[Pramāṇa](https://indike.org/pramana/) › [Studies](https://indike.org/pramana/ins) › [Science, technology and education](https://indike.org/pramana/ins.science) › [The Science of the Shulba: A Study in Early Hindu Geometry](https://indike.org/pramana/ins.science.datta-1932) › c7 # The Science of the Shulba: A Study in Early Hindu Geometry c7: Chapter II. Commentators The Science of the Śulba: A Study in Early Hindu Geometry c7 - **Language:** English ### Chapter II. Commentators ### [c7.2](https://indike.org/pramana/ins.science.datta-1932.c7.2) There are now available several commentaries on the \*Śulbas\*. The more important manuals are found to have been commented upon by more than one writer. Thus we have two commentaries on the \*Śulba\* of Baudhāyana. One of them is by Dvārakanātha Yajvā and is named \*Śulba-dīpikā\* ("The Light of the \*Śulba\*"). The other, called \*Śulba-mīmāṁsā\* ("The investigation into the \*Śulba\*"), is by Veṅkaṭeśvara Dīkṣita. On the \*Āpastamba Śulba\*, there are as many as four well-known commentaries: (1) \*Śulba-vyākhyā\* ("The Explanation of the \*Śulba\*") by Kapardisvāmī, (2) \*Śulba-pradīpikā\* ("The Light of the \*Śulba\*") by Karavindasvāmī, (3) \*Śulba-pradīpa\* ("The Light of the \*Śulba\*") by Sundararāja and (4) \*Āpastambīya Śulba-bhāṣya\* ("The commentary on the \*Śulba\* of Āpastamba") by Gopāla, son of Gārgya Nṛsiṁha Somasuta. Sundararāja's work is also called \*Sundara-rājīya\* ("The work of Sundararāja") after the name of the author, as is usual in Sanskrit. I have come across two commentaries on the \*Kātyāyana Śulba\*, namely, \*Śulba-sūtra-vṛtti\* ("The Explanation of the \*Śulba-sūtra\*") of Rāma or Rāmacandra, son of Sūryadāsa and \*Śulba-sūtra-vivaraṇa\* ("The Exposition of the Śulba-sūtra") by Mahīdhara. ### [c7.3](https://indike.org/pramana/ins.science.datta-1932.c7.3) The dates of most of the commentators of the \*Śulbas\*, more particularly of the notable ones, have not as yet been ascertained, even approximately. Nor is it easy to do so. The periods to which some of them can be assigned from the reference by them to anterior writers and from the reference to them by writers posterior lie within such widely varying limits as to be of no tangible value. We shall begin here with the notice of those commentators whose times are known either definitely or very nearly so. ### [c7.4](https://indike.org/pramana/ins.science.datta-1932.c7.4) We find from the colophon that Mahīdhara completed his commentary on the \*Kātyāyana Śulba\* in the Saṁvat year 1646 (= 1589 A. D.), at Benares. It is also stated there that that commentary is based on the \*Śulba-sūtra-vṛtti\* of Rāma. Mahīdhara wrote as many as seventeen works on various subjects. His \*Mantramahodadhi\* was completed in 1589 A. D. and \*Viṣṇubhakti-Kalpalatā-prakāśa\* in 1597. ### [c7.5](https://indike.org/pramana/ins.science.datta-1932.c7.5) The commentator Rāma was an inhabitant of Naimiṣa (near modern Lucknow). He seems to have been the author of several works such as \*Karma-dīpikā\*, \*Kuṇḍākṛti\* (with commentary), \*Śulba-vārttika\*, \*Śāṅkhyāyana Gṛhya-paddhati\*, \*Samara-sāra\* and its commentary, \*Samara-sāra-saṁgraha\* and the commentaries on the \*Kātyāyana Śulba\* and \*Śāradā-tilaka Tantra\*. The date of composition of the \*Kuṇḍākṛti\* is given as 1506 Vikrama Saṁvat (1449 A.D.). In his commentary on the \*Kātyāyana Śulba\*, Rāma has quoted copiously from his \*Śulba-vārttika\* ("The Critical Annotation of the \*Śulba\*") and also from his commentary on the \*Śāradā-tilaka\*. There is also a quotation from the \*Triśatikā\* of Śrīdhara (\*c.\* 750 A.D.).¹ In this work we notice some new contributions from him. To construct a right-angled triangle having a given leg (a), Rāma suggests the employment of a new rational rectangle (a, 8a/15, 17a/15) in addition to those taught in the \*Śulbas\*.² But the most notable contribution of him is a correction to the well-known \*Śulba\* value of √2, viz., ### [c7.6](https://indike.org/pramana/ins.science.datta-1932.c7.6) ¹ \*KŚl\*, i. 30 (\*com.\*). The quoted passage is the Rule 47 of the \*Triśatikā\* of Śrīdhara, but there is no mention of any name. ### [c7.7](https://indike.org/pramana/ins.science.datta-1932.c7.7) ² \*KŚl\*, i. 15 (\*com.\*). ### [c7.8](https://indike.org/pramana/ins.science.datta-1932.c7.8) $√2 = 1 + ⅓ + 1/3.4 - 1/3.4.341$ ### [c7.9](https://indike.org/pramana/ins.science.datta-1932.c7.9) Rāma shows that a more accurate (\*sūkṣmatara\*) value will be given by ¹ ### [c7.10](https://indike.org/pramana/ins.science.datta-1932.c7.10) ¹ \*Ibid\*, ii. 13 (\*com.\*). The rationale of this is stated to have been given in the \*Śulba-vārttika\* whence this and the preceding matters have been taken. ### [c7.11](https://indike.org/pramana/ins.science.datta-1932.c7.11) $√2=1+⅓+1/3.4-1/3.4.34-1/3.4.34.33+1/3.4.34.34. (2)$ ### [c7.12](https://indike.org/pramana/ins.science.datta-1932.c7.12) Turned into decimal fraction, the expression (1) gives √2=1·4142156863 and (2) yields √2=1·414213502 According to modern calculation, √2=1·41421356 So that Rāma's value for √2 is correct up to seven places of decimals whereas the \*Śulba\* value is up to five places. ### [c7.13](https://indike.org/pramana/ins.science.datta-1932.c7.13) It should perhaps be noted that for certain inconsistency with the \*Śrauta-sūtra\* of Kātyāyana, Rāma suspects that the \*Śulba\* attributed to Kātyāyana might be written by a different person.² But the inconsistency is so minor that we cannot subscribe to the opinion of Rāma in this matter. It can be reasonably explained in other ways. ### [c7.14](https://indike.org/pramana/ins.science.datta-1932.c7.14) ² Compare \*KŚl\*, ii. 8 (\*com.\*). ### [c7.15](https://indike.org/pramana/ins.science.datta-1932.c7.15) Śivadāsa, son of Nārada, a resident of the city of Benares, wrote a commentary on the \*Mānava Śulba\*. His younger brother, Śaṅkarabhaṭṭa is the commentator of \*Maitrāyaṇīya Śulba\*. Both the brothers quote from Rāma Vājapeya, who is no other than the commentator of the \*Kātyāyana Śulba\*. Śivadāsa has quoted the second Bhāskara's (1150 A.D.) \*Rule of Three\*, and also from his \*Līlāvatī\*, by name. He must have been posterior to the celebrated Sāyaṇa (1320-1380 A.D.) whom he quotes. ### [c7.16](https://indike.org/pramana/ins.science.datta-1932.c7.16) Śivadāsa observes : ### [c7.17](https://indike.org/pramana/ins.science.datta-1932.c7.17) "The study of the \*Śulba\* should be begun after having finished the study of the science of mathematics; otherwise there cannot be a thorough knowledge of the \*Śulba\*." ### [c7.18](https://indike.org/pramana/ins.science.datta-1932.c7.18) Of the known commentaries of the \*Āpastamba Sulba\* the earliest one is, I have good reasons to believe, that of Kapardisvāmī. This writer is known to have commented also on \*Āpastamba-Śrauta-sūtra\*, \*Āpastamba-sūtra-paribhāṣā\*, \*Darśapaurṇamāsa-sūtra\*, \*Bhāradvāja Gṛhya-sūtra\*, etc. He is quoted by Śūlapāṇi, Hemādri, Nīlakaṇṭha and others. Now Śūlapāṇi lived near about 1150 A.D. He was the teacher of the famous Ṣaḍguruśiṣya (1143-1193 A.D.), the author of the \*Vedārthadīpikā\*. Hemādri was the minister of King Mahādeva (1260-71) of Devagiri and of his nephew and successor Rāmachandra (1271-1309). So Kapardisvāmī lived before the twelfth century of the Christian era. He has generalised a method taught in the \*Sulbas\* for finding the rational right-angled triangles having a given leg. He says : ### [c7.19](https://indike.org/pramana/ins.science.datta-1932.c7.19) "The added portion is divided into as many parts as the number obtained by dividing the (given) leg with the added portion by half the added portion ; (put) the \*nirañchana\* mark by diminishing the added portion by one part." ### [c7.20](https://indike.org/pramana/ins.science.datta-1932.c7.20) Let a be the given leg and suppose it to be increased by adding a portion a/m, where m is any rational integer. Dividing the increased length by half the increment, we get $(a+a/m)÷a/2m=2(m+1).$ ### [c7.21](https://indike.org/pramana/ins.science.datta-1932.c7.21) So that the added portion a/m shall have to be divided into 2(m+1) parts. Then the \*nirañchana\* mark is to be made at a distance $a/m-a/m÷2(m+1)=a/m-a/(2m(m+1))=((2m+1)a)/(2m(m+1)).$ ### [c7.22](https://indike.org/pramana/ins.science.datta-1932.c7.22) So that $a²+((2m+1)/(2m²+2m))²a²=((2m²+2m+1)/(2m²+2m))²a².$ ### [c7.23](https://indike.org/pramana/ins.science.datta-1932.c7.23) This method is equally available even when m is a rational fraction; that is, when the given leg is increased also by a multiple of it, instead of by only a sub-multiple of it. But this further generalisation seems to have escaped the notice of Kapardisvāmī. At any rate, his statement does not expressly show that he meant both the cases by his generalisation. Karavindasvāmī is, however, very explicit to leave no doubt in our mind in this respect. ### [c7.24](https://indike.org/pramana/ins.science.datta-1932.c7.24) He says :¹ ### [c7.25](https://indike.org/pramana/ins.science.datta-1932.c7.25) ¹ \*ĀpŚl\*, i. 2 (\*com.\*). ### [c7.26](https://indike.org/pramana/ins.science.datta-1932.c7.26) " In case of all additions, as many times the added portion as the sum of the given side and the added portion is, into twice so many parts the added portion is divided; make the mark there (\*i.e.\*, in the added portion, at a distance) less by one such part. For instance, in case of adding to the given side its half, (consider) that half as one part; the given side contains two such parts. So the given side with its increment contains three parts like the increment. Dividing the added portion into twice as many parts, that mark will be (at a distance) less by one-sixth the added portion. So in case of increasing the given side by itself, the increment is one part ; the given side has one part like it. So the given side with its increment has two parts. On dividing the increment into twice as many parts, it will be divided into four parts; then the mark will be (at a distance) less by the fourth part. Similarly in case of adding the third part, the added portion is one part; the given side contains three such parts. On dividing the added portion into twice as many parts, the mark will be (at a distance) less by its one-eighth part. In the same way in case of adding the fourth and other parts, the sum of the given side and its increment should be divided into parts in the same way and the mark should be made (at a distance) less by one such ### [c7.27](https://indike.org/pramana/ins.science.datta-1932.c7.27) part. Now, when the increment happens to be equal to the given side, how is then the given side to be divided? How also the mark (should be made)? How also in the case when the increment happens to be greater (than the given side)? There also the method is exactly the same, we say. But the given side with its increment should then be reduced to common denominators; the added portion should then be divided into twice the number of parts (thus obtained) and the mark should be made in it (at a distance) less by one such part. For instance, in case of adding twice as much, the increment is one part and (in terms of it) the given side is a half part. Then on adding together the increment and the given side after reduction to common denominators, there will be three halves. On dividing the increment into twice that number of parts, there will be six halves in the denominator; so the mark will be (at a distance) less by one of these parts. In case of adding three times, the increment is one part; the given side is the third part (of that); there the increment contains three third parts. So the given side and the increment together contain four third parts. On dividing the increment into twice as many parts there will be eight third parts in the denominator. Then the mark will be (at a distance) less by one of these parts. In the cases of adding four times, etc., the divisions and the marks should be made in the same way." ### [c7.28](https://indike.org/pramana/ins.science.datta-1932.c7.28) Let a be the given side; let it be increased by its mth part. ### [c7.29](https://indike.org/pramana/ins.science.datta-1932.c7.29) $(a + a/m) ÷ a/m = m + 1$ ### [c7.30](https://indike.org/pramana/ins.science.datta-1932.c7.30) Then $a/m ÷ 2(m + 1) = a/(2m(m + 1))$ ### [c7.31](https://indike.org/pramana/ins.science.datta-1932.c7.31) $a + a/(2m(m + 1)) = ((2m² + 2m + 1)/(2m² + 2m))a$ ### [c7.32](https://indike.org/pramana/ins.science.datta-1932.c7.32) $a/m-a/(2m(m+1))=((2m+1)/(2m²+2m))a$ ### [c7.33](https://indike.org/pramana/ins.science.datta-1932.c7.33) So it follows $a²+((2m+1)/(2m²+2m))² a²=((2m²+2m+1)/(2m²+2m))² a² (1)$ ### [c7.34](https://indike.org/pramana/ins.science.datta-1932.c7.34) Or let the given side a be increased n times it. $(a+na)÷ na=(n+1)/n$ $na÷ 2((n+1)/n)=(n² a)/(2n+2)$ $a+(n² a)/(2n+2)=((n²+2n+2)/(2n+2))a$ $na-(n² a)/(2n+2)=((n²+2n)/(2n+2))a$ ### [c7.35](https://indike.org/pramana/ins.science.datta-1932.c7.35) So it follows $a²+((n²+2n)/(2n+2))² a²=((n²+2n+2)/(2n+2))² a² (2)$ ### [c7.36](https://indike.org/pramana/ins.science.datta-1932.c7.36) The two results can be combined into one $a²+((r²+2r)/(2r+2))² a²=((r²+2r+2)/(2r+2))² a²$ where r is any rational number integral or fractional. ### [c7.37](https://indike.org/pramana/ins.science.datta-1932.c7.37) Karavindasvāmī, indeed, wrote a commentary on the whole of the \*Śrauta-sūtra\* of Āpastamba. He is known to be the author of a few other works also. His time is still very uncertain. He is found to have quoted, without any mention of name, certain passages from the \*Āryabhaṭīya\* (499 A.D.) of Āryabhaṭa I (born 476).¹ So he undoubtedly flourished after the fifth century of the Christian era. Though we are not in a position to fix or even suggest any closer upper limit to his time, this limit seems to us to be too earlier. There is, however, one passage in his commentary on the \*Āpastamba Śulba\* which might lead one to take him as belonging to a very early age. He has referred to a certain treatise on mathematics which gives an incorrect formula for the calculation of the area of a segment of a circle :¹ ### [c7.38](https://indike.org/pramana/ins.science.datta-1932.c7.38) ¹ \*ĀpŚl\*, iii. 5 (com.); the reference is to the \*Āryabhaṭīya\*, ii. 9. ### [c7.39](https://indike.org/pramana/ins.science.datta-1932.c7.39) ¹ “ \*\*शरार्द्ध प्रमाणेन धनुरर्द्ध मध्यस्य धनुषो फलावगम इत्यादि गणितशास्त्रादवगन्तव्यमिति ।\*\* ”—\*ĀpŚl\*, iii. 5 (\*com.\*). But on a different occasion (\*ĀpŚl\*, vii. 14-15, \*com.\*), he says “ \*\*शराहतं तु कोदण्डदलितौ धनुषः फलम् ।\*\* ” This indeed gives accurately the area of the sector of the circle. ### [c7.40](https://indike.org/pramana/ins.science.datta-1932.c7.40) $Area of a segment = arc/2 × arrow/2.$ ### [c7.41](https://indike.org/pramana/ins.science.datta-1932.c7.41) This formula is not found in any known treatise on Hindu mathematics and we further know that from the time of Śrīdhara (\*c.\* 750), the Hindu mathematicians used a more approximate formula for the calculation of the area of the segment of a circle. Does it then follow that Karavindasvāmī lived in an age before the time of the discovery of that formula, that is, before 750 A.D.? It may be noted that the other formulæ in connexion with the mensuration of the segment of a circle have been stated as correctly as we find in the works of Brahmagupta (628) and other early Hindu mathematicians. ### [c7.42](https://indike.org/pramana/ins.science.datta-1932.c7.42) We are equally uncertain about the time of Sundararāja. This much we are sure that he lived before the fourth quarter of the sixteenth century of the Christian era. For it appears from the post-colophon that the copy of the manuscript of his commentary on the \*Āpastamba Śulba\* now in the possession of the State Library of Tanjore (No. 9160) was made in Saṁvat 1638 (=1581 A.D.), and that in the Government Collection of the Asiatic Society of Bengal in Saṁvat 1645 (=1588 A.D.). Sundararāja is found to have quoted from Dvārakanātha Yajvā's commentary on the \*Baudhāyana Śulba\* a few passages dealing with the transformation of a square into a rectangle having a given side, the correction to the Śulba formula for squaring the circle and \*vice versa\*, enlargement of an altar and certain other matters. ### [c7.43](https://indike.org/pramana/ins.science.datta-1932.c7.43) Dvārakanātha must be posterior to Āryabhaṭa I (499) whom he quotes.¹ He proves with the help of illustrative examples that the methods taught in the \*Śulbas\* for the squaring of a circle and \*vice versa\* do not lead to an accurate result as compared with that obtained by the method of Āryabhaṭa. If 2\*a\* be the side of the square equivalent to the circle of radius \*r\*, then according to the \*Śulba\*, ### [c7.44](https://indike.org/pramana/ins.science.datta-1932.c7.44) ¹ \*BŚl\*, i. 60 (\*com.\*); the quoted passages are \*Āryabhaṭīya\*, ii. 7, 10. ### [c7.45](https://indike.org/pramana/ins.science.datta-1932.c7.45) $r=a+a/3(√2-1),$ ### [c7.46](https://indike.org/pramana/ins.science.datta-1932.c7.46) $a=r-r/8+r/8.29-r/8.29.6+r/8.29.6.8.$ ### [c7.47](https://indike.org/pramana/ins.science.datta-1932.c7.47) These lead to π=3·0883, 3·0885, respectively. Dvārakanātha Yajvā emends them to ² ### [c7.48](https://indike.org/pramana/ins.science.datta-1932.c7.48) ² \*BŚl\*, i. 60 (\*com.\*). ### [c7.49](https://indike.org/pramana/ins.science.datta-1932.c7.49) $r=a+a/3(√2-1)(1-1/118),$ ### [c7.50](https://indike.org/pramana/ins.science.datta-1932.c7.50) $a=(r-r/8+r/8.29-r/8.29.6+r/8.29.6.8)(1+½·3/133).$ ### [c7.51](https://indike.org/pramana/ins.science.datta-1932.c7.51) These will work out π=3·141109, 3·157991. ### [c7.52](https://indike.org/pramana/ins.science.datta-1932.c7.52) Dvārakanātha Yajvā states that ### [c7.53](https://indike.org/pramana/ins.science.datta-1932.c7.53) $10/(√3) angulis=5 angulis 27½ tilas,$ ### [c7.54](https://indike.org/pramana/ins.science.datta-1932.c7.54) $12/(√3) angulis=6 angulis 32 tilas,$ ### [c7.55](https://indike.org/pramana/ins.science.datta-1932.c7.55) $15/(√3) angulis=8 angulis 23 tilas.$ ### [c7.56](https://indike.org/pramana/ins.science.datta-1932.c7.56) From these we get ### [c7.57](https://indike.org/pramana/ins.science.datta-1932.c7.57) According to modern calculation 1/√3=·5773 ### [c7.58](https://indike.org/pramana/ins.science.datta-1932.c7.58) It should be noted that there were also other commentators of the \*Śulbas\* anterior to those who are known to us now. Kapardisvāmī, the earliest known commentator of the \*Āpastamba Śulba\* has referred to at least one such anterior commentator.¹ ### [c7.59](https://indike.org/pramana/ins.science.datta-1932.c7.59) ¹ “केचिदत्र सदृशशब्दश्रवणादतिशयस्तृतीयेनेति वर्णयन्ति चैवसप्तत्या, तदनिस्चितमिति वाचार्य्यस्याङ्गं व्यायामेनेति वदत...।” \*ĀpŚl\*, vii. 10 (\*com.\*). ## Linked texts (6) - [Āryabhaṭīya](https://indike.org/pramana/in.astro.siddhanta.early.aryabhatiya.2.9) — Āryabhaṭa, Sanskrit - [Āpastambaśrautasūtra](https://indike.org/pramana/in.vedanga.kalpa.srauta.apastamba) — Āpastamba, Sanskrit - [Līlāvaī](https://indike.org/pramana/in.lit.kavya.prakrit.lilavai) — Kouhala, Māhārāṣṭrī Prakrit - [Bhāradvājagṛhyasūtra](https://indike.org/pramana/in.vedanga.kalpa.grhya.bharadvaja) — Sanskrit - [Āpastambaśulbasūtra](https://indike.org/pramana/in.vedanga.kalpa.sulba.apastamba) — Āpastamba, Sanskrit - [Triśatikā (Pāṭīgaṇitasāra)](https://indike.org/pramana/in.astro.ganita.patibija.trisatika) — Śrīdhara (mathematician), Sanskrit --- Source: [https://indike.org/pramana/ins.science.datta-1932.c7](https://indike.org/pramana/ins.science.datta-1932.c7) · Pramāṇa