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Articles / Pioneers / C.G.Rajan

C.G.Rajan @ C.Govinda Rajan

A remarkable exponent & achiever who gave birth to the mordern indian astrology.
Never waited for help to arrive. He decided to fix the inaccuracies!

Mr.Rajan acknowledged every book, every material, every research paper acquired by him at various places in this preface with a wonderful thanking note in section (6), a must read!

Preface from Raja Jothida Ganitham (1933 A.D.) as it is!

(1) I was drawn to the amateur study of Hindu astrology about two decades ago as the result of the amateur study and practice of the same by my father and eldest brother, who were trained in the art of prediction by a maternal uncle of my father. Till a few years afterwards, I was a staunch believer in the accuracy of indigenous Vakya Ganitha Panchang, following in my belief the footsteps of my relatives referred to above. My faith in it was suddenly shaken when to my astonishment, I noticed, on one evening in the western horizon, the two bright moving luminaries of Jupiter and Venus occupying positions* relatively to themselves contrary to the positions given in the Vakya Ganitha Panchang which was followed in my family till then. I had consequently taken to the study of Hindu Astronomy, prompted by a desire to calculate planetary positions myself. My study of European Astronomy as a part of the University Syllabus for the Mathematics course of B.A. Degree afforded me ample opportunities to study the ground work of astronomy and this, combined with the facilities and help that I had secured then and there from the authorities of the observatories of Madras and Washington in the United States of America in respect of the right kind of books dealing with the Modern Astronomical tables, had enabled me to make a comparative study of Hindu astronomical tables and European astronomical tables. I was convinced thereby that the Hindu Astronomical tables are far inferior to the European astronomical tables in point of their accuracy. The ever-growing tendency now-a-days even among the most orthodox communities of Indians to go in for western tables at least for horoscope-reading and social functions (though not for such religions services as appear to be impregnable citadels founded upon strong and age-long Vakya Panchang traditions) and the tendency of the Vakya and Siddhanta almanac-makers to embody in their Panchangs the data about the occurrence of eclipses as calculated from European tables are a proof positive in themselves that the Indian astronomical tables are defective and require recasting and improvement in the light of numerous astronomical discoveries made since the days of the theories on which the Siddhantas (from which the Vakya Ganitha Tables were evolved) were formulated and written. The fundamental concepts and theories of Hindu Astronomy and European astronomy are so divergent that it is a Herculean, if not a well-nigh impossible, task to raise Hindu astronomy to the same level of Modern European Astronomy in respect of accuracy by making scientific corrections which will be tenable for some centuries at least and without running the risk of seriously jeopardizing Hindu astronomy in its identify. There are some European astronomical tables which enable the calculation of planetary positions correct even to the second decimal place of seconds of arc, but so far as I know, they are either very costly (the cost coming up to about one hundred rupees) or in the German and French languages which even many of the English knowing population do not know. It has therefore been my desire for some years past to produce a cheap and convenient book of planetary tables in the wide-spread language of English, and also in Tamil to enable the public to calculate easily planetary positions correct to the nearest minute of arc without going through the laborious, tedious and yet inaccurate methods of the Siddhantas and to set at rest once for all, if possible, the controversy that very often rages in the most orthodox temples about the exact beginning and ending moments of a thithi to determine when a religious festival should commence. These I consider to be the only justification fo the introduction of this book to the public.
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* Note entirely due to Parallax

(2) The list of the important books that I have consulted is given at the end of this book. As regards the Sun, Mars, Mercury, Jupiter, Venus and Saturn I have adopted the Equations given in the Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac Vols. VI and VII by Professor Simon Newcomb and George William Hill* for the Mean longitudes of the Sun and its perigee and for the mean longitudes of the other planets and of their nodes, and perigees with Ross's correction to Newcomb's equation whichever necessary. I have also pressed into my service the equations given in them for some of the important inequalities including those caused by "Planetary Perturbations" and for the Reduction to the ecliptic, keeping always in view my object of aiming at accuracy to the nearest minute of arc. The arguments for the tables of the Equations of the Centre, Secular Variations, Logarithm of Radius Vector and its Secular variations are expressed in those volumes in terms of days but I have given my arguments in terms of degrees of the mean anomaly to suit my purpose of securing uniformity and convenience. I have constructed altogether original tables from the equations employed by me. I have framed an equation for the Great Inequality (or Long Period Inequality) of Jupiter and Saturn from the equations given from them in the Astronomical Tables containing the Tables of Jupiter Saturn and Uranus by M.A.Bouvard (a French Book). For ready use, I have calculated the values of the Great Inequalities of Jupiter and Saturn for the years from 1800 A.D. to 2100 A.D and given them in Table No.8. For other years outside this limit, the values have to be calculated from the equations given at the foot of Table No.8. To determine the inequality caused by the perturbation of Saturn by Uranus, I have employed the equation given by Simon Newcomb for the mean longitude for Uranus. As for the Moon, I have employed, the equations given for the mean longitudes of the Moon, its Node and Perigee by Mr.E.W.Brown in his "Tables of the Motions of the Moon" and extracted in the Nautical Almanac and Astronomical Ephemeris for the year 1925. As Mr.Brown's book* costing over four pounds sterling was not available to me for loan, I have adopted the equations given in P.A.Hansen's Tables De La Lune (Tables of the Moon with Neisson's corrections to Hansen's Tables) for the "Perturbations" including Equation, Variation and Annual Equation caused to the Moon. With the help of the equations referred to above, I have constructed original tables. European Almanac or Ephemeris usually gives only the mean longitude of the Nodes of the Moon, while I have added an equation to calculate their true longitudes from their mean longitudes and shown also how to calculate them from the first principles.
As regards Nutation, it is made up of several independent motions of the Earth's axis and the most important of them is the "Lunar Nutation". The total quantity of the Lunar Nutation in longitude ranges from +17".639 to -17".639 and the total quantity of the Solar Nutation due to the action of the Sun ranges from +1".454 to -1".454. I have taken into account the greatest component quantity of -17".234 Sin Omega which is the Nutation in Longitude due to the action of the Moon and rejected the other component quantities which are very small. The error in doing so will range between the negligible limit of +1".454 and -1".454, in as much as the total Nutational corrections ranges between +19".093 and -19".093 according to the equations employed by Simon Newcomb. I have prepared table No. 8 of the Sun from the quantity of -17".234 Sin Omega and employed the correction for Nutation in the case of Sun and Rahu and Kethu only. I have not embodied, in the illustrative examples for the other planets, the corrections for Nutation which is -12" as worked out in the Example for the Sun. On the whole, the corrections for Nutation can be safely ignored for all the planets if the reader or computer does not require accuracy to the nearest second of arc and wants to avoid additional labor of calculation. If, however he wants to employ the correction for Nutation also, he has to calculate it from table 8 of the Sun as shown in the case of the Sun and add it "algebraically" to the mean longitude denoted by L in Part I of the illustrative examples for the several planets. I have also added two sections, one on "The Sidereal Time" and the other on the occurrence of eclipses. To calculate sidereal time sufficiently accurate astrological purposes, I have evolved the necessary tables for the period from 1800 A.D. To 3100 A.D. from the equation given by Simon Newcomb in the Astronomical Papers. For other periods, the table NO. (1) in this section can be constructed from the equation given in the first paragraph of this section. I have ignored Nutation in the calculation of the Sidereal Time as the error thereby does not exceed 1.168 (being equal to the Nutation in longitude multiplied by the cosine of the obliquity and reduced to time i.e. = 19".093 x 0.9174 / 15) seconds of time in Right Ascension or 1.168 seconds of time in Sidereal Time. If however the computer wants to embody the Nutational correction also, he has to calculate it from table 8 of the Sun as before, and multiply it by 0.9174 and divide the product by 15 to get Sidereal time in seconds of time and add it algebraically to the final result got for Sidereal time for any given moment for any place. In the illustrative example for sidereal time, -12" x 0.9174 / 15 = -0.73 seconds of time has to be added algebraically to the final result of 23 hrs.0ms.24.53 seconds of Sidereal Time. If we do so, we get 23 hrs.0ms.23.8 seconds. In omitting small quantities on the score that they are negligible, I have always had in my view the object of simplifying calculation and also the purpose for which, or the degree of accuracy to which, we require our calculation and in this, I have followed the precedent of no less an authority than Simon Newcomb who says thus:- "in the following expressions(i.e. expressions of Nutation) I have included all the terms of which the coefficients exceed 0.006. Below this limit, it does not seem necessary to go, as no astronomical result will be practically affected by a small error in the assumed nutation". To preclude the contingency of my tables becoming approximate for distant periods, I have given, where ever necessary, tables of Secular Variation which will keep the tables sufficiently accurate even to distant dates. As regards the occurrence of eclipses, I have consulted the "Recurrence of Solar Eclipses in the Astronomical Papers Vol.I" by Simon Newcomb and R.Buchanan's Mathematical Theory of Eclipses and could not consult Oppolzer's "Kannon der Finsternisse" as it is in German which I don't know and as I could not get it also being very costly. Though Buchanan's book gives the ecliptic limits with reference to the Moon's latitude, I have calculated the ecliptic limits with reference to the difference in longitude between the Moon and its Node and given them along with the ecliptic limits in terms of the Moon's latitude to enable indigenous. Panchang makers and chronologists to determine the occurrence of eclipses with reference to longitude. The tables in this book cover the period from 3200 B.C. to 3100 A.D. So that this book will be useful to chronologists and almanac-makers. As regards the Heliacal Rising and Setting of a heavenly body, they belong to Old Astronomy rather than to Modern Astronomy. The discovery of telescope has made them a thing of the past. They depend on the conditions of the optical power of the sight of the observer's eyes and the atmospheric conditions determining the clearness of the sky. These two factors are very uncertain and varying. Hence the heliacal rising and setting are not exact astronomical phenomenon, I have not therefore dealt with them in this book. They depend also on the latitude of the observer and the elongation of the planets from the Sun. Those who are interested in them are recommended to follow the directions given in chapter IX and X of "Surya Siddanta" which are extracted and given in the foot-note for easy reference in the case of the planets only*. I have added three appendices containing information which will be very useful to astrologers, such as the Conversion of Right Ascension and Declination into Celestial Longitude and Latitude, Apparent time, Meantime, Local time and Sidereal time, the rising and setting of Heavenly bodies including the Sun and the planets and the rising and setting of Zodiacal Sings and the calculation of the Lagna (Ascendant) and the Bhavachakra. In this connection, I have to state that I have great pleasure to acknowledge my indebtedness to the books refered to above as having been consulted by me.
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* The other books, except Surya Siddhanta by Burgess mentioned in the Bibliography, being my own copies, are with me and I shall feel highly obliged if anyone can sell me even a second hand copy of Surya Siddhanta by Burgeis and Siddanta Sironmani of Bashkara by Lancelot Wilkinson.
* The tables of Newcomb and Hill in the Astronomical Papers of the American Ephemeris and Nautical Almanac are still being used for the preparation of Greenwich Nautical Almanac-Vide pages 757 and 766 of Greenwich Nautical Almanac for 1933
* The planets are visible to the naked eye when their elongations (i.e. their distances from the Sun in degrees of time) are less than the figures given below. The heliacal setting can be calculated from these limits and the calculations will be sufficiently accurate for estimating the effects of Asthangatha planets for doing the process of Asthangatha Haranam and for determining ... etc.
The Moon 12 |Mars 17 | Mercury 12 or 14| Venus 8 or 10 | Jupiter 11 | Saturn 15

(3) The books referred to above stop with the calculation of the heliocentric positions of Mars, Mercury, Jupiter, Venus, Saturn (Uranus and Neptune) and do not proceed further with the calculations of their geocentric positions. To facilitate the computer of planetary geocentric positions, I have produced a separate book entitled "Conversion of Heliocentric Co-ordinates into Geocentric Co-ordinates - containing tables for converting Heliocentric longitude and latitude into Geocentric Tropical longitude and latitude and into Indian Sidereal longitude" and it is bound with this book for the sake of convenience. The attention of the reader is invited to the preface (of that book) in which it is stated that (1) that book is specially designed to be useful both to those who know the application of logarithmetic tables and to those who are ignorant of logarithm and that that book contains also four ready-reckoner tables to convert Tropical longitude and latitude into Indian Polar Longitude and Polar Latitude and to find Hindu Kranti (i.e, Hindu Declination) and European Declination. In the case of Mars, Mercury, Jupiter, Venus and Saturn, the computer, after arriving at their Heliocentric longitude, latitude and logarithm of Radius Vector or simply Radius Vector and also the geocentric longitude of the Sun and its Radius Vector - all according to the rules of this book - has to follow the instructions given in the separate book to compute their geocentric positions. In following this method, I have followed the method of European astronomers.

(4) From two-fold considerations, I have adopted the method of arriving at the tropical (i.e. Sayana) longitude and tropical latitude first instead of getting the Nirayana longitude first, by making the necessary correction for Ayanamsa in the several tables giving the mean longitudes. To take the Tropical (or Sayana) mean longitude first and then to apply the necessary corrections ensures much more accuracy - a circumstance which no less an Indian astronomer than the famous Bhaskara of Siddhanta Sironmani had admitted as being more accurate. Further, the question of Ayanamsa is still considered a moot one and several schools of thought follow different quantities of Ayanamsa. I have therefor purposely followed the Sayana measure first to deduce from it the Nirayana measure subsequently so that the accuracy of calculation may not be seriously affected and so that the several schools of Ayanamsa may apply their own Ayanamsa quantity and get at the Nirayana longitude which they consider to be correct. I have devoted also, in the separate referred to above, a section to the question of Ayanamsa, determining in it, from the first principles in some cases, the several quantities of Ayanamsa which have their own votaries or protagonists. The Ayanamsa Tables 1b and 1c in it are so designed that they can be sued by all schools of thought about Ayanbamsa.

(5) I shall now enable the reader to have an idea of the degree of accuracy of my tables by instituting a comparison with European Almanacs, the accuracy of which is undoubtedly beyond question as admitted by many. In the illustrative examples, I have calculated the positions of the planets at 05.30 PM Standard Time on 16-12-1924 A.D at Madras.

..... Illustrative Table of Comparison

The two comparative tables clearly show that our results agree fairly well with the results of the Nautical Almanac and that our tables are sufficiently accurate. In a small volume of this kind, it is absolutely impossible to make our results agree with the results of the European Almanac to the nearest second, as the European Almanac employ innumerable equations resulting in the laboriously calculated tables to arrive at their results.
My object is to make my book popular and useful both to an expert calculator and to the mechanical worker who does not know much of mathematics beyond four fundamental rules of Arithmetic namely addition, subtraction, multiplication and division and at the same time to aim at a moderate degree of accuracy which is sufficient for astrological and chronological purposes. To this and in view, I have given also many tables to reduce the calculation to one of a mechanical nature securing considerable case in practical working; for, otherwise, any book is bound to get into the limbo of oblivion except with the investigating scholars.

(6) There now remains the pleasant duty of gratefully acknowledging the kind help that I have received from various quarters without which I venture to say that I would not have produced this book.In the first place, I have to thank the staff of the Madras observatory and in particular Messrs.C.Chengalvaraya Mudaliar and A.A.Narayana Ayyar B.A (now pensioner) who were exceedingly kind to me in allowing me free access to the observatory of Madras from about 1920 A.D. to consult books on the subject whenever necessary. I have next to thank with very great pleasure the invaluable help and instructions that I have been off and on receiving from 1920 A.D. from the staff and the Superintendent of the United States Naval Observatory, Washington, who were kind enough to clear my doubts whenever I had difficulty to understand the Astronomical Papers of the American Ephemeris and who were also kind to send me free of cost Astronomical Papers Vol. III parts I and V and Vol.V parts I and II. I take this opportunity to thank also some of my well-wishers whose kind correspondence with me has equipped me with such experience as has been utilized by me in the preparation of this book.

Last but not least is the task of thanking those who were responsible for bringing this book out of the press, despite the special difficulties of printing a book on this nature involving figure-work and enormous labor and expenditure. I heartily thank my eldest brother Mr.C.Appaswamy Mudaliar (a Government pensioner) and his son Mr.C.Viswanatha Mudaliyar for their having corrected the proof and pushed the work through the press and for their invaluable help in the construction and preparation of the tables. The readers are specially requested to carry out the corrections given in the Errata list before they proceed to read the book proper; otherwise, they may find it difficult to follow the illustrative examples given. Lastly, I crave the indulgence of the reader to excuse clerical errors and print-mistakes in the book and also any slip of the hand in working out the illustrative examples and to take into account only the methods for guidance in the illustrative examples even if there be any slips of the hand in them.

I have chosen to style this work a Siddhanta and a Graha Karana as well, as I consider that it partakes of the character of both a Siddhanta and a Karana. I have dedicated this work to the loving memory of the ancient and modern astronomers of the East and the West in token of my humble admiration of their intellectual qualities which are par excellence. The under signed, (C.G.Rajan whose full name is C.Govinda Rajan, a Vellala Mudaliar of Sigamani Maharishi Gotram, aged thirty six years) ventures to place this work before the public, in the hope that the almighty will make it useful to astronomers, astrologers, Panchang-makers and chronologists and make him realize that he has done his duty to his countrymen. Salutation to ancient and modern astronomers.

C.G.RAJAN

7, Venkatesa Maistry Street,
Near Krishnappa Naick Tank,
Sowcarpet Post,
MADRAS,
9th February 1933, A.D.