Hindu Geometry - Part 2

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Aniruddha Avanipal

Hindu Geometry - Part 2

 

Continued from Part 1...

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Last week we saw how geometry as we know of today, took its birth in

the ancient culture of Vedic Hindus of India and that it preceded its

Greek counter part by at least thousand years. We also saw that the

Vedic Hindus were not only expert at two dimensional or planar

geometry but also were quite fluent in the theory and application of

three dimensional geometry as well.

 

In this article, I will provide a brief overview of some of the most

important work on the shulba or the science of geometry by ancient

Indians.

 

Baudhyana: The shulba of Baudhyana is divided into three chapters.

The first chapter contains 116 sutras or aphorisms of which two are

used as an introduction to basic planar geometry. Sutras 3-21 define

the various measures ordinarily employed in the science of Sulba.

These sutras deal with many basic theorems of planar geometry. Sutras

22-62 give the more important of the geometrical propositions

necessary for the construction of the Vedic altars. These sutras make

use of the general geometric theorem explained in the earlier

section. Sutras 63-116 of the shulba of Baudhyana deal mainly with

the relative positions and spatial magnitudes of the various vedis or

altars.

 

The second chapter consists of 86 sutras of which major portion,

sutras 1-61 is devoted to the description of the spatial relations in

the different constructions of the Agnis or large fire altars made of

bricks. The remaining portion, sutras 62-86 elaborates the

construction of the two most basic altars the Garhapatya-citi (The

House-holder's fire altar) and Chandas-citi (or the altar made of

mantras in stead of bricks). In case of the Chandas-citi, the altar-

builder draws on the ground the altar of the prescribed shape as

suggested by the geometrical construction of the specific Shulba. He

then goes through the whole process of construction imagining all the

while as if he is placing every brick in its proper place with the

appropriate mantras. The mantras are in deed recited, but the bricks

are not actually laid. Hence the name Chandas-citi, or altar made up

of chandas or Vedic Mantars in stead of bricks.

 

The third chapter of the shulba of Baudhyana contains 323 sutras.

They describe the construction of as many as seventeen different

kinds of geometrical structure many of which are three dimensional in

nature. While describing the process of construction, Baudhyana also

states the relevant theorems and provided detailed proof for each one

of them.

 

Apastamba: The Shulba by Apastamba is broadly divided into

six 'patalas' (or sections). Of these the first, third and the fifth

are each subdivided again into three Adhyayas (or chapters) and each

of the remaining sections into four chapters. So altogether the work

contains 21 chapters and 223 sutras. The first section of the manual

(chapters 1-3) gives the important geometrical propositions required

for the construction of altars.

 

The second section (chapters 4-6) of the describe the relative

positions of the altars and their spatial magnitudes. While

describing the spatial attributes of the altars, Apastambha also

explains the underlying geometrical theorems and gives analytical

proofs. He also provides detailed description of the methods of the

construction. The remaining section of the shulba of Apastamba

details the construction of 'Kamya Agnis' or altars built for

attaining definite objects. The type of Kamya Agnis described by

Apastamba serve different purpose than those described by Baudhyana

 

Katyanana: The Shulba by Katyanana is also known as the Katyana-

Shulba-parishista and is divided into two parts. The first part is

composed in the style of the aphorisms while the second part is

composed in verses. The earlier part is again subdivided into seven

Kandikas (or short-sections) containing 90 sutras. It primarily deals

with explanation of two and three dimensional geometrical theorems

and their proofs which were not covered in the shulbas of Baudhyana

or Apastamba. The second part of the Katyanana shulba deals with

descriptions of various kinds of measuring tape (rajju in Sanskrit)

and their usage. It also gives a detail account of the attributes of

an expert geometrician and a few general rules of his/her conduct.

 

Manu: The Shulba of Manu is a small treatise composed in both prose

and verse. It is divided in seven 'khandas' or parts. The first part

of this shulba gives four methods for determining cardinal

directions. Part 4-6 of the Manav Shulba explains corollaries of some

of the most important geometric theorems mentioned by Baudhyana and

Apastamba via the examples of Pakyajniki, Maruti and Varuni altars.

The last section of the Manav-shulba furnishes us with the

application of these geometrical theorems and corollaries in

construction of housing complexes.

 

The Maitrayaniya and the Varaha Shulbas are very closely related to

the Shulba of the Manu as all of them belong to the Krishna Yajur

Veda school. They base their work on the geometric postulates

described in Manav Shulba and formulate other corollaries from them.

The Varaha Shulba also describes application of these geometrical

concepts in building navigational instruments and appliances.

 

The Shulbas of Hindus are living proof of the fact the Vedic

civilization was highly advanced in the field of pure Geometry and

its application. The dating of the origin of Shulbas also prove that

Hindus in India were dealing with sophisticated geometrical theorems

during a time when the Greeks, the forefathers of the so

called 'advanced' European civilization, did not even exist.