Vedang Period -3

 Astronomical Mentions in Other अङ्गः of Vedas

कल्पः(Manual of Rituals)

कल्पः has various सूत्र (सूत्र is a large collection of such aphorisms in the form of a manual, it is a distinct type of literary composition, based on short aphoristic statements, generally using various technical terms.सूत्र forms a school of Vedic study).

 Types of कल्पःसूत्र are

1. श्रौतासूत्र has यज्ञ Rituals

2. गृह्यासूत्र- has Domestic Ceremonies

3. धर्म सूत्र-has -Religious and Social Laws

4. शुल्बसूत्र-has Geometry and Mensuration 

सूत्र with some important astronomical mentions are

अश्वलायन श्रौतासूत्र

Names of months based on nakshatra and seasons are found in this text.  The reference to seasons indicates that spring was considered the year's first season.

Tithi is not mentioned specifically ,however use of words चतुर्दशी, अष्टमी indicate Tithis. Like in this verse 

मार्गशीर्ष्यं प्रत्यवरोहणं चतुर्दस्यम्  - गृह्यासूत्र 2-3-1, mentions Month and Tithi

ध्रुवमरुन्धतिं सप्तर्षिनिति दृष्ट्वा वचम्विसर्जेता |   गृह्यासूत्र 1-7-22

This gives out the names of the Pole star (Dhruva), The Great Bear (Saptrishi) and Arundhati (Alcor)

Definite instructions are given for doing certain rites on auspicious Nakshatras,for example,fields should be ploughed on उत्तरा भद्रपद, फल्गुनि, रोहिणी नक्षत्र.

Without any weather forecast, but with a sound knowledge of the onset of seasons, it is a communication to the general public to commence sowing and other agriculture activities.

नक्षत्र names are mentioned to carry out यज्ञ Rituals, thread ceremonies, etc. However, the auspicity/ gender of the नक्षत्र is not mentioned.

Taittiriya Sruti gives out the list of नक्षत्र with their controlling Deities' gender and number, which is still followed as of date for all rituals.

परास्कारःगृह्यासूत्र

This सूत्र is of a later date than अश्वलायन सूत्र and appears to be an updated/revised सूत्र. This सूत्र gives out the list of नक्षत्र auspicious for marriage. These are उत्तरा हस्ता चित्रा उत्तराषाढ श्रावण धनिष्ठा उत्तराभद्रपद रेवती अश्विनी.

However, these do not match with the current prevalent list of marriageable नक्षत्र. The नक्षत्र for marriage and that for agriculture are not the same. The reason seems to be obvious.

This सूत्र describes the results of a childbirth on मूल/मूळ नक्षत्र and inauspicious nature of अश्लेषः नक्षत्र. Is the inauspicious nature of the नक्षत्र is declared because of the timing of the childbirth conflicting with other works? A more rationale thought is required on this.

Interestingly there is a difference of views among the various सूत्र on the effects of नक्षत्र for different purposes.

The names of 12 Rashis, days of the week,वार yog and Karana are not found in this sutra

Other सूत्र

The other सूत्र contains almost all the aspects mentioned above. All the सूत्र define the spring season as composed as चैत्र ,वैशाख OR मधु ,माधव months.

बौधायन शुल्बसूत्र

This सूत्र appears to be of a later epoch and has the mention of 12 Rashi as seen from the verse 

मासद्रये यवाप्येकराशिं संक्रमेतादित्यस्तत्राद्यो मलिम्लुच शुदधोन्यः |     मैत्रेय सूत्र

"When the Sun is found to cross only one Rashi in two months the former is called मलिम्लुच ( intercalary month) and the latter a proper month" 

This सूत्र also gives out the working of the dimensions of the यज्ञ कुण्ड and यज्ञभूमिकाः, depending upon the type of ritual being performed,with respect to Nakshatra

दीर्घचतुरस्रस्याक्ष्णया रज्जुः पार्श्वमानी तिर्यग् मानी च यत् पृथग् भूते कुरूतस्तदुभयं करोति ॥

"The diagonal of a oblong(rectangle) produces the combined area of the two separate sides."

This verse also mentions the method to find the square root of 2.

समस्य द्विकरणी । प्रमाणम्तृतीयेन वर्धयेत्

तच चतुर्थेनात्मचतुस्त्रिंशोनेन सविशेषः -   बौधायन शुल्बसूत्र 2:12

"The diagonal of a square. The measure will be increased by a third and by a fourth decreased by the 34th. That is it's diagonal approximately"

${\displaystyle \sqrt{2}\approx 1+\frac{1}{3}+\frac{1}{3\cdot 4}-\frac{1}{3\cdot 4\cdot 34}=\frac{577}{408}\approx 1.414216,}$

Were these verses a precursor to the Pythagoras theorem? 

This सूत्र gives out many such formulas for the construction of temples / ritual places. (Squaring a circle and Circling a square) Some of them are concerned with the position of the nakshatras.

Dr Subhask Kak explains in detail, the astronomical code used for designing the यज्ञ कुण्ड and यज्ञभूमिकाः, in his book mentioned below.

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References

1. SB Dixit " Bhartiya Jyotish Shastra" Part 1

2. William Brennand " Hindu Astronomy"

3. Ancient Indian Astronomy by RN Iyengar.

4. Ancient Astronomy" and"Vedang Jyotish of Lagadha" by Prof TS Kuppanna Shastry

5. Indian Astronomy by BV Subbarayappa and KV Sarma

6. Astronomy in India: A Historical Perspective by Thannu Padmanabhan

7. https://vedicheritage.gov.in

8. The Astronomical Code of Rgved by Subhash Kak

9. Ancient Hindu Science  by Alok Kumar

10. Vedic Irina and the Rann of Kutch by Iyengar, Radhakrishna, Mishra