Geometry in Ancient and Medieval India c8.s2: 3.6. The trapezium and the trapezoidal solid

Geometry in Ancient and Medieval India c8.s2

Language: English

3.6. The trapezium and the trapezoidal solid

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The universe, according to the Jainas, is in the form of three trapezoidal solids piled one upon another. As such the

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trapezium must have been a familiar geometrical figure in their literature from very early times. In the Jyotiṣkaraṇḍaka (10th Prābhṛta V. 190-91) we find a simple rule for calculating the diameter at any height of a mountain shaped like the frustum of a cone. The rate of increase of the diameter is given by $(diameter at base—diameter at top)/height$ Hence the height at which the diameter is required should be multiplied by this ratio and added to the diameter at the top to get the diameter at that height. The geometrical fact that corresponding elements in similar figures are proportionate, known to the authors of the *Sulbasūtras* and the *Brāhmaṇas*, who increased the size of the *vedis* without altering their shape or the proportions of their parts, is apprehended in a more comprehensive and scientific way here. The *Tiloyapaṇṇatti*, which from the quotations from and the references to it in other works must be an early work, has a fuller treatment of trapezia and trapezoidal solids and a fuller application of the properties of similar figures. But since the extant version of the *Tiloyapaṇṇatti* belongs to a much later period, the work is reserved for treatment in a later chapter.

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**3.7.** An important branch of Jaina religio-computational literature, viz., the *karaṇa-gāthās*, are either entirely lost or are yet to be brought out of the obscurity of forgotten private libraries. Stanzas from such works are quoted in almost all Jaina commentaries. What is more, even Bhāskara I, the great exponent of Āryabhaṭa I’s mathematics and astronomy, quotes three Prākṛt *gāthās* in his commentary on the *Āryabhaṭīya*. Amongst the mathematicians mentioned by Bhāskara four bear the rather unusual names Maskari, Pūraṇa, Mudgala and Pūtana. The passages in which these names occur are—

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एतदेकैकस्य ग्रन्थलक्ष्या न मस्करि-पूरण-मुद्गलप्रभृतिभि: आचार्यैर्निबद्धं कृतं, स कथं अनेनाचार्येण अल्पेन ग्रन्थेन शक्यते¹

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¹ ^1R. 14850 Govt. Or. Mss. Lib. Madras, p. 2.

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(This has not been composed as a treatise by the *ācāryas* like Maskari, Pūraṇa and Mudgala with even one lakh verses for each (topic). How can this *ācārya* manage it within a short treatise ?)

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² ^2Ibid, p. 74.

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गणितविदो मस्करीपूरण-पूतनादयः सर्वेषां क्षेत्राणां फलम् आयतचतुरश्रक्षेत्रे प्रत्याययन्ति¹।

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¹ Ibid p. 74.

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(Mathematicians like Maskari, Pūraṇa and Pūtana show the rationale of the areas of all figures in rectangular figures.) One of these names, that of Pūraṇa,² occurs in the *Vyākhyāprajñapti* (*3rd Śataka, Uddeśa 2*). Does this circumstance taken along with the fact that Bhāskara quotes from Jaina works, warrant the assumption that these mathematicians were Jainas? Be that as it may, it is a pity that mathematicians almost on a par, according to Bhāskara, with Āryabhaṭa and responsible for such voluminous outputs in the field of mathematics should remain mere names for us.

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² One *Purāṇa* is mentioned as one of the sages who surrounded Bhīṣma. (*Mahābhārata, Śāntiparvan, Rājadharmānuśāsana* Ch. 47, 12).

Cites (7)

Joisakaraṃḍaga — Māhārāṣṭrī Prakrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Mahābhārata — Vyāsa, Sanskrit
Śulbasūtra
Brāhmaṇa — Sanskrit
Viyāhapannatti — Sudharman, Ardhamāgadhī Prakrit
Tiloyapaṇṇatti — Yativṛṣabha, Śaurasenī Prakrit