पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

XVI.11 XVI. SAURA : MEAN PLANETS 295

2-9 To get mean Jupiter, multiply the days from epoch by 100, and divide by 4,33,232. Revolutions etc. are got. Deduct 10‴ per revolution. Add 8ʳ 6° 20″, the mean at epoch (This is called kṣepa.) (A bīja correction is given by VM, to this, for which see verses 10-11, below.) To get Mean Mars, divide the days by 687. Revolutions etc. are got. Add 14‴ per revolution. Add 2ʳ 15° 35′ 0″, the mean at epoch. (See verses 10-11, below, for bīja correction.) To get mean Saturn, multiply the days by 1000 and divide by 1,07,66,066. Revolutions etc. are got. Deduct 5‴ per revolution. Add 4ʳ 2° 28′ 49″, the mean at epoch. (See verses 10-11, below, for bīja correction). To get the Śīghra of Mercury, multiply the days by 100 and divide by 8797. Revolutions etc. are got. Add 4½‴ per revolution. Add 4ʳ 28° 17′ 0″, the Śīghra at epoch. (See verses 10-11, below, for bīja correction.) To get the Śīghra of Venus, multiply the days by 10 and divide by 2247. Revolu- tions etc. are got. Add 10½″ per revolution. Add 8ʳ 27° 30′ 39″, the śīghra at epoch. (See verses, 10-11, below, for bīja correction).

[Apparatus Criticus — Column 1]

1a A. ॰र्धेवत्पां; B. धैवत्यां b. A. निर्णेर्किसिधांते; B. गणकसिद्धान्ते c. A. महमार्केण; B. मध्यमाकेसा 2a. B. जिवस्य B2. शताभ्यासं b. A1.C. ॰यमाग्निचिसागरैर्विभजेत्; B. ॰विभजेन d. A1. हतं A1. ॰ष्ट्र्द्धक्तं 3a. B. repeats words from previous verse: सौम्यस्य सप्ताभ्यस्तं द्वित्रियमाग्निसामरैः सहस्रगुणा a-b. D. गुणमृत् b. A. दतुरससून्युर्तु; B. रुतु॰ B. खैकः d. B1.2.D. शेषा मध्या 4a. B1.3. दशांश भगणे and one भगणे om by haplography b. B. ॰ध्यास्तसराः c. B. नमवः कुकुक्षु देया d. B1.3. शनैश्च B3. विशोध्य A. ॰ख्रु; B. स्युः 5a. B. ॰मंशं b. B3. ॰शतिवसु C.D. समेता c. B. ॰वेदाक्षलिप्ताः d. A. ॰धनेर्मध्यमास्येव; B. शने मध्यमस्त्वेयम् 6a. A. भामा लिप्त; B. मागाः लिप्त b. A.C.D. र्तवः; B. तवः; A. खमक्षोगरौ (A2. ॰क्षौ॰); B. तवः शेषसौ गरु विः

[Apparatus Criticus — Column 2]

C. खमक्षो गुरौ; D. खपछो गुरोः c. B. क्षेवः A. जमतिथि; B. यमतितिथि d. B. त्रिशद्य 7a. B. गुणितं c. B. ॰र्धपंचमौस्त॰ (B3. ॰स्तस. d. A. हतः; B. हतक्षिपा; C.D. क्षेप्याः 8a. B. गुणीते b. A. ॰वाश्वियमैः c. A1. अर्कैका॰; A2. 'अर्कैका d. A. विलिप्ता 9a. A. सिंहस्य; B. सिंहेस्य b. A. खरेन्वो; B. खरे देवो विलिप्तिका c. A. शोसितस्य; B3. शोषितस्य B. विकला d. A.B. पक्षा गुणा दहनाः (A. ताः) 10a. A.B. क्षेप्या B. विकला b. A.B. वर्षमाध्यम A.B. क्षितिजो 11a. A.B 1.2 पञ्चद्वयो; B3. ॰द्वयोः b. A.B. ॰स्ताश्वि B. चन्द्रयुक्ताः c. A.B. विकालिकाः d. A.B.C.D. om स्युः B. सुर —— प्जतिस्य (B3. पू). A.B.C.D. मध्यात् In D. ch. XVII of the Mss. and of C is continued as part of ch. XVII. with verse numbers duly altered.

296 PAÑCASIDDHĀNTIKĀ XVI. 11 Note 1. I follow TS's emendations, except in verse 6, where I have read khamakṣau as khapakṣau instead of their khamakṣo, makṣo being meaningless. But their meaning, 20, is all right. In 7, the word kṣepa can stand, and need not be emended as done by them. Note 2. The word madhya with reference to Mars, Jupiter and Saturn is mean planet in modern parlance, and śīghra with reference to Mercury and Venus, is mean planet according to modern terminology. Note 3. How to get the days from epoch has already been explained, and it should only to be brought to the mid-night following to be used here. Example 1. Find the mean Mars at 1,20,553 days from the midnight following the Romaka epoch, which is the epoch given for star-planets. 1,20,553 ÷ 687 = 175 revolutions and = 5ʳ 21° 52' 40" The revolution correction = 175 × 14" = + 41" Kṣepa or mean at epoch = 2ʳ 15° 35' 0"

Mean Mars at required date = 8ʳ 7° 28' 21" Example 2. Find the Śīghra Venus at 1,20,553 days for epoch. 1,20,553 × 10 ÷ 2247 = 536 revolutions and 6ʳ 2° 19' 23" Revolution Correction: 536 × 10½" = + 1° 33' 48" Śīghra at epoch = 8ʳ 27° 30' 39"

Śīghra of Venus at 1,20,553 days = 3ʳ 1° 23' 50" Note 4. The rules give to find the mean planets etc. depend on the fact that there are approxi- mately 100 revolutions of Jupiter in 4,33,232 days, one revolution of Mars in 687 days, 1000 revolutions of Saturn in 1,07,66,066 days, 100 śīghra (truly mean) revolutions of Mercury in 8,797 days and 10 of Venus in 2247 days. The revolution corrections make these exact. The epoch constants are the means at epoch. Note 5. From the rules given we can reconstruct the yuga cycles of the original Saura-siddhānta of which the Saura of the PS is a Karaṇa, and from these the epoch constants. These we shall do now. The yuga days of the original Saura are 1,57,79,17,800, as computed from the short Saura yuga given in I.14, from which it can be computed that in 1,80,000 years there are 6,57,46,575 days, since the yuga is 43,20,000 years, being 24 times the short yuga. We might now verify by calculation, the yuga revolutions (yuga-paryaya) and epoch constants (kṣepa) of the several planets. Jupiter: Yuga revolutions 1,57,79,17,800 × 100 ÷ 4,33,232 = 3,64,220, rev. 0ʳ 17° 25' 1" Revolution correction = 3,64,220 × 10''' = – 16° 51' 43"

∴ The number of rev. etc. in the yuga = 3,64,220 rev., 0ʳ 0° 31' 18"

XVI. 11 XVI. SAURA : MEAN PLANETS 297 The error in the karaṇā method is 31' 18" in 43,20,000 years, which is negligible when we consider that the rule is given in a karaṇa, which is not intended to be used for such a long period. The yuga revolutions 3,64,220, is indeed that given in the original, as seen from the Ārdharātrika-pakṣa and Bhaṭṭotpala’s Pauliśa and Khaṇḍakhādyaka. Epoch Constant (kṣepa) for Jupiter The epoch is 427 Śaka i.e. 427 + 3179 = 3606 years from zero Kali, i.e. midnight, − 3 nāḍīs, 9 vināḍis. For 3606 years, the motion is 3,64,220 × (1/1200 + 1/(1200 × 600)) = 304 rev., 0ʳ 8° 6' 36" Subtracting the motion for 3 nāḍīs, 9 vināḍīs, − 16" Jupiter’s epoch constant got = 0ʳ 8° 6' 20" This is exactly what is given above in verse 6. Saturn: Yuga revolutions 1,57,79,17,800 × 1000 ÷ 1,07,66,066 = 1,46,564 rev., 0ʳ 3° 26' 12" The cycle correction = 1,46,564 × 5''' = − 3° 23' 34" ∴ The Yuga cycles got = 1,46,564 rev., 0ʳ 0° 2' 38" This is indeed the yuga cycles given in the Ārdharātrika-pakṣa etc. neglecting the small error of 2' 38" accumulating in 43,20,000 years, due to the karaṇa roughness. Epoch constant for Saturn 1,46,564 (1/1200 + 1/(600 × 1200)) = 122 rev., 4ʳ 2° 28' 5" Deducting for 3 nāḍīs, 9 vināḍis − 6" The epoch constant got = 4ʳ 2° 28' 49" Mars: Yuga revolutions 1,57,79,17,800 ÷ 687 = 22,96,823 rev. 6ʳ 29° 4' 59" Rev. Correction = 22,96,823 × 14''' = + 4ʳ 28° 52' 5" ∴ Yuga-cycles = 22,96,823 rev. 11ʳ 27° 57' 4" = 22,96,824, in round numbers, being short only by 2° 3', negligible in the long period. We see agreement with the original. Epoch constant for Mars The epoch constant is 22,96,824 (1/1200 + 1/(600 × 1200)) = 1917 rev., 2ʳ 15° 36' 43".2

298 PAÑCASIDDHĀNTIKĀ XVI.11 Deduction for 3 nāḍīs, 9 vināḍīs − 1′ 39.5″ Less 1917‴ ÷ 5 − 6″ The epoch constant = 2ʳ 15° 34′ 58″ There is agreement. Mercury: Yuga revolutions 1,57,79,17,800 × 100 ÷ 8997 = 1,79,36,998 rev. 11ʳ 21° 41′ 33″ Rev. Correction = 1,79,36,99 × 4½‴ = + 1ʳ 13° 41′ 15″ The Yuga cycles = 1,79,37,000 Rev. 0ʳ 5° 22′ 48″ There is fair agreement with the Ārdharātrika-pakṣa etc. with an excess of 5° 22′ 48″ in the yuga, which need not be considered great in a karaṇa rule. 4 7/16 instead of 4½‴ would have taken this difference also into account. Epoch constant for Mercury 1,79,37,000 ( 1/1200 + 1/(600 × 1200) ) = 14,972 rev., 4ʳ 28° 30′ 0″ Subtracting for the excess 3 nāḍīs, 9 vināḍīs − 23′ 53″ For 1/16 repeat the correction + 16″ Epoch constant = 4ʳ 28° 17′ 23″ Here the constant seems to have been given to nearest minute. Venus: Yuga revolutions 1,57,79,17,800 × 10 ÷ 2247 70,22,331 rev. 1ʳ 8° 55′ 54″ Revolution correction = 70,22,331 × 10½ = 56 Rev. 10ʳ 21° 47′ 52″ Yuga cycles = 70,22,388 0ʳ 0° 43′ 46″ There is a small error of 43′ 46″, negligible in the long period of yuga, owing to the karaṇa rule. 10 85/178 would have been very correct. Epoch constant for Venus Epoch constant, 70,22,388 ( 1/1200 + 1/(600 × 1200) ) = 5861 rev. 8ʳ 27° 35′ 38″.4 Less for 3 nāḍīs, 9 vināḍīs − 5′ 2″ Extra in the correction + 2′ 4″ Epoch correction (in full agreement) = 8ʳ 27° 30′ 39″ In the Sun, Moon, Rāhu, and Moon’s apogee too we see much exact agreement with Ārdharāt- rika-pakṣa, Khaṇḍakhādyaka and Bhaṭṭotpala-quoted Paulīśa, from which we can conclude that the source of VM’s Saura is the Old Saura-siddhānta.

XVI.11 XVI. SAURA : MEAN PLANETS 299 VM’s Bīja corrections 10-11. Add 17" per year to mean Mars. Deduct 10" per year from mean Jup- iter. Add 7¹/₂" per year to mean Saturn. Add 120'' per year to the śīghra (‘mean’ according to modern parlance) of Mercury. Subtract 45" per year from the śīghra (modern ‘mean’) Venus. In addition, subtract 1400" or 23' 20", constant from Jupiter’s mean. Note. 1 I follow TS’s corrections. Note 2. These corrections are obviously VM’s own, to secure agreement with observation, because VM sees the Saura used widely for almanac making, (besides himself being its follower) and uses these bīja corrections to the Saura. Being VM’s own, we cannot verify the numbers used, but we can compare these corrections with those given by the followers of the Āryabhaṭīya belonging nearly to his time. Note how close they are, and commend the tendency to observe and correct, instead of blindly following the masters. The Kerala school following the Āryabhaṭīya gives the well-known vāgbhāva correction: vāgbhāvōnāc chakābdād dhanaśatalayāhān mandavailakṣyarāgaiḥ prāptābhir liptikābhir virahitatanavaś candratattuṅgapātāḥ | śobhānīrūḍhasamvidgaṇakanarahatān māgarāptāḥ kujādyāḥ saṁyukytā jñārasaurāḥ suragurubhṛgujau vajitau bhānuvarjam || (Kaṭapayādi notation is used here.) According to this the corrections per annum are for Mars

• 11.5", for Mercury + 105", for Jupiter – 12", for Venus – 39", and for Saturn + 5". See that these compare well with VM’s. Example: Give the bīja corrections for Mars and Venus at 1,20,553 days from epoch. This is 330 years. The correction for Mars = 330 × 17" (positive) = + 1° 33' 30". The correction for Venus = 330 × 45 (negative) = – 4° 7' 30". Thus bīja-corrected mean Mars of date is 8ʳ 9° 1' 51", and bīja-corrected Venus, 2ʳ 27° 16' 20". [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां सौरसिद्धान्ते मध्यमानयनं नाम षोडशोऽध्यायः ||] Thus ends Chapter Sixteen entitled ‘Saura-Siddhānta – Mean Planets’ in the Pañcasiddhāntikā composed by Varāhamihira

1. Col.: A.B.D. सूर्यसिद्धान्ते मध्यगतिः (B. शनि for गति) (D. gives this as a section colophon). C. इति सूर्यसिद्धान्ते मध्यगतिर्नाम षोडशोध्यायः

Chapter Seventeen SAURA-SIDDHĀNTA — TRUE PLANETS १७. सप्तदशोऽध्यायः सौरसिद्धान्तः — ताराग्रहाणां स्फुटीकरणम् As stated earlier, the computation of the Saura star-planets is continued in ch. XVII, the topics treated being, the True planets, their heliacal risings and their latitudes. [स्फुटकर्म] शीघ्राख्योऽर्कोऽन्येषां भौमादीनां तु परिधयो द्विगुणाः । 'पक्षस्वरा' (श्च खं षड्यमाः)' 'खकृता' श्चकुजादीनाम् ॥ ३ ॥ Epicycles of the planets

1. For the other planets (i.e. other than Mercury and Venus, viz., for Mars, Jupiter and Saturn), the Sun is their Śīghra. The epicycles of equation of the apsis of Mars etc. are twice, 35°, 14°, 16°, 7°, and 30°, (i.e., of Mars 70°, of Mercury 28°, of Jupiter 32°, of Venus 14° and of Saturn 60°.) Note 1. I follow TS's emendation in pañcatriṁśanmanavaḥ. But I read surāḥ as svarāḥ and not śarāḥ, like TS because svarāḥ is nearer the given reading surāḥ, and also 14° is the epicycle given in the Ārdharātrika-pakṣa etc. Five mātrās are wanting in the last foot, and it must be supplied with some such words as bhāgāḥ, as all numbers are already given. But TS make it ṣaḍyutās-triṁśāḥ and, strangely enough, translate it as 24, confusing the addition mentioned by themselves for subtrac- tion. (However, on page XXIII of the Introduction Thibaut gives the correct 30° × 2 = 60°.) Note 2. The first foot is to be read with verses 7 and 8 of chap. XVI where the śīghra of Mercury and Venus have already been given. Properly speaking, the matter in the foot should have been given in chap. XVI. 'रस-भव-वसु-वेदा-र्का' विंशतिगुणिताः कुजस्य दशको (ना) : । मन्दगति नाम (?) भागाः कुजबुधगुरुशुक(सौ)राणाम् ॥ २ ॥
2. 6, 11, 8, 4, 12 multiplied by 20, Mars's being less by 10°, (i.e. 110°, 220°, 160°, 80°, and 240°) are the apogee positions of Mars, Mercury, Jupiter, Venus and Saturn. 1a. B. र्को d. A. सुरास्त्रिंशाः || B. ष्ट्यः सुरास्त्रिंशा ||; C. ष्ट्यः b. ०नां नु परि; D. तु [मन्द] परिधयः [स्युः] | शराः षड्युतास्त्रिंशाः ||; c. D. द्विगुणाःपञ्च० A.त्रिंशत्सनवो D. ष्ट्यः स्वस्त्रिंशा [श्च] ||

XVI.6 XVII. SAURA : TRUE PLANETS 301 Note 1. I follow TS's emendations. Mandagatināmabhāgāḥ does not make any sense. But the meaning is obvious, it must mean apogee positions. Some drastic emendation of the word can be made to give this meaning, but I am against such as emendation. Note 2. These positions agree with those given in the Ārdharātrika-pakṣa etc., as also in the Āryabhaṭīya. The correct positions according to modern astronomy are 128°, 234°, 170°, 290°, and 244°, respectively. Note 3. The apogee 80° for Venus and the epicycle 14° are the same as given for Sun. The apogee position 80° given is near the perigee position of modern astronomy, so far away. We shall explain this under verses 10-11a, below. शीघ्रपरिधावंशाः ‘कृतगुणप(क्ष)’-‘द्विवह्निशीतकराः’ । ‘पक्षस्वरा’ (श्व खंषड्य माः)’ ‘खकृता’ श्वकुजादीनाम् ॥ ३ ॥ 3. The degrees of epicycles of conjunction of Mars is 234, of Mercury 132, of Jupiter 72, of Venus 260, and of Saturn 40. Note 1. I have generally adopted TS's corrections. But the text is corrupt in the third foot, and TS's correction itself wants one mātrā. I would read the third and fourth feet thus: pakṣasvarāś ca kham ṣaḍyamāḥ khakṛtāś ca kujādīnām. This would follow the original work. Note 2. The values agree with the Ārdharātrika-pakṣa, Khaṇḍakhādyaka, and Bhaṭṭotpala-quoted- Pauliśa group, as to be expected. [स्फुटग्रहाः] शीघ्रान्मध्यमहीनाद् (रा)शित्रितये गतैष्यदं(श)ज्ये । भुजकोटी तत्परतः षड्(भिः) प(ति)ते स एव विधिः ॥ ४ ॥ स्वपरिधिगुणिते भाज्ये ‘खर्तुगुणै’ [स्ते] विप(रिण)ते तच्च । कोटिफलं व्यासार्धे मृगकर्क्यादौ चयापच(यम्) ॥ ५ ॥ तद्भुज[कृति]योगपदैर्भा(ज्यं भुजजं ‘ख)सूर्य’घ्नम् । तच्चापार्धं मन्दे हानिधनं शीघ्रकेन्द्रवशात् ॥ ६ ॥ 2a. B.रससंवत्सुवेदाकर्कों (B2.रसंस०) 3a. B.परिधा यथाशा b. B.गुणिता A.क्रजस्य A.दशकोणाः; B.दशकोणस्पणाः b. B.ततदगुण० A.B.C.D.पक्षा द्वि. B1.2.वाहि (B2.°ष्यणाः; B2.स्पदणाः) c. B.om पक्षस्वराः A.स्वराऽऽऽऽ खंवद्य B.खषद्य c. D.गतीनां भागाः B.नाम लाधवं ||कुज d. A.B.मा ख (B3.माःख) C.D.कृताःस्युःकुजादीनाम् d. A.शुक्रसोराणां०

302 PAÑCASIDDHĀNTIKĀ XVII.7 True planets The first step 4. Deduct the mean from the śīghra. If the remainder (called śīghra-kendra) is within 90°, sin. śīghra-kendra is called bhuja, and sin (90° – śīghra-kendra) is called koṭi. If śīghra-kendra is more than 90° and less than 180°, subtract it from 180°. (Taking this as the śīghra-kendra, sin. śīghra-kendra is bhuja and sin (180° – śīghra-kendra) is koṭi. If śīghra-kendra is more than 180° and less than 270°, deduct 180° from it and take this as śīghra-kendra. Sin. śīghra-kendra is bhuja and sin (90° – śīghra-kendra) is koṭi. If śīghra-kendra is from 270° to 360°, deduct it from 360° and take its sine as the bhuja and sin 90° – śīghra-kendra is the koṭi. (The bhuja of manda-kendra is to be found in the same way using manda-kendra in the place of śīghra-kendra)¹ 5-6. The bhuja and koṭi must be multiplied by the planet’s epicycle of conjunc- tion and divided by 360. Thus transformed, they are called bhuja-result and koṭi-result pertaining to the equation of conjunction. If the śīghra-madhya is from 270° to 90°, the koṭi-result is to be added to 120 (the R. of the PS). If śīghra-madhya is from 90° to 270°, the koṭi-result is to be subtracted from 120. Square this and add it to the square of the bhuja-result. Find its square root, and by this divide 120 × bhuja-result. Find arc-sine of this. Subtract half this from the longitude of apsis if the śīghra-kendra is from 0° to 180°. Add if from 180° to 360°. स्फुटयित्वैवं मन्दं मध्याच्च विशोध्य तस्य भुजम् । परिणाम्य कार्मुकार्धं तन्मन्देनैव धनहानी ॥ ७ ॥ Second step 7. Half rectifying the apogee position thus, deduct it from the mean. The result is to be used as the anomaly of the apsis in the second step. As we find the bhuja of the anomaly of conjunction (śīghra-kendra) so find the bhuja of the anomaly of apsis. Multiply the bhuja by the manda epicycle and divide by 360. and get the transformed bhuja-result of the apsis. (This is sine equation of the 4.5. Quoted by Utpala on BS, 2. pp. 44-45 d. A1. षड्भ्याः; B.C.D.U. षड्भ्यः A.B. पतते 4a. U. मध्यविहीनाद् 5b. A. गुणैर्विपगते तच्च; B. गुणे वियुगतक्ष; b. A. वाशित्रितये A.B. गतैष्व (B2.3. ष्य) C.D. गुणैर्विप [रि] णते तच्च (D. णते ते ततश्च) A.B. ॰दंशे ज्ये; U. ॰दंशज्या d. B. कर्कादौ A.B. चयापचयाः; U. चयापचयः c. B. कोटि ———————————————————————————————————————————————————————————————— ¹ In modern usage, for all the above we can simply say sin. śīghra-kendra is the bhuja and cos. śīghra-kendra is the koṭi, without taking into account the sign + or −.)

XVII.9 XVII. SAURA: TRUE PLANETS 303 centre). Find its arc-sine. Add half this arc to the half rectified longitude of apogee if the anomaly of apsis is from 0° to 180° and subtract if 180° to 360°. Thus the apogee is rectified completely. मध्यात् पु(न)र्विशोध्य (त)स्माद्बा[हुर्न] तस्य यच्चापम् । तन्मध्यमे क्षयधनं कर्तव्यं मन्दकेन्द्रवशात् ॥ ८ ॥ Third step 8. Substract this rectified apogee from the mean and thus get the anomaly of apsis. Find its bhuja and multiply it by the epicycle of the apsis and divide by 360°. The bhuja-result, (this is the equation of the centre), is got. Find the arc- sine of this, and subtract the whole of this arc from the mean if the anomaly of apsis is from 0° to 180°, and add it from 180° to 360°. The result is rectified mean. एवं स्फुटमध्याख्यं शीघ्रात् संशोध्य पूर्वविधिनैव । आदिवदा(प्तं) चापं स्फुटमध्या(ख्ये) चयापच(यम्) ॥ ९ ॥ Fourth step 9. Deduct the rectified mean from the śīghra. The anomaly of conjunction is got. Find the bhuja and koṭi of this in the same manner as we did in the first step. Multiply the bhuja by the epicycle of conjunction and divide by 360°. Sine anomaly of conj. is got. Multiply the koṭi, i.e., cos. anomaly of conjunction, by the epicycle of conj. and divide by 360°. The related cosine is got. Add this to 120 if the anomaly is from 270° to 90° and subtract from 120 if from 90° to 270°. Square this, add the square of the bhuja (i.e. equation of conjunction) and find the square root. Divide the equation of conj. × 120 by this square root. The arc sine of this is the result. Add this result to the rectified mean if the anomaly of conj. is from 0° to 180°. Subtract otherwise. The geocentric true planet is got. 6a. A1. तद्भुज; B. तद्भुजयौग             d. B. धनहानिः; C.D. धनहानि b. A. भाजयेन्नभू (A2. भु) जखं;             B 1.2.3. repeat the verse twice and   B. भाजयेन्नभुजखं; C. विभजेद् भुजं फलं खं;       give them two consecutive verse   D. भाजयेत्ततो भुजं ख               numbers.   B. सूर्यघ्न्यः (B2. घ्नः)              8a. B. मध्यासुरो A. पुरो विशोध्य; C.D. पुनर्विशोध्यः c. A.B. तज्झापा (B. या) र्धं             b. A.B.C.D. तस्माद् बाहुं न d. B. शीघ्रं B. वशातात्               9a. a. मध्याख्यां; C. मध्याख्यान् 7a. B. स्फुटयि (B1. पि, B3. यी) त्वैव मन्द          b. D. संशोध्यं b. C. विशोधितस्य; D. विशोध्यं तस्य           c. A.B.D. आदिवदाप्ते B. चाल्पं c. B2. परिणाम B3. कामुकार्धं             d. A.B. मध्याख्योप (B. च) चयापचयः (B. पापचयः)

304 PAÑCASIDDHĀNTIKĀ XVII.9 Note 1. In verse 4, I follow TS's reading, except that I have emended ṣaḍbhyaḥ into ṣaḍbhiḥ, instead TS's ṣaḍbhyaḥ, because my reading allows us subtraction or addition, as is wanted. In verse 5, I follow TS's except in the second foot, where I give the Bṛhatsaṃhitā reading. Either reading gives the same sense. In verse 6, I follow TS's except in the second foot, where I have given bhājyam for vibhajet, as being more likely. But the meaning is the same. In verse 7, the text required no emending, and TS's dhanabāni is unnecessary. In verse 8, like TS I have corrected puro into punaḥ but I have also corrected bāhum into bāhur which is required by grammar. In verse 9, I have corrected madhyākhyām into madhyākhyam, which is the reading of some of the manuscripts. Otherwise I follow TS. Note 2. VM, here, as elsewhere in the PS, uses his tabular sine where R is 120', as given in chap IV. So we must use his tabular values to get the R sines and R cosines. Of course, we may use the modern table, or the Siddhāntic table with R = 3438. But then the R, 1 for the modern tables, and 3438 for the Siddhāntic tables is to be used instead of 120' which is instructed here. (VM uses bhuja to mean sine, and koṭi to mean cosine, instead using the word jyā). Note 3. The method is the same as what is found in the Later Sūrya Siddhānta, with some changes for convenience. But in the matter of the number or order of the steps, the Āryabhaṭīya and the Siddhānta-Śiromaṇi differ. This is because, correctly speaking, the first two steps are useless, and the last two steps alone are necessary. In essence, the third serves to get the true heliocentric position, and the fourth to convert the heliocentric position into geocentric. The earlier steps are in the fond hope of getting correct positions agreeing with observation, while the real trouble is in the inexact parameters followed by the Siddhāntas. Note 4. The second and third steps are merely akin to finding the equation of the centre and applying to the mean. The first and fourth steps are conversion of heliocentric to geocentric posi- tions, neglecting the latitude, which is small and does not affect the result much. The work can be illustrated thus: Epicycle of Conjunction Planet Bhuja Anomaly of Conjunction Q Koti Sun East <— 120' Chapter XVII. Fig. 1

XVII.9 XVII. SAURA: TRUE PLANETS 305 The fig, is for Mercury and Venus, having the Sun as the mean planet. For Mars, Jupiter and Saturn, interchange the planet and the Sun. The geocentric position in both cases is mean planet plus Q. If Q is formed below the horizontal line, it is so much negative and must be subtracted from the mean. Example 1. Find the true, i.e. geocentric, Mars at 1,20,553 days from epoch. Given: Mean Mars, already found with bīja corr. 8ʳ 9° 2′ Śīghra Mars = mean Sun of date = 0ʳ 17° 18′ Aphelion (apogee) of Mars assumed, 3ʳ 20° (for 120° given) Epicycle of apsis = 70°, Apsis of conj. = 234°. First step. Anomaly of conj. = Śīghra − mean = 17° 18′ − 8ʳ 9° 2′ = 128° 16′. This is more than 90° and less than 180°. So, subtracting from 180°, bhujāṃśa is 51° 44′, Koṭiaṃsa = 38° 16′ Bhuja = 94′ 2″. Koṭi = 74′ 17″ Bhuja-result = 94′ 2″ × 234° ÷ 360° = 61′ 14″ Koṭi-result = 74′ 17″ × 234° ÷ 360° = 48′ 17″ As anomaly of conj. is between 90° and 270°, this is subtractive from 120′. 120′ − 48′ 17″ = 71′ 43″. 120′ × bhuja-result ÷ √(71′ 43″² + _bhuja-result_²) = 120′ × 61′ 14″ ────────────────────── √(71′ 43″² + 61′ 14″²) = 77′ 55″. Arc for 77′ 55″ = 40° 30′ ½ arc = 20° 15′ Subtracting from aphelion (since anomaly of conj. is from 0° to 180°), 110° − 20° 15′ = 89° 5′, which is the half-corrected aphelion. Second step Anomaly of apsis = 249° 2′ − 89° 45′ = 150° 17′ This is more than 90° and less than 180°. So, Bhuja degrees = 20° 43′. Bhujā = 42′ 24″. Bhuja-result (i.e. eq. of the centre) = 42′ 24″ × 70°/360° = 8′ 15″ Arc for this = 3° 56′; ½ arc = 1° 58′ This is additive because An. of apsis is between 0° and 180°. Half rectified aphelion + 1° 58′ = 91° 43′ = full rectified aphelion. Third step The corrected anomaly of apsis = 249° 2′ − 91° 43′ = 157° 19′. The bhuja degrees = 22° 41′. Bhuja = 46′ 17″. The Bhuja-result = 46′ 17″ × 70°/360° = 9′ 0″ . Arc of 9′ 0″ = 4° 18′, deductive because an. of apsis is between 0° and 180°. Mean − arc = 249° 2′ − 4° 18′ = 244° 44′ = mean corrected (for eq. centre).

306 PAÑCASIDDHĀNTIKĀ XVII.9 Fourth step An. conj = 17° 18′ − 244° 44′ = 132° 34′. Bhujāṁśa = 180° − 132° 44′ = 47° 26′, Koṭyaṁśa = 42° 34′. Bhuja = 88′ 20″. Koṭi = 81′ 8″. Bhuja-result = 88′ 20″ × 234° ÷ 360° = 57′ 25″. Koṭi-result = 81 8″ × 234° ÷ 360° = 52′ 44″. As an. of conj. is in 270° to 90°, this is deductive from 120′. So, 120′ − 52′ 44″ = 67′ 16″. 120′ × bhuja-result ÷ √(bhuja-result² + 67′ 16″²) = 77′ 32″. Arc of this taken as sine = 46° 16′, additive because an. conj. is from 0° to 180°. So, corrected mean arc = 244° 44′ + 40° 16′ = 285° 0′ = geocentric true Mars. Example 2. Find the geocentric Venus at 1,20,553 days from epoch. Given: The śīghra of Venus = 2ʳ 27° 16′ 20″ = 87° 16′ Mean Venus = Mean Sun = 17° 18′ Aphelion of Venus = 2ʳ 20° = 80°. Epicycle of Conj. of Venus = 260°. Epicycles of the apsis = 14°. First step An. Conj = 87° 16′ − 17° 18′ = 69° 58′. The Bhuja degrees = 69° 58′. Koṭi degrees = 20° 2′. Bhuja = 112′ 42″. Koṭi = 41′ 5″. Bhuja-result = 112′ 42″ × 260° ÷ 360° = 81′ 24″. Koṭi-result = 41′ 5″ × 260° ÷ 360° = 29′ 40″. An. Conj. is between 270° and 90°. So the Koṭi-result is additive to 120′. So, 120′ + 29′ 40″ = 149′ 40″. Bhuja-result × 120′ ÷ √(81′ 24″² + 149′ 40″²) = 57′ 20″. Arc 57′ 20″ = 28° 33′. Half arc = 14° 17′, subtractive to aphelion (as An. conj. is from 0° to 180°) 80° − 14° 17′ = 65° 43′ = Half corrected aphelion. Second step An. of apsis = mean − half cor. aphelion = 17° 18′ − 65° 43′ = 311° 35′. The Bhuja degrees are 48° 25′.. Bhuja = 89′ 44″. Bhuja-result = 89′ 44″ × 14° ÷ 360° = 3′ 29″. Arc. 3′ 29″ = 1° 40′. Half arc = 50′, subtractive, as an. conj. is from 180° to 360°. Corrected aphelion = Half-corrected aphelion − 50′ = 65° 43′ − 50′ = 64° 53′. Third step An. of apsis = mean-corrected aphelion = 17° 18′ − 64° 53′ = 312° 25′ Bhuja degrees = 47° 35′. Bhuja = 88′ 33″. Bhuja-result = 88′ 33″ × 14° ÷ 360° = 3′ 27″. Arc sine 3′ 27″ = 1° 39′, additive as an. of apsis is 180° to 360°. So, mean Venus + arc = 17° 18′ + 1° 39′ = 18° 57′, is the eq. cent. corrected mean.

XVII.10 XVII. SAURA: TRUE PLANETS 307 Fourth step An. conj. = Śīghra-corrected mean = 87° 16′ − 18° 57′ = 68° 19′. Bhuja degrees = 68° 19′. Koṭi degrees = 21° 41′ Bhuja = 111′ 29″. Koṭi = 44′ 19″. Bhuja-result = 111′ 29″ × 260° ÷ 360° = 80° 31′ Koṭi-result = 44′ 19″ × 260° ÷ 360° = 32′ 0″. As an. conj is from 270° to 90°, additive to 120°. So, 32′ 0″ + 120′ = 152′ 0″. Bhuja-result × 120 ÷ √(Bhuja-result² + 152²) = 62′ 12″. Arc sine = 62′ 12″. Half arc = 31° 14′, additive as an. conj is 0° to 180°. Geocentric true Venus = 18° 57′ + 31° 14′ = 50° 11′. In verse 11 below, VM requires us to subtract 67′ or 1° 7′ constant, as bīja-correction, after all work is over. So, geocentric True Venus = 50° 11′ − 1° 7′ = 49° 41′. [बुधशुक्रयोः विशेषक्रिया] सर्वे स्फुटाः स्युरेवं ज्ञस्य तु शीघ्राद्विहाय रविमन्दम् । रविपरिधिनतं बाहुं (बुधेऽर्कवत्) क्षयधने कुर्यात् ॥ १० ॥ शुक्रस्य सप्त(षष्टि)र्लिप्ताः शोध्याः स्फुटीकृ(तस्यै)व । Special work for Mercury and Venus 10. All star-planets are (geocentrically) made true in the above manner. But in the case of Mercury, this additional work is to be done: Subtract its apogee from the śīghra and, using the Sun’s epicycle, find the bhuja-result and apply it to the mean Mercury (which, of course, is the same as the Sun’s), with the addition or subtraction done, as the Sun’s bhuja-result is additive or subtractive. 11 a-b. From Venus, subtract 67′, constant, after all the earlier sphuṭa work instructed has been done. Note 1. The reading kṣayadhane is better as it is, and TS need not have corrected it into kṣayadhānam. The reading budhavat is deficient by two syllables, Lalla’s reading budherkavat supplies these and makes the meaning more clear. So I have adopted it. TS’s emendation budhaphalavat, with phala added for the two mātrās wanting, is not different in meaning from budhavat. Being an arbitrary rule, we cannot decide which gives the original meaning, budherkavat or budhavat. But since Lalla’s reading is not defective, at least as far as the mātrās are concerned, I have adopted it. 10a. B. omस्फूटः B. कोरेवं           11a. A. व्यष्टिः; B. व्यष्टि b. A2. ज्ञस्य पुर्शी; B. ज्ञेक्यं धुशीघ्रा; D. ज्ञेइयेषु शी    b. A.B. शोध्या A. स्फुटिकृतस्यैव; B. स्फुटितस्यैव c. A2. बाहु; B. वादं              (B2.3. ॰त्त तस्यैव) d. A.B. बुधवक्ष; C. बुध्फलवत्; D. बुधे (क) व(ौ) क्षयधनं

308 PAÑCASIDDHĀNTIKĀ XVII.11 Also, it is clear that it is not a substitute for any of the four steps because, if so the separate epicycle for Mercury will be useless. Note 2. It is clear that the rules given here are VM's own, to secure, in his opinion, better agree- ment with observation, because they are not given in the Ārdharātrika-pakṣa etc. and the original four steps are all in line with them, as also the modern Sūrya-Siddhānta and the Siddhānta Śiromaṇi. Note 3. The whole work of finding the true positions, especially of the star-planets is defective in Hindu astronomy, in that the equation of the centre of Hindu astronomy neglects the second, third, etc. terms, which is considerable in the case of the Moon, Mars, Saturn and Mercury, in which last case the second term is as large as 3°. In the case of Mercury and Venus it is applicable to the Sun, instead of their Śīghra which is really their mean. In the equation of conjunction, the Sun's true distance from the earth and true longitude should be used, instead of the mean distance and mean longitude, as is done in Hindu astronomy. On account of these defects, computation does not agree with observation, and all sorts of hotch-potch rules are given in different astronomical works. The disagreement among themselves would itself show that they are beside the mark. When these defects are remedied, the third and fourth steps alone would be necessary, the third step giving the heliocentric true planet; and the fourth step converting the heliocentric position to the geocentric. Note 4. In the case of Venus, there is another kind of defect. Its maximum eq. of cent. being small, it is confused with the Sun's, and the Sun's epicycle and apogee are given to Venus also. While its aphelion position is 290°, according to modern astronomy, its apogee is given as 80°, the same as the Sun's. Table of Heliocentric Star-planets at epoch. (For mutual comparison)

• This needs explanation: Perhaps the reading is śūnyāśvi in Sūrya Siddhānta I.31, which will reduce the degrees by 12. But the commentator Raṅganātha takes it as śūnyartuḥ.

XVII.12 XVII. SAURA: TRUE PLANETS 309 Table of Synodic periods of the Star-planets

310 PAÑCASIDDHĀNTIKĀ XVII.13 Heliacal rising of the planets 12. The heliacal rising and setting of the Moon, Mars, Mercury, Jupiter, Venus and Saturn are when their elongation (from the true Sun) are 12°, 17°, 13°, 11°, 9°, and 15°. Note 1. I generally adopt TS's readings. But Śaśi is extra, and evidently a mistake which has crept into the reading. To make up for this they have removed rudra which is necessary, and this emen- dation has spoiled the correct agreement with other siddhāntas. Note 2. These are time-degrees, i.e. time expressed in degrees (kālabhāga) and are arbitrary in essence, and depend on the keennees of the observer's eyesight, as also the atmospheric conditions. The later Sūrya-Siddhānta gives 10° and 8° for Venus at superior and inferior conjunctions, and 14° and 12° for Mercury, respectively, while the Sūrya-Siddhānta here and some others give the mean of each. (The Mahābhāskarīya gives even 4° or 4 1/2° for Venus at inferior conj. and 8° at superior conj.) [ग्रहविक्षेपाः] मन्दग्रहान्तरज्या स्वाष्टांशयुतार्कजीवशुक्राणाम् । सौम्यान्ययोः पदो(ना) विक्षेपोऽन्यश्च शीघ्रविधौ ॥ १३ ॥ गुरुभूतनयाऽऽस्फुजितां पादोना (ज्ञयमयोस्तु सा)ष्टांशाः । त्रिज्याघ्नी कर्णाप्ता (वि)यो(गाश) विक्षेपः ॥ १४ ॥ Latitudes of planets 13. Add one eighth of itself to the R (120') sine of (mean planet − apogee), in the case of Saturn, Jupiter and Venus. For, the two others, (i.e., Mercury and Mars), subtract one fourth of itself. (This is one part of latitude). There is another part of latitude using the Anomaly of conjunction. 14. From the R sine anomaly of conjunction of Jupiter, Mars and Venus sub- tract one fourth of itself. From that of the rest, (viz., Mercury and Saturn) add an eighth. Add both algebraically and note the direction, north or south. Multiply this by R (i.e. 120') and divide by the hypotenuse got in the last step. The latitude is got, its direction being that of the noted direction. 12a. A. स्फुट दिनकरांतरांतरांशाः; c. B. सौम्यपन्योः; B. स्फुरदिनकरांतरांतराशाः D. सौम्यारयोः A. पदोनां; B 1.2. पदनां b. A. च दर्शनी ज्ञेयाः; B. व दर्शज्ञेयाः d. A. विक्षेपोप्यश्च c. A.B. विशातिरु (B. रू) ना 14a. B. गुरुभूततया d. A.B.C. वसु शशिशिखि A. रुद्रेदियैः; B. रुद्रेद्रिणैः; b. B. पदोना A. ज्ञयममयोमुशांष्टांशाः; B. ज्ञयमयोमुष्ट्यंशां C. नवकेन्द्रियैः C. ज्ञमयोश्च C.D. साष्टांशा 13a. B. ग्रहातरज्या (B2. नरज्या, B3. तरुज्या) c. B. कर्णप्ता A. नियोगयोशम विक्षेपः; b. B. स्वाष्टांशसुर्किजीव०; D. युता कुजेज्यशुक्राणाम् B. नियोगशास विक्षेपः; C. वियोगजाशः स विक्षेपः; D. वियोगयोगः स विक्षेपः

XVII. 14 XVII. SAURA: TRUE PLANETS 311 Note 1. This is a peculiar primitive way of finding the latitude of the star-planets. It is not found in the allied Khaṇḍakhādyaka and the quoted part of the Bhaṭṭotpala-quoted Pauliśa. It is found in Āryabhaṭa’s Ārdharātrika-pakṣa given in the Mahābhāskarīya (VII. 28-33). But there are some differ- ences between the two, and we cannot decide which follows the original Saura here, and which has slightly modified the original. They both mention two kinds of latitudes for each star-planet which are to be added algebraically. But there is a difference in the maximum latitudes, and in the ascending nodes to be subtracted from the mean longitudes or śīghras. VM’s Saura implies the max. latitude 90′, 90′, 135′, 135′, and 135′, for Mars, Mercury, Jupiter, Venus and Saturn, respectively, to be multiplied by sine anomaly of conjunction, and 90′, 135′, 90′, 90′, and 135′ to be multiplied by sine anomaly of conjunction, no separate node being given, which means that the apogee itself is the node for one kind of latitude, and the mean planet itself for the other. But the Ārdharātrika gives only one set of maximum latitudes for both, viz., 90′, 120′, 60′, 120′, 120°. It gives the nodes, 20°, 40°, 70°, 260°, and 150° for the former and 20°, nil, 70±, 260° and 150° for the latter. Govindasvāmi's Bhāṣya on the Mahābhāskarīya, being meagre, does not help us. Note 2. By implication, we had better take the arguments of the eq. cent. used in the third step for the former, and the anomaly of conj. used and hypotenuse obtained in the fourth step for the latter. Example: Find the latitude of Mars at 1,20,553 days from epoch. In the third step of the earlier example, the sine of the argument of eq. conj. is 42′ 24″. As it is Mars, deducting quarter of itself, the latitude is 31′ 48″, north, as this argument is between 0° and 180°. In the fourth step, the sine of the argument of conj. is 88′ 20″. For Mars one fourth is to be subtracted. So, the latitude due to this is 66′ 15″, again north, since the argument is from 0° to 180°. Adding, 31′ 48″ + 66′ 15″ = 98′ 3″, north. The hypotenuse obtained there, in the fourth step, is 88′ 52″. 98′ 3″ × 120 ÷ 88′ 52″ = 132″ north, is the true latitude of Mars for the day. (As it is, this is far from the latitude obtained from using the later Siddhāntas.) Note 3. The treatment of Saura is taken up by VM in XVIII. 57-60, where the latitude found here is used to correct the mean elongation given in verse 12, for the heliacal setting and rising. The exposition of these verses is given in XVIII. [इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां ताराग्रहस्फुटीकरणं नाम सप्तदशोऽध्यायः ||] Thus ends Chapter Seventeen entitled ‘Saura-Siddhānta – True Planets’ in the Pañcasiddhāntikā composed by Varāhamihira Col.: A.B.D. ताराग्रहस्फुटीकरणं षोडशोऽध्यायः; C. इति ताराग्रहस्फुटीकरणं नाम सप्तदशोऽध्यायः

Chapter Eighteen (VĀSIṢṬHA-) PAULIŚA-SIDDHĀNTA — RISING AND SETTING OF PLANETS अष्टादशोऽध्यायः वासिष्ठ-पौलिशसिद्धान्तौ — ग्रहोदयास्ताधिकारः Introductory Chapter XVIII of PS (NP’s ch. XVII) deals primarily with the star-planets according to the Vāsiṣṭha and Pauliśa siddhāntas. The true motion of the planets is traced from one heliacal rising to the next. The method of getting the true anomaly of the equation of the centre is similar to that of the Moon given by the Vāsiṣṭha in ch. II and based on the same theory of the rate of motion, forming a linear zigzag. Even the same technical term pada is used here. All these are reminiscent of the Babylonian astronomy of the Selucid period. The text as available too is not very pure and this too has made the interpretation of this chapter difficult for TS. In my paper on ‘Some misinterpretations and omissions of Thibaut and Sudhakara Dvivedi in the PS of VM’ (VIJ., 11 (1973) 107-18) I have indicated the errors occurring in the said publication. NP have improved upon TS’s interpretation in some places, but have committed worse mistakes in other places. TS and NP have also failed to understand how the equation of the centre has been computed and applied to the equation of con- junction of Jupiter and Saturn. Since they had to interpret, in some way or other, the related verses, they have altered the verses, in all sorts of ways, to yield what they thought the meaning might be. Even in the case of Venus, where computation has been made simpler by neglecting the small equation of the centre, they have committed several errors which have been pointed out in the exposition of these verses given below. [शुक्रचारः] हित्वा ‘(जलमुनि) चन्द्रा (न्)’ द्युगणाद् ‘वेदाष्ट-भूत’हृतलब्धाः | शुक्रोदया गुणाप्तैः सार्धाः पञ्चालिनो भोगाः || १ || कन्यांशा [न्] षड्विंशतिमित्वा शुक्रोऽपरेण या [त्यु] दयम् | उदयैकादशभागान् दिनेषु दत्वा तत [श्चा] राः || २ || Motion of Venus

1. Subtracting 174 from the days from epoch, the quotient got by dividing the remainder by 584 are the heliacal risings of Venus. Its motions during the periods is 7ʳ 5° 30' 20" each.
2. Having gone to 26° of Virgo, i.e. at, 5ʳ 26°, Venus rises in the west (for the first time after epoch). Adding an eleventh of the quotient (given in verse 1) to the remaining days, the motions (are to be taken from the Table given in verses 3-5).

XVIII.3 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 313 174 days after epoch, Venus rises heliacally in the west. We have emended munijala as jalamuni because 147 will not agree with the longitude of Venus at rising given as 5ʳ 26°. If Venus is 5ʳ 26°, the sun must be 5ʳ 18°, to satisfy the 8° given for Venus's heliacal risng in verse 58. But since at epoch the sun is in the neighbourhood of 358°, it can be only near 4ʳ 25° after 147 days, i.e., 38° from Venus. After 174 days from epoch, the Sun would be near 169° according to the Vāsiṣṭha- Pauliśa. (Vide my paper ‘The epoch of the Romaka etc.’, Indian Journal of History of Science, 13 (1978) ii. 155-58). This gives the elongation as 7° instead of the required 8°. But this small discrepancy can be put to an accumulated error in computing from the original. TS have not felt the need for this emendation because they have emended kanyāṃśān into kālāṃśān and omitted the necessary kṣepa viz. the constant equal to 5ʳ 26°, for beginning the motion. They have not understood the meaning of the word kālāṃśān which they have brought in. It is the same as the 8° mentioned above, for Venus. NP have interpreted kanyāṃśān correctly. But since they have kept 147 days to be deducted intact, they find a serious discrepancy expressed by them on page 124 of Part II. However, they derive satisfaction from the fact that the September 10th position of the sun would agree with the time of Venus's rising and their longitudes. This at least must have shown them that the rising takes place only more than 20 days later. They have made all sorts of unnecessary emendations, but they have failed to do this necessary one. We can infer from the instruction to add an eleventh of the quotient, that one synodic revolution takes 583 10/11 days. In this period the sun has moved 1 revolution 7ʳ 5° 30′ 20″ and Venus, 2 revolutions 7ʳ 5° 30′ 20″. From this we can infer that the sun takes 365-15-25 days for a sidereal revolution. From the methods given for the other planets also we can see that the sidereal period of the sun used is c. 365-15-30, which is an evidence for the system given here being connected with the Pauliśa also, as in the case of the Moon. TS have unnecessarily emended bhogāḥ into bhāgāḥ and taking sārdhāḥ to mean ‘together with’, instead of ‘with half’, have given a motion of 7ʳ 5° 20′ per synodic period. They are unaware that this would make the sidereal year c. 365-22 days, so wrong. The error of 10′ 20″ in Venus would accumulate by 1° every nine years. NP interpret sārdhāḥ correctly, but take guṇāptaiḥ to mean ‘with 1/3 degree’ and given 7ʳ 5° 50′, which would make the sun's sidereal year c. 365-3-0, so very wrong. By this the error in Venus would accumulate by 1° in every 5 years. Next the days for segments of motion in the synodic cycle are given by verses 3-5. षष्टि[र]येण ῾वेदाग्नि-यम-᾿युतामंशसप्ततिं भुङ्क्ते | (अर्था)ष्ट[कैर्द्वा] (त्रिंशतं) विंश(त्या) [वि]त्रिभिः सपा(दार्थान्)᾽ || ३ ||

1. A.B.C.D.मुनिजल (B. जाल). A.B. चंद्रा 2a. A.D. कन्यांशाः; B. कन्याशाः;. C. कालांशाः b. B. ॰ष्टमूत. A2. दूत corrected to हत; B. हत B. षड्वति c. A. शुक्रेन्द्या. C. गुणांशौ:; D. गुणांशाः b. A1. यात्पदयं; A2. यात्पदयं d. A. सार्द्धा; B. सार्धा c. B. उदैका. C.D. भागं B1.3. पंचलिनो; B2. पञ्चालिनो. C. भागाः; d. A.C. सतस्ताराः;. D. ततश्चवारः D. भोगः

314 PAÑCASIDDHĀNTIKĀ XVIII.5 वक्रमतस्तिथिभिर्द्वौ पञ्चभिरेवं ततोऽपरास्तमितः । दशभिः प्रागुदितः स्यान्नखैश्च जलधीन् मि(तान्) गत्वा ॥ ४ ॥ अनुवक्री परिगत्वा विपरीतमस्तमैत्यै(न्द्रयाम्) । षष्ट्यांशपञ्चसप्ततिमित्वाऽपरतो भृगुर्दृश्यः ॥ ५ ॥ ॥ वसिष्ठसिद्धान्ते शुक्रः ॥ Days for segments of motion in the synodic cycle 3. In three periods of 60 days Venus moves 74°, 73°, 72°, respectively. In 40 days it moves 32° and in 17 days, 5 1/4° 4. From here retrograde motion (begins). In 15 days these are 2°; in 5 days the same, i.e., 2°. Then, setting in the west, it rises in the east after ten days. Venus is in follow-up retrograde for 20 days, moving 4°. 5. Then continuing the (direct) motion round, in the order of days and motions reversed, Venus sets in the east. Then moving 75° in 60 days, it becomes visible in the west. The argument to be used in the above table of motions are the days left over, together with the eleventh of the quotient, as mentioned. It can be seen that in my emendations of some of these I have done very little violence to the text. I have been guided in these by the actual motion that must have been observed, putting it to observational or other error, where the numbers are clear, but deviate from the actual. The ratio of Venus's distance from the sun to the earth is c. 0.72 given by all Hindu and modern astronomy, and this I have used to compute the segments of actual motion for comparison. (See Table on next page) Another guide is that the days and motions from the superior to the inferior conjunction must add up to half of the whole, i.e., 292 days, and 287 3/4°, since the equation of the centre has been dispensed with. 72° motion for the third 60 days is about 4° in excess. For the next 40 days the motion has to be 36°, and I could have filled up the lacuna by (khaissat) instead of (khairdvā) to get this. But the Siddhānta seems to have compensated the earlier 4° excess by the 4° defect here, which of course is an error. From this we can see that the emendation of 'dviṁśat' into 'triṁśat' is necessary lest the motion be reduced to 26°, which is too small. 3a. A.B.C.D. षष्टित्रयेण B. सेप्ताग्नि B. मंशसप्त | ति; B. साप्तति भुक्ते c. A1. अर्धाष्टकेन विंशति (A1. om केन); B. अर्धाष्टकेन विंशातं (B2. विंशति) C. अर्धाष्टकेन सप्त; D. अर्धाष्टकविंशत्या d. A.B. विंशत्यैस्त्रिभिः सपादांशं; C. सप्तत्यंशांस्त्रिभिः सपादांशम्; D. विंशत्यं [शक्र] स्त्रिभिः सपादांशम् 4a. B. वक्रमतं क्षितिभिः d. B. स्रान्तरवैश्च A. मिता गत्वा; B. मिता गता; D. सितो for मितान् 5. B. verse missing 5a. C. दन्तक्वैः for परिगत्वा b. A1. मस्तमत्यैड्य्यां; A2. मस्तमत्यैड्यं; C. रवशरयमानस्तमेत्यैन्द्रयाम्; D. विपरीतं चास्तमेत्यैन्द्रयाम्