A History Of Mathematics

Victor J. Katz

Chapter 10 Mathematical Methods in the Renaissance - all with Video Answers

Educators

Chapter Questions

Problem 1

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.)

Megan Mcfarland

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01:33

Problem 2

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

Pammi Eswari

Numerade Educator

01:10

I have 25 pounds of silver alloy which contain 8 ounces of pure silver per pound and 16 pounds which have 9 I/2 ounces of silver per pound. How much copper must be added to the total so that I can make coins containing $71 / 2$ ounces of silver per pound?

Ankit Gupta

Numerade Educator

04:27

Problem 4

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. ${ }^{39}$

Isaac Chettiath

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02:24

Problem 5

This problem and the next two are from the work of Piero della Francesca. Three men enter into a partnership. The first puts in 58 ducats, the second 87; we do not know how much the third puts in. Their profit is 368 , of which the first gets 86 . What shares of profit do the second and third receive and how much did the third invest?

Zach Steedman

Numerade Educator

01:37

Problem 6

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?

Hubert Agamasu

Numerade Educator

02:01

Problem 7

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?

Alison Rodriguez

Numerade Educator

02:43

Problem 8

Solve this problem from the work of Antonio de" Mazzinghi. Find two numbers such that multiplying one by the other makes 8 and the sum of their squares is 27 . (Put the first number equal to $x+\sqrt{y}$ and the second equal to $x-\sqrt{y}$; then the two equations are $x^2-y=8$ and $2 x^2+2 y=27$.)

Swati Agarwal

Numerade Educator

Problem 9

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. Antonio set the parts $u, v$ equal to $5+x$ and $5-x$ respectively and derived an equation in . r.)

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03:12

Problem 10

Maestro Dardi gives a rule to solve the fourth degree equation $x^4+b x^3+c x^2+d x=e$ as $x=\sqrt[1]{(d / b)^2+e}-$ $\sqrt{d / b}$. 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.

Nick Johnson

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Problem 11

Piero della Francesa gives the problem to divide 10 into two parts such that if their product is divided by their difference, the result is $\sqrt{18}$. To solve this he uses a rule for solving the fourth degree equation $a x+b x^2+c x^4=d+e x^3$, namely, $x=\sqrt[4]{(b / 4 c)^2+(d / c)}+(e / 4 c)-\sqrt{a / 2 e}$. Show that this formula works in this case, but not in general. How did Piero derive the formula?

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01:35

Problem 12

The equation $6 x^3=43 x^2+79 x+30$ is solved in the Summa of Luca Pacioli as follows: "Add the number to the cose to form a number, and then you get one cubo equal to $71 / 6$ censi plus $181 / 6$. after you have reduced to one cubo [divided all the terms by 6]. Then divide the censi in half and multiply this half by itself, and add it onto the number. It will be $311 / 144$ and the cosa is equal to the root of this plus $37 / 12$. which is half of censi. ${ }^{100}$ Show that Pacioli's answer is incorrect. What was he thinking of in presenting his rule?

Julie Silva

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01:27

Problem 13

Carry Chuquet's approximation procedure for $\sqrt{6}$ further. That is, since $24 / 9<\sqrt{6}, 25 / 11>\sqrt{6}$, and $29 / 20>\sqrt{6}$, the next approximation is $213 / 29$. Continue the procedure until you reach Chuquet's final value of 289/198.

Wendi Zhao

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01:10

Problem 14

Use Chuquet's approximation procedure to calculate his values for $\sqrt{5}$. namely $2161 / 682$.

Carson Merrill

Numerade Educator

01:13

Problem 15

Find two numbers in the proportion $5: 7$ such that the square of the smaller multiplied by the larger gives 40 .

Rylie Howey

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06:00

Problem 16

Find a number which, when multiplied by 20 and then having 7 added to the product. has the sum in the proportion $3: 10$ with the number formed by multiplying the original number by 30 and subtracting 9 . (Chuquet notes that the problem is impossible. Why?)

Himanshu Kushwaha

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05:47

Problem 17

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 one 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 are open?

Hong Joo Ryoo

Numerade Educator

06:50

Problem 18

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?

Oluwadamilola Ameobi

Numerade Educator

00:24

Problem 19

Express $\sqrt{27+\sqrt{200}}$ as $a+\sqrt{b}$.

James Kiss

Numerade Educator

02:49

Problem 20

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?

Lauren Shelton

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01:02

Problem 21

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

Sanchit Jain

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04:34

Problem 22

In the sequence of odd numbers, the first odd number equals $1^5$. After skipping 1 number, the sum of the next 4 numbers $(5+7+9+11)$ equals $2^5$. After skipping the next 3 numbers, the sum of the following 9 numbers $(19+21+23+25+27+29+31+33+35)$ equals $3^{\text {6 }}$. 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.

Naresh Bagrecha

Numerade Educator

03:34

Problem 23

The basis of Stifel's procedure for finding higher order roots (as well as that of Scheubel and others of his time) was the appropriate binomial expansion. or. more specifically. the entries in the appropriate row of the "Pascal" triangle. For example, to find the fourth root of $1.336,336$. one first notes that the answer must be a two-digit number beginning with 3 . One then subtracts $30^4=810,000$ from the original number to get remainder 526,336 . Recalling that the entries in the fourth row of the triangle are $1,4,6,4,1$, and guessing that the next digit is 4 . one checks this by successively subtracting from that remainder $4 \times 30^3 \times 4=432,000,6 \times 30^2 \times 4^2=86,400$, $4 \times 30 \times 4^3=7680$, and $4^4=256$. In this case, the result is 0 , so the desired root is 34 . Write a brief report explaining this procedure in detail and use it to calculate the fourth root of 10.556 .001 .

John Gehad

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00:23

Problem 24

There is a certain army composed of Dukes, Earls, and soldiers, Each Duke has under him twice as many Earls as there are Dukes. Each Earl has under him four times as many soldiers as there are Dukes. The two hundredth part of the number of soldiers is 9 times as many as the number of Dukes. How many of each are there?

Elizabeth Xu

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02:19

Problem 25

A gentleman, willing to prove the cunning of a bragging Arithmetician, said thus: I have in both hands 8 crowns. But if I count the sum of each hand by itself severally and add to it the squares and the cubes of the both, it will make in number 194. Now tell me, what is in each hand? ${ }^{\text {? }}$

Dale Sanford

Numerade Educator

01:15

Problem 26

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

Hoan Nguyen

Numerade Educator

01:15

Problem 27

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 $x^3+3=8 x$.

Hoan Nguyen

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02:04

Problem 28

Prove that the equation $x^3+c x=d$ always has one positive solution and no negative ones.

Rakesh Kumar Sharma

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00:46

Problem 29

Use Cardano's formula to solve $x^3+3 x=10$.

K B

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01:26

Problem 30

Use Cardano's formula to solve $x^3=6 x+6$.

Carson Merrill

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06:03

Problem 31

Consider the equation $x^3=c x+d$. Show that if $(c / 3)^3>$ $(d / 2)^2$ (and thus that Cardano's formula involves imaginary quantities), then there are three real solutions.

Nick Johnson

Numerade Educator

02:00

Problem 32

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 $y$.

Vishal Parmar

Numerade Educator

Problem 33

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 $-2 b x^2+b^2$ to both sides. Determine the cubic equation which $b$ 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?

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Problem 34

The dowry of Francis' wife is 100 aurei more than Francis' own property and the square of the dowry is 400 more than the square of his property. Find the dowry and the property.

Julie Silva

Numerade Educator

05:16

Problem 35

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

Aamir Mithaiwala

Numerade Educator

01:18

Problem 36

Divide 8 into two parts $x, y$ such that $x y(x-y)$ is a maximum.

Adrian Co

Numerade Educator

Problem 37

It is obvious that 3 is a root of $x^3+3 x=36$. Show that the Cardano formula gives
$$
x=\sqrt[3]{\sqrt{325}+18}-\sqrt[3]{\sqrt{325}-18}
$$
Using Bombelli's methods, show that this number is in fact equal to 3.

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01:10

Problem 38

Express $\sqrt[3]{52+\sqrt{-2209}}$ in the form $a+b \sqrt{-1}$.

Amy Jiang

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04:11

Problem 39

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 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.

James Kiss

Numerade Educator

01:05

Problem 40

Given the product of two numbers and their ratio, 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.)

Carson Merrill

Numerade Educator

02:10

Problem 41

Given the difference between two numbers and the difference between their cubes, find the numbers. Let $E$ be the sum of the numbers, $B$ the difference between them, and $D$ the difference between the cubes. Show that $E^2=\frac{\frac{1 D-B^3}{3}}{3}$. Once $E^2$ is known, so is $E$ and then the numbers themselves. Find the solution when $B=6$ and $D=504$. (Diophantus has the same problem twice, once in Book IV with these numerical values and once in Book B.)

Raushan Kumar

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00:37

Problem 42

Show that if $x^3+b^2 x=b^2 c$, then there are four continued proportionals, the first of which is $b$, the sum of the second and fourth being $c$, and the unknown $x$ being the second. Use this result to solve $x^3+64 x=2496$.

Amrita Bhasin

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02:59

Problem 43

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

Nick Johnson

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06:07

Problem 44

Given the two numbers 237 (1) 5 (1) 7 (2) 8 (3) and 59 (0) 7 (1) 3 (2) 9 (3), subtract the second from the first.

Anas Venkitta

Numerade Educator

Problem 45

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?

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01:39

Problem 46

Outline a lesson introducing the study of complex numbers via the problems with Cardano's formula giving a real root as the sum of two complex values. Discuss the merits of such an approach.

Juhi Singh

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00:41

Problem 47

Compare the various notations for unknowns used by the mathematicians discussed in the text. Write a brief essay on the importance of a good notation for increasing a student's understanding in algebra.

Sneha Ravi

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Problem 48

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 (note 39).

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Problem 49

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?

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Problem 50

Compare the symbolism of Jordanus and Viète. In what way is Viète's work an advance on that of Jordanus?

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02:07

Problem 51

Explain why mathematicians of the sixteenth century equated the new algebra with the Greek analysis as described by Pappus.

Ankit Gupta

Numerade Educator