Title
| Title | Title |
|---|---|
| Author(s) | nidhi Nidhi Bhutada |
| Platform | Bharatiya Jnana Sarita |
| Categories | Mathematical, Physical and Astronomical Sciences |
| Year | 2026 |
| Type of Article | Original Article - Short Form |
| Difficulty Level | Beginner |
| Abstract | Abstract |
Geometry :
Before understanding any knowledge system understanding कः, किम्, कुतः, कुत्र, कदा, किमर्थम्, केन, कथम्, कीदृशम्, कस्मिन् all this questions has to be answered, What is Ancient Indian Geometry ? What is the relevance? Why should we learn about it ? What is Sulbā ? Why were they used ? Who used it ? When was it in use? How can we learn from it ? These are general questions we come up with in mind when we learn about Indian geometry , and to answer these questions we will further deal with different concepts from Indian Geometry. Understanding Geometry from Ancient Indian Perspective in Most Important point to note in Modern mathematics. What is more fascinating about those concept are Construction were done with ROPE i.e Śulba and NAIL i.e. Sanku.
History of Geometry is divided into two Vedic Period and Post Vedic Period. The subject treated in the Hindu Gaṇita of the early renaissance period consisted of the following :
परिकर्मं व्यवहरो रज्जु रासी कलासवर्णा या । यावत्तावति वर्गो घनो ततः वर्गोवर्गो विकल्पो ता ।। (Sthānāṅgasūtra Sūtra 747)
परिकर्मं Fundamental operations , व्यवहर Determinations , रज्जु Rope meaning Geometry, रासी Rule of Three, कलासवर्णा operations with fractions, यावत्तावत as many as meaning simple equations, वर्ग Square meaning Quadratic equations, घन Ghana Cube meaning cubic equations वर्ग वर्ग Biquadratic equations and विकल्प Permutations and Combinations.
What was the Purpose to making Geometry ?
All the construction was made for the making of Cities. For the purpose of worship three types of agnis or fire sheltering them in certain altars of special design. The Agnis were called दक्षिण , गार्हपत्य , आहवनीय । दक्षिण = Semi Circular , आहवनीय = Square System , गार्हपत्य = Square in one system and Circular in other system . Therefore, for this construction sulba was used.
Area of each had to be the same and equal to one square व्याम {१ व्याम = ९६ अङ्गुल = 2 yards} To construct this three alters involved three geometrical operations : 1) To construct a square on a given straight line 2) To circle a square and vice versa 3) To double a circle 4) Doublinga square and then circling it. or Evaluation of the surd root 2 {१ पुरुष = १२० अङ्गुल = 2½ yards} Vedic geometry also treated problems of Application of areas.
Kalpa - sūtras are broadly divided into two classes. Gṛhya sūtras : “ The rules for ceremonies relating to family or domestic affairs” such as marriage , birth , etc Śrauta - sūtras : “The rules for ceremonies ordained by Veda” such as the preservation of sacred fires, performance of the sacrifices , etc. Each school of Veda had its own Śrauta-Sūtra and hence probably its own Śulba. Thus it seems that there were numerous manuals of geometry in ancient India. But most of them are now lost. BB Dutta lists at present we know however only 7 Śulba-sūtras : Baudhāyana, Āpastamba, Kātyāyana, Mānava, Maitrāyaṇa, Vārāha and Vādhula. Other two which are not available now but known : Maśaka Śulba and Hiraṇyakeśi Śulba Kṛṣṇa Yajur-veda = Baudhāyana, Āpastamba, Mānava, Maitrāyaṇa and Vārāha Śulba-sūtras Śukla Yajur-veda = Kātyāyana Śulba-sūtras.
What is Śulba ?
शुल्ब माने : to measure
भावव्युत्पत्ति शुल्बनम् शुल्बः । = The act of measuring
कर्मव्युत्पत्ति शुल्बयते इति शुल्बः ।= to do i.e entity or result of measuring
करणव्युत्पत्ति शुल्बयत्यनेन इति शुल्बः । = instrument i.e instrument of measuringशुल्बनम् = शुल्बः (शुल्ब् + घञ्) ।
In Sanskrit Śulba or Rajju have got identical significance i.e. Ordinarily rope or cord. Mention of a linear measure called rajju in Śulba sūtra , Arthaśāstra , Śilpa sāstra. In Ancient India there were three kinds of measure : Linear , Superficial , and Voluminal. Opening sūtra of Kātyāyana Śulba parisista i.e. : रज्जु समा समम् वक्ष्याम । {I shall speak of the collection of rules regarding the rajju } In work by Jainas (500 - 300 BC) Rajju started a branch of mathematics.
Śilāṅka(862 A.D.) : रज्जु रिति रज्जुगणितम् क्षेत्रगणित मित्यर्थ । Abhayadeva Sūri : रज्ज्वा यत् सम्ख्यानम् तद् रज्जु रभिधीयते त्त्च क्षेत्रगणितम् । Rajju types by Jainas : सूची रज्जु (Needle like) , प्रतर रज्जु (Superficial) , घन रज्जु (Cubic)
Understanding Important Geometrical Constructions Mentioned in BSS, KSS, ASS like :
To divide a line into any number of equal parts to divide a circle into any number of equal areas by drawing diameters {Bss 2.73-4 and Ass 7. 13-14}
To divide a triangle into a number of equal and similar areas {Bss 3.256} to draw a straight line at right angles to a given line. {Kss 1.3}
To draw a straight line at right angles to a given straight line from a given point on it. To construct a square on given side {Ass 8.8-10}
To construct a rectangle on the given side {Bss 1. 36-40} to construct an isosceles trapezium of given altitude , face and base. {Bss 1.41 Ass 5. 2-5}
To construct a parallelogram having given sides at a given inclination. {Ass 19 .5}
To construct a square equal to the sum of two different squares. {Bss 1. 51-52}
To construct a square equivalent to two given triangles.
To construct a square equivalent to two given pentagons. {Bss 3.68,288 ; Kss 4.8}
To construct a square equal to a given rectangle{Bss 1.58 ; Ass 2.7 ; Kss 3.2-3}
To construct a rectangle having a given side and equivalent to a given square{Ass 3.1;Bss 1.53}
To construct an isosceles trapezium having a given face and equivalent to a given square or rectangle. {Bss 1.55 }
To construct a triangle equivalent to a given square {Bss 1.56}
To construct a square equivalent to a given isosceles triangle.{Kss 4.5}
To construct a rhombus equivalent to a given square or rectangle.{Bss 1.57; Ass 9.9; Kss 4.4}
To construct a square equivalent to a given rhombus. {Kss 4.6} Value of Pi and Root 2 Famous Baudhayana Sulba Sutra {BSS 1.12 , 1.13}
And Many More,
Postulates and Theorems of Constructions are also added. In the Post Vedic Period Ganitasārasangraha and Lilavati are Used for understanding calculations , questions , and methods of Geometry. Other References used , Book by TA Sarasvati Amma Geometry in Ancient Medieval India. Examples and Method from , Chapter 7 of Ganitasārasangraha and Lilavati Chapter क्षेत्रगणितव्यवहारः are Used and Explained thoroughly.
Word Count: 1258