孫子算經（選）孙子算经（选）

Master Sun says: calculation is the warp and weft of heaven and earth, the head of all living things, the root and branch of the five constant virtues, the father and mother of yin and yang, the giver of names to the stars and constellations, the outside and inside of the three lights, the level of the five phases, the beginning and end of the four seasons, the ancestor of the ten thousand things, and the guiding thread of the six arts. With it we examine how all kinds gather and disperse, study how the two breaths rise and fall, follow the turning of cold and heat, measure how far and near differ, observe the subtle beginnings of the way of heaven, survey the lengths of the land from north to south and east to west, discover where the spirits dwell, and test to the end the signs of success and failure. It reaches to the principles of the Way and virtue, and fathoms the nature of life. It sets up the compass and square, makes circles and squares true, governs the laws and measures, fixes feet and yards, establishes the steelyard and balance, levels heavy and light, divides the hair's breadth and splits the smallest grain. It will last through hundreds of millions of years without decay, and reach the eight ends of the earth without limit. Spread out, it can never be exhausted; gathered in, it does not fill the palm of a hand. Those who turn towards it will be rich and have more than enough; those who turn their backs on it will be poor and in want. To an open mind it is clear even in childhood; to a closed mind it remains hard to master even when the hair is white. Whoever wishes to study it must measure his abilities, know himself, and keep his will fixed on it. If he does this, how could he fail?

Book Three#

Suppose there are some things whose number is not known. Counted in threes, two are left over; counted in fives, three are left over; counted in sevens, two are left over. How many things are there? Answer: twenty-three.

Method: counted in threes with two left over, put down one hundred and forty; counted in fives with three left over, put down sixty-three; counted in sevens with two left over, put down thirty. Add these together to get two hundred and thirty-three, subtract two hundred and ten, and you have the answer. In general, for each one left over when counting in threes, put down seventy; for each one left over when counting in fives, put down twenty-one; for each one left over when counting in sevens, put down fifteen. If the sum is one hundred and six or more, subtract one hundred and five, and you have the answer.

Suppose pheasants and rabbits are in the same cage. Above there are thirty-five heads; below there are ninety-four feet. How many pheasants and how many rabbits are there? Answer: twenty-three pheasants and twelve rabbits.

Method: put down thirty-five heads above and ninety-four feet below. Halve the feet to get forty-seven. Subtract the smaller from the larger, twice over: the three above removes three below, the five above removes five below; then the one below removes one above, the two below removes two above, and you have the answer.

Another method: put down the heads above and the feet below. Halve the feet, take the heads away from the feet, then take what remains of the feet away from the heads, and you have the answer.