing by the right ascension of the next sign is obtained the tropical (sayana) longitude of the meridian ecliptic point." Sankaranarayana further says, "How is the longitude of meridian ecliptic point to be obtained when the traversed or untraversed part of the Sun s sign, while being subtractive, is less than the asus intervening Between the time of geocentric conjunction of the Sun and Moon, falling near noon, and the time of midday ? There, the difference, in asus, between the times of g cocen * tric conjunction and midday is itself multiplied by 30 and divided by the right ascension o f the sign occupied by the Sun : the quotient subtracted from or added to the Sun's longitude according as the time of geocentric conjunction occurs in the forenoon or afternoon gives (the longitude of) the meridian ecliptic point." It may be pointed out that in the above determination of the meridian ecliptic point, use is to be made of the Sun's tropical longitude, because the signs of the zodiac, whose right ascensions are made use of in the obove pro- cess, are tropical (sayana). The resulting longitude of the meridian ecliptic point is also tropical. 76 THE SOLAR ECLIPSE [CH. V A rule relating to the determination of the celestial latitude from the tropical longitude of the meridian ecliptic point obtain- ed by the above rule : 4(ii). From that (tropical longitude of the meridian ecliptic point) diminished by the longitude of the Moon's ascending node calculate the celestial latitude, north or south, (as in the case of the Moon). A rule relating to the determination of the drkksepa for the time of geocentric conjunction of the Sun and Moon : 5-7(i). Take the sum of the declination of the meridian ecliptic point and the celestial latitude (calculated from the tropical longitude of the meridian ecliptic point), and of the (local) lati- tude when they are of like directions and the difference when they are of unlike directions, the direction of the remainder (in the latter case) being that of the minuend. (The Rsine of the sum or difference is) the madhyajya. By that multiply the Rsine of the bhuja of the tropical longitude of the rising point of the eclip- tic and divide (the product) by the (local) divisor (defined in stanza 1). Square whatever is thus obtained and subtract that from the square of the madhyajya. The remainder is the square of the Rsine of the drkksepa. A rule relating to the determination of the drggatijya for the time of geocentric conjunction : 7-(ii)-8(i). Having added that (square of the dikksepajya) to the square of the Rsine of the instantaneous altitude (of the Sun), subtract that from the square of the radius : (the result is the square of the drggafijya). The drkksepajya and drggatijya obtained above, are neither precisely those for the Sun nor those for the Moon. 2 They would have been for the Sun, had the author not taken into account the celestial latitude calculated » MBh, v. 14. 2 The Sun's drkksepajya is the Rsine of the zenith distance of that point of the ecliptic which is at the shortest distance from the zenith; and the Sun's drggatijya is the distance of the zenith from the plane of the secondary to the ecliptic passing through the Sun. (Contd. on the next page footnote) 77 VSS. 8-10] LAMBANA from the longitude of the meridian ecliptic point while finding the madhyajya; whereas they would have been for the Moon, had the author, while calculat- ing the value of the drkksepajya, also taken into account the celestial latitude due to the rising point of the ecliptic (more correctly, the rising point of the Moon's orbit). See MBh, v. 1 3-23. The intention of the author seems to find such values of the drkskepajya and drggatijya as may roughly correspond to both the Sun and the Moon. The artifice adopted for the purpose by him, however, is not mathematically correct. It would have been better if he had omitted the use of the celestial latitude calculated from the longitude of the meridian-ecliptic point. See Paramesvara's commentary on LBh, v. 1 1-12. A rule relating to the determination of the lambana-nadh for the time of apparent conjunction of the Sun and the Moon: 8-10. Having divided the square root thereof by 191, further divide the quotient by 4 and a half: the result m nadh is the time known as lambana in the case of a solar eclipse. It is subtracted from the time of (geocentric) conjunction if the latter occurs in the forenoon, and is added to that if that occurs in the afternoon. To get the nearest approximation for the lambana {i.e., the lambana for the time of apparent conjunction of the Sun and Moon), one should similarly perform the above operation again and again with the help of the time of (geocentric) con- junction. The term lambana means the difference between the parallaxes in longi- tude of the Sun and Moon. The above rule aims at finding the lambana in terms of time, for the time of apparent conjunction (in longitude) of the Sun and Moon. Buta ? f thlS lambana depends on the time of apparent conjunction of the Sun and Moon itself, which is unknown, so recourse is taken to the method of successive approximations prescribed in the text. To begin with, the time of geocentric conjunction of the Sun and Moon is taken as the first approximation to the time of apparent conjunction, and The Moon's drkksepajya is the Rsine of the zenith distance of that point of the Moon's orbit which is at the shortest distance from the zenith; and the Moon's drggatijya is the distance of the zenith from the plane of the secon- dary to the Moon's orbit passing through the Moon. 78 the corresponding lambana in glo!ऽ is obtained by the formula :- The second approximation to the time of apparent conjunction is then obtained by the application of the formula : time ofapparent conjunction = time ofgcocentric conjunction + lambara in time for the time ofapparent conjunction. (2) The text prescribes the use of + or - sign in this formula according as the time ofgeocentric conjunction falls in the aftermo n or in the foremoon But this is incorrect; the correct procedure is to use + or - sign according as the Sun and Moon at the time of apparent conjunction lie to the west or to the east of the central ecliptic point The second approximation to the time of apparent conjunction of the Sum and Moon having been thus found, the above process is repeated again and again until the nearest approximation to the lamburla for the time of apparent conjunction is arrived at The rational८ of formula (1) is as follows: Moon's parallax in longitude dgatjya x Earth's semi-diameter in y0jar1075 minutes of arc. Moon's true distance in yojards But Moon's true distance in 9ja7105 Moon's mean daily motion in y9jara5 x 3438 Moon's true daily m0tion in minutes of arc Earth semi-diameter in 19jaras Moon's true distance in 199jar105 (1) M1001's mean daily motion in 1ygjurus x R. x (Moon's true daily motion in minutes of arc) 525 (Moon's true daily motion in minutes of arc) 79058x34383 Moon's true daily motion in minutes of arc 15x3438 vs. 11] Therefore Similarly, Sun's parallax in longitude Moon's paralax in longitude ="(Moon's true daily motion in minutes०farc), minutes of 15x343४ 84 4(Sun's true daily motion in minutes), minutes of arc. 15x 3438 Therefore AT (Moon's true daily motion in minutes of arc 15x3438 -Sun's true daily motion in minutes of arc) 3438 x 4 79 The usual Hindu method for deriving this formula is to apply the follow ing proportion

• When the dggatiya amounts to the radius (= 3488), the laाbora is

equal to 4 ghats; what them would be the value of the artbarma when the dgatjya has its calculated value?” The gha s of the lamba for the time of apparent conjunction having been thus determined, the time of apparent conjunction is obtained by using formula (2) above A rule relating to the determination of the 10t for the time of apparent conjunction af the Sun and Moon 11 . Multiply the Rsine of the d:ge#0 obtained by the method of successive approximation31 (i.e., multiply the Rsine of the dkkg2८ for the time of apparent conjunction) by the 1 While finding the nearest approximation to the lamburu for the time of apparent conjunction by the method of successive approximations, the Rsines of the dkksha obtained by the method of successive approximations is here meant the value of the Rsine of the dksha calculated at the last stage, which corresponds to the time of apparent conjunction. 80 difference between the daily motions (of the Sun and Moon) and divide by 51570 : the result is (the 10t) in minutes of arc, etc The nati means the difference between the parallaxes in latitude of the Sum and Moon. We have (uide MBl, v. 28) Moon's parallax in latitude drk:$920 x Earth's semi-diameter in 19jaras Moon's true distance in ygjaras But, as before, Parth's semi-diameter in 19jाmas Moon's true distance in y१jamas Therefore Moon's parallax in latitude 15 x 3438 Similarly, Sun's parallax in latitude dkg०)0 x Sun's true daily motion 15 x 3438 ence d. *}60 [Moon's true daily motion- 15 x 3438 525 79058 x 3438 Moon's true daily motion in minutes of arc 15 x 3438 ०i. Sum and Moon) 51570 525 ] In the above rationale we have assumed that 7905*83 525 1ikewise taken But this is incorrect, because Iminutes. 51770 Sun's true daily motion]]. vss. 13-14] STHITYARDIIAS 81 A rule relating to the determination of the Moon's true latitude (i.e., the Moon's latitude corrected for parallax) for the time of apparent conjunction : 12. (The nati) and the Moon's latitude for that instant should be added if they are of like directions and subtracted if they are of unlike directions : thus is obtained the true latitude (of the Moon) in the case of a solar eclipse. 1 "For that instant" means "for the time of apparent conjunction". A rule relating to the determination of the spar'sa- and moksa- sthityardhas : 13-14. From half the sum of the diameters of the Sun and the Moon and from the Moon's true latitude (for the time of apparent conjunction), calculate the sthityardha* as before. 3 (Sub- tracting that from and adding that to the time of apparent conjunc- ion, find the gross values of the times of the first and last contacts). Then find out the lambanas 2 and the (Moon's) true latitudes for the times of the first and last contacts, applying the respective rules only once. Then add the difference of the lambanas 2 (for the times of the first contact and apparent conjunction at one place and for the times of apparent conjunction and the last con- tact at another place) to the sthityardha? the results should be announced as the true values of the {spar'sa- and moksa-)sthityardhas. 2 (Then subtracting the sparsa-sthityardha 2 from the time of appar- ent conjunction, find the time of the first contact; and adding the moksa-sthityardha 2 to the time of apparent conjunction, find the time of the last contact.) 4 The valanas (for the times of the first contact, apparent con- junction, and the last contact) should be obtained as before. 5 » Gf. MBh, v. 31. 2 In ghatls, etc.

• Gf. MBh, v. 34.
• Cf. MBh, v. 35-36.

5 This last sentence is the translation of "pragvat valanakarma ca", which occurs in the end of verse 13. We have shifted its translation to this place, because this is the most appropriate place for it. 82 The term stliyardha means “halfthe duration (of the eclipse)”. The shar50 ऽthiyardlla, in the case of a solar eclipse, is the time-interval between the first contact and apparent conjunction; and the moksa-stlijyardha is the time-inter val between apparent conjunction and the last contact. The ऽthijuardia is obtained as in the case of the lunar clipse by the formula where or denotes half the sum of the diameters of the Sun and Moon, 3 the Moon's true latitude, and d the difference between the true daily motions of The sparsa- and m0:0- stliyardlhas obtained by the above rule give their approximate values only. To obtain the nearest approximations to the exact values one should apply the method of successive approximations. See Condition for the impossibility of a solar clipse: 15. When the minutes of the (Moon's) true latitude (ob tained above) are equal to the minutes of half the sum of the diameters of the Sun and the Moon, then the Moon does not hide the disc of the Sun, wh0se rays are the destroyers ofdarkness1.

• C. MBl, v. 33. CHAPTER VI

VISIBILITY, PHASES, AND RISING AND SETTING OF THE MOON A rule relating to the visibility correction known as aksa-drk' karma : 1-2. Multiply the Rsine of the Moon's latitude by the Rsine of the (local) latitude and divide (the product) by the Rsine of the colatitude. Whatever is thus, obtained should be subtracted from the Moon's longitude in the case of rising of the Moon (i.e., in the eastern hemisphere) and added to that in the case of set- ting of the Moon (i. e., in the western hemisphere), provided that the Moon's latitude is north. When the Moon's latitude is south, the above correction is applied reversely in the cases of rising and setting (both). 1 A rule relating to the visibility correction known as ayana-drk- karma : 3-4. Multiply the (Moon's) instantaneous latitude by the Rversedsine (of the Moon's longitude) as diminished by three signs and then by the Rsine of the (Sun's) greatest declination and divide that (product) by the square of the radius. The result- ing minutes of arc should be subtracted from the longitude of the Moon when the latitude and ayana (of the Moon) 8 are of like directions. In the contrary case, they should always be added to the longitude of the Moon. 3 The two corrections stated in the foregoing stanzas are known as dar- sana-safnskara or drkkarma {"visibility corrections"). The first correction, stated in stanzas 1-2, is known as aksa-drkkarma. 1 Cf. MBk, vi. 1-2U). 2 The Moon's ayana is north or south according as the Moon is in the half- orbit beginning with the tropical (sayana) sign Capricorn or in that beginning with the tropical (sayana) sign Cancer. 8 Cf. MBh, vi. 2(ii)-3. 84 Suppose that a planet is rising on the eastern horizon or setting on the western horizon. Then the portion of the ecliptic lying between the hour circle of the planet and the horizon is defined as the al:$0-alau of the planet; and the portion of the ecliptic lying between the hour circle and the circle of longitude is defined as the gy070-0alu0 of the planet. The true longitude of a planet calculated in accordance with the rules stated in chapter II above denotes the longitude of that point of the cliptic where the planet's circle of longitude meets it. The object of the visibility corrections is to obtain the longitude of that point of the ecliptic which rises or sets with the planet. This has been done in two steps by the successive application of the nksa- and gy५070- dkarmas. The natural order, however, is to apply the gyna-dkarma first and.the aka-dkarाma next. Generally this matural order of correction has been followed by the Hindu astronomers The formulae for the alk०- and gyana- dkarmas for the Moon stated in the text are : 0rm's latitude) ................ , ९sin xR in ( Rversirm (14-90°) x Rsin 6 x Moon's latitude where M is the Moon's (tropical) longitude, 2 the latitude of the place, and 6 the Sun's greatest declimation. e rationale and discussion of these formulae, the reader is refered t my notes on MBh, wi. 1-3 Minimum distance of the Moon from the Sun, in terms of deg rees of time, at which she becomes visible : 5. When the Moon obtained by applying these (two visi bility) corrections is found to be twelve degrees (of time) distant from the Sun, she shall be (just) visible in clear cloudless sky.१ What the author really means is that : Rversin {{bhuja (M- 90")} x Rsin e x Moon's latitude ( 70 = R* Cf. MB, wi. 4(ii)- 50(i). [While consulting my edition of the MBh, read “time ofsetting' in place of“obligue ascension' in line 21, p. 186, and “set ing” in place of “obligue ascension' in line 31, p. 188. Similarly, the word “agu७” occuring in lines 5 and 7, p. 192, should be changed into “29us of setting' , and that occurring in line 9, p. 192, into “45us of rising”. The last sentence of that paragraph should be deleted] VS. 6-7] ILLUMINATED PART OF THE MOON 85 360 degrees of time are equivalent to 60 ghaiis or 21600 asus, so that one degree of time is equivalent to 1/6 of aghatl or 60 asus. Thus 12 degrees of time are equivalent to 2 ghatis. On the fifteenth lunar day of the dark half of the month, the Moon comes near the Sun from behind and is lost in his splendour. After about two days she is beyond the limit of invisibility and is again seen in the sky af .er sunset being in advance of the Sun. In order to see whether the Moon will be visible on the first or second lunar day of the light half of the month, one should calculate the (tropical) longitude of the Sun for sunset on that day and also for the same time the (tropical) longitude of the Moon as corrected for the visibility corrections. If the portion of the ecliptic lying between the Sun and the Moon thus obtained sets at the local place in two ghaiis or more, the Moon will be visible after sunset on that day, otherwise not. Similarly, in order to see whether the Moon will be visible before sunrise on the fourteenth or fifteenth lunar day of the dark half of the month, one should calculate the (tropical) longitude of the Sun for sunrise on that day and also for the same time the (tropical) longitude of the Moon as corrected for the visibility corrections. If the part of the ecliptic lying between the Sun and Moon thus obtained rises at the local place in two ghatis or more, the Moon will be visible before sunrise on that day, otherwise not. A rule relating to the determination of the measures of the illumi- nated and unilluminated parts of the Moon : 6-7. (In the light half of the month) the Rversed-sine of the difference ( between the longitudes of the Moon and the Sun) multiplied by the true diameter of the Moon and divided by 6876 gives the measure of the illuminated part (of the Moon). When the difference exceeds a quadrant, one should add the radi- us to the Rsine of the excess and from that (find) the measure of the illuminated part. In the dark half of the month, one should obtain in the same way, the unilluminated part (of the Moon) with the help of the Rversedsine (of the difference between the longitudes of the Moon and the Sun diminished by 6 signs) and from the Rsine (of the excess of that difference over a quad- rant). 1 'Gf . MBh, vi. 5(ii)-7. 86 That is, if the longitude of the Moon minus the longitude of the Sun be demoted by D, then (1) If the !ight half9f the month, the illuminated part of the Moon 6876 if D) < 3 signs; and [[R-+ Rsim (D- 90०)] x Moon's true diameter 6876 (ii) in the dark half of the month, the unilluminated part of the Moon Rversin (D-180°) x Moon's tr॥e diameter 6876 [R-+Rsin (D-270°)] x Moon's true diameter 6876 ifD) > 9 signs.

• Moon's true diameter' means “Moon's. angular diameter in minutes '

See supra, dth. TV, starm2a 5 For the rationale of these formulae, see my notes on MBh, wi. 5 (ii)-7 Verses 8-17 relate to the elevation of the horms of the Moon in (१uarter of the lumar month. the first A rule regarding the determination of the Moon's saikuagra at 8. From the 25us intervening between the Sun and Moon (corrected for the visibility corrections) and from the Moon's earthsine and ascensional (Moon's) altitude; and from that find out the (Moon's) 5aikugra, which is always south (ofthe rising-setting line of the Moon). The 05us intervening between the Sun and the Moon (corrected for the visibility corrections) are the ८5us to elapse before moonset. To obtain these 4945, one should increase the above longitudes of the Sun and the Moon both by six signs and find the obligue ascension of the portion of the ecliptic lying between the two positions thus found The Moon's earthsine is the portion of the Moon's diurnal circle intercep ted between the local and e१uatorial horizons. The Moon's ascensional differ vs 12(ii)-17] EXHIBITION OF LUNAR HORNS 87 ence is the corresponding time, i.e., the time that the Maon take? in moving from the equatorial horizon to the local horizon. The Moon's sahkvagra is the distance of the foot of the perpendicular dropped from the Moon on the plane of the horizon, from the rising-setting Jine of the Moon. The methods of finding the Moon's earthsine, ascensional difference, alti- tude and sahkvagra are similar to those for the Sun. A rule relating to the determination of the Moon's true decli- nation and the Moon's agra : 9-10. The Rsine of the difference or sum of the (Moon's) latitude and declination according as they are of unlike or like directions is (the Rsine of) the Moon's true declination. 1 From that (Rsine of the Moon's true declination) determine her day- radius, etc. Then multiply (the Rsine of) the Moon's (true) declination by the radius and divide by (the Rsine of) the colati- tude : then is obtained (the Rsine of) the Moon's agra. 2 The true declination of the Moon means the declination of the centre of the Moon's disc. The Rsine of the Moon's agra is the distance between the east-west line and the Moon's rising-setting line. A rule relating to the determination of the base (bahu) : ll-12(i). If that (Rsine of the Moon's agra)- is of the same direction as the (Moon's) sahkvagra, take their sum ; otherwise, take their difference. Thereafter take the difference of (the Rsine of) the Sun's agra and that (sum or difference) if their directions are the same, otherwise take their sum : thus is obtain- ed the base {bahu). 3 Construction of the figure exhibiting the elevation of the lunar horns in the first quarter of the month at sunset : 12(ii)-17. Lay that (base) off from the Sun in its own direc- tion. (Then) draw a perpendicular line passing through the head 1 Cf. MBh, vi. 8. a Gf. MBh, vi. 10-1 l(i). 8 Cf. MBh, vi. ll(ii)-12. 88 VISIBILITY, PHASES AND RISING OF THE MOON [CH. VI and tail of the fish-figure constructed at the end (of the base). (This) perpendicular should be taken equal to the Rsine of the Moon's altitude and should be laid off towards the east. The hypotenuse-line should (then) be drawn by joining the ends of that (perpendicular) and the base. TheMoonis (then) constructed with the meeting point of the hypotenuse and the perpendicular as centre; and along the hypo- tenuse (from the point where it intersects the Moon's circle) is laid off the measure of illumination towards the interior of the Moon. The hypotenuse (indicates) the east and west directions : the north and south directions should be determined by means of a fish-figure. (Thus are obtained the three points, viz.) the north point, the south point, and a third point obtained' by lay- ing off the measure of illumination. (Then) with the help of two fish-figures constructed by the method known as tri'sarkaravidhana draw the circle passing through the (above) three points. Thus is shown, by the eleva- tion of the lunar horns which are illumined by the light between two circles, the Moon which destroys the mound of darkness by her bundle of light. 1 Exhibition of the lunar horns in the second quarter of the month : 18. (When the Moon is) in the eastern half of the calestial sphere, the true base should be found out with the help of the rising point of the ecliptic and the Moon's agra, etc.; and the un- mentioned element (i.e., the upright) should be laid off towards the west 2 The true base here corresponds to the base of stanza 11. A rule for finding the duration of visibility of the Moon in the light half of the month : 19. The nadts (of oblique ascension of the portion of the ecliptic) intervening between the Sun and the Moon 3 (at 1 Cf. MBh, vi. 13-17. 2 Cf. MBh,v. 19. 3 Corrected for the visibility corrections. vs. 19] moon's visibility in the light fortnight §9 moonset), both increased by six signs, calculated by the method of successive approximations, give the duration of visibility of the Moon in the light half of the month. 1 The process of successive approximations may be explained as follows : Compute the (tropical) longitudes of the visible Moon 2 and the Sun for sunset and increase both of them by six signs Then find out the asus (A^ due to ob- lique ascension of the part of the ecliptic lying between the two positions thus obtained. Then A 1 asus denote the first approximation to the duration of the Moon's visibility at night. Then calculate the displacements of the Moon ar d the Sun for A x asus and add them respectively to the longitudes of the visible Moon and the Sun for sunset, and increase the resulting longitudes by six signs; and then find out the asus {A 2 ) due to the oblique ascension of the part of the ecliptic lying between the two positions thus obtained. Then A 2 asus denote the second approximation to the duration of the Moon's visibility at night. Repeat the above process successively until the successive approxima- tions to the duration of the Moon's visibility agree to vighath. The time thus obtained is in civil reckoning. If, however, use of the Moon's displacement alone be made at every stage, the time obtained would be in sidereal reckoning. According to theinterpretation of the commentator Sankaranarayana, the translation of the text would run as follows : "The riadis (of oblique ascension of the portion of the ecliptic) lying between the Sun as increased by six signs and the Moon (at moonrisej calculated by the method of successive approxi- mation give the time of moomise (before sunset) in the light half of the month." The process of successive approximations in this case would be as follows: Calculate the longitudes of the Sun and the visible Moon for sunset, and increase the former by six signs. Then find out the asus (flj due to oblique ascension of the part of the ecliptic lying between the two positions thus ob- tained. Then B l asus denote the first approximation to the time between moonrise and sunset. Then calculate the displacements of the Moon and the Sun for B t asus, and subtract them respectively from the longitudes of the visible Moon and the Sun for sunset, and, as before, increase the latter by six signs; and then find out the asus {B 2 ) due to the oblique ascension of the part of the ecliptic lying between the two positions thus obtained. Then B 3 2 C£MBk,vi. 27. 3 i.e., the Moon corrected for visibility corrections. 90 VISIBILITY j PHASES AND RISING OF THE MOON [CH. VI asus denote the second approximation to the time between moonrise and sun- set. Repeat the above process successively until the successive approximations to the time between moonrise and sunset agree. The time finally obtained, gives the time of moonrise before sunset. This being subtracted from {he du- ration of the day gives the time of moonrise as measured since sunrise, A rule relating to the time of rising of the Moon on the full moon day : 20-21- If (at sunset) on the full moon day the longitude of the Moon (corrected for the visibility corrections) agrees to mi- nutes with the longitude of the Sun (increased by six signs), then the Moon rises simultaneously with sunset. If (the longitude of the Moon is) less (than the other), the Moon rises earlier; if (the longitude of the Moon is) greater (than the other), the Moon rises later. (In the latter cases) multiply the minutes of the difference by the asus of the oblique ascension of the sign occupied by the Moon and divide by the number of minutes of are in a sign, and on the resulting time apply the method of successive approxima- tions (and get the nearest approximation to the time to elapse at moonrise before sunset or elapsed at moonrise since sunset). 1 When the longitude of the Moon is less than the longitude of the rising point of ecliptic (at sunset), the process of successive approximations will be similar to that explained under stanza 19 above while dealing with Sankara- narayana's interpretation; when the longitude of the Moon is greater, the pro- cess of successive approximations will be similar to that explained below in stanzas 23-25. A rule relating to the determination of the shadow of the gnomon due to the Moon: 22- From the asus { of the oblique ascension of the portion the ecliptic) lying between the rising point of the ecliptic and

the Moon (corrected for the visibility corrections) or from those

(taken in setting at the local place by the portion of the ecliptic) lying between the setting point of the ecliptic and the M 1 Cf. MBh, v. 22. VS 23-25] MOON IN THE DARK FORTNIGHT 91 (corrected for the visibility corrections) (according as the Moon is above the eastern or western horizon), and from the Moon's day radius, etc , determine (the Rsine of) the (Moon's) altitude and zenith distance and therefiom the shadow' of the gnomon (due to the Moon) 1 . A rule for finding the time of moonrise in the dark half of the month: 23-25. Multiply the minutes of arcof the rising sign to be traversed by therising point of the ecliptic at sunset by the oblique ascension of thatsign and divide by the number of minutes of arc in a sign: thus are obtained the asus (of the oblique ascension of that part of the rising sign which is below the horizon). Adding thereto the asus (of the oblique ascension) of the succeeding portion of the ecliptic traversed by the Moon calculated for sunset up to the last minute of arc (of her longitude), find out the Moon's motion corresponding to that time by proportion, and add that to the longitude of the Moon. Then by repeating the above process again and again find the nearest approximation to the time between sunset and moonrise. After the lapse of that time during night, in the dark half of the month, is seen to rise the Moon who by her rays of light has destroyed the mound of darkness. 3 The time obtained above is in sidereal reckoning. If the use of the Sun's displacement is also made at every stage, the resulting time would be in civil reckoning. 1 See supra, Chapter iii, stanzas 7-10, 11. 2 Corrected for the visibility corrections, 3 Cf. MBh, vi. 28-31. CHAPTER VII VISIBILITY AND CONJUNCTION OF THE PLANETS Minimum distances of the planets from the Sun at which they become visible: 1-2. If Venus corrected for the visibility corrections is 9 degrees (of time) distant from the Sun, it is visible. Jupiter, Mercury, Saturn, and Mars are visible in the clear sky when their distance (from the Sun) are nine degrees increased succes- sively by twos (i.e., when they are respectively at the distances of 11, 13, 15 and 17 degrees of time from the Sun). 1 The degrees of time multiplied by 10 are known as vinadikas. Since 360 degrees of time are equivalent to 60x60 vinadikas, therefore one degree of time is equivalent to 10 vinadikas. A rule relating to the determination of the degrees of time be- tween the Sun and a planet: 3. (When the planet is to be seen) in the east," (its) visibility should be announced by calculating the time (of rising of the part of the ecliptic between the Sun and the planet 3 ) by using the oblique ascension of that very sign (in which the Sun and the planet are situated); (when the planet is to be seen) in the west, (its) visibility should be announced by calculating the time (of letting of the part of the ecliptic between the Sun and the pla- net 3 ) by using the oblique ascension of the seventh sign. 4 A rule relating to the determination of the common longitude of two neighbouring planets when they are in conjunction in lon- gitude: 4-5. Divide the difference between the longitudes of the two given planets by the sum or difference of their daily motions 1 Gf. MBh, vi. 44. Also cf. A, iv. 4; ATT, (Sengupta), vi. 6.

• Cf. MBh, vi. 46(i).

3 Corrected for the visibility corrections. « Cf. MBh, vi. 46(H). vss. 6-9 (i)] LATITUDES OF PLANETS 93 according as they are moving in unlike or like directions: then are obtained the days, etc. (elapsed since or to elapse before the time of conjunction of the two planets) - 1 The longitude of those two neighbouring planets should then be made equal up to minutes of arc by subtracting from or adding to their longitudes their motions (corresponding to the above days, etc) obtained by proportion with their true daily motions. 2 To obtain the nearest approximation to the desired "result, the above process should be repeated again and again. A rule relating to the determination of the latitudes of the two planets which are in conjunction in longitude : 6-9 (i) . In the case of Mercury and Venus, subtract the longi- tude of the ascending node from that of the sighrocca: (thus is obtained the longitude of the planet as diminished by the longi- tude of the ascending node). 3 The longitudes (in terms of deg- rees) of the ascending nodes of the planets beginning with Mars are respectively 4, 2, 8, 6, and 10 each multiplied by 10. 4 The greatest latitudes, north or south, in minutes of arc, (of the planets beginning with Mars) are respectively 9, 12, 6, 12, and 12, each multiplied by 10. 5 (To obtain the Rsine of the latitude of a planet) multiply (the greatest latitude of the planet) by the Rsine of the longitude of the planet minus the longitude of the ascending node (of the planet) (and divide by the "divisor" defined below). 6 The product of the mandakarna and the slghrakarna divided by the radius is the distance between the Earth and the planet : this is defined as the "divisor". 7 i' Cf. MBh, vi. 49-50 (i). 2 Cf. MBh, vi. 51(i). 3 Cf. MBh, vi. 53(H)- Also see SiSi, II, vi. 23(i).

• Cf. MBh, vii. 10(i).

6 Cf. MBh, vii. 9. 8 Cf. MBh, vi. 52. 7 Cf. MBh, vi. 48. 94 VISIBILITY AND CONJUNCTION OF PLANETS [CH. VII Thus are obtained the minutes of arc of the latitudes (of the two planets which are in conjunction in longitude). Two things deserve mention here. One is that the revolution-numbers of the nodes of Mercury and Venus, stated in Hindu works on astronomy, as says Bhaskara II 1 , are those increased by the revolution-nnmbers of their res- pective sighra-kendras. The result is that when we subtract the longitude of the ascending node of Mercury or Venus from the longitude of its sighrocca, we obtain the longitude of the planet (Mercury or Venus) as diminished by the longitude of its ascending node. The second is that in finding the celestial latitude of a planet we should use the heliocentric longitude of the planet and not the geocentric longitude. Brahmagupta (628 A. D.) and other Hindu as- tronomers have, therefore,prescribed the use of the true-mean longitude in the case of Mars, Jupiter and Saturn, and that of the longitude of the planet's sighrocca as corrected for the planet's mandaphala in the case of Mercury and Venus. 2 A rule relating to the determination of the distance between the two planets which are in conjunction in longitude : 9-10. From those latitudes obtain the distance between those two given planets by taking their difference if they are of like directions or by taking their sum if the are of unlike directions. 3 The true distance between the two planets, in minutes of arc, being divided by 4 is converted into ahgulas^ Other things should be inferred from the colour and bright- ness of the rays of the (two) planets or else by the exercise of one's own intellect. 5 1 See SiSi, II, viii. 23. 8 See BrSpSi, ix. 9. Also see SuSi, ii. 56-57 ; SiSe, xi. 15 ; and SiSi, II, vi. 20-25(i). a Cf. MBh, vi. 54. 4 Gf. MBh, vi. 55. 5 Set SuSi, vii. 18(ii)-23(i). Longitudes of the junction-stars ! (20diacal asterisms) : CONJUNCTION OF A PLANET AND A STAR 1-4. Eight, eighteen, tem, fourteen, twelve, eight, twenty two, thirteen, mine, fourteen, thirteen, thirteen, nineteen, twelve, twelve, fifteen, tem, six, thirteen, thirteen, twelve, eighteen, eleven, twelve, twenty-one , seventeerm, and fifteen- each of these numbers being increased by (the sum of) the preceding numbers, in the order in which they have been stated above, are to be taken as the degrees of the longitudes of the junction-stars of the (twenty seven) 10k90tr25. To the longitudes of (the junction-stars of) Puirvasadha, Sravapa, Mula, Magha, Dhanistha, Bharapi, and Uttarāsadha (thus obtained), one should further add thirty mi nutes (of arc). * 1. The longitudes of thejunction-stars stated above are, in some cases, slightly different from those given in the author's bigger work, the Maha-Bhaskarya. The differences are exhibited by the following table : 2. Junction-star of Differences between the longitudes of the junction-stars A5vin Bharam of the twenty-seven 10:20traऽ 15 Longitude given in 278 6* 18 269 6० 30' 30 । 1 The junction-stars (yogatāra) of the maksatraऽ are the prominent stars of the makyatras which were used in the study of the conjunction of the planets, especially the Moon, with them.

• Cf. MBh, i. 63-66(i). 96

4. 7. 8. 10. 11. 13. 17. 19. 22. 23. 24. 25. 26. Junction-star Rohin Pumarvasu॥ Pusya Magha Prvā Phalgunl CONJUNCTION OF A PLANET AND A STAR. Fasta Anurādhā Mula Sravar1a Dhanistha 5atabhisakः Purva Bhadrapad Uttara Bhadrapada | | 23 35 28 10० 49 39 15० 2 35 24 79 Longitude given in 45 21 79 2० 85 5 23० 8० 109 2० 12 1० 85 27० 99 15० 99 26 7० 109 28० 119 15° 30' 19 20 25 25 10 35 35 15 45 39 24 2 69 2 45 21 75 53 23० 8० 85 5 73 12 2० 79 18० 109 85 14० 1० 83 26० 95 14० 95 25 7० 109 28० 11: 15 30' 30' 30' 30' 30' 30' 30 30' 30 vs.6-9] Conjunction (in longitude) of a planet with a star': 5. All planets whose longitudes are equal to the longitude of the junction-star ofa malkpatra are seen in conjunction with that star. (Of a planet and a star) whose longitudes are unegual, the time of conjunction is determined by proportion. 5. 6. Latitudes of the junction-stars of the twenty-seven 70920tras: 6-9. North, tem, twelve, five; south, five, tem, eleven; north, six, zero; south, seven, 2ero; m0rth, twelve, thirteen; south, seven, two; north, thirty-seven; south, one and a half, three, four, eight and a half, seven, seven; north, thirty, thirty-six; south, eighteen Iminutes of arc; north, twenty-four, twenty-six, and Zero-these have been stated by the learned to be the degrees (unless other wise stated) of the latitudes of the junction stars of the 10ksatraऽ beginning with A5vini in their serial order. 10. 13. The latitudes stated above are being exhibited below in the tabular form: Mrga5irā Ardra Puऽya of Magha Hasta Celestial 1atitude 12-N 5°N 10*S Es OF JUNCTION-STARS 7०S 12०N 13°N 2°S 1 Cf. AMBh, i. 70(ii)

• Cf. MBh, i. 66(ii)-70().

| 22. 24. 25. 26. 27. Junction-star of Sravana 97 Satablhisalk Prva-Bhadrapada Uttara-Bhadrapada Revat Celestial latitude 1°30'S 4-S 8-30'S 7°S 30" 18'S 24०N 26"N 98 CONJUNCTION OF A PLANET AND A STAR [CH. VIII In the Maha-Bhaskariya, the latitudes of Mula and Uttarasadha are stated to be 8°20'S and 7°20'S respectively. Definition of absolute conjunction of the Moon with a star : 10. The Moon is in (absolute) conjunction with a junction- star when her longitude and celestial latitude both in magnitude and direction, are the same as the longitude and celestial latitude, in magnitude and direction, of the star. Latitudes of the Moon when she occults some of the prominent stars of the zodiac : 11-16. When the Morn attains 160 minutes (of arc) of north latitude, she clearly covers the junction-star of the naksatra Krttika (i.e., the Pleiades). 1 Having attained her maximum northern latitude, the Moon covers with her disc the central star of the naksatra Magha. 2 With her latitude 60' (south), the Moon clearly occults the cart of RohirJ (i.e., the V-shaped constellation of Hyades); and with latitude 256' south, she covers the junction-star (of Rohiru) (i.e., Aldebaran). 3 With her latitude 95 (minutes of arc) south, (the Mcon covers the junction-star of) the naksatra Citra (ie., Spica) ; with 150 (minutes of arc) south, (the junction-star of) the. naksatra Anuradha 4 ; anol with 200 (minutes of arc) (south), (the junction- star of ) the naksatra Jyestha (i.e., Antares) . 5 With latitude 87 (minutes of arc south), the Moon clearly occults (the brighter of ) the two northern stars of the naksatra Visakha; with 24 (minutes of arc) south, (the junction-star of) the naksatra Satabhisak (i.e., x Aquarii). 6 i Cf. MBh, iii. 74(i). 8 Cf. MBh, iit. 74(ii). 3 Cf. MBh, iii. 71(ii)-72(i).

• According to H. T. Colebrooke and E. Burgess, it is 5 Scorpii. Accor-

ding to Bentley, it is jS Scorpii. 6 Cf. MBh, Hi. 72(H) -73$. 6 Cf. MBh, iii, 73. vs. 17] AN ASTRONOMICAL PROBLEM 99 The Moon, situated at her ascending node, occults (the junction-stars of) Pusya and Revati (i.e., £ Piscium). 1 The above occultations (bheda) of the stars by the planet (Moon) are based on the minutes of latitude determined from actual observation by means of the instrument (called) Yasti. 8 An astronomical problem on indeterminate equations : 17. The sum, the difference, and the product increased by one, of the residues of the revolution of Saturn and Mars— each is a perfect square. 3 Taking the equations furnished by the above and applying the method of such quadratics obtain the (simplest) solution by the substitution of 2, 3, etc. successively (in the general solution). Then calculate the ahargana and the revolutions performed by Saturn and Mars in that time together with the number of solar years elapsed. Let * and y denote the residues of the revolution of Mars and Saturn respectively. Then we have to find out two numbers x and y such that each of the expressions * ±y t x-y, and xy±l may be a perfect square. Let *+jr=4< 2 , and *-j>=40 2 , so that JC =2< 5! -f2j3 2 and y=2°(*-2$ Therefore xy+l = (2a< 2 - l) 2 +4(c( 2 - 0*). Hence the condition that xy+l be a perfect square is that Consequently, we have

• =2(p ,4 4-P 2 )

and ^=2(j3 4 -jS 2 ), 1 Cf. MBh, iii. 73(H). 2 Cf. MBH, iii. 75(i). 3 According to Parames'vara's interpretation, the first half of this stanza means: "The sum, the difference, and the product of the residues of the revolution of Saturn and Mars, each increased by cJne, is a perfect 100 CONJUNCTION OF A PLANET AND A STAR [CH. VIII where (3=2, 3, 4, .... neglecting the case in which * or y is zero. 1 Putting j3=2, 3, 4, we see that *=40 and ^=24 is the least solution. Assuming now that the residues of the revolution of Saturn and Mars are 24 and 40 respectively, we have to obtain the ahargana and the revolu- tions performed by Saturn and Mars. To obtain the ahargana and the revolutions performed in the case of Saturn, we have to solve the equation 3664 1«-24 _ #i) 394479375 where u and v denote the ahargana and the revolutions performed respectively. Applying the rules given in the Maha-BKaskarlya (i. 41-45), the general solution of the above equation is found to be a = 394479375/4/346688814, and 3664U4- 32202, where t = 0, 1, 2, ... . The least solution corresponds to t = 0. To obtain the ahargena and the revolutions performed in the ease of Mars, the equation to be solved is 191402^-40 {9 . 131493125 Z and w denoting the ahargana and the revolutions performed by Mars respec- tively. The general solution of this equation is

• =131493125*+ 118076020,

«;=191402j+-171872, 1 This solution was given by the Hindu Mathematician Narayaria (1356 A. D.). See GK, i. 47. The Hindu mathematician Brahraagupta (628 A.D.), who was a contemporary of Bhaskara I, had given the following solution ; y = A(p-y% i A (P 8 4-r a )-HP*-r 2 ) wn rc A ~ iHiF+y 2 )-iP-yw ' which reduces to Narayana's solution by taking y =*1. The commentator Udayadivakara has given a unique method for solving the above multiple equations. His method has been discussed by me in a paper entitled •"Acarya Jayadeva, the mathematician". Sec Canita, Vol. 5, No. 1, June 1954, pp. 18-19. VS. 18] ANOTHER ASTRONOMICAL PROBLEM 101 where s=0, 1,2 i=0 gives the least solution. 1 Another astronomical problem on indeterminate equations : 18. The residue of the minute of Mars multiplied by the cube of two and increased by one yields a square-root (without remainder) ; that square number multiplied by seven and then further increased by one is again a perfect square. Having as- certained the residue from this (hypothesis) one who can find out ihe longitude of Mars and the ahargana together with the number of solar years elapsed is (indeed) the foremost amongst the intelligent mathematicians on this earth girdled by the oceans. Let x denote the residue of the minute of Mars, trten we have to solve the equations 8*+ !=/, say, (i) 7y4-l=* 2 , say, (2) Eliminating j> between (1) and (2), we get, 56*+8=c 2 . (3) Evidently *=1, £=8 is a solution of this equation, so that we may take 1 as the residue of the minute for Mars. 2 Let « be the ahargana corresponding to this residue of Mars. Then 165371328m- 1 Ml =o, (4) 5259725 v ; where c denotes the revolutions performed by Mars is u days. Solving (4) we get « = 1863192 days, © = 2712 revolutions, sign, 25°, 31', which agrees with the solution given by the commentator Sankaranarayana. The commentator Sankaranarayana has also given an alternative inter- pretation of the text. According to that interpretation the above stanza 1 The results obtained above agree with those given by the commentator Sankaranarayana. It may be noted that there is no ahargana which may satisfy both equations (1) and (2) above. For, if we take u=£, then we get 39447937f«-131493125.r+228612794*0, which is impossible. 2 According to the commentator Udaya Divakara, one should first find the value of y by solving (2) and then substituting this value in (1) find x. 102 CONJUNCTION OF A PLANET AND A STAR [CH. VIII would run as follows : “The residue of the minute of Mars multiplied by the cube of 2 yields a square root (without remainder); that souare root being increased by one, then multiplied by 7 and then increased by one is again a perfect 59uare. Having ascertained the residue from this (hypothesis), one who can find out the longitude of Mars and the alhargarm2 together with the number of years elapsed is (indeed) the foremost amongst the intelligent mathematicians on this earth girdled by the oceans.” so that where P and s are integers. 7 (.78R +1) + 1=s, say, Putting ऽ=0, 1, 2, ..., we see that only ऽ=6 and s=8 give integral values to R, the corresponding values being 2 and 8 respectively. Thus the residue of the minute of Mars is either 2 or 8. Let us take R८=2. Then to find out the required alhargao we have to solve the equation 1653713289-2 । 5259725 where x denotes the alhargapa and y the total number of minutes traversed by The general solution of this equation is

• =5259725t +4386086,

y=1653713281-+137903192, where t=0, 1, 2, If we take R=8, we shall get 4alhargan4 = 52597255-+3726384 , Sarikaranārāyapa gives the alhangap0 as equal to 3726384 or 4386086 . The former corresponds to s=0, and the latter to t=0. told by Bhaskara. Object, scope, and authorship of the book : 19. For acquiring a knowledge of the true motion of the planets by those who are afraid ofreading voluminous works, APPENDIX 1 THEORY OF THE PULVERISER As applied to Problems in Astronomy by BHATTA GOVINDA 1. The following twenty-twostanzas dealing with the theory of the pulveriser as applied to problems in astronomy have been quoted by Saikara Narayapa (in his commentary on LBl, wi, 18) from certain astronomical work (called Govinda-kti) of Acarya Bhatta Govinda. These throw new light on the subject and will, it is hoped, be of interest to historians of mathematics. 2-1 . 22. THEORY OF THE PU,VERSER As applied to Problems in Astronomy Introduction to the subject : i.e., “Although the entire working of the pulveriser has been described (by previous writers), but it is not clearly understood. So here I explain the theory of the pulveriser more fully.' यद्यप्युक्त सकलं तथापि नैतत् प्रतीयते कर्म । अत इह कुट्टाकार गणितं सम्यक् प्रवक्ष्यामि ।। १ ।। ८ The two kinds of the pulveriser :

.e., “The pulveriser is of two varieties, residual and non resअंdual. Of these, the non-residual pulveriser will be explained by me first. स पुन: कुट्टाकारो द्विविधस्तावन्निरग्रसाग्रतया । तत्र निरग्र वाच्यः कुट्टाकारो मया पूर्वम् ।। २ ।। An indeterminate cguation of the type

• yadyapyuktah: 5akalaih: tathat

muitat ratyat० karma | (1) 104 THEORY OF THE PULVERISER is called a pulveriser (ku!akara). The pulveriser of the type (1) is called a non-residual pulveriser (miragra-ku!!akara), and that of the type 2) is called a residual pulveriser (ठagra ku!!akara) The difference between the two types will become clearer by the follow ing examples, of which the first relates to the non-residual pulveriser and the second to the residual pulveriser: Ex. 1. “8 is multiplied by some number and the product is increased by 6 and then the sum is divided by 13. If the division be exact, what is the (unknown) multiplier and what the resulting guotient ?” Bx. 2. *What is that number, O mathematician, which yields 5 as re mainder when divided by 12, and 7 when divided by 31 ?” The rules given in the following starm2as relate to the non-residual pulve riser, which is of the type (1). It may be mentioned that in equation (1), ८ is called the “dividend', b the *divisor', and ८ the *interpolator'. When the interpolator is negative, it is technically called gata ; and when the inter polator is positive, it is called guntarya 2:3. Preliminary operation : गुणकारभागहारौ विभजेदन्योन्यभक्तशेषेण । तौ तत्र भाज्यहारौ दृढ़ावाप्तौ विनिर्दिष्टौ ।। ३ ।। अन्योन्यशेषभक्तं गतगन्तव्यं यदा निरवशेषम तत्रेष्टाभ्यां कार्य कुट्टनमन्यत्र दृढ़ाभ्याम् ।। ४ ।। .८., “Divide out the dividend (1it. multiplier) and the divisor by the (non-Zero) remainder of their mutual division. The re resulting dividend and divisor are then said to be prime to each other When the gata (i.e., negative interpolator) or garntaya (i.८ positive interpolator) is found to be exactly divisible by the (non-Zero) remainder of the mutual division, (it should be understood that the given interpolator corresponds to the true non-abraded values of the dividend and divisor, and 50) onc should proceed with the actual (non-abraded) values of the THEORY OF THE PULVERISER 105 dividend and divisor in solving a pulveriser. In the contrary case, (it should be understood that the given interpolator corres ponds to the abraded values of the dividend and divisor, and so) one should proced with their abraded values. Let ). be the greatest common multiple of 0 and b; and let 2= ).4 and b=).B. If ८= XC, then according to the above rule, we have to solve the (0) .B 2-4. 4* + C If८ is not divisible by ), then we should solve the pulveriser 4x + ८ 105 n general, a pulveriser is said to be wrong when the interpolator is not divisible by the greatest common multiple of the dividend and the divisor But in the present case, as will be seen from the following rule, the author while enunciating the above rule has in his mind a particular astronomical problem in which the dividend denotes the number of revolutions of a planet, the divisor the number of civil days, and the interpolator the residue of the revolution of the planet. And in such an astronomical problem, the residue of the revolution depends upon whether it has been obtained by धsing the actual values of the revolution-number and the civil days or by using their abraded values. Hence the justification of the above rule. It is presumed that the given problem is in no case incorrect. The method of solving a pulveriser: भाज्यं निधाय तदधो हारं च पुनः परस्परं छिन्द्यात् लब्धमधोऽधः प्रथमावाप्तस्याधस्ततोऽप्यन्यत् ।। ५ ।। विभजेदेवं यावद् भाजकभाज्यावशून्यरूपौ स्तः । मतिकल्पना च विधिना समे पदे व्यत्ययाद्विषमे ।। ६ ।। भाज्याद्भाज्याहृतगतशेषोनाद् भाजकाभिहतदेहात् । गतसहिताद् भाज्याप्तं गतस्य हानौ मतिर्भवति । ७ ।। रूपानहारगुणताद्गन्तव्याप्तस्य भाज्यलब्धस्य । हारहृतस्य च शेषं योगे हारो मतिरशेषे ।। ८ ।। मतिहतभाज्याच्छोध्यं गतमगतं योजयेत्ततो विभजेत् । हारेण मतिं वल्ल्याऽधोऽधो निधायाप्तमप्यस्याम ।। ९ ।। 106 THEORY OF THE PULVERISER उपरिष्ठमुपान्त्यहतं युतमन्त्येनैवमेव परतश्च । एव तावत् कुर्याद्यावद् द्वावेव तौ राशी ।। १० ।। उपरिस्थो हर्तव्यो हारेणाध:स्थितश्च भाज्येन । शेषं दिनादि चक्रादि च तत् स्याद्यच्च तेनाप्तम् ।। ११ ।। .८., “Set down the dividend and underneath that (dividend set down) the divisor, and then perform their mutual division. Write down the quotients (of mutual division) one below (the other: the second one under the first, the third one under the second, and so on. Carry on the mutual division till the (reduc ed) dividend and the (reduced) divisor are different from 2ero. If the number of quotients (thus obtained) is even , obtain the (number called) 70at in accordance with the (following) rule; and if the number of quotients is odd, obtain the mat con trarily: When the interpolator is negative, divide the interpolator by the (reduced) dividend (bhajalhtta-gata), then subtract the resulting remainder from the (reduced) dividend (5८507d bha ylt), then multiply the remainder obtained by the (reduced) divisor (bl:jakabhthatadehat), then increase the resulting product by the interpolator (gata5ahitat), and then divide the resulting gata5alifad blaiyapturiः gata9ya harau matibhar0at ॥7 | 42aris!harmularlyalhataोः yutamargyermaium८0 barata5८ ।

• That is, assuming the dividend as the divisor, the divisor as the divid

end, and the positive (or negative) interpolator as the negative (or positive) interp०lator THEORY OF THE PULVERISER 107 sum by the (reduced) dividend (bhija2ta7) : the tained) is the mat 107 uotient (ob

• When the interpolator is positive, diminish the (reduced)

(/07alharaguritat garntagya), divide that by the (reduced) divid end (a24asya bltiyalabalhaya), and then divide the (resulting) quotient by the (reduced) divisor (hralhta5ya) : the remainder (obtained) is the mat. In case the remainder is 2ero, the divi sor itself is the 70t;

• Multiply the (reduced) dividend by the matt; then sub

tract the gator ( ८., negative interpolator) from or add the garat0 90 (i.e., positive interpolator) to that (product); and then divide that (difference or sum) by the (reduced) divisor. Write down the 720t under the chain (of(uotients), and underneath that (72ati) write down the quotient (obtained) als0. “By the permultimate number (of the chain of quotients) multiply the upper number and (to the product) add the last (i.८., lowermost) number. (After doing this rub out the last number). Repeat this process again and again until there are lef only two numbers in the chain. 36641.४-24 394479375 (Of these two numbers) divide the upper number by the divisor and the lower number by the dividend (if it is possible) The remainders (obtained) denote (respectively) the days, etc ., and the revolutions, etc., which are the requisite quantities.' The above rule would be clear by the following example: Ex. 3. The residue of the revolution (bhagap0-ssa) of Saturn is 24 ; find the days (alhagapa) and the revolutions performed by Saturn, given that the formula for the Sun's revolutions for 4 days is 366414/394479375 Let ४ be the unknown days and y the unknown revolutions performed by Saturn in x days. Then we have to solve the pulveriser We see that the numbers 36641 and 39479375 are already prime each other, so we proced with these numbers. to 108 THEORY OF THE PULVERISER Mutually dividing 36641 and 39479375 until the remainder is 1 (i.८., monzero), and writing down the successive quotients one below the other, we get 10766 15 Writing down the mats The reduced dividend and reduced divisor are 1 and 3 respectively. Since the number of quotients obtained is even, and the interpolator is nega tive, we follow the rule for the negative interpolator and thus obtain 27 for the mat. Multiplying 1 by 27 and subtracting 2+ from the product, we get 3 which divided by the reduced divisor 3 yields I as the quotient. 22 and this quotient under 1076 15 22 27 the chain of quotients, Reducing the chain, we successively obtain 10766 1076 10766 10766 10766 310804439 (multiplier) 15 15 15 15 28689 288689 (uotient) 2 18665 18665 7 8714 8714 22 1237 1237 55 55 27 Dividing 310804 39 by 39479375, and 288689 by 36641, we get 346688814 and 32202 respectively as remainders. Therefore =346688814,5y=32202. These are the least integral values satisfying the equation. कृत्वा वा कर्तव्यः कुट्टाकारस्तु रूपयुतिवियुती । गुणकारो लब्धं च स्यातां तदुपर्यधःशेषौ ॥ १२ ॥ THEORY OF THE PULVERISER 109 गुणकारगुणे शेषे लब्धगुणे हारभाज्यसंहृतयोः । शेषौ तत्र क्रमशो दिनचक्रादी भवेतां तौ ॥ १३ ॥1 etc 2.८., “or (alternatively), solve the pulveriser by taking +! (if the given interpolator is positive) or - 1 (if the given inter p0lator is negative). The remainders (resulting in this way) from the upper and lower numbers (of the reduced chain) are the (corresponding) multiplier and quotient (respectively) Multiply the (given) interpolator severally by these multiplier and quotient and divide (the products thus obtained) by the divisor and the dividend (respectively). The remainders (thus obtained) are respectively the days, etc., and the revolutions To get a solution of Ex. 3 by this method, we solve the pulveriser 36641 x - ॥ 394479375 by the previous method, and get 109 ४=113065211 (multiplier), y=10502 (पuotient) Now multiplying 113065211 by 2+ and dividing the product by 394479375, we get 346688814 as remainder. These are the reguired days. Again multiplying 10502 by 24 and dividing the product by 36641, we get 32202 as remainder. These are the required revolutions of Saturn. This method is based on the consideration that if*=4, y८ B be a solu tion of (८४:+1)| b=y, then ४=64, 1y =cB will be a solution of (८४+८)|b= y The importance of this method lies in the fact that any astronomical problem like the one considered above may be solved by talking recourse to the table of solutions of the euations (0.४:+1)/b =y, for different values of a and b. 110 THEORY OF THE PULVERSISER 26. When the residue of the revolution is given in terms 01 signs, degrees, etc राश्यादावुद्दिष्टे राश्यादेर्भागहारसंगुणितात् राश्यादिमानलब्ध स्याच्छेषं मण्डलादीनाम ।। १४ ।। ]

८., “When the residue (of the revolution) is given in terms

of signs, etc., multiply those signs, etc., by the divisor and divide (the product) by the number of signs, etc., in a revolution : the quotient obtained is the residue of the revolution.” Suppose, for example, that the residue of thoc given to be 4 signs, 28 degrees, and 20 minutes. । Since 4 signs, 28 degrees, and 20 minutes=8900', therefore we multiply 8900 by 210389 (the divisor in this case)* and divide by 21600 (the number of minutes in a revolution). In this way we get 8688 as the quotient. This 576 ४-86688 To find the days and the revolutions performed by the Sun we will now have to solve the pulveriser 210389 revolution of the Sun is 2-7. When the residue of the the residue of the revolution : Following the above method we can also obtain the residue if it be given in terms of degrees, minutes, etc.

• The formula for the Sun's

5764/210389 यच्छेषं युतहीनं तज्जातीयं सदा भवति भाज्यम् । इति राश्यादेः शेषे भाज्यो राश्यादिमानहतः ॥ १५ ॥ राश्याद्याहतभाज्यो हारेण यो भवत्यदृढ़रूपः । तत्रेष्टाभ्यां ताभ्यां शेषवशात्कर्म कर्तव्यम् ।। १६ ॥ । if this case, sign, etc., is given, and revolutions corresponding of the sign not to 4 days is प्रदृढं वा कर्तव्यं शुद्धया तेनापवत्र्य शेषं च । क्रियते मतिर्मतिमता तया पुनः कर्म कर्तव्यम् ।। १७ ।। दृढ़वासरे च गुणिते भाज्ये च तयोरदृढ़ता स्यात् । ताभ्यां दृढीकृताभ्यामेव तदा कर्म कर्तव्यम् ।। १८ ।। 2.८., “The dividend should always be of the same denomina tion as the interpolator which has been added or subtracted. So when the interpolator is the residue of the sign, etc., then the dividend should be multiplied by the number of signs, etc., (in a revolution) When the dividend as thus multiplied by the number of signs, etc., (in a revolution) is not prime to the divisor. then the process of the pulveriser should be performed with their actual (non-abraded) values, depending on the value of the residue (interpolator) (i.e., provided that the interpolator is completely divisible by the greatest common factor of the dividend as multi plied above and the divisor). In that case the multiplied divid end and the divisor as also the residue (interpolator) should be Imade prime to each other by dividing them by the (non-टero) remainder (of the mutual division of the first two). The intel ligent person should (in this case also) find out the mut and pro ceed further with it (in the mammer explained heretofore) (In case the interpolator is not exactly divisible by the great est common factor mentioned above, the following rule should be applied:) If the abraded number of days (viz. the abraded divisor) and the (abraded) dividend as multiplied (by the num ber of signs, etc., in a revolution) are found to be non-prime to each other, then the process of the pulveriser should be perform ed after having made them prime to each other 112 Verses 15 to 17 relate to the case when the given interpolator corresponds to the actual values of the dividend and divisor, and verse 18 to the case when the given interpolator corresponds to the abraded dividend and abrad ed divisor 2-8. When the dividend is greater than the divisor: Ex. 4. हारादधिके भाज्ये हाराप्तं भाजितं पृथक्कृत्य । वल्युपहारान्तं पूर्वोक्त कर्म निष्पाद्य ।। १९ ।। तत्रोपरिराश्याहतपृथक्स्थसहितो भवेदधोराशिः एष विशेषो गदितः परमपि तुल्यं पुरोक्तन ॥ २० ॥1 i.e., “when the dividend is greater than the divisor, divide the dividend by the divisor and set down the quotient (obtained) in a separate place. Then (treating the remainder of the divi sion as the new dividend) having carried out the aforesaid ope rations ending in the reduction of the chain (of quotients), increase the lower number (of the reduced chain) by the prod uct of the upper number (of the reduced chain) and the quotient written in a separate place. This has been stated to be the difference (in this case); the other things are the same as stated before ' Solve the pulveriser Since the dividend 23 is greater than the divisor 7, we divide 23 by 7 Thus we get 3 as quotient and 2 as remainder. Treating 2 as the new divid end, we solve the pulveriser 2-1 The chain of quotient thus obtained is Adding to the lower number 1 the product of the upper १u5otient obtained in the beginning, we get Hence

• ४=4, y=13.

13 29. When the residue of the sign, or degree, etc., is given. (472 alter72tue Pr0685): केचिद्गृहादिशेषे (ज्ञाते) तन्मण्डलादिशेषस्य । तन्मानं चानयते भाज्यस्थाने तु परिकल्प्य ।। २१ ।। कृत्वा कुट्टाकारं मण्डलशेषेण तत्र लब्धेन । भगणानां च दिनानामानयने कुर्वते भूयः ।। २२ ।। number 4 and the t.८., *When the residue of the sign, etc., is known, some (writers),assuming the number of signs, etc., in a revolution as the dividend and applying the process of the pulveriser, first find out the residuce of the revolution, and then from the residue of the revolution obtain the revolutions (performed by the planet) and the days (i.e., allarga10) by applying the same process again ." The following example will illustrate this rule. Bx. 5. The residue of the sign of the Sun is 154168; find (alkarga79) and the revolutions and signs of the Sun's longitude. Here we first solve the pulveriser 12u - 154168 , where a denotes the residue of the revolution the Sun's longitude. Thus we get 14=82977 , the days of the Sun, and u the signs of 114 Now we solve the pulveriser 576 x-82977 210339 where * denotes the days (alkargura) and the revolutions of the Sun's longitude. Wo get ४=176564, 9=5800. When the residue of the minute is given, then, according to the above rule, we first find the residue of the degree, then the residue of the sign, then the residue of the revolution, and then the days (ahargad) and the revo PASSAGES FROM THE LAGHU-BHASKARIYA 9UOTED OR ADOPTED IN LATER WORKS (2) PAssAGEs QUoTED Passages from the Loglu-Bhaskarya occur as quotations in the following commentaries : (1) Karavinda Svāml's commentary on the bastamb0-5ulb0 (2) The Prgy0g0-ra6ara, an anonymous commentary on the (3) Suryadeva's commentaries on the publtatya and the (4) Yalaya's commentary on the Aryablhaya. (5) Nilakantha's commentary on the Aryabhay0. (6) Raghumatha Raja's commentary on the Aryabhayu. (7) (Commentary on the 7aratra-60aigraha of NIlakantha. (8) (Govinda Somayaji's commentary, entitled Da5d on the Brhgjataka of Varahamilhira. (9) Vispu Sarma's commentary on the Vidya-madlujya. Below we refer briefly to these passages and to the places where they occur as quotations. 1. Passage quoted by Karavinda Svāmi. Passage uoted: LBh, i. 1. 90uoted under: 20startba-sulbu-stra, patala 1, khanda 1, stra 1. 2. Passages quoted in the Prgyoga-70and Passages quoted : LB}, i. 19-21, 22. Quoted under: 1MBh, iv. 1-2. 1 This passage shows that Karavinda Swami lived after Bhaskara , i.e., after A. D. 629. In this connection see B. Datta, sinc८ १f the Sulba, Calcutta (1932), pp. 16-17 116 3. Passages quoted by Suryadeva. Suryadeva has quoted a number of passages from the Laghu Bhaskarya, which are arranged below in the tabular form. (i) Passages uoted in the commentary on the Aryabhatya. Passages quoted Quoted under LBh, i. 9(ii). 2. 1. 2; i. 6 LBh, i. 10-11(i) LBh, i. 12(ii)-13 () LBh, i. 14(i) 2, 1i. 6 LBl, i. 14(ii) LBl, i. 15-17 LBh, i. 26 24, i. 22.

• LBh, i. 3(ii)

LBh, i. 6-7(i) LBl, iv. 2-3 24, iv. 41 (ii) Passages quoted in the commentary on the Laghu-manasa. Passages quoted 9uoted under LBl, i. 49, 15-16 opening remarks LBl, i. 10(i) LBl, i. 12(ii), 10(i) LBl, i. 11(i), 14(i) LBl, i. 19-21 LBl, i. 8 LBh, li. 8; i. 23; i.20 LBh, i. 5-6 LMa, iv. 2. LBh, i. 17-20 LBh, iv. 2-5 LBh, iv. 2, 3, 7 LBh, i. 6-7(i) LBh, iv. 11, 12, 14. LBl, vii. 2(i) 4. 5. 6. 7. 8. 9. Passages quoted by Yalaya. LBh, i. 15-17 LB, i. 6-7(i) LB, i. 14(ii) LBl, i. 9(ii) Passages quoted by NIlakantha. Passages quoted LBh, i. 8, 15(ii), 9-10, 14-15(ii) LB, i.29 LBl, i. 14 (ii) Quoted in his comm. on Quoted in his comm. on , i. 11. 24, i. 22-25 24, i. 32-33 Passages quoted by Raghumatha Raja. Passages quoted Quoted in his comm. on LBh, i. 9, 10, 11(i) LB, i. 12, 13(i), 14(i) 4, i. 3 LBl, i. 14(ii) Passage quoted in the commentary on the Tamtra-56aligrald. Passage quoted: LB, iv. 9. (Quoted under : TS, iv. 20(ii)-21(i). Passage quoted in the Duऽadlyy. Passage quoted: LBl, i. 5 f. Quoted under : B3, i. 19 Passages quoted by Vispu Sarma. Passages quoted : LB, v. 2-4(i) | LB}, wi. 1-5 9uoted under : Wil4a, i. 13 | W.Ma, xiv.5

• This passage has been cited by S. Dvivedi in his Gagaka-taraigit p. 14. 1.

The following verses occurring in the 772tra-७८igrah८ (“A collection of ta12traऽ') of Nilakantha (1500 A. D.) are either verbatin reproduction or reproduction with verbal alterations of the verses found in the Laga-Bhaskar)ya: (6) 2. Passages Adopted 3. (i) Version of the Tantra-5aigrala. देशान्तरघटीक्षण्णा मध्याभक्तिद्यचारिणाम । षष्ट्या भक्तमृणं प्राच्यां रेखायाः पश्चिमे धनम् । देशान्तरघटीक्षुण्णा मध्याभुक्तिद्यचारिणाम् । षष्ट्या भक्तमृणं प्राच्यां रेखायाः पश्चिमे धनम् । ।11 Both the versions of the same. (ii) Version of the Loglu-Bhaskary0 (i) Version of the 7antra-sagraha. श्वस्तनेऽद्यतनाच्छुद्ध वक्रभोगोऽवशिष्यते । विपरीतविशेषोत्थः चारभोगस्तयोः स्फुट: ।। 2 (ii) Version of the Lagha-Bhaskarya. श्वस्तनेऽद्यतनाच्छुद्ध वक्रभोगः प्रकीर्तितः । विपरीतविशेषोत्थश्चारभोगस्तयोः स्फट: 13 (i) Version of the 7artra-50igraha. उदक्स्थेऽर्के चरप्राणाः शोध्यास्स्वें याम्यगोलयोः । व्यस्तमस्ते तु संस्कार्या न मध्याह्नार्धरात्रयोः । ।'

• de5antaraghatksuppā madhya bhuktirdyucāripam ।

$asya bhaktamrmain pracyati rekhayah pa5cime dhanam ॥ (LBh, i. 31).

• 5vastane'dyatanachuddhe vakrabhogo'vasisyate ।

viparltavisesotthalh carabhogastayoh sphutah ।। (TS, i. 68) 3 5vastanc'dyatanāchuddthe vakrabhogah prakirtital।। viparitavisesottha5cārabhogastay०h sphutah ।। (LBh, i. 41).

• udaksthe'rke caraprapah 5odhyassvai yamyagolayoll।

vyastamaste tu sainslkarya ma madyahmardharatray०h ।। (TS, i. 29). PASSAGES QUOTED OR ADOPTED 1 19 (ii) Version of the Laghu-Bhaskarlya. 4. TS, ii. 53-56 and LBh, ii. 25-28 are also the same. 1 udaggolodaye sodhyadeya yamye viVasVati 1 vyatyayo'stasthite karya na madhyahnardhatatraybl* II (LBh, ii. 20) (GLOSSARY of Terms used in the Laghu - Bhaskary. 6e6e 4ka (अक्ष) Latitude. [The term 4ks2 is an abbreviation of the complete meaning “the inclimation of the (earth's) (to the plane of the celestial hori - 201) ”, .e., the latitude of the place. 4k50 = axis , umat inclimation.] latitude 4k50-ita (अक्षजीवा) The Rsine of latitude 4:4-jya (अक्षज्या) The Rsine of latitude 4:2090 0ala707 (अक्षस्य वलनम्) See 4g(ata (अगत) Untraversed por tion; portion to be traversed. 4grt (अग्नि) Three 4gra (अग्र) (1) End. (2) 4gra. 4grd (अग्रा) The arc of the celes tial horizon lying between the east point and the point where the heavenly body concerned rises; or the Rsime thereof, which is equal to the distance between the east west line and the rising- setting line of the 4lig0 (अङ्ग) Six 4igula (अङ गुल) heavenly Finger-breadth. defined by the breadth of eight barley corms. 4dri (अद्रि) Seven. 4dllimaऽ८ (अधिमास) Intercalary month The intercalary months (excess of the (synodic) months over the solar months. Thus intercalary months in a yuga =lumar months in a y490 minus solar months in a 9ugu true intercalary Sum does not pass from one sign into the next. 47updata (अनुपात) Proportion. 4ltul070 (अनुलोम) Direct. A planet is said to be (direct, i.e., from west to east 47ul070ga (अनुलोमग) A planet having direct motion, t. ८., Imoving from west to east. 47aturala (अन्तराल) Interval GLOSSARY 121 Antya-jya~ (srjt^tt) The current Rsine-difference, i.e., the Rsine-difference correspond- ing to the elementary arc occupied by a planet. In Hindu trigonometry a quad- rant of a circle is divided into 24 equal parts, called elementary arcs. Antya-maurvi {^hh) Same as Antya-jya'. Apakrama, (stt^t) Declination. Apakrama-dhanu The arc of declination, or simply declination. Apakranti (apnnfNr) Declination. Apakranti-capa {m^te^m) The arc of declination. Apakranti- bhaga (arc^rfarm) Decli- nation. Apama (sptit) Declination. Apamo gunah {3PT*ff m:) The Rsine of declination. Apara (w) (1) West. (2) Apa- ra~hna or afternoon. 4par3 ianr^T) West. ApaTdhna (anrcr^:) Afternoon. (srfssr) Four. Abhuktamsa (apr^r) Untraversed portion. Abhya~sa (spRmr) Multiplication. Amrtatejas (artr^srcr) The Moon.

Ambara (arrsn;) Zero.

Ambhodhi (3F*ftfa) Four. 4kwk& (apw) The northward or southward course of a planet, particularly the Sun. The ayana of a planet is north or south according as the planet lies in the half-orbit begin- ning with the tropical sign Capricorn or in that begin- ning with the tropical sign Cancer. Ayuta (arp) Ten thousand. Arka (arf) ( I ) The Sun. (2 ) Twelve. Arkaja (appsr) Saturn. Arka-suta (ar&pr) Saturn. Arka'gra' (mto) The Sun's Agra. See Agra. Arkodaya (st^t) Sunrise. Ardha-pancama {3R<T^w) Four and a half (4£). Literally, five minus half. Ardharatra (ankra-) Midnight. ' Avanat'i (anr^fa) Moon's true lati- tude as corrected for paral- lax. Avamaf&ira (wrr) Omitted lunar days, or omitted titkis. Avisista (arf^f^) Obtained by applying the method of suc- cessive approximations. Avisesa-karma (arfaSl'SHH) Method of successive approxima- tions. Avisesa-kalakarija (3ri%%^5rr^w) The distance (lit. hypotenuse) of a planet, in minutes, obtained by the method of successive approximations. 122 GLOSSARY Avi'sesana (arfaSraT) Same as Avi- sesa-karma. Asvi (srfar) Two. A'svin (stfonr) Two. A'svim (aren't) Name of the first naksatra. As ft (srfe) Sixteen. Asita (arfer) (1) Asita-paksa, i.e., the dark half of a lunar (synodic, month. (2) The measure of the unilluminated part of the Moon. Asu (arg) A unit of time equal to four sidereal seconds. Asia (arer) (1) The setting of a heavenly body. (2) Asta- lagna, i.e., the setting point of the ecliptic. Astamaya (3R^m) Setting. Astodqyagra-rekha (ar^ft^T^F) The rising-setting line. Ahan (sr?*) Day. Ahargana (ar^Fr) The number of mean civil days elapsed since the beginning of Kali- yuga (or any other epoch). Ahoratra (sr^tm) (1) A day and night, a nychthemeron. (2) The day-radius, i.e., the radius of the diurnal circle. Ahora'tra-dala (srfTTT^^r) Same as Ahordtrardha-viskambha. AhorZtrHsu (arfftl^Tg) The number of asus in a day and night, i.e., 21600. Ahoratr'Srdha (s^T^r) Same as Ahoratrardha-viskambha. Ahorfitrcirdha-viskambha (anftTPTFT- f<p»frj:?T) Semi-diameter of the diurnal circle (of a heavenly body, particularly the Sun), i.e., the day- radius. Aditya (srrfcc^) The Sun. Apya (arr^r) The naksatra Purva- sadha, which is presided over by Apa. A'sH (amir) Direction. Indu{^%){) The Moon.(2) One. Inducca (? f ^ 5 ^) The Moon's apo- gee, i.e., the remotest point of the Moon's orbit. Indvagra (l^?r) Moon's agrTi. See Agra. Isu (^|) Five. Ista {^) (1) Given, desired, or chosen at pleasure. (2) Ista-graka, i.e., desired or given planet. Ista-kala (^z-m^) Desired time or given time. Ista-graha Desired or given planet. Istasu (^T^) Given asus. Ucca (3^) Ucca (apex) is of two kinds: (1) Mandocca (apex of slowest motion), and (2) Sighrocca (apex of fastest motion). The mandocca is that point of a planet's orbit which is at the remotest distance and where the motion of the planet is slowest. In the case of the GLOSSARY 123 Sun or Moon, it is the apo- gee; and in the case of the other planets it is the apogee or aphelion, the geocentric longitude of the apogee being equal to the heliocentric longitude of the aphelion. The sighrocca of a superior planet (Mars, Jupiter, or Saturn) is defined as the mean Sun; that of an inferior planet (Mercury or Venus) is an imaginary body which is supposed to more in such a way that its direction from the Earth is always approxi- mately the same as that of the actual planet from the Sun. Utkrama (3^r) (1) Reverse order. (2) Utkrama jya. See Utkramajya. Utkramajiva (^Tsftar) Same as Utkramajya. Utkramajya (sfWfrr) Rversedsine. Rversin = Radius X (l—cos0). U tkramajya-phala (^cw^iw ) Result (or correction) deriv- ed with the help of the utkramajya of a certain arc. Utkramabhava jiva (scWJTTT ^fter) Same as Utkramajyi . Uttara (^rr) North. Udak (^P) North. Udaggola ( sswfta" ) Northern hemisphere. Udaya (357) ( 1 ) The rising of a planet on the eastern hori- zon. (2) Heliacal rising of a planet. (3) Udaya-lagna, i.e., the rising point of the eclip- tic. (4) Addition, as in Ksayodayau (subtraction and addition). Upapluti (3T^fa) Eclipse. Usnadldhiti (ssw^fafr) The Sun. Rtu (^5) Six. Ekadikka (^feR>) Same direction. Aindnifcil) The east, eastern direction (presided over by Indra). Ojn (afar) Odd. Kakubh (vf*r) Ten. Karana (*PTT) The name of one of the five principal elements of the Hindu Calendar. Karkata The sign Cancer. Karna (^) (1) The hypote- nuse of a right-angled triangle. (2) The distance of a heavenly body. Karnabhukti (^ofaftrr) True daily motion of a planet derived with the help of the planet's distance. Karnasutra (^"T^) The hypoten- use-line. Kala (^tt) Minute of arc. 124 GLOSSARY Kalpa (^t) Addition. Ktirmuka (^ufo) Arc. Kila (^) Time. Kilabhaga (tmvm) The degrees of time. One degree of time is equivalent to 60 asus or 10 viriadh. KisjhS (sftest) Direction. Ku ($) Earth. Kuja (fsr) Mars. Krta (f?r) Four, iffft" Square. Krttikfi (ffirar) The naksatra .Krttika. iiWra (^S) (1) Anomaly. The Ag«</ra is of two kinds : ( 1 ) manda-kendra, and (2) sighra- kendra. The manda-kendra of a planet is equal to "the longitude of the planet minus the longitude of the planet's mandocca (apogee)" and the sighra-kendra of a planet is equal to "the longitude of the planet's sighrocca minus the longitude of the planet." (2) Centre. Kendra-bhukti (%^fw) The daily motion (or change) of a planet's anomaly (hendra) ; or anomalistic motion. Koti (^fc) (1) The upright of a right-angled triangle. (2 ) The Koti correspond- ing to a planet's anomaly. If be the anomaly (or any arc or angle) then the corres- ponding koti is equal to 90° — 0, 0—90°, 270° -0, or g—270 o according as < Q < 90°, 90° < <180°, 180° <0<27O°,or 27O°<0<36O°. Kotiphala {^tfvm) The result obtained by multiplying the Rsineof koti due to a planet's kendra by the tabulated epi- cycle and dividing the pro- duct by 80. Kotisadhana (^faiT«PT) Same as Kotiphala. Kofisutra (*fftfe^r) The thread or line denoting the upright of a right-angled triangle; a perpendicular line. Krama (^3T) Serial order. Kramajya (^>*twt) Same as JyH. Kramabhava jtva (^fpt^t sfkr) Same as Kramajya. Kranti (^>rfar) Declination. Kriya (for) The sign Aries. Ksapa~bhartr (STTTtq ) Moon. * Ksaya (spt) Subtraction. Ksitija (fiarfira) Mars. Ksitijya (fcrf^TT) Earthsine. The distance between the rising-setting line and the line joining the points of intersection of the diurnal circle and the six o'clock circle. Ksiti-suta (fsrfagcr) Mars. 8it-titikal? (क्षिप्तिलिप्तिका:) The minutes of celestial latitude . 8820 (क्षेप) Used for Wikge}}0 (celestial latitude) Galhakardha (ग्राहकार्ध)Half the dia Imeter of the eclipsing body Graly0 (ग्राह्य) The eclipsed body Gralyur-bamab0 (ग्राह्यबिम्ब) The disc of the eclipsed body 125 (G07tta (गणित) Calculation, com- circle of the eclipsed body putati011 Garit0-7akrja (गणितप्रक्रिया) (Cal - (Gha (घटी) A unit of time equi Culat101, Computat101 (Gata (गत) Traversed, elapsed , Ghata (घात) Product, multiplica past, preceding Gat (गति) Motion. Generally Cakra (चक्र) (Circle, twelve signs, used in the sense of “daily । (01 360 Cakralia (चक्रलिप्ता) The number Guyantara (गत्यन्तर ) 0f minutes of arc in a circle difference. (Gantarya (गन्तव्य) To be traversed , .Cakrardlla (चक्रार्ध) Half of a circle to come, succeeding Gup0, (गुण) (1) Multiple or mul - Cakraih:Suka (चक्रांशक) The num tiplication. (2) Rsine ber of degrees in a circle, i.८ Guru (गुरु).Jupiter 360 (G0 (गो) The sign Taurus Cartdru (चन्द्र) The Moon Camudraka710 (चन्द्रकर्ण) The dis or southern hemisphere tance of the Moon (Grula (ग्रह) (1) Planet . (2) (Cardhamas (चंद्रमस्) The Moon. Eclipse (Ca20 (चर) Ascensional difference Gralha70 (ग्रहण) Eclipse It is defined by the arc of the ( ग्रहमध्य ) The celestial eguator lying bet middle of an eclipse ween the six 0'clock circle Grul0durtm0 (ग्रहसद्वत्र्म) True a1d the hour circle of a motion of a planet heavenly body at rising (Gras0 (ग्रास) The eclipsed por- Carajuardl0 (चरजीवार्ध) The RSime of the ascensional diff Graluk0 (ग्राहक) The eclipsing ference body, the eclipser 2.6., 180० 126 GLOSSARY Carasu {^%) The asus of ascen- sional difference. Cala-kendra Slghra-kendra. See Kendra. Cala-kendra-phala (xM*n!4M) Sighra- phala. Calocca (^fr 5 ^) Slghrocca. Capa (=*iq) Arc. Capa-bhaga (^ttthi) An element of arc, or elementary arc (i.<r., one of the twenty-four equal divisions of a quadrant, the Rsine-differences for which have been tabulated by Aryabhata I) . Capita (*TTfWj Converted into (or redsced to) the correspond- ing arc. Ccirabhoga (^r^ftT)' Direct motion. Caitra (#^) The name of the first month of the year. Chaya (WTr) (1) Shadow. (2) The Rsine of the zenith dis- tance. Chaya -dairghya ( ) The length of the Earth's shadow, i.e., the distance of the vertex of the Earth's shadow from the Earth's centre. Chayb-vidha'na (qreifwir) The method of shadow. Cheda (&s) Divisor. Jaladhi (srsrfa) Four. Jim (f«H) Twenty four. Jha (<#rr) Same as Jyi. Jhabhukti (afcmgftfi) True daily motion derived with the help of the table of Rsine-differ- ences. Jiika (^>) The sign Libra. Jya~ {^n) (1) Rsine (= Radius* sine). (2L The Rsine-differ- ences corresponding to the twenty four equal divisions of a quadrant. Jyotis (5*fricrcr) A heavenly body. Tama (?pt) The section of the cone of the Earth's shadow where the Moon crosses it, by a plane perpendicular to the axis of the shadow cone; briefly called "the shadow". Tamomurti (cRtofa ) The Moon's ascending node. Tamovyasa lajffarnr) The diame- ter of the shadow. See Tama. Taraka (circ^r) Star. Tara-samagama (cTRTWTT) Same as Togabhaga. Tigmatejas i ftpw^srcr) The Sun. Tigmarasmi (fartfN) The Sun. Tigmamsu (f%nrtw) The Sun. Tithi (farfa) (1) Lunar day (called tithi). See notes on LBh, ii, 27. (2) Time of conjunction or opposition of the Sun and Moon (parva- tithi). (3) Time of beginning middle, or end of an eclipse. (4) Fifteen. Tithi -varga (%f«T^f) 15 2 , or 225. GLOSSARY 127 Tithyardhahira ( fassrsr|rT ) A divisor which is equal to half of that used in calcula- ting the tithi, i.e., 360. Tiryak (fcnfa) Oblique. Tula {^fT) The sign Libra. Tulyatva (3?zic?) Equality. Tulyadik (^rfep) Like direction, or same direction. Tulyadiktva (^Hi«w j Likeness or sameness of direction. Trijya (fV^TT) Radius or 3438'. Literally, the Rsine of three signs. Trimaurvt "(fsnrHf) Same as Trijya'. Trira'si ( f%Trf% ) Three signs. Trisarkara-vidhana ( fasr^ tt-rsr ) The method of cons- tructing a circle through three given points has been called trisarkara-vidhana by Bhaskara I. Trairasika (^TTfere>) The rule of three. Tvastra (r^f) The naksatra Gitra which is presided over by Daksina (sfa<»r) (I) South. (2) Southern hemisphere (dak- nna-gola). Daksinasa (^ferrim) The southern direction. Darsana-samskara ( 5sfa-w**FR ) usually called Drkkarma ?++4f) Visibility corrections. There are two visibility correc- tions : ( 1 ) Aksa-drkkarma, which is the measure of the arc of the ecliptic lying between the hour circle and the circle of posi- tion of the planet concerned, and (2) Ayana-drkkarma> which is measured by the arc of the ecliptic lying be- tween the circle of celestial longitude and the hour circle of the planet concerned. These corrections having been applied to the true longitude of a planet, we obtain the longitude of that point of the ecliptic which rises on the local horizon simultaneously with the actual planet. Dala Half. Dasra Two. {Dasra is a synonym of A'svi). Dik (ftv)(l) Direction. (2) Ten. Dikka (fere?) Direction. Dim (fer)'(l) Day. (2) Fifteen. Dinagana (ffc*Wl ) Same as Ahargana. DinapuwH parZrdha iGwftrtmnf ) Forenoon and afternoon. Dinantodayalagna (ferRikWT) The rising point of the eclip- tic at sunset. Diri&rdha (few) Midday, Dis(fe$) (1) Direction. (2) Ten. 128 GLOSSARY Di'sa (ferr) Direction. Drkksepa (mm) The drk- ' ksepa is the zenith distance of that point of a planet's orbit which is at the shortest distance from the zenith. This term is sometimes also used for the Rsine of that zenith distance. DrkksepajyH (^fosrr). The Rsine of the drkksepa. See Drkksepa. Dr'sya-Kila (^m^) Duration of visibility. Desintara ( forarc: ) The lon- gitude of a place. It is either the distance of the place from the prime meri- dian, or the difference be- tween the local and standard times. Desantara-ghati Desin- tara, in ghath, i.e., the ghath of the difference between the local and standard times. Dyuganai^Ji^Scime as Ahargana. Dyucarin (er^for) Planet. Drasti (sTgst) Observer. Dhana («pt) Addition. Dhanu Arc. Dhanurbhaga (<*3*k) The ele- ment of arc, or elementary arc (i.e., one of the twenty- four equal divisions of a quadrant, the Rsine-dif- ferences of which have been tabulated by Aryabhata I;. Dhanus (vqs) (1) Arc. (2) 225- Dhara (to) The Earth. Dhisnya (fe^T, Star. Dhrti Eighteen. Naga Seven. Mata (•TrT) Meridian zenith dis- tance. Natabhaga (sTcPror) Meridian ze- nith distance. Nati (*far) ( 1 ) Meridian zenith distance, or the Rsine of that. (2) Difference between the parallaxes in latitude of the Sun and Moon. Nabha (*rx) Zero. JVabhaso madhya (*nnsft wi) The meridian of the place. Lite- rally, the middle of the sky Nadika (TTfofiT) Same as Ghatt. Nddl Same as Ghatt. Nirdksajah (asavah) (Ptoststt: Asus of right ascension, or the time in asus of rising at the equator. MsZ (ftm) Night. Ni'sakara (ftSTre*:) (1) The Moon. (2) One. Ni akrt (fa-JiTTfrcr) The Moon. MsanStha (fom*W) The Moon. Paksa (1ST) Lunar fortnight, i.e., the period from new moon to full moon, or from full moon to new moon. The period from new moon to full moon 129 is called the light fortunight are to clapse at sumrise on (or the light half of a lumar that day. Or, in other words, mmonth) and that from full the time in mळ ऽ which is to m001 to new moon is called elapse at sumrise before the the dark fortunight (or the time of conjunction or opp0 dark half of a lumar month) sition of the Sun and M00n Pada (पद) (1) (9uadrant. (2) Pala (पल) Latitude Souare r00t Palajya (पलज्या) The Rsine० Padminbandhu (पद्मिनीबन्धु) The Sun. latitude declimation of the Sun, 2.6., the obliquity of the ecliptic Pa5cind (पश्चिम) West Paramma-5it (परमक्षिप्ति) Greatest Pata (पात) The ascending mode celestial latitude (of the Moon), of a i.e., planet's orbit (on inclimation of the the M00n's orbit ecliptic) Pata-bhaga (पातभाग) The degrees (परमापक्रम) Same of the longitude of the ascen ding mode. Paramahakram0gupal (परंमापक्रमो गुणः) Pinya (पित्र्य) The Rsine of the Sun's great The mokatra Magha, which is presided est declimation Paridhi (परिधि) (1) Circumference, Pagkara (पुष्कर) Three." periphery. (2) 5picycle Pary990 (पर्यय) Same as Bhaga70. Parua(पर्व) (1) Time of conjunction 'Pirudharayata (पूर्वापरायत) Direc or opposition of the Sun and ted east to west the Moon. (2) Full moon or Imew moon titlhi. (3) An ec Pau910 (पौष्ण) The mak9atra Revati, which is presided over by Pus. Par0ata (पर्वत) Seven Pailt (पंक्ति) Tem Pralrt (प्रकृति) Eight of an eclipse. Pralkge2 (प्रक्षेप) Addition Par0anadi (पर्वनाडी) Prakrya (प्रक्रिया) Process The ra45 of Prograsa (प्रग्रास) The beginning the full moon or new m001 1 There are three pukaras. See Wacas tilhi (also called board) which 201yum, p. 3374, under Thuskara. 130 of an eclipse the first contact Prati20d (प्रतिपद्) The first tilli of either halfofalumar month is called Prat}}}0d. Pratilommu0 (प्रतिलोम) Retrograde A planet is said to be }rat rograde Prabha (प्रभा) The shadow Pak-54ala(प्राक्कपाल) The eastern hemisphere. Paguitagm0 (प्राग्विलग्न) The rising point'of the ecliptic Prap0 (प्राण) Same as 451. Phala (फल) Result or correction. B00 (ब) The name of the first being one of the five impor tant elements of the Hindu Calendar Bala (बाहु) (1) The base of a right-angled triangle. (2) The balu (or blugja) corres ponding to a planet's an10m - aly (or to any arc or angle) . If 60 be the an0maly of a planet (or any arc or angle whatever ), then the corres ponding ballu is 60, 180°-0, 8-180", or 360" 0, accord ing as 0<0-<90°. 90°<0< 180°, 180°<60-<270", or 270°<6< 360० (3) The Rsine of the blut (of a planet's anomaly) Baluplhala (बाहुफल) Correction due to the ॐghr02ca of a planet. The formula for the baluhala is balagiya x tabulated epicycle 80 Bindu (बिन्दु) Point Binto (बिम्ब) Disc of a planet Buddha (बुध) Mercury Bha (भ) Sign Bhaga10 (भगण) The revolution number of a planet the number of revolutions that a planet performs around the Earth in a certain period. The revolutions given by Bhaskara I correspond t0 a period of43,20,000 years Bhara (भव) Eleven Bhaga (भाग) (1) Part, fraction (2) Division. (3) Degree (*) Bhagalara (भागहार) Divisor Blamu Blarg0 (भार्गव) Venus . Bhaskara (भास्कर) (1) The Sun (2) Twelve Bhasuat (भास्वत्) The Sun Bhim0-dik (भिन्नदिक्) Unlike direc Bhima-dika (भिन्नदिक्क) direction Unlike Blukta (भुक्त) Traversed, passed Blukt-90ga (भुक्तियोंग) Sum of (daily) motions Bhukt-utॐ८50 (भुक्तिविशेष) Motion - difference Blugja (भुज) Same as Balu Blugjjya (भुजज्या । The Rsine of Bhajo (Bhagja or Bळhat) Bhaga-lhala (भुजाफल) Blugja-70ur (भुजामौर्वी) Same Bhujळ (भूज्या) as Same as Blti-chaya-digly0(भूच्छायादैघ्र्य) Same Same as hitijya. The distance between the Earth and a star-planet Bludia (भूदिन) Civil days Blimi (भूमि) Earth mi-pasa (भूमिव्यास) The dia Imeter pf th0e Earth cumference of the Earth Bhed, (भेद) (Occultation of a star . Bloga (भोग) Motion

• There are five elements (blita),

wi2. earth, water, sacrificial fre ether, and air 131 Mal:०ra (मकर) Capricorn Maga (मघा)The makऽ0tra Magha. मध्यस्थतारकम्) The central star Mardala (मण्डल) (1) Circle. (2) Revolution centre of a circle 1Mapdatardha (मण्डलार्ध). Half of a revolution, i.e., six signs, or 180० Matsya (मत्स्य) Fish-figure Madly0-6ha)7 (मध्यच्छाया) The midday shadow (of the gno 14adly0-ita (मध्यजीवा) The Rsine of the 2emith distance of the meridian-ecliptic point Madhygja (मध्यज्या ) Same as daily) motion Madly970 (मध्यम) (1) Mcan. (2) Mean planet (170dly070 4ardlyurma blukti? (मध्यमा भुक्ति: ) Mean (daily) motion. Madly0-tagma (मध्यलग्न) Meridian ecliptic point. 14adly blukti (मध्या भुक्तिः) Mean (daily) motion Madlyalhna (मध्याह्न) Midday 132 Mallbalhru0-6alaya (मध्याह्नच्छाया ) The midday shadow (of the gnommon) Manda (मन्द) (1) Mard06८. (2) Mard-20ridha (m07d0 epi cycle) Mardocca (मन्दोच्च) The apoge of a planet. See U८0 Mard0-5arra(मन्दोच्चकर्ण) Mard Mard0८0-2lhala (मन्दोच्चफल) Cor- rection due to a planet's Mandalisa (मन्दांश) The longj- tudes of the apoges of the planets in terms of degrees Mum? (मुनि) Seven Mga (मृग) The sign Capticorn M८dit (मेदिनी) Earth M630 (मेष) The sign Aries Matra (मैत्र) The 70lgatra Amu rādha, which is presided over by Mitra M090 (मोक्ष) The separation of the eclipsed body after an eclipse, the last contact, or the end of an eclipse 7am20 (यम) (1) Saturn. (2) Two 7amala (यमल) Two ?ata (यात) Blapsed 17ळmp५0 (याम्य) (1) The south direction which is presided over by Yama (2) The southern hemisphere ( my0 g010) Bharani, which is presided over by Yama (याम्योत्तर) The local 2agddhika (युगाधिक) Intercalary Imonths in a yuga . 17agmu0 (युग्म) Even 12ati (युति) Union, junction 70ga (योग) (1) Conjunction in longitude of two heavenly bodies. (2) Addition 170ga-tara (योगतारा).Junctio These are those prominent stars of the twenty-seven 10kऽ0tra5 which were used by th_e Hindu astronomers for the study of the conjunction of the planets, especially the 20g0-blळg0 (योगभाग) The degrees of longitudes of the junction Stat s 70jun0 (योजन) The 90 is a unit of distance. The length of a y94720 has differed at different places and at differ ent times. The wojarta of Arya bhata I and Bhaskara I is roughly eguivalent to' 7}} (योजनकर्ण) The dis tance of a planet in terms of 1791070-0as0 ( योजनव्यास ) diameter 1]] Ramdhra (रन्ध्र) Nine Par (रवि) (1) The Sun. Twelve ally, the son Rast-5८90 (राशिशेष ) of the sign of the The of (2) Past-ala (राशिकला) The number of minutes 1800 king The residue Rakha (रेखा) (1) Line. (2) Prime Lag10 (लग्न) The rising point of the ecliptic L८ikळ (लंका) A hypothetical place on the equator where the meridian of Ujjain inter Sects 11t Laik0d490 (लंकोदय) Times of ris ing (of the signs) at Larika 1 There were three persons called Rama, Parasurama, Balarāma or right ascensions (of the signs) Lambuka (लम्बक) The Rsine of the colatitude Limbulka-gu10 (लम्बकगुण) Same as Lamboarma (लम्बन) Paralaxin long tude or, in particular, the difference between the paral laxes in longitude of the Sun Lipt (लिप्ता) Minute of arc. लिप्ताव्यास) Diameter (of a planet) in minutes. L}}1-5ega (लिप्ताशेष) The residue of the minute. //nkratu0 (वक्रत्व) Curvature. Wakrathoga (वक्रभोग) Retrograde 133 7akraramabha (वक्रारम्भ) Commence Iment of retrograde motion Watsara (वत्सर) Year Warga (वर्ग) Square Wargd-ridhi (वर्गविधि) Method of solving a quadratic egua Warg0-7asi //artaman0 (वर्गराशि) (वर्तमान) A sguare Present /artamana-gu70 (वर्तमानगुण) The present (or current) Rsine difference of the elementary are 0ccupied by a planet 134 Warturnam0-grulb0 (वर्तमान-ग्रह) The longitude of a planet (at sum rise) on the current day Warturrdin0d0 (वर्तमानोदय) Time of rising of the sign lying, at the present moment, on the Wartm0 (वत्र्म) Path, locus 70artm01to, (वत्र्मवृत्त) The circle denoting a path (or locus) years, or simply years 7ul010 (वलन) (lit. deflection) 700 relates to an eclip sed body. It is the angle subtended at the body by the arc joining the north point of the celestial horiz01 and the morth pole of the ecliptic (i.e., the angle between the circle of position and the circle of celestial longitude of the eclip sed body). Woulurat is generally divided into two components and (2) is the angle subtended at the body by the 2 0c the north point of the celestial horizon and the north pole of the celestial euator (i.e. the angle between the circle of position and the hour circle of the eclipsed body). The 49070-ularl0 is the angle sub tended at the body by the arc joining the north poles of the equator and the ecliptic (i.e., the angle between the hour circle and the circle of celestial longitude of the eclipsed b0dy) as follows : The great circle of which the eclipsed body is the pole is called the horizon of the clipsed body. Suppose that the prime vertical, egua tor, and the ecliptic intersect the horizon of the eclipsed body at the points A, B and C respectively towards the east of the eclipsed body. Then 00 , arc B07 is called the 490-ul0, and the arc AC is also Walun-arma (वलनकर्म) called Calcula Gemini (Literally, “the lute holder') Wu54 (वसु) Bight Wusundlora (वसुन्धरा) Earth Walert (वह्नि) Three. Waru (वार) Day called Wळrur because it is presided over by Woru10. /50 (वासव) The 10:070 Dhanistha, which is presided over by Wa.5 Wiki?t (विक्षिप्ति) Celestial lati tude W:20, (विक्षेप) (Celestial latitude . Wik20-jya (विक्षेपज्या) The Rsine Iminutes of celestial latitude . Wik98}ो 50 (विक्षेपांश) The degrees Wi! (वित्) Mercury Widikku (विदिक्क) (Contrary direc - 101 7imali (विनाडिका) A unit of time, cuivalent to 24 seconds /imurdardl20 (विमर्दीर्घ) Half the duration of totality of an eclipse Wy06 (वियत्) Zero. Wal}}tika (विलिप्तिका) 72 a 7070,(विवर) Difference, interven 1g Spa0Ce [7u50ut (विवस्वत्) The Sun. Same 2S 7.5490, (विश्लेष) Difference W:500 (विश्व) Thirteen Wiguogija (विषुवज्या) The Rsine of the latitude (of a place) Wiu00ddin0 (विषुवद्दिन) The day of the cuinox . ) The cui m0ctial midday shadow Wikambha (विष्कम्भ) Diameter Wkortblb0-dalo, (विष्कम्भदल) Semi Radius diameter, radius. Wistrt (विस्तृति) Radius. Wtt0 (वृत्त) (1) A circle or its cirumference. (2) Epicycle 7८d, (वेद) Four Wadlblt (वैधृत) An astronomical phenomenon. See LBh, i.29 Watitut (वैश्व) The 10lksutrt Uttara alha, which is presided over by Visve Devalh Waiu0 (वैष्णव) The mukgthu. which is presided over by Wi$7u। An astrono Imical phenomenon. See LBः i. 29 7900०lb८du (व्यवच्छेद) Divisor in 135 terms of y90 5, 136 (व्योम) Zero &lळbult (शकाब्द) The years of the Saka era kru, (शक्र) Fourteen. which is presided over by Indra. (Sakra=Indra) Suku (शंकु) (1) (Gnommon The Rsime of altitude (of a heavenly body) Suil:0gr0 (शंक्वग्र) The distance of the projection of a heavenly body on the plane of the celestial horizon, from the rising-setting line of the heavenly body Sunt (शनि) Satur1 Suru (शर) Five Su50-luk7ma(शलक्ष्मा) The Moon . 5asi(शशि) (1)The Moon.(2)One Su90 (शश्युच्च) Moon's apoge 6;. lit (शिशिरदीधिति)TheMoon" Sgbrt-end}0 (शीघ्रकेन्द्र) The Sgh10 1 There are fourteern Indras (Sakra) corresponding to the fourteen Sighu00 (शीघ्रोच्च) See U८ (शीघ्रन्या $gthrog}}hould. See [(P + 2 sin ;)* + (Rsin b)*]*** Suk70 (शुक्र) Friday 5090, (शून्य) Zero Sigon0ut (शृङ्गोन्नति) The elevation of the Mo0m's hdrns (or cusps) S30 (शेष) Remainder, residue Sutlo, (शैल) Seven. Salk:t (सकृत्) By the application of the rule only once (3.6 ., without the application of the Imethod of successive approxi mations) Som0ukuld (समकल) Two planets are said to be summukould when they are either in conjunction or opposition in longitude Summupirudharu (समपूर्वापर) Same शंकुः) The Rsine of the prime vertical altitude (of the Sum) S0720072011ala (सममण्डल) The prime vertical Samurkha समरेखा) The meridian Samual}}}termult (समलिप्तेन्दु) The longi time of opposition or conjuc So7207:0 (सम्पर्क) The sum of the Contact. Used in the sense of “the sum of the diameters of the clipsed and eclipsing bodies . Sam}}07:0-dala (सम्पर्कदल) Same as Sam107kardlh0 (सम्पर्कार्ध) Half the sum of the diameters of the eclipsed and eclipsing bodies . Sal:057ditSu (सहस्रांशु) The Su Sagara (सागर) Four Sa720705tak4 (सार्पमस्तक) Name of astronomical phenome- Sauitra (सावित्र) Pertaining to the Sita (सित) (1) The measure of the illuminated part of the M00m's disc; the phase of the Moon. (2) The lightlhalf of a 1umar mouth (sta-20aka). (3) 137 of the illuminated part of the Moon's disc. Sutradlt}}20 (राधिप) Fourteen. Sy0 (सूर्य) (1) The Sum (2) Twelve. Surya (सौम्य) (1) North. (2) The northern (hemisphere) (3) Mercury Saur (सौरि) Saturn Stliyurdlb0 (स्थित्यर्घ) Half the dura tion (of an eclipse) Half the duration (of arm Stlbild, (स्थूल) (Gross, approximate S2050 (स्पर्श) (Contact Shlbutta (स्फुट) True, corrected Shuta-grolb0 (स्फुटग्रह) True planet. Shuta-bl५kti ( स्फुटभुक्ति । True (daily) motion Shuta-blb00g0 ( स्फुट-भोग ) True (daily ) motion Slbuto-710dly0 ( स्फुटमध्य ) True mean; the true-mean planet; the true-mean longitude of a planet Slbut0-9ja70-010 (स्फुटयोजनकर्ण) The true distance (of a planet) in terms of99juras Shu40-urta (स्फुटवृत्त) '[rue corrected epicycle Sudasa-blbumt-uttu (स्वदेशभूमिवृत्त) The local circumference of the Earth, .e., circumference of the local circle of latitude 0" 138 Sud८50-6b0dgya (स्वदेशभोदय) Times of rising of the signs at the local place, or obligue ascen - sions of the signs. Suad८5aka (स्वदेशाक्ष) The latitude of the local place. Sund८50dya (स्वदेशोदय) Same as S0d८50-blb0d490. Suablbotta (स्वभूवृत्त) Same as Sura (स्वर) Seven. ant (हन्ति) (Occults. Harja (हरिज) Horizon.

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