०१

In the first two chapters Baudhāyana has given a summary of geometrical knowledge and some results of mathematical interest required for the construction of sacrificial altars. How the knowledge was used in connection with the measuring of grounds and placing of different layers of bricks has been discussed in detail in subsequent chapters. This chapter deals specifically with the units of measurements of altars, methods of construction of squares and rectangles, application of surd numbers, and the theorem of square on the diagonal of a rectangle.

अथेमेऽग्निचयाः १

The various constructions of sacrificial fires are now given.

अथेमेऽग्निचयाः १

तेषां भूमेः परिमाणविहारान्व्याख्यास्यामः २

We shall explain the methods of measuring areas of their (different) figures (drawn) on the ground.

तेषां भूमेः परिमाणविहारान्व्याख्यास्यामः २

अथाङ्गुलप्रमाणं चतुर्दशाणवः,
चतुस्त्रिंशत्तिलाः पृथुसंश्लिष्टा इत्यपरम्
(~ ४-अङ्गुलायतम् /४))।
दशाङ्गुलं क्षुद्रपदम् । द्वादश प्रादेशः । पृथोत्तरयुगे त्रयोदशिके । पदं पञ्चदश । अष्टाशीतिशतमीषा । चतुःशतमक्षः । षडशीतिर्युगम् । द्वात्रिंशज्जानुः । षट्त्रिंशच्छम्याबाहू । द्विपदः प्रक्रमः । द्वौ प्रादेशावरत्निः । अथाप्युदाहरन्ति पदे युगे प्रक्रमेऽरत्नावियति शम्यायां च मानार्थेषु याथाकामीति । पञ्चारत्निः पुरुषो व्यामश्च । चतुररत्निर्व्यायामः ३

Now, the measure of an aṅgula is 14 anus (grain of Panicum milliaceum); according to others, (it is) 34 tilas (sesamum indicum) placed broad side on. One small pada is 10 aǹgulas; one prādeśa 12 aṅgulas; one pṛthā and one uttara- yuga 13 aṅgulas each; one (big) pada 15 aṅgulas. One îșă measures 188 aṅgulas; one akṣa 104 aṅgulas; one juga 86 aṅgulas; one jānu 32 aṅgulas; one samyā and one bāhu 36 aṅgulas each. One prakrama equals 2 padas (30 aṅgulas); one aratni 2 prādeśas (24 aṅgulas). But there are also instances of pada, yuga, prakrama, aratni and śamyā having different measures when these (words) are used as units of measurement. 5 aratnis (120 aṅgulas) make one purușa; one vyāma also has the same measure (5 aratnis); and 4 aratnis (96 aṅgulas) make one vyāyāma.

1.3. Baudhayana’s table of units of measurements runs as follows:

1 aṅgula = 14 aņus 1 small pada = 10 aṅgulas; 1 prādeśa = 12 aṅgulas; 1 pada = 15 aṅgulas; 1 iṣā = 188 aṅgulas; 1 akṣa = 104 aṅgulas; 1 yuga = 86 aṅgulas; 1 janu = 32 aṅgulas; 1 śamyā = 36 aṅgulas; 1 bāhu = 36 aṅgulas; 1 prakrama = 2 padas; 1 aratni = 2 prādeśas = 24 aṅgulas; I puruşa = 5 aratnis = 120 aṅgulas; 1 vyāma = 5 aratnis;
1 vyāyāma = 4 aratnis; 1 aṅgula = \(\frac{3}{4}) inch¹ (apprpx.).

Āpastamba (Aśl. 6.5, 15.4) has prescribed the same values of Baudhāyana for iṣā, akṣa, yuga, puruṣa, vyāyāma, aratni and prādeśa. These units of Āpastamba have been used by both Kātyāyana (Kśl. 2.1, 5.9) and Mānava (Mśl. 2.1, 4.4). The unit pada has been made equal to 12 aṅgulas by Kātyāyana (Kśl. 5.9). The term vitasti has been used in place of prādeśa by these two latter śulbakāras as well as by Kautilya in his Arthaśāstra², while its value remains the same. Mānava (Mśl. 4.2- 4.4) has supplied some more units, e. g. 6 tuṇḍa=1 bāla of 3 years old calf; 3 bālas = $\frac{1}{2}$ mustard seed; 2 mustard seeds=1 yava³; 1 aṅgula=6 yavas; and 1 prādeśa=10 vitastis.

The units like aṅgula, pada, prakrama, prādeśa, bāhu, aratni carry a long tradition and have been used earlier in the Samhitas and Brāhmaṇic literature in the same. sense as these have been used in the Šulbasūtras.⁴

अथाङ्गुलप्रमाणं चतुर्दशाणवः चतुस्त्रिंशत्तिलाः पृथुसंश्लिष्टा इत्यपरम् । दशाङ्गुलं क्षुद्रपदम् । द्वादश प्रादेशः । पृथोत्तरयुगे त्रयोदशिके । पदं पञ्चदश । अष्टाशीतिशतमीषा । चतुःशतमक्षः । षडशीतिर्युगम् । द्वात्रिंशज्जानुः । षट्त्रिंशच्छम्याबाहू । द्विपदः प्र-क्रमः । द्वौ प्रादेशावरत्निः । अथाप्युदाहरन्ति पदे युगे प्रक्रमेऽर-त्नावियति शम्यायां च मानार्थेषु याथाकामीति । पञ्चारत्निः पुरुषो व्यामश्च । चतुररत्निर्व्यायामः ३

चतुरश्रं चिकीर्षन्यावच्चिकीर्षेत्ताव-तीं रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । लेखामालिख्य तस्या मध्ये शङ्कुं निहन्यात् । तस्मिन्पाशौ प्रतिमुच्य लक्षणेन म-ण्डलं परिलिखेत् । विष्कम्भान्तयोः शङ्कू निहन्यात् । पूर्वस्मि-न्पाशं प्रतिमुच्य पाशेन मण्डलं परिलिखेत् । एवमपरस्मिंस्ते यत्र समेयातां तेन द्वितीयं विष्कम्भमायच्छेत् । विष्कम्भान्तयोः शङ्कू निहन्यात् । पूर्वस्मिन्पाशौ प्रतिमुच्य लक्षणेन मण्डलं परिलिखेत् । एवं दक्षिणत एवं पश्चादेवमुत्तरतस्तेषां येऽन्त्याः संसर्गास्तच्चतुरश्रं सम्पद्यते ४

Having desired (to construct) a square, one is to take a cord of length equal to the (side of the) given square, make ties at both ends and mark it at its middle. The (east-west) line (equal to the cord) is drawn and a pole is fixed at its middle. The two ties (of the cord) are fixed in it (pole) and a circle is drawn with the mark (in the middle of the cord). Two poles are fixed aɩ both ends of the diameter (east-west line). With one tie fastened to the eastern (pole), a circle is drawn with the other. A similar (circle) about the western (pole). The second diameter is obtained from the points of intersection of these two (circles); two poles are fixed at two ends of the diameter (thus obtained). With two ties fastened to the eastern (pole) a circle is drawn with the mark. The same (is to be done) with respect to the southern, the western and the northern (pole). The end points of intersection of these (four circles) produce the (required) square.

चतुरश्रं चिकीर्षन्यावच्चिकीर्षेत्ताव-तीं रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । लेखामालिख्य तस्या मध्ये शङ्कुं निहन्यात् । तस्मिन्पाशौ प्रतिमुच्य लक्षणेन म-ण्डलं परिलिखेत् । विष्कम्भान्तयोः शङ्कू निहन्यात् । पूर्वस्मि-न्पाशं प्रतिमुच्य पाशेन मण्डलं परिलिखेत् । एवमपरस्मिंस्ते यत्र समेयातां तेन द्वितीयं विष्कम्भमायच्छेत् । विष्कम्भान्तयोः शङ्कू निहन्यात् । पूर्वस्मिन्पाशौ प्रतिमुच्य लक्षणेन मण्डलं परिलिखेत् । एवं दक्षिणत एवं पश्चादेवमुत्तरतस्तेषां येऽन्त्याः संसर्गास्तच्चतुरश्रं सम्पद्यते ४

अथापरम् । प्रमाणाद्द्विगुणां रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । स प्राच्यर्थः । अपरस्मिन्नर्धे चतुर्भागोने लक्षणं करोति । तन्न्यञ्चनम् । अर्धेंऽसार्थं । पृष्ठ्यान्तयोः पाशौ प्रतिमुच्य न्यञ्चनेन दक्षिणापायम्यार्धेन श्रोण्यंसान्निर्हरेत् ५

Now another (method). Ties are made at both ends of a cord twice the measure and a mark is given at the middle. This (halving of the cord) is for the east-west line (that is, the side of the required square). In the other half (cord) at a point shorter by one-fourth, a mark is given; this is the nyañcana (mark). (Then) a mark is given at the middle (of the same half cord) for purposes of (fixing) the corners (of the square). With the two ties fastened to the two ends of the east-west line (pṛṣṭhyā), the cord is to be stretched towards the south by the nyañcana (mark); the middle mark (of the half cord) deter- mines the western and the eastern corners (of the square).

1.4-1.5. Square. Baudhāyana has described here two methods of construction of squares. First Method. Let XY be the given cord and U a mark at its middle (Fig. 1 (a)); EW, the prācī of the figure = XY; O the middle point of EW obtained corresponding to U of XY, where a pole is fixed.

A circle with O as centre and OE as radius is drawn (Fig. 1(b)). Then EW is a diameter of the circle along east-west line. Two other circles with E and W as centres and EW as radius are separately drawn. The points of intersection of these two circles are denoted by N’ and S’ The line N’S’ fixes the second diameter NS of the circle whose centre is 0. Again fastening the two ties once at E, W, N and S and drawing arcs, the points A, B, C, D are fixed. Then ABCD gives the required square.

Second Method. Let XS, the given measure (pramāņa) be a and XY, the increased cord, 2a (Fig. 2(a)). S is the mark at the middle of XY; then XS measures the length for prāci.

T is the nyañcana mark, so that ST-a-la-la. U is another mark at the middle of SY.

XT, the diagonal (akṣṇayā) = $a+\frac{1}{4}a=\frac{5}{4}a$

TY, the breadth (tiryanmāni) = $2a-\frac{5}{4}a=\frac{3}{4}a$

Clearly, $a^2+(\frac{3}{4}a{)}^2=(\frac{5}{4}a{)}^2$

In other words, $X{Y}^2+Y{T}^2=X{T}^2.$

… XYT is a right-angled triangle (Fig.2(b)).

For the construction of any geometrical figure intended in the śulbasūtra XY is always stretched along east-west line, known as pārśvamāni, YT along north-south line, known as tiryañmānī, and XT along the diagonal known as akṣṇayārajju. Now it is easy to see how the right-angled triangle XYT has been used for the cons- truction of the square.

The corners L and K (śroņi points), M and N (amsa points) are fixed with the help of the point T of the triangle XYT [Fig.2(c)]. Then by using the half-cord UY, the points C, B, D and A are marked such that $WC=WB=ED=EA.$ The figure ABCD gives the required square.

अथापरम् । प्रमाणाद्द्विगुणां रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । स प्राच्यर्थः । अपरस्मिन्नर्धे चतुर्भागोने लक्षणं करोति । तन्न्यञ्चनम् । अर्धेंऽसार्थं । पृष्ठ्यान्तयोः पाशौ प्रतिमुच्य न्यञ्चनेन दक्षिणापायम्यार्धेन श्रोण्यंसान्निर्हरेत् ५

दीर्घचतुरश्रं चिकी-र्षन्याव-च्चिकीर्षे त्तावत्यां भूम्यां द्वौ शङ्कू निहन्यात् । द्वौ द्वावेकै-कमभितः समौ । यावती तिर्यङ्मानी तावतीं रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । पूर्वेषामन्त्ययोः पाशौ प्रतिमुच्य लक्षणेन दक्षिणापायम्य लक्षणे लक्षणं करोति । मध्यमे पाशौ प्रतिमुच्य लक्ष-णस्योपरिष्टाद्दक्षिणापायम्य लक्षणे शङ्कुं निहन्यात् । सॐऽस एतेनोत्त-रॐऽसो व्याख्यातस्तथा श्रोणी ६

When (the construction of) a rectangle is desired, two poles are fixed on the ground at a distance equal to the desired length. (This makes the east-west line). Two poles one on each side of each of the (two above mentioned) poles are fixed at equal distances (along the east-west line). A cord equal in length to the breadth (of the rectangle) is taken, its two ends are tied and a mark is given at the middle. With the two ties fastened to the two end poles (on either side of the pole) in the east, the cord is stretched to the south by the mark; at the mark (where it touches the ground) a sign is given. Both the ties are now fastened to the middle (pole at the east end of the prăcî), the cord is stretched towards the south by the mark over the sign (previously obtained) and a pole is fixed at the mark. This is the south-east corner. In this way are explained the north-east and the two western corners (of the rectangle).

दीर्घचतुरश्रं चिकी-र्षन्याव-च्चिकीर्षे त्तावत्यां भूम्यां द्वौ शङ्कू निहन्यात् । द्वौ द्वावेकै-कमभितः समौ । यावती तिर्यङ्मानी तावतीं रज्जुमुभयतः पाशां कृत्वा मध्ये लक्षणं करोति । पूर्वेषामन्त्ययोः पाशौ प्रतिमुच्य लक्षणेन दक्षिणापायम्य लक्षणे लक्षणं करोति । मध्यमे पाशौ प्रतिमुच्य लक्ष-णस्योपरिष्टाद्दक्षिणापायम्य लक्षणे शङ्कुं निहन्यात् । सॐऽस एतेनोत्त-रॐऽसो व्याख्यातस्तथा श्रोणी ६

यत्र पुरस्तादंहीयसीं मिनुयात्तत्र तदर्धे लक्षणं करोति ७

When the eastern side is desired to be of shorter measure, a mark is given at half (the tiryaṁmāni).

1.6-1.7. Rectangle and isosceles trapezium. Baudhāyana’s method of construction of rectangle with the help of a cord runs as follows:

Let XT be a piece of cord taken equal to the desired breadth of the rectangle [Fig. 3 (a)];

S, a mark at the middle of XY;

E, W, the prācī poles;
P and Q, G and H, poles at equal distances apart on both sides of each of the prācī poles.

O, the mark assigned by the middle mark S when ties at X and Y are fixed at P and Q and stretched by S; and

A, the point designated by the middle mark S, when both ties at X and Y are fixed at E and stretched.

In a similar way, the other corner points, B, C, D, are traced [Fig. 3(b)]. Hence ABCD is the required rectangle.

In the rule (Bśl. 1.7), Baudhāyana hints at the method of construction of an isosceles trapezium shorter in one side. For this purpose, a mark on the cord accor- ding to desired length is to be given and the rest is similar to that of Bśl. 1.6.

यत्र पुरस्तादंहीयसीं मिनुयात्तत्र तदर्धे लक्षणं करोति ७

अथापरम् । प्रमाणादध्यर्धां रज्जुमुभयतः पाशां कृत्वापरस्मिंस्तृतीये षड्भागोने लक्षणं करोति । तन्न्यञ्छनम् । इष्टेंऽसार्थम् । पृष्ठ्यान्तयोः पाशौ प्रतिमुच्य न्यञ्छनेन दक्षिणापाय-म्येष्टेन श्रोण्यंसान्निर्हरेत् ८

Now another (method). Ties are made at both ends of a cord of length equal to the measure increased by its half (so that the whole length of the cord is divided into three parts of half the measure each). In the third (extended) part on the western side a mark is given at a point shorter by one-sixth (of the third part); this is the nyañcana. Another mark is made at the desired point for fixing the corners. With the two ties fastened to the two ends of the east-west line (pṛṣṭhyā), the cord is stretched towards the south by the nyañcana, and the western and eastern corners (of the square) are fixed by the desired mark.

1.8. Square. Baudhāyana describes here another method of construction of a square as follows:

Let XS be the cord of given measure a [Fig. 4(a)] ;
$XY=1\frac{1}{2}a=\frac{3}{2}a$ ;

$SY=\frac{1}{2}a$ ;

T the nyañcana mark.

.. $XT=a+(\frac{1}{3}.\frac{1}{2}a-\frac{1}{6}.\frac{1}{2}a)=a+\frac{1}{12}a=\frac{13}{12}a$

and $TY=\frac{1}{2}a-\frac{1}{12}a=\frac{5}{12}a$

The relation $a^2+(\frac{5}{12}a{)}^2=(\frac{13}{12}a{)}^2$ holds,

i.e., $X{Y}^2+Y{T}^2=X{T}^2.$

Fig. 4. (a) Cord. (b) Right-angled triangle. (c) Square having given side.

The right-angled triangle XYT [Fig. 4(b)] has been used to fix the corner points L, M, N, and K of the required construction. The points C, D, A and B are fixed such that $WC-ED-EA-WB=\frac{a}{2}$ [Fig. 4(c) ]. ABCD gives the required square.

अथापरम् । प्रमाणादध्यर्धां रज्जुमुभयतः पाशां कृत्वापरस्मिंस्तृतीये षड्भागोने लक्षणं करोति । तन्न्यञ्छनम् । इष्टेंऽसार्थम् । पृष्ठ्यान्तयोः पाशौ प्रतिमुच्य न्यञ्छनेन दक्षिणापाय-म्येष्टेन श्रोण्यंसान्निर्हरेत् ८

समचतुरश्रस्याक्ष्णयारज्जुर्द्विस्तावतीं भूमिं करोति ९

The diagonal of a square produces double the area (of the square).

समचतुरश्रस्याक्ष्णयारज्जुर्द्विस्तावतीं भूमिं करोति ९

प्रमाणं तिर्यग्द्विकरण्यायामस्तस्याक्ष्णयारज्जुस्त्रिकरणी १०

The breadth (of a rectangle) being the side of a given square (pramāņa) and the length the side of a square twice as large (dvikaraņi), the diagonal equals the side of a square thrice as large (tṛkaraņi).

प्रमाणं तिर्यग्द्विकरण्यायामस्तस्याक्ष्णयारज्जुस्त्रिकरणी १०

तृतीयकरण्येतेन व्याख्याता नवमस्तु भूमेर्भागो भवतीति ११

Thereby is explained the side of a square one-third the area of given square (trtiyakaraṇī). It is the side of a square one-ninth the area of the square (explained in the preceding rule, that is, of the square on the tṛkarani).

1.9-1.11. Here Baudhāyana states that in a square $ABCD,A{C}^2=2A{B}^2[sinceAB=BC]orAC=\sqrt{2}AB$, where AC is known as the dvikarani of the measure AB. If $AB=a,AC=\sqrt{2}a$ , where a is the measure. The result is sometimes considered by scholars as a particular case of the more generalized rule given by Baudhāyana in Bśl. 1.10. But Baudhāyana gave no such hint. On the other hand, he has tried to establish a more generalized result on the basis of this statement. According to him, when the measure of the side of a square is a, its diagonal is $sqrt2a.$ Then again the measure of the diagonal of a rectangle having sides a and $\sqrt{2}a,$ is $\sqrt{3}a,$ for (\a^2+ (\sqrt{2}a)^2 = (\sqrt{3}a)^2 ; \sqrt{3a}\) is known as the trkarani. This result has been extended

Fig. 5. Square on the diagonal. Fig. 6. Producer of tṛkaraṇi and tṛtiyākaraṇī.

to obtain the value of tṛtiyākaraṇi by both Apastamba (Ãśl. 2.2 and 2.3) and Kātyā- yana (Kśl. 2.10 and 2.11). The commentators Kapardisvāmī, Sundararāja and Rāma have expressed in identical terms the value as well as meaning of this term. According to them, a square on the producer AB (=√3a) when divided into nine equal parts by means of three parallel lines drawing from both sides, produces the square, EBGF which is one-ninth of the square ABCD (Fig. 6). Then $E{B}^2=\frac{1}{9}A{B}^2$
or $EB=\frac{1}{3}AB$

$\frac{1}{3}\sqrt{3}a$

$\frac{1}{\sqrt{3}}$

The producer EB is known as tṛtiyākaraṇi = $\frac{1}{\sqrt{3}}a,$ where a is the side of the original square.

तृतीयकरण्येतेन व्याख्याता नवमस्तु भूमेर्भागो भवतीति ११

दीर्घ-चतुरश्रस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति १२

The areas (of the squares) produced separately by the length and the breadth of a rectangle together equal the area (of the square) produced by the diagonal.

दीर्घ-चतुरश्रस्याक्ष्णयारज्जुः पार्श्वमानी तिर्यङ्मानी च यत्पृथग्भूते कुरुतस्तदुभयं करोति १२

तासां त्रिकचतुस्कयोर्द्वादशिकपञ्चिकयोः पञ्चदशिका-ष्टिकयोः सप्तिकचतुर्विंशिकयोर्द्वादशिकपञ्चत्रिंशिकयोः पञ्चदशिकषट्त्रिंशिकयोरित्येतासूपलब्धिः १३

This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7and 24, 12 and 35, 15 and 36.

1.12-1.13. The theorem states that in a rectangle $ABCD,A{C}^2=A{B}^2+B{C}^2$ (Fig. 7). This is a most general statement and is enunciated first by Baudhāyana. The proposition is stated almost in identical language by Āpastamba (Āśl. 1.4), Kātyāyana (Kśl. 2.7) and Mănava (Mśl. 10.10). Baudhayana further says that the theorem is easily verified from the following relations:

$3^2+4^2=5^2$

${12}^2+5^2={13}^2$

${15}^2+8^2={17}^2$

$7^2+{24}^2={25}^2$

${12}^2+{35}^2={37}^2$

${15}^2+{36}^2={39}^2$ .

No proof of this theorem is given by Baudhāyana and other śulba writers, since it is beyond their tradition to do so. Zeuthen, Cantor, Vogt, Cajori and Heath have expressed the view that the general statement was possibly the result of an induction from a small number of cases of right-angled triangles having sides in rational numbers known to them. But this is not the actual case. Our discussions on rational rectangles and construction of geometrical figures amply justify that the general character of the theorem was rightly understood by the śulbakāras.

A number of conjectures by Heath, Bürk, Müller, Thibaut, Datta and others as to the way the proof of the theorem could have been arrived at are available. A few of them are discussed in what follows.

(i) According to Heath⁵, the problem of transformation of a square into a rectangle given by Baudhayana in Bśl. 2.3. formed the basis of the proof. For, square ABCD drawn on the diagonal AD of the right-angled triangle AGD is equivalent to four equilateral triangles, while its sides GD and AG produce two each (Fig. 16). This has also established Bürk’s hypothesis.⁶

(ii) The combination of two different squares as described by Baudhāyana in Bśl. 2.1 (Fig. 12) might have laid the foundation of the general statement of the theorem. This is according to Müller.⁷

(iii) Thibaut⁸ opined that the śulbakāras were observant of the fact that the square on AD is equivalent to four equal triangles, one of which is equal to half of the square on OA or OD, i.e. the squares on OA and OD together are equivalent to four equal triangles (Fig. 8). This pattern of arrangement of equilateral triangles are actually found in the first layer of construction with bricks in the vakrapakṣaśyenacit as described by Baudhāyana.

(iv) According to Datta⁹ , the construction of the paityki vedi established the theorem of square on the diagonal. The altar is mentioned in the Satapatha Brāhmaṇa (XIII.8.1.5) as a square with its corners pointed towards the cardinal directions. It is referred to by Baudhāyana (Bśl. 3.11) and also by Kātyāyana (Kśl.2.2), where the method of its construction in detail has appeared. The square EGWH obtained by joining the middle points of a square ABCD (of area 2 sq. purușas) is the paitṛki vedi and is half (in area) of the original square (Fig. 29). The original square ABCD is a square on its east-west line EW. EW is again the diagonal of the newly formed square EGWH. This is undoubtedly a convincing proof (since $E{W}^2=2E{G}^2)$.

(v) The knowledge of dvikaraņī, tṛkaraṇī, discussed by Baudhāyana (Bśl. 1.9— 1.11,) led in a way to the theorem of square on the diagonal.

(vi) Both Āpastamba (Ãśl. 3.7) and Kātyāyana (Kśl. 3.7) gave an ingenious method for calculating the area of a square or rectangle, thereby establishing the theorem of square on the diagonal. According to this method, if there are p units in AB and q units in BC, then the rectangle ABCD has pq square units, which can be obtained by drawing number of parallel lines through p units of AB and q number of parallel lines through 9 points of BC (Fig. 9). This proves directly that $A{C}^2=A{B}^2+B{C}^2.$

(vii) Kātyāyana (Kśl. 2.4 and 2.5) has considered a rectangle of breadth 1 pada and length 3 padas, whose diagonal is a 10 fold producer. According to Datta¹⁰, this justifies the statement of the theorem of square on the diagonal, as may be seen from Fig. 10. In the square $ABCD,DH=CG=CK=BF=AE1$ pada.

Now sq. ABCD
sq on BK + sq. on DH + 4 tr. AEH
$A{H}^2+A{E}^2+4tr.AEH.$
Again sq $ABCD=E{H}^2+4tr.AEH$

… $A{H}^2+A{E}^2=E{H}^2$

(viii) Datta¹¹ has given another proof of this theorem. Four rectangles each. equal to a given rectangle having breadth a, length b and the diagonal c are so constructed that the diagonal of each rectangle forms the side of a square (Fig. 11).

Then
$c^2=(a+b{)}^2—4(\text{(}\frac{1}{2}ab)$)

Or, $c^2=a^2+b^2.$

Although such specific constructions and arguments are not supplied as proofs by the sulbakāras, ample evidence is left by them in their details of constructions to believe that the proofs of the theorem of square on the diagonal of a rectangle were known to them.

Pythagorean Theorem in other Culture Areas.

The theorem of square on the diagonal of a rectangle is usually known as Pythagorean theorem after the name of Pythagoras (c. 540 B.C.). In fact, the relation $3^2+4^2=5^2$ and some such relations have been used by Pythagoras, but evidence of any general statement regarding this is not yet available. Actual proof was first given by Euclid (c. 300 B.C.). Proclus (c. 460 A.D.), the commentator of Euclid’s Elements¹² remarked: “For my part, while I admire those who first observed the truth of the theorem, I marvel more at the writer of the Elements, not only because he made it first by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in the sixth book.” Heath has quoted this with the remark:¹³ “It is difficult for us to be more positive than Proclus was”.

It is fairly certain that the practical use of the theorem was current in old Babylonian times (c. 1800-1600 B.C.). The evidence for this is found in certain Babylonian cuneiform tablets. No general theorem was found to have been men- tioned. It has been conclusively proved by Neugebauer that Pythagoras derived his “Number theorem of Universe” as well as the so-called Pythagorean theorem from cuneiform tablets.¹⁴ The Chinese knew of a similar relation which appeared in Chou Pei (4th century B.C.), but it really became well known from the time of its first commentator Chao Chun Chhinge¹⁵ (3rd century A.D.). A proof of the theorem was given by Bhāskara II¹⁶ (1150 A.D.). According to Needham, Bhaskara II’s treatment was derived from the Chou Pei.¹⁷ This is not true, for the proof of Bhaskara II and that given in Chou Pei can readily be deduced from a number of constructions described already in the Sulbasūtras.

तासां त्रिकचतुस्कयोर्द्वादशिकपञ्चिकयोः पञ्चदशिका-ष्टिकयोः सप्तिकचतुर्विंशिकयोर्द्वादशिकपञ्चत्रिंशिकयोः पञ्चदशिकषट्त्रिंशिकयोरित्येतासूपलब्धिः १३