"[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity which is the kernel of modern classical analysis."

"The radius into the sine divided by the cosine is the first quote: this multiplied by the square of the sine, and divided by the square of the cosine, is the second quote; this second, and those obtained continually in the same way, multiply and divide by the square of the sine and the square of the cosine respectively: divide the quotes in order by 1, 3, 5, 7,11, etc. respectively, and the difference of the sum of the first, third, fifth, etc. and of the second, fourth, sixth, etc., will be the arc whose sine was taken."

"Square the diameter and multiply the product by 12, and extract the root of this product; the root obtained will be the modulus of odd quotes, which if you divide by 3, the quotient will be the modulus of even quotes. Divide each modulus continually by 9, and the quotient thus obtained from the former, divide by double the numbers 1, 3, 5, 7, 9, etc. minus 1 respectively, and the quotient obtained by the latter, by double the number 2, 4, 6, 8, 10, etc. minus 1 respectively, add up the new obtained quotes, and subtract the sum of those gotten from the even from the sum of those gotten from the odd modulus, the remainder is the circumference of the circle. Square the diameter and multiply the product by 12, and extract the root of this product; this root divide continually by 3, and the quotients thus obtained by 1, 3, 5, 7, 9, 11, etc., and subtract the sum of the second, fourth, sixth, eighth of the last obtained quotes from the sum of the first, third, fifth, seventh, ninth, etc. If you do thus, and measure the diameter of a great circle by 100000000000000000 equal parts, the circumference will be equal to 314159265358979324 of such parts."

"The diameter multiplied by four and divided by unity (is found and saved). Again the products of the diameter and four are divided by the odd numbers like three, five, etc., and the results are subtracted and added in order."

"Why is it that the actual value is left out and this very near value stated? Let me say. It is impossible to state the actual value. Why? That unit which leaves no remainder when the diameter is measured will leave a remainder if used again for measuring circumference. Likewise, the unit which leaves no remainder in the measure of the circumference will leave a remainder in the diameter if measured by the same unit. Hence if both (the diameter and circumference) are measured by the same unit, a remainderless state is never attained. Even if this is carried out farther to a great extent only diminution of the remainder can be obtained but absence of remainder can never be obtained— this is the meaning."

"One has to realize that the five siddhantas [i.e. astronomical systems] had been correct at a particular time. Therefore, one should search for a siddhanta that does not show discord with actual observations (at the present time). Such accordance with observation has to be ascertained by (astronomical) observers during times of eclipses etc. When siddhantas show discord, that is, when an earlier siddhanta is in discord, observations should be made of revolutions etc. (which would give results in accord with actual observations) and a new siddhanta enunciated."

"One has to accept that [each of ] the five siddhantas had been authoritative at one time [though they might not be so now]. Therefore one has to look for a system which tallies with observation. The said tallying has to be verified by contemporary experimenters at the time of eclipses etc."

"A part of a circle is of the form of a bow, so it is called the ‘bow’ (dhanu). The straight line joining its two extremities is the ‘bow-string’ (jiva). It is really the ‘full-chord’ (samasta-jya). Half of it is here (called) the ‘half-chord’ (ardha-jya), and half that arc is called the ‘bow’ of that half-chord. In fact the Rsine (jya) and Rcosine (kotijya) of that bow are always half chords. [24]"