Earliest Evidence of Numbers

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.