91Ó°ÊÓ

The gold florin is worth 5 lire 12 soldi, 6 denarii in Lucca. How much (in florins) are 13 soldi, 9 denarii worth? (Note that 20 soldi make 1 lira and 12 denarii make 1 soldo.)

This problem is from the Treviso Arithmetic, the first printed arithmetic text, dated 1478: The Holy Father sent a courier from Rome to Venice, commanding him that he should reach Venice in 7 days. And the most illustrious Signoria of Venice also sent another courier to Rome, who should reach Rome in 9 days. And from Rome to Venice is 250 miles. It happened that by order of these lords the couriers started their journeys at the same time. It is required to find in how many days they will meet, and how many miles each will have traveled \({ }^{37}\)

Of three workmen, the second and third can complete a job in 10 days. The first and third can do it in 12 days, while the first and second can do it in 15 days. In how many days can each of them do the job alone?

A fountain has two basins, one above and one below, each of which has three outlets. The first outlet of the top basin. fills the lower basin in two hours, the second in three hours, and the third in four hours. When all three upper outlets are shut, the first outlet of the lower basin empties it in three hours, the second in four hours, and the third in five hours. If all the outlets are opened, how long will it take for the lower basin to fill?

I am owed 3240 florins. The debtor pays me 1 florin the first day, 2 the second day, 3 the third day, and so on. How many days does it take to pay off the debt?

There is a certain army composed of dukes, earls, and soldiers. Each duke has under him twice as many earls asthere are dukes. Each earl has under him four times as many soldiers as there are dukes. The 200th part of the number of soldiers is 9 times as many as the number of dukes. How many of each are there? (This problem and the next two are from Recorde's The Whetstone of Witte.)

Show that if \(t\) is a root of \(x^{3}=c x+d\), then \(r=t / 2+\) \(\sqrt{c-3(t / 2)^{2}}\) and \(s=t / 2-\sqrt{c-3(t / 2)^{2}}\) are both roots of \(x^{3}+d=c x\). Apply this rule to solve \(x^{3}+3=8 x\).

Use Cardano's formula to solve \(x^{3}=6 x+6\)

Use Ferrari's method to solve the quartic equation \(x^{4}+\) \(4 x+8=10 x^{2}\). Begin by rewriting this as \(x^{4}=10 x^{2}-\) \(4 x-8\) and adding \(-2 b x+b^{2}\) to both sides. Determine the cubic equation that \(b\) must satisfy so that each side of the resulting equation is a perfect square. For each solution of that cubic, find all solutions for \(x\). How many different solutions to the original equation are there?

Write \(13.395\) and \(22.8642\) in Stevin's notation. Use his rules to multiply the two numbers together and to divide the second by the first.