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
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
Appendix 1: Application of the sixteen sutras to the present system
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
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,
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