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Mathematics in Ancient and Medieval India c6.s12: Euclid's Elements in India

Mathematics in Ancient and Medieval India c6.s12

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Euclid's Elements in India

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Prince Maḥmud Shāh Bahamani ( 2nd half of 14th century A. D. ) studied Taḥrīr - au - Uqlīdas, Ar-bian version of Euclid's *Elements*.¹ Abul Fazl (c. 1590 A. D.) in his *Ain-i-Akbari* has referred to some of the propositions of the Euclids' *Elements*. He has compiled it in a way which shows that he had thorough acquaintance with the work.² Abul Fazl got his education in India, so he must have learnt Euclid's *Elements* here, of course, in Arabic or in Persian translation. This hints that the work was known in India at least in the 14th century A. D. We do not know exactly, when Euclid's *Elements* came to India but we are sure that in the beginning it remained confined to the circle of Moslem scholars in India.

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¹ *Ferishta*, translated by John Briggs, 2, p. 296.

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² 2,p. 415-6; 3, p. 24 ( Jarrett's edition ).

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The earliest trace of its influence is found in the Sanskrit work, *Siddhāntatattvaviveka* ot Kamalākara, the court astronomer of the emperor Jahangir of Delhi. Some of the passages of *Siddhāntatattvaviveka* are evidently copied from the *Elements*.³ Other instances of resemblance

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³ The definition and classification of a line is given in the following terms, which is very akin to that of Elements of Euclid; exist and (which is) very fine. That line should be known by the intelligent as of two kinds : curved and uncurved. Of these the uncurved is called straight......".

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*dairghaṃ yasyāḥ sadaivāsti vistāro naiva labhyate |*
*atisūkṣmā ca sā rekhā jñeyā budhimatā dvidhā ||*
*avakrā vakragā tatra'vakrā tu saralābhidhā. |*
( *Siddhāntatattvaviveka*, ch. 3, v. 22-23½ )

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"Of which there is always the length, the breadth does not can also be identified with particular propositions of *Elements* without much difficulty.¹ On the whole, there is absolutely no doubt that Kamalākara had knowledge of Euclid's *Elements*. The first complete translation of this work in Sanskrit was made in 1718 A. D. under the title *Rekhāgaṇita* ("Calculation with lines") by Samrāṭ Jagannātha at the command of his patron king Jayasiṃha of Jaipur. It was rendered from the Arabic translation of Euclid by Naṣir-ed-dīn aṭ-ṭusi (c. 1250 A. D.). The *Rekhāgaṇita* contains 15 *adhyāyas* or Books. Books I-IV are devoted to plane geometry and Books XI-XIV treat of solid geometry. The remaining books deal with laws of proportion, principle of numbers and other related problems,

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¹ For instance, compare proposition of III. 38, 45, 46, 48 with those of Euclid's *Elements* – I. 21; VI. 8; I. 15; I, 19 respectively.

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In the *Rekhāgaṇita*, we find quite a large number of Sanskrit words coined by the author for translating Euclid's *Elements* from its Arabic version. A list of these terms is tabulated below with their English equivalents :

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( a ) Technical terms of *Rekhāgaṇita* connected to plane figures :

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| Sanskrit terms | English Equivalents |
| 1. *abhīṣṭa-rekhā* | Straight line of desired length. |
| 2. *āyata* | Oblong. |
| 3. *cāpa-karṇa* | Chord. |
| 4. *kodaṇḍa* | Segment of a circle. |
| 5. *koṇa* | Angle. |
| i *adhikakoṇa* | Obtuse angle. |
| ii) *alpakoṇa* | Acute angle. |
| iii) *samakoṇa* | Right angle. |
| iv) *viṣama-koṇa* | Angle other than a right angle. |
| 6. *kṣetra* | Figure. |
| i ) *dharātalakṣetra* | Superficies. |
| ii) *sama-dharātalakṣetra* | Plain superficies. |
| iii) *viṣama-dharātalakṣetra* | Crooked superficies. |
| | N. B. – From one point to another number of crooked lines may be drawn, but one right line which is the shortest may be drawn. So infinite number of crooked superficies may be drawn from one line to another, but only one plain superficies may be drawn which is shortest. |
| 7. *nyāsa* | Construction. |

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| Sanskrit Terms | English Equivalents. |
| 8. *nyunakoṇa* | An acute angle. |
| 9. *pāli* | Circumference. |
| 10. *rekhā* | Line |
| i) *samānāntara-rekhā* | Parallel lines. The definition omits the most important thing that parallel lines must be in the same plane. |
| ii) *sarala rekhā* | Right line or straight line. |
| iii) *viṣama-rekhā* | Crooked line. |
| 11. *vṛtta-kṣetra* | Circle. |

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( b ) Technical terms for solid figures :

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| Sanskrit Terms | English Equivalents |
| 1. *ghana-koṇa* | Solid angle. |
| 2. *ghana-kṣetra* | Solid figure. |
| i) *aṣṭaphalakaghanakṣetra* | Octahedron. |
| ii) *cheditaghanakṣetra* | Prism. |
| iii) *ghanahastakṣetra* | Parallelepiped. |
| iv) *gola-kṣetra* | Sphere. |
| v) *samabhujadvādaśaphalakakṣetra* | A dodecahedron. |
| vi) *samānāntara-dharātalaghanakṣetra* | Parallelepiped. |
| vii) *samatalamastakapari-dhirūpaśaṅkukṣetra* | Cylinder. |
| viii) *sucīphalakaghanakṣetra* | Pyramid. |
| 3. *mastaka.paridhi* | Altitude. |
| 4. *śaṅku* | Cone, cube. |
| i) *catuṣphalakaśaṅku* | Tetrahedron. |
| ii) *trasraphalakaśaṅku* | Pyramid having a triangle as its base. |
| 5. *tala* | Base. |

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