The Later Pauliśsiddhanta

The Later Pauliiasiddhiinta by

DAVID PING=*

GaneSdya namo namah

The original Pauliiasiddhfinta was evidentiy written in c. 400 A.D., presumably at Ujjayini during the reign of the Gupta monarch Candra- gupta 11. The author’s name-PauliSa-may well be an Indianization of the Greek IIaiXoS, though al-Biriini’s attempt to connect him with Alexandria (P 1) is founded on a misreading. Indian tradition generally considers the name of the siddhanta to be derived from the personal name PuliSa, which is sometimes identified with the name of the Vedic rji Pulastya.

Al-Biriini’s erroneous Paul of Alexandria has often been identified with the Greek astrologer Paulus Alexandrinus, the second edition of whose Eioaywyfi was published in 3811. I have elsewhere2 given evidence for the impossibility of this identification. I can now add that another Paul of Alexandria is known to have practiced astrology in the early fifth century; in Abii Macshar’s Kit& aZ-mawCZid 15 (Bodleian Huntington 546 f. 64) Paul of Alexandria is associated with a native who was born 4 June 429 and who died in 445. I hasten to add that there is as little reason for identifying this Paul of Alexandria with PauliSa as there was for so identifying his predecessor.

The earliest version of the PauliJasiddhGnta was revised by LBtadeva in c. 505. This revision is known to us from the summary of it in VarSt- hamihira’s ParicasiddhtintikG (c. 550) and from the references to it in Brahmagupta’s BrGhmasphutasiddhcTnta (629). A full discussion of what is known of it will be found in the new edition of the Paiicasiddhcintikd being prepared by Professor Neugebauer and myself.

The later PauliSasiddhGnta was published in western or northwestern India, probably at SthBniSvara, between c. 700 and 800. It is not known to Brahmagupta; it may have been influenced by his Khacdakhridyaka (665); and it is first cited by PflhiidakasvBmin in his commentary on the

Oriental Institute, University of Chicago.

Ccntuurw 1969: POI. 14: DO. 1: pp. 172-241

The Later Paulifasiddhcinta 173

Khandakha'dyaka (864). Those authors who knew it directly all belonged to the western or northwestern portions of India: PythiidakasvBmin to SthBniSvara, Utpala (966) to KBSmira, al-BirGni's pandit (1030) to the Panjab, and AmarBja (c. 1200) to Gujarat. It fell out of use in the thir- teenth century, and no complete copies have come down to us.

We do have, however, a number of fragments, which are collected here. They show that the later Pauliiasiddha'nta, like the KhandakMdyaka (its principal rival in northwestern India), basically represents the iird- hara'trika system of h a b h a t a ; in some respects, however, (e.g., in the calculation of the length of a KaIpa and in the value of n) it depended rather on the Aryabhafiya. Aside from the Sisyadhivrddhidatantra of Lalla it is the only siddha'nta of the eighth century of which much is known today.

The fragments of the later Pauliiasiddha'nta are of two sorts. Those found in Sanskrit sources generally quote his ipsissima verba (in iiryii meter), and are completely reliable; those found in al-Biriini's several works3 have come to us only through the feeble understanding of al- Biriini's interpreter. In fact, it is clear from the material gathered here that al-BirGni never examined personally the Sanskrit manuscript; that he was, in fact, virtually unable to translate himseIf a Sanskrit technical work on astronomy into Arabic. Much of the confusion noted in the commentary to several of the fragments given below was due to the inadequacy of the interpreter.

The Cry6 verses of the later Pauliiasiddha'nta that have been recovered are the following":

1-11. See P 3. sat prBnBs tu vinBdi tatsastyg nadika dinam sasfyB / etiisZim tattrimSan masas tair dvadasabhir abdah // SasfySi tu tatparanam vikala tatsastir api kalB tBsBm / saStyBmSas te trimgad rBSis te dvBdaSa bhacakram //

111. See P 5. astBcatvBrimSat pBdavih-nB kramiit kmdinBm / abdas te 4atagunitB grahatulyayugam tadekatvam // IV-VII b. See P 59; IV also in P 14. parivarttair ayutagunair dvitrikytair bhaskaro yugam bhuiikte / rasadahanahutavahZinalaSaramunipavanendriyaiS candralj //

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vedHhivasurasZiiikalocanadasrair avanislsmuh / ambaragaganaviyanmuniguqavivaranagendubm SaSisutaS ca // BkFiSalocaneksanasamudra~a?kiinalair jivah / as?avasuhutavahekgapayamakhanagair bhiirgavd &pi // kWasaSarartumanubhih sauro budhabhiirgavau divakaravat /

VIII. See P 14. khakhiisfarnunir8rnaSvine~iista~~ara~p~ / bhiinm caturyugenaite parivarttiih prakirtitah //

IX. See P 14. dasriirthabiinatithayo laksahaah savanena te divasah / viyadastakhacatu+katririkhaSodaSa candramlnena.

X. See P 14; X a-b also in P 15, X c-d in P 59. vasusaptarfipanavamuninagatithayah SatagunaS ca saurena / iirksena khtistakhatraya~addasragunilnilaSaSgiikiih //

XI. See P 59; XI c-d also in P 14. caturyugena ciindram8sebhyo y2vanto 'tiricyante / adhimiisakah qadagnitrikadahanacchidraiararfipiih //

XII. XI1 a-b in P 16, XI1 c-d in P 14. cilndraih siivanaviyuw pracayas tair upacayo "rkadinahinah / tithilopah khavasudvikadasriisfakaiiinyahrapaksiih //

XIII. See P 14. savanam a m m ciindram sih-yendusamiigamiin diniiflya / sauram bhtidinaraib SaSibhaganadiniini niiksatram //

XIV. See P 21. yugavatsara prayacchati yadi manacatustayam kim ekena / yad avaptam te divasah vijiieyah savaniidinam //

XV. See P 24. uj jay in i rohI takakuruyamuna~~v~same~narn / delantaram na kilryarp tallekhtimadhyasamsthadeSesu //

The La~er Paulihiddhrlnta 175

XVI c-XVII b. See P 33. bhaumiidimagdakarmaqi paridher nicoccavfltasya / saptatir astiivimlatir aiviguniih samanavah khasad bhiigiih //

XVIII. See P 36. tryagniyamii radaSaSino dvinagiih kharasiiivina sadalanandaguniih / trijyiighnii bhiimSahmS caliintajiviih kujiidiniim //

XIX c-d. See P 14. bhaganiintaragesam yat samiigamiis te dvayor grahayoh //

XX a-b. See P 49. sarve jayina udaksthii daksinadikstho jay? bhavati Sukrah /

XXI. See P 51. vfiti3 cakravad acalii nabhasy apiire vinirmitii dhiitrii / paiicamahiibhiitamayi tanmadhye merur amariniim //

XXII. See P 52. tasyopari dhruv& khe tadbaddhaqi pavanaraSmibhiS cakram / pavanaksiptam bhiiniim udayiistamayam paribhramati //

XXIII-XXXVI. See P 59. yugamiisiidhikamiisiih sabhamiisi y o j a m h so 'dhvii / priigyiiyiniim grahiin- tasmiit kaksyii bhaganabhaktiih // vedarasiisfaviyadvasukharavinagiislendavah sahasraghnih /

khatrayasfiryanaviimbaracandrabaOamahidhar / candrarasiistakamaiigalacandrii miirgam idam nabhasah pravadanti // gkiiSaSiinyatithigunadahanasamudrair budhiirkaSukriiniim / indob sahasragur.&a& samudranetrignibhiS ca syit // bhfisiinor muniriimacchidrartusamudraSaiivasubhih / rudrayamiignicatuskavyoma5aSiiiikair budhoccasya // jivasya vedasafkasvaravisayanagiigniSitakiraniirtha* / Sukroccasya yamiinalasafkasamudrarturasadasra~ // bhramano 'rkajasya navaSikhimunindunagasafkamu~s~~ / ravikhaviyannavavasunavavisayeksanayojanah bhakaksyiiya // isfagrahakaksyiibhyo yal labdhaxp candrakaksyayii bhaktvii /

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tii madhy amii grahgniim saurgdiniim kaliiS candry ah // paficadaSahat.2 yojanasaiikhya tatsafiguno 'rdhaviskambhah / yojanakarno 'dhahsthad bhiiyojanakarnavidhin3 v5i // k~artuSiinyavediiivipaficartukhaniSiikarah / VasvagnirandhrendukhakhagunZh karngrdham ambaram // vasumuniguniintariisfaka$atkair dinan3thaSukrasaumyZnZm / dvZidaSadalasafkendriyaSas8iikabhiitair ajanikartuh // dasraSvisatkarasanavalocanacandrair dharanisiinoh / riipatriSiinyasatk8stibhir m i t w tad budhoccasya // astavasurasasanmuniSaiiiiikavasubhis tu jivasya / vasuvasuSiinyiibdhidvikavedair api bhsrgavoccasya // ekiirnavZirthanavaSaSidahanakhadasrai ravisutasya / trivasurasadvirasiinalaSaiivedair rksakarnardham //

XXXVII. See P 59. yojanam asfau kroSah kroSa9 catvari karasahasrani I hastah iaiikudvitayam dv2daSabhih so 'iigulaih Saiikuh //

P 1. Al-Biriini, India, ed. pp. 118-119, trans. vol. 1, p. 153. They (ie., the zijes of the Indians) are five. One of them is the SCrya-

SiddhGnta, attributed to the Sun; Ltita wrote it. The second is the Va- siqthasiddh5nta attributed to one of the stars in Ursa Maior; Visnucandra made it. The third is the PuZiSasiddhZnta attributed to Paulus the Greek from the city of Sayntra, which I think is Alexandria; PuliSa made it. The fourth is the RomakasiddhGnta attributed to the Romans; Sri9ena made it. And the fifth is the BrahmasiddhZnta attributed to Brahma; Brahmagupta the son of Jisnu made it in the city of Bhillamda, which is between Miiltiin and Anahilwarah 16 yojanas. There is a dependence of all of them on the book of Paitimaha attributed to the first father, who is Brahma.

Commentary In this passage al-Biriini is describing the contents of VarrIhamihira's

PaiicasiddhGntikd on the basis of Byhatsumhit5 2 (p. 22), but he supple- ments this with various pieces of misinformation derived from his pandit's as yet incomplete version of the Pauliiasiddhfinta and from the Brtihma-

The Later PaulihiddhZnta 177

sphu?asiddhlrnta of Brahmagupta; these three works were his main sources of information regarding Indian astronomy when he wrote the India.

There were or are several versions of each of the five siddhiintas. Varahamihira names them (Pakasiddhiintikii 1,3) as the PauliSa, Romaka, VBsistha, Saura, and Paiamaha, and states that the first two were commented on by Latadeva; in the next verse he assigns them to three ranks of accuracy: the Savitra at the highest, the PauliSa and Romaka in the middle, and the Vasistha and Paitiimaha at the lowest. Though all of these texts in the versions used by Variihamihira are lost, his summaries justify his classification.

Al-Biriini’s first text, the Siiryasiddhiinta, evidently originally was written in the fifth century A.D., but it was revised in c. 505 so as to conform to the iirdhariitrika system of kyabhafa. It is this revised form which Variihamihira has used; Varahamihira’s summary is the basis of the Bhiisvati composed by SatZinanda in 1099. Al-Biriini would make this revision of c. 505 to be the work of LBtadeva (probably from LBta, the modem Southern Gujarat), the commentator on the Paulis‘asiddhiinta and on the Romakasiddhiinta; he is probably right in asserting this. The modern Siiryasiddhiinta, which has been published only in the arrange- ments of ParameSvara (early fifteenth century) and of Raiiganatha (seventeenth century), is very different. It probably does not antedate the eleventh century; and the earliest extant commentary, that of CandeSvara, is dated 1178. Utpala in the tenth century knew a Siiryasiddhiinta different from the modern version, but it is unclear whether this was the version of c. 505 or some intermediate redaction.

Vasistha is the name of one of the seven Vedic rjis who are considered to constitute Ursa Maior (Saptarsi), though it was in his role as a sage rather than in that as a star that he was supposed to have composed works on various subjects including jyotihs’iistra. A Vasigha(siddhiinta) is first referred to by Sphujidhvaja (269/70) in his Yavanajfitaka (79,3); this work, like that summarized by Varshamihira, was evidently largely based on Babylonian astronomy. At present we know of two Vasi~~hasiddh~nta~, of which one is virtually identical with one of the modern versions of the Romakasiddhcinta. Utpala quotes as Vaskfha’s verses which occur in neither of these modem siddhintas and not in the VasiJfhasamhitii; again it is not clear whether they are from the version summarized by Vara- hamihira, or represent an intermediate stage in the development of the text such as that of Visnucandra. Visnucandra is referred to by Brahma-

178 David Pingree

gupta (Br&maspu{asiddhrfnta 11,50) as having compiled the Viisisfha from the same elements that SriSena used in compiling the Romaka; both Visnucandra and Srisena, then, are to be assigned to the late sixth century.

The original Paulisasiddhrfnta was probably written in the early fifth century; the version summarized by Varahamihira was due to LZifadeva. Al-B-Eiui’s statement here that Paulus comes froin Sayntra, which he believes to be Alexandria, is due to his misreading of his pandit’s un- pointed Arabic; the pandit had surely written T.n.y.s.r (SthBniSvara) rather than S.y.n.t.r. One finds the correct reading in P 25 and P 41. The later Pauliiasiddhrfnta, then, like Pythiidakasvfimin, was associated with SthiiniSvara; SthBniSvara, incidentally, is identical with the Kuru mentioned in vs. XV.

The original Romakasiddhznta, like the original Pauliiasiddhrfnta, was probably written in the early fifth century and was commented on by LBtadeva; Varahamihira used LBtadeva’s version. Romaka, of course, does mean Roman. There are two versions of a Romakasiddhznta extant in manuscripts today; one is very similar to one of the versions of the Vusighasiddhcinta, and the other is largely based on Muslim sources. The attribution of the Romakasiddhrfnta to Srisena is found in Brrfhma- sphufasiddhcinta 1 1,48-50b:

“Srisena, taking the mean Sun and Moon, the lunar apogee and node, and the mean Mars, Mercury’s conjunction, Jupiter, Venus’ conjunction, and Saturn from LBfa, the revolutions in the lapsed years of the yuga from the Vrfsijlha whose pzda was composed by Vijayanandin, and the mandocca epicycles, nodes, method of computing true longitudes, and so on, from k a b h a t a , made the Romaka, which was a clothes-knot into a rag.”

Sri$ena, then, must be dated to the latter half of the sixth century. Incidentally, it is some such passage as that quoted here that al-BirBni had in mind when he noted the shortcomings of the PauliSa and Romaka in AI-Qcinzin al-Masciidi, vol. 1, p. 268.

The BrahmasiddhEnta or Paitcimahasiddhanta utilized by Varahamihira was a text similar to the Jyot@avedrfiiga of Lagadha, but using as epoch 80 A.D. Another Paitrfmahasiddhznta, incorporated in the Vi$nudhar- mottarapurrfga, was written in the first half of the fXth century; this was used by kyabhafa in his &yabhat@a (499 A.D.) and by Brahmagupta in his Brcihmasphutasiddhcinta (629 A.D.). It is this last work which al-

The Later Pauliiasiddhfinta 179

Biriini refers to; Brahmagupta was the son of Jisnugupta and did live at Bhillamiila, the modern Bhinmal, between Miilasthiina (Miiltan) and Anahilapattana (Anahilwiirah). A fourth Brahmasiddhrinta, which is extant, is alleged to belong to the Srikalyasamhitd.

P 2. AEBiriini, India, ed. p. 119, trans. vol. 1, p. 154. There has not come to me any manuscript except that for Puliia and

that for Brahmagupta, nor is the (whole) extent of their translation completed for me.

Commentary When the India was written, then, in 103 1, al-Biriini had before him only

incomplete translations of the later Pauliiasiddh&zta and of the Brchma- sphutasiddhrinta of Brahmagupta. The interpreter or pandit, or his translation, is mentioned in P 6, P 11, P 29, P 34, and P 38. It appears, therefore, that al-Birtini never personally examined the Sanskrit manuscript.

P 3. Utpala on Byhatsamhitri 2 (p. 24). Thus in the Pauliia (vss. I and 11): “Six prinas equal one vinridi; sixty of these one nridikri; sixty of these

one day; thirty of these one month; and twelve of these one year. Sixty tatparas are one vikalz; sixty of these one kali; sixty of these one degree; thirty of these one zodiacal sign; and twelve of these the zodiac.”

Commentary So Paulida represents the sexagesimal divisions of time and of the

circumference of a circle. The divisions of time are correct only if one assumes that a “day” is a saura day-i.e., the time required for the mean Sun to travel I”. But a ncidikd is normally defined as a sixtieth of a 24-hour day. However, as will be apparent from P 14, in the PauZiSa- siddhrinta the normal meanings of sivana and saura are interchanged. PauIiSa is more consistent than is Brahmagupta (Brd~masphuCasiddhrinta 1,5-6).

P 4. Al-Biriini, India, ed. pp. 282-283, trans. vol. I, p. 335. Puliia says that the minutes of the sphere, which are 21,600, are at

the times of the equinoxes like the normal breaths of a man in perfect

12.

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health; the sphere revolves one minute (of arc) while the time of one breath elapses.

Commentary Since we know from P 3 that 6 prdnas equal a vincidi, 60 vinddis a

nddiki, and 60 ncidikcs a “day”, it is easy to understand why al-BXini assumes that PauliSa means that the sphere revolves one minute of arc in one prdpa. But he can only do so by ignoring the fact that PauliSa’s “day” is not a sa’vana day, but a saura day.

P 5 . PrthiidakasvBmin on Brdhmasphu?asiddha’nta 1,9 and Utpala on Byhatsamhita’ 2 (p. 23).

Thus are read (verses) with the divine measure (of time) in the Pauliia (Puliiasiddhdnta in Utpala) (vs. 111):

“(The measures) of the Krtayuga and so on are successively forty-eight minus a fourth; these years are multiplied by 100; their sum is the Yuga for the equal (longitudes) of the planets.”

Commentary The divine measure of time signifies that 360 earthly years equal one

divine year. So 48 successively diminished by a fourth (i.e., 12) gives the sequence 48, 36, 24, 12. If we multiply each of these numbers by 100 and again by 360, we get the familiar:

Krtayuga 1,728,000 Tretdyuga 1,296,000 DvEparayuga 864,000 Kaliy uga 432,000 Caturyuga 4,320,000

P 6, Al-Birfini, India, ed. pp. 314-317, trans. vol. 1, pp. 374-377. And as for what is reported according to PuliSa, in his SiddhEnta he

does not stop determining rules for the numbers; some are approved and some are despised. For the rule for the Yugas he put down 48 as a base and subtracted a fourth from it; there remains 36. He subtracts exactly the same (amount) from it, because he made it (12) the base for

The Later Pauli.6asiddhdnra 181

subtraction, and there remains 24. He subtracts it again from it and there remains 12. Then he multiplies each one of the remainders by 100 and there result the years of the Yugus in divyu years.

But if he had made 60 (the base)-for there is a greater scope for operations according to it-and had made a fifth of it the base for subtraction, or if he had made the subtraction by the successive fractions receding from a fifth-1 mean: he should subtract from 60 a fifth, from the remainder a fourth, from the remainder after this a third, then from the remainder a half-there would result for him what first resulted.

It is possible that this is an account of one of several opinions of his, not that which is particularly his; for the emergence of his book in its entirety into Arabic is not agreed upon because of the dogma which is evident in its procedural purposes.

PuliSa deviates from what appears of the rule when he desires to compute what years in our years have lapsed of the life of Brahma before our Kulpu. This is, in the calculation of his (Brahma’s) years, 8 years 5 months and 4 days, which are, in the calculation of Kulpas, 6,068. First he makes them Cuturyugas by multiplying them by the number of Cutur- yugas in a Kulpu, which, according to him, is 1,008; there results 6,116,544. Then he makes them Yugus by multiplying by 4; there results 24,466,176. He makes them years by multiplying them by the years in one Yuga, which, according to him, are 1,080,000. There results 26,423,470,080,000, which are the years that elapsed from the life of Brahma before our Kulpu.

It is possible that it will occur to the followers of Brahmagupta that he (Puliia) does not convert Caturyugus into Yugus, but simply converts Caturyugus into fourths and them multiplies the fourths by the years in one fourth (of a Caturyuga). We do not ask him about the advantage of converting them into fourths, since they do not have fractions which require this process of assimilation; the multiplication of the number of complete Cuturyugas by the years of one complete one, which are 4,320,000, would be sufficiently long. But we say to him: it is conceivable that one should do this if one did not desire to add what have passed of the years of our Kalpa to it.

One multiplies the complete lapsed Manvantarus by 72, as his doctrine (has it), and what results by the years in a Caturyuga; there results 1,866,240,000 years. One multiplies the number of complete lapsed Cutur- yugus of the fractional Munvuntara by the years in one of them; there

182 David Piirgree

results 116,640,OOO. There has passed of the fractional Caturyuga three of the Yugas; their years according to him are 3,240,000, which number is three fourths of a Caturyuga. He used it similarly with respect to the position within a week by means of the days of the cited (years). If he believed in the above-mentioned rule, he would have used it in the necessary place and he would have taken three Yugas to be nine tenths of a Caturyuga.. .

We have mentioned (in P 8) the measure of a Caturyuga in sa’vana days according to the three (of them). The excess of the opinion of PuliSa over that of Brahmagupta is 1,350 days, but the number of years in a Caturyuga is the same in both of them; so the days in a solar year according to PuliSa are certainly greater than those according to Brahma- gupta. But, by the computation of his (Brahmagupta’s) account of Lyabhata, his opinion is less than that of PuliSa by 300 days, and greater than that of Brahmagupta by 1,050 days. Therefore, the days in a solar year according to him (kyabhata) are greater than those according to Brahmagupta and less than those according to PuliSa.

Commentary Obviously al-Biriini had before him a translation of vs. 111, but was

unsure of what else relevant to the subject might be found in the PauZiSa- siddhrinta. The tradition concerning the length of the life of Brahma recorded as PauliSa’s by al-Biriini here and in P 9, P 12, and P 13 is based on the Aryabhatiya, wherein (Ddagitikd 3 ) a Kalpa is taken to consist of 1,008 Caturyugas and a Caturyuga of four equal Yugas of 1,080,000 years each. A Kalpa is a half-day of Brahma; therefore, 8 years 5 months and 4 days in Brahma’s reckoning do equal 6,068 KaIpas on the assumption that a year consists of 360, a month of 30 days exactly. Brahmagupta and his followers are mentioned as he is a severe critic of kabhata’s Kalpa and Yugas (Bra’hmasphutasiddhGnta 1,9 and 12).

Al-Biriini justly criticizes PauliSa for the inconsistency in using Lyab- hata’s audayika system in computing what has lapsed of the life of Brahma before the beginning of our KaZpa and in computing what has lapsed of our Kalpa before the beginning of the present Kahyuga, when he elsewhere (e.g., P 5 and P 7) follows the traditional smrti system in dividing the present Caturyuga into Yugas. PauliSa’s rule for determining what has elapsed of our KaZpa, like kyabhata’s, assumes that

The Later PaulL6asiddhdnta 183

there have passed 6 Manvantarus, each of which contains 72 Cuturyugus, and 27 Cuturyugus plus 3 Yugus; the total is 1,986,120,000 years. Brahma- gupta (BrZhmusphu?usiddhCntu 1,2627) also says that 6 Manvantarus, 27 Cuturyugus, and 3 Yugus have elapsed from the beginning of our KuZp till the beginning of our Kuliyugu; but in his calculation this equals 1,972,944,000 years (see The Thousands of Abzi Mucshur, p. 34). As Brahmagupta remarks (1,28), the difference between his calculation and habhata’s is 3 1/20 Caturyugus.

The civil days in a Cuturyugu according to the three are:

Pauliia (vs. X a-b): 1,577,917,800

) : 1,577,917,500 DuSagitikZ 1 together with Kdukriya’ 5 k a b h a t a (

Brahmagupta (3r~hmusphutasiddhcta 1,22): 1,577,9 16,450.

PauliSa (Crdhura‘triku): 6,5;15,31,30 days h a b h a t a (uuduy iku) : 6,5; 15,3 1,15 days Brahmagupta (bra‘hmu): 6,5; 15,30,22,30 days.

The corresponding year-lengths are:

P 7 . Al-Biriini, India, ed. p. 314, trans. vol. 1, pp. 373-374. Then Brahmagupta says of h a b h a t a that he thinks concerning the

four Yugas that the four in a Caturyuga are equal; so he differs from what we say after smyti, and he who differs is an enemy. He says:

“But as for Puliia, he is praised for what he does, since he does not differ from smrti, because he subtracts a fourth from 4,800, which belongs to the Kytuyugu, and does not stop subtracting it from what results until the Yugus come out corresponding to smyti; but they do not have sandhis or sundhyutpius. Since the Romans are foreign to the tradition of smyti, they do not measure time in Yugus, Manvantarus, and Kalpus.”

This is what he says.

Commentary Brahmagupta in fact says (Br~hmusphutusiddha’ntu 1,12-13): “ h a b h a t a desires that the sandhi for a Munvunturu should be a

Cuturyugu; therefore his Manvantam contains 72 (instead of 71) Cutur-

184 David Pingree

yugas, and his Kalpa is 1,008 Caturyugas or 4,354,560,000 years. The divisions of time, which are Yugas, Manvantaras, and Kalpas, are described in smyti; as they are not in the Romaka, the Romaka is outside of smyti.” and (1 1,lO):

“(kyabhata’s) Manvantaras, Yugas, and Kalpas, (his time) that has elapsed from the beginning of (our) Kalpa, and (his time) that has elapsed from the beginning of (our) Kytayuga are not the same as those enunciated by smyti. Hence h a b h a t a does not know mean motion.”

Al-Birfini’s “quotation” from Brahmagupta is really an interpretation by his pandit of these two passages.

P 8. Al-Birfini, India, ed. pp. 311-312, trans. vol. 1, p. 370. According to k a b h a t a the great (al-kabir) and PuliSa a Manvantara

consists of 72 Caturyugas and a Kalpa of 14 Manvantaras. They do not at all interpose in their construction a sandhi. Therefore it is known that the number of Caturyugas in a Kalpa according to them is 1,008. The years in a Kulpa in divya years are 12,096,000, in human years 4,354,560,000. And PuliSa says about the sa‘vana days in a Caturyuga that they are 1,577,917,800; and the days in a Kalpa in his opinion are 1,590,541,142,400. Thus does he use them.

I have not found anything of the books of k a b h a t a , but what I know of his method is through Brahmagupta’s quotations from him. He says about him in his chapter al-intipid ‘alZ al-zija’t (= TantraparikTZ) that the days of a Caturyuga according to him are 1,577,917,500 by the subtraction of 300 days from those which there are according to Puliia; then, by the computation of this quotation, the days in a Kalpa according to him are 1,590,540,840,OOO.

According to them both the KaZpa and the Caturyuga begin at midnight after (read before) the day from whose beginning they both start according to Brahmagupta.

Commentary “The great” k a b h a t a is a curious designation, the reason for which

is not at all clear. The basic information given in this fragment is derived from Briihmasphufasiddha‘nta 1 ,I 2-1 3 and from the TantraparikTZ-the

The Later PauliJasiddhdnta 185

eleventh chapter of Brahmagupta’s BrrThmusphutusiddhltu. There, however, only k a b h a t a is spoken of. Brahmagupta does speak of the difference of 300 days in Brlihmasphu?usiddh&zta 1 1,5 (cf. 1 1,13):

“When it is said that there are 4,320,000 revolutions of the Sun in a Yugu, that is the correct Yugu for both (the auduyiku and the cirdhurdtriku systems); then why is there a difference between them of 300 risings of the Sun?,

It is true that PauliSa was an adherent of the cirdhurZtriku system, but al-Biriini can only have gained this knowledge from his pandit, not from Brahmagupta.

Similarly, it is true that Lyabhata in his lirdhmitriku system (see Brdhmasphutasiddhlintu 11,14) and PauliSa begin a quarter of a day earlier than Brahmagupta does in the BrrihmusphuCusiddhdntu, which is audayiku; but elsewhere al-Biriini has been talking only of the Jryubhutiyu, which also is uuduyiku. He has again been misled by his pandit, and has displayed his own inability to read Sanskrit. Indeed, were he able to read Sanskrit, he would have noticed Brahmagupta’s criticism in Briihma- sphupuiddhdntu 11,ll of k a b h a t a for beginning the Kulpu with a Thursday rather than a Sunday.

It is true that 1,577,917,800 x 1,008 = 1,590,541,142,400 and that 1,577,917,500 x 1,008 = 1,590,540,840,000, but Brahmagupta nowhere gives these numbers.

P 9. Al-Birfini, India, ed. p. 344, trans. vol. 2, pp. 4-5. He (Brahmagupta) and PuliSa agree that the KuZpus in the life of

Brahma (that have elapsed) before our Kulpu are 6,068, but they disagree about the Cuturyugus; according to PuliSa they are 6,116,544, and according to Brahmagupta 48,544 fewer. If we apply the opinion of PuliSa, according to which a Munvunturu is 72 Cuturyugus without any sandhi, and a KuZpu is 1,008 Cuturyugus, and each Yugu is a fourth (of a Catur- yuga), there has passed of the life of Brahma to the time of our example (400 Yazdijird = A.D. 1031) 26,425,456,200,000 (years), and of the KuZpu 1,986,124,132 (years), and of the Munvunturu 119,884,132 (years), and of the Cuturyugu 3,244,132 (years).

As for what is after the Kuliyugu, there is no difference about the whole years, but there are according to both of them from the Kuliyuga 4,132 (years), which is the Kulikdu, and from the Bhlirutu battle, which is the PrTnduvukdla, 3,479 (years).

186 David Pingree

Commentary Most of this discussion can be easily understood by reference to P 6.

Note, however, that the number 26,425,456,200,000 years is not that which has passed of Brahma’s life till A.D. 103 1, but only till the beginning of the Kaliyuga. PuliSa’s calculation places the Mahiibhiirata war in -2448; for his reference to this battle see also P 41.

P 10. Al-Biriini, India, ed. p. 281, trans. vol. 1, p. 333.

(31 1,O4O,OOO,OOO,OOO years) is a day of Purusa; but it is said that a day of Purusa is a Pariirdhakulpa (this is 864,000,000,000,000,0,000 years).

In the Pulijasiddhiinta it is said that the life of Brahma

Commentary As will be seen from P 11, the parilrdha for PauliSa is the seventeenth

decimal place. Since a KaIpa is half a nychthemeron of Brahma, and Brahma “lives” for a hundred “years” of 360 “nychthemera” apiece, Brahma’s life is 3 1 1 ,04O,OOO,O00,0OO years. The Pariirdhakalpa given here extends to the twenty-first decimal place, and is two Kulpas (a nychthemeron) multiplied by 100,000,000,000-i.e., it is not based on the normal “life” computations, and defines parilrdha in a manner different from Pauliia’s. Only the first statement in this fragment, therefore, can be attributed to the Pauliriasiddhfinta.

P 11. Al-Biriini, India, ed. pp. 138-139, trans. vol. 1, p. 177. It is translated for us from the Puliiasiddhiinta that after the fourth

(decimal place), sahasram, are uyutam as fifth, niyutam as sixth, prayutam as seventh, koti as eighth, arbudum as ninth, kharva as tenth, and after this according to what is in the previous table.

Commentary The previous table occurs in India ed. p. 137, trans. vol. 1, p. 175:

1. ekam 10. padma 2. daSa 1 1. kharva 3. iatam 12. nikharva 4. sahasram 13. muhipadma 5 . ayuta 14. Saiiku 6. Iuksa 15. samudra

The Later PaulihsiddhEnnta 187

7. prayuta 16. madhya 8. koti 17. anrya 9. nyarbuda 18. parcrdha

P 12. Al-Biriini, India, ed. pp. 355-356, trans. vol. 2, p. 18. Similarly we derive from the revolutions in a Caturyuga according to

PuliSa the revolutions in a Kalpa, assuming that it is 1,OOO Caruryugas and that it is 1,008. We put them down in this table:

Names Yugas according to PuliSa Revolutions Revolutions in a Revolutions in a

Kalpa of 1,008 in a Caturyuga Kalpa of 1,OOO Sun Moon Lunar Apogee Lunar Node Mars Mercury Jupiter Venus Saturn

4,320,000 57,753,336 488,219 232,226

2,296,824 17,937,000 364,220

146,564 7,022,388

4,320,000,000 57,753,336,000 488,2 19,000 232,226,000

2,296,824,000 17,937,000,000

3 64,220,000 7,022,388,000 146,564,000

4,354,560,000 58,215,362,688 492,124,752 234,083,808

2,315,198,592 18,080,496,ooO 367,133,760

7,078,567,104 147,7363 12

Commentary The Pauliiasiddhdnra’s revolutions in a Caturyuga are precisely those

of the drdhardtrika system. See also vss. IV-VII. These parameters imply the following mean daily motions:

Saturn Jupiter Mars Sun Venus’ conjunction Venus’ anomaly Mercury’s conjunction Mercury’s anomaly Moon Moon’s apogee Moon’s node

0;2,0,22,41,. . .’ 0;4,59,8,48,. . . 0;3 1,26,27,47, . . . 0;59,8,10,10,. . . 1;36,7,44,13,. . . 0;36,59,34,3,. . . 4;5,32,17,45,. . . 3;6,24,7,35,. . . 13; 10,34,52,2,. . . 0;6,40,59,29,. . .

-0;3,10,44,7,. . .

David Pingree 188

P 13. Al-BTriini, India, ed. pp. 368-370, trans. vol. 2, pp. 31-33. But we shall do this according to the opinion of PuliSa, because, even

though it is in Caturyugas, the operation resembles (that) in Kalpas. For the time of our example there have passed according to him 3,244,132 years of the Caturyuga; the solar days are 1,167,887,520. If we multiply their months by the adhim&zs in a Caturyuga or by the substitute multiplier, and divide the product by its saura months or by the substitute divisor, there results 1,961,525 (read 1 , 196,525) adhimzsas, and there

44,837 45,000’ remains - There are in these (months) 1,203,783,270 tithis. If we

multiply these by the avamas in a Caturyuga and divide the product by 598,055 2,226,389’ its tithis, there result 18,835,700 avamas, and there remains

And there come out by this 1,184,947,570 Slivana days from the beginning of the Caturyuga, which is what we wanted to find.

w e transmit a passage from the PuliSasiddhGnta in which he operates similarly to the way we do, in order to make the meaning clearer, and firmly estabkhed in its essence. Pu%a says:

“We put down what has passed of the life of Brahma before (our) Kalpa, which is 6,068 Kalpas, and we multiply these by the number of Caturyugas in a Kalpa, which is 1,008; there results 6,I 16,544. Then (we multiply) by the number of Yugas in a Caturyuga, which is 4; there results 24,466,176. Then (we multiply) by the years in one Yzrga, which are 1,080,OOo; there results 26,423,470,080,000, which are the years before our Kalpa. We multiply this by 12; there result 317,081,640,960,000 (saura) months. We put this down in two places, and we multiply one of them by the number of adhimcisas in a Caturyuga, which is 1,593,336, or by the number which we have mentioned as standing in its place, and we divide the product by the saura months in a Caturyuga, which are 5 1,840,OoO; there results 9,745,709,750,784 adhimzsas. We add these to the second place; there results 326,827,350,710,784. We multiply this by 30; there comes out 9,804,820,521,323,520, which are tithis. We put these down in two places, and multiply one of them by the avamas in a Caturyuga, which are the difference between the sZvatza days and the tithis, and we divide the product by the ritfiis; there results 153,416,869,240,320; these are the avamas. We subtract them from the second place, and there remains 9,65 1,403,652,083,200, which are the

The Later Pauliiasiddhznta 189

lapsed days from the life of Brahma before our Kalpa, I mean the days in 6,068 Kalpas, each of which has 1,590,541,142,400. If you subtract sevens from these days, there will not be any of them left; so it ended on a Saturday, and this Kalpa began on Sunday.”

It is known (from this) that it is necessary that the life of Brahma also began on a Sunday.

He says: “There has passed of the fractional Kalpa 6 Manvantaras, each one of which contains 72 Caturyugas, and each Caturyuga has 4,320,000 (years); so the sum of their years is 1,866,240,000. We do with these as we did previously in the other (example), and there come out 68 1,660,489,600 days in 6 complete Manvantaras. If you subtract sevens, there remains 6; so it ended on Friday, and the beginning week-day (of our Manvantara) is a Saturday. There have passed of it 27 Caturyugas, whose days are, by the example of the previous operation, 42,603,780,600. They end on a Monday, and the twenty-eight begins with Tuesday. There have passed of it three Yugas; the sum of their years is 3,240,000. By the example of what went before, their days are 1,183,438,350, which require (an ending) on Thursday; and the Kaliyuga begins on Friday. The days which have passed of the KaZpa are 72,544,770,800 (read 725,447,708,550), and the days which have passed of the life of Brahma up to the beginning of the KaIiyuga in which we are is 9,652,129,099,791,750.”

Commentary

The interval found by al-Biriini is that from the beginning of our Caturyuga to the beginning of the CaitrcTdi year which began in A.D. 1031. This interval is known from P 9 to be 3,244,132 years. The method of computation is I of Brahmagupta (see the commentary on Yacqiib ibn

66,389 2,160,000’ TBriq T 4). The substitute, is from the parameters of P 12

1,593,336 I 24 5 1,840,OOO ’ 24’

-- (cf. P 18):

PauliSa uses the same method to find the ahargapa from the beginning of the life of Brahma to the beginning of our KaZiyuga; for his first interval see also P 6. Note that 6 Manvantaras contain 432 Caturyugas of 4,320,000 years, and that of the present Manvantara 27 (for the 27 nakFatras?) Caturyugas have passed. In summation:

190 David Pingree

6,068 Kalpas 9,65 1,403,652,083,200 6 Manvantaras 681,660,489,600 27 Caturyugas 42,603,780,600 3 Yugas 1 ,1 83,438,550 4,132 years 1,509,020 Total uhurguna 9,652,129,101,300,970

P 14. Utpala on Byhatsayhitti 2 (pp. 25-27). Thus in the Puliiasiddhlinta (vs. XIII): “The slivunu day is natural (unmade); for the ca‘ndru, one converts the

conjunctions of the Sun and Moon into days; the saura is the sum of earthly days; and the nciksatru the days of the revolutions of the Moon.”

“The sa’vana day is natural”-that is, itself perfect. As many as there are revolutions of the Sun in a Catwyugu, so many are there years by the saura measure. What we call the suura measure is called sa’vana by PuliS3cZrya. There the years of a Caturyugu are 4,320,000. These multiplied by 12 are the sa‘vunu months, 51,840,000; so many sa;kr&tis of the Sun are there in a Caturyuga. These multiplied by 30 are the days of the sa‘vana measure, 1,555,200,000; so many degrees does the Sun pass through in a Caturyuga. There, the time it takes the Sun to pass through one degree is a perfect day of the sa’vana measure.

“For the cundra, one converts the conjunctions of the Sun and Moon into days”. Thus (vs. XV a-b):

“The Sun passes through a Yugu with 432 multiplied by 10,OOO revolutions.”

The revolutions of the Sun are 4,320,000. (vs. IV c-d): “The Moon with 57,753,336.” The revolutions of the Moon are 57,753,336. The difference between

the revolutions of the Sun and Moon is 53,433,336. (For one) converting these conjunctions of the Sun and Moon into days by multiplying (them) by 30, there result 1,603,000,080; so many are the days of the ca‘.ndru measure in a Caturyugu. The ca’ndra measure (day) is a tithi.

“The suura is the sum of earthly days.” The sum of earthly days is the revolutions in a Cuturyugu; its computation is by means of the aharguna. The (sum) of the sa‘vana months has been seen before to be 51,840,000; the adhimcisus in a Cuturyugu are described (vs. XI c-d):

“The adhima‘sas are 1,593,336.”

The Later PauliSasiddhanta 191

By adding these together there result 53,433,336; these are the conjunctions of the Sun and Moon. Whence is it said (vs. XIX c-d):

“Whatever is the remainder of taking the difference between their revolutions, these are the conjunctions of the two planets.”

By multiplying these by 30 there results 1,603,000,080; these are the days of the cindru measure. The iinara‘trus (omitted tithis) of a Caturyugu are described thus (vs. XI1 4):

“The omission of tithis is 25,082,280.” By subtracting these there results 1,577,917,800; this sum of earthly

days is the uhargapz of a Caturyuga, and so many are the days of the saura measure in a Caturyuga. For our scivana is Pulifiiciirya’s suuru measure; as much as is from midnight to midnight or from sunrise to sunrise is (his) day of the saura measure.

“The ncikzatra the days of the revolutions of the Moon.” In this the revolutions of the Moon are 57,753,336; these multiplied

by 30 and converted into days are 1,732,600,080. So many are the days of the niksutru measure in a Cuturyugu. Here the day defined by progress through a nukjatru is a thirtieth part of a revolution of the Moon; thus, as much time as it takes the Moon to pass through 12”, so much time is a ncikjatru day. A ncikjatru day is defined by Bhagaviin Vasistha and others by the Moon’s progress through a naksatru, and a (sidereal) month is defined by 27 days of the ncikjatru measure. Thus the nikjatru measure of PuliSiic3rya is different from that of Vasistha and others. It is accepted that a month consists of 30 days in all measures. It is said: “He gives the name sa‘vanu to that which is the sauru measure of ParaSara and others, the name saura to the sa‘vana.” Thus does he (PuliSa) say

“The days of the scivana (measure) are 15,552 multiplied by 100,OOO; those of the ccindru measure are 1,603,000,080; those of the suura (measure) are 15,779,178 multiplied by 100; and those of the Crksa (= ncikpztru) (measure) are 1,732,600,080.

Then the revolutions of the Moon multiplied by 27 are the na‘kptra measure in the opinion of VasiStha and others:

“These are said by the ancients to be the irkja (days) in a Caturyugu: 1,559,340,072.’’

Thus Bhaga Balabhadra: “A saura day is the Sun’s progress through I”; a ccindrumusa is a tithi;

a ncikptru is called the Moon’s progress through a nakgztra; and a

(vss. IX-X):

192 David Pingree

s6vana the time from one rising to the next of a planet or nakzratra. By the n6kJatra measure, a (sidereal) month is 27 days; in the other measures, a month is said to consist of 30 days. The years are taken as consisting of 12 months in all the measures.”

There is another nlikptra measure that is the basis of the siivana measure; these are the (“diurnal”) revolutions of the nakjatras, and from their immobility (with respect to the zodiac is known the sa‘vana measure). For one subtracting the revolutions of a planet from the (“diurnal”) revolutions of the naksatras there results the siivana measure for each planet. Whenever the planet and the nak$atra are together, because of the immobility of the naksatra in which the planet is, the planet goes ahead of the nakSatra day by day by its proper motion and rises and sets as it comes into the line of vision of one situated on the earth. Therefore, (this) niikqatra measure is the “perfector” of the savana.

Thus are the (“diurnal”) revolutions of the naksatras described in the Puliiasiddhlinta (vs. VIII):

“The (“diurnal”) revolutions of the naksatras in a Caturyuga are said to be 1,582,237,800.’’

By subtracting the revolutions of the Sun from these, there result the savana (read saura) days-i.e., 1,577,917,800.

Commentary

Utpala’s explanation is fairly clear. PauliSa has reversed the normal significances of saura and szvana, so that the following definitions of and relations between the four measures pertain (a further discussion will be found in P 16):

1) S6vana days are & of a sidereal year; therefore there are 4,320,000 x 360 = 1,555,200,000 savana days in a Caturyuga, and 4,320,000 x 12 = 51,840,000 saikrcfntis or szvana months.

2) Cdndra days or tithis are & of a synodic month. Since there are 57,753,336 sidereal revolutions of the Moon in a Caturyuga, there are 57,753,336 - 4,320,000 = 53,433,336 synodic months, and 53,433,336 x 30 = 1,603,000,080 tithis. The synodic months (53,433,336) diminished by the siivana months (51,840,000) equal the adhimcisas (1,593,336).

3 ) Saura days are the span of time between two consecutive midnights. They can be described as the difference between the tithis and the Enarb tras; the latter are defined as being 25,082,280. Therefore, there are 1,603,000,080 - 25,082,280 = 1,577,917,800 saura days in a Caturyuga.

The Later PauliSusiddhfin!a 193

4) Ndkgatra days are & of a sidereal month according to PauliSa; 57,753,336 x 30 = 1,732,600,080 ncikgatra days. Utpala quotes a verse from Vasistha (not found in any of the texts currently circulating under his name) in which the normal Indian definition of the ndk;atra day is given, namely & of a sidereal month; for 57,753,336 x 27 = 1,599,340,072 ndk;atra days.

Utpala further quotes a verse from Balabhadra (from his lost commentary on Brahmagupta’s Khandakhddyaka ?) which gives the traditional definitions of all four measures, and then defines another ndk;atra measure from PauliAa as the time it takes for one revolution of the nakgatras about the axis of the earth; I shall call these “constellation” days. The total number of “constellation” days in a Caturyuga, then, equals the saura days (1,577,917,800) plus the revolutions of the Sun (4,320,000), or 1,582,237,800.

P 15. Utpala on Brhajcitaka 8,lO.

that is called a saura day in the Puliiatantra (vs. X a-b): Now as much time as there is from one sunrise to the next is a day;

“Those of the saura (measure) are 15,779,178 multiplied by 100.” This PuliSa knows; wherefore, with the exception of the Puliiatantra,

(a year) of the saura measure in all the siddhdntas is the Sun’s progress through the zodiac.

Commentary See the commentary on P 14.

P 16. Utpala on Byhatsamhitd 2 (pp. 28-30). In this PuliSgc8rya (says) (vs. XI1 a-b): “Thepracaya (epact) (is defined) by the cdndra (days) diminished by

the sdvana (days), the apacaya by those (cdndra days) diminished by the saura days.”

By subtracting the sdvana days from the cGndra days in a Caturyuga, there result the adhimdsaka days in a Caturyuga; dividing the result by thirty, (there result) the adhimdsas in a Caturyuga. Whatever is left when one diminishes these cdndra days by the saura-i.e., sdvana-days, so many are the apacaya days in a Caturyuga (the meaning is iinardtras [omitted tithis]). When one takes the difference between a sdvana month and one of the cdndra measure, whatever i s left is called adhimdsa; and

13 CENTAUIUS. VOL. XIV

194 David Pingree

the saura month subtracted from one of the ccindra measure is the iinara'tras. Thus: by subtracting the days of the sa'vana measure in a Caturyuga- 1,555,200,000-from the ca'ndra days- 1,603,000,080-there results 47,800,080, and so many ca'ndra days in a Caturyuga are in excess over those of the sa'vanu measure. By dividing this number by 30, there is obtained 1,593,336; so many are the adhima'sa in a Cuturyuga. Or else, by dividing the number of sa'vana days and ccindra days (each) by 30, the (respective) months are found; and the difference between them is the adhima'sas.

Then, for the iinara'tras, by subtracting the days of the saura measure - 1,577,917,800-from the days of the ca'ndra measure- 1,603,000,080- there results 25,082,280; so many are the iinara'tras in a Cuturyuga. Thus are described (the elements) of a Caturyuga.

Then there are described (the elements) of a year. In this, the days in a year by the sa'vana measure are 360; in this year, the ca'ndra days are

280'080 We shall describe the genesis of this (number) from the 37 '4,320,000. beginning. By subtracting the days of the sa'vana measure-360-from

so many days is the pracaya (epact) for this there results 11

every year. Then the pracaya days of a year are calculated by the rule of three

(trairk'ika); the calculation is 4,320,000 : 47,800,080 : 1. From this the result is this.

Then the time it takes for an adhimlisu to occur is seen by the rule of

280,080 4,320,000'

280,080 4,320,000

three; the calculation is 11 : 360 : 30. Here, the first number,

when made to have the same denominator, becomes 47,800,080. Because the first number when multiplied by the years in a Caturyuga comes to have the same denominator, therefore the last number also multiplied by the years in a Cuturyuga becomes 129,600,000. Then the divisor is 47,800,080; when divided by the multiplicand- 360-the divisor becomes 132,778. Then the dividend and the divisor become 129,600,000 : 132,778; these again divided by two become 64,800,000 : 66,389. By dividing the

dividend by the divisor there results 976'. when so many sa'vana

days have passed, an adhimlisa occurs. This is seen to be 2 years 8 months

4 336 66,389'

The Later Pauliiasiddhdnta 195

16 days and the fraction; this is the time it takes. The calculation of this by the rule of three is 1,593,336 : 1,555,200,000 : 1; the answer is

Here, by dividing the numerator and denominator of the 976i,593,336* 4,336 fraction by 24, there results - Thus is described the genesis of the

66,389’ udhimdsus.

Then the upucuyus (omitted tithis) are described. In this, by subtracting the suuru days-l,577,917,800-from the ccindru days- 1,603,000,080- in a Cuturyugu, there results 25,082,280; so many are the iinurdtrus in a Cuturyugu. Thus are described (the elements) of a Cuturyugu.

Then are described (the elements) of a year. In this, by a rule which will be mentioned, the days of the suuru measure in a cindru year are

104,064

1,634,388 By subtracting these from the cdndru days in a ccindru year

3544,452,778’ 2,8 18,390 4,452,778’

-360-there results 5 - here the numerator and the denominator

of the fraction are divided by 2 to give 17409,195). So many are the ( 2,226,389 zinurdtrus in a (cdndru) year. To calculate them by the rule of three: the calculation is 1,603,000,080 : 25,082,280 : 360. Here, when the first number is divided by the multiplier- 360- there results 4,452,778; by dividing the number of zinurcitrus in a Cuturyugu by this number, there results 2,8 18,390 . It was known from the previous calculations that so many

54,452,778 are zinuriitrus in a year.

Then it is described by the rule of three in how many days an iinurdtru

occurs. The calculation is 5 29818’390 : 360 : 1. In this, the first number,

when made to have the same denominator, becomes 25,082,280. Because the first number multiplied by 4,452,778 comes to have the same denominator, the last number- 1 -multiplied by this number becomes 4,452,778; this number is the dividend. Then, by dividing the divisor -25,082,280-by the multiplier-360- there results 69,673. By dividing

- in so many days 63,379

the divident-4,452,778- by this there results 63- 69,673’

of the ccindru measure does an iinurdtru occur. In the calculation of this

4,452,778

196 David Pingree

zinara'tra, there is another kind of rule of three: 25,082,280 : 1,603,000,080 : 1. 22,8 16,440 25,082,280 The answer to this is 63( ); here, when the numerator and

63,379 denominator of the fraction are divided by 360, there results - 69,673' It

is known from the previous calculations that so many are the (ccindra) days for an iinardtru.

Commentary These rules apply the relationships mentioned in, e.g., P 14, to the

problems of finding the epact (pracaya) and the zinarcitras in a ccindra year (apacayu). The computations are fairly repetitious of what has gone

is, sexagesimally, 11;3,53,24 tithis. This before. The epact 11

means that there are 12;22,7,46,48 synodic months in a year of 6,5; 15,3 1,30 days. Therefore, a synodic months equals 29;31,50,6,53,. . . days and a tithi equals 0;59,3,40,13,. . . days.

Furthermore, if we divide 30 by the epact of 11;3,53,24 tithis, we find that an adhimcisa occurs once every 2;42,40,39,11,. . . sa'vana years; 2;42,40,39,11 x 6,O = 16,16;3,55,6 sa'vana days. Utpala gives the number

280,080 4,320,000

4 336 66,389

9 7 C ; sexagesimally this is 16,16;3,55,7,. . . scivana days. Cf. P 18.

Finally, there are 29;31,50,6,53 X 12 = 5,54;22,1,22,36,. . . days in 1,634,388

twelve synodic months. Utpala gives the number 3544,452,778 - , sexa- . .

gesimally this is 5,54;22,1,22,35,51,. . . The iinar6tras in a ccindra year, then, are 6,O - 5,54;22,1,22,36 = 5;37,58,37,24 tithis. Utpala gives the

* sexagesimally this is 5;37,58,37,24,. . . tithis. 1,409,195 2,226,389' number 5

P 17. Al-Birfini, India, ed. pp. 493-494, trans. vol. 2, p. 187. According to PuMa the scivana days in a Caturyuga are 1,577,917,800,

the sazira months 51,840,000, the adhimcisas 1,593,336, the tithis 1 ,603,oOo,010 (read 1,603,OO0,080), and the iinarcitras 25,082,280. The number of the Yugas, according to him, is four; and for each one of these numbers a fourth is an integer. So the beginnings of the Yugas,

The Later Pauliiasiddhiinta 197

like the beginning of the Caturyugu, do not depart from the beginning of Caitra and the vernal equinox, but they disagree concerning the week-da y .

Commentary In each Yugu of 1,080,000 years there are 394,479,450 days; this equals

(56,354,207 x 7) + 1 days. Therefore, the Yugus begin on different week-days.

P 18. Al-Biriini, India, ed. p. 362, trans. vol. 2, pp. 24-25. As for what PuliSa (says), in a Caturyugu there are 51,840,000 suura

months, 53,433,336 lunar months, and 1,593,336 adhima'sas, and there are 1,555,200,000 days of sawa months, 1,603,000,080 days of lunar months, and 47,800,080 days of adhimisas. If we wish to diminish these numbers, the common factor of these months is 24; so there comes out 2,160,OOO suuru months, 2,226,389 lunar months, and 66,389 adhima'sus. As for the days, all of them have a common factor of 720; so there comes out 2,160,000 suuru days, 2,226,389 tithis, and 66,389 days of the udhi- m&sus. If we take as an example about them what went before, there comes out for the completion of an adhirna'su 976 suuru days and 1,006

tithis; and each one of these two is followed by the fraction - 66,389'

2 1,465 the scvuna days are 990- These are the bases in the udhim6sus,

66,389' worked out for what follows.

4,336 And

Commentary We have already seen from P 16 that there is one udhimZsu in every

- - 4,336 2,226,389 x 30 66,389 66,389 976- = 16,16;3,55,7,. . . sa'vana days. Clearly,

- 1,577,917,800 2 1,465 66,389

= 990- - 4,336 66,389 1,593,336

1,006- = 16,46;3,55,7,. . . tithis. And

16,30; 19,23,57,28,. . . days. Since the days of any measure are simply the months of that measure

multiplied by 30, it is odd that aI-BirGni bothered to divide them all by 720 (= 24 x 30) when he had already divided the months by 24.

198 David Pingree

P 19. Al-Biriini, India, ed. p. 498, trans. vol. 2, p. 192. 50,663

This (time of 1 CnarZtra) according to Brahmagupta is 62- 55,739

tithis; according to 50,663

saura days or 62 (read 63) - 182 55,739 days or 62- 55,739

274 for the sa‘vana and tithis and - 63,379

€’uliSa the fractions are - 69,673 the saura days.

69,673 for

Commentary The relevant parameters in the Bra’hmasphufasiddhita are:

1,577,91 6,450,OOO sZvana days equal 1,555,2oO,OOO,OoO saura days equal 1,602,999,oOO,OOO tithis equal 25,082,550,000 iinariitrus. One iinaratra,

then, equals 62- (1,2;54,32,9,27,. . .) siivana days; 62- 50,663 182 55,739 55,739

50,663 55,739 (1,2;0,11,45,17,. . .) saura days; and 63- (.1,3;54,32,9,27,. . .) tithis.

The relevant parameters in the Pauliiasiddhijnta are: 1,577,917,800 satlra” days equal l,555,2OO,OOO “scTvanay’ days equal 1,603,OO0,080

63,379 tithis equal 25,082,280 Cnaretras. One iinardtra, then, equals 67- -69,673

c c

(1,2;0,14,9,27,. . .) suuru 274

(1,2;54,34,47,21,. . .) sZvana days; 62- 69,673 63,379 69,673 days; and 63- (1,3;54,34,47,21,. . .) tithis.

P 20. Al-Birfini, India, ed. p. 361, trans. vol. 2, p. 23. 1 only mention this (i.e., al-Faz3ri Z 7 = Yacqiib ibn Tiiriq T 6) be-

cause PuliSa explains about the second of two months having the same name that it is the additional one.

Commentary

rather than before the month whose name it receives. In other words, Pauliia’s usage is to add the intercalary month after

The Later PauliJasiddhdnta 199

P 21. Utpala on Byhatsallrhitci 2 (pp. 37-39). Or else it is described by what follows the PuliJasiddhdnta; there is a

rule for knowing the similarity (vs. XIV): “If there exist four measures for the years of a Yuga, what is it for

one? Whatever is obtained, those days are to be known as being of the sdvana measure and so on.”

If, for each measure, the days spoken of for the years of a Yuga are taken, then how many will they be for one year? The calculation in the rule of three is 4,320,000 : 1,555,200,000 : 1. Of this the answer is 360. Thus it is known that in a year of the sdvana measure there are so many days of the sdvana measure.

The calculation is 4,320,000 : 1,603,000,080 : I . The answer to this is

* in a siivana year there are so many cdndra days. 280,080 4,320,000

The calculation is 4,320,000 : 1,577,917,800 : 1. The answer is 1,117,8OO

in a sdvana year there are so many days of the saura 3654,320,000; measure.

The calculation is 4,320,000 : 1,732,600,080 : 1. The answer is 280,080

* in a szvana year there are so many days of the ncikjatra 4014,~20,000’ measure.

The calculation is 4,320,000 : 1,559,340,072 : 1. The answer is 4,140,072

in a scvana year there are so many (“diurnal” revolutions 3604,320,000; of the) nakjatras.

Then, by dividing the days of the cdndra measure in a Caturyuga - l,603,OOO,O80- by 30, one obtains 53,433,336; so many are the months of the ccindra measure in a Caturyuga. By dividing these by 12, one obtains 4,452,778; so many are the years of the cdndra measure in a Caturyuga. Here the calculation is 4,452,778 : 1,603,000,080 : 1. The answer is 360; so many are the days of the cdndra measure in a year of the cdndra measure.

1,180,478 4,452,778 ’ Calculation: 4,452,778 : 1,555,200,000 : I. The answer is 349

so many are the days of the sdvana measure in a cdndra year.

200 David Pingree

1,634,388 Calculation: 4,452,778 : 1,577,917,800 : 1. The answer is 3544,452,778;

so many are the saura days in a ccindra year. 469,438

4,452,778, Calculation: 4,452,778 : 1,732,600,080 : 1. The answer is 389

so many are the days of the nfikjatra measure in a ccindra year. 867,772

4,452,778; Calculation: 4,452,778 : 1,559,340,072 : 1 . The answer is 350

so many are the (“diurnal” revolutions of the) nakjatras in a ccindra year. Then, by dividing the days of the saura measure in a Caturyuga

-1,577,917,800-by 30, one obtains 52,597,260; so many are the months of the saura measure in a Caturyuga. By dividing these by 12, one obtains 4,383,105; so many are the (saura) years in a Caturyuga.

Calculation: 4,383,105 : 1,577,917,800 : 1 . The answer is 360; so many are the days of the saura measure in a saura year.

3,580,830 Calculation: 4,383,105 : 1,555,200,000 : 1. The answer is ’54

4,383,105’

3,166,755 4,383,105’

1,273,605 4,383,105’

so many are the days of the sfivana measure in a saura year.

Calculation: 4,383,105 : 1,603,000,080 : 1 . The answer is 365

so many are the days of the cfindra measure in a saura year.

Calculation: 4,383,105 : 1,732,600,080 : 1. The answer is 395

so many are the days of the niikjatra measure in a saura year. 3,337,797 4,383,105’ Calculation: 4,383,105 : 1,559,340,072 : 1. The answer is 355

so many are the (“diurnal” revolutions of the) nakjatras in a saura year. Then the days of the mikjatra measure in a Caturyuga are 1,732,600,080.

By dividing these by 30, one obtains 57,753,336; so many are the months of the ncikjatra measure in a Caturyuga. By dividing these by 12, one obtains 4,812,778; so many are the years of the ncikjatra measure in a Caturyuga.

Calculation: 4,812,778 : 1,732,600,080 : 1. The answer is 360; so many are the days of the ncikjatra measure in a year of the nfikjatra measure.

345,006 4,8 12,778’ Calculation: 4,812,778 : 1,603,000,080 : 1. The answer is 333

The Later PauliSasiddhdnta 20 1

so many are the days of the c6ndra measure in a year of the ncikptra measure.

672,706 4,8 12,778’

Calculation: 4,812,778 : 1,555,200,000 : 1 . The answer is 323

so many are the days of the szvana measure in a year of the ncikjatra measure.

4,139,394 4,8 12,778’

Calculation: 4,812,778 : 1,577,917,800 : 1. The answer is 327

so many are the days of the saura measure in a year of the nikjatra measure.

Calculation: 4,812,778 : 1,559,340,072 : 1. The answer is 324; so many are the (“diurnal” revolutions of the) nakjatras in a year of the nikjatra measure.

Then, by dividing the (“diurnal” revolutions of the) nakjatras in a Caturyuga- 1,559,340,072-by 27, one obtains 57,753,336; as has been shown in the previously calculated months of the nfikjatra measure, so many are the nikjatra months in a Caturyuga.

Commentary

be useful, however, to tabulate the results. The computations in this fragment are quite straightforward. I t may

Table 1

“Years“ in a Catiiryrrga “Sdvuna” days Ccindra days “Saura” days “ Ndk ptra”

days Naksatra revolution days

“Sdvana” year

20,0,0,0

6 8

6, I 1 ; 3,53,24

6.5 ; 15,3 1,30

6,41 ; 3,53,24

6,0;57,30,3,3r

Cdndra year ~ ~~

20,36,52,58

5,49; 15,54,23,51

6.0

5,54; 22,1,22,35

6,29 ; 6,19,0,3 1

5,50;1 1,41,34,44

“Sauro” year

20,17,3 I .45

5,54; 49,1,4,16

6.5 ;43,20,58,6

6,O

6,35 ; 17,26,3,25

5,55 ; 45,41,30,2

“Ndk$atra” ycur

22.16.52.58

5,23;8,23,11,18

5.33 ; 4, I8,4,0

5,27;51,36,18,10

6.0

5,24

202 David Pingree

P 22. Al-Biriini, India, ed. p. 496, trans. vol. 2, p. 190. Similarly it (the solar year) according to PuliSa is 365;15,31,30,0; so

the gupakara (numerator) is 1,007 and the bh8gah8ra (denominator) is 800.

Commentary

equals 1:15,31,30; this is the annual increment of 1,007 The fraction -

800 the lord of the year in the Pauliiasiddhlinta.

P 23. Al-Biruni, India, ed. pp. 496497, trans. vol. 2, p. 190. This was dictated by Awlatta (a.w.1.t.) ibn SahBwi, based on the opinion

of PuliSa. This is that one should subtract from the Sakakafa 918, multiply the remainder by 1,007, add to the product 79, divide the result by 800, and diminish what results by integer sevens. The remainder is the base, and the additions to it for each sign by the computation which was previously mentioned are put in the table:

Signs

Aries Taurus Gemini Cancer Leo

Libra Scorpio Sagittarius Capricorn Aquarius Pisces

Virgo

Additions to the base Days Ghatis

1 35 4 33 0 59 4 37 1 6 4 6 6 31 1 23 2 41 (read 51) 4 I0 5 37 0 28 (read 18)

Coniinentary

AwIatta (?) ibn Sahawi is otherwise unknown; but it is clear that he 79

800 flourished in $aka 918 = A.D. 996. The ksepa - shoufd mean that

The Later Pauliiasiddhcinta 203

Awlatta considered the beginning of Saka 918 to be an integer number of days + 0;5,55,30 days from the epoch of an era. In fact, he has computed this on the basis of the epoch of the KaZiyuga being precisely the beginning of a day. For 4,097 years had elapsed between the epoch of the Kaliyuga and the beginning of $aka 918; and 4,097 (= 1,8,17) x 1;15,31,30 = 1,25,57;5,55,30. Moreover, 1,25,57 + 7 = 12,16 + 5; we have seen before (P 13) that the epoch of the Kaliyuga was Friday; therefore, the first day of $aka 918 was a Tuesday, not a Sunday. Part of the rule seems to be missing.

The table to which al-BirGni refers (ed. p. 496, trans. vol. 2, p. 189) is as follows:

Aries 3 days 19 ghatikcs Libra 1 day 14ghaCikii.s Taurus 6 17 Scorpio 3 6;30 Gemini 2 43 Sagittarius 4 35;30 Cancer 6 21 Capricorn 5 54 Leo 2 49 Aquarius 0 30 Virgo 5 49 Pisces 2 11:20

If we compare the differences in the two tables (adding 28)’ we find: Awlatta Other table Awlatta Other table

29;35 3 1 ; 19( -x) 30;25 30;25 30;58 30;58 29;52 29 ; 5 2,30 3 1 ;26 3 1 ;26 29;28 29;28 3 1 ;38 31;38 29;19 29;19,30 3 1 ;29 3 1 ;28 29;27 29;26 31;O 31;O 29;41 29;41,20

Awlatta’s year i s a scant 6,4;18 days; the other table’s is 6,6;2,20. If one reads 2;35 instead of 1;35 as the fist entry in Awlatta’s table, the result is a not unrespectable, though crude, 6,5;18. The other table begins with an epoch value of about 0;45.

P 24. Prthiidakasviimin on Briihmasphutasiddhiinta 1,3638. This verse is also quoted by Pfiiidakasviimin and by Utpala on Khandakhiidyaka 1,14; by Amariija on Khagdakhldyaka 1,13; and by Makkibhatta on SiddhZntaJekhara 2,105.

The prime meridian is thus described in the Pauliia (vs. XV): “No longitudinal difference is to be computed for Ujjayini, Rohitaka,

204 David Pingree

Kuru(kSetra), Yamunii, the HimBlayas, and Meru because they are in the middle of the prime meridian.”

Commentary Kuruksetra, of course, is the locality of Sthiinisvara, elsewhere men-

tioned by PauliSa. These places do not in fact lie upon one and the same parallel of longitude; see P 25.

P 25. Al-Biriini, India, ed. p. 269, trans. vol. 1, p. 316. I have not found concerning this (the prime meridian) any difference

except what is in the book of &yabhala of Kusumapura; these are his words:

“People say that KurukSetra-I mean the plain of Sthgniivara-is on the line which runs from Laiikii to Meru through the city of Ujjayini; they say this on the authority of Puliia. But he is better than that this should be concealed from him. The times of eclipses disprove this, and Pythijdakasvamin says that the difference between the two longitudes is 120 yojanas.”

This is what Aryabhata said.

Commentary This is certainly not what kyabhafa of Kusumapura (= Pgtaliputra)

said, since he wrote the Aryabhatiya in 499, long before either the later PuuliSasiddhcnta or Prthtidakasviimin, whose commentary on Brahma- gupta’s Khapdakh8dyaka was written in 864. The passage is rather taken from a commentary on the KhapdakhLfdyaka later than Prthtidakasva- min’s and (since Utpala does not agree with what al-BirGni reports) the reasonable candidate might seem to be Balabhadra. But Balabhadra is frequently quoted by PythCidakasvZmin in his commentary on Brahma- gupta’s Br8hmasphu[asiddh8nta7 and therefore could not have quoted Pythiidaka. The next commentator on the Khagciakhcidyuku known to us is Varuqa, who wrote in KBSrnira in 1040-after al-BirGni wrote his Tndia. However, though Lalla probably wrote his commentary (now lost) in the eight century, we know that many other commentaries once existed -e.g., that of Durga which is cited by Arnaraja in c. 1200.

The unknown commentator on the Khavdakhcdyaka that a1-Biriini quotes from bases his remarks on 1,1415 in the recension of Pythiidaka.

The Later Paulihsiddh&ta 205

The latter cites verse XV of PuliSa on 1,14 (see P 24; he is in this followed by Utpala), and on 1,15 states: “In Kuruksetra an eclipse was observed to occur one and a haifghutikrSs after the time predicted from the kuruau (the Khundukhctdyuku); from this yojanas are made thus. The circumference of the earth is 4,800; if these are multiplied by 9Opufus and divided by 60 there results 7,200. This comes from multiplying by one and a half ghatik6.s; if one divides it by 60, there are 120 yojanus at Kuruksetra.”

P 26. Al-Biriini, India, ed. p. 398, trans. vol. 2, p. 67.

14 circumference 5,026-.

25

So its (the earth’s) diameter, according to PuliSa, is 1,600 yojunas, its

Commentary 125,664 40,000 The same diameter is given in P 27; cf. P 60. The value of 3t is - =

1 1 1 3- This is also the value of 7c in P 61 and P 62. See Yacqiib ibn

1,250’ T2riq T 1. PauliSa has taken it over from the &uzbhutiyu.

P 27. Al-Biriini, India, ed. p. 266, trans. vol. 1 , p. 312. If the circumference (of the earth) is 4,800 (yojunas), then its diameter

is nearly 1,527; but according to PuliSa it is 1,600, and according to Brahmagupta 1,581 in yojunus-I mean (those) of which each one is eight miles.

Commentary PauliSa does make the diameter of the earth 1,600, but says its circum-

ference is 5,026--. In the 3rrShmas~h~~usiddh~n~a, Brahmagupta makes

the diameter 1,581 (21,32), but says that the circumference is 5,000 (1,36). In the Khua&khctdyaka, however, he says that the circumference of the earth is 4,800 (1,15). See YaCiib ibn Tgriq T 3.

14 25

P 28. AI-Biriini, India, ed. p. 267, trans. vol. 1 , pp. 313-314. PuliSa multiplies these (day-minutes) by the circumference of the earth

and divides the product by sixty, which are the minutes (sixtieth parts)

206 David Pingree

of the diurnal revolution. There result the yojanas between the two localities. This is correct, but the result comes out on the great circle on which LaiikB lies.

Commentary

Pythiidakasv%min quoted in the commentary on P 25. AI-BirGni is right. For an example of the use of this method, see

P 29. Al-Birtini, India, ed. pp. 377-378, trans. vol. 2, pp. 4 1 4 2 . Similar to this is what PuliSa prescribes for it. Put down the partial

months in two places, and multiply one of them by 1 , 1 11 and divide the product by 67,500; subtract what results from the second. Then divide the remainder by 32; there result the adhimcsas, and the remainder is what has passed of the fractional (adhimcisa). If you multiply it by 30 and divide the product by 32, there result its days and what follows it (of fractions).

The reason for this is that if you divide the saura months in a Caturyuga 35,552 by the adhimiisas in it according to him, there results 32- 66,389’ If

divide the months by this, there result the complete adhimcisas in what has passed of the Caturyuga or KaZpa. But he desired to divide by integers only, so he had to subtract something from the dividend just as was done previously in a similar case. The similar dividend in our example was 2,160,000, and the fraction of its one is 35,552, and after these 32;

so first there results 67,500

PuliSa has worked this operation of his with the saura days which have passed from the epoch rather than with months. For he says:

“One puts down these days in two places, and multiplies one of them by 271, and divides the product by 4,050,000, and subtracts what results from the second. Then one divides the remainder by 974; there result the adhimcisas and what follows them of days and their fractions.”

Then he says: “This is because, if you divide the days in a Caturyuga by the adhimcisas,

there results 976, which are days, and there remains 104,064; the common

207 The Larer Paulifasiddh~nta

factor between this and the dividend is 384. If we divide the two of them

4,050,000 by that, there results 271 (read

2,050,000 I suspect in this the copy or the translator as PuliSa is greater that than

he should be inattentive in something like this. This is because the days divided by the udhimrisas are necessarily sauru, and what results from integers are integers and a remainder as was said. The numerator and the denominator have as a common factor 24; the numerator comes out as 4,336 and the denominator as 66,389. If we imitate with respect to the months what preceded, and make similar the measure of the adhimrisas, there results 47,800,000 (read 47,800,080); the common factor of this and its numerator is 16. By this there results that what is multiplied by it is 271, and that which is divided by it is 2,800,000 (read 2,987,505). As for the number which he put down for division, if we multiply it

by the common denominator which was mentioned, which is 384, there results 1,555,2oO,OOO, which are the saura days in a Caturyuga. But it is impossible that, in this part of the operation, this be what is to be divided by that.

Commenatry The ratio of the saura months to the adhimcisas in a Caturyuga is

From this it follows that the 35,552 5 1,840,OOO 24 2,160,OoO 1,593,336 24 66,389 66,389'

= 32- =-

( 67,500 equals ma - 35,552

lapsed saura months (ma) divided by 32- 66,389 1 - and that this number represents the lapsed adhimrisas; it is more

32' difficult to see what advantage is gained by this transformation. In the second rule, where PauliSa does indeed use suura days, his

al- 1 , 5 5 5 , 200,000 104,064 equals 976 procedure is quite erroneous. 1,593,336 1,593,336'

Birfini reduces this latter fraction to - 4y336 The number 47,800,080 is 66,389' that of the number of tithis in the adhimzsas of a Caturyuga; this number divided by 16 is 2,987,505, as 4,336 divided by 16 is 271. Instead of using the tithis in the udhiintisas Pauliia has used the saura days in a Caturyuga;

208 David Pingree

27 1 4,050,000’

which reduces to 104,064 1 ,555,20O,OoO7 thereby he gets the fraction

Compare P 18.

P 30. A1-Biriini, India, ed. pp. 390-391, trans. vol. 2, pp. 58-59. This is what Puliia says also according to another method. That is

that, when the complete revolutions have come out for him, he divides the remainder by 131,493,150; there come out mean signs, He divides the remainder by 4,383,105; there come out degrees. He divides four times the remainder by 292,207; there come out minutes. After this he multiplies the remainder by 60 and divides the product by this last number; there come out seconds. And so on for as long as one wishes. This is the mean (longitude) that is desired.

This is because it is necessary, with respect to the remainder from the revolutions, to multiply it by 12 and divide the result by the days of a Caturyuga, because he uses this; but instead of this he divides by the days in a Caturyuga divided by 12, which is the first of the three numbers. It is necessary, with respect to the remainder of the signs, to multiply it by 30 and divide the product by what is divided by it; but instead of this he divides by the first number divided by 30, which is the second number. In this manner he wishes to divide the remainder of the degrees by the second number divided by 60; but, when one divides it by this, there comes out 73,051 and a remainder of three fourths. So he multiplies the whole by 4 so that the fractions are removed. Because of this he also uses four times the remainder; but when the numbers did not work for him as was indicated at first, he returned to multiplication by 60.

Commentary

and minutes with their sexagesimal fractions. This is a fairly obvious method of reducing fractions to signs, degrees,

P 31. Al-Birijni, India, ed. p. 230, trans. vol. 1, p. 275. The beginning from which Balabhadra starts is what is in the Pauli-

Sasiddhcfnra, when it divides the Sine for a quarter of a circle into twenty- f n r l r Gnp/lninc T h m n h,= EQV,C* ‘‘LEfi- :-*-----&-- - ’ r ‘ - - _

The Later PauliSariddhBnfa 209

a circle, which is 225 minutes; if we derive its Sine, it is (also) 225 minutes. We know from this that the Sines are equal to their arcs in whatever is less than this kardaju.” Since the total Sine, according to PuliSa and h a b h a t a , is according to the ratio of the radius to a circle of 360”, it instilled a delusion in Balabhadra from this numerical equality.

Commentary The ,&yzbhatiyu - does indeed derive the value of R from that of R; -

27 967 4,500,000 1,309 1,309 ’ = 3-. But in P 59, R is defined as 3,437- =

21,600 for

3,438 x 2 191

a value also attested in P 26, P 32, P 61, and P 62. 177

R, then, equals 3- 1,250’

Al-BirGni’s statement concerning PauliSa is not strictly accurate then. Balabhadra’s statement is true in the sense that, if R = 3,438, then

the Sine of 225‘ is 225’, and the Sines of arcs of less magnitude than 225’ are equal to their magnitudes in minutes.

Al-Birfini probably did not have any secure grounds for ascertaining the relative chronology of Balabhadra and PauliSa.

P 32. Al-Biriini, On Transits 28:1429:4. And since the ratio of the circumference to the diameter according to

PuliSa is like the ratio of 3,927 to 1,250, the radius of the circumference of the apogee (-epicycle) by the computation of this ratio will be, according to him: for the Sun 2;13,41; for the Moon 4;56,1. And, on rounding off and truncation, there results (a quantity) like what we said. But he points out in the case of these circumferences that their ratios to the maximum equation are as the ratio of a circumference which is 360 to the total Sine. So if we derive the circumference from the equations of the two luminaries by this ratio, there results for the Sun 14;(1,5)3” and for the Moon 30;59,41”; and these are the circumferences of their apogee (-epicycles).

Commentary

familiar, e.g., from P 31. 177

for 7c is the value 3- 3,927

1,250 The ratio -

1.250 The proportions using maximum equations yield, as circumferences for

14 CENTAURUS. VOL. XIV

210 David Pingree

the apogee-epicycles of the Sun and Moon, respectively 13;59,56,. . . O - 14" and 30;59,54,. . . O * 31". We can be certain, then, that Paulisa's parameters were the standard ones of the &&ara'trika system:

Em of the center Mandaparidhi Sun 2;14" 14" Moon 4;56" 31"

2 14 Al-Birfini's further operation is to form the two proportions: A = X

4,56 57,18 and - = - ; these yield, respectively, 14;1,53,. . . " (not 57,18 - 6 0 X 6,O

14;3") and 30;59,41,. . . ". Al-Biriini, in P 32, P 34, P 35, and P 37 consistently assumes that

Em, = r, the radius of the epicycle, whereas PauliSa more correctly in P 36 says that Sin(Emsx) = r.

P 33. AmarZtja on KhandakhrZdyaka 2,12. Thus in the Pauliiatantra (vss. XVIc-XVIIb): "In computing the manda equation of Mars and the other planets, the

circumferences of the circles through the perigees are (respectively): 70°, 28", 32", 14", and 60°."

Commentary Again these are straightforward cfrdhurcitrika parameters:

Em, of the center Mandaparidhi Saturn 9;36" 60" Jupiter 5;6 32 Mars 1 I;10 70 Venus 2;14 14 Mercury 4;28 28

Cf. P 34.

P 34. Al-Biriini, On Transits 31:15-32:2. And as for what is in the Indian zljes which we have studied, this is

to the utmost of confusion to the extent that its measure is not permitted.

The Later PauliJasiddhcnta 21 1

So suspicion turns toward the copies which have come to us and toward the commentator who dictated to us. And this is that PuliSa explained concerning the measures of these equations that they are: for Saturn 568 minutes, for Jupiter 284 minutes, for Mars 676 minutes, for Venus 134 minutes, and for Mercury 268 minutes. If one multiplies them by 360 and divides the products by 3,438 minutes (the total Sine), there result the circumferences of the apogee: for Saturn (60"), for Jupiter 30", for Mars 70", for Venus 14", and for Mercury 28".

Commentary

correct values see P 33; see further P 35. Al-BiriinT's copy or commentator certainly was confused. For the

- if One can not go from Saturn's Emax, 568'' to 60". But - - Em, 191

we posit - - - - then 191 x 60 = 11,360. If al-Biriini's commen-

tator made a mistake and wrote 11,460, then - - - 568'. In P 35 we

find 9;38" = 578'. From Jupiter's E m u , 284', one gets 29;43"; the mandaparidhi 30" yields

an Em of 4;46,30" as in P 35. The commentator should, of course, have used 32" rather than 30".

From Mars' E-, 676', one gets 70;46"; the mandaparidhi 70" yields an Emax of 11;8,30" as one should read in P 35.

From Venus' Emax, 134', one gets 14;1,53" (see P 32); the mandaparidhi 14" yields an Emax of 2;13,12" as one should read in P 35 (a scribal error due to a misreading of abjEd numbers).

From Mercury's Emax, 268', one gets 28;3"; the mandaparidhi 28" yields an Emax of 4;27,24" as in P 35.

The reason for these erroneous parameters has been given in the commentary to P 32.

360 20 3,438 - 191 '

60 20' 1 1,460

20

P 35. Al-Biruni, On Transits 32:12-33:3. And what result from the circumferences which PuliSa put down are:

for Saturn 9;38", for Jupiter 4;46,30"; for Mars 11;30" (read 11;8,30"), for Venus 2;38,42" (read 2;13,12"), and for Mercury 4;27,24". As for

14.

212 David Pingree

those who make ratios between the equation of the Sun and those equations, such as the commentator on the zy Khapdakhfidyaka, which is known among us as al-Arkand, he claims that the equation of Saturn is four times the sum of the equation of the Sun and (a half of its) seventh; that the equation of Jupiter is twice the equation of the Sun and one time its seventh; that the equation of Mars is five times the equation of the Sun; that the equation of Venus is like its (the Sun’s) equation; and that the equation of Mercury is double its (the Sun’s) equation. And what results by these maximum equations is close to what results from the circumferences which PuliSa put down.

Commentary The computation of the maximum equations of the center here ascribed

to PauliSa has been discussed in the commentary on P 34. The passage in the Khaadakhcidyaka to which al-Birilni refers is 2,6-7, where it is erroneously stated that the manda equations of the planets are certain multiples of that of the Sun. In fact, the circumferences of the epicycles should have been mentioned, to yield, given that the mandaparidhi of the Sun is 14”:

Mandaparidhis Saturn 4(14 + #) = 60” Jupiter 2(14 + 9) = 32” Mars 5 x 14 = 70” Venus 1 x 14 = 14” Mercury 2 x 14 = 28”

Brahmagupta’s mistaken formulation of this relationship, as adapted by al-BirGni, gives the following values of the maximum equations to be compared with those erroneously ascribed to PauliSa (Emax of the Sun is 134‘):

“Brahmagupta’s’’ Em, “PauliSa’s” E, Saturn 9;34,17” 9;38” (or 9;28”?) Jupiter 5;6,17 4;46,30 Mars 11;lO 1 1 ;8,30 Venus 2; 14 2;13,12 Mercury 4;28 4;27,24

The Later PauZiSasiddhcinta 213

P 36. Amargja on KhaadakhZdyaka 2,13-14. Here, in the Pauliia and other (siddhzntas), the degrees of the circum-

ferences of the circles through the Sighranica and the s'ighrocca for Mars and the other (planets) are (vs. XVIII):

"233", 132", 72", 260°, and 39;30". These, multiplied by R and divided by 360°, are the Sines of the maximum equations of Mars and the other (planets)."

233", 132", 72", 260°, 39;30"; these, individually multiplied by R and divided by 360°, result in the (following) Sines of the maximum equations: 9-73 : 55 : 30 : 108;20: and 16;22 (read 16;27). The arcs of these are the degrees of the maximum equations: 40;30", 21;30", 11;30", 46;15", and 6;20".

Commentary The values of the maximum equations here given by h a r g j a as

Pauliia's are again precisely those of the drdhardtrika system. But, as the Sines show, they are computed with a Sine-table in which R = 150'

177 1,250

rather than PauliSa's normal R = 3,438' or 3,437-'. The values of

the maximum equations are in some cases roundings from the arcs actually derived from the Sines; for several of PauliSa's fighraparidhi's differ from the standard Zrdhardtrika values, but not by enough to cause any noticeable divergence:

PauliSa's drdhardtrika Sighraparidhis Sighraparidhis

Saturn 39;30° 40" Jupiter 72 72 Mars 233 234 Venus 260 260 Mercury 132 132

It is difficult to conjure up any motive for this variation.

P 37. Al-Biriini, On Transits 54: 19-55 : 1 I. And as for PuliSa, he put down the maximum equations as the circum-

ferences of the epicycles carrying (the planets) by multiplying the equations by 360 and dividing the product by the total Sine, which is, ac-

214 David Pingree

cording to him, 57 parts and 18 minutes. As for the equation of Saturn, it is 6;22", and the circumference of its epicycle is 40"; as for the equation of Jupiter, it is 11;32" and the circumference of its epicycle is 72"; the equation of Mars is 40;32" and the circumference of its epicycle is 255"; the equation of Venus is 45;15" (read 46;15") and the circumference of its epicycle is 290"; and the equation of Mercury is 21;36" and the circumference of its epicycle is 135".

Commentary Again al-Biriini uses the mistaken identy: Emx = r. By applying the

usual proportion to the Emax's given by him, one can compute the following Sighraparidhis (in col. 4):

Emax PauliSa iighraparidhi PauliSa Saturn 6;22" 6;20" 40" 39;30" Jupiter 11;32 11;30 72;27,38 72 Mars 40;32 40;30 254;39,34 234 Venus 46;15 46;15 290;34,33 260 Mercury 21;36 21;30 135;42,24 132

I do not know the source from which al-Birtini derived his maximum equations. I suspect the bungling hand of his pandit must share the responsibility with al-B-Mini's failure to apply a check to its ramblings.

P 38. Al-Biriini, India, ed. p. 286, trans. vol. 1, pp. 339-340. But the commentator of the Siddhrfnta of PuliSa explains about this

latter opinion (that the muhiirtas are like the seasonal hours) and criticizes anyone who makes the general statement about the length of a muhiirta that it is two ghatis, saying that the number of ghatis in a day differs during the year, but the number of muhllrtas does not. But he contra- dicts himself in his explanation of the length of a muhiirta, and makes it 720 prcinas, wherein a breath is composed of aprfna, which is drawing in the air, and of prdna, which is sending it out; these are also called nihs'v6.w and avaivcisa. However, if one of the two is mentioned, the other is included, as nights (are included) in the mentioning of days, if you mention them. Therefore it (a muhiirta) is 360 inhalations and an equal number of exhalations. Therefore, in the measure of a ghati, it is limited to one of the two kinds; so he makes it in general 360 breaths.

The Later Puuliiasiddhdnta 215

Commentary This and the ensuing confusion, into which Sachau gratuitously injects

the name of Pauliia, is entirely due, as al-Biriini himself states, to the miraculous pandit. A muhiirta is &j of a sHvana day while a ghati is & of 24 hours; therefore it is perfectly true that 2 ghatis do not equal a muhfirta and that 720 prznas also do not equal a muhzirta, but that they do equal 2 ghatis. See P 3.

P 39. Al-Biriini, Al-Qrfniin al-Masciidi, vol. 2, p. 972. PauliSa the Greek says: “If you know the time of the determination of the position, then make

the declination of the Sun and its direction and the declination of the minute (occupied by) the Moon equal to it. Make its latitude corrected by its bhukti. If the latitude of the Moon and the declination of its degree are in the same direction, add them together; if they are in two different directions, take the difference between them and this is the declination of the Moon-in the direction of the declination of the degree if the operation is by adding, and in the greater direction if it is by taking the difference. If you add the latitude of the Moon in order to know its declination, then subtract it from the declination of the Sun; if you subtracted the. latitude of the Moon, then add it to the declination of the Sun. Then look at what is between what results from the declination of the Sun and the Moon; if they are equal, that is the correct time.”

We say about this that most of the operations of the Indians are based on other than solid grounds and that they are sometimes furnished with marvels. All of them add the declination of the degree of a planet to its latitude or take the difference between them even though they are not on the same circle. If we know what is intended, we shall follow a more correct method in this and avoid error, an example of which is their calculation of the latitude of the Moon by multiplying arcs and sines with each other.

What Pauliia instructs in this passage for calculating the latitude of the Moon by its bhukti is that he multiplies the sine of its distance from the node by the Moon’s maximum latitude and divides the result by the total Sine; then he multiplies what comes out by the corrected bhukti of the Moon and divides it by its mean bhukti. There results the latitude of the Moon which he instructs (one to calculate).

216 David Pingree

Commentary This passage evidently comes from that section of the Pauli.iusiddhE?ntu

in which the vyutbcita and vuidhrtu were discussed. The addition of latitudes and declinations upon which al-Biriini remarks so disparagingly is a common feature of Indian textbooks on astronomy. So also is the basic formula for finding the Moon’s latitude (w is the elongation of the Moon from its node and i is the maximum latitude):

Sino x i R B =

The further correction, however, is very curious; it simply makes no sense to multiply B by the true daily progress divided by the mean daily motion of the Moon. Rather I suspect that Pauliia, like Brahmagupta (Khag&khcjdyuka 1,31), gave instructions for finding the latitude and the apparent diameter of the Moon in one verse; the ratio of the true to the mean velocity is used in finding the apparent diameter. Again al- Birfini’s pandit has displayed his ignorance.

P 40. Al-Biriini, Al-Qciniin al-Muscddi, vol. 2, p. 974. PauliSa says: “If the Moon is in Gemini or Sagittarius and its declination is less

than the declination of the Sun, then it is impossible for their declinations to be equal in one direction. In this case he takes the “shadow” of the mean Sun when the sum of the two true longitudes comes to six signs; this is weak in its (astrological) effects. If at that time the Moon is in the two above-mentioned signs and its declination is greater than the declination of the Sun, then it is impossible for their two declinations to be equal in opposite directions. In this case he takes the “shadow” of the mean Moon at that time; (this also is) weak in its (astrological) effects.”

Commentary Here also PauliSa is discussing the occurrence of vyatipa‘tas and

vuidhrtus. A vyatbdta occurs when the sum of the true longitudes of the Sun and Moon is 180” and their declinations are equal on the same side of the equator; a vaidhrtu occurs when the sum is 360” and their declina-

The Later Pauliiusiddhdizta 217

tions are equal on opposite sides of the equator. The first situation described in the fragment is a vyatipa’ta in which the two luminaries are each less than 30” from the solstice, the Moon is behind the Sun, and the Moon’s latitude is southern; the second is a vaidhrta in which the two luminaries are again each less than 30” from a solstice, but in which the Moon has not yet reached its solstice and the Moon’s latitude is northern. Admittedly equality of declinations would be unusual in such situations, but surely if the Moon is close to RBhu in the first situation or to Ketu in the second thepdta might take place. See the Paita‘maha- siddhhta of the Vijnudharmottarapura’na (V 9).

PauliSa’s substitute “shadows” are obvious, and would certainly be weaker in their deleterious effects.

P 41. Al-Birfini, Maqda fi sayr sahmi al-sa%da wa al-ghayb 92v: 5-13 (translation based on that by E. S. Kennedy).

So we say that, for any month we want, we ascertain when the daily circles of the two luminaries unite or become equal (on either side of the equator), each by one of the two kinds of its connection. One of them is by the latitude of the Moon, and the other according to its parallax in order that it agree with the apparent (position). This would be an investigation praiseworthy of anyone who asks it. This was included in the zv of PuliSa the Greek known as the PuliSasiddh&ta. His accounts indicate things about the ancients; he mentioned the divisions of time and the great wars which took place in the plain of TBnishar (SthBniSvara) and other things such as what we have mentioned, like the agreement of the Greeks and the Indians concerning what goes back to one source, and their being in faith like one family, or that the (above-) mentioned PuliSa moved from Alexandria to their land, which is the more improbable of two possibilities.

Commentary The great war in the plain of SthiiniSvara was the battle celebrated in

the Mah&h&ata; see also P 9. Al-Biriini here definitely discounts the story of PauliSa’s Alexandrian origin found in P 1, where it is due to a misreading of the Arabic transcription of SthgniSvara or its form in the local dialect.

21 8 David Pingree

P 42. Al-Biriini, India, ed. p. 51 1, trans. vol. 2, p. 206. Then PuliSa adds together the declinations of the two luminaries in

vyat@dta if they differ in their directions, and in vaidhrta if they agree; he takes the difference between the declinations of the two luminaries in the vyat@Zta if they agree in their directions, and in vaidhrta if they differ. This is the first thing to be retained, which is for the mean time. Then he reduces the minutes of the day to rnfijas, supposing that they are less than a fourth of the day. He computes from the bhuktis of the luminaries and the node their motions, and from these their longitudes by the computation of their situation from the mean time in the past and future; from these he makes the second thing to be retained. In this he knows the condition in the past and future and compares it to the mean time. If the time of the equality of the declinations for both of them is past or future, then the difference between the two retained values is the part of division (bha'gahzra: divisor); if it is past in one of them and future in the other, then the sum of the two retained values is the part of division. Then he multiplies the minutes of the day which have been put down by the first retained value and divides the product by the part of division; there result minutes of the distance from the mean time, understanding that these can be either past or future. So, by the computation of this the time of the equality of the two declinations becomes known.

Commentary The problem which Pauliia wishes to solve is the true time of the

pa'ta. Let AM, As, and ;IR be the true longitudes of the Moon, the Sun, and the node at midnight, the mean time, and d~ and 6s the declinations of the Moon and Sun respectively. Then, if the declinations are both in the same direction, form:

vaidhrta;

if in opposite directions: ..

vaidhrta

(or respectively 6s - d M if dS > dM). The result we will designate Ad.

The Later Paulifasiddhtinta 219

Next, by means of the minutes of time that have elapsed since (or will elapse before) midnight (t) converted into minutes of arc (m&u is an error for ca~aka) and the bhuktis or true daily progresses of the Sun, Moon, and node, one computes A'M, A'S, and A I R , the longitudes of the three bodies at the given time, and I'M, I f ' s , and Z'R, their longitudes at an equal time before (or after) midnight, and the declinations b ' ~ , d's, S"N, and 8's.

Comparison of the three respective declinations of the Sun and Moon will tell one whether, in each case, the moment of equality in declinations occurred before or after midnight. If they are both on one side of midnight, form :

if on opposite sides: S'M + 6's or d " ~ + 6"s;

6 ' ~ - 6"s or 6 " ~ - 6's

(or 6's - 6 " ~ or 6''s - & M if 6s > B M ) .

This result we will designate Ad'.

Then PauliSa states the time of the equality before (or after) midnight (t') thus:

Ad t' = t- AS"

P 43. Al-Biriini, India, ed. p. 513, trans. vol. 2, p. 208. The number of vyut@Htus in the nukJutras is many according to what

PuliSa says on the authority of ParaSara; their origin is accordicg to what he has said. The kinds do not increase, but the individual speci- mens become numerous.

Commentary The idea seems to be that there is a yoga whenever the sum of the

longitudes of the Sun and Moon equals x(13;2Oo). To ParaSara, another Vedic y ~ j , were attributed both astrological (cf. Utpala on ByhatsamhitH, the ByhatpHr~~arahorH~Hstra, etc.) and astronomical (cf. the MahHsidd- hHnta of the second k a b h a t a ) works. It is not yet clear to which Paulila refers.

David Pingree 220

P 44. Al-Biriini, Al-Qfiniin al-Mas%di, vol. 2, pp. 979-980. PauliSa says: “Add together the declination of the Sun and the declination of the

Moon for the obscuration (of the Sun) if their directions are different and take the difference between them if they agree. Reverse the instruc- tion in the case of the obscuration of the Moon; add them together if their directions agree and take the difference between them if they differ. The result is called the first retained value. There previously occurred for you, in the case of opposition or illumination (full Moon), the time of the equality of the two declinations. Determine the time from the known minutes of the day and multiply it by the three bhuktis-I mean, the bhukti of the Sun, that of the Moon, and that of the node. Divide the products by the sum of the bhuktis of the two luminaries; there result their equations. If the time (read waqt for qamar) is in the future, add those of the two luminaries to them and subtract that of the node from it; if the time is in the past, reverse the operation with respect to addition and subtraction. Compute for what results the declinations of the Sun and Moon, and work out from them as was previously done the second retained value. Then look; and if the time is in the future, take the difference between the two retained values; if the time is passed, add the two retained values together. The result from them is the part of division (divisor). Then multiply the determined minutes of time by the first retained value and divide the product by the part of division; there results the period of time till the time of the equality of the two declinations in unity and equalness. Repeat the operation until the time of the obscuration conforms and is correct.

Commentary If we let d~ and 6s be the declination of the Moon and Sun at con-

junction or opposition, t the time before (or after) conjunction or opposition, and bM, bs, and bR the respective bhuktis of the Moon, the Sun, and the node, then form:

SM f 6s as the declinations are in E:z: directions at conjunction,

the same different directions at opposition (or respectively 6s - SM if 6s > S,).

The result we will call Ad.

The Later Pauliiasiddhtinta 221

t(bM) t(bs) , and t(b R)

bM + bs’ Then find the time t’ by forming

and: t’ = PM * bs ’ b R as the time is

b i + bs’ blcl -I- bs future past. b%I + bs

Next compute S’M and 6 ‘s for the time t‘, and form:

8 M & 6’s

as was done previously. This we will call Ad’. The time till equality (t”) is found thus:

(past future. as the time is

AS AS f AS’

t” = t

By a process of iteration find t”’, t“”, etc., until the desired degree of accuracy is attained.

P 45. A-Biriini, India, ed. pp. 406407, trans. vol. 2, p. 74. Therefore PuliSa says: “Multiply the yojanas of the radius of the sphere of the Sun or Moon

by its connected hypotenuse (qup) and divide the result by the total Sine; divide by the quotient 22,278,240 for the Sun and 1,650,240 for the Moon. There result the minutes in the diameter of its body worked our for it.”

These two numbers are the products of the yojanas of the diameters of the two luminaries multiplied by 3,438, which are the minutes of the total Sine.

Commentary Since the hypotenuse (the distance of the heavenly body from the

center of the universe) is measured in parts of R, one must employ the implied proportion to obtain that distance in yojanas (cf. also P 61). Further, as the mean distance in yojunas is to the diameter of the body in yojanas as R is to the diameter of the body in minutes, one should divide the mean distance in yojanas by the product of R (3,438’) and the diameter of the body in yojanas (6,840 for the Sun, 480 for the Moon; see P 47); these are the divisors given here, but one is erroneously told to divide by them the true distances in yojunas.

222 David Pingree

P 46. Al-Biriini, Al-QZntin al-Mas%di, vol. 2, p. 985. PauliSa says: “Add the measure of the Sun to the measure of the Moon; take half

of the sum and call it ‘half of the two measures’. Then multiply this by 60 and divide the product by the difference between the bhuktis of the two luminaries; there result the minutes of decline from the day. Then put down the correct time in two places. Subtract the minutes of decline from the first; there remains the time of the beginning of the obscuration. Add the minutes of descent to the other; there results the time of the completion of the disappearance of the obscuration. The correct time between these two (times) is for the mid-point of it (the obscuration).”

The transit therefwe in an eclipse is what is useful; this is because it fixes the Sun in its orbit at a place which cuts the orbit and the inclined sphere, though it (the Sun) proceeds progressively (through the signs), and the Moon catches up with it as it catches up with it for an eclipse. So the duration of its transit over it (the Sun) comprises the beginning, middle, and release according to the condition of the duration of the eclipse and it is the same in its computation.

Commentary The “correct time” here is the time of true conjunction of the Sun

and Moon, and, as al-BirBni remarks, the problem of the half-duration of an eclipse is assumed to be a problem in transit. The difficulty with PauliSa’s formulation is that it ignores lunar latitude; it will only work if the two centers pass through each other.

P 47. Al-Biriini, India, ed. p. 406, trans. vol. 2, p. 73. As what is found in minutes for the diameter of the Moon is in a ratio

to 21,600, which are the minutes of a circumference, as the ratio of its share of yojanas, which is 480, is to the yojunas of the whole circumference, one operates similarly for what is found in minutes for the diameter of the Sun. Its yojanas, according to Brahmagupta, are 6,522, and, according to PuliSa, 6,480. As there results for PuliSa that the minutes of the body of the Moon are 32, which is a multiple of two, he divides it by two for the planets till (it reaches) one. There results for Venus its half, for Jupiter its fourth, for Mercury its eight, for Saturn a half of its eight, and for Mars a fourth of its eight. He approves of

The Later Pauliiasiddhtinta 223

this order, but the diameter of Venus by observation is not half the diameter of the Moon, and Mars is not half of its (Venus,) eighth.

Commentary See P 48 and Yacqiib ibn Tiiriq T 1 . The yojanas of the orbit of the

Moon are 324,000 (see vs. XXVI); the proportion does indeed yield 32’ as the measure of the Moon’s diameter. The yujunas of the orbit of the Sun are 4,331,500; the Sun’s diameter, then, is 0;32,31,24,. . . ’. The minutes for the remaining planets are: Saturn 2’, Jupiter 8’, Mars l’, Venus 16’, and Mercury 4‘.

It is normal in Indian siddhiintas to use this halving process to obtain the diameters of the planets, but to express them in yojanas rather than in minutes. Thus, in the Paitcimahasiddhcinta of the ViFnudharmotturupurZna (111 8):

Sun Moon Venus Jupiter Mercury Saturn Mars

yojanas kcihmhs

6,500 for visibility

480 12” 240 9 120 1 1 60 13 30 15 15 17

P 48. Al-Birfini, Al-Qiiniin al-MasCiidi, vol. 3, p. 1313. There preceded, in the beginning of the treatise, the method of the

Indians concerning the distances of the planets and what the opinion of PauliSa the Greek requires. Since we related from his book or from someone else’s book the ratios of the diameters of the planets to each other, it is possible to know their bodies according to the example of the methods which we have introduced. PauliSa says that the diameter of the Moon is 32; for Venus it is half of this or 16; for Jupiter it is half of this or 8; for Mercury it is half of this or 4; for Saturn it is half of this or 2; and for Mars it is half of this or 1.

Commentary See P 47.

224 David Pingree

P 49. Utpala on Byhatsimhitci 17,lO and on Khandakhcidyaka in the preface to chapter 8.

Thus (says) PuliSSicSirya (vs. XX a-b): “All the planets are victorious in the north, (but) Venus is victorious

in the south (as well).”

Commentary This refers to the astrological theory of grahayuddha or planetary

transits. In these, that planet which has the more northern latitude “transits” the other; but Venus is the “transitor” whenever she passes another planet.

P 50. Al-BIrtini, India, ed. p. 420, trans. vol. 2, p. 91. Puli5a says: “Double the apogee of the Sun. When the position of the Sun equals

this, it is the time of its (Agastya’s) disappearance.” The apogee of the Sun according to him is two and two thirds signs,

and he puts its double at a third of Virgo, which is the beginning of the nakjatra Hasta; half of the apogee is in a third of Taurus, which is the beginning of the nakjatra Rohini.

Commentary According to the tradition of the brahmapaksa (see the Paitimaha-

siddhinta of the Vijqudharmottarapuriqa 111 30), the longitude of Agastya (Canopus) is 87” and its latitude 76” S. The extreme southern latitude means that it is only visible for a short time after the summer solstice. What we learn of importance from this fragment is that PauliSa pIaced the solar apogee at 80”, which is its position according to the iirdhariitrika system. We may assume, then, that he used tirdharctrika longitudes for the other apogees also:

longitude of apogee Saturn 240” Jupiter 160 Mars 110 Sun 80 Venus 80 Mercury 220

The Later PauliSasiddhdnta 225

The beginning of the nakjatra Hasta is at 160”, the beginning of Rohini at 40”. It is difficult to see what Rohini has to do with Agastya.

P 51. Utpala on Byhatsamhita‘ 2 (p. 57). Thus (is it said) in the Puuliia (vs. XXI): “The earth is formed round like a circle, and is motionless in the

boundless sky; it consists of the five rnuhiibhiitas. In its middle is Meru, (the home) of the Gods.”

Commentary This is the usual confused Indian cosmology. Meru in the middle of the earth presupposes that it is a flat disc, but its being round like a circle (i.e., globe) and motionless in a boundless sky presupposes its being a ball whose only “middle” is that point within it which is equidistant from all points on its surface. See al-Birfini’s discussion in P 53. The five muhabhiiras are enumerated by al-Biriini in P 54: earth, water, fie, wind, and heaven.

P 52. Utpala on Byhatsamhitd 2 (p. 59) and Pythiidakasviimin on Briih- masphutasiddhdnta 21,4.

Thus (is it said) in the Pauliiasiddhfinta (Pauliia Utpala) (vs. XXII): “Above it (Meru) is the (north) pole in the sky; the circle of the con-

stellations, which is bound to it by chords of wind, revolve to their risings and settings as they are driven by the wind.”

Commentary

impetus for the motion of the sphere of the iixed stars. The wind that PauiiSa refers to is the Pravaha, which provides the

P 53. Al-Biriini, India, ed. pp. 221-222, trans. vol. 1, pp. 266-267. Puliia says in his siddhiinta that PuliSa the Greek says in (one) place

that the earth’s form is spherical, and he says in another place that it is like a flat plate. He is right in both of these (places); for there is round- ness in its surface and straightness in its diameter. But (for the fact that) he does not believe about it that it is other than spherical there are many demonstrations (to be made) from his own words. . .

As for its (the earth‘s) place, it is in the middle (of the universe). Half

226 David Pingrce

of it is clay and half water. Mount Meru is in the dry half, the residence of the Devas, the angels (al-maZZika); above it is the North Pole. In the half flooded by water, under the South Pole, is Vadavilmukha, which is dry land like an island; the Daityas and Nggas, who are relatives of the angels who (live) on Meru, inhabit it, and therefore it is also called Daityiintara. The line dividing the two halves of the earth, the dry and the wet, is called nirakja, that is, that which has no latitude; it is the equator.

In its (the earth’s) four quarters are four large cities: in the East Yamakoti, in the South LaHkB, in the West Romaka, and in the North Siddhapura. The earth is regulated by the two Poles, and the axis holds it. If the Sun rises at the line (parallel of longitude) running through Meru and LaiikB, then it is noon at Yamakoti, midnight for the Romans, and sunset at Siddhapura.

Commentary There is clearly some corruption in the opening phrase of the fragment.

One should probably read: “The translator of PuliSa’s siddhanta says that PuliSa the Greek.. .” For the cosmological ideas expressed by PauliSa see W. Kirfel, Kosmologie, pp. 173-177 and also P 56. For the four cities at quadrants along the equator see, as well as P 57, The Thousands of Abii Macshar, p. 45, and Yacq6b ibn TBriq Z 3 and Z 4 (= al-Faz9ri Z 20 and Z 21).

P 54. Al-Birfini, India, ed. pp. 182-183, trans. vol. 1, p. 224. PuliSa says in his Siddhrintu that the entire world is a combination of

earth, water, fire, wind, and heaven, which was created in what is beyond the darkness. Heaven appears to be blue in color because of the deficiency of the rays of the Sun so that it is illuminated by them as are the watery but not fiery spheres (I mean the bodies of the planets and the Moon) of which, if the rays of the Sun come to them and the shadow of the earth does not reach them, the blackness vanishes and the figures appear in the night. The illuminator is one, and the rest of them (the planets) are illuminated by it.

In this passage (fa;Z) he alludes to the utmost limit that can be reached and calls it heaven; he puts it in darkness because of what he says of its

The Later Paulihzsiddhdnta 221

being in a place which the rays (of the Sun) do not reach. A discussion of the blue-gray color that is seen would be very lengthy.

Commentary For the five mahcbbhiitas see vs. XXI. I know of no parallel in Sanskrit

to the cause PauliSa gives for the blueness of the sky; but vs. XXV gives in yojanas the circumference of heaven within which the Sun dispels darkness, and, therefore, beyond which darkness must exist.

P 55. Al-Birfini, India, ed. pp. 230-231, trans. vol. 1, p. 276. As for what PuliCa says about the earth, that an axis holds it, he does

not mean by this that, if (really) an axis were not there, the earth would fall. How could he say this when he believes that four cities around the earth are inhabited, and this has as its cause the falling of weights toward the earth from all directions? But he believes concerning this that the motion of what is at the periphery is the cause of the non-motion of what is in the center. Motion in a circle is not possible without two poles and a line running between them, which is the axis in the imagination.

Commentary

and motion as was the Hellenized al-Birfini. It is doubtful that PauliCa was as concerned with problems of mechanics

P 56. Al-Biriini, India, ed. pp. 232-233, trans. vol. 1 , pp. 278-279. PuliSa says: “The wind turns the sphere of the fixed stars, and the

two Poles keep it in place. The inhabitants of Mount Meru see its motion, which is to the West, as being from left to right; the inhabitants of Vadavgmukha see it as being from right to left.”

He says in another place: “If an interrogator asks about the direction of the motion of the stars with what one sees of their rising from the East and their rotating towards the West till they set, let him know that their motion towards the West which we observe is different depending on the perception of the people from where they live. The inhabitants of Mount Meru see it (the motion) from left to right, the people of the island Vadaviimukha find it the opposite of this, from right to left, and the inhabitants of the equator towards the West only; and everyone (else), wherever he is between these places, (sees it) as depressed according

15’

228 David Pingree

to the latitudes of their abodes. (This motion) in its entirety is caused by the wind which turns the spheres until the planets and the other (stars) are compelled to rise from the East and set in the West. (This is) by accident; but by essence their motion is towards the East. This is the motion which is from al-Sharafayn towards al-Bufayn, since al-Bufayn is in an easterly direction from al-Sharafayn. But if the interrogator does not know the mansions of the Moon and is not able from reference (to them to understand) the easterly motion, let him observe the Moon itself in its elongation from the Sun a first time and a second; then (how) it comes near it similarly until it is in conjunction with it. Let him imagine from this the second motion.

Commentary For the celestial winds see vs. XXII; for the inhabitants of the two

poles, P 53. The opposite manners in which the Devas and the Asuras observe the revolutions of the stars is a commonplace of Indian astronomical literature; see, e.g., Briihmasphutasiddhiinta 21,4. PauliSa, however, would have said nothing about the difference between accidental and essential motion.

Al-Sharafayn is the first manzil of the Moon, identified with the nakSatra ASvini; al-Bufayn is the second, identified with Bharani

P 57. AI-Bjriini, India, ed. pp. 223-224, trans. vol. 1, p. 268. hyabhata and PuliSa and Vasisfha and Lgfa agree that if it is noon

at Yamakofi, it is then midnight among the Romans, dawn at Laiik2, and sunset at Siddhapura; and this would not be except for the circularity (of the earth). SimilarIy the times of eclipses would not proceed with regularity except for that.

Commentary For kyabhafa see A);vabhatiya, Gola 13 and Paiicasiddhiintikd 15,20;

for Lgfa Paficasiddhiintikii 15,18; and for PauliSa P 53. I do not know the source of al-Biriini’s statement regarding Vasisfha.

P 58. ALBiriini, India, ed. p. 401, trans. vol. 2, pp. 69-70. Then I did not find a treatise on this subject, but there occur within

the books some references to corrupt numbers about this (planetary

The Later Pauliiasiddhiinta 229

distances), such as the reply of PuliSa to someone who objected to his conclusion that the circumference of the sphere of every planet is 21,600 and its radius 3,438, while Variihamihira says that the distance of the Sun is 2,598,900 (read 259,890,012) and that of the fixed stars 321,362,683, that the first (i.e., PuliSa’s) is in minutes and the second (i.e., Varahami- hira’s) in yojunus; while he says that the distance of the fixed stars is 60 times that of the Sun, and it is necessary that the distance of the fixed stars be 155,934,000 (read 15,593,400,720).

Commentary Al-B-Mini’s translator has apparently again led him into serious con-

fusion. PauliSa does say, and quite correctly, that the circumference of the sphere of every planet is 21,600 minutes, and that its radius is

967 1,309

3,437- minutes (cf. P 31), but he goes on to compute the yojunus in

each planet’s orbit and in the orbits’ radii on the basis of the assumption that every planet traverses an identical number of yojunus in a Cuturyuga, that a minute in the Moon’s orbit is 15 yojunus, and that the distance of the nukjutrus from the center of the earth is equal to sixty times that of the Sun; see P 59. Al-Biriini, in fact, knew the true situation; see P 60 and P 61.

That the circumference of the nukjutrus’ orbit is 259,890,012 yojunus is found in the Pauliiusiddhintu (P 59), but nowhere in the works of Varahamihira. The number 15,593,400,720 is simply 259,890,012 x 60. The other radius of the nuk;utrus’ orbit that al-Biriini ascribes to Varii- hamihira- 321,362,683 yojunm-remains obscure to me.

P 59. Utpala on Byhutsuphiti 2 (pp. 48-55). The knowledge of the diameters of the orbits is read in PuliSa and the

rest; their computation will be described by us here. This is the knowledge of the length of a yojuna (vs. XXXVII):

“A yojunu is 8 kroius; a kroiu is 4,000 hands (kuru); a hand (hustu) is two gnomons (iuzku); a gnomon consists of 12 digits (uzgulu).”

Then the measure of the yojanm of travel of the planets in a Cuturyuga

“The travel (of each planet) is equivalent to the months and udhimfisus plus the “constellation” months in a Yugu converted into yojunus; there-

(vs. XXIII):

230 David Pingree

fore, the orbits of the planets which travel towards the east are equal to this divided by their revolutions.”

As many as are the revolutions of the Sun in a Caturyuga, so many are the years of the sa‘vuna measure in a Caturyuga (vs. IV a-b):

“The Sun passes through a Yuga with 432 multiplied by 10,000 revolutions.”

These years in a Caturyuga multiplied by 12 are 51,840,000; these are the sEvana months in a Caturyuga. These are to be added to the adhimcisus (vs. XI c-d):

“The adhimcisas are 1,593,336.” Added to these they become 53,433,336; these are the months of the

ccindra measure in a Caturyuga. “The ‘constellation’ months” are to be taken here as the excess (vs. XI a-b):

“As many as are in excess over the crindra months in a Caturyuga.” Here the days of the nGkSatra measure in a Caturyuga are (vs. X c-d): “In the ncikjatra measure 1,732,600,080.” But dividing these by 30, one obtains 57,753,336; so many are the

months of the ncikSatra measure in a Caturyuga. They are also the revolutions of the Moon. In one revolution of the Moon there is a month of the nzksatra measure; therefore the czindra months are to be subtracted from the months of the ncikpztra measure. There results from the subtraction 4,320,000; so many ncikjatra months are in excess of the ca’ndru months in a Caturyuga. Therefore, by adding these constellation^' months to the sum of the solar months and adhimcisas-53,433,336- there results 57,753,336. This sum of the months, udhima’sus, and “constellation” months in a Yuga is equivalent to the revolutions of the Moon. Therefore it will be said (vs. IV c-d):

“The Moon with 57,753,336.” Then the revolutions of the Moon are to be converted into yojunus.

How? it is asked. In the orbit of the Moon, one minute (@fa‘) consists of 15 yojanas; by converting the revolutions of the Moon into minutes and multipIying (the result) by 15 they obtain the number of yojanas. It will be said (vs. XXXI a):

“The number of yojanas is (this) multiplied by 15.” Here the sum of the months, adhim&as, and “constellation” months

in a Yuga-57,753,336-multiplied by 12 results in 693,040,032; these are the zodiacal signs (traversed) by the Moon in a Caturyuga. These multiplied by 30 are 20,791,200,960; these are the degrees (traversed) by the

The Later Pauliiasiddhdrita 23 1

Moon in a Cuturyuga. Multiplied by 60 they are 1,247,472,057,600; these are the minutes (traversed) by the Moon in a Cuturyuga. These multiplied by 15 result in 18,712,080,864,000; so many yojunus does each planet travel in a Cuturyuga. The teacher will say (vs. XXIV a-b):

“18,712,080,864 multiplied by 1,000.” “So many yojunus must each planet situated on its orbit travel to the

east in a Cuturyugu”: that is the sense of “the planets which travel towards the east.”

“Therefore this divided by their revolutions.” Here, with regard to the measure of the orbit of heaven (&C.iu), these yojunus of travel in a Cuturpga- 18,712,080,864,000-are involved. Since a KaZpa is 1,008 Cuturyugus, these, multiplied by 1,008, result in 18,861,777,510,912,O; this is the orbit of heaven within which the Sun produces an absence of darkness. This is read (vs. XXV):

“They say that the path of heaven is this: 18,861,777,510,912,000.” Then the revolutions of the planets are (vss. IV-VII b): “The Sun passes through a Yuga with 432 multiplied by 10,OOO revo-

lution; the Moon with 57,753,336; Mars with 2,296,824; Mercury with 17,937,000; Jupiter with 364,220; Venus with 7,022,388; and Saturn with 146,564. Mercury and Venus are like the Sun.”

By dividing the (given) number by these in order, (one obtains) the measures of the orbits:

the orbit of the Sun is 4,331,500; the orbit of the Moon is 324,000; the orbit of Mars is 8,146,937; the orbit of Mercury’s conjunction is 1,043,211 ; the orbit of Jupiter is 51,375,764; the orbit of Venus’ conjunction is 2,664,632; the orbit of Saturn is 127,671,739; the orbit of the nakgztrus is the orbit of the Sun with its fraction

“The Sun is a sixtieth part of the constellations;’’ this (orbit of the naksutrus) is 259,890,012.

multiplied by 60. So, in the Bruhrnusiddhtinta:

Thus PuliSBdrya (vss. XXVI - XXIX): “The orbit of Mercury, the Sun, and Venus is 4,331,500; that of the

Moon is 324 multiplied by 1,000; that of Mars 8,146,937; that of Mercury’s apogee 1,043,211; that of Jupiter 51,375,764; that of Venus’ apogee

232 David Pingree

2,664,632; that of Saturn 127,671,739; and that of the constellations 259,890,012.”

Then the calculation of the diameters (of the orbits). Here he first describes the calculation of the lunar minutes which are useful for it

“The quotient of the division of the orbit of any planet by the orbit of the Moon is the mean lunar minutes of the planets beginning with Saturn.”

By dividing the yojanus of the orbit of any planet by the yujunas of the Moon’s orbit, one obtains the lunar minutes of that planet; “beginning with Saturn” is said in the sense of the magnitude of the orbits. They are the lunar minutes of the mean planets beginning with Saturn. Here, in order to shorten the work, the dividend and divisor of every planet are divided by the yojanus of the Moon’s orbit; in this the dividend of every planet is the revolutions of the Moon, the divisor its own revolutions. This work arises in this: the revolutions of the Moon are divided by the revolutions proper to a planet, and the result is its lunar minutes. The lunar minutes of all the planets, which are arrived at by this rule,

(vs. XXX):

are written down: the Sun

1,593,336 134,320,000;

the Moon 1;

Mars

Mercury

Jupiter

Venus

Saturn

332,736 252,296,824;

3,942,336 17,937,000; 206,576 364,220’ 1,574,232

87,022,388; 7 120

158-

39*4*

Then he describes the computation of the diameters of the orbits of the planets by means of the lunar minutes (vs. XXXI):

“(This) multiplied by 15 is the number of yojunus; R multiplied by that is the radius in yojanas (of the planet’s orbit); or it is (expressed) in earth-radii.”

The Later PauliSasiddhdnta 23 3

These lunar minutes multiplied by 15 become yojanas; they are written down in order as yojanas:

the Sun

The Moon

Mars

Mercury

Jupiter

Venus

Saturn

Nakjatras

2,300,040 2004,320,00a;

15;

3372,296,824 397’392 (read

5,324,040 48 17,937,000;

184,880 364,220’ 2,546,3 16

lZ37,022,388; 106 800

5,91-; 146,564 20,412 21,600’

2,378-.

12,031-

These are the yojanas in a minute for the orbit (of each) of the planets.

“R multiplied by that.” R, the radiu~-3,437--multiplied by the

yojanas in the orbits of every planet is the yojunas of the radius of each

orbit. Here R-3,437--when multiplied by the yojanas in a minute

of the orbit of heaven which have previously been computed -873,230,440,320 [ 18,861,7773 10,912 f 21,60O]-becomes

967 1,309

967 1,309

224 3,001,938,106,524,064 really 3,M)1,938,106,524,064-]. 1,309 ’ this is the

radius of the orbit of heaven. By so many yojanas is the orbit of heaven above that place which is dug 800 yojanas beneath the surface of the earth; the origin of all the radii of the orbits is to be measured from that place. This is read in the PuZiiasiddh&zta (vs. XXXII):

“The radius of heaven is 3,001,938,106,524,064.” Then R multiplied by the yojanas in a minute of the orbit of the

nakjatras, which have previously been computed. The multiplicand is

12,031- * if the multiplicand and the multiplier are made to have 20,4 1 2 21,600’

234 David Pingree

the same denominator, the orbit of the nakJatras results-259,890,012. This orbit of the nakjatras is to be multiplied by R made to have the same denominator-4,500,000; one should divide this by the denominator of R- 1,309-multiplied by 21,600, which is 28,274,400; the product of the two denominators multiplied by each other is the divisor. When a ratio is made of the multiplier and divisor-4,500,000 : 28,274,400-there results 7,200, and the multiplier is 625. The orbit of the nakjatras is multiplied by this; there results 162,431,257,500. This number is to be divided by the divisor divided by the denominator (7,200), which is 3,927; the quotient is 41,362,683. That is the radius of the orbit of the naksatras.

Then the yojanas in a minute of the orbit of Saturn-5,910--

are made to have the same denominator- 866,300,040-and multiplied by R made to have the same denominator-4,500,000. The dividend of every planet is 3,898,350,180,000,000 [324,000 x 15 x 4,500,000]. This is divided by the revolutions proper to each planet multiplied by the denominator of R-IY309-aand the result is the radius of the orbit of each (planet). Therefore the yojanas in a minute of the orbit of every planet, when converted into a fraction, is 866,300,040.

R converted into a fraction is the multiplier of this number for all the planets, and the denominator of R- 1,309-multiplied by the planets’ revolutions is the divisor. The revolutions of all the planets multiplied by 1,309 are written down here; they will be the divisors:

106,800 146,564

the Sun 5,354,880,000; the Moon is not included as in its orbit converted into minutes each minute is 15 yojanas; Mars 3,006,542,616 Mercury 23,479,533,000 Jupiter 476,763,980 Venus 9,192,305,892 Saturn 191,852,276

By dividing the dividend for all the planets- 3,898,350,l 80,000,0~- by the divisor of the Sun-5,354,880,000-one obtains 689,378; that is the radius (of the orbit) of the Sun.

The Later Pauliiasiddhdnta 235

967 1,309’

R is 3,437--* multiplied by 15, the yojanas in a minute of the orbit,

Thus, by dividing the dividend for all by the divisor of each, all it becomes 51,566; this is the radius of the Moon’s orbit.

(the radii) are computed:

the radius of the Sun is 689,378; the radius of the Moon is 51,566; the radius of Mars is 1,296’622; the radius of Mercury’s conjunction is 166,031; the radius of Jupiter is 8,176,688; the radius of Venus’ conjunction is 424,088; the radius of Saturn is 20,319,541; the radius of the nakgztras is 41,362,683.

Thus PuIiiZicZirya (vss. XXXIII-XXXVI): “The radius of the Sun, Venus, and Mercury is 689,378; that of the

Moon is 51,566; that of Mars is 1,296,622; that of the apogee of Mercury is 166,031; that of Jupiter is 8,176,688; that of the apogee of Venus is 424,088; that of Saturn is 20,319,541; and that of the nakqatras is 41,362,683.”

“Or it is (expressed) in earth-radii,” a part of the rule whose purpose is to demonstrate a different method, is not commented on.

Commentary The basic assumptions for these lengthy computations have been

enumerated in the commentary on P 58. If y represents the yojanas of any orbit, x the number of any planet’s revolutions in a Caturyuga, and r the radius of any orbit, and if subscript m, n, and p represent the Moon, the nakJatras, and any planet, then:

1) yn = Xm (21,600 X 15) and

Yn

XP 2) yp = -.

The simplest way to derive the radius of any orbit would be by the R formula:

236 David Pingree

PauliSa substitutes:

4) rp = lSR(E). Ym

yn Xm

Ym XP and, since - = -, Utpala uses:

5) r p = 15R(:).

PauliSa employs a number of crude roundings in his calculations: see P 61 for the true values of yp and rp from 2) and 3). PauliSa assumes .~

i.e., 3~ = 3- 177) and that the diameter of the that R = 3,437- 1250 1,309 967 (-

earth is 1,600 yojanas.

P 60. Al-Biriini, AZ-Qfiniin aZ-Masciidi, vol. 3, p. 1302. PauliSa uses with respect to the world-days (only) a part. The civil

days according to him are 1,577,917,800 and the revolutions of the Moon in these are 57,753,336. If you multiply by the degrees of revolution and then by 60, there result the minutes of motion of the Moon in the totality of this period. They (the Indians) agree that the distance of each minute in the measure of the Moon is 15 yojanas, and this name is applied to eight of our miles-I mean to 32,000 cubits. Then the number of the motion of the Moon (multiplied) by this above-mentioned measure -1 mean the product of its minutes multiplied by 15-is 18,712,080,864,000, which is the motion of each planet in these (days). So, when this number is divided by the revolutions of a planet in this period, there results the measure of its mean revolution in its sphere measured in the above-mentioned measure (i.e., yojanas). The revolutions of Saturn in this (period) according to him are 146,564, the revolutions of Jupiter 364,220, the revolutions of Mars 2,296,824, the revolutions of Venus 7,033,388 (read 7,022,388), and the revolutions of Mercury 17,937,000. If the circumference is known, then the diameter is known because the circumference is in a ratio to the diameter according to him of 3,927 to 1,250; this ratio is not very different from that used according to the opinion of Archimedes. The diameter of the earth in the above- mentioned measure is, according to him, 1,600.

The Later Pauliiasiddhfinta 237

Commentary

mile and an Indian kroSa. See P 59, P 26, and P 27. Al-Biriini assumes the identity of an Arab

P 61. Al-BWini, India, ed. pp. 404-405, trans. vol. 2, pp. 72-73. Because PuliSa uses the Caturyuga, he multiplies the extent of the

circumference of the sphere of the Moon by its revolutions in it (a Ca- turyuga) (to get) 18,712,080,864,000; he calls this the “yojanas of heaven,” which is what the Moon passes through in each Caturyuga. According to him, the ratio of the diameter to the circumference is 1,250 to 3,927. So, when one multiplies the circumference of the sphere of each planet by 625 and divides the product by 3,927, there results the distance of the planet from the center of the earth. We operated with this as before and put down the results according to his opinion in a table also. As for the radii, we eliminate the fractions less than a half in them and restore the excess to an integer; but we do not do this with respect to the circumferences, but we verify them because they are needed for the (planetary) motions. This is because, if the yojanas of heaven in a KaZpa or Caturyuga are divided by its savana days, there results 11,858 and a remainder of

for PuliSa. This is what the Moon 209,5 54

for Brahmagupta and - 25,498 35,419 292,207 passes through in one day; as the rate of motion (of all the planets) is the same, this is what each planet travels every day, and its ratio to the yojanas of the circumference of its sphere is like the ratio of its unknown motion to a circumference taken as being 360. So, when one multiplies the travel common to all of the planets by 360 and divides the result by the yojanas of the circumference of the intended planet, there results its mean bhukti, which is its mean daily motion.

Planet Yojanas of the circum- Yojanas of their ferences of the spheres distances from the

of the planets center of the earth Moon 324,000 5 1,566

Mercury

Venus

573 1,043,211-

1,993 90,232 585,199

2,664,632-

166,033

424,089

238 David Pingree

Sun

Mars

Jupiter

Saturn

4,661 ,SO@ 690,295 (read 4,331,5w)

i 8 , m 8,146,937-

95,701 4,996

51,375,7643-3i

1,296,624

8,176,689

27,301 36,641

127,67 I ,739- 20,319,546

Fixed stars on the assump- 259,890,012 41,417,700 tion that the distance of the Sun is one sixtieth of their distance

Commentary According to the BrZ/zmasphu?asiddhZnta (1,16) the Moon revolves

57,753,300,000 times in a KaZpa. The circumference of heaven, then, is 18,712,069,200,000 yojanas. The number of savana days is 1,577,916,450,000; the division of the number of yojanas by the number

25,498 of days does result in 11,858- The rest of the fragment is clear 35,419' enough.

P 62. Al-Biriini, India, ed. p. 132, trans. vol. 1 , pp. 168-169. 177

1,250 As for Puliia, he employs the ratio: as 1 is to 3- (for n). This also

is smaller than a seventh by the amount by which it is smaIler in the opinion of h a b h a t a .

This is derived from an ancient opinion which YacqGb ibn Tgriq mentions in his Tarkib al-aJZk on the authority of the Indian concerning the yojanas of the circumference of the zodiac, that they are 1,256,640,000 (read 125,664,000farsakhs), and that its diameter in yojanas is 400,000,000

56,640,000 400,000,000

(read 40,000,000 farsakhs). This is the ratio: as 1 is to 3

57664'000). Both (numerator and denominator) contain in (read 3 4 ~ , ~ ~ , ~ common 360,000 (read 32,000); so there results (after division) the

The Later PauliSasiddhdnta 239

numerator 177 and the denominator 1,250, which is what PuliSa adheres to.

Commentary S e e P 26 and P 27; YaCqiib ibn Tiiriq T 1 , T 2, T 3, and al-Fazgri Z 17.

125,664,000 40,000,000

The value for n expressed by the fraction is precisely Pauliia’s

62 832 as al-B-Mini says; it is also equivalent to kabhata’s - But 177 3-

1,250’ 20,000’ the number of farsakhs in the circumference of the. zodiac equals 251,328,000 yojanas rather than Pauliia’s 259,890,012. This implies that the circumference of the Sun’s orbit is 4,188,466g yojanas, that the circumference of heaven is 18,094,176,000,000 yojanas, and that the Moon makes 55,846,2225 revolutions in a Caturyuga. This last parameter is an utter impossibility; hence it is obvious that the circumference of the zodiac- 125,664,000farsakhs-is derived directly from habha ta ’ s value of n, and has nothing to do with the normal Indian theory of the distances of the planets.

ADDENDUM

This additional fragment was discovered only after the rest of the article was already in proof. Its proper position is after P 5.

P 63. Pythiidakasvamin on Br6hmasphufasiddh&ta 1 1,4. In this (matter) Puliia (says) (vs. XXXVIII): “When 648,000 years have gone from the Krtayuga, on that day is

the beginning of the Yuga for the equality (in mean longitudes) of the planets.’’

Commentary Mean conjuctions of the planets (exclusive of the lunar apogee and

node) are assumed to occur at the beginnings of each of the Yugas. If the division of the Caturyuga into Yugas given in P 5 is correct, this assumption requires that the numbers of revolutions of each of the

240 David Pingree

planets in a Cuturyuga be divisible by 10 as the Kahyuga is a tenth of a Caturyuga. Inspection of the table of these numbers in P 12 demonstrates that this is not the case. But the numbers are divisible by four, and therefore will fit the & ~ b h ~ [ i y ~ ’ s division of the Caturyuga into four equal Yugas of which each consists of 1,080,000 years. A mean conjunction, then, occurs 648,OOO years before the end of a Krtayuga consisting of 1,728,000 years.

Verse XXXVIII. See P 63. ayuttini catuhsastih Satiiny aSitiS ca y2tam abdtinam / krtayugato yatra dine grahatulyayugasya tatradih //

B I B L I O G R A P H Y Primary sources Abii MaCshar, Kit& ul-muwcilid. Manuscript: Bodleian Library, Huntington 546. Amaraja, KhanhakhddyakavrSsancrbhrTSya. ed. B. Misra, Calcutta 1925. Aryabhata, Aryabhatiya, ed. H. Kern, Leyden 1874. English translation by W. E. Clark,

Al-Biriini, Al-QdnGn al-Maflfidi, 3 vols., Hyderabad 1954-1956. Chicago 1930.

- OR Transits, ed. as part 3 of Rasd’il al-Birrini, Hyderabad 1948. English translation

- India, Hyderabad 1958. English translation by E. C. Sachau, 2 vols., London 1879. - Muqdlu f i say‘ sahrni al-su%du wu ul-ghayb. Manuscript: Bodleian Library, Selden

M. Saffouri and A. Ifram with a commentary by E. S. Kennedy, Beirut 1959.

A. 11. Brahmagupta, Brrihmasphutasiddhdnta, ed. S . Dvivedin, Benares 1902. - KhandakhcTdyakn, ed. B. MiSra, Calcutta 1925; ed. P. C. Sengupta, Calcutta 1941.

Al-FazBri. See D. Pingree, “The Fragments of the Works of al-Faari,” to appear in JNES

Makkibhatta, Siddhdntaiekharabhdsya, ed. B. MiSra, 2 vols., Calcutta 1932-1947. Paitdmahasiddhdnta. See D. Pingree, “The Pait~mahusiddhdnto of the Visnudharmuttara-

purdna,” to appear in BrahmavidycT. PrthtidakasvBmin, Brrihmaspu~asiddhdntavivuruna. Manuscripts: an apograph in my pos-

session of manuscript 178 1 in the VibveSvarBnanda Vedic Research Institute, Hoshi- arpur, and India Office Library. Sanskrit 2769.

English translation by P. C. Sengupta, Calcutta 1934.

28, 1969.

- Khu~~akhdd~akavivarana, ed. P. C. Sengupta. Calcutta 1941. Sivaraja, Jyotirnibandha, Poona 191 9. Utpala, ByhajjcTtukolikd, Bombay 1864. - Byhatsarphitdvivyti, ed. S . Dvivedin, 2 vols., Benares 1895. - Khay&khddyakafikd. Manuscripts: apographs in my possession of two manuscripts

a t the Scindia Oriental Institute, Ujjain.

The Later PaulifasiddhBnta 24 1

Varahamihira, Paiicasiddhdntikd, ed. G . Thibaut and S. Dvivedin, Benares 1889; new

Yacqiib ibn THriq. See D. Pingree, “The Fragments of the Works of Yaeqiib ibn Tlriq,” edition by 0. Neugebauer and D. Pingree in preparation.

JNES 21, 1968, 97-125.

Secondary sources W. Kirfel, Die Kosmographie der Inder, Bonn-Leipzig 1920. D. Pingree, “Astronomy and Astrology in India and Iran,” Isis 54, 1963, 229-246. - The Thousands of Abli M&shar, London 1968.

NOTES A N D REFERENCES

1. Edited by E. Boer with astronomical notes by 0. Neugebauer, Leipzig 1958. 2. Isis 54, 1963, 231 n. 63. 3. Al-Biriini, AI-Qdnlin al-MasCPdi, vol. 2, p. 128 is not from the later Pauliiasiddhdnta

4. Some other verses on astrology are ascribed to PauliSa by Sivarlja in his Jyotirni- as it employs different parameters.

bandha; as they are in anu,s{ubhs, they cannot be from the later Pauliiasiddhdnta.

p. 24. PauliSah: brlhmam daivam manor manarp pai t rap sauram ca slvanam / nakgatram ca tatha candram jaivam rnanani vai nava // e$u SHvanasaurarkSaclndraih sygt sarvakarrnanam / siddhir atratha varvasya phalam jaivena grhyate //

bharagy2rdrP1 tath%Ie$ marutam SHkravaruge / $ad a i t h y ardhabhoglni sampiirnlnitarani ca // bhamanatrilavo dvighnah svasmrtyamiayuto hatah / bhasankhyaya phalaih sthalam bhahinam siiksmatam iylt // evam ViSvarkvaparyantam pausne Siinyamukhlntyatab / karnlntam prlgvad atrastham upantye Sravanadvaye I / vaiSvark$asya caturth2xpSas tatha paficadaS8rpSakah / Sravaqasya ca tattulya abhijidbhoganldiklb //

sphutagatyl yatha candro ravimandalanemigah / tadarhvam sankramo bhanor miisah sa syan malirnlucah //

p. 66. PauliSah:

p. 82. PauliSasiddhBnte:

16 CENTAURUS. VOL. XIV

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The Later Pauliśsiddhanta