Earliest Evidence of Numbers
The usage of numbers would indicate the earliest sign of Gaṇitā. In the Vedic era, there is abundant evidence of usage of the decimal number system along with enumeration of powers of tens that indicate an active representation of very large numbers. A study of the Vedic works reveals that our ancestors were well-versed in the use of numbers. They knew all the fundamental operations of arithmetic including: addition, subtraction, multiplication, division, squaring, cubing, square-root and cube-root. They were also well-versed in the use of fractional numbers and surds, mensuration and construction of planar geometric figures, and could solve some algebraic problems also.
In arithmetic, they were masters of numbers and could use large numbers. They had developed an extremely scientific numeral terminology based on the scale of 10. In the Yajurveda-saṃhitā (Vājasaneyi, XVII.2) we have the following list of numeral denominations proceeding in the ratio of 10:
|eka (1), daśa (10), śata (100), sahasra (1000), ayuta (10000), niyuta(10 5 ), prayuta (10 6 ), arbuda (10 7 ), nyarbuda (10 8 ), samudra (10 9 ), madhya (10 10 ), anta (10 11 ), and parārdha (10 12 ).|
The same list occurs in the Taittirīya-saṃhitā (IV.40.11.4 and VII.2.20.1), and with some alterations in the Maitrāyaṇī (II.8.14) and Kāṭhaka (XVII.10) Saṃhitās and other places.
The numbers were classified into even ~ yugma, literally meaning ‘pair’ and odd ~ ayugma, literally meaning ‘not pair’. In two hymns of the Atharvaveda (XIX.22, 23), there seems to be a reference to the zero, as well as to the recognition of the negative number. The zero has been called kṣudra (trifling). The negative number is indicated by the term anṛca, while the positive number by ṛca.
Development of Geometry
Geometry developed significantly in the very ancient Vedic age in connection with the construction of the altars for the Vedic sacrifices. As described in the Vedic literature, the sacrifices were of various kinds. The performance of some of them was obligatory upon every Vedic Hindu, and hence they were known as nitya meaning “obligatory” or “indispensable”. Other sacrifices were to be performed each with the purpose of achieving some special object. Those who did not aim at the attainment of any such object had no need to perform any of them. These sacrifices were classed as kāmya meaning “optional” or “intentional”.
As per injunctions of the scriptures, each sacrifice must be made in an altar of prescribed shape and size. It was emphasised that even a slight irregularity and variation in the form and size of the altar would nullify the object of the whole ritual and might even lead to an adverse effect. So the greatest care had to be taken to secure the right shape and size of the altar. In this way there arose in ancient India problems of geometry and also of arithmetic and algebra.
Let us take for instance the three primary altars: Gārhapatya, Āhavanīya and Dakṣiṇa. These were altars in which every Vedic Hindu had to offer sacrifices daily. The Gārhapatya altar was prescribed to be of the form of a square, according to one school, and of a circle according to another. The Āhavanīya altar had always to be square and the Dakṣiṇa altar semi-circular.
The catch is this: the area of each had to be the same and equal to one square vyāma. As per the units of measures as documented in the literature of Vedic era, 1 vyāma = 96 aṅgulis or “finger breadths”. In the modern parlance, this is about 2 yards.
So the construction of these three altars involved three geometrical operations:
(i) to construct a square on a given straight line
(ii) to circle a square and vice versa and
(iii) to double a circle.
It is important to note that the last problem is the same as the evaluation of the surd √2. We shall see more about the algorithms, methods, approaches and calculations involved through further articles in the series.
There were altars of the shape of a falcon with straight or bent wings, of a square, an equilateral triangle, an isosceles trapezium, a circle, a wheel (with or without spokes), a tortoise, a trough and of other complex forms all having the same area.
Again at the second and each subsequent construction of an altar, it was necessary to increase its size by a specified amount, usually one square puruṣa (1 puruṣa = 120 aṅgulis = ~2.5 yards). But the shape was always kept similar to that of the first construction. Thus there arose problems of equivalent areas and transformation of areas. Hence the Vedic geometers elaborately treated problems of ‘application of areas’.
Recognizing that manuals would be greatly helpful in constructing such altars, the vedic priests have composed a class of texts called Śulba-sūtras. The key aspects of this text are very fascinating. We shall learn in detail about Śulba-sūtras in our next article.