9th-century-indian-jains

"O! tender girl, out of the swans in a certain lake, ten times the square root of their number went away to Manasarovara when the rainy season arrived, one eighth of that number went away to the Sthala Padmini forest. Three pairs of swans remained in the tank, sporting in the water. What is the total number of swans?"

"The first known mention of the geometric series is due to Mahavira (850 CE) in his Ganitasarasangraha (ii.93): पदमितगुणहतिगुणितप्रभवस््स्ययाद्णगु धनं तदाद्यूनम।् एकोनगुणविभक्तं गुणसङ्कलितं विजानीयात्॥ which translates to The first term when multiplied by the continued product of the common ratio, taken as many times as the number of terms, gives rise to the gunadhana [the (n+1)th term of the geometric progression]. And it has to be understood that this gunadhana, when diminished by the first term and then divided by the common ratio lessened by one, becomes the sum of the series in the geometric progression."

"Beginning with one and increasing by one, let the numbers going up to the given number of things be written down in regular order and in the inverse order (respectively) in an upper and a lower (horizontal) row. (If ) the product (of one, two, three, or more of the numbers in the upper row) also taken fromright to left, (the quantity required in each such case of combination) is (obtained as) the result."

"The science of numbers is used in all transactions whether earthly or Vedic, meditation or business, in the science of love, in the art of cooking, in economics and statecraft, in music and medicine, in theatre, architecture and all such things. (9-10) In prosody, in poetics and poetry, in logic, grammar and such like, in the presentation of the qualities of various arts, mathematics is held in esteem. For the movements of the sun and stars, for eclipses and planetary motion, for the three significant questions and the course of the moon, for all these connections it is used (11-12). The number, diameter and perimeter of islands, of oceans and mountains, for the dimensions of rows of buildings of the earth’s inhabitants, of the space between the worlds, of the world of light, for the worlds of devas, for the dimensions of the dwellers in hell; for measurements of all sort and voluminous evidence, all these are made out by means of computation (13- 14). The organization of living beings, the length of their lives, their eight attributes and other similar things, their travel and cohabitation and such other things, all depend on mathematics. What more can be said? All there is in the three worlds, moving or unmoving, all that cannot be described without mathematics (15-16)."