Search Loci: Convergence:
Mathematics is not a deductive science -- that's a cliche. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
I Want to Be a Mathematician, Washington: MAA Spectrum, 1985.
The Great Calculation According to the Indians, of Maximus Planudes
In regard to astronomy we have then the need of six types of operations13, of which the first is that referred to as numeration, 14 and then addition, subtraction, multiplication and division, while the sixth is the extraction of the square root15 of each (given) number. There has already been mention made of the representations, so let us turn our attention to the method of addition.
We begin then with 3 and 7; 3 and 7 gives 10, write the cipher above, and carry17 the unit, which indicates ten. Again, 4 and 8 gives 12, add in18 also the unit which you carried and we get19 13. We again take the unit which indicates ten and write the 3 above. Now 3 and 6 gives 9, add in the unit you carried and we get 10. Write the cipher above and keep the unit; now say 2 and 5 makes 7, add in the unit you held and we get 8 and write this above. Thus the sum of five thousand six hundred and eighty seven and two thousand three hundred and forty three is eight thousand and thirty.
For the sake of clarity, let us demonstrate on the previous example. We say that 8 and 3 is 11, take away 9 leaves [2. Now 5 and 6 is 11, take away 9 leaves 2, 2 and 8 is 10, take away 9 leaves]21 1. And 1 and 7 is 8 and so write 8 in turn in the middle row. Again 2 and 3 is 5, 5 and 4 is 9, take away 9 and the resulting number 3 remains. Write this in turn after the numbers added together. Now 3 and 8 is 11, take away 9 and 2 is left over. You then have the number equal to the number from the previous check and so the addition is correct.
So much for addition.