9. GEOMETRY FOR AN ORAL TRADITION - details

This book presents direct, immediate and easily understood proofs. These proofs are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). Starting from these first principles it leads to theorems on elementary properties of circles.. It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience. 1999; Author: A. P. Nicholas. 130 pages, A4 size, paperback. ISBN 1 902517 05 9. Price 15 pounds.

"Geometry for an Oral Tradition" - Contents

Introduction 
Preliminaries 
Provisions 
Definitions 
PROPOSITIONS 
Part A: Congruence, magnitudes and Lines 
Part B: Angles, parallels, triangles and quadrilaterals 
Part C: Concerning area equalities and similar triangles 
Part D: Elementary properties of a circle 
COMMENTARY 
Part I: Some basics 
Part II: Language and reason 
Part III: Comparisons with Euclid's Elements 
Part IV: Movement in geometry 
Part V: The valid use of figures 
Summary and Conclusions 
References 
APPENDICES 
Appendix 1: Application of the sixteen sutras to the present system of geometry 
Appendix 2: Alternative proofs and sequences in Part D 
Appendix 3: Further definitions

"Geometry for an Oral Tradition" - Introduction

How this book arose
The inspiration behind this book was two-fold: First, Tirthaji's reconstruction of vedic mathematics, of which a brief historical account follows later in the Introduction. Being an oral tradition, little or no ancient material is extant, and the one surviving book by Tirthaji contains only a handful of examples in geometry.

Secondly, I wondered if the elementary properties of a circle can be demonstrated simply, in such a way that we can see why they hold good. Having satisfied myself that they can, the next issue was to investigate what prior steps, such as definitions and axioms, would serve to establish these demonstrations as part of a system of geometry suitable for an oral tradition.

An oral tradition in geometry
The reader might like to consider, what might such a system be like? As for me, an image comes to mind of an exposition being given in a sheltered nook on a beach, figures being sketched in the sand, and the rest of the exposition being spoken. Proofs would generally need to be brief and to the point. Qualities such as effortlessness, simplicity, brevity and clarity would be highly prized. The aim of this book is to provide a text suitable for such an exposition.

Two other aspects of an oral tradition are worthy of mention. First, the use of verse as an aide memoire. It is much easier to memorise material in rhyming verses - but this idea has not been used here. Secondly, there is the use of sutras, as in the vedic tradition, a sutra being a terse statement of some important point of principle (literally, a sutra is a thread). The material of this book was developed without reference to Tirthaji's sutras, but their application to this system is investigated in Appendix 1.

Anyone who has read through the first three of the thirteen books of Euclid's 'Elements' will have encountered the theorems on circles given here. That the present material covers the ground more swiftly is partly because less ground is covered, partly because these methods are generally much simpler and briefer.

B.K. Tirthaji's reconstruction of vedic mathematics
Ancient India's oral vedic tradition began to be written down about 1600 or 1700 B.C., according to western scholars. Over a period of about 1000 years the four vedas were written down: the Rig - veda, the Yajur - veda, the Sama - veda, and the Atharva - veda.

Tradition had it that the vedas were the embodiment of all knowledge. Yet when nineteenth century scholars examined the vedas there were some puzzles. Consider the Atharva - veda, for example, which deals with architecture, engineering, mathematics, and other topics. The material supposed to be on mathematics comes under the heading of 'Ganita sutras', i.e. mathematics sutras. Under this heading came statements such as, "In the reign of King Kamsa, arson, famine and unsanitary conditions prevailed". The scholars could make nothing of it: there appeared to be no connection with mathematics.

However, a brilliant south Indian scholar, later known as Shri Bharati Krishna Tirthaji, was convinced that there was something in the ancient tradition. By persistence he obtained a clue (he tells us), and after that things began falling into place. In due course he concluded that the whole of mathematics, pure and applied, in all its branches, comes under sixteen sutras. He wrote sixteen volumes on the subject, which subsequently were all lost.

Tirthaji was born in 1884. His key work on vedic mathematics appears to have been done between the years 1911 and 1918. In 1921 he was made Shankaracharya of Puri (Hindu India being led by four Shankaracharyas, a bit like having four popes). Shortly before this he became a renunciate, i.e. he renounced his former life. This, and his considerable religious duties as Shankaracharya, are no doubt the reasons why he did not turn his attention to vedic mathematics again until the 1950s, only to realise that the sixteen volumes were lost. He decided to rewrite them all, and as a preliminary step wrote another book, Vedic Mathematics, to introduce the whole series. Owing to ill - health he got no further, and died in 1960, His introductory book, the only one by him surviving on the subject, was published in 1965.

A further issue
At the outset mathematics divides into two branches, based on number and form: arithmetic (from which stems algebra) and geometry. Tirthaji's introductory book deals mainly with arithmetic and algebra: geometry is scarcely addressed. Furthermore, the handful of examples he gives on geometry are unlike anything here. This material is something new. The present study does not simply arise out of his book. Yet it does use a mental approach. Can it be considered to belong to Tirthaji's system? If it does it complements his introductory material on vedic mathematics.

Geometry and the nature of an oral tradition
Imagine a society with an oral tradition, and willing to allow its understanding of subjects such as geometry to develop; willing to incorporate fresh insights into the tradition. It would be in their interests to do so, and it would happen quite naturally through teachers mastering the current understanding, and in some cases developing it. Such a society would probably have a fairly pragmatic outlook, having respect for the tradition but not regarding its current version as a perfect system, necessarily faultless, but rather as reflecting the current understanding, and as such subject to amendment, be it correction or further refinement.

Perhaps this is the nature of an oral tradition in geometry. Certainly the writing of the present book has been a bit like that, fresh insights constantly changing the material and the format, and it would be no surprise if it could usefully benefit from further insights and amendments, etc.

A word about the Preliminaries
An earlier draft of this book began in much the same way as Euclid's Elements. Subsequently it became clear that an earlier starting point was needed. The normal thing is to begin at the beginning, and then go to the end. (Of course this is mathematics, so perhaps all normality is suspended!) But what kind of activity is it that begins at the beginning and then goes backwards? Somewhere Bertrand Russell says that it is the philosophy of mathematics, adding that once established it becomes mathematics.

The Preliminaries outline the new starting point, and the reasons for it.