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varuna2
Joined: 20 Feb 2012 Posts: 319 Location: Lemuria

Posted: Sat Mar 09, 2013 11:30 pm Post subject: Astronomical Applications of Vedic Mathematics 


Oftentimes contemporary humans (who have not actually studied all of the available and unavailable, ancient astronomy treatises and maths) make categorical statements about the ineptitudes and astronomy errors and ignorance of the ancients (merely based on astronomy errors of people such as AbuMashar), and so one sometimes gets the impression the ancients were barely capable of making grunting sounds.
The Arabs called maths: hindisat. Apollonius of Tyana praised the Indians for their maths. Various Indian texts describe a heliocentric solar system, including as early as the rg veda (+7000 b.c.), and the Srimad Bhagavatam, and the Shatapatha Brahmana 8.7.3.10 (3800 b.c.), and at least one siddhantic text, and as late as the vishnu purana 2.8 (1st century ad), and also Aryabhata. The Chaldeans borrowed a Sanskrit word for month and the Sumerians likely learned sexagesimal maths from the Indians, since the sexagesimal maths are implied in the rg veda. Varahamihira (c.a. 1st century bc or 2nd century ad?) calculated the precession rate in his day at 50.32 seconds, and the modern precession rate has changed and is currently at 50.27.
Since there are various people here, interested in the technical side of astrology, I would like to present an interesting method of doing Vedic maths, oftentimes more efficiently than modern maths. The techniques used in this text, are quite different from what was taught in our primary schools and includes techniques on how to solve math problems with many digits, easily and efficiently, which is something the ancient Indians were also known for.
The text is titled: Astronomical Applications of Vedic Mathematics by Kenneth Williams
Here are a few excerpts from the text:
“In this section an equation is derived for calculation of the time of a total eclipse for an observer on the eclipse path. The formula, which can only be solved easily with the aid of one of the Vedic sutras, has an error for the example chosen of just 5 seconds. By contrast it is shown that using the standard method, due to Bessel, the result obtained differs by 7 minutes from the first iteration, and by 12 seconds after the second iteration. At least three iterations of Bessel’s method are therefore required to give a better result than that shown here.” p. 20
“As can be seen Bessel’s method is long and tedious: it involves many formulae, and consequently many calculations, and requires access to the table of elements and a calculating device. It is however, capable of giving a very accurate result. By contrast, the formula put forward in this section (which is in error by only 5 seconds) requires knowledge of just eight quantities including the observer’s latitude and longitude, all of which are readily available, and which give an equation that can be solved quickly without the aid of any artificial calculating device.” p. 26
“So the problem is to find E given M and e in the equation M=E  e sin E
Kepler’s equation looks very simple, but as we require E the equation is a transcendental one, which means E cannot be made the subject of the equation without having an infinite number of terms on the righthand side.
Many methods have been proposed to solve this equation including one by Kepler himself. The Vedic method which follows is extremely efficient, using each digit of the answer as they are obtained to get the next digit. Modern calculating devices make the solution rapid, using the NewtonRaphson, or some other, iterative technique. But their methods, though extremely fast are not efficient and there is also a need for quick pencil and paper solutions.
Before an example of the solution of Kepler’s equation a simpler transcendental equation will be solved. This will show the method more clearly.” p. 32
“Though the methods shown in this chapter show that it is possible to calculate positions they are probably of little practical use as all the information is readily available in Ephemerides and Almanacs. But they do show that it is possible to predict positions of heavenly bodies without too much effort and without a calculator. If a lesser degree of accuracy was permissible calculations could of course be further reduced.” p. 68
“These equations are somewhat complex and are not easy to remember or apply. However, when translated into triple form simple patterns emerge which enable us to solve spherical triangles much more easily.” p. 71
“The diagrams used in this chapter show that there are simple Vertical and Crosswise patterns behind the otherwise complexlooking formulae. These patterns could be used to make computer programmes run more efficiently.
Since also any angle can be represented by a perfect triple and the angle in a perfect triple can be found to any desired degree of accuracy, the methods shown here have a general application to the solution of spherical triangles. But with the widespread use of calculators and computers nowadays these would not usually be appropriate techniques. However, they can be used to easily give approximate answers and provide checks. The methods being simple they also provide an elementary introduction to the subject of spherical triangles and their solution. And there is something more satisfying about obtaining an exact solution using a simple method.” p. 97
“A 3dimensional equivalent to triples can be used to define a direction in 3dimensional space, just as ordinary triples can define any direction in 2dimensional space. This leads to the notion of ‘quadruples.’
These can be defined and developed along similar lines to the triples and we will see that they have useful applications in astronomy.” p. 98
“It appears then that this addition and subtraction method for quadruples would have useful astronomical applications: when, for example, a body in an inclined orbit (inclined to some reference plane) advances in its orbit by a certain amount, and we want to know its new position relative to the same reference plane.” p. 110
“In fact the inclination described by the code numbers 23,d,10 is [inverse tangent of 10 divided by 23] which is 23*30’ and not 23*27’. The following method can be used to obtain the code numbers for any inclination to any desired degree of accuracy.” p. 113 

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varuna2
Joined: 20 Feb 2012 Posts: 319 Location: Lemuria

Posted: Sat Mar 30, 2013 8:57 pm Post subject: 


"The picture below is that of Large Samrat Yantra (Onehalf) in Jantar Mantar in New Delhi. It is not functional. Sometime during the last several years, a natural water spring developed and submerged lower portion of quadrants on either side of Gnomon (triangular structure). Then concrete was filled to control the flow of water from the natural spring on either side. Thus it became unusable. It is calibrated for every 2”. Such is the precision that can be achieved."
The above quote is from saptarishisastrology.com volume 9 article section and article number 118 titled: Time Measurement Systems in Ancient India by Pidaparty Hariprasad
http://www.saptarishisastrology.com/articles_archive_vol9.php
The author of this article gives a list of reference texts for study of ancient time measurement systems in India. This particular observatory in New Delhi is a later one, but there were earlier observatories also.
Compare these ideas with this:
"Nonetheless, even though our knowledge of the history of ancient astronomy is extremely incomplete, there are scholars who believe that they can uncover important parts of this history by speculative reconstruction. One example of this is a paper entitled ' The Recovery of Early Greek Astronomy from India,' by David Pingree (PG). In order to indicate the complexities and pitfalls of the speculative process, we will examine the key argument of this paper in detail. This will involve the use of a number of technical astronomical terms, but we will explain these as we go along. Our method will be to first present Pingree's theory, and then give his reasons for accepting this theory as true. Then step by step we will show the fallacies in his reasoning and present an alternative theory that is in better agreement with the facts..." p. 182 Vedic Cosmography and Astronomy by Richard Thompson
"...Let us suppose that Aryabhata did this, and that he then computed his parameters using his observed longitudes rather than longitudes copied from a Greek table. This leads to a reconstruction of his parameters based on modern calculation of the differences between mean longitudes and the sun's mean longitude. The longitudes and resulting parameters for this reconstruction are listed in the last two columns of Table A2.3, and the errors in this reconstruction are listed in column (5) of Table A2.1. As we can see, these errors are zero, except for Mercury, where the error is equal to that in Pingree's reported reconstruction (see columns (1) and (2)). Thus, the hypothesis of observation yields better results than the hypothesis of copying from Greek tables..." p. 193 Vedic Cosmography and Astronomy by Richard Thompson


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varuna2
Joined: 20 Feb 2012 Posts: 319 Location: Lemuria

Posted: Sat Mar 30, 2013 9:47 pm Post subject: Happy Eostre or something 


Since there has been much talk about there being originally 8 houses or signs in the allegedly Babylonian system  the Babylonians who were using a Sanskrit word for month or season, I would like to add some more information regarding this topic.
Aditi with the 'a' prefix means 'without bounds' or 'limitless'.
"Aditi is a sky goddess and mother of the gods (devamatr) from whose cosmic matrix the heavenly bodies were born." http://en.wikipedia.org/wiki/Aditi
Aditi is associated with the sky and is the mother of the Adityas, and dates to the rg veda period which has been astronomically dated to +9000 bp. I decided to refer to the vedic period with the bp, meaning years before present, in the spirit of the archeologists who study the allegedly prehistoric periods of human history.
Keep in mind Aditi is the boundless sky and the Adityas are the progeny, i.e. representing subsections of the boundless sky into alternatively 8 or 12.
"In the Rigveda, the Ādityas are the seven celestial deities, sons of Āditi, headed by Varuna, followed by Mitra (Vedic personification of Surya) :
Varuna
Mitra (Surya)
Aryaman
Bhaga
Anśa or Aṃśa
Dhatri
Indra
The eighth Āditya (Mārtanda) was rejected by Aditi, leaving seven sons. In the Yajurveda (Taittirīya Samhita), their number is given as eight..." http://en.wikipedia.org/wiki/%C4%80dityas
"In the Shatapatha Brahmana, the number of Ādityas is eight in some passages, and in other texts of the same Brahmana, twelve Adityas are mentioned." http://en.wikipedia.org/wiki/%C4%80dityas
Keep in mind the age of the Shatapatha Brahmana 6.2.2.18: "18. And furthermore, at the Phâlguna (full moon), for that full moon of Phâlguna, that is, the second (Phâlguna) 1, is the first night of the year; and that first (Phâlguna) is the last (night of the year): he thus begins the year at the very mouth (beginning)." The year in this context is referring to the winter solstice and the Moon is in uttaraphalguni and is opposite the Sun. Therefore, this verse dates to at least 5800 bp (before present).
"Who knows the one wheel with twelve fellies and three axles [i.e. foci]? Therein are set together the three hundred and sixty like spokes moving and unmoving" (rg veda I.164.4 c.a. +9000 years bp (before present)
Happy Eostre!
"Eostre is a goddess in the true European Germanic tradition and religion who, by way of the Germanic month bearing her name (Northumbrian: Ēosturmōnaþ; West Saxon: Ēastermōnaþ; Old High German: Ôstarmânoth), is the namesake of the festival of Easter." http://en.wikipedia.org/wiki/%C4%92ostre


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