tag:blogger.com,1999:blog-78366078311305208412024-02-02T15:02:32.279-08:00Astrology SoftwareAbout Vedic Astrology Software & the Computation of Sidereal LongitudesGovind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.comBlogger119125tag:blogger.com,1999:blog-7836607831130520841.post-26628036323343907912024-01-04T17:04:00.000-08:002024-01-04T17:04:47.520-08:00OF WORLD PEACE
It is heartening to note,
That the Indian PM's name,
Hath been recommended,
For establishing World Peace !
"My chariots shall not be bloody,
Till the earth records my name",
Said the conscientious Buddha,
As he renounced War.
Milton averred " Peace hath her victories,
No less renouned than War " !
Man's essential nature is Peace,
War is only an abnormality !
How can a Nation which declares Peace,
Aum Shanti, Shanti, advocate War ?
From the ethical, moral angle,
Peace gets Fame, and War notoriety !
Love cognised is Truth
Love in action is Non VIolence
Love as pure Being is Wisdom
Love as Feeling is Peace.
Yet, many believe that war is a necessity biologic,
An outlet for the human desire to compete and excel !
Conquest by force is only half victory,
True victory is conquest by Love !
Harmony has to be established,
And perfection worked out.
We have to effectuate the Divine,
In a conflicting and contradicting world !
We have been instructed,
To rise from Non Being to Being,
Asado Ma Sad Gamaya,
And to rise from Darkness to Light.
In other words,
From the Negative to the Positive.
But we have a big problem here,
As we are confronted by qualities negative !
We have seen Death,
But not Immortality !
Mrutyor Ma Amritam Gamaya,
Appears to be difficult !
Death, Non Being, Ignorance,
Darkness, adversity, disease,
All these we have seen,
But never experienced the Positive !
Qualties Negative,
Are merely the first terms of the Formula,
Unintellible to us,
Till we work out its esoteric terms !
These qualities negative,
Are merely initial discords
Of the Musician's symphony,
Didnt He create a world miserable ?
That too deliberately flawing perfection,
Of His Own Creation ?
The discords of the world are Lord's discords,
To unfold the Godhead within us?
The tormenting Evil is He,
As well as the Redeeming Grace !
Oru Kai Praharikkave
Maru Kai kondu thalodum Easwaran !
Only when we have an integral vision,
Can we see beyond all masks,
The serene and lovely face,
Of the All Blissful Godhead !
His tests test our imperfection,
True friemd is He and Heart, the Teacher !
Mind is the greatest Guru,
And world, the true Book !
Let us be what the Seer Poets recommended,
What Lord Krishna recommended to Arjuna,
Realised souls viewing Becoming,
Under the metaphor of Being !
Let us rise from Death to Self Realisation,
From the Darkness of Nescience to Light Supernal,
From the mirage of Non Being to True Being,
Thus fulfilling that eternal Prayer Asato Ma Sadgamaya !
Did not Bhagavan Aurobindo,
Say that Strive and thou shalt have ?
Trust and thy trust shall be justified ?
That the Divine Crown is within our reach ?
That we have to overcome the Negative,
And reach the Positive !
And Implement Positivity,
Then only shall the world be redeemed !
The Kingdom of Heaven,
Is both worldly and transworldly
To implement this Rama Rajya,
India, US and Russia exist !Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-46405479046481831282014-02-25T04:56:00.002-08:002014-02-25T04:58:36.668-08:00Our software, Zodiac VII <div dir="ltr" style="text-align: left;" trbidi="on">
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg43amrUscH5bfHgUBde2vuQiaTzEzMQsiwdBeI3b0HQL8b8u4Nl-wtqG2MqnS4TaW4auYpno-d7jsaiJB3Cu2hiL6go1z3MRwbi8e69F0i-pE5NAxXOHQHBD5Dd3NTsiR-egcrO3BozhU/s1600/ksam.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg43amrUscH5bfHgUBde2vuQiaTzEzMQsiwdBeI3b0HQL8b8u4Nl-wtqG2MqnS4TaW4auYpno-d7jsaiJB3Cu2hiL6go1z3MRwbi8e69F0i-pE5NAxXOHQHBD5Dd3NTsiR-egcrO3BozhU/s1600/ksam.jpg" height="287" width="320" /></a></div>
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<b>Ours, Zodiac 7, is one of the best softwares in the world, but now the market is flooded with softwares of all descriptions. When I started in 1999, there were only some competitors. Now there are many.</b><br />
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<b>But professional softwares will stand out from the rest. In India, only some, we find, are accurate.</b><br />
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<b>The KSAM algorithms still hold good. They are as good as the Western algorithms !</b><br />
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Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com1tag:blogger.com,1999:blog-7836607831130520841.post-17308522092812777542011-11-03T20:46:00.001-07:002011-11-03T20:46:58.644-07:00The Alpha Numeric System of Mathematica Indica<iframe width="480" height="315" src="http://www.youtube.com/embed/viR5g6qnv44" frameborder="0" allowfullscreen></iframe><br /><br /><br />Known as Paral Sankhya, the Alpha Numeric System was developed by Vararuchi. It is also known as Katapayadi, as 1 represents the letters Ka, Ta, Pa and Ya, whose corresponding digit is 1. <br /><br /><br />Number Letters<br /><br />1 ka ta pa ya<br />2 kha tta pha ra<br />3 hu da bha la<br />4 gha da ba va<br />5 nga na ma sha<br />6 cha tha sha<br />7 chha dha sa<br />8 ja dha ha<br />9 jha ddha<br />0 nja na ksha <br /><br />The Value of Kaa ki ku ke and ko was 1<br /><br /><br />31 was represented by Kala <br /><br />Kala is ka + la = 31 ( The digits are counted from the right )<br /><br />and 351 by Kamala<br />Kamala = ka + ma + la = 351Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com1tag:blogger.com,1999:blog-7836607831130520841.post-67181212190582627582011-11-01T20:58:00.000-07:002011-11-01T21:03:38.936-07:00Elemental Numbers in Mathematica Indica<iframe width="480" height="315" src="http://www.youtube.com/embed/ZJoukCNNky4" frameborder="0" allowfullscreen></iframe><br /><br /><br /><br />Sanskrit poets used Elemental Numbers in their poems. As eyes are 2, two represents eyes. As the Vedas are four, four represents the Vedas. Netra Netra ( eyes , eyes ) means 22 and Veda Veda means 44. <br /><br />Here we give the correspondence between numbers and the Elements<br /><br /><br />1 - Earth, Moon<br />2. Eyes, hands, feet, ears<br />3. Worlds, trinitarian unity, three yogas, three vedas.<br />4. Vedas <br />5. Sense organs.<br />6 Rithus or the six seasons.<br />7. Rishies<br />8 Directions, eightfold prosperity<br />9 Digits, numbers<br />0 Ether<br /><br /><br />10 - Ego and its ten heads.<br />11 - Rudras<br />12 Months, Zodiacal Signs<br />13 Universe<br />14 Vidyas, Manus<br />15 Lunations, thidhis<br />16 King<br />17 Athushti<br />18 Treatises astronomical, mythological<br />19 Athi Dhruthi<br />20 Nails<br />21 Uthkruthi <br />22 Akrithi<br />23 Vikruthi<br />24 Gayathri<br />25 Principles<br />27 Constellations<br />33 Deities<br /><br /><br />The digits for the letters are written, counting from right to left<br /><br />For example,<br /><br />Bhanandagni<br /><br />Bham = Stars = 27<br />Nandan = 9<br />Agni = 3<br /><br />Bhanandagni = 3927Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-45088288434788179512011-10-30T03:01:00.001-07:002011-10-30T03:01:24.692-07:00Of Scholarship<iframe width="480" height="315" src="http://www.youtube.com/embed/905toTTplI8" frameborder="0" allowfullscreen></iframe><br /><br /><br />Scholarship is great and is like an expanding flower<br /><br /><span style="font-style:italic;">Pandityam nava pallavam vikasitham pushpam prayogajnatha</span><br /><br />In order to have scholarism graced, one should strive hard to gain it. It is said that the hard work done by the scholar is known only to the scholar ! How can others know it ?<br /><br />The scholar and the world, the endless strife<br />The discord in the harmonies of Life<br />The love of Learning, the sequestered nooks<br />And the sweet serenity of Books<br />The marketplace, eager love for gain<br />Whose aim is vanity, whose end is pain !<br /><br /><span style="font-style:italic;"><br />Vidwaneva Vijanathi<br />Vidwajjana parishramam<br />Na hi vandhya vijanathi<br />Gurveem prasava vedanam !</span><br /><br />How can others know the work hard <br />Which the scholars have indulged in ?<br />Like a barren woman who cannot know<br />The pain of delivering !Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-23710999499481086642011-10-28T20:20:00.000-07:002011-10-28T20:31:11.484-07:00Names of big Indian numbers ( contd )<iframe width="480" height="315" src="http://www.youtube.com/embed/ZHu0ryjnkYw" frameborder="0" allowfullscreen></iframe><br /><br />There are greater numbers, than the ones mentioned below. <br /><br />Maha Padma = 10^15<br />Kshoni = 10^16<br />Parardha = 10^17<br />Sankha = 10^18<br />Maha Sankha = 10^19<br />Kshithi = 10^20<br />Maha Kshiti = 10^21<br />Kshobha = 10^22<br />Maha Kshobha = 10^23<br /><br />Neethi Lamba = 10^27<br />Sarva Bala = 10^45<br />Thallakshana = 10^83Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-32668832451936227392011-10-27T20:26:00.001-07:002011-10-27T20:51:54.431-07:00Indian names for big numbers<iframe width="480" height="315" src="http://www.youtube.com/embed/FMlisMg4VPo" frameborder="0" allowfullscreen></iframe><br /><br /><br />Ganitha Kerala, the Kerala School of Astronomy and Maths, has given a lot of names for big numbers.<br /><br />Like the Western <br /><br />Myriad = 10^4<br />Million = 10^6<br /><br />Indian names are <br /><br />Shata = 10^2<br />Sahasra = 10^3<br />Ayutha = 10^4<br />Laksha = 10^5<br />Dasa Laksha = 10^6 ( Million ) <br />Koti = 10^7<br />Dasa Koti = 10^8<br />Vrinda = 10^9<br />Kharva = 10^10<br />Nikharva = 10^11<br />Mahapadma = 10^12<br />Mahakharva = 10^13<br />Padma = 10^14<br /><br />In the Western, we have<br /><br />Million = 1000^2<br />Billion = 1000^3<br />Trillion = 1000^4<br /><br />or <br /><br />Nillion = 1000^(n+1)<br /><br />where n is the number. Tri meaning 3 and trillion = 1000^(3+1)!Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com1tag:blogger.com,1999:blog-7836607831130520841.post-13198361797991853542011-10-17T23:10:00.000-07:002011-10-17T23:13:35.613-07:00Of Mathematica Vedica<iframe width="480" height="315" src="http://www.youtube.com/embed/gIo88Rt1JtI" frameborder="0" allowfullscreen></iframe><br /><br /><br /><br />By one more than the one before<br /><br />Ekadhikena Purvecha is Sanskrit for " One more than the previous one". This <br />sutra can be used for multiplying or dividing algorithms.<br /><br />You can use this formula to compute the squares of numbers.<br /><br />For example.<br /><br />35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25<br /><br />By the sūtra, multiply "by one more than the previous one."<br /><br />35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-54373535875091469572011-09-20T20:51:00.001-07:002011-09-21T01:57:35.001-07:00India will overtake Japan by 2012<iframe width="480" height="315" src="http://www.youtube.com/embed/XRem2vMXSd8" frameborder="0" allowfullscreen></iframe><br /><br /><br />India may overtake Japan by 2012 and become the third largest economy by 2012. According to Pricewatherhouse Coopers, India's GDP PPP will overtake Japan by 2012. Such a forecast seems to be probable, as Jupiter will be in the House of Gains for India next year. Jove in the 11th will bring handsome gains. <br /><br />China will become the world's largest economy by 2020. Within one year, Indian GDP PPP will become $4.32 trillion, overtaking Japan's economy, estimated at $1000^4 ( 4.32 tr to be exact ). <br /><br />“While the exact date is open to doubt, it seems highly likely that, by 2030, China will clearly be the largest economy in the world on PPP,” John Hawksworth, head of macroeconomics at PwC, wrote in the report.<br /><br />The Decoupling theory, that India and China will not be affected by the global meltdown, still holds true. China will overtake Japan as the second largest economy this year, as Japan is facing a severe crisis !.<br /><br />Yesterday the Sensex grew by 353 points and closed about 17 K. Despite the secondary reactions, the Stock Market seems to be on a growth track.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com1tag:blogger.com,1999:blog-7836607831130520841.post-931250357350633982011-09-18T04:25:00.001-07:002011-09-18T04:25:18.396-07:00The Churning of the Milk Ocean at Bangkok Airport<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR0zR2y0ylHxP11-DWpsVn1pynmuCrdze967AerbdtpSDFJbazLNbNympLhoEb8Fp-5XGgd0h85DA5zgbHeAleg3uU-yOWYxbXgEuvXJm8UeTXlpmqJSU8qCTLlo5JU_w8-9ACcozB0YQ/s1600/DSCN1414.JPG"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 240px; height: 320px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgR0zR2y0ylHxP11-DWpsVn1pynmuCrdze967AerbdtpSDFJbazLNbNympLhoEb8Fp-5XGgd0h85DA5zgbHeAleg3uU-yOWYxbXgEuvXJm8UeTXlpmqJSU8qCTLlo5JU_w8-9ACcozB0YQ/s320/DSCN1414.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5653653569580739138" /></a><br /><br /><br /><br />We were amazed to see Palazhi Madhanam, the Churning of the Ocean, depicted at <br />Thai Suvarnabhoomi Airport.<br /><br />The Churning of the Ocean is allegorical, symbolising the great fight in the human bosom between Virtue and Vice. The forces of Virtue, the celestials, the devas fight the forces of Vice, the demons, the asuras. In this fight Immortality is the end result and Immortality is nothing but Self Actualisation !<br /><br />P N Oak's theory that the base Civilization was Vedic from which everything began seems to be proved right !<br /><br /><a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhe7hNBC-ZCDzM9AkoI3jno_m_Hsp6c0lhmNrFhpd-WyoG4j3Z24JxrBWnaBvWLaR4AVO39mJ0_-hEy6cAZwCc65TJE_YGBnCIwZ-OJ8_u2Rcu0BXbL1FRieqW55kllcdDgmDgETmZk5w/s1600/DSCN1407.JPG"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 223px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhe7hNBC-ZCDzM9AkoI3jno_m_Hsp6c0lhmNrFhpd-WyoG4j3Z24JxrBWnaBvWLaR4AVO39mJ0_-hEy6cAZwCc65TJE_YGBnCIwZ-OJ8_u2Rcu0BXbL1FRieqW55kllcdDgmDgETmZk5w/s320/DSCN1407.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5653652210285581858" /></a><br /><br /><br />Dr Kenneth Chandler averrs that the Vedic Civilizations is 10,000 years old.<br /><br />He writes in his thesis " The Origins of Vedic Civilization" .... <br /><br />How Ancient is Vedic Civilization?<br /><br />Astronomical References in the Rig Veda and Other Evidence<br /><br />Evidence from other sources known since the late nineteenth century also tends to<br />confirm the great antiquity of the Vedic tradition. Certain Vedic texts, for example,<br />refer to astronomical events that took place in ancient astronomical time. By calculating<br />the astronomical dates of these events, we thus gain another source of evidence that can<br />be used to place the Rig Veda in a calculable time-frame.<br /><br /><br />A German scholar and an Indian scholar simultaneously discovered in 1889 that the<br />Vedic Brahmana texts describe the Pleiades coinciding with the spring equinox. Older<br />texts describe the spring equinox as falling in the constellation Orion. From a<br />calculation of the precision of the equinoxes, it has been shown that the spring equinox lay in Orion in about 4,500 BC.<br /><br /><br /><br />The German scholar, H. Jacobi, came to the conclusion that the Brahmanas are from a<br />period around or older than 4,500 BC. Jacobi concludes that “the Rig Vedic period of<br />culture lies anterior to the third pre-Christian millennium.”22 B. Tilak, using similar astronomical calculations, estimates the time of the Rig Veda at 6,000 BC.23<br />More recently, Frawley has cited references in the Rig Veda to the winter solstice<br />beginning in Aries. On this basis, he estimates that the antiquity of these verses of the Veda must go back at least to at least 6,500 BC.24 The dates Frawley gives for Vedic civilization are:<br /><br />Period 1. 6500-3100 BC, Pre-Harappan, early Rig Vedic<br />Period 2. 3100-1900 BC, Mature Harappan 3100-1900, period of the Four Vedas<br />Period 3. 1900-1000 BC, Late Harappan, late Vedic and Brahmana period<br /><br />Professor Dinesh Agrawal of Penn State University reviewed the evidence from a<br />variety of sources and estimated the dates as follows:<br /><br />• Rig Vedic Age - 7000-4000 BC<br />• End of Rig Vedic Age - 3750 BC<br />• End of Ramayana-Mahabharat Period - 3000 BC<br />• Development of Saraswati-Indus Civilization - 3000-2200 BC<br />• Decline of Indus and Saraswati Civilization - 2200-1900 BC<br />• Period of chaos and migration - 2000-1500 BC<br />• Period of evolution of syncretic Hindu culture - 1400-250 BC<br /><br />The Taittiriya Samhita (6.5.3) places the constellation Pleiades at the winter solstice, which correlates with astronomical events that took place in 8,500 BC at the earliest.<br /><br />The Taittiriya Brahmana (3.1.2) refers to the Purvabhadrapada nakshatra as rising due<br />east—an event that occurred no later than 10,000 BC, according to Dr. B.G.Siddharth<br />of India’s Birla Science Institute. Since the Rig Veda is more ancient than the<br />Brahmanas, this would put the Rig Veda before 10,000 BC.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-2154816082512267662011-09-17T22:11:00.001-07:002011-09-18T02:36:31.197-07:00Ramayana Garden in Thailand<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn6YtCcmebELu4WChRyL6TDxGlrsqy3SMN_q0IfucJJLbM4oRu-CYJhw4lt-aN2oL4HAZMZQYmf9GfNKWT6hRUSzyQZbJEWkXekZqBuy8f8BgCUWFsZX-q8NjOBpwrBu5AWgPdVFsOtn4/s1600/DSC01570.JPG"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn6YtCcmebELu4WChRyL6TDxGlrsqy3SMN_q0IfucJJLbM4oRu-CYJhw4lt-aN2oL4HAZMZQYmf9GfNKWT6hRUSzyQZbJEWkXekZqBuy8f8BgCUWFsZX-q8NjOBpwrBu5AWgPdVFsOtn4/s320/DSC01570.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5653561094138696338" /></a><br /><br /><br />This is the photo of the Ramayana Garden in Thailand. <br /><br />The first poem and the greatest was Ramayana, opined Aurobindo. When the poet Valmiki saw two birds mating and a native killing one of the two said " Ma Nishada ", meaning " Dont do that, for it is Sin, it is Evil, it is against the Law of Conscience"!<br /><br /><span style="font-style:italic;">Ma nishada pratishtantva nigama shashvathi sama<br />Yat krouncha midhunath eva avadhim kamamohitam<br /></span><br />Ramayana is an allegorical and epic poem. As the Aryan Invasion Theorists argue, it is the not onslaught of the " Aryans" against the " Dravidians". The Aryan Invasion Theory has been proved wrong by scholars like Dr Konrad Elst, Dr Dennis Chandler, Aurobindo and Vivekananda. <br /><br /><iframe width="480" height="315" src="http://www.youtube.com/embed/DwVZeCLEWrE" frameborder="0" allowfullscreen></iframe><br /><br />Man's dual aspect is highlighted in this poem. The Ego is the imperfect term in Man and Self or the Divine in the perfect term. The Ego is the lower imperfect term of our being; Self is the higher, perfect term. <br /><br />The Ego is symbolised as Ravana, the ten headed demon king. His ten heads represent Mind, Intelligence, Processing Mind, Ego, Lust, Greed, Anger, Avarice, Jealousy, Gluttony and Sloth.The Self is symbolised by Rama, who goes in search of Peace, Seetha, his beloved consort. The great struggle for supremacy between good and evil, light and darkness, prosperity and adversity, happening in the human bosom, is the Ramayana, where the forces of the Ego clash with the forces of the Self. <br /><br />Ramayana is part of Buddhist lore and the Ramayana Garden is a testimony to this fact. This strengthens P N Oak's theory that the mother civilization of all was Vedic.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-43151304342762225912011-09-15T21:06:00.000-07:002011-09-15T21:07:03.394-07:00Saturn in Thailand<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRsPvdThSVw3HLNWiTCzGHYwMRtRUHxaH4g9rONglLWSJeAYFh9BLTU82F8e0NuL6NyFzNqThpXXo2aX0_7NdvtNMlv6mwSYCVMX2TaN0N-vrZBIlIqpuH_oU1aHnq7Wif9mPIbyDc_EM/s1600/DSC01547.JPG"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 320px; height: 240px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRsPvdThSVw3HLNWiTCzGHYwMRtRUHxaH4g9rONglLWSJeAYFh9BLTU82F8e0NuL6NyFzNqThpXXo2aX0_7NdvtNMlv6mwSYCVMX2TaN0N-vrZBIlIqpuH_oU1aHnq7Wif9mPIbyDc_EM/s320/DSC01547.JPG" border="0" alt=""id="BLOGGER_PHOTO_ID_5652796927030247714" /></a><br /><br />Shani, the deity of Justice, Destiny and Retribution, has His presence in Thailand. In ancient Siam, one can see his statue.<br /><br />Saturn, the melancholy planet, was rising at the birth of Dante. He is prominent in the horoscopes of Goethe and other poets and philosophers like Sankara. Without His Grace, liberation is not possible. <br /><br />The most ancient Civilization was the Vedic and Dr Kenneth Chandler puts the origin at 4900 BC. According to him, the Aryan Invasion Theory is not correct. <br /><br />Will Durant opined that India " is the mother of us all, through Sanskrit, the mother of Europe's languages".<br /><br />Dr Kenneth Chandler writes<br /><br />"The original theory proposed by the early historical linguistics who considered these issues was that Vedic Sanskrit conserved the original sound system of the “proto-Indo-European” language most closely, and that Iranian and European languages underwent a systematic sound shift, creating break-away or daughter languages spoken by the people who populated India and Europe. According to this theory, Vedic Sanskrit was put at near the trunk of the proto-Indo-European language tree, if not the trunk itself.<br /><br />This theory has been challenged and hotly debated in recent years, most especially by<br />computer linguists.<br /><br />Since the 1990s, it is now common for computer linguists to hold<br />that Sanskrit is not so near the root of the Indo-European language tree, but a<br />subsequent branch. A currently dominant theory is that the original Indo-European<br />language stemmed from an Indo-European proto-language that has since been lost.<br />The first languages to break off from the proto-Indo-European root, according to the<br />dominant contemporary linguistic theories, was Anatolian (the language of what is now<br />central Turkey), followed by Celtic (a language found in nearby Thrace in northeastern Greece, and also Ireland suggesting that there was a commerce or colonization between Ireland and early Thrace), then Greek, and then Armenian. According to these theories, the Indian and Iranian language groups are still later branches off the proto-Indo-European “root.”<br /><br />The linguistic evidence appears to imply migrations of people from the Black Sea area<br />into India, and yet there is no anthropological evidence to support either a migration into northern India, or an invasion. Evidence from skeletal remains, as we saw, as well as pottery and other artifacts, show no cultural replacement at any time in north Indian Thomas V. Gamkrelidze and V. V. Ivanov, “Family Tree of the Indo-European Languages,” Scientific American, March, 1990, p. 110 and following.<br /><br />Dr. Don Ringe and Dr. Ann Taylor, two linguists at the University of Pennsylvania, with the help of computer scientist Dr. Tandy Warnow, developed a computer algorithm to sift through the Indo- European languages and look for grammatical and phonetic similarities between them. Their work, published in 1996, has thrown up four possible family trees. “We have come up with a favorite,” says Dr. Warnow. The tree shows that the first breakaway language was Anatolian, an ancient group of languages once spoken in Turkey. Celtic was quick to follow, spawning Irish, Gaelic, Welsh and Breton. Armenian and Greek then developed from proto-Into-European. Strangely enough, one of the later branches to emerge, according to the runs of the computer programs, was Sanskrit.<br /><br />It is interesting that the Celts settled in Thrace in northern Greece, just a short distance from Anatolia. Thrace was the birthplace of the Orphic mysteries which swept into Greece in the sixth century BC. Celtic is one of the earliest languages, along with Anatolian and Greek, to break off from the Indo European proto-language. The technique for self-knowledge described by Socrates were said to have come from Thrace. The Anatolians of central Turkey occupied the area near where the pre-Socratic tradition sprang up in the sixth century BC. This suggests that a technique was passed from India into the Celtic language.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-26532751443464776172011-09-01T21:24:00.001-07:002011-09-01T21:25:45.393-07:00Of Intercalary months, Adhi Masas<iframe width="480" height="345" src="http://www.youtube.com/embed/ZlYeWO5wbhk" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />The solar month is 30.438030202068 days
<br />
<br />A lunar month = 29.5305881 days
<br />
<br />It need not be added that a lunation or synodic month means the interval between two consecutive full moons or new moons. Conjunction ( New Moon ) is 0 degrees and Opposition ( Full Moon ) is 180 degrees
<br />
<br />Hence a solar year does not have a whole number of lunar months ( about 12.37 lunations ) So a thirteenth embolismic or intercalary month is inserted.
<br />
<br />It was observed that 19 solar years or 19*12 = 228 solar months = 235 lunations and hence 7 Adhi Masas were found in every 19 years. An intercalary or 13th month had to be inserted in a 19 year cycle and 19/7 was the ratio. .
<br />
<br />They are called <span style="font-style:italic;">Adhi Masas </span>in Indian Astronomy and they were computed using the Theory of continued fractions. The Theory of contiued Fractions is attributed to Euler. This 19 year old cycle is called the Metonic Cycle, named after the Greek astronomer, Meton.
<br />
<br />But then the Indian mathematicians correctly computed the <span style="font-style:italic;">Adhi Masas</span>, centuries before Meton or Euler ! The Indian National Calender is lunisolar, whose dates both indicate the solar year and the moon phases and the next date when the New Moon or Full Moon will occur. The length of the synodic month is given as 29.5305879 days in the <span style="font-style:italic;">Surya Siddhanta</span>, which is correct to six decimals. <span style="font-style:italic;">Surya Siddhanta</span> stated that there are 15933396 Adhi Masas in 51840000 solar months ! Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-90916673277700620392011-08-30T20:43:00.000-07:002011-08-31T00:33:01.914-07:00Of Vedic Maths<iframe width="480" height="345" src="http://www.youtube.com/embed/kZKOPKIHsrc" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />Consisting of 16 basic aphorisms or Sutras, Vedic Mathematics is a system of Maths which prevailed in ancient India. Composed by Bharati Krishna Thirtha, these 16 sutras help one to do faster maths.
<br />
<br />The first aphorism is this
<br />
<br />"Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)"
<br />
<br />When computing the square of 9, as the nearest power of 10 is 9, let us take 10 as our base. As 9 is 1 less than 10, we can decrease it by the deficiency = 9-1 =8. This is the leftmost digit
<br />On the right hand put deficiency^2, which is 1^2.
<br />
<br />Hence the square of nine is 81.
<br />
<br />For numbers above 10, instead of looking at the deficit we look at the surplus.
<br />
<br />
<br />
<br />For example:
<br />
<br />
<br />11^2 = (11+1)*10+1^2 = 121
<br />
<br />12^2 = (12+2)*10+2^2 = 144
<br />
<br />14^2 = ( 14+4)*10+4^2 = 196
<br />
<br />25^2 = ((25+5)*2)*10+5^2 = 625
<br />
<br />35^2= ((35+5)*3)*10+5^2 = 1225
<br />
<br />
<br />
<br />
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com2tag:blogger.com,1999:blog-7836607831130520841.post-69576473172606024062011-08-27T21:42:00.001-07:002011-08-27T21:42:45.053-07:00Indian Maths, on Argive heights divinely sang !<iframe width="480" height="345" src="http://www.youtube.com/embed/x22o8TUdOuw" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />In India, mathematics is related to Philosophy. We can find mathematical
<br />concepts like Zero ( Shoonyavada ), One ( Advaitavada ) and Infinity
<br />(Poornavada ) in Philosophia Indica.
<br />
<br />The Sine Tables of Aryabhata and Madhava, which gives correct sine values or values of
<br />24 R Sines, at intervals of 3 degrees 45 minutes and the trignometric tables of
<br />Brahmagupta, which gives correct sine and tan values for every 5 degrees influenced
<br />Christopher Clavius, who headed the Gregorian Calender Reforms of 1582. These
<br />correct trignometric tables solved the problem of the three Ls, ( Longitude, Latitude and
<br />Loxodromes ) for the Europeans, who were looking for solutions to their navigational
<br />problem ! It is said that Matteo Ricci was sent to India for this purpose and the
<br />Europeans triumphed with Indian knowledge !
<br />
<br />The Western mathematicians have indeed lauded Indian Maths & Astronomy. Here are
<br />some quotations from maths geniuses about the long forgotten Indian Maths !
<br />
<br />In his famous dissertation titled "Remarks on the astronomy of Indians" in 1790,
<br />the famous Scottish mathematician, John Playfair said
<br />
<br />"The Constructions and these tables imply a great knowledge of
<br />geometry,arithmetic and even of the theoretical part of astronomy.But what,
<br />without doubt is to be accounted,the greatest refinement in this system, is
<br />the hypothesis employed in calculating the equation of the centre for the
<br />Sun,Moon and the planets that of a circular orbit having a double
<br />eccentricity or having its centre in the middle between the earth and the
<br />point about which the angular motion is uniform.If to this we add the great
<br />extent of the geometrical knowledge required to combine this and the other
<br />principles of their astronomy together and to deduce from them the just
<br />conclusion;the possession of a calculus equivalent to trigonometry and
<br />lastly their approximation to the quadrature of the circle, we shall be
<br />astonished at the magnitude of that body of science which must have
<br />enlightened the inhabitants of India in some remote age and which whatever
<br />it may have communicated to the Western nations appears to have received
<br />another from them...."
<br />
<br />Albert Einstein commented "We owe a lot to the Indians, who taught us how to count,
<br />without which no worthwhile scientific discovery could have been made."
<br />
<br />The great Laplace, who wrote the glorious Mechanique Celeste, remarked
<br />
<br />"The ingenious method of expressing every possible number
<br />using a set of ten symbols (each symbol having a place value and an absolute
<br />value) emerged in India. The idea seems so simple nowadays that its
<br />significance and profound importance is no longer appreciated. Its
<br />simplicity lies in the way it facilitated calculation and placed arithmetic
<br />foremost amongst useful inventions. The importance of this invention is more
<br />readily appreciated when one considers that it was beyond the two greatest
<br />men of antiquity, Archimedes and Apollonius."
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-56435446213003836392011-08-26T19:55:00.001-07:002011-08-26T19:55:37.610-07:00The Infinite Series of the Pi function of Madhava<span style="font-style:italic;">By means of the same argument, the circumference can be computed in another way too. That is as (follows): The first result should by the square root of the square of the diameter multiplied by twelve. From then on, the result should be divided by three (in) each successive (case). When these are divided in order by the odd numbers, beginning with 1, and when one has subtracted the (even) results from the sum of the odd, (that) should be the circumference. ( Yukti deepika commentary )
<br /></span>
<br />
<br />This quoted text specifies another formula for the computation of the circumference c of a circle having diameter d. This is as follows.
<br />
<br />
<br />c = SQRT(12 d^2 - SQRT(12 d^2/3.3 + sqrt(12 d^2)/3^2.5 - sqrt(12d^2)/3^3.7 +.......
<br />
<br />
<br />As c = Pi d , this equation can be rewritten as
<br />
<br />
<br />Pi = Sqrt(12( 1 - 1/3.3 + 1/3^2.5 -1/3^3.7 +......
<br />
<br />
<br />This is obtained by substituting z = Pi/ 6 in the power series expansion for arctan (z).
<br />
<br />
<br />Pi/4 = 1 - 1/3 +1/5 -1/7+.....
<br />
<br />
<br />This is Madhava's formula for Pi, and this was discovered in the West by Gregory and Liebniz.
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-74580835208531892932011-08-25T21:20:00.001-07:002011-09-16T03:50:49.947-07:00Arctangent series of Madhava, Gregory and Liebniz<iframe width="480" height="345" src="http://www.youtube.com/embed/Xw8PEH8aFTk" frameborder="0" allowfullscreen></iframe><br /><br /><br />The inverse tangent series of Madhava is given in verse 2.206 – 2.209 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar . It is also given by Jyeshtadeva in Yuktibhasha and a translation of the verses is given below.<br /><br /><span style="font-style:italic;"><br />Now, by just the same argument, the determination of the arc of a desired sine can be (made). That is as follows: The first result is the product of the desired sine and the radius divided by the cosine of the arc. When one has made the square of the sine the multiplier and the square of the cosine the divisor, now a group of results is to be determined from the (previous) results beginning from the first. When these are divided in order by the odd numbers 1, 3, and so forth, and when one has subtracted the sum of the even(-numbered) results from the sum of the odd (ones), that should be the arc. Here the smaller of the sine and cosine is required to be considered as the desired (sine). Otherwise, there would be no termination of results even if repeatedly (computed).</span><br /><br />Rendering in modern notations<br /><br />Let s be the arc of the desired sine, bhujajya, y. Let r be the radius and x be the cosine (kotijya).<br /><br />The first result is y.r/x <br />From the divisor and multiplier y^2/x^2 <br />From the group of results y.r/x.y^2/x^2, y.r/x. y^2/x^2.y^2/x^2 <br />Divide in order by number 1,3 etc<br />1 y.r/1x, 1y.r/3x y^2/x^2, 1y.r/5x.y^2/x^2.Y^2/x^2<br /><br />a = (Sum of odd numbered results) 1 y.r/1x + 1y.r/5x.y^2/x^2.y^2/x^2+......<br /><br />b= ( Sum of even numbered results) 1y.r/3x.y^2/x^2 + 1 y.r/7x.y^2/x^2.y^2/x^.y^2/x^2+..... <br /><br />The arc is now given by <br />s = a - b <br /><br />Transformation to current notation<br /><br />If x is the angle subtended by the arc s at the Center of the Circle, then s = rx and kotijya = r cos x and bhujajya = r sin x. And sparshajya = tan x <br /><br />Simplifying we get <br /><br />x = tan x - tan^3x/'3 + tan^5x/5 - tan^7x/7 + .....<br /><br />Let tan x = z, we have<br /><br />arctan ( z ) = z - z^3/3 + z^5/5 - z^7/7<br /><br />We thank <a href="http://www.wikipedia.org">www.wikipedia.org</a> for publishing this on their site.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-7124635140716553582011-08-24T21:32:00.001-07:002011-08-24T21:32:54.153-07:00The Madhava cosine series<iframe width="480" height="345" src="http://www.youtube.com/embed/D8zZZZzppIM" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />Madhava's cosine series is stated in verses 2.442 and 2.443 in Yukti-dipika commentary (Tantrasamgraha-vyakhya) by Sankara Variar. A translation of the verses follows.
<br />
<br /> <span style="font-style:italic;">
<br />
<br />Multiply the square of the arc by the unit (i.e. the radius) and take the result of repeating that (any number of times). Divide (each of the above numerators) by the square of the successive even numbers decreased by that number and multiplied by the square of the radius. But the first term is (now)(the one which is) divided by twice the radius. Place the successive results so obtained one below the other and subtract each from the one above. These together give the śara as collected together in the verse beginning with stena, stri, etc.
<br /></span>
<br />
<br />
<br />Let r denote the radius of the circle and s the arc-length.
<br />
<br />The following numerators are formed first:
<br />
<br />s.s^2,
<br />s.s^2.s^2
<br />s.s^2.s^2.s^2
<br />
<br />These are then divided by quantities specified in the verse.
<br />
<br />1)s.s^2/(2^2-2)r^2,
<br />2)s. s^2/(2^2-2)r^2. s^2/4^2-4)r^2
<br />3)s.s^2/(2^2-2)r^2.s^2/(4^2-4)r^2. s^2/(6^2-6)r^2
<br />
<br />
<br />As per verse,
<br />
<br />sara or versine = r.(1-2-3)
<br />
<br />Let x be the angle subtended by the arc s at the center of the Circle. Then s = rx and sara or versine = r(1-cosx)
<br />
<br />Simplifying we get the current notation
<br />
<br />1-cosx = x^2/2! -x^4/4!+ x^6/6!......
<br />
<br />which gives the infinite power series of the cosine function.Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-76074475029402026872011-08-23T22:30:00.001-07:002011-08-23T22:30:15.007-07:00The Madhava Trignometric Series<iframe width="480" height="345" src="http://www.youtube.com/embed/jFImjntTtts" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />The Madhava Trignometric series is one one of a series in a collection of infinite series expressions discovered by Madhava of Sangramagrama ( 1350-1425 ACE ), the founder of the Kerala School of Astronomy and Mathematics. These are the infinite series expansions of the Sine, Cosine and the ArcTangent functions and Pi. The power series expansions of sine and cosine functions are called the Madhava sine series and the Madhava cosine series.
<br />
<br />The power series expansion of the arctangent function is called the Madhava- Gregory series.
<br />
<br />The power series are collectively called as Madhava Taylor series. The formula for Pi is called the Madhava Newton series.
<br />
<br />One of his disciples, Sankara Variar had translated his verse in his Yuktideepika commentary on Tantrasamgraha-vyakhya, in verses 2.440 and 2.441
<br />
<br /><span style="font-style:italic;">
<br />Multiply the arc by the square of the arc, and take the result of repeating that (any number of times). Divide (each of the above numerators) by the squares of the successive even numbers increased by that number and multiplied by the square of the radius. Place the arc and the successive results so obtained one below the other, and subtract each from the one above. These together give the jiva, as collected together in the verse beginning with "vidvan" etc.</span>
<br />
<br />
<br />Rendering in modern notations
<br />
<br />Let r denote the radius of the circle and s the arc-length.
<br />
<br />The following numerators are formed first:
<br />
<br />s.s^2,
<br />s.s^2.s^2
<br />s.s^2.s^2.s^2
<br />
<br />These are then divided by quantities specified in the verse.
<br />
<br />1)s.s^2/(2^2+2)r^2,
<br />2)s. s^2/(2^2+2)r^2. s^2/4^2+4)r^2
<br />3)s.s^2/(2^2+2)r^2.s^2/(4^2+4)r^2. s^2/(6^2+6)r^2
<br />
<br />Place the arc and the successive results so obtained one below the other, and subtract each from the one above to get jiva:
<br />
<br />
<br />Jiva = s-(1-2-3)
<br />
<br />When we transform it to the current notation
<br />
<br />If x is the angle subtended by the arc s at the center of the Circle, then s = rx and jiva = r sin x.
<br />
<br />
<br />Sin x = x - x^3/3! + x^5/5! - x^7/7!...., which is the infinite power series of the sine function.
<br />
<br />
<br />By courtesy www.wikipedia.org and we thank <a href="http://www.wikipedia.org"> Wikipedia</a> for publishing this on their site. Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-17091993616129222852011-08-21T01:18:00.001-07:002011-08-21T04:26:05.397-07:00Calculus, India's gift to Europe<iframe width="480" height="345" src="http://www.youtube.com/embed/RazMSHy258M" frameborder="0" allowfullscreen></iframe>
<br />
<br />
<br />The Jesuits took the trignometric tables and planetary models from the Kerala School of Astronomy and Maths and exported it to Europe starting around 1560 in connection with the European navigational problem, says Dr Raju.
<br />
<br />Dr C K Raju was a professor Mathematics and played a leading role in the C-DAC team which built Param: India’s first parallel supercomputer. His ten year research included archival work in Kerala and Rome and was published in a book called " The Cultural Foundations of Mathematics". He has been a Fellow of the Indian Institute of Advanced Study and is a Professor of Computer Applications.
<br />
<br />“When the Europeans received the Indian calculus, they couldn’t understand it properly because the Indian philosophy of mathematics is different from the Western philosophy of mathematics. It took them about 300 years to fully comprehend its working. The calculus was used by Newton to develop his laws of physics,” opines Dr Raju.
<br />
<br />
<br />
<br />The Infinitesimal Calculus: How and Why it Was Imported into Europe
<br />
<br />By Dr C.K. Raju
<br />
<br />It is well known that the “Taylor-series” expansion, that is at the heart of calculus, existed in India in widely distributed mathematics / astronomy / timekeeping (“jyotisa”) texts which preceded Newton and Leibniz by centuries.
<br />
<br />Why were these texts imported into Europe? These texts, and the accompanying precise sine values computed using the series expansions, were useful for the science that was at that time most critical to Europe: navigation. The ‘jyotisa’ texts were specifically needed by Europeans for the problem of determining the three “ells”: latitude, loxodrome, and longitude.
<br />
<br />How were these Indian texts imported into Europe? Jesuit records show that they sought out these texts as inputs to the Gregorian calendar reform. This reform was needed to solve the ‘latitude problem’ of European navigation. The Jesuits were equipped with the knowledge of local languages as well as mathematics and astronomy that were required to understand these Indian texts.
<br />
<br />The Jesuits also needed these texts to understand the local customs and how the dates of traditional festivals were fixed by Indians using the local calendar (“panchânga”). How the mathematics given in these Indian ancient texts subsequently diffused into Europe (e.g. through clearing houses like Mersenne and the works of Cavalieri, Fermat, Pascal, Wallis, Gregory, etc.) is yet another story.
<br />
<br />The calculus has played a key role in the development of the sciences, starting from the “Newtonian Revolution”. According to the “standard” story, the calculus was invented independently by Leibniz and Newton. This story of indigenous development, ab initio, is now beginning to totter, like the story of the “Copernican Revolution”.
<br />
<br />The English-speaking world has known for over one and a half centuries that “Taylor series” expansions for sine, cosine and arctangent functions were found in Indian mathematics / astronomy / timekeeping (‘jyotisa’) texts, and specifically in the works of Madhava, Neelkantha, Jyeshtadeva, etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.
<br />
<br />The connection is provided by the requirements of the European navigational problem, the foremost problem of the time in Europe. Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. Navigation, however, was both strategically and economically the key to the prosperity of Europe of that time.
<br />
<br />Accordingly, various European governments acknowledged their ignorance of navigation while announcing huge rewards to anyone who developed an appropriate technique of navigation. These rewards spread over time from the appointment of Nunes as Professor of Mathematics in 1529, to the Spanish government’s prize of 1567 through its revised prize of 1598, the Dutch prize of 1636, Mazarin’s prize to Morin of 1645, the French offer (through Colbert) of 1666, and the British prize legislated in 1711.
<br />
<br />Many key scientists of the time (Huygens, Galileo, etc.) were involved in these efforts. The navigational problem was the specific objective of the French Royal Academy, and a key concern for starting the British Royal Society.
<br />
<br />Prior to the clock technology of the 18th century, attempts to solve the European navigational problem in the 16th and 17th centuries focused on mathematics and astronomy. These were (correctly) believed to hold the key to celestial navigation. It was widely (and correctly) held by navigational theorists and mathematicians (e.g. by Stevin and Mersenne) that this knowledge was to be found in the ancient mathematical, astronomical and time-keeping (jyotisa) texts of the East.
<br />
<br />Though the longitude problem has recently been highlighted, this was preceded by the latitude problem and the problem of loxodromes. The solution of the latitude problem required a reformed calendar. The European calendar was off by ten days. This led to large inaccuracies (more than 3 degrees) in calculating latitude from the measurement of solar altitude at noon using, for example, the method described in the Laghu Bhâskarîya of Bhaskara I.
<br />
<br />However, reforming the European calendar required a change in the dates of the equinoxes and hence a change in the date of Easter. This was authorised by the Council of Trent in 1545. This period saw the rise of the Jesuits. Clavius studied in Coimbra under the mathematician, astronomer and navigational theorist Pedro Nunes. Clavius subsequently reformed the Jesuit mathematical syllabus at the Collegio Romano. He also headed the committee which authored the Gregorian Calendar Reform of 1582 and remained in correspondence with his teacher Nunes during this period.
<br />
<br />Jesuits such as Matteo Ricci who trained in mathematics and astronomy under Clavius’ new syllabus were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand the local methods of time-keeping (‘jyotisa’) from the Brahmins and Moors in the vicinity of Cochin.
<br />
<br />Cochin was then the key centre for mathematics and astronomy since the Vijaynagar Empire had sheltered it from the continuous onslaughts of Islamic raiders from the north. Language was not a problem for the Jesuits since they had established a substantial presence in India. They had a college in Cochin and had even established printing presses in local languages like Malayalam and Tamil by the 1570’s.
<br />
<br />In addition to the latitude problem (that was settled by the Gregorian Calendar Reform), there remained the question of loxodromes. These were the focus of efforts of navigational theorists like Nunes and Mercator.
<br />
<br />The problem of calculating loxodromes is exactly the problem of the fundamental theorem of calculus. Loxodromes were calculated using sine tables. Nunes, Stevin, Clavius, etc. were greatly concerned with accurate sine values for this purpose, and each of them published lengthy sine tables. Madhava’s sine tables, using the series expansion of the sine function, were then the most accurate way to calculate sine values.
<br />
<br />Madhava's sine series
<br />
<br />sin x = x - x^3/3! + x^5/5! - x^7/7!+......
<br />
<br />
<br />The Europeans encountered difficulties in using these precise sine values for determining longitude, as in the Indo-Arabic navigational techniques or in the Laghu Bhâskarîya. This is because this technique of longitude determination also required an accurate estimate of the size of the earth. Columbus had underestimated the size of the earth to facilitate funding for his project of sailing to the West. His incorrect estimate was corrected in Europe only towards the end of the 17th century CE.
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<br />Even so, the Indo-Arabic navigational technique required calculations while the Europeans lacked the ability to calculate. This is because algorismus texts had only recently triumphed over abacus texts and the European tradition of mathematics was “spiritual” and “formal” rather than practical, as Clavius had acknowledged in the 16th century and as Swift (of ‘Gulliver’s Travels’ fame) had satirized in the 17th century. This led to the development of the chronometer, an appliance that could be mechanically used without any application of the mind.
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-36486875734377122422011-08-20T05:23:00.001-07:002011-08-20T05:23:17.337-07:00The Idea of Planetary Mass in India<iframe width="480" height="345" src="http://www.youtube.com/embed/IP2SDbhFbXk" frameborder="0" allowfullscreen></iframe>
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<br />
<br />Many ancient cultures have contributed to the development of Astro Physics.
<br />
<br />Some examples are
<br />
<br />The Saros cycles of eclipses discovered by Egyptians
<br />The classification of stars by the Greeks
<br />Sunspot observations of the Chinese
<br />The phenomenon of Retrogression discovered by Babylonians
<br />
<br />In this context the Indian contribution to Astro Physics ( which includes Astronomy, Maths and Astrology ) is the the development of the ideas of planetary forces and differential equations to calculate the geocentric planetary longitudes, several centuries before the European Renaissance.
<br />
<br />Natural Strength is one of the Sixfold Strengths,<span style="font-style:italic;"> Shad Balas </span>and goes by the name <span style="font-style:italic;">Naisargika Bala</span>. It is directly proportional to the size of the celestial bodies and inversely proportional to the geocentric distance. (<span style="font-style:italic;"> Horasara</span> ).
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<br /><span style="font-style:italic;">Naisargika Bala</span> or Natural Strength is used to compare planetary physical forces. When two planets occupy the same, identical position in the Zodiac at a given instant of time, such a phenomenon goes by the name of planetary war or <span style="font-style:italic;">Graha Yuddha</span>,happening when two planets are in close conjunction. The <span style="font-style:italic;">Karanaratna</span> written by Devacharya explains that the planet with the larger diameter is the victor in this planetary war. This implies <span style="font-style:italic;">Naisargika Bala.
<br /></span>
<br />The <span style="font-style:italic;">Surya Siddhanta</span> says " The dynamics or quantity of motion produced by the action of a fixed force to different planetary objects is inversely related to the quantity of matter in these objects"
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<br />This definition more or less equals the statement of Newton’s second law of motion
<br />
<br />M = Fa
<br />or
<br />a = F/M
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<br />So it strongly suggests that the idea of planetary mass was known to the ancient Indian astronomers and mathematicians. Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-64069083364142055712011-08-18T21:33:00.001-07:002011-08-18T21:41:30.628-07:00Differential Equations used in Siddhantas<iframe width="480" height="345" src="http://www.youtube.com/embed/e08l0rxo9RU" frameborder="0" allowfullscreen></iframe>
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<br />
<br />
<br />Motional strength is one the sixfold strengths, known as Cheshta Bala. This motional strength is computed by the formula
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<br />Motional Strength = 0.33 ( Sheegrocha or Perigee - geocentric longitude of the planet ). This motional strength is known as Cheshta Bala.
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<br />
<br />Differential Calculus is the science of rates of the change. If y is the longitude of the planet and t is time, then we have the differential equation ,dy/dt.
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<br />During direct motion, we find that dy/dt > 0 and during retrogression dy/dt < 0. During backward motion of the planet ( retrogression) y decreases with time and during direct motion y increases with time. When there are turning points known as Vikalas or stationary points, we have dy/dt = 0 ( where planets like Mars will appear to be stationary for an observer on Earth ).
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<br />The quantity in bracket is the Sheegra Anomaly, the Anomaly of Conjuction, the angular distance of the planet from the Sun. This Anomaly or Cheshta Bala is maximum at the center of the Retrograde Loop. Cheshta Kendra is defined as the Arc of Retrogression and is the same as Sheegra Kendra, Kendra being an angle in Sanskrit. During Opposition, when the planet is 180 degrees from the Sun, Cheshta Bala is maximum and during Conjunction, when the planet is 0 degrees from the Sun, it is minimum
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<br />The Motional Strength is given in units of 60s, Shashtiamsas.
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<br />Direct motion ( Anuvakra ) 30
<br />Stationary point ( Vikala ) 15
<br />Very slow motion ( Mandatara ) 7.5
<br />Slow motion ( Manda ) 15
<br />Average speed ( Sama ) 30
<br />Fast motion ( Chara ) 30
<br />Very fast motion ( Sheegra Chara ) 45
<br />Max orbital speed ( Vakra ) 60
<br />(Centre of retrograde)
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-69591096927796515332011-08-17T21:04:00.001-07:002011-08-17T21:04:18.917-07:00The Nine Oribtal Elements<iframe width="480" height="345" src="http://www.youtube.com/embed/PzgXE1XU0Gg" frameborder="0" allowfullscreen></iframe>
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<br />
<br />Mean and true planetary longitudes in the Zodiac is computed by Nine Orbital Elements, in Indian Astronomy.
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<br />Mean longitude of Planet, <span style="font-style:italic;">Graha Madhyama </span>, M
<br />Daily Motion of the Mean Longitude, <span style="font-style:italic;">Madhyama Dina Gathi</span>, Md
<br />Aphelion, <span style="font-style:italic;">Mandoccha</span>, Ap
<br />Daily Motion of Aphelion, <span style="font-style:italic;">Mandoccha Dina Gathi</span>, Apd
<br />Ascending Node, <span style="font-style:italic;">Patha</span>, N
<br />Daily Motion of Ascending Node, <span style="font-style:italic;">Patha Dina Gath</span>i, Nd
<br />Heliocentric Distance,<span style="font-style:italic;"> Manda Karna</span>, radius vector, mndk
<br />Maximum Latitude, L,<span style="font-style:italic;"> Parama Vikshepa</span>
<br />Eccentricity,<span style="font-style:italic;"> Chyuthi</span>,e
<br />
<br />
<br />In Western Astronomy, we have six orbital elements
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<br />Mean Anomaly, m
<br />Argument of Perihelion, w
<br />Eccentricity, e
<br />Ascending Node, N
<br />Inclination, i, inclinent of orbit
<br />Semi Major Axis, a
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<br />With the Nine Orbital Elements, true geocentric longitude of the planet is computed, using multi step algorithms.
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<br />There is geometrical equivalence between both the Epicycle and the Eccentric Models.
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<br />The radius of the Epicycle, r = e, the distance of the Equant from the Observer.
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-9874408192239998082011-08-14T21:21:00.000-07:002011-08-14T22:30:00.533-07:00Astronomical Units of Time Measurement<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeBk3_ZHIGvHJhPNk81d1ZecJWkPwQvJuwPY5czcmgSbwGwJqtdEQrOaUDUZcdJJw2iBBfd0UgAy3bMvsFR3ymfeVg4JmzxK2zFMc7iNbTUqPBk3GRWmINNChuBLUEmIlbBjgTbx4r5LI/s1600/mahamanvantara.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width:100px; height: 520px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeBk3_ZHIGvHJhPNk81d1ZecJWkPwQvJuwPY5czcmgSbwGwJqtdEQrOaUDUZcdJJw2iBBfd0UgAy3bMvsFR3ymfeVg4JmzxK2zFMc7iNbTUqPBk3GRWmINNChuBLUEmIlbBjgTbx4r5LI/s320/mahamanvantara.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5640933366165505746" /></a>
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<br />We find <span style="font-style:italic;">Yuga </span>cylces mentioned not only in astronomical works, but also in mythological works in India.
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<br /><span style="font-style:italic;">Kali Yuga</span> began on the midnight of 17th Feb 3102 BCE and the duration of this <span style="font-style:italic;">Kali</span> <span style="font-style:italic;">Yuga</span> is said to be 4.32 K solar years. <span style="font-style:italic;">Dwapara</span> is 2*Kali Yuga years. <span style="font-style:italic;">Treta</span> is 3*K Y and <span style="font-style:italic;">Krita Yuga</span> is 4*K Y.
<br /><span style="font-style:italic;">
<br />
<br />Krita Treta Dwaparascha Kalischaiva Chaturyugam
<br />Divya Dwadasabhir varshai savadhanam niroopitham
<br /></span>
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<br />Thus an Equinoctial Cycle, <span style="font-style:italic;">Mahayuga</span> is equal to 4+3+2+1 = 10 KYs.
<br />
<br />E C = 10 KYs.
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<br />A Greater Equinoctial Cycle ( <span style="font-style:italic;">Manvantara</span> ) = 71 Equinoctial Cycles
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<br />There are cusps happening in between<span style="font-style:italic;"> Manvantaras,</span> each equal to a <span style="font-style:italic;">Krita Yuga</span> in duration. A Krita is equal to 4 KYs or 2/5 of a Maha Yuga. Since there will 15 such cusps happening amongst the Fourteen Manvantaras, they are equal to 15*2/5 = 6 Mahayugas.
<br />
<br />Hence 14*71+6 = 1000 <span style="font-style:italic;">Mahayugas</span> = 4.32 Billion Years
<br /><span style="font-style:italic;">
<br />Sahasra yuga paryantham
<br />Aharyal brahmano vidu
<br />Ratrim yugah sahasrantham
<br />The Ahoratra vido janah ( The Holy Geetha ).</span>
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<br />This is one Cosmological Cycle, called <span style="font-style:italic;">Brahma Kalpa. </span>
<br />
<br /><span style="font-style:italic;">Chaturyuga sahasram indra harina dinam uchyathe
<br /></span>
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<br />From one second, it can be logarithmically shown, upto 10^22 seconds. This is what the above diagram shows. This diagram is by courtesy of <span style="font-style:italic;"><a href="http://www.wikipedia.org">Wikipedia.</span></a>
<br />
<br />From 10^0 it goes upto 10^22 seconds. One day of <span style="font-style:italic;">Brahma</span> is 4.32 billion years and 100 years of <span style="font-style:italic;">Brahma</span> therefore is 311.04 trillion years, which is shown logarithmically above.
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<br />One <span style="font-style:italic;">Asu</span> is 4 seconds, one Vinadi is 24 seconds and one Nadi is 24 minutes. 60 such Nadis make up one day. This is the Sexagesimal division of a day into 60 Nadis ! In Astronomy, one degree is sixty minutes and one minute is sixty seconds. Hence sexagesimal division is justified ! 365.25 such days constitute a year and Hindu calculation goes upto 311.04 trillion years !
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<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0tag:blogger.com,1999:blog-7836607831130520841.post-90340621842949496192011-08-13T23:07:00.001-07:002011-08-14T02:07:52.911-07:00The Geometric Model of Paramesvara<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDvmTxdnBq4q0VRDsm3PP-P_RDy4nYbpu_7gySjit36_TQKQbTldhMrqRXBQef7cF368pXmGh28QcsVrWNtnw8LRGV32X3vpLwJxX7yxKHM1OO_tPj8L5wz3kTiYSsJ3w9dvaoqWpOZ4I/s1600/parameswara.jpg"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 400px; height: 400px;" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDvmTxdnBq4q0VRDsm3PP-P_RDy4nYbpu_7gySjit36_TQKQbTldhMrqRXBQef7cF368pXmGh28QcsVrWNtnw8LRGV32X3vpLwJxX7yxKHM1OO_tPj8L5wz3kTiYSsJ3w9dvaoqWpOZ4I/s320/parameswara.jpg" border="0" alt=""id="BLOGGER_PHOTO_ID_5640587671602943954" /></a>
<br />
<br />
<br />
<br />The Indian astronomers were interested in the computation of eclipses, of geocentric longitudes, the risings and settings of planets,which had relevance to the day to day activities of people.
<br />
<br />Did not Emerson say?
<br />
<br />"Astronomy is excellent, it should come down and give life its full value, and not rest amidst globes and spheres ".
<br />
<br />They were not bothered about proposing Models of the Universe and gaining publicity. But then they did discuss the geometrical model, the rationale of their computations.
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<br />The above diagram explains the Geometric Model of Parameswara, another Kerala astronomer. Paramesvara and Nilakanta modified the Aryabhatan Model.
<br />
<br />By <span style="font-style:italic;">Sheegroccha</span>, he meant the longitude of the Sun." Sheegrocham Sarvesham Ravir bhavathi ", he says is his book <span style="font-style:italic;">Bhatadeepika </span>. For the interior planets, the longitude of the Sheegra correction is to be deducted from the Sun's longitude,<span style="font-style:italic;"> Ravi Sphuta</span> to get the Anomaly of Conjunction.
<br />
<br />The <span style="font-style:italic;">Manda Prathimandala </span>is the mean angular motion of the Planet, from which the trignometric corrections are given to get the true, geocentric longitude.
<br />Govind Kumarhttp://www.blogger.com/profile/04461087078133954772noreply@blogger.com0