1.10: Women and the Media
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Chapter 10: Women and the Media
Chapter Summary
- Overview
- Profile: Ann Moore: Leveraging the Value of Women
- Project: Women’s Edition
- Questions
- Additional Resources
Chapter 10 discusses the progress that women have made in organizing against exclusion in media. Women who demanded coverage from media outlets were originally categorized as misfits or insane, as they were perceived as departing from their traditional domestic roles. Throughout the 1970s, media outlets and journals covering a range of feminist issues emerged around the world, including Isis International Bulletin in Italy and the Philippines and Manushi in India. Women have been gaining influence in media both within the United States and across the world. Ann S. Moore was the CEO of Time Inc., the largest publisher in the U.S., and steered the business from the print age to the media age during the 2000s. The Women’s Edition program brings together women journalists across the global South to report on various issues from their respective regions and countries.
Key Terms
- Ann S. Moore
- Gender and Media Diversity Journal
- International Federation of Journalists
- International Women’s Media Foundation
- Isis International Bulletin
- Manushi
- Ms. Magazine
- Population Reference Bureau (PRB)
- Revolution
- Women’s Edition
- Woodhull & Clafin’s Weekly
- World Association of Christian Communicators (WACC)
Overview
By Carolyn M. Byerly
W omen brought a gendered analysis of the mass media to the global stage in the 1970s when a multipart critique was first presented at the 1976 Mexico City conference, which opened the U.N. Decade for Women. Much of the substance of that critique remains relevant today. But women’s fight for equal representation in the media began much earlier.
History of Exclusion and Stereotypes
Women’s exclusion from the serious news of the day was raised as early as the 18th century by women suffragists and women’s rights activists in Europe and North America. The early suffrage leaders needed the attention of the news media to carry their ideas and activities to wider publics, but male-run newspapers and magazines largely ignored the women activists. The news outlets that did cover women frequently trivialized their goals. Women who departed from the social norms of passivity and deference to male authority, and the traditional roles of wife and mother, risked being characterized as inappropriate, insane or misfits. If they demanded equality with men, the media depicted them either as curiosities or as loud, militant and aggressive. Such characterizations would continue into the early days of modern feminism (Epstein, 1978).
Not only were women’s issues and leaders excluded from the media, but bias against women was practiced in reporting women’s issues and leaders. Such treatment inspired women in many countries to establish their own magazines, newspapers and book publishing houses during the late 19th and early 20th centuries. The post-Civil War Woodhull & Claflin’s Weekly had as its aim to make Victoria Woodhull the first woman president, while the Lily had a broad women’s rights agenda, and the Una championed the rights of immigrant and poor women. Elizabeth Cady Stanton and Susan B. Anthony’s short-lived but important newspaper the Revolution addressed a spectrum of issues related to women’s discrimination, including low wages of working women and the right to vote.
A New Era of Women’s Rights
By the late 20th century, women across the globe focused on enacting political and legal reforms to extend women’s equality and access to social institutions and to ensure protection of their rights. It was a new era for women’s rights. Many women became politicized during independence movements, as countries broke from colonial powers. The legacy of that activism carried over into women’s media like Ms. magazine, founded by U.S. feminists in the early 1970s; Manushi , an Indian feminist journal founded in the mid-1970s; and Isis International Bulletin , published first in Rome, then later in Manila.
Some feminist leaders were motivated by the enduring problems of exclusion and misogynistic representation in mainstream media to establish their own publishing houses, today numbering many dozens (see www.wifp.org/DWM/publishers.html). Women’s organizations like the South African group Gender Links have assumed dual missions of establishing their own journals, like Gender and Media Diversity Journal , as well as undertaking training for journalists in order to address persistent patriarchal messages in news, advertising, films and television programs (Gender Links, www.genderlinks.org.za/page/publications ).
Another modern media concern was women’s lack of access to media professions. Women were severely underrepresented in newsrooms, television and radio stations, film production and ownership of media outlets. More women on the inside, it was argued, would help resolve many of women’s other problems with the media. Women such as Ann S. Moore (Time Inc.), profiled in this chapter, acknowledge the importance of women in their media operations.
Underrepresentation in news production arose through the U.N. Decade for Women (1976-1985), with leaders pushing the United Nations to fund women’s news and feature services in the 1970s and 1980s to increase global news flow from progressive women’s perspectives. They also gained funding for research on women and media and generated their own research. Two examples are the Brussels-based International Federation of Journalists and the World Association of Christian Communicators (WACC). The latter of these is among advocacy groups that sponsor research aimed at enabling strategy-building for women’s equality in the media. WACC’s (Canada) periodic study Who Makes the News? focuses on women’s representation in news worldwide, while the International Women’s Media Foundation (United States) conducts research on women’s status in news organizations. IWMF also recognizes women journalists for courage in reporting with an annual “Courage in Journalism” award.
Such groups offer workshops to teach media professionals how to include gender angles in news. Women have made slower progress in communications governance and policymaking, at national and international levels, so these remain important areas for critique and action. Programs such as USAID-funded Women’s Edition have given women strong foundations for journalism careers.
Carolyn M. Byerly is a professor in the Department of Journalism, School of Communications, Howard University, in Washington, D.C. She researches communications policy related to women and minority ownership, media and gender, race, sexuality and nationality. She is the co-editor of Women and Media: Global Perspectives (Blackwell, 2004), co-author of Women and Media: A Critical Introduction (Blackwell, 2006) and principal investigator for the study Global Report on the Status of Women in News Media , a 59-nation study sponsored by the International Women’s Media Foundation (2011).
PROFILE: Ann Moore – Leveraging the Value of Women
Women who have achieved top management positions in media corporations are few. Ann S. Moore rose to the top of one of the world’s most influential news organizations, Time Inc., through perseverance, a willingness to take calculated risks and a canny perception of the future of media.
T he impressive thing about Ann S. Moore, who ran Time Inc. from 2002 to 2010, is not her global profile as the first woman to head the legendary company that boasts 115 international magazine titles and some 137 million monthly readers. It’s not even her down-to-earth, straight-talking style, the friendships with influential policymakers and A-list celebrities or her habitual appearance on every Most Powerful Woman list ever devised. Rather, the impressive thing, as you listen to Moore review her rise to the top, is the out-and-out glee with which she did the job.
Revamping Time Inc. for the Digital Age
It was far from easy. As chair and CEO, Moore arguably led Time Inc. through its greatest trials and transformations — and emerged victorious. She successfully steered the largest magazine publisher in the United States out of its fierce attachment to print and straight into the digital age. She streamlined its multilayered, old-boy centralized structure into more nimble brand clusters, making managers diverse and more accountable. “We were facing a crisis,” she acknowledges today. “It was not easy to completely transform an industry and drag everybody, kicking and screaming, into the 21st century.”
Then again, Moore could draw on deep experience in setting goals and tackling the challenges. A keen observer and an unabashed fan of Time Inc.’s unique role in the media landscape — the company has influence that streams from Main Street to Wall Street and from Pennsylvania Avenue to Hollywood and Vine — Moore enjoyed a career at the company that spanned 32 years. “I know every inch of this business,” she declares, not particularly bragging, but merely stating the facts.
At a meeting just before her departure is officially announced, Moore is comfortably settled into a plump armchair in the spacious, thickly carpeted executive suite on the 34th floor of Rockefeller Center’s landmark Time-Life Building. The sweeping views of midtown Manhattan highlight her success. Moore, at 60, looks back with pride and gusto. Clearly, she has thrived on the risks as much as the wins. “I work with really smart people and we produce really amazing products,” she says. “It’s easy to stay working somewhere for 32 years when you’re not bored.”
Born in Biloxi, Mississippi, the oldest of five kids, Moore spent her formative years on a series of military bases. “My father was in the Air Force until I was in about sixth grade,” she says. “I moved all the time when I was young and I have nothing but fabulous memories of every move.” She credits her dad, who was a pilot, with shaping her attitudes toward work. “I always knew I could do the job of CEO, but it wasn’t my lifelong ambition,” Moore explains. “My father retired from the military and went on to a whole second career in aviation.  And you don’t necessarily have to aspire to being the CEO to be successful.’” That outlook served her well as she climbed the ladder. “I always had a lot of confidence,” she says. “You have to not fear failure.”
Drawn to Sports and Publishing
After high school in McLean, Virginia, Moore went on to Vanderbilt University in Nashville, Tennessee, and then earned an MBA from Harvard Business School in 1978. While her classmates headed for Wall Street, Moore wasn’t interested — “I always wondered exactly what they do there,” she says, jokingly. Instead, Moore went to Time Inc. “I was a big sports fan and that’s why I joined the company,” she says. “I turned my hobby of reading Sports Illustrated into my career.”
Through the 1980s, after starting at Sports Illustrated , Moore put in stints at Fortune , Money and Discover , moving up with every jump. By the early ’90s, back at Sports Illustrated as associate publisher, she began to make her mark on the company. Fittingly, Moore earned her claim to fame by tutoring Time Inc. about the value of women.
“I changed the equation at the company because I’m the one who began diversifying into women’s and children’s magazines,” says Moore. She launched Sports Illustrated Kids in 1989. “It was the first kid magazine and we hadn’t launched anything successful since 1974.” At the time, she says, “we thought you couldn’t make money targeting women, so even when I moved to People in 1991, we thought it was a dual-audience magazine.”
People and InStyle : Women as an Important Market
Moore was working with then-editor Lanny Jones. The pair transformed People into a newsmagazine for women, first moving from black and white to color pages. Next, they changed delivery from Monday to Friday. “News is like bread. The fresher it is, the more exciting it is.” Moore also inaugurated People special issues, such as its now-famous “Sexiest Man Alive,” “Best Dressed, Worst Dressed” and others. “It was a license to steal,” she laughs, looking back. “Once you determined this is really a woman’s magazine, you could see what you needed to do with People to unleash its potential.” Nowadays, as it has been for years, People is the company’s most profitable title and, as Moore likes to point out, People.com leads in online entertainment news, with 13 million unique visitors monthly.
The rest remains Moore’s groundbreaking history. She launched InStyle in 1991, which is, today, the nation’s largest-circulation fashion and beauty magazine — “ahead of Vogue,” says Moore. InStyle is Time Inc.’s third-most-profitable title ( Sports Illustrated is second). Real Simple came next, in 2000.  “Time is the single most precious commodity to American consumers, especially for a woman,” she says. “So that was the whole idea behind Real Simple . We would get you organized and you would have an extra hour a day.”
Moore smiles, leaning back in her comfortable chair, and confides her secret to launching successful magazines. “It was such a simple concept, but it solved a problem. That’s where you find holes in the marketplace, and that’s what Time Inc. was particularly good at. We invented most of the categories we publish in.”
What’s in store for Moore as she moves on? She’s not saying. With her 26-year-old son, Brendan, enrolled at Harvard Business School and her husband, Donovan Moore, continuing work as a private wealth manager at Bessemer Trust, her horizon looks wide open, especially considering her dad’s second-act role model.
And what advice would she give young women who’d like to shadow her dramatic footsteps? True to form, Moore speaks forthrightly: “I think it’s all about self-assessment. Who are you? What are you good at? What do you like to do? Then use that to find a match. Turn your hobby into your occupation. You have to take responsibility for your career. I also say to young women, learn how to fill out your dance card. I’m the chairman of Time Inc. because I filled out my dance card better than anyone else. I’m the chairman because I’ve been here for 32 years and I’ve launched more magazines than [Time Inc. founder] Henry Luce. That’s why I’m in Luce’s office.”
“And I was very patient.”
Joanna L. Krotz is a multimedia journalist and speaker whose work has appeared in the New York Times , Worth , Money and Town & Country and on MSN and Entrepreneurship.org . She is the author of The Guide to Intelligent Giving and founder of the Women’s Giving Institute, an organization that educates donors about strategic philanthropy.
PROJECT: Women’s Edition
By Deborah Mesce
Funding for women’s news and feature services opened the door to journalism careers for many women worldwide in recent decades. The Women’s Edition program shows how women may fruitfully collaborate to gain global perspectives on women’s issues, and bring that knowledge to their writing.
A round the table sat 12 women journalists from Africa, Latin America, Asia and Eastern Europe, discussing the status of women in their cultures. The Africans said women in their countries have many babies, often too many to care for adequately. Reporters from India, the Philippines and Peru said families were somewhat smaller in their countries. Then the Romanian journalist surprised them all: “You know, in my country the government pays women to have children,” she said, explaining Romania’s strategy to reverse its population decline.
This conversation took place at Women’s Edition, a program that brings together small groups of veteran women journalists from influential media houses across the developing world to examine and report on a range of issues related to women’s health and development. They gain a global perspective on these issues by learning how countries both similar and different from theirs handle the same issues. As a Nepali reporter put it after several years in the program, “Now I think globally and write locally.”
Women’s Edition, funded by the U.S. Agency for International Development, takes a long-term view of working with journalists. Since 1994 when the program began, 62 journalists have participated. There was little turnover in the early years, but later a two-year participation limit was set. During their tenure in the program, the journalists attend weeklong seminars twice a year in locations around the world. Seminars focus on health and development issues. Each journalist takes away new data and research on specific topics, the insights of experts and memorable experiences from field visits that illuminate the issues. Following each seminar, each journalist prepares a special supplement, a series of articles or a broadcast program for her media house on the seminar topic in the context of her country.
The Population Reference Bureau (PRB), a nongovernmental organization in Washington that runs the project, solicits applications from women editors, reporters and producers every two years. Journalists hear about this from national and international journalism associations, schools and websites. As many as 200 candidates apply. PRB invites around 12 to participate. PRB looks for seasoned journalists who demonstrate a strong interest in women’s health and development issues and who have editorial influence in their newsrooms. To maintain geographic diversity, usually just one journalist is selected from a country.
Women’s Edition grew from an earlier PRB project, Global Edition, which brought together senior editors from the developing world to focus and write on population and the environment. Similarly, Women’s Edition’s mission is to strengthen and increase reporting on women’s health and development and, in doing so, stimulate discussion on these issues among the public and policymakers in developing countries.
In organizing the seminars, PRB seeks input from the journalists in selecting a topic and then links the topic with a relevant venue. For example, a seminar on trafficking was held in New Delhi, where the journalists visited a brothel in the city’s largest red-light district and talked to Nepali sex workers there who had been trafficked years before. For a seminar on violence against women, Women’s Edition met in South Africa, which has one of the world’s highest rates of rape but also some of the most innovative programs to deal with the problem. Some seminars have been held in conjunction with international conferences and other events, such as the biennial AIDS conferences and special U.N. sessions. Other seminar themes have included links between gender and the environment, women’s empowerment and women’s reproductive health.
Sometimes a journalist’s report prompts action. After a magazine cover story on the health and social problems child brides face in India, the Tamil Nadu state government launched awareness campaigns in villages where child marriage is common. More often, the journalists receive calls from ministry officials, parliamentarians and NGO leaders who want more information. A Malawian journalist wrote about an innovative rape crisis center her Women’s Edition group visited in Johannesburg. She received a call from the Malawian president’s daughter, who wanted to become involved in local efforts to replicate it. Sometimes journalists take action on their own: A seminar visit to a rape crisis center in New York so inspired an Indian journalist that she persuaded a physician friend to open such a center in Mumbai.
Women’s Edition has a lasting impact on the journalists themselves. They become the experts in their newsrooms on women’s issues. They gain confidence in their knowledge and abilities, which helps them to lobby for coverage of women’s issues. They receive job promotions and gain more influence over what issues are considered newsworthy. “Gone are the days when health and women’s issues were a once-a-week affair,” said Ropa Mapimhidze of Zimbabwe, who was assistant editor at the Herald when she participated in Women’s Edition. She is now features editor at Newsday , a new independent newspaper.
Deborah Mesce is program director for international media training at the Population Reference Bureau and has been coordinator of Women’s Edition since 2001. Before joining PRB, she worked for more than 20 years as a reporter and editor for the Associated Press in the Connecticut state bureau and on the national staff in Washington.
Multiple Choice Questions
Questions
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Which factors inspired women in many countries to establish their own magazines?
- Male-run publications largely ignored women activists
- The news outlets that did cover women trivialized their goals
- Women that departed from the norms of domesticity and passivity were labelled as misfits or insane
- Women who demanded equality with men were depicted as militant and aggressive
- All of the above
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The first woman to lead Time Inc. was….
- Ann Moore
- Ariana Huffington
- Jane Fonda
- Diane Sawyer
-
People and InStyle demonstrated that…
- Women consumers are mostly interested in fashion and celebrity news
- Women are an important demographic market in news media
- Women were already influential in traditional media networks
- There was no need to develop women-specific media
- None of the above
-
Women’s Edition serves the following purpose(s)…
- Brings together groups of women journalists
- Examines and reports on a range of women’s issues
- Enables participants to gain a global perspective
- Enables participants to learn from each other in handling similar issues
- All of the above
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The objective of Women’s Edition is to…
- Advance the political interests of women at the cost of men
- Increase reporting on women’s health and development
- Begin discussion among the public and policy-makers in developing countries
- Organize with influential celebrities and policy-makers on issues of gender and development
- None of the above
Answers
- The correct answer is E (all of the above).
- The correct answer is Ann Moore (answer A). Ariana Huffington is the co-founder of the Huffington Post, so answer B is incorrect. Jane Fonda is an American actress and co-founder of the Women’s Media Center (answer C). Diane Sawyer is an American television journalist and co-anchor of ABC News’ morning news program.
- The correct answer is that women are an important market in the news media (answer B). The fact that the first Time magazines that publicists deemed “women-targeted” were fashion and celebrity news does not indicate that women consumers are most interested in fashion and celebrity news (answer A). Women were not yet influential in news networks, so Time Inc. and its subsidiaries were an anomaly (answer C). Answer D is incorrect because there was a clear need to develop women-specific media.
- The correct answer is E (all of the above).
- The correct answer is both B and C. Women’s Edition aims to increase reporting on women’s health and development (answer B) and begin discussions among the public and policymakers in developing countries (answer C). The objective is not to advance the interests of women at the cost of men, as gender equality is not zero-sum (answer A). The mission is not to simply engage with celebrities and policymakers, so answer D is incorrect.
Discussion Questions
- How are women and men depicted differently in media? Has the history of women’s exclusion from media influenced these representations? Why or why not?
- What are some impacts of the portrayal of women in media on women’s success?
- Select an election in a jurisdiction of your choice. What were the genders of the candidates? How were they represented by media?
- Do you think that social media provides a greater platform than traditional media for women’s representation and influence? Why or why not?
- How are transgender women portrayed in media?
- Use the additional resources, your experiences, or the experiences of people you know to reflect on how transgender people experience portrayals of gender and transgender people in media in a designated case study. For instance, you might choose Canadian media, American Media or French Canadian media for your examples.
Essay Questions
- Review the “2015 Status of Women in U.S. Media” report. Identify areas where there has been progress and where equal representation along lines of gender and race have fallen behind.
- What are the most common stereotypes of women perpetuated in media? In what ways are these representations dependent on race and class? Will having more women in media play a role in decreasing reliance on stereotypes of women?
- How has the use of social media influenced women’s social and political movements?
Additional Resources
Billingsley, A.
“Technology and Narratives of Continuity in Transgender Experiences.” Feminist Philosophy Quarterly 1(1), 1 – 24: (2015).
Explores transgender narratives in media, social science disciplines, and among the family.
http://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=1009&context=fpq
Glantz, J.
“Women in Popular Music Media: Empowered or Exploited?”
The Spectrum: A Scholars Day Journal
2(5)
,
3 – 38: (2011).
Investigates women’s attitude toward women in media and popular music through a feminist theoretical lens.
http://digitalcommons.brockport.edu/cgi/viewcontent.cgi?article=1018&context=spectrum
Jia, S. Lansdall-Welfare, T. Sudahar, S. Carter, C. & Cristianini, N.
“Women Are Seen More than Heard in Online Newspapers.”
PLOS One
11(2: (2016).
Quantitative analysis of the representation of gender and women’s views in online newspapers.
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0148434
Kamerick, M.
“Women Should Represent Women in Media.”
TEDxABQ.
(2011).
TED Talk on the representation of women as victims; and victim blaming culture in media.
Lang, N.
“AfterEllen and the Digital Media Bust: Queer Spaces for Women are Disappearing, Even on the Internet.”
Salon.
(2016).
Suggests that the lower buying power of queer women is influencing the reduction of media platforms dedicated to queer women’s issues and lived experiences.
Truitt, J.
“Laverne Cox’s Actress Emmy Nod Puts Trans People in Bigot’s Living Rooms.”
Guardian
(2014).
Analysis of Laverne Cox’s influence in pop culture in the wake of her role in the Netflix original series, “Orange is the New Black.”
Truitt, J.
“On Jill Soloway, Caitlyn Jenner, and the Trans Representation the Media Wants.”
Feministing.
(2015).
A critique of the media’s representation of transgender people, focusing on the discourse surrounding Caitlyn Jenner and her position in terms of race and class.
Women’s Media Center.
“Status of Women in the U.S. Media.” (2015).
Features research on the gender gap in broadcast news, internet, print journalism and wire services.
www.womensmediacenter.com/pages/2015-statistics
Sandberg, S.
“So We Leaned In… Now What?”
TED Talks.
(2014).
Sheryl Sandberg, Chief Operating Office of Facebook, discusses the reactions to “Lean In” and the ways in which women continue to struggle with success.
https://www.youtube.com/watch?v=YraU52j3y8s
Striphas, S.
“A Dialectic with the Everyday: Communication and Cultural Politics on Oprah Winfrey’s Book Club.”
Critical Studies in Media Communication
20, 295 – 316: (2010).
Examines the gendered communication strategies within Oprah Winfrey’s Book Club.
http://dx.doi.org/10.1080/07393180302773
Yasmin, M., Sohail, A. & Mangrio, R. A.
Myths Broken or Sustained: Representation of Women Victims in Pakistani Media.
Open Journal of Social Sciences
3(7), 209 – 219: (2015).
Highlights asymmetry in Pakistani media of descriptions of female and male victims and perpetrators of violence (female victims described with their marital status, male victims and perpetrators described using their profession.)
http://www.scirp.org/journal/PaperInformation.aspx?paperID=57967


2.3: Using Percent Slope to Determine Tree Height
2.3 Using Percent Slope to Determine Tree Height
When using percent slope to determine tree height, the tree is the rise, and the horizontal distance from the tree along the ground is the run (Figure 2.3). We can easily measure our horizontal distance from the tree, and we have instruments for measuring the percent slope to the top of a tree. So, with those two measures (run and %slope) we can solve for rise.
%slope
To solve the %slope equation for “rise” we do the following:
1. Multiply both sides of the equation by “run” to cancel out run on the left side of the equation
(run)(%slope)
2. Divide both sides by “100” to cancel out 100 on the left side of the equation
(%slope)
That leaves us with the following equation:
(%slope) where rise =height
Notice that the %slope multiplier (100) becomes the denominator. This remains a constant. The run and %slope values are measurements, so they will change with each tree.
Here are some examples using %slope for tree height:
Example 1:
Georgia walks out a horizontal distance of 50 feet from a tree (Figure 2.4). She looks through her clinometer to determine the %slope from her eye to the top of the tree. Looking up, she reads “+124%.” Using our formula to determine the “rise:”
(%slope)
(+124) = +62 or 62 feet
Therefore, the height of the tree from her eye to the top is 62 feet. Next, we use the same procedure to determine the height from Georgia’s eye level down to the stump. Georgia takes a reading down to the stump. She reads -16%.
(%slope)
(-16) = -8 or 8 feet
Georgia measured 62 feet from her eye to the top of the tree, and 8 feet from her eye to the stump of the tree. We add those two measures together to get the total height of the tree.
62 ft (top) + 8 ft (base) = 70 ft.
Remember that a negative slope measurement simply means you are looking downhill. It is important to recognize that the -8 feet is a drop in elevation, not a negative value .
We can do the two calculations in one step as follows:
(%slope)
(124+16) = 70 or 70 feet
Notice Georgia put “+16” in the formula, even though the reading to the stump was a negative number. She wants to add in the bottom height, not subtract it. Again, a negative slope simply means we looked downhill. You have to think about your eye position in relation to the tree, and what the readings actually mean. Using %slope symbols algebraically can be misleading.
Example 2:
Tobias walks out a horizontal distance of 100 feet from the same tree (Figure 2.5). He looks to the top of the tree and reads “+62%” on his clinometer. He looks down to the stump and reads “-8.”
(%slope)
(62+8) = 70 or 70 feet
Notice that by walking farther from the tree (having a longer run), Tobias’ slope readings were smaller than Georgia’s , even though they both ended up with the same height. The angles at which Tobias was looking at the tree were smaller (less acute).
Also notice that when Tobias walked out 100 feet from the tree, his run equaled the denominator, 100, cancelling it out and leaving the tree equal to the readings from the clinometers. This “shortcut” makes it much easier and faster to determine tree height by reducing the number of conversions that have to be made.
(62+8) = 70 or 70 feet
What happens at a horizontal distance of 150 feet? The slope readings are even smaller.
(%slope)
(42+5) = 70 or 70 feet
Thus, the farther one walks from the tree, the better the perspective for seeing the tree top. Being too close to the tree can result in an obscured top, as side branches will be in the way (Figure 2.6).
In the photo on the left, the tree’s top can clearly be seen. The right photo however, illustrates a foreshortened view of the top where side branches could be mistaken for the top.


But, courage! for beside that boisterous Brook
The mountains have all open'd out themselves,
And made a hidden valley of their own.
No habitation there is seen; but such
As journey thither find themselves alone
With a few sheep, with rocks and stones, and kites
That overhead are sailing in the sky.
It is in truth an utter solitude,
Nor should I have made mention of this Dell
But for one object which you might pass by,
Might see and notice not. Beside the brook
There is a straggling heap of unhewn stones!
And to that place a story appertains,
Which, though it be ungarnish'd with events,
Is not unfit, I deem, for the fire-side,
Or for the summer shade. It was the first,
The earliest of those tales that spake to me
Of Shepherds, dwellers in the vallies, men
Whom I already lov'd, not verily
For their own sakes, but for the fields and hills
Where was their occupation and abode.
And hence this Tale, while I was yet a boy
Careless of books, yet having felt the power
Of Nature, by the gentle agency
Of natural objects led me on to feel
For passions that were not my own, and think
At random and imperfectly indeed
On man; the heart of man and human life.
Therefore, although it be a history
Homely and rude, I will relate the same
For the delight of a few natural hearts,
And with yet fonder feeling, for the sake
Of youthful Poets, who among these Hills
Will be my second self when I am gone.
Upon the Forest-side in Grasmere Vale
There dwelt a Shepherd, Michael was his name.
An old man, stout of heart, and strong of limb.
His bodily frame had been from youth to age
Of an unusual strength: his mind was keen
Intense and frugal, apt for all affairs,
And in his Shepherd's calling he was prompt
And watchful more than ordinary men.
Hence he had learn'd the meaning of all winds,
Of blasts of every tone, and often-times
When others heeded not, He heard the South
Make subterraneous music, like the noise
Of Bagpipers on distant Highland hills;
The Shepherd, at such warning, of his flock
Bethought him, and he to himself would say
The winds are now devising work for me!
And truly at all times the storm, that drives
The Traveller to a shelter, summon'd him
Up to the mountains: he had been alone
Amid the heart of many thousand mists
That came to him and left him on the heights.
So liv'd he till his eightieth year was pass'd.
And grossly that man errs, who should suppose
That the green Valleys, and the Streams and Rocks
Were things indifferent to the Shepherd's thoughts.
Fields, where with chearful spirits he had breath'd
The common air; the hills, which he so oft
Had climb'd with vigorous steps; which had impress'd
So many incidents upon his mind
Of hardship, skill or courage, joy or fear;
Which like a book preserv'd the memory
Of the dumb animals, whom he had sav'd,
Had fed or shelter'd, linking to such acts,
So grateful in themselves, the certainty
Of honorable gains; these fields, these hills
Which were his living Being, even more
Than his own Blood—what could they less? had laid
Strong hold on his affections, were to him
A pleasurable feeling of blind love,
The pleasure which there is in life itself.
He had not passed his days in singleness.
He had a Wife, a comely Matron, old
Though younger than himself full twenty years.
She was a woman of a stirring life
Whose heart was in her house: two wheels she had
Of antique form, this large for spinning wool,
That small for flax, and if one wheel had rest,
It was because the other was at work.
The Pair had but one Inmate in their house,
An only Child, who had been born to them
When Michael telling o'er his years began
To deem that he was old, in Shepherd's phrase,
With one foot in the grave. This only son,
With two brave sheep dogs tried in many a storm.
The one of an inestimable worth,
Made all their Household. I may truly say,
That they were as a proverb in the vale
For endless industry. When day was gone,
And from their occupations out of doors
The Son and Father were come home, even then,
Their labour did not cease, unless when all
Turn'd to their cleanly supper-board, and there
Each with a mess of pottage and skimm'd milk,
Sate round their basket pil'd with oaten cakes,
And their plain home-made cheese. Yet when their meal
Was ended, LUKE (for so the Son was nam'd)
And his old Father, both betook themselves
To such convenient work, as might employ
Their hands by the fire-side; perhaps to card
Wool for the House-wife's spindle, or repair
Some injury done to sickle, flail, or scythe,
Or other implement of house or field.
Down from the cicling by the chimney's edge,
Which in our ancient uncouth country style
Did with a huge projection overbrow
Large space beneath, as duly as the light
Of day grew dim, the House-wife hung a lamp;
An aged utensil, which had perform'd
Service beyond all others of its kind.
Early at evening did it burn and late,
Surviving Comrade of uncounted Hours
Which going by from year to year had found
And left the Couple neither gay perhaps
Nor chearful, yet with objects and with hopes
Living a life of eager industry.
And now, when LUKE was in his eighteenth year,
There by the light of this old lamp they sate,
Father and Son, while late into the night
The House-wife plied her own peculiar work,
Making the cottage thro' the silent hours
Murmur as with the sound of summer flies.
Not with a waste of words, but for the sake
Of pleasure, which I know that I shall give
To many living now, I of this Lamp
Speak thus minutely: for there are no few
Whose memories will bear witness to my tale,
The Light was famous in its neighbourhood,
And was a public Symbol of the life,
The thrifty Pair had liv'd. For, as it chanc'd,
Their Cottage on a plot of rising ground
Stood single, with large prospect North and South,
High into Easedale, up to Dunmal-Raise,
And Westward to the village near the Lake.
And from this constant light so regular
And so far seen, the House itself by all
Who dwelt within the limits of the vale,
Both old and young, was nam'd The Evening Star.
Thus living on through such a length of years,
The Shepherd, if he lov'd himself, must needs
Have lov'd his Help-mate; but to Michael's heart
This Son of his old age was yet more dear—
Effect which might perhaps have been produc'd
By that instinctive tenderness, the same
Blind Spirit, which is in the blood of all,
Or that a child, more than all other gifts,
Brings hope with it, and forward-looking thoughts,
And stirrings of inquietude, when they
By tendency of nature needs must fail.
From such, and other causes, to the thoughts
Of the old Man his only Son was now
The dearest object that he knew on earth.
Exceeding was the love he bare to him,
His Heart and his Heart's joy! For oftentimes
Old Michael, while he was a babe in arms,
Had done him female service, not alone
For dalliance and delight, as is the use
Of Fathers, but with patient mind enforc'd
To acts of tenderness; and he had rock'd
His cradle with a woman's gentle hand.
And in a later time, ere yet the Boy
Had put on Boy's attire, did Michael love,
Albeit of a stern unbending mind,
To have the young one in his sight, when he
Had work by his own door, or when he sate
With sheep before him on his Shepherd's stool,
Beneath that large old Oak, which near their door
Stood, and from it's enormous breadth of shade
Chosen for the Shearer's covert from the sun,
Thence in our rustic dialect was call'd
The CLIPPING TREE, a name which yet it bears.
There, while they two were sitting in the shade,
With others round them, earnest all and blithe,
Would Michael exercise his heart with looks
Of fond correction and reproof bestow'd
Upon the child, if he dislurb'd the sheep
By catching at their legs, or with his shouts
Scar'd them, while they lay still beneath the shears.
And when by Heaven's good grace the Boy grew up
A healthy Lad, and carried in his cheek
Two steady roses that were five years old,
Then Michael from a winter coppice cut
With his own hand a sapling, which he hoop'd
With iron, making it throughout in all
Due requisites a perfect Shepherd's Staff,
And gave it to the Boy; wherewith equipp'd
He as a Watchman oftentimes was plac'd
At gate or gap, to stem or turn the flock,
And to his office prematurely call'd
There stood the urchin, as you will divine,
Something between a hindrance and a help,
And for this cause not always, I believe,
Receiving from his Father hire of praise.
While this good household thus were living on
From day to day, to Michael's ear there came
Distressful tidings. Long before, the time
Of which I speak, the Shepherd had been bound
In surety for his Brother's Son, a man
Of an industrious life, and ample means,
But unforeseen misfortunes suddenly
Had press'd upon him, and old Michael now
Was summon'd to discharge the forfeiture,
A grievous penalty, but little less
Than half his substance. This un-look'd-for claim
At the first hearing, for a moment took
More hope out of his life than he supposed
That any old man ever could have lost.
As soon as he had gather'd so much strength
That he could look his trouble in the face,
It seem'd that his sole refuge was to sell
A portion of his patrimonial fields.
Such was his first resolve; he thought again,
And his heart fail'd him. "Isabel," said he,
Two evenings after he had heard the news,
"I have been toiling more than seventy years,
And in the open sun-shine of God's love
Have we all liv'd, yet if these fields of ours
Should pass into a Stranger's hand, I think
That I could not lie quiet in my grave."
"Our lot is a hard lot; the Sun itself
Has scarcely been more diligent than I,
And I have liv'd to be a fool at last
To my own family. An evil Man
That was, and made an evil choice, if he
Were false to us; and if he were not false,
There are ten thousand to whom loss like this
Had been no sorrow.
When I began, my purpose was to speak
Of remedies and of a chearful hope."
"Our Luke shall leave us, Isabel; the land
Shall not go from us, and it shall be free,
He shall possess it, free as is the wind
That passes over it. We have, thou knowest,
Another Kinsman, he will be our friend
In this distress. He is a prosperous man,
Thriving in trade, and Luke to him shall go,
And with his Kinsman's help and his own thrift,
He quickly will repair this loss, and then
May come again to us. If here he stay,
What can be done? Where every one is poor
What can be gain'd?" At this, the old man paus'd,
And Isabel sate silent, for her mind
Was busy, looking back into past times.
There's Richard Bateman, thought she to herself,
He was a parish-boy—at the church-door
They made a gathering for him, shillings, pence,
And halfpennies, wherewith the Neighbours bought
A Basket, which they fill'd with Pedlar's wares,
And with this Basket on his arm, the Lad
Went up to London, found a Master there,
Who out of many chose the trusty Boy
To go and overlook his merchandise
Beyond the seas, where he grew wond'rous rich,
And left estates and monies to the poor,
And at his birth-place built a Chapel, floor'd
With Marble, which he sent from foreign lands.
These thoughts, and many others of like sort,
Pass'd quickly thro' the mind of Isabel,
And her face brighten'd. The Old Man was glad.
And thus resum'd. "Well I Isabel, this scheme
These two days has been meat and drink to me.
Far more than we have lost is left us yet.
—We have enough—I wish indeed that I
Were younger, but this hope is a good hope.
—Make ready Luke's best garments, of the best
Buy for him more, and let us send him forth
To-morrow, or the next day, or to-night:
—If he could go, the Boy should go to-night."
Here Michael ceas'd, and to the fields went forth
With a light heart. The House-wife for five days
Was restless morn and night, and all day long
Wrought on with her best fingers to prepare
Things needful for the journey of her Son.
But Isabel was glad when Sunday came
To stop her in her work; for, when she lay
By Michael's side, she for the two last nights
Heard him, how he was troubled in his sleep:
And when they rose at morning she could see
That all his hopes were gone. That day at noon
She said to Luke, while they two by themselves
Were sitting at the door, "Thou must not go,
We have no other Child but thee to lose,
None to remember—do not go away,
For if thou leave thy Father he will die."
The Lad made answer with a jocund voice,
And Isabel, when she had told her fears,
Recover'd heart. That evening her best fare
Did she bring forth, and all together sate
Like happy people round a Christmas fire.
Next morning Isabel resum'd her work,
And all the ensuing week the house appear'd
As cheerful as a grove in Spring: at length
The expected letter from their Kinsman came,
With kind assurances that he would do
His utmost for the welfare of the Boy,
To which requests were added that forthwith
He might be sent to him. Ten times or more
The letter was read over; Isabel
Went forth to shew it to the neighbours round:
Nor was there at that time on English Land
A prouder heart than Luke's. When Isabel
Had to her house return'd, the Old Man said,
"He shall depart to-morrow." To this word
The House—wife answered, talking much of things
Which, if at such, short notice he should go,
Would surely be forgotten. But at length
She gave consent, and Michael was at ease.
Near the tumultuous brook of Green-head Gill,
In that deep Valley, Michael had design'd
To build a Sheep-fold, and, before he heard
The tidings of his melancholy loss,
For this same purpose he had gathered up
A heap of stones, which close to the brook side
Lay thrown together, ready for the work.
With Luke that evening thitherward he walk'd;
And soon as they had reach'd the place he stopp'd,
And thus the Old Man spake to him. "My Son,
To-morrow thou wilt leave me; with full heart
I look upon thee, for thou art the same
That wert a promise to me ere thy birth,
And all thy life hast been my daily joy.
I will relate to thee some little part
Of our two histories; 'twill do thee good
When thou art from me, even if I should speak
Of things thou caust not know of.—After thou
First cam'st into the world, as it befalls
To new-born infants, thou didst sleep away
Two days, and blessings from thy Father's tongue
Then fell upon thee. Day by day pass'd on,
And still I lov'd thee with encreasing love."
Never to living ear came sweeter sounds
Than when I heard thee by our own fire-side
First uttering without words a natural tune,
When thou, a feeding babe, didst in thy joy
Sing at thy Mother's breast. Month follow'd month,
And in the open fields my life was pass'd
And in the mountains, else I think that thou
Hadst been brought up upon thy father's knees.
—But we were playmates, Luke; among these hills,
As well thou know'st, in us the old and young
Have play'd together, nor with me didst thou
Lack any pleasure which a boy can know.
Luke had a manly heart; but at these words
He sobb'd aloud; the Old Man grasp'd his hand,
And said, "Nay do not take it so—I see
That these are things of which I need not speak.
—Even to the utmost I have been to thee
A kind and a good Father: and herein
I but repay a gift which I myself
Receiv'd at others' hands, for, though now old
Beyond the common life of man, I still
Remember them who lov'd me in my youth."
Both of them sleep together: here they liv'd
As all their Forefathers had done, and when
At length their time was come, they were not loth
To give their bodies to the family mold.
I wish'd that thou should'st live the life they liv'd.
But 'tis a long time to look back, my Son,
And see so little gain from sixty years.
These fields were burthen'd when they came to me;
'Till I was forty years of age, not more
Than half of my inheritance was mine.
"I toil'd and toil'd; God bless'd me in my work,
And 'till these three weeks past the land was free.
—It looks as if it never could endure
Another Master. Heaven forgive me, Luke,
If I judge ill for thee, but it seems good
That thou should'st go." At this the Old Man paus'd,
Then, pointing to the Stones near which they stood,
Thus, after a short silence, he resum'd:
"This was a work for us, and now, my Son,
It is a work for me. But, lay one Stone—
Here, lay it for me, Luke, with thine own hands.
I for the purpose brought thee to this place."
Nay, Boy, be of good hope:—we both may live
To see a better day. At eighty-four
I still am strong and stout;—do thou thy part,
I will do mine.—I will begin again
With many tasks that were resign'd to thee;
Up to the heights, and in among the storms,
Will I without thee go again, and do
All works which I was wont to do alone,
Before I knew thy face.—Heaven bless thee, Boy!
Thy heart these two weeks has been beating fast
With many hopes—it should be so—yes—yes—
I knew that thou could'st never have a wish
To leave me, Luke, thou hast been bound to me
Only by links of love, when thou art gone
What will be left to us!—But, I forget
My purposes. Lay now the corner-stone,
As I requested, and hereafter, Luke,
When thou art gone away, should evil men
Be thy companions, let this Sheep-fold be
Thy anchor and thy shield; amid all fear
And all temptation, let it be to thee
An emblem of the life thy Fathers liv'd,
Who, being innocent, did for that cause
Bestir them in good deeds. Now, fare thee well—
When thou return'st, thou in this place wilt see
A work which is not here, a covenant
'Twill be between us—but whatever fate
Befall thee, I shall love thee to the last,
And bear thy memory with me to the grave.
The Shepherd ended here; and Luke stoop'd down,
And as his Father had requested, laid
The first stone of the Sheep-fold; at the sight
The Old Man's grief broke from him, to his heart
He press'd his Son, he kissed him and wept;
And to the House together they return'd.
Next morning, as had been resolv'd, the Boy
Began his journey, and when he had reach'd
The public Way, he put on a bold face;
And all the Neighbours as he pass'd their doors
Came forth, with wishes and with farewell pray'rs,
That follow'd him 'till he was out of sight.

Both parents read them with rejoicing hearts.
So, many months pass'd on: and once again
The Shepherd went about his daily work
With confident and cheerful thoughts; and now
Sometimes when he could find a leisure hour
He to that valley took his way, and there
Wrought at the Sheep-fold. Meantime Luke began
To slacken in his duty, and at length
He in the dissolute city gave himself
To evil courses: ignominy and shame
Fell on him, so that he was driven at last
To seek a hiding-place beyond the seas.
There is a comfort in the strength of love;
'Twill make a thing endurable, which else
Would break the heart:—Old Michael found it so.
I have convers'd with more than one who well
Remember the Old Man, and what he was
Years after he had heard this heavy news.
His bodily frame had been from youth to age
Of an unusual strength. Among the rocks
He went, and still look'd up upon the sun.
And listen'd to the wind; and as before
Perform'd all kinds of labour for his Sheep,
And for the land his small inheritance.
And to that hollow Dell from time to time
Did he repair, to build the Fold of which
His flock had need. 'Tis not forgotten yet
The pity which was then in every heart
For the Old Man—ands 'tis believ'd by all
That many and many a day he thither went,
And never lifted up a single stone.
There, by the Sheep-fold, sometimes was he seen
Sitting alone, with that his faithful Dog,
Then old, beside him, lying at his feet.
The length of full seven years from time to time
He at the building of this Sheep-fold wrought,
And left the work unfinished when he died.
Three years, or little more, did Isabel,
Survive her Husband: at her death the estate
Was sold, and went into a Stranger's hand.
The Cottage which was nam'd The Evening Star
Is gone, the ploughshare has been through the ground
On which it stood; great changes have been wrought
In all the neighbourhood, yet the Oak is left
That grew beside their Door; and the remains
Of the unfinished Sheep-fold may be seen
Beside the boisterous brook of Green-head Gill.


2.1: Data For This Chapter
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This chapter, like every subsequent chapter in this book, is filled with exercises. Processing real data is the best way to learn ERP data analysis! Your first step in each chapter will therefore be to download the data for the exercises in that chapter.
The N400 data we’ll be using for the exercises in this chapter can be found in the Chapter_2 folder in the master folder: https://doi.org/10.18115/D50056 . In this chapter, we’ll be looking at the data from only one of the 40 participants (Subject #6). Go ahead and download this folder and all its contents (and make sure to name the folder Chapter_2 ).
I recommend organizing the data for the various chapters of the book on your computer just as they're organized online, with a separate folder for each chapter.


8.1: Equivalent Sets
We have seen several interesting examples of equivalence relations already, and in this section we will explore one more: we’ll say two sets are equivalent if they have the same number of elements. Usually, an equivalence relation has the effect that it highlights one characteristic of the objects being studied, while ignoring all the others. Equivalence of sets brings the issue of size (a.k.a. cardinality) into sharp focus while, at the same time, it forgets all about the many other features of sets. Sets that are equivalent (under the relation we are discussing) are sometimes said to be equinumerous 1 .
A couple of examples may be in order.
- If \(A = \{1, 2, 3\}\) and \(B = \{a, b, c\}\) then \(A\) and \(B\) are equivalent.
- Since the empty set is unique – \(∅\) is the only set having \(0\) elements – it follows that there are no other sets equivalent to it.
- Every singleton set 2 is equivalent to every other singleton set.
Hopefully, these examples are relatively self-evident. Unfortunately, that very self-evidence may tend to make you think that this notion of equivalence isn’t all that interesting — nothing could be further from the truth! The notion of equivalence of sets becomes really interesting when we study infinite sets. Once we have the right definition in hand we will be able to prove some truly amazing results. For instance, the sets \(\mathbb{N}\) and \(\mathbb{Q}\) turn out to be equivalent. Since the naturals are wholly contained in the rationals this is (to say the least) counter-intuitive! Coming up with the “right” definition for this concept is crucial.
We could make the following:
(Well . . . not quite.) For all sets \(A\) and \(B\), we say \(A\) and \(B\) are equivalent , and write \(A ≡ B\) iff \(|A| = |B|\).
The problem with this definition is that it is circular. We’re trying to come up with an equivalence relation so that the equivalence classes will represent the various cardinalities of sets (i.e. their sizes) and we define the relation in terms of cardinalities. We won’t get anything new from this.
Georg Cantor was the first person to develop the modern notion of the equivalence of sets. His early work used the notion implicitly, but when he finally developed the concept of one-to-one correspondences in an explicit way he was able to prove some amazing facts. The phrase “one-to-one correspondence” has a fairly impressive ring to it, but one can discover what it means by just thinking carefully about what it means to count something.
Consider the solmization syllables used for the notes of the major scale in music; they form the set \(\{\text{do}, \text{ re}, \text{ mi}, \text{ fa}, \text{ so}, \text{ la}, \text{ ti}\}\). What are we doing when we count this set (and presumably come up with a total of \(7\) notes)? We first point at ‘do’ while saying ‘one,’ then point at ‘re’ while saying ‘two,’ et cetera. In a technical sense, we are creating a one-to-one correspondence between the set containing the seven syllables and the special set \(\{1, 2, 3, 4, 5, 6, 7\}\). You should notice that this one-to-one correspondence is by no means unique. For instance, we could have counted the syllables in reverse — a descending scale, or in some funny order – a little melody using each note once. The fact that there are seven syllables in the solmization of the major scale is equivalent to saying that there exists a one-to-one correspondence between the syllables and the special set \(\{1, 2, 3, 4, 5, 6, 7\}\). Saying “there exists” in this situation may seem a bit weak since in fact there are \(7! = 5040\) correspondences, but “there exists” is what we really want here. What exactly is a one-to-one correspondence? Well, we’ve actually seen such things before – a one-to-one correspondence is really just a bijective function between two sets.
For all sets \(A\) and \(B\), we say \(A\) and \(B\) are equivalent, and write \(A ≡ B\) iff there exists a one-to-one (and onto) function \(f\), with \(\text{Dom}(f) = A\) and \(\text{Rng}(f) = B\).
Somewhat more succinctly, one can just say the sets are equivalent iff there is a bijection between them.
We are going to ask you to prove that the above definition defines an equivalence relation in the exercises for this section. In order to give you a bit of a jump start on that proof we’ll outline what the proof that the relation is symmetric should look like.
To show that the relation is symmetric we must assume that \(A\) and \(B\) are sets with \(A ≡ B\) and show that this implies that \(B ≡ A\). According to the definition above this means that we’ll need to locate a function (that is one-to-one) from \(B\) to \(A\). On the other hand, since it is given that \(A ≡ B\), the definition tells us that there actually is an injective function, \(f\), from \(A\) to \(B\). The inverse function \(f^{−1}\) would do exactly what we’d like (namely form a map from \(B\) to \(A\)) assuming that we can show that \(f^{−1}\) has the right properties. We need to know that \(f^{−1}\) is a function (remember that in general the inverse of a function is only a relation) and that it is one-to-one. That \(f^{−1}\) is a function is a consequence of the fact that \(f\) is one-to-one. That \(f^{−1}\) is one-toone is a consequence of the fact that f is a function.
The above is just a sketch of a proof. In the exercise, you’ll need to fill in the rest of the details as well as provide similar arguments for reflexivity and transitivity.
For each possible finite cardinality \(k\), there are many, many sets having that cardinality, but there is one set that stands out as the most basic – the set of numbers from \(1\) to k. For each cardinality \(k > 0\), we use the symbol \(\mathbb{N}_k\) to indicate this set:
\(\mathbb{N}_k = \{1, 2, 3, . . . , k\}.\)
The finite cardinalities are the equivalence classes (under the relation of set equivalence) containing the empty set and the sets \(\mathbb{N}_k\). Of course, there are also infinite sets! The prototype for an infinite set would have to be the entire set \(\mathbb{N}\). The long-standing tradition is to use the symbol \(ℵ_0\) 3 for the cardinality of sets having the same size as \(\mathbb{N}\), alternatively, such sets are known as “countable.” One could make a pretty good argument that it is the finite sets that are actually countable! After all, it would literally take forever to count the natural numbers! We have to presume that the people who instituted this terminology meant for “countable” to mean “countable, in principle” or “countable if you’re willing to let me keep counting forever” or maybe “countable if you can keep counting faster and faster and are capable of ignoring the speed of light limitations on how fast your lips can move.” Worse yet, the term “countable” has come to be used for sets whose cardinalities are either finite or the size of the naturals. If we want to refer specifically to the infinite sort of countable set most mathematicians use the term denumerable (although this is not universal) or countably infinite . Finally, there are sets whose cardinalities are bigger than the naturals. In other words, there are sets such that no one-to-one correspondence with \(\mathbb{N}\) is possible. We don’t mean that people have looked for one-to-one correspondences between such sets and \(\mathbb{N}\) and haven’t been able to find them – we literally mean that it can’t be done; and it is has been proved that it can’t be done! Sets having cardinalities that are this ridiculously huge are known as uncountable .
Exercises:
Name four sets in the equivalence class of \(\{1, 2, 3\}\).
Prove that set equivalence is an equivalence relation.
Construct a Venn diagram showing the relationships between the sets of sets which are finite, infinite, countable, denumerable, and uncountable.
Place the sets \(\mathbb{N}\), \(\mathbb{R}\), \(\mathbb{Q}\), \(\mathbb{Z}\), \(\mathbb{Z} × \mathbb{Z}\), \(\mathbb{C}\), \(\mathbb{N}_{2007}\) and \(∅\); somewhere on the Venn diagram above. (Note to students (and graders): there are no wrong answers to this question, the point is to see what your intuition about these sets says at this point.)


3.3: Triple Integrals
Our definition of a double integral of a real-valued function \(f (x, y)\) over a region \(R\) in \(\mathbb{R}^2\) can be extended to define a triple integral of a real-valued function \(f (x, y, z)\) over a solid \(S\) in \(\mathbb{R}^ 3\). We simply proceed as before: the solid \(S\) can be enclosed in some rectangular parallelepiped, which is then divided into subparallelepipeds. In each subparallelepiped inside \(S\), with sides of lengths \(\Delta x, \Delta y \text{ and }\Delta z\), pick a point \((x∗, y∗, z∗)\). Then define the triple integral of \(f (x, y, z)\) over \(S\), denoted by \(\iiint\limits_S f (x, y, z)dV\), by
\[\iiint\limits_S f (x, y, z)dV=\lim \sum\sum\sum f (x∗, y∗, z∗)\Delta x \Delta y \Delta z \label{Eq3.7}\]
where the limit is over all divisions of the rectangular parallelepiped enclosing \(S\) into subparallelepipeds whose largest diagonal is going to 0, and the triple summation is over all the subparallelepipeds inside \(S\). It can be shown that this limit does not depend on the choice of the rectangular parallelepiped enclosing \(S\). The symbol \(dV\) is often called the volume element .
Physically, what does the triple integral represent? We saw that a double integral could be thought of as the volume under a two-dimensional surface. It turns out that the triple integral simply generalizes this idea: it can be thought of as representing the hypervolume under a three-dimensional hypersurface \(w = f (x, y, z)\) whose graph lies in \(\mathbb{R}^ 4\). In general, the word “volume” is often used as a general term to signify the same concept for any \(n\)-dimensional object (e.g. length in \(\mathbb{R}^1\), area in \(\mathbb{R}^2\) ). It may be hard to get a grasp on the concept of the “volume” of a four-dimensional object, but at least we now know how to calculate that volume!
In the case where \(S\) is a rectangular parallelepiped \([x_1 , x_2] \times [y_1 , y_2] \times [z_1 , z_2]\), that is, \(S = {(x, y, z) : x_1 ≤ x ≤ x_2 , y_1 ≤ y ≤ y_2 , z_1 ≤ z ≤ z_2}\), the triple integral is a sequence of three iterated integrals, namely
\[\iiint\limits_S f (x, y, z)dV = \int_{z_1}^{z_2} \int_{y_1}^{y_2} \int_{x_1}^{x_2} f (x, y, z)\,dx\, d y\, dz \label{Eq3.8}\]
where the order of integration does not matter. This is the simplest case.
A more complicated case is where \(S\) is a solid which is bounded below by a surface \(z = g_1(x, y)\), bounded above by a surface \(z = g_2(x, y)\), \(y\) is bounded between two curves \(h_1(x) \text{ and }h_2(x)\), and \(x\) varies between \(a \text{ and }b\). Then
\[\iiint\limits_S f (x, y, z)dV = \int_a^b \int_{h_1(x)}^{h_2(x)} \int_{g_1(x,y)}^{g_2(x,y)}f (x, y, z)\,dz\, d y\, dx \label{Eq3.9}\]
Notice in this case that the first iterated integral will result in a function of \(x\) and \(y\) (since its limits of integration are functions of \(x \text{ and }y\)), which then leaves you with a double integral of a type that we learned how to evaluate in Section 3.2. There are, of course, many variations on this case (for example, changing the roles of the variables \(x, y, z\)), so as you can probably tell, triple integrals can be quite tricky. At this point, just learning how to evaluate a triple integral, regardless of what it represents, is the most important thing. We will see some other ways in which triple integrals are used later in the text.
Example 3.7
Evaluate \( \displaystyle \int_0^3 \int_0^2 \int_0^1 (x y+ z)\,dx\, d y\, dz\).
\[\nonumber \begin{align} \int_0^3 \int_0^2 \int_0^1 (x y+ z)\,dx\, d y\, dz &= \int_0^3 \int_0^2 \left ( \dfrac{1}{2}x^2y+xz \big |_{x=0}^{x=1} \right )\,dy\,dz \\[4pt] \nonumber &=\int_0^3 \int_0^2 \left ( \dfrac{1}{2}y+z \right )\,dy\,dz \\[4pt] \nonumber &= \int_0^3 \left ( \dfrac{1}{4}y^2+yz \big |_{y=0}^{y=2} \right )\,dz \\[4pt] \nonumber &= \int_0^3 (1+2z)\,dz \\[4pt] \nonumber &=z+z^2 \big |_0^3 = 12 \end{align} \]
Example 3.8
Evaluate \(\displaystyle \int_0^1 \int_0^{1-x} \int_0^{2-x-y} (x+ y+ z)\,dz\, d y\, dx\).
Solution
\[\nonumber \begin{align} \int_0^1 \int_0^{1-x} \int_0^{2-x-y}(x+ y+ z)\,dz\, d y\, dx &= \int_0^1 \int_0^{1-x}\left ((x+ y)z + \dfrac{1}{2}z^2 \big |_{z=0}^{z=2-x-y} \right )\,dy\,dx \\[4pt] \nonumber &=\int_0^1 \int_0^{1-x} \left ( (x+ y)(2− x− y)+ \dfrac{1}{2} (2− x− y)^2 \right )\,dy\,dx \\[4pt] \nonumber &= \int_0^1 \int_0^{1-x}\left ( 2− \dfrac{1}{2}x^2 − x y− \dfrac{1}{2}y^2 \right ) \, dy\,dx \\[4pt] \nonumber &= \int_0^1 \left ( 2y− \dfrac{1}{2} x^2 y− x y− \dfrac{1}{2} x y^2 − \dfrac{1}{6}y^3 \big |_{y=0}^{y=1-x} \right )\,dx \\[4pt] \nonumber &=\int_0^1 \left ( \dfrac{11}{6}-2x+\dfrac{1}{6}x^3 \right )\,dx \\[4pt] \nonumber &= \dfrac{11}{6}x-x^2+\dfrac{1}{24}x^4 \big |_0^1 = \dfrac{7}{8} \end{align}\]
Note that the volume \(V\) of a solid in \(\mathbb{R}^ 3\) is given by
\[V= \iiint\limits_S 1dV \label{Eq3.10}\]
Since the function being integrated is the constant 1, then the above triple integral reduces to a double integral of the types that we considered in the previous section if the solid is bounded above by some surface \(z = f (x, y)\) and bounded below by the \(x y\)-plane \(z = 0\). There are many other possibilities. For example, the solid could be bounded below and above by surfaces \(z = g_1(x, y)\text{ and }z = g_2(x, y)\), respectively, with \(y\) bounded between two curves \(h_1(x)\text{ and }h_2(x)\), and \(x\) varies between \(a \text{ and }b\). Then
\[\begin{align*} V &=\iiint\limits_S 1dV \\[4pt] &= \int_a^b \int_{h_1(x)}^{h_2(x)} \int_{g_1(x,y)}^{g_2(x,y)} 1\,dz\, d y\, dx \\[4pt] &= \int_a^b \int_{h_1(x)}^{h_2(x)} (g_2(x, y)− g_1(x, y)) \,d y\, dx \end{align*} \]
just like in Equation \ref{Eq3.9}. See Exercise 10 for an example.


16.2: Sources of Sunshine- Thermal and Gravitational Energy
By the end of this section, you will be able to:
- Identify different forms of energy
- Understand the law of conservation of energy
- Explain ways that energy can be transformed
Energy is a challenging concept to grasp because it exists in so many different forms that it defies any single simple explanation. In many ways, comprehending energy is like comprehending wealth: There are very different forms of wealth and they follow different rules, depending on if they are the stock market, real estate, a collection of old comic books, great piles of cash, or one of the many other ways to make and lose money. It is easier to discuss one or two forms of wealth—or energy—than to discuss that concept in general.
When striving to understand how the Sun can put out so much energy for so long, scientists considered many different types of energy. Nineteenth-century scientists knew of two possible sources for the Sun’s energy: chemical and gravitational energy. The source of chemical energy most familiar to them was the burning (the chemical term is oxidation ) of wood, coal, gasoline, or other fuel. We know exactly how much energy the burning of these materials can produce. We can thus calculate that even if the immense mass of the Sun consisted of a burnable material like coal or wood, our star could not produce energy at its present rate for more than few thousand years. However, we know from geologic evidence that water was present on Earth’s surface nearly 4 billion years ago, so the Sun must have been shining brightly (and making Earth warm) at least as long as that. Today, we also know that at the temperatures found in the Sun, nothing like solid wood or coal could survive.
Just a word about the units we are using. A watt (W) is a unit of power , which is energy used or given off per unit time. It is measured in joules per second (J/s). You know from your everyday experience that it is not just how much energy you expend, but how long you take to do it. (Burning 10 Calories in 10 minutes requires a very different kind of exercise than burning those 10 Calories in an hour.) Watts tell you the rate at which energy is being used; for example, a 100-watt bulb uses 100 joules (J) of energy every second.
And how big is a joule? A 73-kilogram (160-pound) astronomy instructor running at about 4.4 meters per second (10 miles per hour) because he is late for class has a motion energy of about 700 joules.
Conservation of Energy
Other nineteenth-century attempts to determine what makes the Sun shine used the law of conservation of energy . Simply stated, this law says that energy cannot be created or destroyed, but can be transformed from one type to another, such as from heat to mechanical energy. The steam engine, which was key to the Industrial Revolution, provides a good example. In this type of engine, the hot steam from a boiler drives the movement of a piston, converting heat energy into motion energy.
Conversely, motion can be transformed into heat. If you clap your hands vigorously at the end of an especially good astronomy lecture, your palms become hotter. If you rub ice on the surface of a table, the heat produced by friction melts the ice. The brakes on cars use friction to reduce speed, and in the process, transform motion energy into heat energy. That is why after bringing a car to a stop, the brakes can be very hot; this also explains why brakes can overheat when used carelessly while descending long mountain roads.
In the nineteenth century, scientists thought that the source of the Sun’s heat might be the mechanical motion of meteorites falling into it. Their calculations showed, however, that in order to produce the total amount of energy emitted by the Sun, the mass in meteorites that would have to fall into the Sun every 100 years would equal the mass of Earth. The resulting increase in the Sun’s mass would, according to Kepler’s third law, change the period of Earth’s orbit by 2 seconds per year. Such a change would be easily measurable and was not, in fact, occurring. Scientists could then disprove this as the source of the Sun’s energy.
Gravitational Contraction as a Source of Energy
Proposing an alternative explanation, British physicist Lord Kelvin and German scientist Hermann von Helmholtz (Figure 16.2), in about the middle of the nineteenth century, proposed that the Sun might produce energy by the conversion of gravitational energy into heat. They suggested that the outer layers of the Sun might be “falling” inward because of the force of gravity. In other words, they proposed that the Sun could be shrinking in size, staying hot and bright as a result.
To imagine what would happen if this hypothesis were true, picture the outer layer of the Sun starting to fall inward. This outer layer is a gas made up of individual atoms, all moving about in random directions. If a layer falls inward, the atoms acquire an additional speed because of falling motion. As the outer layer falls inward, it also contracts, moving the atoms closer together. Collisions become more likely, and some of them transfer the extra speed associated with the falling motion to other atoms. This, in turn, increases the speeds of those atoms. The temperature of a gas is a measure of the kinetic energy (motion) of the atoms within it; hence, the temperature of this layer of the Sun increases. Collisions also excite electrons within the atoms to higher-energy orbits. When these electrons return to their normal orbits, they emit photons, which can then escape from the Sun (see Radiation and Spectra).
Kelvin and Helmholtz calculated that a contraction of the Sun at a rate of only about 40 meters per year would be enough to produce the amount of energy that it is now radiating. Over the span of human history, the decrease in the Sun’s size from such a slow contraction would be undetectable.
If we assume that the Sun began its life as a large, diffuse cloud of gas, then we can calculate how much energy has been radiated by the Sun during its entire lifetime as it has contracted from a very large diameter to its present size. The amount of energy is on the order of 10 42 joules. Since the solar luminosity is watts (joules/second) or about joules per year, contraction could keep the Sun shining at its present rate for roughly 100 million years.
In the nineteenth century, 100 million years at first seemed plenty long enough, since Earth was then widely thought to be much younger than this. But toward the end of that century and into the twentieth, geologists and physicists showed that Earth (and, hence, the Sun) is actually much older. Contraction therefore cannot be the primary source of solar energy (although, as we shall see in The Birth of Stars and the Discovery of Planets Outside the Solar System, contraction is an important source of energy for a while in stars that are just being born). Scientists were thus confronted with a puzzle of enormous proportions. Either an unknown type of energy was responsible for the most important energy source known to humanity, or estimates of the age of the solar system (and life on Earth) had to be seriously modified. Charles Darwin, whose theory of evolution required a longer time span than the theories of the Sun seemed to permit, was discouraged by these results and continued to worry about them until his death in 1882.
It was only in the twentieth century that the true source of the Sun’s energy was identified. The two key pieces of information required to solve the puzzle were the structure of the nucleus of the atom and the fact that mass can be converted into energy.


11.3: The Effects of pH on Microbial Growth
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- Ying Liu
- City College of San Francisco
Learning Objectives
- Illustrate and briefly describe minimum, optimum, and maximum pH requirements for growth
- Identify and describe the different categories of microbes with pH requirements for growth: acidophiles, neutrophiles, and alkaliphiles
- Give examples of microorganisms for each category of pH requirement
Yogurt, pickles, sauerkraut, and lime-seasoned dishes all owe their tangy taste to a high acid content (Figure \(\PageIndex{1}\)). Recall that acidity is a function of the concentration of hydrogen ions [H + ] and is measured as pH. Environments with pH values below 7.0 are considered acidic, whereas those with pH values above 7.0 are considered basic. Extreme pH affects the structure of all macromolecules. The hydrogen bonds holding together strands of DNA break up at high pH. Lipids are hydrolyzed by an extremely basic pH. The proton motive force responsible for production of ATP in cellular respiration depends on the concentration gradient of H + across the plasma membrane (see Cellular Respiration ). If H + ions are neutralized by hydroxide ions, the concentration gradient collapses and impairs energy production. But the component most sensitive to pH in the cell is its workhorse, the protein. Moderate changes in pH modify the ionization of amino-acid functional groups and disrupt hydrogen bonding, which, in turn, promotes changes in the folding of the molecule, promoting denaturation and destroying activity.
The optimum growth pH is the most favorable pH for the growth of an organism. The lowest pH value that an organism can tolerate is called the minimum growth pH and the highest pH is the maximum growth pH. These values can cover a wide range, which is important for the preservation of food and to microorganisms’ survival in the stomach. For example, the optimum growth pH of Salmonella spp. is 7.0–7.5, but the minimum growth pH is closer to 4.2.
Most bacteria are neutrophiles, meaning they grow optimally at a pH within one or two pH units of the neutral pH of 7 (see Figure \(\PageIndex{2}\)). Most familiar bacteria, like Escherichia coli , staphylococci, and Salmonella spp. are neutrophiles and do not fare well in the acidic pH of the stomach. However, there are pathogenic strains of E. coli, S. typhi, and other species of intestinal pathogens that are much more resistant to stomach acid. In comparison, fungi thrive at slightly acidic pH values of 5.0–6.0.
Microorganisms that grow optimally at pH less than 5.55 are called acidophiles. For example, the sulfur-oxidizing Sulfolobus spp. isolated from sulfur mud fields and hot springs in Yellowstone National Park are extreme acidophiles. These archaea survive at pH values of 2.5–3.5. Species of the archaean genus Ferroplasma live in acid mine drainage at pH values of 0–2.9. Lactobacillus bacteria, which are an important part of the normal microbiota of the vagina, can tolerate acidic environments at pH values 3.5–6.8 and also contribute to the acidity of the vagina (pH of 4, except at the onset of menstruation) through their metabolic production of lactic acid. The vagina’s acidity plays an important role in inhibiting other microbes that are less tolerant of acidity. Acidophilic microorganisms display a number of adaptations to survive in strong acidic environments. For example, proteins show increased negative surface charge that stabilizes them at low pH. Pumps actively eject H + ions out of the cells. The changes in the composition of membrane phospholipids probably reflect the need to maintain membrane fluidity at low pH.
At the other end of the spectrum are alkaliphiles, microorganisms that grow best at pH between 8.0 and 10.5. Vibrio cholerae , the pathogenic agent of cholera, grows best at the slightly basic pH of 8.0; it can survive pH values of 11.0 but is inactivated by the acid of the stomach. When it comes to survival at high pH, the bright pink archaean Natronobacterium , found in the soda lakes of the African Rift Valley, may hold the record at a pH of 10.5 (Figure \(\PageIndex{3}\)). Extreme alkaliphiles have adapted to their harsh environment through evolutionary modification of lipid and protein structure and compensatory mechanisms to maintain the proton motive force in an alkaline environment. For example, the alkaliphile Bacillus firmus derives the energy for transport reactions and motility from a Na + ion gradient rather than a proton motive force. Many enzymes from alkaliphiles have a higher isoelectric point, due to an increase in the number of basic amino acids, than homologous enzymes from neutrophiles.
Survival at the Low pH of the Stomach
Peptic ulcers (or stomach ulcers) are painful sores on the stomach lining. Until the 1980s, they were believed to be caused by spicy foods, stress, or a combination of both. Patients were typically advised to eat bland foods, take anti-acid medications, and avoid stress. These remedies were not particularly effective, and the condition often recurred. This all changed dramatically when the real cause of most peptic ulcers was discovered to be a slim, corkscrew-shaped bacterium, Helicobacter pylori . This organism was identified and isolated by Barry Marshall and Robin Warren, whose discovery earned them the Nobel Prize in Medicine in 2005.
The ability of H. pylori to survive the low pH of the stomach would seem to suggest that it is an extreme acidophile. As it turns out, this is not the case. In fact, H. pylori is a neutrophile. So, how does it survive in the stomach? Remarkably, H. pylori creates a microenvironment in which the pH is nearly neutral. It achieves this by producing large amounts of the enzyme urease, which breaks down urea to form NH 4 + and CO 2 . The ammonium ion raises the pH of the immediate environment.
This metabolic capability of H. pylori is the basis of an accurate, noninvasive test for infection. The patient is given a solution of urea containing radioactively labeled carbon atoms. If H. pylori is present in the stomach, it will rapidly break down the urea, producing radioactive CO 2 that can be detected in the patient’s breath. Because peptic ulcers may lead to gastric cancer, patients who are determined to have H. pylori infections are treated with antibiotics.
Exercise \(\PageIndex{1}\)
- What effect do extremes of pH have on proteins?
- What pH-adaptive type of bacteria would most human pathogens be?
Key Concepts and Summary
- Bacteria are generally neutrophiles . They grow best at neutral pH close to 7.0.
- Acidophiles grow optimally at a pH near 3.0. Alkaliphiles are organisms that grow optimally between a pH of 8 and 10.5. Extreme acidophiles and alkaliphiles grow slowly or not at all near neutral pH.
- Microorganisms grow best at their optimum growth pH . Growth occurs slowly or not at all below the minimum growth pH and above the maximum growth pH .


49 .
You have been assigned a semester-long research project covering a topic of your choosing. This topic must be something that directly impacts your college experience. For example, you might choose to research how students purchase books for classes, or student opinions of campus dining facilities. To get started, search for an appropriate business template and create a proposal for your research. Use the template as a guide to lay out the plan for the research project. Keep in mind that you will need to construct a report at the end of the project that includes the following sections: introduction, research question/problem, literature review (background information on the topic), methods used to collect data, results of the research, key recommendations, and conclusion.


15.6: Personalized vs. Standardized Service
- Compare and contrast personalized and standardized service
Now that we discussed different customer profiles, let’s turn to different types of customer service. During The Industrial Age, the advent of mass merchandising introduced the concept of ready-to-wear garments to the fashion industry. Before industrialization, clothing was handmade. Each dress, shirt, pant, or suit was individually measured, cut, and sewn by a tailor to fit one person. This process was both time-consuming and relatively expensive for most people. Shortly after the end of World War II, ready-to-wear, or “off-the-rack,” clothing made in factories became very popular, in part, because clothing manufacturers could make standardized outfits that were affordable and did not require any timely alterations. Most Americans viewed the standardization of the apparel industry as an advancement in service. (Source: Design Quarterly .)
Service Standardization
A recent study from the Center for Leading Innovation & Cooperation suggests that standardizing services will continue to play an important role in further developing the service economy, which includes retail. By making similar services with different characteristics and structures comparable, service standardization promotes reliability, effectiveness, trust, and improved economic efficiency. The CLIC defines standardization as a process of unification, especially in terminology, capabilities of personnel, technology, and organizational processes. In essence, “standardization simplifies everyday life.” (Source: CLIC – Service Standardization .)
Retailers often employ standardization strategies in store layouts, purchasing, labeling, branding, and a myriad of other operational activities. Retailers standardize customer service specifically to ensure uniform quality of treatment of all customers, both internal and external. Some examples of customer service standardization within the retail industry include:
- ensuring product availability,
- emphasizing customer courtesy,
- safeguarding branding consistency,
- practicing pricing accuracy,
- demonstrating promotional responsiveness,
- and enabling overall service efficiency. (Source: Customer Service Experts .)
Personalized Service
Ironically, many 21st century retailers might appear to be headed back in time, returning to the days of less service standardization and instead more personalized service. This, however, is not a business regression but instead another method of service differentiation. Personalization is a method of adding value to customer choice. In fact, personalization today often relies on technology to more effectively and efficiently implement a new range of services that would otherwise be impossible.
For example, the Container Store, a leading specialty retailer of storage and organization products in the United States, introduced a new “next generation” architectural design, merchandising strategy, and proprietary digital resources to its flagship store specifically to help customers accomplish their projects, maximize their space, and make the most of their homes on a personal basis. According to company officials, the technology components were chosen after many rounds of customer testing, which showed that customers’ biggest hurdle in beginning a project was feeling overwhelmed. The store features 18 digital screens supporting customers as they shop, featuring everything from inspiration and tips, to an interactive design tool and even a new proprietary digital experience called The Organization Studio. This technology allows customers to upload a photo or video of their organizational challenge online, describe the challenge, and set an in-store appointment to meet with a store Organization Expert. The Organization Experts then present the personalized solution they developed for the customer free of charge and with no purchase commitment.
“We know that countless retailers are building digital tools and using them to innovate the shopping experience, but combining the human element with technology is when things really get powerful,” said Val Richardson, Vice President of Real Estate at The Container Store. “The Organization Studio offers the convenience of an online experience and marries it with a personalized in-store engagement that offers a curated solution created by a real person.” (Source: containerstore.com .)
Contributors and Attributions
- Personalized vs. Standardized Service. Authored by : Wade Hyde. Provided by : Lumen Learning. License : CC BY: Attribution


16.5: Central projection
The central projection is analogous to the projective model of hyperbolic plane which is discussed in Chapter 17 .
Let \(\Sigma\) be the unit sphere centered at the origin which will be denoted by \(O\) . Suppose that \(\Pi^+\) denotes the plane defined by the equation \(z=1\) . This plane is parallel to the \(xy\) -plane and it passes thru the north pole \(N =(0,0,1)\) of \(\Sigma\) .
Recall that the northern hemisphere of \(\Sigma\) , is the subset of points \((x,y,z)\in \Sigma\) such that \(z>0\) . The northern hemisphere will be denoted by \(\Sigma^+\) .
Given a point \(P\in \Sigma^+\) , consider the half-line \([OP)\) . Suppose that \(P'\) denotes the intersection of \([OP)\) and \(\Pi^+\) . Note that if \(P=(x,y,z)\) , then \(P'=(\dfrac{x}{z},\dfrac{y}{z},1)\) . It follows that \(P\leftrightarrow P'\) is a bijection between \(\Sigma^+\) and \(\Pi^+\) .
The described bijection \(\Sigma^+ \leftrightarrow \Pi^+\) is called the central projection of the hemisphere \(\Sigma^+\) .
Note that the central projection sends the intersections of the great circles with \(\Sigma^+\) to the lines in \(\Pi^+\) . The latter follows since the great circles are intersections of \(\Sigma\) with planes passing thru the origin as well as the lines in \(\Pi^+\) are the intersection of \(\Pi^+\) with these planes.
The following exercise is analogous to Exercise 17.2.1 in hyperbolic geometry.
Let \(\triangle_sABC\) be a nondegenerate spherical triangle. Assume that the plane \(\Pi^+\) is parallel to the plane passing thru \(A\) , \(B\) , and \(C\) . Let \(A'\) , \(B'\) , and \(C'\) denote the central projections of \(A\) , \(B\) and \(C\) .
- Show that the midpoints of \([A'B']\) , \([B'C']\) , and \([C'A']\) are central projections of the midpoints of \([AB]_s\) , \([BC]_s\) , and \([CA]_s\) respectively.
- Use part (a) to show that the medians of a spherical triangle intersect at one point.
- Hint
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(a). Observe and use that \(OA' = OB' = OC'\).
(b). Note that the medians of spherical triangle ABC map to the medians of Euclidean a triangle \(A'B'C'\). It remains to apply Theorem 8.3.1 for \(\triangle A'B'C'\).