Astronomical Applications of Vedic Mathematics
Posted: Sat Mar 09, 2013 11:30 pm
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 Abu-Mashar), 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 right-hand 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 Newton-Raphson, 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 complex-looking 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 3-dimensional equivalent to triples can be used to define a direction in 3-dimensional space, just as ordinary triples can define any direction in 2-dimensional 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
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 right-hand 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 Newton-Raphson, 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 complex-looking 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 3-dimensional equivalent to triples can be used to define a direction in 3-dimensional space, just as ordinary triples can define any direction in 2-dimensional 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
