Short answer

No. The famous claims do not come from the words of the Vedas. The "speed of light" is a number in a 14th-century commentary, and the verse itself gives no number. The "Vedic aeroplanes" come from a text made in the early 1900s. Engineers at the Indian Institute of Science found that none of its planes could fly. The rules of "Vedic Mathematics" are not in the known texts, as its own editor said. India's real mathematicians, such as Aryabhata, deserve honour, but their work is their own. Test the hymns as this site tests the Qur'an, by their plain words. They teach no such science, and one hymn blames a demon for an eclipse. The Qur'an is not a science book either. But it repeats no such error, and it points to the one God (﷾).

Where each claim really comes from
ClaimWhere it comes fromWhenWhat was found
The speed of lightA remembered saying in Sayana's commentary, not the verse14th centuryIt speaks to the sun, and it matches only with one pairing of units
Vedic aeroplanesThe Vaimanika Shastra, spoken by Pandit Subbaraya ShastryBetween 1900 and 1922None of the planes could fly
Vedic MathematicsA book by Bharati Krishna Tirtha1965Its rules do not appear in the known appendices of the Atharva Veda
Time scalesManu and later books, which are smritiAfter the hymnsSagan said the match is no doubt by accident

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