A History Of Mathematics

Victor J. Katz

Chapter 4 Mathematical Methods in Hellenistic Times - all with Video Answers

Educators

Chapter Questions

Problem 1

Calculate $\operatorname{crd}\left(30^{\circ}\right)$, crd( $\left.15^{\circ}\right)$, and $\operatorname{crd}\left(7 \frac{1}{2}^{\circ}\right)$ using the half-angle formula of Hipparchus. beginning with the fact that $\operatorname{crd}\left(60^{\circ}\right)=R=60$. Also calculate $\operatorname{crd}\left(120^{\circ}\right)$. $\operatorname{crd}\left(150^{\circ}\right), \operatorname{crd}\left(165^{\circ}\right)$, and $\operatorname{crd}\left(172 \frac{1}{2}^{\circ}\right)$ using his formula for $\operatorname{crd}\left(180^{\circ}-\alpha\right)$.

Rakesh Kumar Sharma

Numerade Educator

00:30

Problem 2

Use Theon's method to calculate $\sqrt{4500}$ to two sexagesimal places. The answer is 67:4.55.

Tony Ni

Numerade Educator

03:44

Problem 3

Prove the sum formula,

$$
\begin{aligned}
120 \operatorname{crd}(180-(\alpha+\beta))= & \operatorname{crd}(180-\alpha) \operatorname{crd}(180-\beta) \\
& -\operatorname{crd} \alpha \operatorname{crd} \beta
\end{aligned}
$$

using Ptolemy's theorem on quadrilaterals inscribed in a circle.

Srilakshmi E K

Numerade Educator

01:26

Problem 4

Use Ptolemy's difference formula to calculate crd(12*) and then apply the half-angle formula to calculate crd $\left(6^{\circ}\right)$. $\operatorname{crd}\left(3^{\circ}\right), \operatorname{crd}\left(1 \frac{1}{2}^{\circ}\right)$, and $\operatorname{crd}\left(\frac{3}{4}^{\circ}\right)$. Compare your results to Ptolemy's.

Kimberly Waterbury

Numerade Educator

Problem 5

Write a computer program to do the calculations (in sexagesimal notation) of exercise 4.

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01:58

Problem 6

Compare the derivation of the half-angle formula of Hipparchus to the method used by Archimedes in lemma 2 in Measmement of a Circle.

Stanley Enemuo

Numerade Educator

01:42

Problem 7

. Prove that $\operatorname{crd} \beta: \operatorname{crd} \alpha<\beta: \alpha$ or, equivalently, that $\frac{\sin \beta}{\sin \alpha}<e_a^\beta$ for $0<\alpha<\beta$.

Hast Aggarwal

Numerade Educator

01:20

Problem 8

Derive the law of cosines for the case where $\gamma$ is acute by a method analogous to that of Ptolemy in his algorithm for finding the direction of the sun.

Ahmad Reda

Numerade Educator

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Problem 9

. Calculate, using Ptolemy's methods, the length of a noon shadow of a pole of length 60 at the vemal equinox at a place of latitude $40^{\circ}$ and one of latitude $23 \frac{1}{2}^{\circ}$.

Victor Salazar

Numerade Educator

03:37

Problem 10

Explain why the angle $\epsilon$ between the equator and the ecliptic can be determined by taking half the angular distance between the noon altitudes of the sun at the summer and winter solstices. (See Fig. 4.32.)

Sarah Lewites

Numerade Educator

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Problem 11

Calculate the shadow lengths at the summer and winter solstices of a pole of length 60 for latitudes $23 \frac{1}{2}^{\circ}$ and $36^{\circ}$. (In Fig. 4.33, G represents the position of the sun at noon on the summer solstice, $B$ at noon on the equinox, and $L$ at noon on the winter solstice. According to exercise 10 , arc $G B=$ $\operatorname{arc} B L=23 \frac{1}{2}^{\circ}$,

Victor Salazar

Numerade Educator

05:54

Problem 12

Calculate the declination and right ascension of the sun when it is at longitude $90^{\circ}$ (summer solstice). $120^{\circ}$, and $45^{\circ}$, By symmetry, find the declination at longitudes $270^{\circ}$, $240^{\circ}$, and $315^{\circ}$.

Sheryl Ezze

Numerade Educator

01:38

Problem 13

Write a computer program to calculate rising times $p(\lambda, \phi)$ for any values of the longitude $\lambda$ and the geographic latitude $\phi$.

Lucas Finney

Numerade Educator

06:09

Problem 14

Calculate the rising times $\rho(\lambda, \phi)$ for $\phi=45^{\circ}$ and $\lambda=60^{\circ}$ and $90^{\circ}$.

Abhishek Kumar

Numerade Educator

05:54

Problem 15

Calculate the length of daylight on a day when $\lambda=60^{\circ}$ at latitude $36^{\circ}$. Calculate the local time of sunrise and sunset. Calculate the length of daylight at latitude $45^{\circ}$ on a day when $\lambda=60^{\circ}$. Calculate the local time of sunrise and sunset in that case.

Sheryl Ezze

Numerade Educator

04:46

Problem 16

Suppose that the maximum length of day at a particular location is known to be 15 hours. Calculate the latitude of that location and the position of the sun at sunrise on the summer and winter solstices.

Sheryl Ezze

Numerade Educator

04:29

Problem 17

The formula $\sin \sigma=\tan \delta \tan \phi$ only makes sense if the right-hand side is less than or equal to 1 . Since the maximum value of $\delta$ is $23 \frac{1}{2}^{\circ}$, show that the right-hand side will be greater than I whenever $\phi>66 \frac{1}{2}^{\circ}$. Interpret the formula in this case in terms of the length of daylight.

Eric Mockensturm

Numerade Educator

Problem 18

. Calculate the angular distance of the sun from the zenith at latitude $45^{\circ}$ when $\lambda=45^{\circ}$ and $90^{\circ}$.

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Problem 19

. At approximately what dates is the sun directly overhead at noon at a place whose geographical latitude is $20^{\circ}$ ?

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04:46

Problem 20

Calculate the sun's maximal northerly sunrise point for latitudes $45^{\circ}, 36^{\circ}$. and $20^{\circ}$. At approximately what date does the "midnight sun" begin at latitude $75^{\circ}$ ?

Sheryl Ezze

Numerade Educator

06:31

Problem 21

Show how to calculate the distance between two inaccessible points $A, B$ by the use of similar triangles. (Assume, for example, that the two points are on the bank of a river opposite your position.)

Eric Mockensturm

Numerade Educator

01:38

Problem 22

Calculate the area of a triangle with sides of lengths 4, 7. and 10 using both of Heron's methods.

Teresa Fuston

Numerade Educator

04:16

Problem 23

In Heron's formula $A_3=\frac{13}{30} a^2$ for the area of an equilateral triangle with side $a$, what approximation has he used for $\sqrt{3}$ ? Derive this value by his square root algorithm.

Anurag Kumar

Numerade Educator

01:57

Problem 24

. Derive a formula for the area $A$ s of a regular pentagon with side $a$ (using plane geometry). Discuss the differences between Heron's formula $A_5=\frac{5}{3} a^2$ and your formula.

Ryan Pollard

Numerade Educator

01:21

Problem 25

Heron derived his formula for the area $A_7$ of a regular heptagon of side $a, A_7=\frac{43}{12} a^2$, by assuming that $a=\frac{7}{8} r$, where $r$ is the radius of the circumscribed circle. Use this approximation to derive Heron's result. What square root approximation is necessary here?

Jay Patel

Numerade Educator

Problem 26

Derive $17 / 6$ as an approximation to $\sqrt{8}$ to complete the proof of Heron's formula for $A_9$.

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04:10

Problem 27

Using trigonometry, derive a general formula for the area $A_n$ of a regular $n$-gon of side $a$.

Andrew Bassila

Numerade Educator

13:06

Problem 28

. Derive Heron's formula for the volume $\frac{1}{3} \sqrt{2} a^3$ of a regular octahedron of edge length $a$.

Bobby Barnes

University of North Texas

04:24

Problem 29

Outline a trigonometry course following Ptolemy's order of presentation. Namely, derive the major formulas as tools for producing a sine table. Discuss the advantages and disadvantages of this approach compared to the standard textbook approach today.

Tommy Erdos

Numerade Educator

05:23

Problem 30

Ptolemy must have been aware of a method of trisecting angles by the use of conic sections. Such a method would have enabled him to construct the chord of $\frac{1}{2}^{\circ}$ given that he knew the chord of $1 \frac{1}{2}^{\circ}$. Why would Ptolemy not have considered this to be a construction by "geometrical methods"? Can one use such a construction to calculate the chord of $\frac{1}{2}^e$ numerically?

Gabriela Sanchez

Numerade Educator

01:47

Problem 31

Discuss the potential for including some spherical trigonometry in courses on trigonometry, following the general lines of Ptolemy's approach.

Nick Johnson

Numerade Educator

01:04

Problem 32

Outline a lesson using the basic formulas of spherical trigonometry to calculate some simple astronomical phenomena.

Zach Steedman

Numerade Educator

03:52

Problem 33

What observations would have convinced the Greeks that the radius of the celestial sphere was so large that the earth could in effect be considered a point with respect to that sphere?

Linda Winkler

Numerade Educator

00:29

Problem 34

List evidence that convinces you that the earth (a) rotates on its axis once a day and (b) revolves around the sun once a year. Would this evidence have convinced the Greeks? How would you refute the reasons Ptolemy gives for the earth's immovability?

Morgan Thompson

Numerade Educator

03:09

Problem 35

Look up in an astronomy work the "equation of time," and discuss why the times of sunrise and sunset calculated via the methods in the text are likely to be incorrect by several minutes.

Sarah Mccrumb

Numerade Educator

01:32

Problem 36

. Read the 1990 article in Mathematics Magazine to learn why the extrema of sunrise and sunset times do not occur on the solstices, even though those days are in fact the longest and shortest of the year. ${ }^{20}$ Prepare a brief report to explain this surprising phenomenon.

Carson Merrill

Numerade Educator