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Saturday, June 8, 2013

Vedic Mathematics

Astronomical Applications of Vedic Mathematics.
Kenneth Williams
2000 139p. Contents: Introduction to Vedic Mathematics. Prediction of Eclipses: Prediction of the Times Of Contact of the Moon’s Penumbral and Umbral Shadows With the Earth. Partial Phase. Total Phase. Approximate Position of the Eclipse Path. Time of Total Eclipse For an Observer on Earth. Bessel’s Method, Early Eclipse Prediction, Solution of Eclipse Equation. Kepler’s Equation: A Transcendental Equation, Solution of Kepler’s Equation. Introduction to Triples: Notation & Combination, Triple Addition, Quadrant Triples, Rotations, Subtraction, Half-Angle Triple. Triple Code Numbers, Angles in Perfect Triples. Prediction of Planetary Positions: Heliocentric Position, Mean Anomaly, Geocentric Position, Geocentric Longitude, Geocentric Correction, Planet Finder. Spherical Triangles Using Triples: Triple Notation & Formulae for Spherical Trigonometry, Cosine Rule to Find an Angle, Sine Rule to Find Angle, Cotangent Rule to Find Angle, Polar Cosine Rule. Right Angled Spherical Triangles. Spherical Triangles Using Code Numbers. Determinants. Quadruples. Addition of Perpendicular Triples, Rotation About A Coordinate Axis. Quadruples & Orbits, Inclination of Orbit, Angle Between Two Directions. Inclination of Planetary Orbits. Calculation of Radius Vector.

Applications of Vedic mathematics


K. R. Williams; Rs. 125.


A. P. Nicholas, K. P. Williams, J. Pickles; Rs. 225.

ASTRONOMICAL APPLICATIONS OF VEDIC MATHEMATICS: Kenneth Williams; Rs. 195. All the three books have been published by Motilal Banarsidass Publishers Private Limited, 41, U.A. Bungalow Road, Jawahar Nahar, Delhi-110007.

THE PATH-BREAKING book Vedic Mathematics written by Bharati Krishna Tirthaji, a former Sankaracharya of Puri, was published posthumously by the Hindu Vishwavidyalaya of Varanasi in 1965, at the instance of Manjula Devi, to whom the manuscript was entrusted by Tirthaji. Soon the book was published by M.B. publishers and it was in hibernation till it caught the attention of the British savants who have hailed the contribution of Tirthaji by discovering newer facets. The outcome is the availability of a dynamic and vibrant choice of approach in applying Vedic mathematics effectively in various branches of mathematics. What continues to be in vogue can be seen to be static and sleepy through routine methods of handling problems. Books published abroad in U.K. for British institutions using Vedic Mathematics have now been brought out by the renowned publishers of Indology books as Indian editions.

Steering clear of the controversy which is yet to be settled, the mathematical content of the books under review opens vistas missed by our curricular experts and textbooks writers, teachers and paper setters. To clinch the issue, how many are even today, familiar with negative digits in numerals and interpretation of identities in mental calculation?

It is rare to find religious heads evincing interest in mathematics, leave alone expounding with passion the unconventional strategies that empower one to put aside paper and pencil and get at the answers to problems of calculation mentally. Unlike abacus-dependent calculation skills, they turn out to be promotive of self-esteem and self-confidence in the practitioner. As varying treatments are evoked, it is not akin to robot like behaviour.

Tirthaji enumerates Sutras or Aphorisms, 16 in number and 13 Upasutras or corollaries and succeeds in showing how they go beyond the capabilities of calculator devices, the use of which has a debilitating effect on the learners and users. Their role in effecting a tie-up of mathematical processes in the entire gamut of mathematics carries with it a novelty that is not known to majority of mathematics educators. British scholars show that they lend themselves to extensions not conceived by the propounder himself. According to Tirthaji, "Vedic mathematics" that he has chosen for the title of his book is Vedic-inspired. It will help to assess his stand if one could refer to the book Mathematics As Known To The Vedic Samhita by M.D. Pandit published in 1993 by Sri Satguru Publications.

Narinder Puri of Roorkee University brought out Indian editions of Vedic Mathematics and related literature by non-Indian authors in the 1990's and popularised the contributions in India and abroad, organising conferences involving the authors and conducting courses.

In the book, The Natural Calculator, the author shows how natural processes of the mind are associated with mental calculation. Since multiplication operation reveals remarkably the properties of number, the book deals mainly with it. Prodigies are recognised by their ability to give instantly products of large numbers. The book is so written that it can be gone through, in any desired sequence. Algebraic proofs justifying the approaches are given at the end of the book. Problem-solving skills get a boost. The objective is to show how reliance on the calculator is harmful, as it deprives the mind from exercising its capabilities which get frozen through repetitive endeavour. Major part of mathematics education can be directed to give one-line response to basic operations, singly or combined.

The experience is exhilarating and delightful, once one gets settled in the Vedic way. Availability of choice and the judgement that it entails provides one with expertise to shun routineness, the bane of curricular learning today. As Tirthaji affirms, there is flexibility, innovativeness and creativity in mathematical computation and this brings mathematics to life. The criteria for looking upon an activity as natural are spelled out by advent of increasing speed and accuracy. To put it in other words, calculation requiring pencil and paper is objective and external, whereas when resorted to mentally is subjective and internal and as vouchsafed by transcendental meditation is deeper. Instructional standards today centre round general methods but Vedic mathematics emphasises that every problem is unique with its own singularly arrived at solution. To ignore this tantamounts to underestimating children's capabilities to hold and remember.

There are nine chapters, each of which is flagged with a bordered version of the experiences in brief of child prodigies who during the last four centuries exhibited extraordinary powers of mental calculation and baffled the audiences. Some of them blossomed into professors of mathematics like Aitken. The book is presented in prescribable format with models and exercises for practice with answers given at the end. It closes with general exercises with hints for two sections. A list of references is also provided.

In the second book the authors deal exclusively with the versatility of the single sutra "Vertically and crosswise" (urdhwa thryagbhyam). They point out that while the applications of this sutra are extraordinarily diverse and wide-ranging, they cannot claim that they have exhausted its applicability. The methods get started in a big way in chapter I with coverage gathering momentum in the subsequent chapters. Applications embrace basic calculations, logarithms, exponents, trigonometric functions and solutions of simultaneous, transcendental polynomial and differential equations. The link between arithmetic and algebra is revealed through treatment of polynomial on the analogy of place value notation.

In Astronomical Applications of Vedic Mathematics the author affirms that his is a personal contribution to new application of Vedic mathematics. He professes no thorough or complete treatment of the topics taken up, admitting that the ideas given can lend themselves to further development and continuance of application to other areas of astronomy. To make the material intelligible to as wide a readership as possible, rigour is curtailed. Besides chapters on prediction of eclipses and planetary positions, the book carries illustrations, four appendices and planet finder circles, three in number. Answers and exercises of chapter V on spherical triangles are provided at the end. Books of reference are mentioned, followed by glossary and Vedic sutras with English rendering.

The printing and get-up are excellent. Proof-reading has been done well . The books will be a weighty addition to mathematics education section of school and college libraries, if not included already. One looks forward to further improved editions of these books in the near future.


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