The Śulba Theorem in Mānava-śulbasūtra

In the pre­vi­ous arti­cle, we have looked at the less­er known fact that a clear enun­ci­a­tion of the so-called ‘Pythagore­an’ the­o­rem, which must right­ly be named the Śul­ba The­o­rem (called bhu­jā-koṭi-karṇa-nyāya in the lat­er lit­er­a­ture) has been described in Baud­hāyana-śul­basū­tra. In this edi­tion we shall look at fur­ther break­throughs and con­tri­bu­tions found in the ancient texts of Śul­ba-sūtras.

The Śul­ba The­o­rem in Māna­va-śul­basū­tra

The pre­sen­ta­tion of the the­o­rem in Māna­va-śul­basū­tra dif­fers from Bod­hāyana Śul­basū­tra both in form and in style. Here it is giv­en in the form of a verse as fol­lows:आयामं आयामगुणं विस्तारं विस्तरेण तु।

समस्य वर्गमूलं यत् तत् कर्णं तद्विदो विदुः॥

Phrase in Māna­va-śul­basū­traMean­ingआयामं आयामगुणंthe length mul­ti­plied by itselfविस्तारं विस्तरेण तुand indeed the breadth by itselfसमस्य वर्गमूलंthe square root of the sumतत् कर्णंthat is the hypotenuseतद्विदो विदुःthose versed in the dis­ci­pline say so

Using mod­ern nota­tion the result may be expressed as:

āyāmā2 + vistārā2 = karṇa

In no uncer­tain terms, it is clear that the state­ments in both Māna­va-śul­basū­tra and Bod­hāyana Śul­basū­tra pre­date the peri­od of Pythago­ras. With­out well-rea­soned under­stand­ing this the­o­rem con­tin­ues to be cred­it­ed to Pythago­ras ( c. 570 – c. 495 BC) and even tout­ed as a “Greek mir­a­cle” by cer­tain short-sight­ed claims.

A list of the­o­rems in Śul­basū­tras

The fol­low­ing the­o­rems are either express­ly stat­ed or the results are implied in the meth­ods of con­struc­tion of the altars of dif­fer­ent shapes and sizes:

1. The diag­o­nals of a rec­tan­gle bisect each oth­er. They divide the rec­tan­gle into four parts, two and two (ver­ti­cal­ly oppo­site) — all of which are equal in all respects.

2. The diag­o­nals of a rhom­bus bisect each oth­er at right angles.

3. An isosce­les tri­an­gle is divid­ed into two equal halves by the line join­ing the ver­tex to the mid­dle point of the base.

4. The area of a square formed by join­ing the mid­dle points of the sides of a square is half that of the orig­i­nal one.

5. A quadri­lat­er­al formed by the lines join­ing the mid­dle points of the sides of a rec­tan­gle is a rhom­bus whose area is half that of the rec­tan­gle.

6. A par­al­lel­o­gram and rec­tan­gle on the same base and with­in the same par­al­lels have the same area.

7. The square on the hypotenuse of a right angled tri­an­gle is equal to the sum of the squares on the oth­er two sides.

8. If the sum of the squares on two sides of a tri­an­gle be equal to the square on the third side, then the tri­an­gle is right-angled.

Con­struc­tion knowl­edge repos­i­to­ry

As explained in ear­li­er arti­cles, the Śul­ba-sūtras, which form a part of the Vedic lit­er­a­ture, deal with the con­struc­tion of fire altars for yajña pur­pos­es. Their con­struc­tion requires a thor­ough knowl­edge of the prop­er­ties of the square, the rec­tan­gle, the rhom­bus, the trapez­i­um, the tri­an­gle and the cir­cle. The knowl­edge preva­lent of those times are astound­ing and are hard­ly known to Indi­ans and stu­dents of sci­ence around the world. The geo­met­ri­cal knowl­edge repos­i­to­ry in these texts have been well researched by seri­ous schol­ars and involved in the con­struc­tions are the fol­low­ing:

1. To divide a line into any num­ber of equal parts.

2. To divide a cir­cle into any num­ber of equal areas by draw­ing diam­e­ters.

3. To divide a tri­an­gle into a num­ber of equal and sim­i­lar areas.

4. To draw a straight line at right angles to a giv­en line.

5. To draw a straight line at right angles to a giv­en straight line from a giv­en point on it.

6. To con­struct a square on a giv­en side.

7. To con­struct a rec­tan­gle of giv­en sides.

8. To con­struct an isosce­les trapez­i­um of giv­en alti­tude, face and base.

9. To con­struct a par­al­lel­o­gram hav­ing giv­en sides at a giv­en incli­na­tion.

10. To con­struct a square equal to the sum of two dif­fer­ent squares.

11. To con­struct a square equiv­a­lent to two giv­en tri­an­gles.

12. To con­struct a square equiv­a­lent to two giv­en pen­tagons.

13. To con­struct a square equal to a giv­en rec­tan­gle.

14. To con­struct a rec­tan­gle hav­ing a giv­en side and equiv­a­lent to a giv­en square.

15. To con­struct an isosce­les trapez­i­um hav­ing a giv­en face and equiv­a­lent to a giv­en square or rec­tan­gle.

16. To con­struct a tri­an­gle equiv­a­lent to a giv­en square.

17. To con­struct a square equiv­a­lent to a giv­en isosce­les tri­an­gle.

18. To con­struct a rhom­bus equiv­a­lent to a giv­en square or rec­tan­gle.

19. To con­struct a square equiv­a­lent to a giv­en rhom­bus.

It is indeed enthralling to know in depth about the rich sci­en­tif­ic her­itage of the Indi­an civ­i­liza­tion. In the forth­com­ing arti­cles we shall delve more into greater under­stand­ing of Jyotiṣa and Gaṇitā in India across dif­fer­ent ages.