Siddhantas, authored either by Rishis in the 18 Siddhanta era or Manuja in the early Classical era, do not include a single scientific proof in their texts. Hence, some historians have claimed in the past, that mathematical proofs are purely a Greek invention.

In the 20th century, GH Hardy frequently expressed a similar frustration with Ramanujan’s apathy regarding proofs.

It is only with Bhaskaracharya in the 12th century that an effort to provide proofs begins in the Siddhantas themselves. Later, in the 16th century, the Kerala mathematician, Jyeshtadeva, dealt with proofs extensively in his Malayalam book, Yuktibhaasha.

But much earlier, the commentators of Siddhantas, have discussed and explained proofs of mathematical statements in their bhashyaas.

Proof by Argument

Strangely, the only scientific explanation of an astronomical phenomenon, which can be understood even by an illiterate person with common sense, needs no mathematical expertise, simply but logically overturns the old religious superstition, and is verifiable by any observer, is provided by Varahamihira in his Brihat Samhita.

I refer to the first few slokas in the chapter on Raahuchaara, which explains how eclipses are caused, not by mythological snakes Rahu and Ketu, but by the shadows of the earth on the moon and the moon blocking the sun.

Remarkably, unlike Brahmagupta who severely condemns and insults earlier astronomers like Aryabhata and Vishnuchandra, though in enjoyably poetic hyperbole, Varahamihira is respectful of religious tradition and makes no personal attacks whatsoever.

Varahamihira uses four arguments — Geometry, Direction, Timing, and Size Difference — to explain the eclipses as shadows and debunk the snake explanation.

In Pancha Siddhantika, Chapter 15 Jyotishopanishad, verse 10, Varahamihira mentions that he has explained this proof in his work Brhat Samhita.

Additionally, in verse 5, he states that those living at or near Meru (north pole) never see an eclipse, because the sun and moon aren’t high enough above the horizon.

वृक्षस्य स्वच्छाया यथैकपार्श्वे भवति धीर्घचया ।

निशि निशि तद्वद्भूमेरावरणवशाद्दिनेशस्य ॥

Transliteration: vRukshasya (of tree) svacchaayaa (self shadow) yatha (how) eka paarshvE (on one side) bhavathi (becomes) dheerghachayaa (increases)

nishi (night) nishi (night) tadvad (so too) bhumEH (of Earth) aavaraNavashaath (covers) dinEshasya (the Sun’s)

Meaning: As tree’s shadow grows longer on one side, so the Earth’s shadow elongates on one side, because of the Sun. This is a simple, obvious, but profound, metaphor and explanation, to show that the Earth casts a shadow in space, like a tree casts a shadow on the ground!

भूच्छायां स्वग्रहणे भास्करमर्कग्रहे प्रविशतिन्दुः ।

प्रग्रहणमतः पश्च्चान्नेन्दोर्भानोश्च पूर्वार्द्धात् ॥

Transliteration bhu (Earth) chaayaam(shadow) sva grahaNE (in self eclipse, i.e. lunar eclipse) bhaaskaram (Sun’s disc) arka grahE (in solar eclipse) pravishath (enters) indu (the Moon)

pragrahaNam (Eclipse) athaH (hence) paschaath (from west) na (not) indu (Moon) bhaanu (Solar) cha (and) poorvaardhaath (from east)