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Shulbasutras (Samskrit: शुल्बसूत्राणि) are manuals for the construction of yajnas. They are sections of the Kalpasutras, associated in particular with the Shrautasutras. Each Shrautasutra seemed to have their own Shulbasutra section based on literary data; however, in the present days, only seven of these sutra works, Baudhayana, Apastamba, Katyayana, Manava, Maitayana, Varaha and Vadhula are available.  

Shulbasutras (Samskrit: शुल्बसूत्राणि) are manuals for the construction of yajnas. They are sections of the Kalpasutras, associated in particular with the Shrautasutras. Each Shrautasutra seemed to have their own Shulbasutra section based on literary data; however, in the present days, only seven of these sutra works, Baudhayana, Apastamba, Katyayana, Manava, Maitayana, Varaha and Vadhula are available.  

== Introduction ==

== परिचयः ॥ Introduction ==

Recognized as the oldest and earliest treatises on mathematical problems, Shulbasutras give us a glimpse of the knowledge of geometry that the vedic people possessed. Incidentally they furnish us with a few other subjects of much mathematical interest.<ref name=":0">Datta. Bibhutibhusan, (1932) ''The Science of the Sulba. A Study in Early Hindu Geometry.'' Calcutta: The University of Calcutta</ref>

Recognized as the oldest and earliest treatises on mathematical problems, Shulbasutras give us a glimpse of the knowledge of geometry that the vedic people possessed. Incidentally they furnish us with a few other subjects of much mathematical interest.<ref name=":0">Datta. Bibhutibhusan, (1932) ''The Science of the Sulba. A Study in Early Hindu Geometry.'' Calcutta: The University of Calcutta</ref>

Mahidhara in his commentary on Katyayana shulbasutra succintly describes the qualities of a Shulbakara.<ref name=":1">Prof. K. Ramasubramaniam's Lectures - ''Vedas and Sulbasutras, Parts 1 and 2''</ref><blockquote>सङ्ख्याज्ञः परिमाणज्ञः समसूत्रनिरञ्छकः । समसूत्रौ भवेद्विद्वान् शुल्बवित् परिपृच्छकः।।</blockquote><blockquote>शास्त्रबुद्धिविभागज्ञः परशास्त्रकुतूहलः। शिल्पिभ्यः स्थपतिभ्यश्चाप्याददीत मतीः सदा।।</blockquote><blockquote>तिर्यङ्मान्याश्च सर्वार्थः पार्श्वमान्याश्च योगवित्। करणीनां विभागज्ञः नित्योद्युक्तश्च सर्वदा।।</blockquote>A shulbakara must be versed in arithmetic, versed in mensuration, must be an inquirer, quite knowledgeable in one's own discipline, must be enthusiastic in learning other disciplines, always willing to learn from practicing sculptors and architects... and one who is always industrious.

Mahidhara in his commentary on Katyayana shulbasutra succintly describes the qualities of a Shulbakara.<ref name=":1">Prof. K. Ramasubramaniam's Lectures - ''Vedas and Sulbasutras, Parts 1 and 2''</ref><blockquote>सङ्ख्याज्ञः परिमाणज्ञः समसूत्रनिरञ्छकः । समसूत्रौ भवेद्विद्वान् शुल्बवित् परिपृच्छकः।।</blockquote><blockquote>शास्त्रबुद्धिविभागज्ञः परशास्त्रकुतूहलः। शिल्पिभ्यः स्थपतिभ्यश्चाप्याददीत मतीः सदा।।</blockquote><blockquote>तिर्यङ्मान्याश्च सर्वार्थः पार्श्वमान्याश्च योगवित्। करणीनां विभागज्ञः नित्योद्युक्तश्च सर्वदा।।</blockquote>A shulbakara must be versed in arithmetic, versed in mensuration, must be an inquirer, quite knowledgeable in one's own discipline, must be enthusiastic in learning other disciplines, always willing to learn from practicing sculptors and architects... and one who is always industrious.

== Etymology ==

 

== व्युत्पत्तिः॥ Etymology ==

The word Shulba (शुल्ब) means a 'cord', 'a rope', 'a string'. A Sutra refers to a short rule. The commentators refer to this subject matter as Shulba as the true name, Shulba-parishista, Shulbi Kriya etc.

The word Shulba (शुल्ब) means a 'cord', 'a rope', 'a string'. A Sutra refers to a short rule. The commentators refer to this subject matter as Shulba as the true name, Shulba-parishista, Shulbi Kriya etc.

ॐ पूर्णमदः पूर्णमिदं पूर्णात् पूर्णमुदच्यते । पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥

ॐ पूर्णमदः पूर्णमिदं पूर्णात् पूर्णमुदच्यते । पूर्णस्य पूर्णमादाय पूर्णमेवावशिष्यते ॥

== Relationship between Shulba and Yajnas ==

=== Fractions ===

Yajnas are of two main classes: Nitya (obligatory) and Kamya (optional or performed with a special intent). Nitya yajnas are indispensible and not performing them will incur papam (sin) to the yajamana. The Kamya yajnas are, however, optional and those who do not have any specific aims to achieve any desired objects need not perform them. Every yajna must be made in an altar of prescribed shape and size. These sutras give a compilation of principles in geometry that were used in designing the altars (called vedi or citi) where the yajnas were to be performed. The platforms of the altars were built with burnt bricks and mud mortar. It is stated that even a slight irregularity and variation in the form and size of the chiti (Citi) will nullify the object of the whole ritual and may even lead to an adverse effect, accuracy was the key. So ancient adhvaryus took utmost care in building a chiti of right shape and size.<ref name=":0" />

These names of fractions referred in context with bargaining for a price of material (Soma) to be purchased. The price of the material is increased step by step. <blockquote>वसो ह्येष विन्दते यः सोमं क्रीणाति ॥ सोमविक्रयिन्सोतिमं ते क्रीणानि महान्तं बह्वर्हं बहु शोभमानं , कलया ते क्रीणानि, कुष्टया ते क्रीणानि, शफेन ते क्रीणानि, पदा ते क्रीणानि ॥ (Mait. Samh. 3.7.7)<ref>Maitrayani Samhita ([https://sa.wikisource.org/wiki/%E0%A4%AE%E0%A5%88%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BE%E0%A4%AF%E0%A4%A3%E0%A5%80%E0%A4%B8%E0%A4%82%E0%A4%B9%E0%A4%BF%E0%A4%A4%E0%A4%BE/%E0%A4%95%E0%A4%BE%E0%A4%A3%E0%A5%8D%E0%A4%A1%E0%A4%82_%E0%A5%A9/%E0%A4%AA%E0%A5%8D%E0%A Kanda 3 Prapathaka 7])</ref></blockquote>Thus we see naming of fractions and operations like sum, subtractions and fractions were quite common at those times.<ref>Jayashankara, K. (2007) Ph. D. Thesis: ''Sulba sutra, A Critical Analysis.'' Mangalore University ([http://hdl.handle.net/10603/132298 Chapter 8 Page 8])</ref>

 

== Components of Yajna Vedis ==

[[Yajna (यज्ञः)|Yajnas]] are of two main classes: Nitya (obligatory) and Kamya (optional or performed with a special intent). Nitya yajnas are indispensible and not performing them will incur papam (sin) to the yajamana. The Kamya yajnas are, however, optional and those who do not have any specific aims to achieve any desired objects need not perform them. In this section the knowledge of mathematics, the instruments and units of measurements, the bricks used for construction of the altars - as mentioned in Shulbasutras, are discussed.

 

Every yajna must be carried out in a [[Chiti (चितिः)]] (a fire altar) also called as Vedi, of prescribed shape and size. These sutras give a compilation of principles in geometry that were used in designing the altars where the yajnas were to be performed. The platforms of the altars were built with burnt bricks and mud mortar. It is stated that even a slight irregularity and variation in the form and size of the chiti (Citi) will nullify the object of the whole ritual and may even lead to an adverse effect, accuracy was the key. So ancient adhvaryus took utmost care in building a chiti of right shape and size.<ref name=":0" />

The chitis had rich symbolic significance and their designs were often intricate. For instance, the Syenacit has the shape of a falcon in flight (a symbolic representation of the aspiration of soaring upward); the Kurmachit is shaped as a tortoise, with extended head and legs, the rathacakracit as a chariot wheel with spokes, and so on.<ref>A. K. Dutta and M. S. Sriram. ''[https://pdfs.semanticscholar.org/bb84/9b5ff23bc22e2056f0aab069c92c05f7af0c.pdf Mathematics and Astronomy in India before 300 BCE.]'' </ref>

The chitis had rich symbolic significance and their designs were often intricate. For instance, the Syenacit has the shape of a falcon in flight (a symbolic representation of the aspiration of soaring upward); the Kurmachit is shaped as a tortoise, with extended head and legs, the rathacakracit as a chariot wheel with spokes, and so on.<ref>A. K. Dutta and M. S. Sriram. ''[https://pdfs.semanticscholar.org/bb84/9b5ff23bc22e2056f0aab069c92c05f7af0c.pdf Mathematics and Astronomy in India before 300 BCE.]'' </ref>

== Subject-matter of Shulbasutras ==

== Subject-matter of Shulbasutras ==

|Describing the shapes of syenachiti

|Describing the shapes of syenachiti

|}

|}

A brief explanation of a few topics dealt in the shulbasutras is given below

 

A brief explanation of a few mathematical topics dealt in the shulbasutras is given below

=== Approximation of Surds ===

=== Approximation of Surds ===

Now, to answer the question about the necessity of such an experiment, instead of simply looking at the sunrise or sunset to determine the cardinal directions, the commentator Mahidhara observes:<blockquote>... तस्य उदयस्थानानां बहुत्वात् प्रतिदिनं भिन्नत्वात् अनियमेन प्राची ज्ञातुं न शक्या। तस्मात् शङ्कुस्थापनेन प्राचीसाधनमुक्तम्। दक्षिणायने चित्रापर्यन्तमर्कोऽभ्युदेति। मेषतुलासङ्क्रात्यहे प्राच्यां  शुद्धायामुदेति। ततोऽर्कात् प्रचीज्ञानं दुर्घटम्।</blockquote>Meaning: Since the rising points are many, varying from day to day, the (cardinal) east point cannot be known (from the sunrise point). Therefore it has been prescribed that the east be determined by fixing a Shanku (शङ्कु)... therefore simply looking at the sun and determining the east is difficult.  

Now, to answer the question about the necessity of such an experiment, instead of simply looking at the sunrise or sunset to determine the cardinal directions, the commentator Mahidhara observes:<blockquote>... तस्य उदयस्थानानां बहुत्वात् प्रतिदिनं भिन्नत्वात् अनियमेन प्राची ज्ञातुं न शक्या। तस्मात् शङ्कुस्थापनेन प्राचीसाधनमुक्तम्। दक्षिणायने चित्रापर्यन्तमर्कोऽभ्युदेति। मेषतुलासङ्क्रात्यहे प्राच्यां  शुद्धायामुदेति। ततोऽर्कात् प्रचीज्ञानं दुर्घटम्।</blockquote>Meaning: Since the rising points are many, varying from day to day, the (cardinal) east point cannot be known (from the sunrise point). Therefore it has been prescribed that the east be determined by fixing a Shanku (शङ्कु)... therefore simply looking at the sun and determining the east is difficult.  

Having obtained the prachi, getting udichi (the north-south line) correctly is extremely important for the construction of various vedis having bilateral symmetry.<ref name=":1" />

Having obtained the prachi, getting udichi (the north-south line) correctly is extremely important for the construction of various vedis having bilateral symmetry. Two methods have been described for obtaining the perpendicular bisector of a given straight line:<ref name=":1" />

 

 

== References ==

== References ==