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Geometry in Ancient and Medieval India c7.s8: 2.10. Transformation of Figures

Geometry in Ancient and Medieval India c7.s8

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2.10. Transformation of Figures

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The votive fire-altars were prescribed different shapes according to the specific benefit sought for — the *śyenacit* (the fire-place in the form of a falcon) for attaining heaven, the *praugacit* (the fire-place in the form of an isosceles triangle) for destroying enemies and so on. But all these different shapes had to have strictly the same area, viz. 7½ sq. *puruṣas*. Hence were evolved methods for transforming one geometrical figure into another, more especially the square into other equivalent geometrical figures. These constructions are given below.

2.10.1. To convert a square into a circle

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No geometrical method can achieve this exactly. What the *Śulbasūtras* do is to give approximate constructions. The centre O of the square is joined to a vertex A and the circle is drawn with half the side of the square combined with ⅓ the excess of OA over half the side of the square,¹ i.e. if ‘a’ is the side of the square and ‘r’ the radius of the circle. r & = a/2 + (√2 · a/2 - a/2)/3 / & = a/2 × ((2 + √2))/3 The value of π calculated from this is only about 3.088. But according to some of the commentators, the last sentence of this rule, namely *Sānityā maṇḍalam*, is to be split as *Sā anityā maṇḍalam*, when it will mean that Āpastamba and the other authors of *Śulbasūtras* as well were aware that this was an approximate method only. Thibaut and Bürk, understandably, do not accept this explanation.

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¹ *Āp. Sl.* III. 2; *B. Sl.* I. 58 and *K. Sl.* III.

2.10.2. To convert a circle into a square

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All the three important *Śulbas* direct us to divide the diameter into 15 parts and to take 13 of these parts as the side of the square i.e., if d is the diameter of the circle² and a the side of the equivalent square, $a = 13/15 d, whence π = 3.004$ Baudhāyana gives a slightly better approximation too.

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² *Āp. Sl.* III. 3; *B. Sl.* I. 60 and *K. Sl.* 14

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मण्डलं चतुरश्रं चिकीर्षन् विष्कम्भमष्टौ भागान् कृत्वा भागमेकोनत्रिंशता विभज्य अष्टाविंशति भागानुद्धरेत् भागस्य च षष्ठं अष्टमभागोनम् । (*B. Sl.* I. 59)

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(Wishing to convert a circle into a square one should divide the diameter into 8 parts, divide one of these parts again into 29 parts and subtract 28 of these (29th parts) together with one-sixth of one of these parts, the latter being diminished by one-eighth of that (one-sixth part) ).

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$i. e., a=d (1-28/8.29-1/6.8.29+1/6.8.29.8 )$

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This value is based on an inversion of the relation between r and a given in connection with the problem of circling the square. How exactly the value was brought to the form of this long and complicated fractional expression is a matter for speculation, but may not be of geometrical interest.

2.10.3. To convert a rectangle into a square

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Āpastamba's rule is:

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दीर्घचतुरश्रं समचतुरश्रं चिकीर्षन् तिर्यङ्मान्यापच्छिद्य शेषं विभज्योभयत उपदध्यात्। खण्डमागन्तुना संपूरयेत्। तस्य निर्ह्रास उक्तः।
(*Āp. Śl.* II. 7)

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(Wishing to turn a rectangle into a square, one should cut off a part equal to the transverse side and the remainder should be divided into two and juxtaposed at the two sides (of the first segment). The bit (at the corner) should be filled in by an imported bit. The removal of this has been explained already).

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Fig. 14

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If ABCD is the rectangle, a square AB₁ C₁ D with side AD, is cut off from it. The remaining rectangle B₁ BCC₁ is divided into two equal strips B₁ B₂ E C₁ and B₂ BCE. The strip B₂BCE is cut off and applied to the side of the square DC₁. Now we get a square of side AD₁ with a small square C₁ E C₂ F unfilled up at one corner. The larger square is completed and the imported square in the corner is removed by the method of removing a square from a square, (given in 2. 11. 4.).

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Baudhāyana (I. 54) and Kātyāyana (III. 2) give the same method. Though this method works with any rectangle,¹ Kātyāyana provides for a very long rectangle with a separate *sūtra*.

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¹ If the length is greater than thrice the breadth, the *āgantu* or imported square will be the bigger square.

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अतिदीर्घं चेत्तिर्यङ्मान्या॑पच्छिद्य अपच्छिद्यैकसमासेन समस्य शेषं यथायोगमुपसंहरेत् ।
(*K. Sl.* III. 3)

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(If the rectangle) is very long, cut off repeatedly by the transverse side (breadth) and join the squares so formed into one big square, and then the remainder of the rectangle should be joined to this square as it fits (to form a square).

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The method is no improvement over the general method, since here no side of the remainder rectangle will be equal to the side of the bigger square to which its strips are to be joined.

2.10.4. To convert a square into a rectangle

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समचतुरश्रं दीर्घचतुरश्रं चिकीर्षन् यावच्चिकीर्षेत् तावर्ती पार्श्वमानीं कृत्वा यदधिकं स्यात् यथायोगमुपदध्यात् ।
(*Āp. Sl.* III. 1)²

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² Also *B. Sl.* I. 53.

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(Wishing to convert a square into a rectangle one should make the lateral side as long as is desired and the excess should be joined suitably.)

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We are not told how exactly the excess is to be joined. Thibaut and Bürk suggest that this was achieved by repeated slicing and joining. If A B C D is the square and the side of the rectangle is to be A D₁, a rectangle A B by A D₁ is sliced off first. From the remainder a rectangle with length equal to A D₁ can be obtained. This is sliced and joined to A B C₁ D₁ as shown. The remaining square is to be sliced and put together suitably so as to get a narrow strip with length = A D₁, which is then again joined to A F₁ F D₁. This procedure is not merely empirical but also highly unsatisfactory, since there is no guarantee that the remainder square always yields a rectangle of the requisite length.

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Fig. 15

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Āpastamba's commentator Sundararāja, explaining this *sūtra*, gives a purely geometrical and exact construction.

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यावदिच्छं पार्श्वमान्यौ प्राच्यौ वर्धयित्वा उत्तरपूर्वां कर्णरज्जुमायच्छेत्, सा दीर्घचतुरश्रमध्यस्थायां समचतुरश्रतिर्यङ्मान्यां यत्र निपतति तत उत्तरं हित्वा दक्षिणांशं तिर्यङ्‌मानीं कुर्यात्, तद् दीर्घचतुरश्रं भवति ।

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(Producing the sides of the square eastward to the desired length of the lateral side, one should draw the north-eastern diagonal. The part of the transverse side to the north of the point where the diagonal cuts it is to be discarded and its southern part is to be made the transverse side of the rectangle. That will be the rectangle.)

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Fig. 16

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Let A B C D be the given square. Produce A D and B C to F and E so that A F = B E = the required side of the rectangle. Complete the rectangle A B E F and join the diagonal B F cutting C D in G. Through G draw a st. line IH parallel to the sides of the square. Then IBEH is the required rectangle.

c7.s8.t4.11

For, F A B &= F E B / I G B &= G C B / and F D G &= F H G / Hence rect. A I G D &= rect. G C E H / \therefore rect. I B E H &= rect. I B C G + rect. G C E H / &= rect. I B C G + rect. A I G D / &= sq. A B C D

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Here the case where the given side of the rectangle is greater than the side of the square only is dealt with. But with slight change the method is applicable to the case where the side is less also.

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Baudhāyana and Kātyāyana give an easier method.

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समचतुरश्रं दीर्घचतुरश्रं चिकीर्षन् मध्येऽक्ष्णयाऽपच्छिद्य विभज्येतरत्पुरस्तादुत्तरतश्चोपद्ध्यात् । विषमं चेद्यथायोगं उपसंहरेदिति व्यासः। (*K. Sl.* III. 4)¹

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¹ Also *B. Sl.* I. 52.

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(Wishing to transform a square into a rectangle one should cut diagonally in the middle, divide one part again and place the two halves to the north and east of the other part. If the figure is a quadrilateral one should place together as it fits. This is the distribution). Let A B C D be the square. It is cut along AC to form two right triangles. The A B C is again halved along the altitude B E. The two halves B E A and B E C are then removed to the positions D F C and D G A.

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*Fig.* 17

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Then, obviously rect. A C F G = sq. A B C D. The defect of this method is that the rectangle cannot be given any desired side.

2.10.5. To convert a rectangle or square into a trapezium with the shorter parallel side given

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Baudhāyana deals with this problem. चतुरश्रमेकतोऽणिमच्चिकीर्षन् अणिमतः करणीं तिर्यङ् मानीं कृत्वा शेषमक्ष्णया विभज्य विपर्यस्येतरत्रोपद्ध्यात्। (*B. Sl.* I. 55)

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(If one wishes to make a square or rectangle shorter on one side, one should cut off a portion by the shorter side. The remainder should be divided by the diagonal, inverted and attached on either side).

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If A B C D is the given rectangle, let the rectangle A F E D be cut off so that A F = D E = the given shorter side. The remaining rectangle E F B C is to be cut diagonally along B E and the portion B E C is to be inverted and attached to the side A D of the rectangle in the position E′ A D. Then D E′ B E is the equivalent trapezium.

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*Fig.* 18

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The *Śatapatha Brāhmaṇa* seems to give another method for this conversion.¹

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¹ *Ś. Br.* X 2. 1. 3-4.

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चतुर्विंशत्यंगुलिभिर्मिमीते । चतुर्विंशत्यक्षरा वै गायत्री, गायत्रोऽग्निर्यावानग्निर्र्यावत्यस्य मात्रा तावतैवैनं तान्मिमीते । स चतुरंगुलमेव उभयतोऽन्तरत उपसमूहति, तावद् व्यूहति । तन्नाहेवातिरेचयति नो कनीयः करोति ।

c7.s8.t5.8

(He measures by 24 *aṅgulis*. For *Gāyatrī* has 24 letters and *agni* is of *Gāyatrī*. As much as is the *agni*, as much as is its measure, by that much he measures them. He contracts 4 *aṅgulis* inwards on both sides. He stretches 4 *aṅgulis* outwards on both sides. as much as he contracts, so much he stretches. Thus he does not make it exceed (the right measure) nor does he make it smaller).

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Fig. 19

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Datta takes this to mean² that the face of the square (A B C D) is shortened on either side by a small length (D D' = C C') and the base is lengthened on either side by the same length to A' and B'. Then the area of trapezium A'B'C'D' = the area of square ABCD. and obviously so, since $Δ P D D′ Δ P A A′$ and Δ Q C C′ Δ Q B B′

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² *Science of the Śulba*, p. 91.

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The other *Śulbasūtras* do not deal with this problem.³

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³ B.B. Datta says (*Science of the Śulba*, p. 92) that *Āp. Śl.* XV-9 ff deal with this problem, but in the edition I am using the said *sūtras* do not deal with this problem.

2.10.6. To convert a trapezium into an equivalent rectangle

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Āpastamba tackles the converse problem of converting an isosceles trapezium into an equivalent rectangle. It is not given as a general prescription but rather as a means of finding out the area of the trapezium of the Mahāvedi.

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दक्षिणस्मादंसाद् द्वादशसु दक्षिणस्यां श्रोण्यां निपातयेत्, छेद विपर्यस्य उत्तरत उपदध्यात् । सा दीर्घा चतुरस्रा । तथायुक्तां संचक्षीत ।

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(From the southern top corner one should drop a perpendicular on the southern botton corner at a distance of 12 (*padas* from the *pṛṣṭhyā*). The removed bit should be placed inverted at the northern side. That is the rectangle. One should study it thus joined).

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Fig. 20

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That is, to convert the trapezium A B C D (parallel sides 24 ft. and 30 ft.), C B′ is drawn perpendicular to A B, B′ being 12 padas away from PQ, the *pṛṣṭhyā*. The triangle C B′ B is then placed in the position A D D′. Then rect. A B′ C D′ = trap. A B C D

2.10.7. To construct an isosceles triangle equal in area to a given square and vice versa

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Conversion of a square into an equivalent triangle, being necessary for the construction of the *Praugacit*, is tackled by all the three important *Śulbasūtras* and all of them give the same prescription.

c7.s8.t7.2

यावानग्निसारत्निप्रादेशो द्विस्तावतीं भूमि चतुरश्रं कृत्वा, पूर्वस्याः करण्या अर्धात् श्रोणी प्रत्यालिखेत् । सा नित्या प्रउगम् ।
(*Ā. Śl.* XII. 5)¹

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¹ See also *B. Śl.* I. 56 and *K. Śl.* IV. 5.

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(Making an area which is double as much as the fire-altar with the *aratnis* and *prādeśas*, into a square, one should draw lines from the middle point of the eastern side towards the bottom corners. That is the equivalent *prauga* (isosceles triangle).

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Fig. 21

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Let A B C D be a square of twice the required area. Let E be the middle point of C D. E A and E B are joined. Then A E B is the required triangle. For if the altitude EF is drawn, the square is divided into 2 equal rectangles A F E D and F B C E.

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{rcl@{ }l} & A F E & = 1/2 rect. A F E D & / and & F B E & = 1/2 rect. F B C E & / \therefore Whole & A E B & = 1/2 sq. A B C E. &

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This construction leads to the formula, area of a triangle = ½ base × altitude.

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The converse problem is treated by Kātyāyana only.

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प्रउगं चतुरश्रं चिकीर्षन् मध्ये प्राञ्चमपच्छिद्य विपर्यस्येतरत उपधाय दीर्घचतुरश्रसमासेन समस्येत् । (IV.7)

c7.s8.t7.11

(Wishing to convert an isosceles triangle into a square, one should cut in the middle towards the east, place on the other side inverted and manipulate by the method of conversion of a rectangle into a square.)

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Fig. 22

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Let A B C be the triangle. Let it be cut into two halves along the altitude C D. Let the part D B C be applied inverted to the side A C. Then rect. A D C E = △ A B C. Now the rectangle can be converted into an equivalent square by the method already given.

2.10.8. To construct a rhombus of given area

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तावदेव दीर्घचतुरश्रं विहृत्य पूर्वापरयोः करण्योरर्धात् तावति दक्षिणोत्तरयोर्निपातयेत् सन्त्योभयतःप्रउगम् । (*Āp. Śl.* XII. 9)¹

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¹ Also *B. Śl.* I. 57 and *K. Śl.* IV. 6.

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(Drawing a rectangle of the same area (i.e. of the area of the square for the *prauga*), one should draw lines from the middle points of the eastern and western sides to the middles of the southern and northern sides. That is the rhombus of the same area).

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Fig. 23

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Let A B C D be a rectangle of twice the area of the rhombus. Let E, F, G, H be the middle points of the sides. E F, F G, G H, H E are joined. Then rhombus E F G H = ½ rect. A B C D.

2.10.9. To transform a rhombus into a rectangle

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This converse construction occurs in Kātyāyana only. उभयतःप्रउगं चेन्मध्ये तिर्यंगपच्छिद्य पूर्ववत् समस्येत् । (K. *Śl.* iv. 8) (If it is an *ubhayataḥ prauga* one should cut transversely along the middle and join together as before.)

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*Fig. 25*

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The process is exactly the same as for the *prauga*. The rhombus is first divided into two isosceles triangles by joining a diagonal and again into 4 right triangles by cutting along their altitudes. The four triangles are joined together to form a rectangle.

2.11.1. Combination of areas and the converse

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By the application of the theorem of the square of the diagonal the authors of the *Śulbasūtras* combine any number of squares to form another square.

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For, combining two equal squares i.e. for doubling a square Āpastamba's rule is: समस्य द्विकरणी । (I. 5) (The diagonal of the square is the double-maker.) Hence if a square is drawn on the diagonal of the given square, it will produce double the area. The diagonal will therefore be √2 a where a is the side of the square. It is noteworthy that the *Śulbasūtras* give a very close approximation to the value of √2 प्रमाणं तृतीयेन वर्धयेत्तच्चतुर्थेन आत्मचतुस्त्रिंशोनेन सविशेषः । (Āp. *Śl.* I. 5 and B. *Śl.* I. 61-62)¹ (The measure should be increased by one-third of itself, which again is increased by its one-fourth and diminished by 1/34 th of that (second) increment. This is the *saviśeṣa*.)

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¹ Also *K. Śl.* II. 13.

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i.e. √2=1+⅓+1/3.4-1/3.4.34

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(This result must have been arrived at by Rule of Three and the method of repeated correction.¹ For trebling a square

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¹ Thibaut's hypothesis about the deduction of this expression (*J.A.S.B.*, 1875) is ingenious, but one fails to understand the preference for 12² + 12² = 288 = 17² - - 1 when the earlier 2² + 2² = 8 = 3² - 1 is equally suitable.

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प्रमाणं तिर्यक् द्विकरण्यायामः तस्याक्ष्णयारज्जुस्त्रिकरणी । (*Āp. Sl.* II. 2).

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(The breadth is the measure (of the side of the given square) and the length is the double-maker. The diagonal (of such a rectangle) will be the treble-maker.)

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In this way proceeding step by step one can combine any number of equal squares.

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**2.11.1b** For combining a large number of squares, Kātyāyana gives an ingenious method in one step.

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यावत्प्रमाणानि समचतुरश्राण्येकीकतुं चिकीर्षेत् एकोनानि तानि भवन्ति तिर्यक् द्विगुणान्येकत एकाधिकानि । त्र्यस्रिर्भवति तस्येषुस्तत्करोति ।
(*K. Sl.* VI. 7)

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The verse is not easy to interpret. The only logical meaning assignable is what Dr. B. B. Datta gives², viz. "As many squares (of equal side) as you wish to combine into one, the transverse line will be (equal to) one less than that; twice a side will be (equal to) one more than that. It will be a triangle. Its arrow (i.e. altitude) will do that". That is, if n squares of side a are to be combined, we have to construct an isosceles triangle A B C with (n -- 1) a as base and ((n+1) a)/2 as sides. AD the altitude is drawn. Then AD is the side of the square whose area will be n a².

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² *Science of the Śulba*. pp. 72-73.

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Fig. 25

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$For B D=½ B C=((n-1)a)/2$

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And from the rt-angled △ ADB

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AD² & = AB² - BD² \\ & = {((n+1)a)/2}² - {((n-1)a)/2}² \\ & = (a²)/4{(n+1)² - (n-1)²} \\ & = (a²)/4× 4n = n.a² \end{aligned}

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Where the number is expressible as the sum of two squares the first method itself can be shortened. e.g.: 10 = 3²+1 40 = 6²+2² In such cases we can construct a rectangle with sides 3 a and a or 6a and 2a and then the diagonal will be the side of the combined square.¹ Or, in general, if n = p²+q², one has to construct a rectangle of sides pa and qa. Then the square on the diagonal will be & p² a² + q² a² / = & a²(p²+q²) = n a².

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¹ *K. Sl.* II. 8-9.

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**2.11.².** Methods are also given for getting squares which are fractions of a given square. Since the *Sautrāmaṇī vedi* is to be ⅓ of the *Saumikī*, the Śulbasūtras deal in detail with the construction of a square whose area is ⅓ that of a given square. And the method can be extended to any fraction. Kātyāyana's instructions are clearest.

c7.s8.t10.21

तृतीयकरण्येतेन व्याख्याता । प्रमाणविभागस्तु नवधा करणीतृतीयं नवभागः । नवभागास्त्रयः तृतीयकरणी ।
(*K. Sl.* 15-18)²

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² *Āp. Sl.* II. 3 and *B. Sl.* I. 47.

c7.s8.t10.23

(The one-third-maker is explained by this. But the original measure (of the area) is to be divided into nine. The third part of the side (produces) a ninth part (of the area). Three ninth parts will give the one-third-maker.)

c7.s8.t10.24

Here we are directed to divide the square into 9 equal parts by dividing the pair of opposite sides into 3 equal parts by lines parallel to the other pair of sides, 3 of the small squares so formed are to be combined into a square, the side of which will then be the one-third-maker.

c7.s8.t10.25

The commentators give an alternative explanation also. The triple square is first to be obtained, which is then to be divided into 9 equal parts as above. These parts will be ⅓ of the original square.

2.11.3. To combine two unequal squares

c7.s8.t11.1

ह्रसीयसः करण्या वर्षीयसो वृद्ध्रमुल्लिखेत् । वृद्ध्रस्याक्ष्णयारज्जुर्भ समस्यति ।
(*Āp. Śl.* II. 4)¹

c7.s8.t11.2

¹ Also *B. Śl.* I. 52 and *K. Śl.* II. 22.

c7.s8.t11.3

(With the side of the smaller one a segment of the bigger one should be cut off. The diagonal cord of the segment will combine the two squares).

c7.s8.t11.4

Fig. 26

c7.s8.t11.5

In effect, a rectangle with sides equal to the sides of the squares is constructed. If a and b are these sides, the square on the diagonal of the rectangular segment = a²+b².

2.11.4. To draw a square equal to the difference of two squares

c7.s8.t12.1

चतुरश्राच्चतुरश्रं निर्जिहीर्षन् यावन्निर्जिहीर्षेत् तस्य करण्या वृद्ध्रमुल्लिखेत् ।
वृद्ध्रस्य पार्श्वमानीं अक्ष्णया इतरत् पार्श्वं उपसंहरेत् सा यत्र निपतेत्तदपच्छिन्द्यात् ।
(*Āp. Śl.* II. 5)²

c7.s8.t12.2

² Also *B. Śl.* I. 51 and *K. Śl.* III. 1.

c7.s8.t12.3

(Wishing to deduct a square from a square one should cut off a segment by the side of the square to be removed. One of the lateral sides of the segment is drawn diagonally across to touch the other lateral side. The portion of the side beyond this point should be cut off.)

c7.s8.t12.4

Fig. 27

c7.s8.t12.5

Let A B CD be the larger square, and A E the side of the square to be removed. The segment A E F D is cut off. A D is drawn diagonaliy across with A fixed, till D touches E F at P. Then E P is the side of the required square. For, from the right triangle A E P,

c7.s8.t12.6

$EP² = AP² - AE² = AD² - AE².$ This explanation is given by Āpastamba himself in the next *sūtra*.

Cites (4)