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**Typed and Proof-read by Smt. R. Shakuntala, India**

The earliest book in antiquity on Hindu Astronomy is Vedanga Jyothisha. The Hindu astronomers refrained from making alterations of some of the readings of Vedanga Jothisha, though necessary, lest they may be labelled irreligious. But by using the pseudonyms like Brahma Siddhanta, Varaha Mihira in his Pancha Siddhantika, spoke lightly of Vedanga Jyothisha.

Perhaps Varaha took recourse to this indirect attack to avoid the calumny of the Vedic priests. Brahma Gupta later openly criticised Vedanga Jyothisha with impunity under the impression that the authors of Jyothisha Samhita and Vedanga Jyothisha were no divine personages. Had the Hindu astronomers considered the subject-matter of Vedanga Jyothisha in the ancient past, from the mathematical stand-point, setting aside the religious prejudices, they could have made the emendations of all the verses and supplied the ellipses (adhya haras) in their interpretations, wherever needed. Unfortunately that was not done.

Europeans devoted themselves to critical study of Vedanga Jyothisha as they did of the Vedas. Some of them were Webber, Whitney, Colebrooke and others. Dr. Thibaut brought out his own contribution of Vedanga Jyothisha in 1877. Among the prominent English-educated Hindu interpretors were Shankara Balakrishna Dikshit and Lala Chottelal (non deplume, Barhaspatya). Pandit Sudhakara Dvivedi contributed his Sanskrit commentary and edited Somakara Bhashya of Yajusha Jyothisha. Vedanga Jyothisha does not contain any account of Graha gananam or Grahana gananam.

It gives us an account of lunar days, parvanas, semesters, seasons etc. It is simply called the positional astronomy. Still dissensions of interpretations do exist, each claiming supremacy over the other. Vedanga Jyothisha was brought to light in two texts viz., Aarcha Jyothisha and Yajusha Jyothisha the authors of which are not known. But from the second verse of Aarcha, we come to know the exponent of Vedanga Jyothisha was Lagadha Maharshi. The Aarcha text contains 36 verses whereas Yajusha text contains 43 verses. Since 36 verses are practically common to both, these two texts are conjectured to have been authored by one and the same person.

Now let me present before esteemed readers a few verses of Vedanga Jyothisha to show how our Hindu scholars strangely differed in their interpretations.

** पंच संवत्सरमयं युगाध्यक्षम् प्रजापतिम्। दिनऋतु अयन मासाम्गम् प्रनम्य शिरसा शुचिः॥ paṁca saṁvatsaramayaṁ yugādhyakṣam prajāpatim | dinaṛtu ayana māsāmgam pranamya śirasā śuciḥ ||**

This is the first verse of both the texts. Pandit Sudhakara Dvivedi made an emendation of the above as

**पंचसंवत्सरमया युगाध्यक्षम् प्रजापतिम्। paṁcasaṁvatsaramayā yugādhyakṣam prajāpatim |**

The sensible meaning implied in the line is the lord of 5 year-Yuga, but not the lord who is 5 years. The verse is unambiguous in the point of the exposition of a luni-solar astronomical cycle of lustrum, and all our past scholars were of this opinion only. But recently an author of Vedic Astronomy tried to establish a luni-solar cycle of 19 years based on Aarcha text. In fact it is a Metonic cycle, since a Greek astronomer by name, Meton discovered it in 432 B.C. which was found defective by our Hindu astronomers.

Let us know the basis of both the cycles. A month has 30 tithis. For 12 months of a lunar year we get 30 x 12 = 360 tithis. The lunar year (12 synodical months) has about 354 days. The solar year has 365.25 days. The difference in days between a solar year and a lunar year is 11.25 (365.25 – 354); this comes to 11.25 tithis. It means 11 tithis are completed and the part of the next tithi is operating. Hence total number of tithis in integers is 12. Thus the total number of tithis in a lunar year is 360 + 12 = 372. So for every solar year, the 12 tithis go on adding and for a lustrum of 5 years, there will be 60 tithis in excess which come to mean two intercalary months in addition (two adhikamasas). But the basis of a 19 year cycle is different. The luni-solar year difference of 11.25, mentioned above is taken as 11 leaving away the fraction 0.25 of the next tithi. So at the rate of 11 tithis per year, the number of tithis for a lustrum will be 5 x 11 = 55. This means in a lustrum of 5 years, the excess is one month and 25 days. So we will get only one intercalary month in addition and an incomplete month of 25 tithis in addition in 5 years. This is a paradox. But for 19 years, we get 11 x 19 = 209 tithis. This would come to 6 months and 29 tithis (days) as intercalary months in addition in a cycle of 19 years. So within a cycle of 19 years no full intercalary months in addition are probable.

Let us see:

For 5 years = 5 x 11 = 1 month 25 days For 10 years = 10 x 11 = 3 months 20 days For 15 years = 15 x 11 = 5 months 15 days For 19 years = 19 x 11 = 6 months 25 days

But in case of luni-solar cycle of lustrum two full Adhikamasas are probable. I leave it to the scholars to decide whether a cycle of 5 years or a cycle of 19 years bears the propriety.

The following is the 6th verse in Yajusha (the 5th in Aarcha):

**स्वरक्रमेते सोमार्कौ अदासकं सवासवौ। स्यात्तदादियुगं माघसप्ताः सुक्लोयानम् ह्युदक्॥ svarakramete somārkau adāsakaṁ savāsavau | syāttadādiyugaṁ māghasaptāḥ sukloyānam hyudak ||**

This verse means that when the Moon and the Sun arrive at the Dhanishta asterism in the heavens (ecliptic), the luni-solar astronomical cycle of lustrum begins in the bright half of Maagha Masa, known as Tapa, the period of summer solstice. The conjunction of the Moon and the Sun need not be close. Somakara supplied the eclipses tatrayadi Brihaspati raste (if there be Jupiter) without quoting any authority. Barhaspatya meant Tapa as winter season. But Sudhakara Dvivedi did not feel its propriety, since Maagha and Tapa mean the same. This means Maagha Masa, known as Tapa. Besides Maagha and Phalguna months constitute winter season according to sruti tapamcha tapasyascha saisira vritoo. However this dissension matters little. But the protagonist of 19 year cycle, retained the following corrupt verse and tried to interpret in his own way.

**स्वरक्रमेते सोमार्कौ अदासकं सवासवौ। स्यात्तदादियुगं माघसप्तासुक्लो दिनांत्यजः॥ svarakramete somārkau adāsakaṁ savāsavau | syāttadādiyugaṁ māghasaptāsuklo dināṁtyajaḥ ||**

His translation of the above verse runs as follows:

When the Sun and the Moon are on both sides (with the Sun) at Vaasavanakshatra in the heaven, the cycle begins with Full Moon day in Maagha month in winter season. But he could not succeed in what he wanted to derive even from the above defective verse for the following reasons. Somarkau and Savasavau are connected both being in dual number. So both the Sun and the Moon must be in Dhanishta in which case Full Moon is unthinkable. The meaning of the Full Moon deduced from Shuklodinantyajah is not correct. It is not Dinantyajah but it may be Dinaantyajah or Dinaantyayah in either case of which meaning would be evening.

The meaning got from Swararakameke is incorrect and incoherent. How Arka (Sun) is connected to Eke is absurd. He cites simple grammatical rule that Eke indicates a dual nominative case. But one should know that Eke is plural when it means ‘some’ and it can never be dual. The singular is Eka and it changes according to gender. Let us go to the next verse.

**प्रपद्येते स्रविष्ठादौ सूर्य चन्द्रमस वुदक्। सर्पार्धे दक्षिनार्कास्य माघ श्रवणयोस्सदा॥ prapadyete sraviṣṭhādau sūrya candramasa vudak | sarpārdhe dakṣinārkāsya māgha śravaṇayossadā ||**

The verse means that when the Sun and the Moon arrive at the beginning of Dhanishta, it is Uttarayana and when they arrive at the middle of Aslesha, it is Dakshinayana. See the Ravibhaganas etc., which are the same in Vedanga Jyothisha and Pancha Siddhantika.

1 Yuga = 5 solar years (Ravibhaganas) Solar months = 60 (5 x 12) Solar days = 1800 (60 x 30) Lunar months = 62 Lunar days = 1860 Kshayahas = 30 The solar days from one ayana to another = 360/2 = 180 Each Nakshatra will be 13°1/3 The number of Nakshatras to one semester (ayana) to another semester (ayana) = 180/131/3 = 3 x 180/40 = 27/2 = 131/2.

So counting from Dhanishta the 131/2 Nakshatras, we arrive at the middle of Aslesha which begins Dakshinayana. Thus it is found that the Sun from the beginning of Dhanishta constitutes Uttarayana and from the middle of Aslesha in Cancer, Dakshinayana. The following is the 9th verse in Yajusha or the 8th verse in Aarcha.

**प्रथमं सप्तमं चहुरया नद्यं त्रयोदश। चतुर्थं दशमं चैव दैर्युग्मद्यं बहुळे प्र्युतौ॥ prathamaṁ saptamaṁ cahurayā nadyaṁ trayodaśa | caturthaṁ daśamaṁ caiva dairyugmadyaṁ bahuḻe pryutau ||**

Dwigunam takes place of Pradhamam in Aarcha text. Somakara explained the above verse taking ‘ritu’ to mean Kaala. Barhaspatya’s explanation agrees with that of Somakara, but Sudhakara Dvivedi gave a sensible meaning of ‘ritu’ as six because of the seasons being six. This is justified by the following explanation. The number of tithis in a solar year is 372 as shown before. So for a semester of six months the number of tithis is 186. This when divided by 30, gives the ritu sesha of 6 tithis which exceed from the tithi to the other, from one semester to another semester. They are 1st, 7th, 13th, 19th, 25th, 1st, 7th, 13th, 19th, 25th. The 4th tithi in the dark fortnight is called the 19th tithi and the 10th tithi in the dark fortnight is the 25th tithi. Dwigunam in the verse of Aarcha does not convey any coherent sense. Dwiryugmam is a better reading than Dwiryugmadyam of one syllable more not needed in Anustup metre. The following is the 13th verse in Yajusha or 4th verse in Aarcha.

**निरेकं द्वादशभ्यास्तं द्विगुणं गतसंयुक्तं। षष्ठ्याषष्ठ्या युतं द्वाभ्यां पर्वानां राशिरुचयते॥ nirekaṁ dvādaśabhyāstaṁ dviguṇaṁ gatasaṁyuktaṁ | ṣaṣṭhyāṣaṣṭhyā yutaṁ dvābhyāṁ parvānāṁ rāśirucayate ||**

Somakara’s explanation of the above verse was accepted for fixing the time (Parva) but it did not stand the test applied according to astronomical calculations. Dikshit explained as follows:

Reduce the number of current year by one. Multiply the remainder by twelve and add the elapsed months of the current year. Double this total which should be further increased by two for every sixty of the total doubled. The result thus obtained will be the total number of Parvanas elapsed from the beginning of the lustrum to the end of the current year.

Suppose we require to know the total number of Parvanas elapsed since the beginning of the lustrum upto and including the 10th month or the 20th Parvana of the 4th year. Then according to Dikshit, the number of Parvanas

= ((4 – 1) x 12 + 10) x 2 + 2 = (3 x 12 + 10) x 2 + 2 = 94

But Barhaspatya supplied the ellipsis ‘parva’ instead of month multiplied by 12 and then by 2 before adding Parvanas. He therefore, remarked, “I need not quarrel with Mr. Dikshit though I consider my own explanation better”. So according to Barhaspatya the number of Parvanas

= (3 x 12 x 2 + 20) + 2 = 92 + 2 = 94

But by this formula, the result differs.

Formula = ((x – 1) x 12 x 2 + Y) (1 + 2/60) x = 4 years, y = 20 Parvanas = ((4 – 1) x 12 x 2 + 20) x (1 + 2/60) = 92 (1 + 2/60) = 92 + (92 x 2/60) = 92 + 3 – 1/15 = 95 + 1/15

But according to Sudhakara Dvivedi the interpretations of Dikshit and Barhaspatya are wrong, since they have wrongly interpreted Shashttya shashttya yutam dwabhyam.

According to Dvivedi (4 – 1) x 12 x 2 + 20 + integral number of 2 (3 x 12 x 2 + 20)/60 = 92 + integral number of 2 x (92/60) = 92 + 3 = 95

But the reading of the above verse in Aarcha text is as follows:

**निरेकंद्वासशार्धाब्दं द्विगुणं गत संञितम् nirekaṁdvāsaśārdhābdaṁ dviguṇaṁ gata saṁñitam**

The word dwadasardhabdam does not qualify any coherent sense in the method of calculating Parvanas. Yet a fanciful interpretation quite independent of the verse has been brought out recently ignoring the valid emendation, dwadasabhyastam. I have presented so far, the dissonant notes on the hymns of Vedanga Jyothisha by sthalee pulaakanyaaya