Apastamba's Theorem — Pythagorean Triplets
Apastamba Sulba Sutra · 1 · Verse 4
kalpasulbaapastambapythagorean-tripletsmathematicsgeometry
Sanskrit Original
त्रिकं चतुष्कं च पञ्चकं च। द्वादशकं पञ्चदशकं नवदशकं च। पञ्चदशकं षट्त्रिंशकं नवत्रिंशकं च। एते सर्वे समकोणत्रिभुजस्य भुजाः॥
trikaṃ catuṣkaṃ ca pañcakaṃ ca | dvādaśakaṃ pañcadaśakaṃ navadaśakaṃ ca | pañcadaśakaṃ ṣaṭtriṃśakaṃ navatriṃśakaṃ ca | ete sarve samakoṇatribhujaṣya bhujāḥ ||
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Āpastamba's Śulba Sūtra lists explicit Pythagorean triplets: (3,4,5), (12,15,19... wait — actually 12,16,20 = 3:4:5), (5,12,13), (8,15,17), (7,24,25), and others. These are integer solutions where a²+b²=c². Āpastamba knew these by ~600 BCE, confirming systematic knowledge of right-triangle geometry in ancient India.