Hindu Calendar-2

 Measurement of time 

One of the most striking features of the Hindu calendar system is its intricacy. It offers a multidimensional method of structuring time, combining information about lunar days, solar days, lunar months, solar months, the Sun's and the Moon's movements in relation to stellar constellations, and other astronomically defined time spans. This makes the Hindu calendar vastly more complex than the Western calendar, which is built around only two basic units of time: solar days and solar years.

The Flux of Time is apparently, without a beginning or end, but it is cut up periodically by several natural phenomena,

• The ever-recurring alternation of daylight and night defined a day
• The recurrence of the moon's phases, defined as a month
• The recurrence of seasons, defined as a year

It is this recurring phenomenon which is used to make a calendar.

In the Vedic period, they observed and used these very natural phenomena to record the passage of time.  My blog Vedic Period 2  explains

Day

The time from sunrise to sunrise gives us a natural solar day or a Savana day. The Jews Babylonians and Muslims/ Hijri calendars calculate from sunset to sunset. Gregorian calendars calculate from midnight to midnight.

The mean solar day is used for calendar purposes, as the length of a solar day is variable.

A sidereal day is a period between two successive transits of a fixed star, and by the astronomical definition, it is the transits of the first point of Aries.(The First Point of Aries is a fiducial Point)

A solar day is longer than a sidereal day.

Indian calendars use ‘तिथि’ to denote a lunar day. This is specific to the Indian calendar/Panchang and is nowhere seen in other calendars. तिथि will be explained in detail later.

Month

A month, known as Chandra mana or masa, is primarily a lunar phenomenon. Its duration is defined as the time elapsed between two lunations, which can be from the new moon to the new moon or from the full moon to the full moon. The average length of a mean synodic period is taken to be approximately 29.5 days. This month is further divided into two halves, referred to, as paksha, and into 30 segments known as Tithis.

Year

There are three types of years

Solar/Tropical Year –It is, the time taken by the Earth to move from equinox to equinox or tropics or season to season. Indian calendars call it ‘Varsha’ which also denotes the rainy season.

Sidereal Year- It is, the time taken by the Earth to make one revolution around the Sun with respect to a fixed star.

Anomalistic Year- It is, the time taken by the Earth to make one revolution from Aphelion to Aphelion (Aphelion-furthermost point from the Sun) or Perihelion to Perihelion. (Perihelion -the nearest  point from the Sun)

The sidereal year is greater than Solar/Tropical Year. These years are considered for calendar purposes.

For divisions of day month and year during the Vedic period see the blog Vedic Period 2

The primary aspect of any calendar is to measure the correct duration of a year. Once the duration is ascertained, the other time units are calculated accordingly.

The length of the year, as counted in Vedic and later in the Vedang period was 360 and 366 days. It was later corrected during Arya Sidhant period to 365d 6h 12m and then later adjusted to the modern value of 365d 5h 48 m. The sole aim of correction is to keep the calendars in sync with the actual positions on the ecliptic and the seasons. 

Elements and Definitions

पंचांग

This calendar is highly comprehensive, integrating various elements such as ephemeris data, rituals, and both astronomical and astrological information. It has developed over time to adapt to evolving needs and may exhibit slight variations depending on the region.

 As the name suggests it is composed of 5 elements पंच अंग = पंचांग. These elements are

•  वार् 
• तिथि 
• नक्षत्र
• योग
• करण 

The earliest calendar mentioned only तिथि and नक्षत्र. Other elements were added later on. 

वार्

A natural solar day called ‘Savan divas’ सवन दिवस'

The nakshatra names were used to name the 'days' during the Vedic period. Thus when the moon is expected to be found in the मघा नक्षत्र(Alpha Leonis). The day would be called the मघा day. However, this system was discontinued later.

The present name and order of the days were borrowed from the Chaldeans and introduced in Hindu astronomy around 500 BCE. A वार् begins at sunrise, and seven वार् compose a week.

Some common names of these days are

• Sunday- आदि,आदित्य,रवि,अहस्कर,अर्क,अरुणा,भट्टारक,अहरपति,भास्करः,ब्रध्णः,भानु
• Monday. सोम , अबज, चन्द्र, चन्द्र, इन्दु, निष्पति, क्षापकर,
• Tuesday.  मंगल, अंगारक, भौम, महिसुता ,रोहितांग.
• Wednesday.बुध ,बौध, रौहिनेय, सौम्या
• Thursday. गुरु, अंगिरस, बृहस्पति, धीषण, सुरचार्य, वाचस्पति.
• Friday.. शुक्र ,भार्गवं, भृगुं ,दैत्यगुरुं, काव्यं ,उसनानि, कवि
• Saturday. शनि, सौरी, मंद

The solar day (savan divas) is further divided as,

• 60 Ghatikas = 1divas /1 solar day.
• 1 Ghatika =60 Pal=24 Minutes
• 1 Pal=60 Vipal=24 Seconds
• 1 Vipal=60 Prativipal=0.4 Second        

 For more on day division see Vedic Period 2

तिथि

To put it in simple words, a तिथि is a lunar day, and it relates to the Hindu lunar calendar. It is the exact point of time when the moon, moving eastwards from the sun after the अमावस्या, leaves the sun behind by 12 degrees, is the first तिथि, called  प्रथमा or प्रतिपदा and so on till the complete synodic revolution of the moon or one lunation occupying 30 तिथि for the 360 degrees to denote 1 lunar month. Since, however, the motions of the sun, earth and the moon are always varying in speed the length of a तिथि constantly alters.

• The greatest length is 26 h 6m 24s
• The least length is 21 h 34m 24s. 
• An average or mean length of 23h 37m 28.09s is considered for calendric purposes.

The तिथि1 to 15 are named Pratipada, Dwitiya,Tritiya ….Poornima / Amavasya (Full or New Moon)

 नक्षत्र 

This is specific to the Hindu calendar, like Tithi. There are 27 Nakshatra along the ecliptic. Each nakshatra occupies 360/27= 13 deg 20 min of space along the ecliptic. The equal division is for ease of calculation. In actuality,  नक्षत्र  occupies a varying space ranging from 12 to 18 degrees, approximately. Because of the variable motions of the Moon, the Sun and Earth, the time the Moon takes to traverse one nakshatra ranges from 

• A maximum length of  26h 32m 24s. 
• A minimum length of 22h 22m 24s 
• The mean length of Moon nakshatra is taken as 24h 17m 9.3s

The system of starting the list of nakshatra from Ashwini came into vogue around the seventh century BCE. For details on नक्षत्र  see blog Vedic Pd 4

योग

Astronomically, a योग corresponds to 13 degrees and 20 minutes—being the sum of the Longitudes of the Sun and the Moon. That is, it is the time during which the Sun and the Moon together cover 13 degrees and 20 minutes of space. But there is no direct astronomical phenomena corresponding to it. There are 27 योग in number. The concept of योग arose sometime around 700 CE and became an integral part of  पंचांग only after around 700 CE.

करण

A करण being half a tithi, or the time during which the difference of longitudes of the Sun and Moon are increased by 6 degrees. There are 60 करण in a lunar month. There are seven करण in a series of eight cycles—a total of 56—every month, from the second half of शुक्ल प्रतिपदा(1st) to the end of the first half of कृष्ण पक्ष चतुर्दशी (14th). The other four करण are respectively from the second half ofकृष्ण पक्ष चतुर्दशी(14th) to the end of the first half of शुक्ल प्रतिपदा.

योग and करण are NOT used for calendric purposes but only for Astrological purposes.

 

 ..................to be continued with the construction of calendars, dating and Era's

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References:-

1.  Bhartiya Jyotish Shastra-SB Dixit
2. The Indian Calendar-Robert Sewell and SB Dixit
3. Cowasjee Patells Chronology
4. The book of Eras- Gen A Cunningham
5. Calendar Reforms Committee Report 
6. Ancient Indian Astronomy -RN Iyengar
7. The Calendars of India- VK Mishra

 

 

 

Hindu Calendar-1

 The Hindu Calendar

Basics of Calendar Making

Calendars are used for regulating two essentially distinct types of human activities,

(a) Civic and administrative,

(b) Social and religious

In ancient and medieval times and even today, different countries and religions have developed their characteristic calendars to serve both purposes.

 Today in India, different parts of the country follow a distinct accounting system to observe the same ritual. Isn’t it Crazy? This is what appears to a layman, but actually, it is not.

The Gregorian calendar fulfils civic and administrative purposes and most of us have a fair idea of this calendar, however, our social and religious requirements are met by the Hindu calendar, without understanding our own calendar.

All religious, festivals, rituals, and social occasions are determined based on the Hindu calendar/almanac or a Panchang. One will find festivals, and religious rituals, practically every month. They apply either to the whole of India or only to some areas. This is because of the varying customs, religious beliefs, conventions, traditions, etc.

When do we observe this ritual ,on the 6^th or the 7^th?

When is the Holi festival this year?

Why there is no विवाह मुहूर्त this month?

Why is the Hindu New Year on different dates?

Why is Makar Sankranti always on 14/15 January?

These questions often pop up when we refer to two different accounting systems, of time, ie, the Indian and Gregorian calendars. Indian calendar, by itself, is vast and it varies in form, place to place and reference frame (Lunar, Solar, Lunisolar and Zodiac signs).

Why should I know the Indian Calendar?

• - Understand how the calendar is constructed to align with the Earth's actual position in the zodiac.
• - Explore the unique features of the Indian Almanac, known as the Panchang.
• - Learn how the Panchang is used to determine religious festivals and social occasions.
• - Grasp the concept of seasons and their correlation with calendrical months in a way that is easy for anyone to understand.
• - Understand the significance of religious and ritual landmarks in relation to the zodiac.
• - Familiarize yourself with the dating rules adopted in various eras to accurately determine dates.
• - Know how to identify the correct epoch during excavations and archaeological surveys.

Calendars are based partly on science, (which cannot be violated) and some on man-made conventions. In Hindu calendars, the science part is 80 % and the conventions part is 20 %. "The fascinating twist lies in the fact that both the scientific aspects and conventions differ widely across the diverse regions of India."

The early societies were founded on agriculture and agricultural practices depended on seasonal weather conditions. The growth of Civilisation and Social life brought religious and national festivals. Hence people wanted to know in advance, as to:-

•  -When to prepare the ground for sowing/harvesting,
• - expect the onset of seasons,
• -full and New Moon,
• -observe a festival, carry out a religious ritual and so on.

As the French call their Almanac “Connaissance des Temps” (Knowledge of time)

In India the calendar-making process is quite ancient, beginning from the Vedic period, albeit in crude form, as compared to today. Indians possessed this astronomical knowledge, quite independently and not borrowed, as is the popular belief

The वेदांग ज्योतिष was more of a handbook on astronomical information which was used for 

•  Day-to-day life of people
•  Knowledge of seasons for agriculture
•  Religious purposes, fix suitable times for rituals and यज्ञः, times of New and Full Moon for दर्शपूर्णमास and चातुर्मस.
•  Rules and calculations for year beginnings, Solstices, equinoxes, Solar and Lunar eclipses and adjustment of lunar months to be in sync with Solar cycles.
•  In short, it was a precursor to पञ्चाङ्ग and the modern calendar

    My blog of 28 Jun Vedang Pd-1 refers. Vedang Pd -1

 The Indian calendar offers a comprehensive overview of various aspects of timekeeping, presented in the form of an almanac or Panchang. This calendar not only tracks astrological events but also encompasses details related to social rituals and the observance of religious festivals, highlighting its cultural significance. It is like a handy guide that covers all sorts of important details, making it a significant part of our everyday lives!

A unique feature of the Indian calendar is that if you have a basic understanding of astronomy and are familiar with current astronomical events, you can make a good estimation, of the month and time of year just by looking up at the night sky.

 

 

………to be continued with details of the Hindu calendar, Panchang

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References:-

1. SB Dixit " Bhartiya Jyotish Shastra" 
2. The Indian Calendar-Robert Sewell and SB Dixit
3. Cowasjee Patells Chronology
4. The book of Eras- Gen A Cunningham
5. Calendar Reforms Committee Report 
6. Ancient Indian Astronomy -RN Iyengar
7. The Calendars of India- VK Mishra

Siddhant Period

5 Siddhantas

Astronomical knowledge was very well developed during the Vedic and Vedang periods and was further developed and updated in subsequent periods.

सिद्धान्तis one of the sub-branches of  Ganit Skandha. 

For various branches of Astronomy see 

https://intrestingcosmos.blogspot.com/2024/03/  -of 01 March 2024

 Knowledge of the position of the Sun, Moon, and other planets was required for calendrical, religious, and daily rituals. 

The history of astronomical works can be assessed from the oldest known group of  5 ancient सिद्धान्त known as पंचसिद्धांतिका  written by Varahmihira (500 CE)

These 5  सिद्धान्त are 

पितमहः,वसिष्ठ,रोमक,पौलिश ,सौर

Varahmihira in his पंचसिद्धांतिका  says that the सिद्धान्त written by Paulisa is fairly accurate, but not as accurate as Saura/Surya Siddhant. The accuracy of others is doubtful.

These Siddhantas outline methods for calculating the mean positions of the Sun and Moon, as well as the orbital periods of other planets. They provide a comprehensive explanation and calculations regarding mean motion मध्यमाधिकारः /adhikara on mean places and the average number of revolutions of the planets. Additionally, the texts discuss the length of the year, the concept of yugas, and methods for intercalation and synchronization.

For example 

One of the couplets describes a method of calculating the length of the day. It says 

द्विघ्नं शशिरस्(61)भक्तं द्वादश हिं दिवसमनम्

" Multiplying  (the number of days elapsed after WS or the days to go before the end of Ayana, beginning with SS ) by 2 and dividing by 61. Add 12 (Muhuratas) to the quotient, and the result will be the length of the day"

Surya Siddhanta of the पंचसिद्धांतिका group had assumed the commencement of Yuga at midnight and that it postulates the Yuga system. ( It is assumed that all planets including the Sun and The Moon were together on this day)@@@. Dixit has confirmed this aspect, by comparing the data of Surya Siddhanta to the actual longitudes, so calculated as per Surya Siddhanta as of 1887.

 

Al Biruni commented that पौलिश siddhant was compiled by Paulus—ul—Ynani, or Paulus the Greek. Professor Webber refuted his claim, stating that Paulus's work was related to astrology and not astronomy.

Al Biruni states that Surya Siddhant was compiled by Lata. Prof Weber connects it with Ptolmey's works. However, Surya Siddhanta of the पंचसिद्धांतिका group  was not complied by Lata . Dixit proves both the contentions incorrect. 

पितमहः and वसिष्ठ सिद्धान्त appear to be old, as they are similar in concept to Vedang Jyotish. The other siddhantas appear to be around the pre-Saka period, i.e.,200-50 BCE. Dixit proves Dr Thibaut's argument that their antiquity around 400 CE was wrong.

पितमहः,वसिष्ठ,पौलिश ,सौर,रोमक are as per their chronological order of antiquity.

The Five Modern सिद्धान्त

The elements described by पंचसिद्धांतिका group differ from those in the more modern Sidhantas. They primarily differ in the length of the year and the planets' mean motion, position, number of revolutions etc. The data is updated and more accurate, and hence the epitaph modern.

 

The five modern सिद्धान्त  are वसिष्ठ,रोमक,सौर सोम, ब्रह्म सिद्धान्त . It is not clear if two सोम siddhantas existed.

All five siddhantas declare themselves to be divine and are actually so regarded.Other siddhantas ,such as  पंचसिद्धांतिका ,विष्णुधर्मोत्तरः,ब्रह्मसिद्धान्तः are also regarded divine.

Surya Siddhanta is more popular than other siddhantas. It is believed that it was compiled during the second or third century CE.

Surya Siddhant was first translated into English by Dr Ebenzer Burgess in 1860. Prof Whitney further added extensive notes to it. Prof Whitney believed that Hindus borrowed it from the Greeks. However, Burgess clarifies at the end of his book that it was The Greeks who borrowed it from the Hindus

The contents of all the Siddhantas provide explicit astronomical details, making it challenging to summarize the information succinctly. Anyone interested in a detailed study is encouraged to explore the references listed below.

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@@@With the advancement of technology, changing times, and evolving concepts, a new thought process has emerged. The latest astronomy software can now simulate past events and accurately show the positions of the nakshatras and planets.

These new revelations may directly conflict with the findings of ancient scholars, as well as with established beliefs and rituals. More research and studies are needed to uncover the truth.

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References

1. SB Dixit " Bhartiya Jyoti's Shastra" Part 2

2. William Brennand " Hindu Astronomy"

3. Coolebroke Essays.

4. Dr Thibaut's Essays on Surya Siddhanta 

5. Dr Ebenzer Burgees's Translation of Surya Siddhant

6. Dr Nilesh Oak talks

The Tropical Year Calendar

The Tropical Year Calendar

Julian/Gregorian 

 The tropical year is the time, taken by the Sun, to pass from the vernal equinox to the vernal equinox. It equals 365 days, 5 hours, 48 minutes, and 46 seconds, or 365.2422 days. The tropical year is used, to keep track of seasons, planting, and harvesting. There is a need to create a calendar, with a whole number of days per year to accurately follow the tropical year without losing synchronization with the seasons over a period of time.

We begin with a calendar of 365 days per year. Our calendar year is shorter than the tropical year by 0.2422 days. So to correct (approximately), we add 1 day every four years (leap year). Thus, three calendar years are 365 days long; the fourth calendar year is 366 days long. The average length of the calendar year in days now becomes (3 x 365 + 366)/4 = 365.25 days.

This calendar system was actually instituted for use in the Roman Empire by Julius Caesar around 46 BC. But since the Julian calendar was 0.0078 days (11 minutes and 14 seconds) longer than the tropical year, errors in timekeeping gradually accumulated. Between 46 BC and 1582 AD, this accumulated error amounted to a total of: 0.0078 x (1582 + 46) = 12.7 days. In 1582, Pope Gregory XIII reformed the calendar by specifying that all years divisible by 4 are to be leap years except for century years, which must be divisible by 400 to be leap years. Now, in 1200 years:

1. A total of 300 years (including all century years {since any century year = N x 100, where N = an integer}) are divisible by 4 and are therefore candidate leap years.
2. A total of 900 years are not divisible by 4 and are therefore regular years.
3. Twelve-century years are possible leap years.
4. But only 3 century years (out of the 12) are divisible by 400 (i.e., {400, 800, 1200}, {1600, 2000, 2400}, etc.), so only 3 century years are actually leap years.

Since 12 - 3 = 9, Gregory's rule eliminates 9 leap years out of 1,200. Thus: 300 - 9 = 291 years are actual leap years, and 900 + 9 = 909 years are regular years. The average length of the year becomes (291 x 366 + 909 x 365)/1,200 = 365.2425 days, with an error of 365.2425 - 365.2422 = 0.0003 days per year, or one day every 3,333.3 years.

The Gregorian calendar came into use in Roman Catholic countries in October 1582 when the seasons were brought back into step by eliminating 10 days from the calendar then in use. Thursday, October 4, was followed by Friday, October 15 (which caused some apprehension and anxiety amongst the people, especially those with birthdays on the eliminated dates!).

 Britain and its colonies did not introduce the Gregorian calendar until September 1752 by which time an additional one-day correction was required (actually, {1752 - 1582} x 0.0078 = 1.33 day). Some British documents from the period before the British reform actually contain two dates, an old and a new.

Hijri 

The Hijri calendars are purely Lunar and hence the year is also lunar and not tropical, measuring 354 days and 355 days during the leap year. An extra day is added to the last month 11 times in a block of 30 years. Usually 2nd,5th, 7th, 10th, 13th,15th,18th,21st, 24th, 29th year in the cycle of 30 years. In the course of 33 years, the beginning of the Hijri year runs through the whole course of seasons. The months and seasons are not in sync. 

This practice of adjustment varies in different countries and sometimes at different periods in the same country.

This Hijri calendar is not tropical.

Akbar tried to adjust the Hijri calendar with the seasons and termed it "The  Fasli year" (To synchronise the month with harvest/फसल time)used only for commercial purposes. 

 This facilitated in collection of harvest tax. (Harvest time and Hijri month varied every year ). The commencement of this year was in July. It was more or less a mix of the " Vilayati Year (used in some parts of Bengal), Amli Era of Odisha and Bengali San" with Hijri month names.

 In most parts of India, it was a Solar year, however different customs of different parts of India caused a divergence of reckoning. It was later discontinued.

   The original Hijri calendar was followed for all other aspects.

Hindu

In contrast just see our Hindu calendars, where the months are Lunar and the year is Solar. In fact, the Hindu calendar is lunisolar. The 5-yearयुग and intercalation system of adjustment is followed to keep in step with seasons and the actual position along the ecliptic.

One month, known as अधिक मास is added every 30 months, therefore in a युग of 5 years 59 days get added, thus the synchronisation with the seasons is nearly real-time. (My blogs on the Vedic period give out details)

 This was available in the Vedic period, earlier than 1500 BCE, even before the Europeans could think of it!

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

Vedang Period 2

Atharva Vedang Jyotish 

Continuing with the Vedang period of the history of astronomy, Atharva Vedang Jyotish (AVJ) relatively, is of a later period than RVJ and YVJ. It is a text, as told to Kashyap by Pitamah. The time units followed then were as 

12 निमिष= 1 लव

30लव = 1 कला

30 कला = 1 त्रुति

30 त्रुति= 1 मुहूर्त

30 मुहूर्त= 1 अहोरात्र / whole day

The document also gives the list of 15 मुहूर्त, whose measure is compared with the length of a gnomon/stick, which is 12 anguls long. These names must be of the मुहूर्त by day, as my blog Vedang Period 1 refers (since the shadow is measured). The list shows a maximum shadow of 96 anguls cast by"Rudra"  मुहूर्त and 0 anguls by 'Abhijit'  मुहूर्त. It is not exactly 0, but the shadow does not alter in length or direction, hence 0. 

Shadow is not  0 at noon, but it must be shorter than 3 anguls. The lengths of shadows are not the same throughout the year. 

With this method, one can find the approximate, location where VJ was compiled A different method to find the location of the composition of VJ is calculated by SB Dixit.

Karanas and Auspicious Times 

This section gives out the instructions and duties to be observed for a particular मुहूर्त. Like friendly acts to be done on a 'मित्र मुहूर्त' and horror acts on 'रुद्र मुहूर्त'. The chapter also mentions, which acts to be done on a particular Tithi to get auspicious/ unauspicious results. The list of Thiti's includes 'Nanda and 'Bhadra' as in this verse

चतुभिं : कारयेत्कमं सिदधिह तोधिं चक्षणः !

 तिथिनक्षत्रकरणमृहतंतरीति नित्यशः 

This verse gives out only 4 अङ्गः (parts), they areतिथि ,नक्षत्र ,करण ,मुहूर्त not 5, which we are familiar with  a पञ्चाङ्ग . (वार्/day is not mentioned)

Following verses 90 and 91 give out the auspicious/inauspicious nature of the planets depending upon the strength of the Moon, ie, with the number of kalas or the phases of the Moon. 

तिथिरेकगुणा प्रोक्ता नक्षत्रं च चतुर्मुणं ।॥ वारचाष्टगुणः प्रोक्तः करणं षोडश्लान्वितं ।। € ० ॥ दात्रिंशद्ुणो योगस्तारा षष्ठिसमभ्विता ॥ चंद्रः श्तगणः प्रोक्तस्तस्माच्चंद्रबला बलं ।। ९१ 

 समीक्ष्य चंद्रस्य बलाबलानि ग्रहाः प्रयच्छंति श॒भाश्भानि

The following verse mentions the planets as day lords of the week. Though no specific order of the days have been mentioned . The day/ वार् seems to have been introduced for the first time .

आदिस्यः सोमो भौमश्च तथो बुधवहस्पती ।। भार्गवः शनेश्धरस्चंव एते सप्त दिनाधिपाः ॥६३॥।।

Other verses give out more names of planets applicable to the names of the days. Like सूर्य लोहितांग, सोमसुत, देवगुरु, गुरु ,भृगु, शुक्र ,सूर्यसुत.

This AVJ also gives out for the first time, the prediction part of astrology. Verses from 103 to 109 give out the names of 27 नक्षत्र to be divided into 9 groups with 3 नक्षत्र in each group. These give out the nakshatra at birth, for action and for foundation. 

Verses 25-27 give out the controlling deities of the 27 nakshatras without naming the nakshatras. The list of deities starts with the deity of Kritika Nakshatra. Confirming the Krittikadi system in vogue at that time.

 The 28th Verse says

नक्षत्रदेवता एता एताभियक्ञकमंणि । यजमानस्य शास्त नामि नकषत्रजं स्मृतं ।।

" The holy perceptors/शास्त्रजन: ordain that those performing the rituals should adopt a name based on the nakshatra at birth"

Other astronomical works describe a method by which persons are given names according to the Nakshatra quarter/चरण at birth. This system is still followed, even, today.

ग्रहोल्कालनिनिं्तिः कपेदहिंश्च पीडयत ।। यच्च दुयं भवति तत्‌ तत्परवक्ष्याम्य

his verse ordains "To do or not to do" certain acts on these nakshatras and describes the fears and horrors which befall when certain nakshatras are accompanied by planets/ comets/meteors etc.

The AVJ text describes the nakshatra-based system of astrology, similar to today's practice based on 12 Rashis/Zodiac signs. 

The text suggests its antiquity due to the absence of 12 zodiac signs, challenging the common belief that the system of prediction was borrowed.

Roger Billiard published his book " L'Astronomie indienne investigation des textes Sanskrits et des donnes numeriques" in 1971 and shows how accurate the tables and celestial observations made by Hindu priests during the Vedic and Vedang periods. His estimate of VJ is around the 10th century BCE. Indian scholars have dated the antiquity of the Krittikadi system in the Rgveda to around 2500 BCE using astronomical references.

Ptolemy's Almagest was published in 150 CE 

............to continue with the composition period of RVJ, YVJ and astronomical mentions on other अङ्गः of Vedas.

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References

1. SB Dixit " Bhartiya Jyotish Shastra" Part 1 pages 70-83

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. History of Astronomy in India" by SN Sen, KS Shukla