the Sulbasutras‑2

Con­text of Devel­op­ment

In Indi­an civ­i­liza­tion, we saw that Geom­e­try orig­i­nat­ed in a very remote age in con­nec­tion with the con­struc­tion of the fire altars for the Vedic rit­u­als. In the course of time, geom­e­try grew beyond its orig­i­nal altar-spe­cif­ic pur­pos­es or the bounds of prac­ti­cal util­i­ty and began to be cul­ti­vat­ed as a sci­ence for its own sake.

This hap­pened in the Vedic Era when dif­fer­ent schools of geom­e­try were found­ed. More notable ones amongst them were the schools of Baud­hāyana, Āpas­tam­ba and Kātyāyana. Though the geo­met­ri­cal propo­si­tions treat­ed in all of them were almost the same, and there were many things com­mon in the meth­ods of their solu­tion, still there were oth­er things to dis­tin­guish one school from anoth­er. Even in the solu­tion of ele­men­tary propo­si­tions such as the con­struc­tion of a square, rec­tan­gle or an isosce­les trapez­i­um, dif­fer­ent schools had pref­er­en­tial lik­ing for dif­fer­en­tial meth­ods. The dif­fer­ence appears most marked in the solu­tion of the prob­lems of the divi­sion of fig­ures.

The word śul­ba (or sul­va) means a cord or rope and sūtra describes the style of the writ­ing, a com­pressed apho­ris­tic pre­sen­ta­tion in which all inessen­tial words (includ­ing, often, the verbs) are dropped. The geom­e­try described in these “Cord Sutras” is the geom­e­try that can be done with (inex­ten­si­ble) cords. The two basic ele­ments out of which the geom­e­try is built up are there­fore the cir­cle (pro­duced by fix­ing one end and tak­ing the oth­er end of a stretched cord round) and the straight line (pro­duced by stretch­ing the cord end to end). In oth­er words, they deal with ‘com­pass and ruler’ geom­e­try.So far sev­en dif­fer­ent Śul­basū­tra texts have been iden­ti­fied by schol­ars.

They are:

1. Baud­hāyana-śul­basū­tra

2. Āpas­tam­ba-śul­basū­tra

3. Kātyāyana-śul­basū­tra

4. Māna­va-śul­basū­tra,

5. Maitrāyaṇa-śul­basū­tra

6. Vāra­ha-śul­basū­tra and

7. Vād­hūla-śul­basū­tra

Of them, the Baud­hāyana-śul­basū­tra is con­sid­ered to be the most ancient one (pri­or to 800 BCE). This assess­ment is based upon the style, com­plete­ness, and cer­tain archa­ic usages that are not that fre­quent­ly found in lat­er texts. Baud­hāyana-śul­basū­tra also presents a very sys­tem­at­ic and detailed treat­ment of sev­er­al top­ics that are skipped in lat­er texts. It is made up of three chap­ters con­sti­tut­ing about 520 sūtras (113 + 83 + 323). It is quite remark­able to just know the con­tents of the Baud­hāyana-śul­basū­tra.

Table: Top­ics cov­ered in the Baud­hāyana-śul­basū­tra

In the last arti­cle, we saw that the large altars of which the fun­da­men­tal one was of the shape of a fal­con, had to be built with 200 bricks. While this was main­ly enabled by Śul­ba-sūtras, it is impor­tant to under­stand some selec­tive math­e­mat­i­cal break­throughs achieved much ahead of the rest of the word by the Vedic civ­i­liza­tion of India. In the Vedic Era, the inter­est­ing geo­met­ri­cal achieve­ments made are the fol­low­ing ﹣

1. Con­struc­tion of rec­ti­lin­ear (square, trapezia, etc.) and curvi­lin­ear (cir­cles, vedis, etc.) geo­met­ri­cal objects

2. Enun­ci­a­tion of geo­met­ric prin­ci­ples and prac­ti­cal appli­ca­tion of them

3. Trans­for­ma­tion of one geo­met­ri­cal object into anoth­er by apply­ing these prin­ci­ples

4. Obtain­ing the val­ue of surds by means of geo­met­ri­cal con­struc­tion

5. Esti­mat­ing the val­ue of surds in the form of a sequence of ratio­nal num­bers

6. Dif­fer­ent meth­ods pro­posed by schol­ars to arrive at these expres­sion for the val­ue of surds ﹣ in par­tic­u­lar √2 and Pi

In the forth­com­ing arti­cles we shall delve deep into the two key math­e­mat­i­cal aspects of this era ﹣the­o­rem of Śul­ba and how √2 and Pi were born.