Chapter 12

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.)

If 8 braccia of cloth are worth 11 florins, what are 97 braccia worth?

I have 25 pounds of silver alloy that contain 8 ounces of pure silver per pound and 16 pounds that have \(9 \frac{1}{2}\) ounces of silver per pound. How much copper must be added to the total so that I can make coins containing \(7 \frac{1}{2}\) ounces of silver per pound?

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?

Divide 10 into two parts such that if one squares the first, subtracts it from 97, and takes its square root, then squares the second, subtracts it from 100 , and takes its square root, the sum of the two roots is 17. (This problem is also from the work of Antonio de' Mazzinghi. Mazzinghi set the parts \(u, v\) equal to \(5+x\) and \(5-x\), respectively, and derived an equation in \(x .\) )

Maestro Dardi gave a rule to solve the fourth-degree equation \(x^{4}+b x^{3}+c x^{2}+d x=e\) as \(x=\sqrt[4]{(d / b)^{2}+e}-\) \(\sqrt{d / b}\). His problem illustrating the rule is the following: A man lent 100 lire to another and after 4 years received back 160 lire for principal and (annually compounded) interest. What is the interest rate? As in the text's example, set \(x\) as the monthly interest rate in denarii per lira. Show that this problem leads to the equation \(x^{4}+80 x^{3}+2400 x^{2}+32,000 x=96,000\) and that the solution found by "completing the fourth power" is given by the stated formula.

In a vessel full of wine there are three taps such that if one opens the largest it will empty the vessel in 3 hours, if one opens the middle tap it will empty it in 4 hours, and if one uses the smallest tap it will empty it in 6 hours. How long would it take to empty the vessel if all three taps were open? (This problem and the next are also from Chuquet's work.)

A man makes a will and dies leaving his wife pregnant. His will disposes of 100 écus such that if his wife has a daughter, the mother should take twice as much as the daughter, but if she has a son, he should have twice as much as the mother. [Sexist problem!] The mother gives birth to twins, a son and a daughter. How should the estate be split, respecting the father's intentions?

Express \(\sqrt{27+\sqrt{200}}\) as \(a+\sqrt{b}\). (This problem and the next two are from Rudolf's Coss.)

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?

Divide 10 into two parts such that their product is \(13+\) \(\sqrt{128}\)

This problem is from Stifel's Arithmetica integra. In the sequence of odd numbers, the first odd number equals \(1^{5}\). After skipping one number, the sum of the next four numbers \((5+7+9+11)\) equals \(2^{5}\). After skipping the next three numbers, the sum of the following nine numbers \((19+21+\) \(23+25+27+29+31+33+35\) ) equals \(3^{5}\). At each successive stage, one skips the next triangular number of odd integers. Formulate this power rule of fifth powers in modern notation and prove it by induction.

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.)

There is a strange journey appointed to a man. The first day he must go \(1 \frac{1}{2}\) miles, and every day after the first he must increase his journey by \(\frac{1}{6}\) of a mile, so that his journey shall proceed by an arithmetical progression. And he has to travel for his whole journey 2955 miles. In what number of days will he end his journey?

Show that if \(r, s\), are two positive roots of \(x^{3}+d=c x\), then \(t=r+s\) is a root of \(x^{3}=c x+d\).

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 \(x^{3}+3=8 x\).

Prove that the equation \(x^{3}+c x=d\) always has one positive solution and no negative ones.

Use Cardano's formula to solve \(x^{3}+3 x=10\).

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

Solve \(x^{3}+21 x=9 x^{2}+5\) completely by first using the substitution \(x=y+3\) to eliminate the term in \(x^{2}\) and then solving the resulting equation in \(y\).

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?

The dowry of Francis's wife is 100 aurei more than Francis's own property, and the square of the dowry is 400 more than the square of his property. Find the dowry and the property. (Note the negative answer for Francis's property; Cardano interpreted this as a debt.)

Find two numbers \(x, y\), with \(x>y\) such that \(x+y=\) \(y^{3}+3 y x^{2}\) and \(x^{3}+3 x y^{2}=x+y+64\). (This problem and the next are from Ferrari's contest with Tartaglia. Tartaglia's solution is $$ x=\sqrt[3]{4+\sqrt{15 \frac{215}{216}}}+\sqrt[3]{4-\sqrt{15 \frac{215}{216}}}+2 $$ while \(y=x-4\).)

Divide 8 into two parts \(x, y\), such that \(x y(x-y)\) is a maximum. (Note that this was posed in the days before calculus.)

Divide 8 into two parts \(x, y\), such that \(x y(x-y)\) is a maximum. (Note that this was posed in the days before calculus.)

Given a right triangle with base \(D\), perpendicular \(B\), and hypotenuse \(Z\), and a second right triangle with base \(G\), perpendicular \(F\), and hypotenuse \(X\), show that the right triangle constructed in the text in Viète's work with base \(D G-B F\), perpendicular \(B G+D F\), and hypotenuse \(Z X\) has its base angle equal to the sum of the base angles of the original triangles.

Given the product of two numbers and their ratio, to find the roots: Let \(A, E\), be the two roots, \(A E=B, A: E=\) \(S: R\). Show that \(R: S=B: A^{2}\) and \(S: R=B: E^{2}\). Viète's example has \(B=20, R=1, S=5\). Show in this case that \(A=10\) and \(E=2\). (Jordanus has the same problem but with different numbers.)

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.

Why is Cardano's formula no longer generally taught in a college algebra course? Should it be? What insights can it bring to the study of the theory of equations?

Why is Cardano's formula no longer generally taught in a college algebra course? Should it be? What insights can it bring to the study of the theory of equations?

The first printed mathematics book is the so-called Treviso Arithmetic of 1478, by an unknown author. Write a brief essay on its contents and its importance. Consult Frank J. Swetz, Capitalism and Arithmetic, from note \(37 .\)

Why was the knowledge of mathematics necessary for the merchants of the Renaissance? Did they really need to know the solutions of cubic equations? What, then, was the purpose of the detailed study of these equations in the works of the late sixteenth century?