04 July 2011

A problem

A gentleman has given his daughter in marriage, and to complete a necklace he needs 100 stones made up of pearls, rubies, sapphires, and balas rubies. He summons his agent and gives him 100 ducats and says: go to Genoa and lay out these 100 ducats in buying pearls, rubies, sapphires, and balas rubies; and make the total number 100 and do not spend more than one-third of a ducat per pearl and one-half per ruby and one per sapphire and three per balas ruby. I ask how many pearls, how many rubies, [how many sapphires,] how many balas rubies he will have.
This is a problem in Piero della Francesca's Trattato d'abaco. Oh, what a problem to have. The answer is 51 pearls, 8 rubies, 22 sapphires, 19 balas rubies. It is solved by the method of double false position, which Piero explains by solving another problem:
There is a plain on which there are two towers, one 40 bracci high, the other 50 bracci, and from one tower to the other it is 100 bracci. And on each tower there is a bird, which birds, flying at the same speed, set off at the same time to drink; and they arrive at the same moment at a fountain that is between one tower and the other. I ask how far the fountain is from the tower that is 40 bracci and how far from the tower that is 50.
What you do is assign a length to the fountain from the first tower (say 60 bracci) and calculate the flight path of the first bird (or rather its square) using Pythagoras' theorem. You then calculate the length to the second tower and the square of the flight path of the second bird. The two squares should be equal, because the birds arrive at the same time, but they probably won't be, because you're guessing.

You note the difference between the squares of the flight paths (1100) and make another guess of the length from the first tower (55 bracci). You then find the difference between the squares again (100) and use the two guesses and the two differences to calculate the correct lengths:
The excesses must be taken one from another; take 100 from 1100, there remains 1000, which is the divisor; now multiply 55 by 1100 [which] makes 60500, and multiply 60 by 100 [which] makes 6000, take this from 60500 there remains 54500, which you divide by 1000 and the result is 54 and one-half. And that is the distance from the fountain tower of 40 bracci, and [so] 45 and one-half from the fountain to the tower of 50 bracci.
This is all standard fare, derived from Islamic sources. Piero's big contribution in the Trattato d'abaco, which he casually introduces with no fanfare (just like how he casually provides the first mathematical proof of perspective in De prospectiva pingendi), is the first rediscovery of Archimedean solids:

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