The Science of the Shulba: A Study in Early Hindu Geometry c12: Chapter VII. Transformation of Areas

The Science of the Śulba: A Study in Early Hindu Geometry c12

Language: English

Chapter VII. Transformation of Areas

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*To transform a rectangle into a square.*

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For this purpose Baudhāyana gives the following rule:

“If you wish to transform a rectangle into a square, make its breadth as the side of a square ; divide the remainder into two parts and changing the place (of the farther one of them) and inverting, add it on the other side of the square. Then adding a (square) portion, fill up that (the empty space in the corner). It has been taught (before) how to deduct it (the added square from the full square thus formed).”¹

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¹ *BŚl*, i. 58.

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Let it be required to change the rectangle *ABDC* into a square.

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Fig. 44

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From the longer side *AC* cut off a portion *CE* equal to the breadth *CD* of the rectangle. Complete the square *CDFE*. Divide the remaining part *ABFE* of the given rectangle into two halves by the line *GH*. Take the remoter half *ABHG* and place it after inversion on the other side of the square *CDFE* in the position *DG'H'F*. Complete the square *CGPG'*, by adding the portion *HPH'F*. The given rectangle is easily found to be equal to the difference of the two squares *G'PG'C* and *HPH'F*. This difference can be found by the method taught before. That is, draw a circle with centre *G'* and radius *G'P* cutting *DH* at *M*. Draw *MN* perpendicular to *G'P*. Then $G'N² = G'M² - MN² = G'P² - HP²$ so that *G'N* is the side of the square which is equivalent to the given rectangle *ABDC*.

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The same method is taught also by Āpastamba¹ and Kātyāyana.² The latter is a little more explicit than the other. He says:

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¹ *ĀpŚl*, ii. 7.

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² *KŚl*, iii. 2.

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"If you wish to transform a rectangle into a square, cut off (from the rectangle a square portion) with its shorter side. Divide the other portion into two parts. Take the farther portion on the east and add it on the south (of the square portion). Complete the (square) figure by introducing a (small square) piece. The method for its deduction has been taught."

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Kātyāyana suggests a different procedure for the transformation of a rectangle whose length much exceeds its breadth:

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"If (the rectangle be) very long, cut it again and again (into squares) by the breadth; combine these squares into one square; add to this the remaining portion (of the rectangle) after transforming it suitably."³

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³ *Ibid*, iii. 3.

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Fig. 45

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This is certainly no improvement on the other method. It will, in fact, require more operations to complete the desired transformation. The first method is very general and is equally available for all cases of transformation of a rectangle into a square.

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*To transform a square into a rectangle.*

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Baudhāyana gives the following rule for transforming a square into a rectangle:

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"If you wish to transform a square into a rectangle, divide it by the diagonal. Divide again one part into two, and add them suitably so as to fit the two sides (of the other half)."¹

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¹ *BŚl*, i. 52.

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Fig. 46

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The same method is also taught by Kātyāyana.² This method is much circumscribed inasmuch as it transforms the given square into a rectangle in which the length is double the breadth and is itself equal to the diagonal of the given square.

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² *KŚl*, iii. 4.

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*To transform a square into a rectangle which shall have a given side.*

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To transform a square into a rectangle which shall have a given side, Āpastamba gives the following rule:

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"If you wish to transform a square into a rectangle, (cut off from it a rectangular segment) by making a side as long as you wish (a side of the transformed rectangle to be). What remains in excess, should be added (to the former) as suitably as to fit.”¹

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¹ *ĀpŚl*, iii. 1.

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A similar rule is given also by Baudhāyana:

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“‘Or else if the square is to be transformed into (a rectangle) of this (*i.e.*, one specified) side, cut off (from the square) a segment by that side. What remains in excess should be added along the other side.’”²

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² *BŚl*, i. 53. Thibaut’s reading of the beginning of this sūtra is incorrect. It should be अपि वैतश्चिचतुरस्रं not अपि चेतश्चिंचतुरस्रं The former reading is given by Caland in his edition of the *Baudhāyana Śrauta Sūtra* and also appears in my copies of the *Baudhāyana Śulba*.

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The latter portion of both the rules is obscure inasmuch as the operations to be employed have not been explained fully. They were doubtless handed down by oral tradition.

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Thibaut,³ followed by Bürk,⁴ thinks the method implied to be this: Let the side of a given square be 7 units long. It is to be transformed into a rectangle whose one side will be, say, 5 units long. Cut off from the given square a rectangle of 5 by 7 units. There will then remain in excess a rectangle of 7 by 2 units. From this cut off a rectangle of 5 by 2 units and add it properly to the other portion. Then remains a square of 2 by 2 units. Change this into a rectangle of 5 by 4/5 units and place it by the side of the previous rectangle so as to fit. Thus we have finally a rectangle of 5 by 49/5 (=7+2+4/5) units.

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³ *Śulba Sūtra*, p. 20.

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⁴ *ZDMG*, LVI, p. 334.

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This explanation is doubtless wrong. For at the final stage it begs the original problem itself. If the *Śulba* writers had really intended a simple arithmetical operation to be followed at the final stage of their method, as is supposed by Thibaut and Bürk, they could have, and very likely would have, directed to adopt it at the preliminary stage, without taking recourse to any kind of geometrical operations at all. The method of Āpastamba and Baudhāyana was of course geometrical, not arithmetical.¹

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¹ It may be noted that Thibaut has discarded on a different occasion certain arithmetical interpretation of a rule of Baudhāyana by Dvārakanātha Yajvā with the remark, "The commentary instead of showing how the desired end could be attained by making use of the geometrical constructions taught in the paribhashā sūtras, employs arithmetical calculation; but this was of course not the method of the sūtrakāra." (*Paṇḍit* O. S., X, p. 73.) Similarly Bürk remarks about certain explanation of the commentator Sundararāja: "This explanation of *jñeya* is incorrect...because the Śulbasūtra deals with geometrical construction, and not with 'numerical calculation.' In commenting *jñeyā* from the standpoint of arithmetic, the commentator falls into a similar error as disclosed by Thibaut in another case (*JASB*, XLIV, 272)." (*ZDMG*, LVI, p. 329.)

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According to the commentators Sundararāja and Dvārakanātha Yajvā, the method implied is similar to this:

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"Produce the northern and southern sides towards the east as much as you wish (a side of the transformed rectangle to be. Complete the rectangle and) draw the diagonal passing through its north-eastern corner. (Find the point) where it cuts the transverse side of the (given) square lying inside that rectangle. Leave off the portion of that side lying to the north of that point and make its southern portion the breadth of a rectangle. That will be the rectangle, (required)."²

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² "यावदिच्छं पार्श्वमानी प्राचीं वर्धयित्वा उत्तरपूर्व्वां कर्णरज्जुमायच्छेत् । सा दीर्घचतुरस्रमध्यस्थायां समचतुरश्रतिर्य्यङ्मान्यां यत्र निपतति तत उत्तरं हित्वा दक्षिणांशं तिर्य्यङ्कानीं कुर्य्यात् । तद्दीर्घचतुरस्रं भवति ।"

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Let *ABCD* be the given square and *M* the given length which is greater than a side of the square.

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Fig. 47

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Produce *DA* and *CB* to *E* and *F* respectively so that DE=CF=M. Join *EF* and complete the rectangle *EFCD*. Draw the diagonal *EC* cutting *AB* at *P*. Then *PB* will be the breadth of the transformed rectangle. Through *P* draw the straight line *HPG* parallel to *ED* or *FC*. Then *HFCG* is the rectangle which is equivalent to the square *ABCD* and whose side *CF* is equal to the given length *M*. For

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$△ EFC=△ EDC,$ $△ EHP=△ EAP,$ $△ PBC=△ PGC.$ $∴ parallelogram HFBP=parallelogram APGD.$ $Hence parallelogram HFCG=square ABCD.$ *Q. E. D.*

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Bürk was led to suspect whether the method explained by the commentators was indeed in view of the Śulbakāras and to discard it ultimately for the simple reason "that this method is so scientific." Thibaut has not directly assigned any reason for his rejection of the interpretation of the commentators. He seems, however, to have been led by an observation of Dvārakanātha Yajvā *anyacca prakāraḥ* ("Also another method") which just precedes his delivery of the method. But the preliminary remark of Sundararāja is *ayamatra prakāraḥ* [“This is the method (taught) here].” The rule has been formulated by the two commentators in identical words. So one has doubtless copied from the other. Difference in the preliminary remark may be explained thus: Sundararāja explains the method just after the above rule of Āpastamba which is its proper place. But Dvārakanātha Yajvā gives it under the imperfect method of Baudhāyana for the transformation of a square into a rectangle which precedes his perfect rule stated above. So it was very natural for him to remark that it is “another method.”

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Why Dvārakanātha Yajvā explains the method at a place other than what is proper for it seems very probably to be this. The process as defined in the rules of Baudhāyana and Āpastamba does not begin, as has been already pointed out by Bürk, with the construction of a greater rectangle (*EFCD* in Fig. 47) as required in the above explanations of the commentators. But it begins very clearly to cut off from the given square a smaller rectangle (*ABFE* in Fig. 48). Or in other words the explanation of the commentators has in view the case in which the given length is greater than a side of the given square, whereas the rule of the Śulbakāras has in view the case in which the given length is smaller than a side of the given square. So Dvārakanātha Yajvā was not wrong in calling the method described by him to be a different one. This case was treated under the unsatisfactory rule of Baudhāyana undoubtedly as a sort of modification and improvement of it. The case of the Śulbakāra has been explained by him in its proper place under the succeeding rule.

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The geometrical process in the case that was in view of the Śulbakāras is this: From the sides *AD* and *BC* of the given square *ABCD* cut off *AE* and *BF* respectively,¹

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¹ We cut off lengths equal to *M* from the sides *AD* and *BC* of the given square but not from the sides *AB* and *CD*, because the rule clearly directs that the given length *M* should be made the *pārśvamānī* of the rectangle cut off.

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Fig. 48

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making each equal to the given length *M* which is smaller than a side of the given square. Join *AF* and produce it to meet *DC* produced at *P*. Complete the rectangle *ADPH*. Produce *EF* to meet *HP* at *G*. Then *AHGE* is the rectangle which is equivalent to the square *ABCD* and which has a side *AE* equal to the given length *M*. The proof is similar to that given before.

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On one occasion Baudhāyana makes a rectangle equivalent to three given squares, one side of the rectangle being half of a side of a square.² This case is a very simple one indeed, but it shows that he knew other methods of transforming a square into a rectangle than that indicated in his imperfect rule.

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² *BŚl*, iii. 255.

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*To transform a square or a rectangle into an isosceles trapezium which shall have a given face.*

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Baudhāyana gives the following rule for the transformation of a square or a rectangle into an isosceles trapezium, or for what they call "shortening of a square or a rectangle on one side."

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"If you wish to make a square or a rectangle shorter on one side, (cut off a rectangular portion) by making the shorter length a side. Divide the remainder by the diagonal and place (the two portions) on either sides (of the portion cut off) after inverting."¹

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¹ *BŚl*, i. 55.

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Fig. 49

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Let ABCD be a given square and P a given line which is shorter than AB. From AB and DC cut off AE and DF respectively making each equal to P. Join EF and EC. Take the triangle CBE and place it after inverting in the position ADG. Then AECG is the isosceles trapezium which is equal to the given square ABCD and whose face AE is equal to the given length P.

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We find a nearly similar rule in the *Śatapatha Brāhmaṇa*.² Let ABCD be a rectangle. Take AE=FB=

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² *ŚBr*, x. 2. 1. 4.

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Fig. 50

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DH=CG. Then it is said that the trapezium EFGH is exactly equal to the rectangle *ABCD*. This method reappears in the *Āpastamba Śulba*.¹

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¹ *ĀpŚl*, xv. 9f.

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*To transform a square or a rectangle into a triangle.*

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"If you wish to transform a square or a rectangle into a triangle, construct a square whose area will be twice as much as the area of the figure (to be transformed). Fix a pole at the middle of its eastern side. Having fastened at it two ties (of two cords), stretch the cords towards the two western corners. Cut off the portions lying beyond these cords." ²

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² *BŚl*, i. 56.

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Fig. 51

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Let *P* be the given rectangular figure to be transformed. Draw the square *ABCD* so that its area will be twice that of *P*. Let *M* be the middle point of *AB*. Join *MD* and *MC*. Then the triangle *MCD* is equal to the rectangular figure *P*. For each is equal to the half of the square *ABCD*.

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*To transform an isosceles triangle into a square.*

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"If you wish to transform an isosceles triangle into a square, cut off its northern half by the middle line ; then place it on the opposite side after inverting, By the method of constructing a square equal to a rectangle, construct the square. This is the method of construction." ¹

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¹ *KŚl*, iv. 5.

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Fig. 52

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Let *ABC* be an isosceles triangle. Draw the median *AD*. Take the half *ADC* and place it after inversion in the position *BGA*. Then the rectangle *AGBD* is equal to the isosceles triangle *ABC*. Now transform the rectangle *AGBD* into a square by the method given before.

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*To transform a square or a rectangle into a rhombus.*

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"If you wish to transform a square or a rectangle into a rhombus, construct a rectangle which shall have an area twice as much as the area (of the figure to be transformed). Fix a pole at the middle of its eastern side. Having fastened at it two ties (of two cords), stretch the cords towards middle (points) of the northern and southern sides (of the rectangle). Cut off the portions lying beyond (these cords). Thereby is also explained the construction of the other triangle." ²

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² *BŚl*, i. 57.

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Let *P* be a rectangular figure. Draw the rectangle *ABCD* so that its area will be double that of *P*. Let *G*, *H*, *E*, *F* be the middle points of the sides *AB*, *BC*, *CD*, *DA* respectively. Join *GH*, *HE*, *EF*, *FG*. Then the rhombus *GHEF* is equal to the rectangular figure *P*.

Linked texts (5)

Āpastambaśulbasūtra — Āpastamba, Sanskrit
Kātyāyanaśulbasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Baudhāyanaśrautasūtra — Baudhāyana, Sanskrit
Baudhāyanaśulbasūtra — Baudhāyana, Sanskrit