The Man of Numbers by Keith Devlin

Author:Keith Devlin
Language: eng
Format: epub
Publisher: Walker Books
Published: 2011-03-25T16:00:00+00:00

Hence, if xn is an approximation of the solution that is too high (respectively, too low), then

is an approximation that is too low (high). It follows that the average of these two approximations

is a better one. Thus, starting with an initial approximation of x0 = 1.5 (say), one calculates a sequence of approximations x0, x1, x2,…, xn,… which fairly quickly reaches an acceptably accurate approximation. Computing to fifteen decimal places of accuracy, this process yields the values:

and the final approximation in this list turns out to be correct to seven decimal places, after just twelve steps.

Leonardo performed the calculation using sexagesimal fractions (i.e., fractions expressed in base 60), following the practice used by astronomers since Ptolemy. In the sexagesimal notation he was using, the answer he obtained was 1022′ 7″42′″ 33IV4V40VI. In this notation, 22′ is 22⁄60, 7″ is 7⁄3600, 42′″ is 42⁄216,000, and so forth, each successive fraction being expressed as a higher power of 60. Expressed in words, as Leonardo presented it to the court, this reads: “One unit, 22 in the first fractional part, 7 in the second, 42 in the third, 33 in the fourth, 4 in the fifth, and 40 in the sixth.”

In modern decimal notation, Leonardo’s solution is 1.3688081075, which is correct to nine decimal places, a result that is far more accurate than the answer to the same problem that had been obtained (using the same method) by Arab mathematicians who had solved it previously (and more accurate than the answer presented above that was obtained using modern algebra and a computer spreadsheet).

The third problem Leonardo solved was the easiest of the three, being a computation where the unknown quantity is not raised to any power. (In modern parlance, it involves only linear equations.) Liber abbaci was full of such problems, though of course Johannes had chosen one that did not appear in Leonardo’s own book.

Three men owned a store of money, their shares being ½, 1⁄3, and 1⁄6. But each took some money at random until none was left. Then the first man returned ½ of what he had taken, the second 1⁄3, the third 1⁄6. When the money now in the pile was divided equally among the men, each possessed what he was entitled to. How much money was in the original store, and how much did each man take?

Leonardo solved the problem using the Direct Method. He began his solution, as subsequently recorded in Flos, by pointing out that

If you take away half of anything, you have an equal half left; similarly, if you take away a third, that third is half of the remaining two-thirds; likewise, if you take away a sixth, that sixth is a fifth of the remaining five-sixths.

Let us use the term res for the amount each man received when the pile of money was divided equally among them. Then it follows that after the three men had returned the given portions of their money, the first one had half of the money in the original store minus res.

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