Shulbasutras (शुल्बसूत्राणि) - Revision history

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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.

== 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>Datta. Bibhutibhusan, (1932) ''The Science of the Sulba. A Study in Early Hindu Geometry.'' Calcutta: The University of Calcutta</ref>

One of the prime occupations of the vedic people, performing yajnas, required altars or yajna-vedis of prescribed shapes and sizes. It was primarily in connection with the construction of altars that problems of geometry and also of arithmetic and algebra presented themselves to these ancient rshis, which led to the development of these texts.

== References ==
[[Category:Vedangas]]
[[Category:Shastras]]

← Older revision		Revision as of 09:16, 16 April 2020
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−	~~{{ToBeEdited}}~~	+	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.

	== परिचयः ॥ Introduction ==		== परिचयः ॥ Introduction ==

@@ Line 213: / Line 213: @@
 |86 Angula
 |}
-It should be remembered that the Shulbasutras of different Sutrakaras have slight variations thus we see versions of the above angula measures.
+It gives us an insight about how our ancients developed units of measurement long before other civilizations mentioned it. However, it should also be remembered that the Shulbasutras of different Sutrakaras have slight variations thus we see versions of the above angula measures. It cannot be more emphasized that later day developments in such areas of ganita are founded on these ancient works.
 == Subject-matter of Shulbasutras ==

@@ Line 127: / Line 127: @@
 * 1/8 = Shapha (शफः)
 * 1/4 = Pada (पदा)
-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>
+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 name=":3">Jayashankara, K. (2007) Ph. D. Thesis: ''Sulba sutra, A Critical Analysis.'' Mangalore University ([http://hdl.handle.net/10603/132298 Chapter 8] )</ref>
 == Components of Yajna Vedis ==
@@ Line 137: / Line 142: @@
 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>
-Units and Measurements
+=== Units of Measurements ===
 == Subject-matter of Shulbasutras ==

@@ Line 3: / Line 3: @@
 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>
@@ Line 81: / Line 81: @@
 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 ==
+Besides having the above qualities, Shulbakaras had to perform several other important duties systematically viz.,
 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.
@@ Line 114: / Line 121: @@
 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" />
+The Maitrayani Samhita (3.7.7) mentions about several fractions
 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 ==
@@ Line 167: / Line 187: @@
 |Describing the shapes of syenachiti
 |}
 A brief explanation of a few topics dealt in the shulbasutras is given below
 === Approximation of Surds ===
@@ Line 180: / Line 202: @@
 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 ==

@@ Line 12: / Line 12: @@
 === List of Shulbasutra Works ===
-Of the available extant texts, Baudhayana Shulbasutra is considered to be the most ancient one. It also presents a very systematic and detailed treatment of several topics that are skipped in later texts.
+Of the available extant texts, Baudhayana Shulbasutra is considered to be the most ancient one. Following is the list of shulbasutras and their commentators.
 {| class="wikitable"
 !Vedas
@@ Line 118: / Line 118: @@
 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>
 == References ==

@@ Line 10: / Line 10: @@
 == Authors and Commentators ==
 Unlike the other Sutra charanas, there are a very few Shulbasutra texts available at present and all of them belong to Yajurveda. The Shulbasutras of Baudhayana, Apastamba, Katyayana and Manava are the main sources of the knowledge of geometry. The Kalpa texts of Vedangas clearly outline that astronomy on one hand and geometry on the other were cultivated under different circumstances by different rtviks based on the duties apportioned to them in the conduct of yajnas.<ref name=":2">Bag, A. K., (1979) ''Mathematics in Ancient and Medieval India.'' Varanasi: Chaukhambha Orientalia. (Pages 103 - 174)</ref><ref name=":0" />
 {| class="wikitable"
 !Vedas
@@ Line 28: / Line 31: @@
 |Purvabhaga (7 Kandikas with 90 Sutras)
 Uttarabhaga (40 or 48 shlokas)
-|Rama or Ramachandra (Shulba-sutra-vrtti)
+|Rama or Ramachandra (Shulba-sutra-vrtti), Shulbasutravivarana (Mahidhara), Karka (Shulbasutrabhashya)
 |-
 ! rowspan="7" |Krishna Yajurveda
 | rowspan="4" |Taittriya
 |'''Baudhayana'''
-|3 Adhyayas
+|3 Adhyayas (about 520 Sutras)
 |Dvarakanatha Yajva (Sulba-dipika), Venkatesvara Dikshita (Shulba-mimamsa)
 |-
 |'''Apastamba'''
 |6 Patalas (21 Adhyayas and 223 Sutras)
-|Kapardisvami (Shulba-vyakhya), Karavindasvami (Shulba-pradipika), Sundararaja (Shulba-pradipa or Sundararajiya),
+|Kapardisvami (Shulba-vyakhya), Karavindasvami (Shulba-pradipika), Sundararaja (Shulba-pradipa or Sundararajiya), Gopala (Shulbabhashya)
 |-
 |Varaha
@@ Line 74: / Line 77: @@
 | colspan="3" |None available
 |}
 == 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.<ref name=":0" />
+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.
-In few other instances the word "rajju (रज्जुः) has been used. B. B. Datta states that the words "shulba (शुल्ब)" "shulva (शुल्व)" and "rajju (रज्जुः) may be used synonymously to mean a rope or cord. However, shulba or shulva is derived from the dhatu "शुल्ब्" in the meaning "to measure" and hence its etymological significance is "measuring" or act of measurement. The terms Shulba and Rajju have four meanings
+The word "shulba or shulva" (sulva) stems from the dhatu "शुल्ब्" used in the meaning "माने । to measure" and hence its etymological significance is "measuring" or act of measurement. The term Shulba can be presented in the following ways<ref name=":1" />
-# mensuration - the act and process of measuring
+# भावव्युत्पत्तिः - शुल्बनम् इति शुल्बः । the act of measuring (mensuration)
-# line - the result obtained by measuring
+# कर्मव्युत्पत्तिः - शुल्बयते इति शुल्बः । the entity/result obtained by measuring (line)
-# measure - the instrument of measuring
+# करणव्युत्पत्तिः - शुल्बयत्यनेन इति शुल्बः । the instrument of measuring
-# geometry - the art of measuring
+In the Shulbas the measuring tape is called "Rajju", so is "a line". One who is well-versed in the science of Shulba (geometry) is called Shulbavijnana, a Shulbavid (expert in geometry), a Sulbapariprcchaka (an inquirer into the Shulba), a Samkhyaajna (expert in numbers), Parimaanajna (expert in measuring). A special term sama-sutra-niranchaka (meaning uniform rope stretcher) is also used, this rope plays a very significant role and we find many references of similar words rajjuka, rakku-grahaka, sutra-grahaka - all used to denote a maharaja's land surveyor. A sutra-grahaka is further described as an expert in alignment, a rekajna, one who knows "the line".<ref name=":0" />
 It is extremely interesting to note that land surveying is an ancient procedure followed by our ancestors.
 == Relationship between Shulba and Yajnas ==
-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 such object need not perform them. Every yajna must be made in an altar of prescribed shape and size.
+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" />
-The Vedic altars 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>
 == References ==

@@ Line 4: / Line 4: @@
 == 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>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>
 One of the prime occupations of the vedic people, performing yajnas, required altars or yajna-vedis of prescribed shapes and sizes. It was primarily in connection with the construction of altars that problems of geometry and also of arithmetic and algebra presented themselves to these ancient rshis, which led to the development of these texts.
 == References ==
 [[Category:Vedangas]]
 [[Category:Shastras]]