ConstructionsMa 330 Spring 2012Ma 330Ancient GeometrySummaryTools and terms.Some Geometric Constructions.Underlying Theorems.Later Developments.Introduction.Since oral tradition was mainly prevalent in scriptures, theancient geometric literature available in Indian traditionconcentrates on construction sacrificial altars and theirproper alignment. This is available in Kaalpas¯utras and thebooks in various traditions are called Shulbas¯utras (theaphorisms of the cord).The altars had complicated shapes, were required to havespecific number of bricks of different types and had to coverprescribed areas. They also had to be aligned properlytoward auspicious directions. This led to geometricconstruction problems as well as algebraic equations to besolved.A pole called´Sa ˙nku was set up to mark directions byshadows or citing to establish a local East-west line. It wasalso used to determine the equinoxes which were vital fordetermination of proper starting days for the rituals.More Terms.The main East-west line is called pr¯ac¯i (East) and anyparallel line is called p¯ar´svam¯an¯i (sideline).A perpendicular line is called tirya ˙nm¯an¯i (crossline) and itwas constructed using a rope and creating an isoscelestriangle with base on the East-west line.The rope has different words dari, rajju, ´sulba or s¯utra.These could be flexible ropes from hemp or grass or tapesmade from bamboo as needed. The cords have lengthmarks by joining cords of definite lengths.Basic unit of length is a ”a ˙ngula (finger)” which isdescribed as 34 sesame seeds in a line. It seems to be about0.75inches. However, for ritual purposes the finger of theyajam¯ana (the host) is used!Needed Constructions.For creating complicated shapes of given areas, severalgeometric constructions were needed. We list a few here.1 Square of a given side.2 Squaring a circle.3 Doubling a circle.4 Area of the square on the diagonal of a rectangle is the sumof areas on the squares on the sides (Pythagoras).5 Construction of a square whose area is the sum (ordifference) of areas of two given squares.Some Interesting Constructions.A square equal to a sum of two squares. One who ismaking a sum of two squares should cut of the bigger bythe side of the smaller. The diagonal of the (so cut )rectangle gives the side of the desired square.A square equal to a difference of two squares. One who ismaking a difference of two squares should cut of the biggerby the side of the smaller. Lay the cutting line across tothe extended side of the large square . The point where itmeets gives the side of the desired square.Square to Rectangle and back.Extend a side of the square to the desired length. Draw thediagonal to the opposite corner of the square. From thepoint where it cuts the square, draw a line parallel to theoriginal extended side. Extend the other side of the squareas well and these two parallel lines form the two sides ofthe desired rectangle.Given a rectangle, we make it as a difference of two squares.Then apply the old trick to make the desired square.To do this, cut off the square on the breadth. Split theremainder into two equal parts parallel to the breadth.Transfer the top to the side of the rectangle. Thus, we havea square with two rectangles attached to adjacent sides.When this is extended to a square, the original rectangle isthe difference of it and the small square on the breadth.Roots.The word for root is karan.¯i. The square root of 2 is thus,dvikaran.¯i Similarly√3 is trikaran.¯i. A rule states, thediagonal of a square makes twice as much area which, inmodern notation would say d2= 2a2or d =√2a if d is thediagonal length of a square of side a.The next rule states that the diagonal of a rectangle withlength equal to the slant diagonal (of measure d) hasmeasure trikaran.¯ii.e.√3a. It goes on to say that a squarewith this as the side has nine times the area (of theoriginal).This shows a fuller understanding of what is popularlyknown as the Pythagoras Theorem. It is clear thatPythagoras came later and there is an interesting storyabout an attempt to hide such roots of integers which areirrational!Pythagoras Theorem.The theorem given in these S¯utras is thus not just for forinteger sides, but even for more general numbers! Thattheorem is indeed stated next as “the diagonal of arectangle makes (as much area) as (the total of what ) thetwo sides separately make.”The Pythagoras theorem has a long and colorful historyand we leave it as an exploratory project, for now.The theorem is said to become explicit or evident et¯asuupalabdhih.in the sample pairs(3, 4), (12, 5), (15, 8), (7, 24), (12, 35), (15, 36).We know that the general formula for giving allPythagorean triples is 2st, (s2− t2), (s2+ t2). The abovelist corresponds to the values(s, t) = (2, 1), (3, 2), (4, 1), (4, 3), (6, 1), (9, 6).Square to Circle.One necessary construction is to find a circle with areaequal to a square. This amounts to solving 4a2= πr2for r,where the square has side 2a.The suggested construction is to draw a circlecircumscribing the square which would have radius√2a.Draw a radius of the circle perpendicular to a side of thesquare. Its part inside the square is a while outside thesquare is (√2 − 1)a.The recommended radius is a(1 +13(√2 − 1)).This evaluates to a(1.138071187).Of course, for a precise answer, we need to calculatep4/π = 2/√π. It will give the answer a(1.128379167).Note that the multiplier differs by .009692020 and thus, itis a reasonable approximation for a practical construction.I observe that a much closer approximation is obtained ifthe13is replaced by 0.31, but that is not so easy to execute.Circle to Square.A correction suggested by an old commentatorDv¯arak¯an¯atha Yajv¯a on the baudh¯ayana S¯utras suggestsmultiplying the value of the radius by117118, which makes theanswer accurate to 5 places, but cannot have a practicaluse!For the reverse problem of converting a circle of radius r toa square of side 2a, a practical method is to takea = r(1 −215), which involves splitting a radius into 15parts. It gives a multiplier which is larger by 0.0195602588.A more refined formula is given, but it is clearly not verypractical:1 −18+18 · 29−78 · 29 · 48= 0.8786817529.However, the first two terms, which give an easier fraction,78are actually closer to the modern answer, than1315.Approximations