The Book of Birds

Paul Klee, "Landscape with Yellow Birds" (1923)

Paul Klee, “Landscape with Yellow Birds” (1923)

It is rather uncommon for a treatise in Mathematics to bear a title seemingly so irrelevant to Mathematics such as Abu Kamil’s “Book of Birds” (Kitab al-tair, 9th century). Abu Kamil (c. 850 – c. 930) was an ingenious muslim mathematician, with significant contributions in expanding the accepted range of numbers into the vast universe of the irrationals. In his “Book of Birds” he confines his study within the integers and examins six variations of the “100 fowls problem”, a type of “bird purchase” problem which can be first traced to Zhang Quijian’s (or Chang Ch’iu Chien) “Mathematical Manual” (a Chinese mathematical text, dating probably from the 5th century) and became widespread in the East during the few next centuries. Variations of the problem were later found in texts of Alcuin of York (8th century), Leonardo of Pisa (better known today as Fibonacci, 12th – 13th century) and the calendar maker Caspar Thierfelder (16th century), though not always involving birds. The problem is about finding all the possible ways in which a certain amount of money (usually 100 units) can be spent for buying a certain number of whole items (such as birds, usually 100 of them) of a few different kinds (usually three or four kinds) if each of these kinds has a different price. The following is the problem 38 in chapter 3 of Zhang Quijian’s text:

If a rooster is worth 5 qian, a hen is worth 3 coins and three chickens together cost 1 qian, then how many roosters, hens and chickens, 100 in total, can be bought for 100 qian?

Problems such as this are reduced to systems of linear Diophantine equations with infinite solutions, though only a specific number of them are acceptable under the conditions that all answers must be positive and the total of birds must be 100. Working on such problems, Abu Kamil was astonished by a specific problem that yielded 2676 solutions and, as he writes in the introduction of “The Book of Birds”, his discovery was accepted with suspicion and arrogance by some. His treatise was an effort to present his calculation in detail (though his Algebra was rhetorical)  “with the purpose of facilitating its treatment and making it more accessible“. One of the more demanding of his 6 problems is the following:

With 100 dirhams buy 100 birds of 4 kinds: geese at 4 dirhams each, chickens at 1 dirham each, pigeons at 1 dirham for two and starlings at 1 dirham for ten

This specific problem possesses 98 solutions, which are discussed here.