A History Of Mathematics

Victor J. Katz

Chapter 5 The Final Chapters of Greek Mathematics - all with Video Answers

Educators

Chapter Questions

04:53

Problem 1

Devise a formula for the $n$th pentagonal number and for the nth hexagonal number.

Willis James

Numerade Educator

04:06

Problem 2

Derive an algebraic formula for the pyramidal numbers with triangular base and one for the pyramidal numbers with square base.

Shaza Hammoud

Numerade Educator

01:32

Problem 3

Show that in a harmonic proportion the sum of the extremes multiplied by the mean is twice the product of the extremes.

Katelyn Chen

Numerade Educator

00:37

Nicomachus defined a subcontrary proportion, which occurs when in three terms the greatest is to the smallest as the difference of the smaller terms is to the difference of the greater. Show that 3, 5, 6 are in the subcontrary proportion. Find two other sets of three terms that are in subcontrary proportion.

Mitchell Cutler

Numerade Educator

00:47

Problem 5

Nicomachus claims that if three terms are in subcontrary proportion, then the product of the greater and mean terms is twice the product of the mean and smaller: for, he notes, 6 times 5 is twice 5 times 3 . Show that Nicomachus is incorrect.

Kara Merfeld

Numerade Educator

Problem 6

Nicomachus defined a "fifth proportion" to exist whenever among three terms the middle term is to the lesser as their difference is to the difference between the greater and the mean. Show that 2, 4, 5 are in fifth proportion. Find two more triples in this proportion.

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01:37

Problem 7

Determine Diophantus age at his death from the epigram at the opening of the chapter.

Melissa Stefan

Numerade Educator

01:29

Problem 8

Solve Diophantus' problem 1-27 by the method of I-28: To find two numbers such that their sum and product are given. Diophantus gives the sum as 20 and the product as 96.

Ashley Volpe

Numerade Educator

02:08

Problem 9

Solve Diophantus' problem II-10: To find two square numbers having a given difference. Diophantus puts the given difference as 60 . Also, give a general rule for solving this problem given any difference.

Lucas Finney

Numerade Educator

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Problem 10

Generalize Diophantus' solution to problem 11-19 by choosing an arbitrary ratio $n: 1$ and the value $(x+m)^2$ for the second square.

Victor Salazar

Numerade Educator

00:48

Problem 11

Solve Diophantus' problem II-13 by the method of the double equation: From the same (required) number to subtract two given numbers so as to make both remainders square. (Take 6,7 for the given numbers. Then solve $x-6=u r^2$. $\left.x-7=r^2.\right)$

Julie Silva

Numerade Educator

Problem 12

Solve Diophantus' problem B-8: To find two numbers such that their difference and the difference of their cubes are equal to two given numbers. (Write the equations as $x-y=$ $a, x^3-y^3=b$. Diophantus takes $a=10, b=2120$.) Derive necessary conditions on $a$ and $b$ which ensure a rational solution.

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Problem 13

Solve Diophantus' problem B-9: To divide a given number into two parts such that the sum of their cubes is a given multiple of the square of their difference. (The equations become $x+y=a, x^3+y^3=b(x-y)^2$. Diophantus takes $a=20$ and $b=140$ and notes that the necessary condition for a solution is that $a^3\left(b-\frac{3}{4} a\right)$ is a square.)

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02:27

Problem 14

Solve Diophantus' problem D-12: To divide a given square into two parts such that when we subtract each from the given square, the remainder (in both cases) is a square. Note that the solution follows immediately from problem II-8.

Julie Silva

Numerade Educator

05:19

Problem 15

Solve Diophantus' problem IV-9: To add the same number to a cube and its side and make the second sum the cube of the first. (The equation is $x+y=\left(x^3+y\right)^3$. Diophantus begins by assuming that $x=2 z$ and $y=27 z^3-2 z$.)

Sarah Lewites

Numerade Educator

01:18

Problem 16

Solve Diophantus" problem V - 10 for the two given numbers 3.9.

Mariam Saleh-Esa

Numerade Educator

02:21

Problem 17

Book VI of the Arithmetica deals with Pythagorean triples. For example, solve problem V1-16: To find a right triangle with integral sides such that the length of the bisector of an acute angle is also an integer. Hint: Use Elements VI-3. that the bisector of an angle of a triangle cuts the opposite side into segments in the same ratio as that of the remaining sides.

P Krishnamurthy

Numerade Educator

03:00

Problem 18

Carry out the analysis of Elements V1-28: To a given straight line to apply a parallelogram equal to a given rectilinear figure and deficient by a parallelogram similar to a given one. Just consider the case where the parallelograms are all rectangles. Begin with the assumption that such a rectangle has been constructed and derive the condition that "the given rectilinear figure must not be greater than the rectangle described on the half of the straight line and similar to the defect."

Nick Johnson

Numerade Educator

01:39

Problem 19

Provide the analysis for Elements XIII-4: If a straight line is cut in extreme and mean ratio, the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment.

James Kiss

Numerade Educator

00:47

Problem 20

Write an equation for the locus described by the problem of five lines. Assume for simplicity that all the lines are either parallel or perpendicular to one of them and that all the given angles are right.

Jay Patel

Numerade Educator

03:36

Problem 21

. Show that a regular hexagon of given perimeter has a greater area than a square of the same perimeter.

Ankit Singh

Numerade Educator

03:07

Problem 22

Find the volume of a torus by applying Pappus" theorem. Assume that the torus is formed by revolving the disk of radius $r$ around an axis whose distance from the center of the disk is $R>r$.

Vikash Ranjan

Numerade Educator

Problem 23

Solve Epigram 116: Mother, why do you pursue me with blows on account of the walnuts? Pretty girls divided them all among themselves. For Melission took two-sevenths of them from me, and Titane took the twelfth. Playful Astyoche and Philinna have the sixth and third. Thetis seized and carried off twenty, and Thisbe twelve, and look there at Glauce smiling sweetly with eleven in her hand. This one nut is all that is left to me. How many nuts were there originally? ${ }^{27}$

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05:20

Problem 24

Solve Epigram 130: Of the four spouts, one filled the whole tank in a day, the second in two days, the third in three days, and the fourth in four days. What time will all four take to fill it? (Note the similarity of shis problem to the problem from Chapter 6 of the Jiushang in the Exercises to Chapter 1.)

Gus Steppen

Numerade Educator

03:13

Problem 25

Solve Epigram 145: A: Give me ten coins and I have three times as many as you. B: And if I get the same from you. I have five times as much as you? How many coins does each have?

Trinity Steen

Numerade Educator

A History Of Mathematics

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