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The Science of the Shulba: A Study in Early Hindu Geometry c9: Chapter IV. Postulates

The Science of the Śulba: A Study in Early Hindu Geometry c9

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Chapter IV. Postulates

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For the geometrical operations described in the *Sulba*, the authors, we find, have tacitly assumed the truth of certain other results without any attempt to describe them beforehand or to indicate how they could be effected. These results we have called here postulates of the *Sulba*. They might not be postulates in the Euclidean sense of the term; but they can certainly be so called in accordance with the meaning given by Aristotle, namely "whatever is assumed, though it is a matter for proof, and used without being proved." Most of the postulates of the *Sulba* are concerning the division of figures, such as straight lines, rectangles, circles and triangles. A few of them are about other matters of importance.

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(*a*) *A given finite straight line can be divided into any number of equal parts.*

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In the geometry of the *Sulba*, it is oftentimes required to divide a given finite straight line into a specified number of equal parts. For example in one instance, the diameter of a given circle is divided into 8 equal parts, each of these again into 29 parts and so on into other number of divisions.¹ There are indeed numerous such instances.² Now it will be naturally asked how it was used to be done. Certainly not arithmetically. At least it is not always possible to do so. For we find instances of division of straight lines which cannot be expressed in terms of commensurate numbers. In circling a square, such a straight line has to be divided into three parts¹ and in another case into twelve parts.² Sometimes the given straight line is such that the parts when expressed arithmetically will contain big fractions. Thus the side of a square of 96 *aṅgulis* has to be divided into 7 equal parts.³

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¹ *BŚl*, i. 59.

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² *BŚl*, i. 60, 68-9, etc.

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¹ *BŚl*, i. 56.

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² *BŚl*, iii. 162; compare also *ĀpŚl*, xix. 7 in which the diagonal of a square of sides = ¼ of a puruṣa is divided into seven equal parts.

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³ *BŚl*, ii. 64.

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(b) *A circle can be divided into any number of parts by drawing diameters.*

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In the *Śulba*, we have several instances of the division of a circle into a specified number of parts. For instance, it is said that the *Dhiṣṇiya* may be square or circular in shape and one of them, *viz*., the *Agnīdhrīya*, has to be divided into nine parts. Now in the case of the square shape of the altar, it is easily divided into 9 smaller squares by drawing cross-lines through the points of trisection of the sides. When it is circular, there is described a small circle about its centre and the annulus

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Fig. 2 Fig. 3.

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is then divided into 8 equal parts by drawing four diameters.⁴ Similarly in an alternative circular shape of the *Mārjālīya* fire, the circle has to be divided into 6 equal parts.¹ A circular annulus has been divided into 32 equal parts;² another annulus into 64 equal parts and then again into two parts each by drawing the mean circle.³

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⁴ *BŚl*, ii. 73-4; *ĀpŚl*, vii. 13-14.

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¹ *BŚl*, ii. 77. Compare *ĀpŚl*, xvii. 21. 3, 4 for instances of division of a circle into 12, 16 and 24 parts.

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² *BŚl*, iii. 200.

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³ *BŚl*, iii. 202.

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(*c*) *Each diagonal of a rectangle bisects it.*

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(*d*) *The diagonals of a rectangle bisect one another and they divide the rectangle into four parts two and two vertically opposite of which are equal in all respects.*

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The description of the division of a rectangle or a square by diagonals is found in the *Sulba* primarily in connexion with the making of bricks of desired size and shape. Certain interesting geometrical theorems have been assumed there. The brick which resembles in shape the portion of a rectangle or of a square divided by its diagonal is termed *ardhyā* (or "the half") and that resembling a portion by the two diagonals is called *pādyā* (or "the quarter"). There are distinguished two kinds of *pādyā* of a rectangle, *viz.*, *dīrgha-pādyā* (or "longish or broader quarter") and *śūla-pādyā* (or "trident quarter").¹ This distinctive nomenclature implies: (1) the halves of a rectangle or of a square by a diagonal are identical in size as well as in shape, and (2) so are also the quarters of a square by its diagonals; and (3) the diagonals of a rectangle divide it into four parts which are equal in area but they are of two kinds as regards their shape. These names perhaps further imply an idea of *obtuse* and *acute* angles. There are bricks which are halves of the quarter bricks by the perpendicular from the vertex on the base. No distinction is found to have been made between the half of a *dīrgha-pādyā* and that of a *śūla-pādyā*, which clearly shows that the Śulba-kāras were aware that those halves were identical. Thus it appears that the early Hindu geometers knew the simple cases of the congruence theorems.

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¹ *BŚl*, iii. 168-9, 178.

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Fig. 4.

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Fig. 5

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Fig. 6.

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Another interesting kind of bricks is formed by the combination of a half of a *dīrgha-pādyā* or a *śūla-pādyā* with another brick. Baudhāyana describes:

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"The eighth parts of them² should be so combined as there will be (a brick having) three corners."³

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² The reference is to a square brick, called *pañcamī*, for each side of it is equal to the fifth (*pañcama*) part of a *puruṣa* and to a rectangular brick one-fifth of a *puruṣa* by one-fifth of a *puruṣa* and its half.

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³ *BŚl*, iii. 122.

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A brick of this kind is technically called *ubhayī*¹ (from *ubhay*, "both") because it is formed by the combination of

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¹ *Ibid*, iii. 129.

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Fig 7

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two bricks of two different kinds. Since there is mentioned only one *ubhayī* though there are distinguished two different kinds of quarter bricks of a rectangle (the *adhyardha*) it follows that Baudhāyana was fully aware, what has been just mentioned, that all the eighth parts of a rectangle are identical. What is much more noteworthy is that in the formation of the *ubhayī* we find the source of the discovery of the later Hindu principle of forming a rational scalene tringle by the juxtaposition of two rational right-angled triangles.²

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² Bibhutibhusan Datta, "On Mahāvīra's Solution of Rational Triangles and Quadrilaterals," *Bull. Cal. Math. Soc.*, Vol. xx, pp. 267-294; see particularly pp. 276-7.

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(*e*) *The diagonals of a rhombus bisect each other at right angles.*

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Fig. 8

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(*f*) *A triangle can be divided into a number of equal and similar parts by dividing the sides into an equal number of parts and then joining the points of division two and two.*

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Baudhāyana on a certain occasion says, "this (triangle) is divided into ten parts."¹ But how to do it he does not explain expressly. We, however, learn it from the commentators that the traditional practice in such a case was

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¹ *BŚl*, iii. 256.

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Fig. 9.

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to divide each side into four equal parts and then to join the points of division two and two as indicated in the Fig. 9.

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(*g*) *An isosceles triangle is divided into two equal halves by the line joining the vertex with the middle point of the opposite side.*² Each of these has again been divided into six parts.³

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² *Ibid*, iii. 258.

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³ *Ibid*, iii. 260.

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Fig. 10

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We shall see later on that figures of more complex shapes had to be divided into a specified number of parts, namely 200, of given forms. And this led to some interesting problems of indeterminate character.

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*(h) A triangle formed by joining the extremities of any side of a square to the middle point of the opposite side is equal to half the square.*

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Fig. 11

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*(i) A quadrilateral formed by the lines joining the middle points of the sides of a square is a square whose area is half that of the original one.*

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Fig. 12

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*(j) A quadrilateral formed by the lines joining the middle points of the sides of a rectangle is a rhombus whose area is half that of the rectangle.*

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Fig. 13.

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*(k) A parallelogram and a rectangle which are on the same base and within the same parallels are equal to one another.*

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Fig. 14

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The assumption of the truth of this theorem forms the basis of the *Sulba* method of the construction of a parallelogram having given sides inclined at a given angle, which will be described later on. It was also known in the time of the *Satapatha Brāhmaṇa*.¹

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¹ X. 2. 1. 5.

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*(l) The maximum square that can be described within a circle is the one which has its corners on the circumference of the circle.*

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In the *Sulba* it is sometimes necessary to draw within a circle "a square as large as possible (*yāvat sambhavet*) ;" but it is not indicated how to do it. From the subsequent descriptions it, however, appears clearly, that the corners of that square are assumed to be on the periphery of the circle. The commentators explain that a side of this square will be equal to √2 times the radius of the circle. In fact, two diameters of the circle are

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Fig. 15

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drawn at right angles to each other. The figure formed by joining the end points of them, is the largest possible square within the circle.

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² *BŚl*, i. 70; *ĀpŚl*, vii. 10; xii. 12. Compare Paṇḍit, O. S., X, p. 166.

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It will be noticed that every one of the above propositions is demonstrable. Branding of them as postulates raises the important question of the character of the early Hindu geometry as regards the matter of demonstration. Of course the propositions of the *Sulba* are not proved after the manner of Euclid by purely deductive reasoning. On the other hand it is not wholly empirical without any semblance of demonstration. In fact we find a kind of proof in case of the propositions of the subtraction of one square from or its addition with another square and the mensuration of an isosceles trapezium. After the enunciation as a general proposition of the theorem of the square of the diagonal, Baudhāyana observes that the truth of it will be "realised" in case of certain rational rectangles enumerated. This is really an attempt for a kind of demonstration. What is much more noteworthy in this connexion is the fact that after a description of the geometrical construction for a proposition the Śulbakāras are often found to have remarked *sa samādhiḥ* or "This is the construction."¹ The significance of such an observation is obvious. It emphasizes that the construction which was required to be made, has been thus actually made, and indeed corresponds to the expression *Quod Erat Faciendum* (or "What it was required to do") occurring at the end of a proposition of Euclid's *Elements*. Further it discloses a rational and demonstrative attitude of the mind of the early Hindu geometer. With reference to a similar remark occurring in the works of the celebrated Hindu mathematician of the twelfth century of the Christian era, Bhāskara II, Hankel observed: "The small word 'see' along with the figure together with the necessary auxiliary lines supplies the Brāhmaṇas with the 'proof of the Greeks' concluding with solemn words 'what was to be proved.' All that a practised mind could recognise by means of assiduous consideration of a figure was admitted as certain."¹

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¹ *ĀpŚl*, i. 2; *KŚl*, ii. 6; iii. 13; etc.

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¹ H. Hankel, *Zur Geschichte der Mathematik in alterthum und mittelatter*, Leipzig, 1874, pp. 205 f.

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