Chapter 10

This problem and the next two are from Alcuin's Propositions for Sharpening Youths. \(^{37} \mathrm{~A}\) cask is filled to \(100-\) metreta capacity through three pipes. One-third of its capacity plus 6 modii flows in through one pipe; one-third of its capacity flows in through another pipe; but only onesixth of its capacity flows in through the third pipe. How many sextarii flow in through each pipe? (Here a metreta is 72 sextarii and a modius is 200 sextarii.)

A man must ferry a wolf, a goat, and a head of cabbage across a river. The available boat, however, can carry only the man and one other thing. The goat cannot be left alone with the cabbage, nor the wolf with the goat. How should the man ferry his three items across the river?

A hare is 150 paces ahead of a hound that is pursuing him. If the hound covers 10 paces each time the hare cover 6 , in how many paces will the hound overtake the hare?

If a chord of length 8 has distance 2 from the circumference, find the diameter of the circle.

The Artis cuiuslibet consummatio claimed that the formula \(A=\frac{3 n^{2}-n}{2}\) gave the area of a pentagon of side \(n\). Show, instead, that it provides a formula for the \(n\)th pentagonal number. Calculate the area of regular pentagons with sides. of length \(n=1,2,3\), and compare your answer to the value of the \((n+1)\) st pentagonal number. How close an approximation does the given formula provide?

Prove this theorem from Levi ben Gerson's Trigonometry: If all sides of any triangle whatever are known, its angles are also known. Start by dropping a perpendicular from one

Prove Proposition 33 of the Maasei Hoshev: $$ \begin{aligned} &(1+2+3+\cdots+n)+(2+3+\cdots+n) \\ &\quad+(3+\cdots+n)+\cdots+n \\ &\quad=1^{2}+2^{2}+\cdots+n^{2} \end{aligned} $$

Prove Proposition 34 of the Maasei Hoshev: $$ \begin{aligned} [(1+2+\cdots+n)+(2+3+\cdots+n)+\cdots+n] \\ \quad+[1+(1+2)+\cdots+(1+2+\cdots+(n-1))] \\ =n(1+2+\cdots+n) \end{aligned} $$

One of the problems from the Maasei Hoshev: A barrel has various holes: The first hole empties the full barrel in 3 days; the second hole empties the full barrel in 5 days; another hole empties the full barrel in 20 hours; and another hole empties the full barrel in 12 hours. All the holes are opened together. How much time will it take to empty the barrel?

Recall that Jordanus used the Pascal triangle in Proposition IX-70 of the Arithmetica to determine series of numbers in continued proportion. Namely, beginning with the series \(1,1,1,1, \ldots\), he derived first the series \(1,2,4\) \(8, \ldots\), and by using those terms, he derived the series \(1,3,9,27, \ldots\) Now use this latter series in the same way to derive the series \(1,4,16,64, \ldots\). Formulate and prove by induction a generalization of this result.

This problem and the next six are taken from the Liber abbaci. One roll of saffron is sold for 3 bezants and \(7 \frac{1}{4}\) mils (where there are 10 mils in a bezant). How much are 17 rolls and \(5 \frac{1}{2}\) ounces worth (where there are 12 ounces to the roll)?

A Genoese solidus is sold for \(21 \frac{1}{2}\) Pisan denarii. How much are 7 Genoese solidi and 5 denarii worth in Pisan money? (Recall that 1 solidus equals 12 denarii.)

If an Imperial solidus is sold for \(31 \frac{1}{2}\) Pisan denarii, and a Genoese solidus is worth \(19 \frac{3}{4}\) Pisan denarii, then how many Genoese pounds will one have for 17 Imperial pounds, 11 solidi, and 5 denarii? (One pound equals 20 solidi. Note, that the exchange rate between Pisan and Genoese money is different in this exercise from that stated in the previous exercise.)

If a lion eats one sheep in 4 hours, a leopard eats one sheep in 5 hours, and a bear eats one sheep in 6 hours, how long would it take the three animals together to devour one sheep? (Begin by supposing that the answer is 60 , the least common multiple of \(4,5,6\).)

Two men have some denarii. The first said to the second, if you will give me one of your denarii, then mine will equal yours. The other responded, and if you will give me one of your denarii, then I will have ten times as many as you. How many does each man have?

Solve this problem discussed in the text: There are five men. with money who have found a purse with additional money. The amount the first has together with the amount in the purse is \(2 \frac{1}{2}\) times the total of the amounts held by the other four. Similarly, the second man's amount together with the amount in the purse is \(3 \frac{1}{3}\) times the total held by the others. Analogously, the fraction is \(4 \frac{1}{4}\) for the third \(\operatorname{man}, 5 \frac{1}{5}\) for the fourth man, and \(6 \frac{1}{6}\) for the fifth man. Find the amounts of money that each man had originally as well as the amount in the purse. (Note that Leonardo found that the first man actually had a debt of \(49,154 .\) )

The Fibonacci sequence (the sequence of rabbit pairs) is determined by the recursive rule \(F_{0}=F_{1}=1\) and \(F_{n}=\) \(F_{n-1}+F_{n-2}\). Show that $$ F_{n+1} \cdot F_{n-1}=F_{n}^{2}-(-1)^{n} $$ and that $$ \lim _{n \rightarrow \infty} \frac{F_{n}}{F_{n-1}}=\frac{1+\sqrt{5}}{2} $$

From the Book of Squares: Find a square number for which the sum of it and its root is a square number and for which the difference of it and its root is similarly a square number. (In modern notation, find \(x, y, z\), such that \(x^{2}+x=z^{2}\) and \(x^{2}-x=y^{2}\). Leonardo began his solution by using the congruous number 24 in solving \(a^{2}+24=b^{2}, a^{2}-24=\) \(c^{2}\); he then divided everything by \(24 .\) )

If the sum of the two quotients formed by dividing the two parts of a given number by two different known numbers is given, then each of the parts is determined. Namely, solve the system \(x+y=a, x / b+y / c=d\). Jordanus sets \(a=10, b=3, c=2\), and \(d=4\).

If the sum of two numbers is given together with the product, of their squares, then each of them is determined. Jordanus's example is \(x+y=9, x^{2} y^{2}=324\).

Show that under the assumptions of the mean speed theorem, if one divides the time interval into four equal subintervals, the distances covered in each interval will be in the ratio \(1: 3: 5: 7\). Generalize this statement to a division of the time interval into \(n\) equal subintervals and prove your result.

From Oresme's Tractatus de configurationibus qualitatum et motuum: Show geometrically that the sum of the series$$ \begin{aligned} &48 \cdot 1+48 \cdot \frac{1}{4} \cdot 2+48 \cdot\left(\frac{1}{4}\right)^{2} \cdot 4+\cdots \\ &+48\left(\frac{1}{4}\right)^{n} \cdot 2^{n}+\cdots \end{aligned} $$ is equal to \(96 .\)

Solve the following problem of Oresme: Divide the line \(A B\) of length 1 (representing time) proportionally to infinity in a ratio of \(2: 1\); that is, divide it so the first part is one-half, the second one-quarter, the third one-eighth, and so on. Let there be a given finite velocity (say, 1 ) in the first interval, a uniformly accelerated velocity (from 1 to 2 ) in the second, a constant velocity (2) in the third, a uniformly accelerated velocity (from 2 to 4 ) in the fourth, and so on (Fig. 10.18). Show that the total distance traveled is \(7 / 4\).

Prove the result of Oresme: \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots\) becomes infinite. (This series is usually called the harmonic series.)

Determine what mathematics was necessary to solve the Easter problem. What was the result of the debate in the Church? How is the date of Easter determined today? (Note that the procedure in the Roman Catholic Church is different from that in the Eastern Orthodox Church.)

Compare Levi ben Gerson's use of "induction" to that of alKaraji. Should the methods of either be considered "proof by induction"? Discuss.