The corpus, every claim citable.

Surya-Siddhanta III.9 (~5th c. CE) gives 600 equinoctial revolutions per mahayuga (4,320,000 years) — ≈50 arcseconds per year, close to the modern 50.3″/yr. Whether the motion is monotonic (progressive precession) or oscillatory (libration/trepidation between ±27°) is ambiguous: III.11-12 suggest libration, III.9 alone reads as progression. Burgess 1860 reads libration; modern scholarship leans toward original progression later edited for libration.

Translation of the Surya-Siddhanta · 400

Surya-Siddhanta I.30 (~5th c. CE) gives Mars's revolutions per mahayuga (4,320,000 years) as 2,296,832. Period = 4,320,000 / 2,296,832 = 1.8809 sidereal years = 687.05 mean solar days. Modern Mars sidereal period (NASA): 686.971 days. Off by 0.08 days = ~2 hours over a 23-month orbit. ~120 parts per million accuracy, pre-telescopic. The same I.29-34 verse block gives every planet to this kind of precision.

Translation of the Surya-Siddhanta · 400

Aryabhatiya I.6 (Dasagitika, 499 CE) gives Earth's axial tilt (the obliquity of the ecliptic) as 24°. Modern measurement is 23.44°. The 0.56° discrepancy isn't a measurement error: Earth's obliquity oscillates between ~22.1° and ~24.5° on a 41,000-year cycle (Milankovitch orbital forcing), and was ~24° several thousand years before Aryabhata wrote — consistent with his value reflecting inherited observational tradition from earlier Indian astronomy.

The Aryabhatiya of Aryabhata · 499

Aryabhatiya I.5 (Dasagitika, 499 CE) gives Earth's diameter as 1,050 yojanas. The yojana is an Indian unit of distance whose conversion to modern units is disputed in Sanskrit-scholarship (estimates 5-8 miles, depending on text and period). With the most-cited mid-period value of ~7.5 miles per yojana, 1,050 yojanas = 7,875 miles — within 0.5% of the modern measurement of 7,917 miles. The Sun, Moon, and planet diameters in the same verse (in ratios to Earth's) are similarly close.

The Aryabhatiya of Aryabhata · 499

Brahmasphutasiddhanta XII.21 (628 CE) gives the exact-area formula for any cyclic quadrilateral with sides a, b, c, d: K = √[(s−a)(s−b) (s−c)(s−d)], where s = (a+b+c+d)/2. First known generalization of Heron's triangle area formula to four sides. Rediscovered in Europe by Carl Strehlke in 1842 — 1,214 years later. Bhaskara II preserved the rule in Lilavati §167 (1150 CE), source of the verbatim quote.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

Surya-Siddhanta I.29-34 (~5th c. CE) gives the revolutions of each celestial body in one mahayuga (4,320,000 years). The sidereal year derives by arithmetic: (asterism revolutions − sun revolutions) / sun revolutions = (1,582,237,828 − 4,320,000) / 4,320,000 ≈ 365.2587 civil days per year. Modern sidereal year: 365.25636 days. The text is 3.5 minutes / ~7 parts per million long, pre-telescopic. Ptolemy's Almagest (~150 CE) gives 365.2467 — off by ~14 minutes.

Translation of the Surya-Siddhanta · 400

Surya-Siddhanta I.32 (Burgess 1860 trans) gives Saturn's revolutions in one mahayuga (4,320,000 years) as 146,568. Dividing yields a sidereal period of 10,765.5 days. NASA's modern measurement: 10,759.22 days. The match is 0.063% — extraordinary for a pre- telescopic observational tradition. The accuracy reflects ~1000 years of accumulated Indian observational records aggregated into the mahayuga period parameters.

Translation of the Surya-Siddhanta · 400

Brahmasphutasiddhanta XVIII.19 (628 CE) gives explicit rules for arithmetic with zero as a number: addition, subtraction, multiplication, square root. Brahmagupta also writes a rule for division by zero — getting it wrong (treats x/0 as finite) but pioneering the question itself. Bhaskara II refined the rule ~500 years later (1150 CE) treating 1/0 as khahara — closer to modern infinity-as-limit.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

The Lalitavistara, a canonical Buddhist life of the Buddha, stages a counting contest: the examiner Arjuna asks the Bodhisattva to count beyond a koti (10⁷), and he recites a centesimal ladder — each name a hundred times the last — through ayuta, niyuta, kaṅkara and onward to tallakṣaṇa = 10⁵³, with further series beyond. High-magnitude number-naming was so culturally prized that a religious biography made it proof of the hero's perfection. Datta & Singh (1938) translate the dialogue.

History of Hindu Mathematics — A Source Book · 1938

Sūrya-Siddhānta iii builds working astronomy from a gnomon: on a leveled surface, a drawn circle and a twelve-digit vertical stick yield the cardinal directions (from the shadow-tip's morning and evening crossings), the observer's latitude (equinoctial noon shadow, iii.17), and time-of-day quantities. Burgess's commentary works the rules for Washington, D.C. and gets its latitude right. The gnomon procedures are the observational ground floor of the siddhānta's parameter system.

Translation of the Surya-Siddhanta · 400

The Vedāṅga Jyotiṣa — the calendar manual among the six Vedāṅgas, late Vedic period — declares gaṇita (mathematics/computation) the highest of the auxiliary sciences: "As the crests on the heads of peacocks, as the gems on the hoods of snakes, so is gaṇita at the top of the sciences known as the Vedāṅga" (trans. Datta & Singh 1938). It is the earliest known text to rank mathematics supreme among the sciences — a cultural charter the Sanskrit mathematical tradition cited for centuries.

History of Hindu Mathematics — A Source Book · 1938

Mahāvīra's Gaṇita-sāra-saṅgraha (850 CE) poses the hundred-birds problem: pigeons at 5 for 3 paṇas, sārasas at 7 for 5, swans at 9 for 7, peacocks at 3 for 9 — buy 100 birds for 100 paṇas: two equations, four unknowns, integer solutions required. The same problem-type appears in Zhang Qiujian's Chinese classic (c. 475 CE, priority) and Alcuin's Latin puzzles (c. 800) — a marker problem for the circulation of mathematics across Eurasia.

History of Hindu Mathematics — A Source Book · 1938

Bhāskara II's Bījagaṇita §139 (1150 CE, Colebrooke trans.) poses (x/8)² + 12 = x and derives both roots, x = 48 and x = 16, noting both satisfy the conditions. Companion problems show the other case: where a root is negative or otherwise inconsistent with the problem, it is declared incongruous and dropped — "people do not approve a negative absolute number." Two-root awareness plus root-validity screening, standard curriculum in 1150; Descartes was still calling negative roots "false" in 1637.

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628

Sūrya-Siddhānta xiii (the "astronomical upaniṣad" chapter) directs the teacher to build an armillary sphere — an earth-globe ringed by the circles of the asterisms and ecliptic — explicitly "in order to the instruction of the pupil," then covers other instruments, especially for timekeeping (xiii.17–25). Burgess notes Indian practice paired a meridian circle with the clepsydra, closely analogous to later Western method: the hardware behind the siddhānta's precision parameters.

Translation of the Surya-Siddhanta · 400

Pañcasiddhāntikā XII (505 CE) preserves the Paitāmaha Siddhānta: a five-year luni-solar calendar of 1,830 civil days, intercalating every 30 months, with its epoch at the asterism Dhaniṣṭhā — the winter-solstice marker of the Vedāṅga Jyotiṣa tradition, datable by precession to c. 1400–1200 BCE. Varāhamihira transmits the system faithfully while ranking it "far from the truth" — the Indian canon documenting and superseding its own oldest astronomy.

The Panchasiddhantika: The Astronomical Work of Varaha Mihira · 575

Pañcasiddhāntikā XIV (505 CE) teaches graphical methods: construct a degree-marked circle of 180 aṅgulis with auxiliary declination circles and strings, from which the ascensional difference for any latitude — and related rising-time quantities — are read directly off the figure. Thibaut's commentary describes the procedure as finding the result "without calculation, by the mere inspection of a kind of diagram": a worked analog-computing device inside a 6th-century astronomy curriculum.

The Panchasiddhantika: The Astronomical Work of Varaha Mihira · 575

The Sadratnamālā of Śaṅkara Varman (Kerala, 1819) gives the circumference of a circle of diameter 10¹⁷ parts as 314,159,265,358,979,324 — π correct to seventeen figures — encoded in one verse via the kaṭapayādi consonant-to-digit cipher and computed with the Kerala school's series methods. Whish reported it to the Royal Asiatic Society in 1834. Europe held longer digit records by then; the claim is the encoding and the unbroken lineage, not the record.

On the Hindu Quadrature of the Circle, and the Infinite Series of the Proportion of the Circumference to the Diameter Exhibited in the Four Sastras · 1500

Classical Sanskrit mathematics and astronomy used bhūta-saṁkhyā ("object numerals"): numbers written as words — candra (moon) = 1, netra (eyes) = 2, agni (fires) = 3, sāgara (oceans) = 4 — arranged by place value. Datta & Singh (1938) document the system's rationale: scientific works were metrical, and word numerals with many synonyms per digit let any number be versified. One number could be written hundreds of ways; the convention remains in use for numbers in Sanskrit verse.

History of Hindu Mathematics — A Source Book · 1938

Sūrya-Siddhānta iv.1 gives the Moon's diameter as 480 yojanas; with the Earth's diameter fixed at 1,600 yojanas (i.59), the implied Moon-to-Earth size ratio is 0.30, against a true value of 0.27 — accurate to about ten percent. The same verse's solar diameter (6,500 yojanas ≈ 4 Earth-diameters) is too small by a factor of ~27: lunar parallax was within naked-eye reach and solar parallax was not. The accuracy tracks the observable.

Translation of the Surya-Siddhanta · 400

The Līlāvatī of Bhāskara II (1150 CE) is an arithmetic and geometry textbook composed in Sanskrit verse, with word problems addressed to a woman — by tradition Bhāskara's daughter Līlāvatī. Colebrooke §54: a swarm of bees splits into fifths and thirds among named flowers, one bee hovers between a jasmine and a pandanus; find the swarm. The book stayed the subcontinent's standard mathematics text for roughly 700 years and was translated into Persian at Akbar's court (Fyzī, 1587).

Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara · 628