ConstructionsMa 330 Spring 2012Ma 330Ancient GeometryAvinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 1 / 11SummaryTools and terms.Some Geometric Constructions.Underlying Theorems.Later Developments.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 2 / 11SummaryTools and terms.Some Geometric Constructions.Underlying Theorems.Later Developments.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 2 / 11SummaryTools and terms.Some Geometric Constructions.Underlying Theorems.Later Developments.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 2 / 11SummaryTools and terms.Some Geometric Constructions.Underlying Theorems.Later Developments.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 2 / 11Introduction.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 properly towardauspicious directions. This led to geometric constructionproblems as well as algebraic equations to be solved.A pole called Śa˙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.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 3 / 11Introduction.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 properly towardauspicious directions. This led to geometric constructionproblems as well as algebraic equations to be solved.A pole called Śa˙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.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 3 / 11Introduction.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 properly towardauspicious directions. This led to geometric constructionproblems as well as algebraic equations to be solved.A pole called Śa˙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.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 3 / 11Introduction.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 properly towardauspicious directions. This led to geometric constructionproblems as well as algebraic equations to be solved.A pole called Śa˙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.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 3 / 11Introduction.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 properly towardauspicious directions. This led to geometric constructionproblems as well as algebraic equations to be solved.A pole called Śa˙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.Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 3 / 11More Terms.The main East-west line is called pr¯ac¯i (East) and anyparallel line is called p¯arśvam¯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, śulba or s¯utra.These could be flexible ropes from hemp or grass or tapesmade from bamboo as needed. The cords have length marksby joining cords of definite lengths.Basic unit of length is a "a˙ngula (finger)" which is describedas 34 sesame seeds in a line. It seems to be about 0.75inches.However, for ritual purposes the finger of the yajam¯ana (thehost) is used!Avinash Sathaye (Ma 330) Summary Geometry Ancient Geometry 4 / 11More Terms.The main East-west line is called pr¯ac¯i (East) and anyparallel line is called p¯arśvam¯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, śulba or s¯utra.These could be flexible ropes from hemp or grass or tapesmade from bamboo as needed. The cords have length