The Science of the Shulba: A Study in Early Hindu Geometry c6: Chapter I. Śulbas

The Science of the Śulba: A Study in Early Hindu Geometry c6

Language: English

Chapter I. Śulbas

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The *Śulbas*, or as they are more commonly known at present amongst oriental scholars, the *Śulba-sūtras*, are manuals for the construction of altars which are necessary in connexion with the sacrifices of the Vedic Hindus. They are sections of the *Kalpa-sūtras*, more particularly¹ of the *Śrauta-sūtras*, which form one of the six *Vedāṅgas* (or "The Members of the Veda") and deal specially with rituals or ceremonials. Each *Śrauta-sūtra* seems to have its own *Śulba* section. So there were, very likely, several such works in ancient times.² At present we know, however, of only seven *Śulba-sūtras*, those belonging to the *Śrauta-sūtra* of Baudhāyana, Āpastamba, Kātyāyana,

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¹ The *Kalpa-sūtras* are broadly divided into two classes, the *Gṛhya-sūtras* (or "The rules for ceremonies relating to family or domestic affairs" such as marriage, birth, etc.) and the *Śrauta-sūtras* ("The rules for ceremonies ordained by the Veda" such as the preservation of sacred fires, performance of the sacrifices, etc.). The *Śulba-sūtras* belong to this latter class.

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² We have it on the authority of Patañjali (150 B.C.), the Great Commentator of Pāṇini's Grammar, that there were as many as 1,131 or 1,137 different schools of the *Veda*.

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"एकविंशतिधा बाह्वृच्यम्, एकशतमध्वर्युशाखाः।
सहस्रवर्त्मा सामवेदः, नवधा चायर्वणो वेदः पञ्चदशभेदो वा॥"

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Or "There were 21 different schools of the *Ṛg-veda*; 101 schools of the *Yajur-veda*; 1,000 of the *Sāma-veda*; and 9 or 15 of the *Atharva-veda*." Each school of the *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.

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Mānava, Maitrāyaṇa, Vārāha and Vādhula.¹ These manuals are also found separately.

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¹ In the commentary of Karavindasvāmī on the *Āpastamba Śulba* (xi. 11), we find reference to two other works, *viz.*, *Maśaka Śulba* and *Hiraṇyakeśi Śulba*, which are not available now. There is also a quotation from the latter work (*ĀpŚl*, vi. 10).

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As related to the different *Vedas*, the *Śulba-sūtras* of Baudhāyana, Āpastamba, Mānava, Maitrāyaṇa and Vārāha belong to the *Kṛṣṇa Yajur-veda*; and the *Kātyāyana Śulba-sūtra* to the *Śukla Yajur-veda*.

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It was perhaps primarily in connexion with the construction of the sacrificial altars of proper size and shape that the problems of geometry and also of arithmetic and algebra presented themselves, and were studied in ancient India, just as the study of astronomy is known to have begun and developed out of the necessity for fixing the proper time for the sacrifice.² At any rate, from the *Śulba-sūtras*, we get a glimpse of the knowledge of geometry that the Vedic Hindus had.³ Incidentally they furnish us with a few other subjects of much mathematical interest.

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² Bibhutibhushan Datta, "The Scope and Development of the Hindu *Gaṇita*," *Ind. Hist. Quart.*, Vol. 5 (1929), pp. 479-512.

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³ There are reasons to believe that side by side with the practical geometry of the *Śulbas*, the Vedic sacrificial priests had also an esoteric geometry as their secret property.

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Of all the extant *Śulbas*, that of the Baudhāyana is the biggest and is also, perhaps, the oldest. It is divided into three chapters. The first chapter contains 116 *sūtras* ("aphorisms") of which the opening two are merely introductory; *sūtras* 3-21 define the various measures ordinarily employed in the *Śulbas*; *sūtras* 22-62 give the more important of the geometrical propositions necessary for the construction of the sacrificial altars; and *sūtras* 63-116 deal briefly with the relative positions and spatial magnitudes of the various *vedis* (or "altars"). The second chapter consists of 86 *sūtras* of which the major portion, *sūtras* 1-61, is devoted to the description of the spatial relations in the different constructions of the Agnis (or "the large Fire-altars made of bricks") in general, and the remaining portion, *sūtras* 62-86, elaborates the construction of the two simplest Agnis, *viz.*, the *Gārhapatya-citi* (or "The House-holder's Fire-altar") and *Chandaściti*¹ (or "The Agni made, as it were, of *mantras* instead of bricks"). The third chapter, in altogether 323 *sūtras*, describes the construction of as many as seventeen different kinds of *Kāmya Agnis* (or "the altars for the sacrifices performed with a view to attain definite objects") of rather complex nature. In case of some, the description is quite elaborate and minute in details, but in other cases it is less so.

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¹ In case of the *Chandaściti*, the *agnicit* ("the Fire-altar-builder") draws on the ground the Agni of the prescribed shape, ordinarily of the primitive shape of the falcon. He then goes through the whole prescribed process of construction imagining all the while as if he is placing every brick in its proper place with the appropriate *mantras*. The *mantras* are, indeed, muttered but the bricks are not actually laid. Hence the name *Chandaściti*, that is, the *citi* or altar made up of *chandas* or Vedic *mantras* instead of bricks or loose mud pieces.

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The *Śulba-sūtra* of Āpastamba is broadly divided into six *paṭalas* (or "sections"). Of these the first, third and the fifth are each subdivided again into three *adhyāyas* (or "chapters") and each of the remaining sections into four chapters. So that altogether the work contains twenty-one chapters and 223 *sūtras*. The first section of the manual, chapters i-iii, gives the important geometrical propositions required for the construction of altars. The second section or the chapters iv-vii, describe the relative positions of the various *vedis* and their spatial magnitudes. Unlike Baudhāyana, Āpastamba here indicates briefly also the methods of their construction. They are of course the particular applications of the general geometrical theorems taught in the earlier section. The remaining sections of the *Āpastamba Śulba-sūtra*, comprising the chapters viii-xxi deal with the construction of the *Kāmya Agnis*. It is noteworthy that almost the same set of geometrical propositions are taught by both Baudhāyana and Āpastamba. But the latter has treated of a smaller number of varieties of the *Kāmyas* than the former. For instance, Āpastamba teaches only one kind of *ratha-cakra-citi* (or "the wheel-shaped altar") whereas Baudhāyana gives two.

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The *Śulba-sūtra* of Kātyāyana, also known as *Kātyāyana Śulba-pariśiṣṭa* or *Kātīya Śulba-pariśiṣṭa*, is divided into two parts. The first part is composed in the style of the *sūtras* or aphorisms, as those noted above, while the second part is composed in verses. The earlier part is again subdivided into seven *kaṇḍikās* (or "short sections") containing altogether 90 *sūtras*. It teaches the geometrical propositions, the different measures employed in the work, and the relative positions and spatial relations for the different constructions of the *Agnis*. This manual does not treat of the construction of the *Kāmya Agnis*. It is because that subject has been treated in a different chapter of the *Kātyāyana Śrauta-sūtra*.¹ The second part comprises nearly about 40 or 48 verses.² It gives mainly a description of the measuring tape (*rajju*), the gnomon, the attributes of an expert altar-builder and also a few general rules for his conduct. Some of the processes of construction described in the earlier part together with a few other new matters, though of comparatively minor importance, also appear there. I think the title *Kātyāyana Śulba-pariśiṣṭa* or ("The Appendix to the *Śulba of Kātyāyana* ") was originally designed for this part and should be kept reserved to it, even now. For it is really a sort of an appendix to the earlier part, the *Kātyāyana Śulba* proper. The commentator Rāma is also of the same opinion as we are. And the same differentiation is found to have been scrupulously maintained by Yājñika Deva, the commentator of the *Kātyāyana Śrauta-sūtra*. Kātyāyana observes that the second part, especially the recapitulations in it, was meant to help those whose intellects are too poor to be able to fully grasp the inner meanings of the compositions in the sūtra style. Compared with the works of Baudhāyana and Āpastamba, the *Śulba* of Kātyāyana presents some interesting features as it exhibits the whole body of geometrical knowledge required for the Vedic altar-builder in a more systematic form.

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¹ *KŚr*, Chap. xvii.

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² There is a bit of uncertainty about the total number of verses in the *Pariśiṣṭa* of the *Kātyāyana Śulba*. The manuscript of it that is preserved in the Library of the India Office, London (No. E 363), has 48 verses, whereas the manuscript in possession of the Bhandarkar Institute, Poona (No. 74 of A 1881-82), shows only 40 verses. The latter MS. also includes the commentary of Mahīdhara on that manual and he counts 43 verses.

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The *Śulba-sūtra* of Manu is a small treatise composed in both prose and verse. It is divided into seven *khaṇḍas* (or "parts," "sections"). In the first section is given a description of the measuring tape, the gnomon, measures, four methods of determining the cardinal directions and also a method of constructing a square on a given straight line. It may be noted that we do not find in the Āpastamba and Baudhāyana *Śulba-sūtras* any method of determining the cardinal directions, though it is essentially necessary for the proper construction of the sacrificial altars to have an accurate knowledge about them. They proceed on the assumption that the cardinal directions are already known. Kātyāyana teaches three methods for the same, while Manu teaches as many as four. The sections ii-vi treat of the relative positions, spatial magnitudes and also the methods of the construction of the the different *vedis*. Here we find mention of certain *vedis*, *e.g.*, the *Pākayājñikī*, *Māruti* and *Vārunī vedis* which are not included in the abovementioned manuals. The last section of the *Mānava Śulba-sūtra* furnishes us with some hints about the sacrificial fees. It also describes the method of the construction of the *Suparṇa-citi*. This *citi* is not found in other *Śulba-sūtras*. But for the head, its spatial magnitudes are the same as those of the most primitive *citi*, the *Saptavidha-sāratni-prādeśa-caturasra-śyena-cit*, described by Baudhāyana and others.

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The *Maitrāyaṇīya Śulba-sūtra* is a different recension of the *Mānava Śulba-sūtra*. They cover almost the same ground and, more than that, many passages of them are identical. But still they should not be mistaken as one and the same work. The arrangement of matter in them is not parallel. And there are also other marks of distinction between them. The *Maitrāyaṇīya Śulba-sūtra* is comprised of four *khaṇḍas* (or “sections”).

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The *Vārāha Śulba-sūtra* is very closely related to the above two works. There are found several repetitions between these works. This will not seem strange if we remember that they belong to the same school of the *Kṛṣṇa Yajur-veda*. Similarly we find in these *Śulba-sūtras* repetition of a few verses of the *Kātyāyana Śulba-pariśiṣṭa*. The *Vārāha Śulba-sūtra* is broadly divided into three parts and each part is again subdivided into several sections.

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As regards their importance, the available *Śulba-sūtras* can sharply be divided into two classes. The first class will include the manuals of Baudhāyana, Āpastamba and Kātyāyana. They give us an insight into the early state of Hindu geometry before the rise and advent of the Jaina Sect (500-300 B.C.).¹ The *Śulba-sūtras* of Mānava, Vārāha, Maitrāyaṇa and Vādhula add practically very little to our stock of information in this respect. So they may be considered to be of minor importance from our point of view.

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¹ For an insight into Hindu geometry after the advent of the Jainas the reader is referred to the author's article, "Geometry in the Jaina Cosmography," in *Quellen und Studien zur Geschichte der Mathematik*, Abteilung B, Bd. 1, 1930, pp. 245-254.

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In the title *Śulba-sūtra*, the word *sūtra* means an "aphorism," "a short rule." It simply describes the style of the composition of the works and has practically no reference to their subject-matter. The science itself is really called the *Śulba*. And that is, in fact, the original title of the manuals. It is by this title that the *Śulba-sūtra* of Āpastamba has been mentioned in his *Śrautasūtra*.² The commentators are oftentimes found to speak of the *Śulba* of Baudhāyana, the *Śulba* of Āpastamba, etc. This will be further confirmed by the commonly known title of the second part of the work attributed to Kātyāyana, namely the *Śulba-pariśiṣṭa* (or "The Appendix to the *Śulba*") and also by the title *Śulbī-kriyā* (or "The Practice of the *Śulba*") given to that appendix in itself. Thus it is proved conclusively that the true name of the subject is *Śulba*. As the *Śulba* deals with the science of geometry and its application as known amongst the early Hindus, we conclude that the earliest Hindu name for geometry was *Śulba*. Geometry was then sometimes also called *Rajju*, as is evident from the opening sūtra of the *Śulba* of Kātyāyana, "I shall speak of the 'Collection of (rules regarding) the *Rajju*.'" There are many other reliable pieces of evidence leading strongly to the same conclusion.³

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² *ĀpŚr*, xvii. 26. 2.

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³ Bibhutibhushan Datta, "Origin and History of the Hindu Names for Geometry," *Quellen und Studien z. Gesch. d. Math.*, Abteil. B, Bd. 1, 1930, pp. 113-9.

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In Sanskrit, the words *sulba* and *rajju* have the identical significance, which is ordinarily "a rope," "a cord." The word *sulba* or *sulva* is derived from the root *sulb* or *sulv* meaning "to measure" and hence its etymological significance is "measuring" or "act of measurement." From that it came to denote "a thing measured" and consequently "a line (or surface)" as well as "an instrument of measurement" or "the unit of measure." Thus the terms *sulba* or *rajju* have four meanings: (1) mensuration—the act and process of measuring; (2) line (or surface)—the result obtained by measuring; (3) a measure—the instrument of measuring; and (4) geometry—the art of measuring. In the ancient literature of the Hindus we indeed find mention of three kinds of measure—linear, superficial as well as voluminal—having the same epithet *rajju*. In the *Śulbas* the measuring tape is called *rajju*. And we further find there the use of the word in the sense of "a line" also. For instance, we have the term *akṣṇayā-rajju* = "diagonal line." Kātyāyana observes:¹ "(The terms) *karaṇī* ('producer'), *tat-karaṇī*² ('that-producer'), *tiryaṅmānī* ('transverse measurer'), *pārśvamānī* ('side measurer'), and *akṣṇayā* ('diagonal') *rajjus* are ('lines')."

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¹ *KŚl*, ii. 7.

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² That is, *dvi-karaṇī*, *tri-karaṇī*, etc.

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In the *Mānava Śulba*³ and *Maitrāyaṇīya Śulba*,⁴ the science of geometry is called the *Śulba-vijñāna* (or "the Science of the *Śulba*").⁵ One who was well versed in that science was called in ancient India as *saṁkhyājña* (or "the expert in Numbers"), *parimāṇajña* ("the expert in measuring"), *sama-sūtra-nirañchaka* ("uniform-rope-stretcher"), *Śulba-vid* ("the expert in the *Śulba*") and *Śulba-paripṛcchaka* ("the inquirer into the *Śulba*").¹ Of these, one term, *viz.*, *sama-sūtra-nirañchaka*, perhaps deserves more particular notice. For we find an almost identical term, *harpedonaptae* ("rope-stretcher"), appearing in the writings of the Greek Democritos (*c.* 440 B.C.). It seems to be an instance of Hindu influence on Greek geometry. For the idea in that Greek term is neither of the Greeks nor of their acknowledged teachers in the science of geometry, the Egyptians, but it is characteristically of Hindu origin.) In the Pāli literature, we find the terms *rajjuka* and *rajju-grāhaka* ("rope-holder") for the king's land surveyor.² The first of these terms appears copiously, in its various case-endings, in the inscriptions of the Emperor Aśoka (250 B.C.). In the comparatively later *Śilpa-śāstras*, the surveyor is spoken of as *sūtra-grāhī* or *sūtra-dhāra* ("rope-holder") and he is further described as an expert in alignment (*rekhā-jña*, lit. "one who knows the line").

Linked texts (9)

Jātaka — Pāli
Āpastambaśrautasūtra — Āpastamba, Sanskrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Kātyāyanaśrautasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Hiraṇyakeśiśulbasūtra — Sanskrit
Mānavaśulbasūtra — Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit
Maitrāyaṇīyaśulbasūtra — Sanskrit