Ancient results, running live

Aryabhata's sine table Heritage only

In 499 CE the Aryabhatiya (Gitikapada 12) gave 24 sine differences for steps of 3¾°, on a circle of radius 3438 — the number of arc-minutes in a radian, rounded. Add them up and you have a sine table. Here it is against the sine your device computes.

Source: W. E. Clark, The Aryabhatiya of Aryabhata (1930), public domain in the US since 2026. Aryabhata's π, 62832/20000, is shown below with Ramanujan's.

Kuttaka — "the pulverizer" Heritage only

Aryabhata's method for whole-number equations (Ganitapada 32–33) is what we now call the extended Euclidean algorithm. Astronomers used it to work back from planetary positions to the number of days elapsed since the epoch. Solve a·x ≡ c (mod m):

Preset: Bhaskara II's Lilavati asks for the multiplier of 221 which, with 65 added, divides exactly by 195 — that is, 221·x ≡ −65 (mod 195).

Kautilya's clock Heritage only

Arthashastra II.20 sets out time from the truti to the day: 2 truti = 1 lava, 2 lava = 1 nimesha, 5 nimesha = 1 kashtha, 30 kashtha = 1 kala, 40 kala = 1 nalika, 2 nalika = 1 muhurta. A day and night is 30 muhurtas. Here is the time since today's sunrise at Ujjain, told his way:

R. Shamasastry's translation (1915), public domain. In Book I ch. 19 the same translation also calls the king's eight daily divisions "nalikas" of 1½ hours — a different use of the word, kept apart here.

Katapayadi — numbers hidden in words Heritage only

Kerala's astronomers encoded numbers as syllables: ka ta pa ya start the groups that count 1, 2, 3…; a vowel alone is 0; in a cluster only the last consonant counts. Numbers were written lowest digit first. Type Devanagari, or try a preset:

The 72 melakarta ragas of Carnatic music are numbered this way from the first two syllables of their names. Earliest dated astronomical use: Haridatta's Parahita system (traditionally 683 CE).

The Saptarshi era Heritage only

Kashmir counted years from about 3076 BCE on the belief that the Seven Sages (the Big Dipper) spend a century in each nakshatra (Vishnu Purana IV.24; Brihat Samhita 13). Kalhana dated his Rajatarangini to Laukika 4224 — 1148–49 CE — which our formula reproduces. The stars do not actually move through the nakshatras this way; the calendar is a tradition, and we say so.

π: Aryabhata to Ramanujan Heritage only

Ramanujan's 1914 series adds about eight correct digits of π with every term. It powers no planetary position — an ephemeris needs only the sixteen digits every computer already has — but it is one of the most beautiful results in mathematics, and its descendant (the Chudnovsky algorithm) is how π records are set.

Aryabhata (499 CE): π ≈ 62832 / 20000 = 3.1416 — about 2.3 parts per million high, and he called it "approximate". Madhava of Sangamagrama (c. 1400) found an infinite series for π more than two centuries before Europe did.

Why calendars cycle: continued fractions The idea behind every lunisolar calendar

A year is not a whole number of months — 12.3683 of them. Continued fractions give the best whole-number cycles that fit, which is why 19 years ≈ 235 months (the Metonic cycle) and why the Hindu lunisolar calendar adds an adhika masa about 7 times in 19 years.