What is Calculus?: The Two Pillars

By Oscar Fernandez

This is the third in a series of short articles exploring calculus. The first article explored the origins of calculus, including the three “big problems” that drove calculus’ development. The second article explored limits, the foundation of calculus. This article discusses how limits help us solve the three “big problems” and introduces two of calculus’ pillars: derivatives and integrals.

In the first article in this series I discussed three Big Problems that drove the development of calculus: the instantaneous speed problem, the tangent line problem, and the area problem. I illustrated these via the figure below.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These problems stumped mathematicians for millennia. (We briefly talked about why in the first article.) But their inability to solve these problems—echoing Morpheus in the movie The Matrix—was not due to the techniques they were using; it was due to their mindset.

How a Dynamics Mindset Solves the Three Big Problems

If you’ve read the second article in this series, you’ll remember my first characterization of calculus: calculus is a dynamics mindset. Yet nothing about the figure above says “dynamics.” Every image is a static snapshot of something (e.g., an area). So let’s calculus the figure. (Yep, I’m encouraging you to think of calculus as a verb.)

The figure below takes each Big Problem from the figure above and adds in the dynamics.

Reprinted, with permission, from Calculus Simplified (Princeton University Press). Click to expand.

These images show apples falling, gray lines approaching a blue tangent line, and areas being swept out. Lots of movement (dynamics)! Moreover, notice that as the central change in each row of the figure gets closer to zero —the quantity ∆t in the first row and ∆x in the second and third rows—the resulting diagram approaches the respective diagram in the first figure in this article. We’ve met this “as ∆t  approaches zero” language before—it’s the language of limits we discussed in the second article! Adding this new revelation to the figure above produces…

 

Reprinted, with permission, from Calculus Simplified (Princeton University Press. Click to expand.

Finally, expressing our result in terms of equations involving limits yields the final piece of the puzzle…

Notice how each row employs a dynamics mindset to recast the Big Problem (contained in the “limiting picture” column) as the limit of a sequence of similar quantities (e.g., speeds) involving finite changes, changes which pre-calculus mathematics can handle. Specifically:

  • Row #1: The instantaneous speed of the falling apple is realized as the limit of its average speeds  ∆d / ∆t (ratios of changes in distance to changes in time) as ∆t —> 0.
  • Row #2: The slope of the tangent line is realized as the limit of the secant line slopes ∆y / ∆x (the gray lines in the figure) as ∆x —> 0.
  • Row #3: The area under the curve is realized as the limit as ∆x —> 0 of the area swept out from x = a up to ∆x  past b.

Introducing…Derivatives and Integrals

The limit obtained in the second row of the last figure is called the derivative of f(x) at x = a, the x-value of point P. The limit obtained in the third row of the Figure is called the definite integral of f(x) between x = a and x = b. Derivatives and integrals round out the three most important concepts in calculus (limits are the third).

You now have a working understanding of what derivatives and definite integrals are, what they measure, and how they arise from the application of a dynamics mindset to pre-calculus mathematics. The next post in this series will explore the derivative in greater details. We’ll discover that it has a nice geometric interpretation and a powerful real-world interpretation. (The last figure above hints to what these are.) Near the end of this series we will return to these interpretations to illustrate the power of derivatives, using them to help us understand phenomena as diverse as the fate of the Universe and, more pragmatically, how to find the best seat in a movie theater. Stay tuned!

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

Noah Webster’s civil war of words over American English

In the United States, the name Noah Webster (1758-1843) is synonymous with the word ‘dictionary’. But it is also synonymous with the idea of America, since his first unabridged American Dictionary of the English Language, published in 1828 when Webster was 70, blatantly stirred the young nation’s thirst for cultural independence from Britain.

Webster saw himself as a saviour of the American language who would rescue it from the corrupting influence of British English and prevent it from fragmenting into a multitude of dialects. But as a linguist and lexicographer, he quickly ran into trouble with critics, educators, the literati, legislators and much of the common reading public over the bizarre nature of his proposed language reforms. These spelling reforms – for example, wimmen for ‘women’, greeve for ‘grieve’, meen for ‘mean’ and bred for ‘bread’ – were all intended to simplify spelling by making it read the way that words were pronounced, yet they brought him the pain of ridicule for decades to come.

His definitions were regarded as his strong suit, but even they frequently rambled into essays, and many readers found them overly aligned with New England usage, to the point of distortion. Surfeited with a Christian reading of words, his religious or moral agenda also shaped many of his definitions into mini-sermons or moral lessons rather than serving as clarifications of meaning. A typical example is one of his expositions of purpose: ‘We believe the Supreme Being created intelligent beings for some benevolent and glorious purpose, and if so, how glorious and benevolent must be his purpose in the plan of redemption!’ Overall, his dictionary was prescriptive rather than descriptive, a violation, if you will, of a central tenet of lexicography that holds that dictionaries should record the way language is used, not the way the lexicographer thinks it should be used. 

Webster’s etymology, meanwhile, which he spent a decade dreaming up, was deeply flawed because of his ignorance of the exciting discoveries made by leading philologists in Europe about the evolution of Indo-European languages from roots such as Sanskrit. His etymologies conform entirely to the interpretation of words as presented in the Bible. He was convinced that ‘the primitive language of man’ spoken by the ‘descendants of Noah … must have been the original Chaldee’.

Webster fought his battles over language not within philology circles but within the larger context of an emerging American dialect (pejoratively dismissed by the British as provincialisms). He believed that increasing immigration, the multiplication of unique American words, the new meanings attaching to English words and the proliferation of slang – or, as the English saw it, vulgar and undisciplined language – made an American dictionary essential to American life.

New words came from several sources. Native Americans contributed wampum, moccasin, canoe, moose, toboggan and maize; from Mexico came hoosegow, stampede and cafeteria; from French, prairie and dime; meanwhile, cookie and landscape came from the Dutch. Existing words were combined to make new ones, for example rattlesnake, eggplant and bullfrog. Settlers of the West borrowed mesa and canyon from Spanish, and came up with robust words and expressions such as cahoots and kick the bucket. There were also entirely new words: gimmick, fudge, notify, currency, hindsight, graveyard, roundabout. Shakespearean and other Old World words returned: gotten (got), platter (plate), mad (angry). There were new spellings, too, a few of them of Webster’s own invention: some of those were preserved – specter (spectre) and  offense (offence) for example – but many more were mocked: wimmen (women), blud (blood), dawter (daughter). Idiomatic ‘tall talk’, as Daniel Boorstin called it in The Americans (1965)– the robust informality and ‘brash vitality’ often attacked by the British as vulgar Americanisms – thrived, especially out West: down-and-out, flat-footed, to affiliate, down-town, scrumptious and true-blue. Not surprisingly, the British worried that, one day, if this mushrooming of Americanisms continued, they would scarcely be able to understand Americans.

That didn’t happen. Because of high mobility and the blending of different cultures and backgrounds in the US, there were far fewer dialects or dramatically different pronunciations than in England, where isolation was more common in spite of the smallness of the country. 

The British thought that Samuel Johnson’s great Dictionary of the English Language (1755) would suffice for America as it did for Britain. Many Americans agreed, but many more wanted their own national dictionary to lend them a type of secular authority that was analogous to the spiritual authority of the Bible. But then there was the question of whose American dictionary would provide such an authority – which consideration instigated the ‘American dictionary wars’. Should Webster’s voice prevail, on behalf of the Americanising of English and the writing of dictionaries that would record such usage? Or would Webster’s great rival Joseph Emerson Worcester (1784-1865) with his more traditional, well-informed and solid scholarship triumph? Their conflict became America’s. What emerged in the country was an adversarial culture concerning language in which Americans fought each other in a civil war of words. It was also partly an ideological war, pitting various sectors of society – political, social, educational, religious – against each other over the direction that American English should take. 

Webster died before these wars were resolved, feeling that he had failed as a lexicographer (and a visionary), and disheartened by poor sales of his dictionaries. His legacy and eventual iconic standing was secured largely by his editors (chiefly Webster’s son-in-law Chauncey Allen Goodrich) and publishers (Charles and George Merriam) who began to remove most of his work from his dictionary while he was living, and continued the process over the 20 years following his death. The Merriams knew that Worcester was the superior lexicographer, but they recognised that Webster was more marketable because of his patriot credentials, so they dedicated themselves to cleaning up his dictionary and defeating Worcester in the marketplace.

Ultimately, the Merriams were the real winners in the American dictionary wars, having made a fortune from Webster’s name. Had Webster returned to see what had happened to his dictionary, he probably would have thought of himself as one of the big losers. Meanwhile, American English would pursue its own inevitable national development, with little help from him.Aeon counter – do not remove

Peter Martin is the author of numerous books, including the acclaimed biographies Samuel Johnson and A Life of James Boswell. He has taught English literature in the United States and England and divides his time between West Sussex, England, and Spain.

This article was originally published at Aeon and has been republished under Creative Commons.

Justin Smith: If reason exists without deliberation, it cannot be uniquely human

Philosophers and cognitive scientists today generally comprehend the domain of reason as a certain power of making inferences, confined to the thoughts and actions of human beings alone. Like echolocation in bats or photosynthesis in plants, reason is an evolved power, but unlike these, the prevailing theory goes, it emerged exactly once in the history of evolution (porpoises and shrews also echolocate, cyanobacteria photosynthesise). Reason is exceedingly rare, a hapax legomenon of nature, and yet this rarity has led to a bind: when pushed to account for its origins, thinkers who champion reason’s human-exclusivity are forced to lean on supernaturalism, while those who contend that reason is a fundamentally natural property have then to concede that ‘lower’ lifeforms are capable of exercising it. The question is – how?  

Most philosophers and scientists who see reason as some sort of inferential ability involving abstract representations will allow that experiments with ‘higher’ animals can yield evidence of some low-level reason-like faculty: for example when apes hide stones in anticipation of future conflicts. But researchers almost always draw the lower limit for such ability in a way that excludes species whose behaviour is not observably similar to ours. The search for reason beyond the bounds of the human species always ends up as a search for beings that remind us of ourselves.

But what if reason is not so much an inferential ability, as simply the power to do the right thing in the right circumstances? Furthermore, what if this power flows automatically, from simply being the sort of creature one is? This is, more or less, the view of the 16th-century diplomat Girolamo Rorario, as captured in the title of his principal, posthumous treatise, That Brute Animals Make Better Use of Reason than Men (1654). Rorario is representative of a little-known but venerable tradition of thought, extending back to Plutarch, that not only holds that reason is natural, but insists that reason is very widespread in nature indeed.

Rorario’s core idea is that human deliberation – the period of hesitancy when we survey our various options and eventually select what appears to be the best of them – far from being an advantage over other beings, is in fact a mark of our inferiority. Animals and plants do not hesitate. They cut right to the chase and, to the extent that they do not examine alternative options in order to choose among them, they are in a sense incapable of being wrong.

This is not to say that they are never foiled, that gazelles always take a path in fleeing the lion that assures their escape, or that vines always creep in the direction that will give them the most sunlight. It is just that, when they are foiled, this cannot be because they failed in their deliberation, since they do not deliberate. And still they seem to be doing just fine for themselves, pursuing their species-specific ends.

Potentially, it’s not just living beings that fall under the scope of this alternative interpretation of reason as the power to move directly to action, rather than the power of making the correct inference. For everything in nature also just does what it does, simply and without deliberation, by virtue of the fact that everything in nature is bound by the same physical laws. Nature just keeps working smoothly. It never, ever breaks down.

Nature itself is a rational order, on this alternative view, both as a whole and in any of its subdomains. Reason is everywhere, with human reason being only an instantiation or reflection, within a very tiny subdomain, of the universal reason that informs the natural world. The regularities of the motions of the heavens (to speak with the ancients) or the laws governing the orbits of the planets (to speak with the moderns) are not there in place of reason. Rather, these regularities or laws are the reflection ‘out there’ in the world of what human thought is ‘in here’ inside our minds.

If such a view seems irretrievably prescientific, note that among the things we find out there in the world are not just natural systems such as galaxies and tidepools, but also artificial systems such as reckoning machines and smartphones. And unlike the attribution of rationality to animals, the idea that machines make better use of reason than men, first fashionable in the 17th century, is still taken very seriously now.

In the 1670s, the German polymath G W Leibniz expressed the sincere hope that machines might soon be able to take over the drudge work of calculation that we undertake when we do mathematics. He also anticipated that, some day soon, formalised natural language, perhaps processed by machines, might help two disputing parties, including diplomats representing warring states, to determine which side had the morally stronger position.

If we have been disappointed recently in our overreliance on algorithms to maintain social harmony, we are nonetheless prepared, for the most part, to see computer programs as, in some sense, ideally rational. And unless we are raving singularitarians, we do not concern ourselves so much with the question of whether the machines that run these programs are approaching self-awareness, the capacity to have qualia, or any of the other usual markers of subjecthood. For most of us, to say that a computer is rational is, as it was for Leibniz, simply to say that it is following down the right pathways for reasons that are predetermined in the program it is running.

But vines follow the right pathways when they move toward the sunlight too, and in the end computers, like plants, are made only of various combinations of minerals and chemical elements. So when we look at our reckoning machines and we detect an instance of reason, we might use this occasion to work our way back into that less familiar understanding of reason, according to which its instances are everywhere in nature.

In answering the where question of reason in this maximally broad way, we are able to preserve the naturalism that philosophy and cognitive science insist upon today, while dispensing with the human-exclusivity of reason. And all the better, since faith in the strange idea that reason appears exactly once in nature, in one particular species and nowhere else, seems, on reflection, to be itself a vestige of pre-scientific supernaturalism.

A basic insight of the scientific revolution had it that whatever is natural is unexceptional. As Leibniz expressed this idea, citing a line from his favourite comic opera: ‘Everywhere and always, it’s the same as it is here.’ We accepted this idea long ago as concerns physical laws. But terrestrial biology is now considered unexceptional too: most biologists today are inclined to believe that exoplanetary life is a statistical near-certainty. So perhaps it is also time to give up the idea of rationality as nature’s last remaining exception.

Irrationality: A History of the Dark Side of Reason (2019) by Justin E H Smith is published via Princeton University Press.Aeon counter – do not remove

Justin E H Smith  is professor of the history and philosophy of science at the University of Paris 7–Denis Diderot. His books include The Philosopher: A History in Six Types (Princeton). An editor at large of Cabinet Magazine, he also writes frequently for the New York TimesHarper’s Magazine, and other publications. Twitter @jehsmith

This article was originally published at Aeon and has been republished under Creative Commons.

What is Calculus?: Limits

This is the second in a series of short articles exploring calculus. The first article explored the origins of calculus. The next few articles explore the mathematics of calculus. This article focuses on the foundation of calculus: limits.

The first article in this series asked the question: What is calculus? I promised then that the second article in the series would explore the substance of calculus, the mathematics of calculus. So let’s dive right in.

Here’s my two-part answer to the “what is calculus?” question:

Calculus is a mindset—a dynamics mindset.
Contentwise, calculus is the
mathematics of infinitesimal change.

The second sentence describes the mathematics of calculus. But I don’t expect you to understand that sentence just yet. That’s where the first sentence comes in. If you ask me, all of calculus flows from a more fundamental—and intuitive—principle, articulated in the first sentence: the notion that calculus is a dynamics mindset. Let me explain.

Calculus: A New Way of Thinking

The mathematics that precedes calculus—often called “pre-calculus,” which includes algebra and geometry—largely focuses on static problems: problems devoid of change. By contrast, change is central to calculus. Calculus is all about dynamics. Example:

  • What’s the perimeter of a square of side length 2 feet? ← Pre-calculus problem.
  • How fast is the square’s perimeter changing if its side length is increasing at the constant rate of 2 feet per second? ← Calculus problem.

Now that I’ve sensitized you to thinking “calculus!” whenever you read about or infer the presence of change, take a quick look at the second sentence in my two-part answer above. What’s the last word? Change. But it’s a new type of change—infinitesimal change—and this requires some explaining. That’s our next stop.

A Philosopher Walks Into a Starbucks

First, a rough definition of “infinitesimal change”:

“Infinitesimal change” means: as close to zero change as you can imagine, but not zero change.

“What?!” I hear you saying. So let me illustrate this definition via my friend, Zeno of Elea (c. 490-430 BC). This ancient Greek philosopher thought up a set of paradoxes arguing that motion is not possible. One such paradox—the Dichotomy Paradox—can be stated as follows:

To travel a certain distance you must first traverse half of it.

Makes perfect sense. Two is one plus one. And one is one-half plus one-half. But don’t be fooled by this seemingly innocent reasoning; it’s a trap! (Admiral Ackbar!) 

To appreciate what’s going on—and connect Zeno’s paradox back to calculus—let’s pretend Zeno is in line at Starbucks, two feet away from the cash register. He’s almost done scanning the menu when the barista yells out “next!” And that’s when poor Zeno panics. He must now walk two feet, but because of his mindset, he walks only half that distance with his first step. He then walks half of the remaining distance with his second step. (Can you imagine how annoying those in line behind him are getting?) The figure below keeps track of the total distance d Zeno has walked and the change in distance Δd after each of his steps.

Fig. 1.2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance with each step. (Reprinted, with permission, from Calculus Simplified.)

Here’s a tabular representation of the action:

Table 1.1: The distance d and change in distance Δd after each of Zeno’s steps. (Reprinted, with permission, from Calculus Simplified.)

Each change d in Zeno’s distance is half the previous one. So as Zeno continues his walk, d gets closer to zero but never becomes zero.[1] If we checked back in with Zeno after he’s taken an infinite

amount of steps—what a patient barista!—the change d resulting from his next step would be . . . drum roll please . . . an infinitesimal change—as close to zero as you can imagine but not equal to zero.

This example, in addition to illustrating what an infinitesimal change is, also does two more things. First, it illustrates the dynamics mindset of calculus. We discussed Zeno walking; we thought about the change in the distance he traveled; we visualized the situation with a figure and a table that each conveyed movement. (Calculus is full of action verbs!) Second, the example challenges us. Clearly, one can walk 2 feet. But as Table 1.1 suggests, that doesn’t happen during Zeno’s walk—he approaches the 2-foot mark with each step yet never arrives. How do we describe this fact with an equation? (That’s the challenge.) No pre-calculus equation will do. We need a new concept that quantifies our very dynamic conclusion. That new concept is the mathematical foundation of calculus: limits.

Limits: The Foundation of Calculus

In modern calculus speak we paraphrase the main takeaway of Table 1.1 this way: the distance d traveled by Zeno approaches 2 as Δd approaches zero. It’s important to note that d never equals 2 and Δd never equals 0. Today we express these conclusions more compactly by writing

read “the limit of d as Δd approaches zero (but is never equal to zero) is 2.” This new equation—and what we take it to mean—remind us that d is always approaching 2 yet never arrives at 2. (Oh, the dynamics!) The same idea holds for Δd: it is always approaching 0 yet never arrives at 0. Said more succinctly:

Limits approach indefinitely (and thus never arrive).

You’ve now met the foundational concept of calculus—limit. You’ve also gotten a glimpse of what infinitesimal change means and how a limit encodes that notion. Finally, you’ve seen many times how a dynamics mindset is at the core of calculus’ new way of thinking about mathematics. In the next article in this series we’ll employ a dynamics mindset and limits to solve the three Big Problems that drove the development of calculus—instantaneous speed, the tangent line problem, and the area problem (discussed in the first post in this series). See you then!

Footnote: [1] Because each d is always half of a positive number.

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

David Grazian: A Sociologist at the zoo, Father’s Day style

Grazian jacketThis Father’s Day, millions of American families with small children will trek to their local zoos for the pleasures of springtime—afternoons of strolling tree-lined paths, watching lions nap and flamingos dance, hand-feeding elk and free-range peacocks their picnic lunch leftovers on the sly, warning signs—DO NOT FEED THE ANIMALS!—be damned. Zoos are not only great places to observe orangutans and chimpanzees but human primates as well, especially when the weather warms and they emerge from their winter hibernation to join the masses seeking to commune with nature in the city’s outdoor public spaces: its waterfronts, public squares, bird sanctuaries, botanical gardens, nature preserves, and of course, its aquariums and animal parks.

It’s all quite a hoot. I even I wrote a whole book about it—American Zoo—to which I have fatherhood itself to thank. As a cultural sociologist and urban ethnographer by training, I had earlier in my career written two books about urban nightlife, the first on Chica­go’s blues scene, the second on the growing social world of res­taurants, nightclubs, and cocktail lounges in and around downtown Philadelphia. These projects required me to spend long hours at late-night music venues, corner taverns, nouveau-fusion eateries, martini bars, high-end speakeasies, dance palaces, and corner taverns from dusk until dawn—until the wee morning hours of April 22, 2006, when my wife gave birth to our son (in the book I call him Scott). Suddenly this nocturnal, free-for-all lifestyle no longer seemed all that tenable. (I would joke that I still hoisted a bottle at 2:30am every morning, but it was a bottle of baby formula, not Budweiser.)

During his infant and toddler days, Scott developed an acute fondness for our neighborhood’s menagerie of leashed puppies, wandering house cats, and (somewhat illegal) backyard chickens. This enthusiasm eventually brought us to the local Philadelphia Zoo nearly every weekend to take in its far more exotic elephants, red pandas, marmosets, pumas, and gorillas. From the helium-filled Channel 6 Zoo Balloon that rose hun­dreds of feet up into the sky to its intricate naked mole-rat exhibit down below, we took it all in: the unforgettable sights, sounds, and smells of the nation’s oldest zoo. (The smells were the most difficult to forget.) With each visit, our curiosity grew about the zoo’s strange creatures, and what makes them tick—and chirp, moo, growl, honk, quack, roar, and squeal.

Yet as Scott and I continued our father-son visits to the zoo, the sociologist in me couldn’t help but wonder about its strange allure, so I spent four years volunteering at two different metropolitan zoos. At one institution I worked primarily in an outdoor children’s zoo where I cleaned enclosures and exhibits, prepared and distributed zoo-prescribed diets to birds of prey and small mammals, managed children in a petting yard filled with goats and sheep, and provided behavioral enrichment to a variety of animals. Along the way I shoveled cow manure and chicken dung, goat pellets and duck droppings. I scrubbed owl and macaw cages and lined them with old issues of USA Today and the Wall Street Journal, clipped a ferret’s toenails, and once got locked inside a bird’s double-caged enclosure. I picked horse and donkey hooves, stuffed frozen feeder mice with vitamin E capsules, bathed tortoises, and exercised overweight rabbits.

At a second zoo, I worked as a docent, or volun­teer educator. I handled and presented a variety of small live animals to a range of zoo audiences, including families, school groups on field trips, children’s birthday parties, and busloads of local nursing-home residents. I learned to handle red-tailed boa constrictors, fat-tailed geckos, black vultures, and an American alligator. (Fortunately, no animals were ever seriously harmed on my watch, although I myself endured bites, scratches, and other humiliations from several domestic rabbits, a bearded dragon, an African gray parrot, and at least one goat.) I also regularly helped prepare diets for most of the animals in the zoo’s collection, which in­cluded giraffes, jaguars, howler monkeys, river otters, peccaries, and Jamaican fruit bats. Much of the food was expired (but still safe) meat, fish, and produce donated by local supermarkets and grocery stores, including whole strip loins, boxes of oranges and kale, and odds and ends of raw salmon and squid for the otters.

The best part about my zoo job was how much I absolutely crushed it on Career Day in Scott’s kindergarten class—unless one of his classmates’ parents turned out to be a firefighter or astronaut, my reputation as the Dad with the Coolest Job was super safe. Of course, when I wasn’t in uniform Scott needed to accompany me on all my zoo visits, since a grown man walking around animal exhibits while taking notes and photographs of other people’s children tends to attract the wrong kind of attention. That was the other highlight of writing the book—trekking across the country to 27 different zoos and aquariums, all with Scott in tow. The only downside is that the repetition of our visits made him immune to the charms of zoos altogether. (“Oh, another Siberian tiger? Yawn.”) It’s a good thing the book came out by the time he turned nine, since he is no longer as susceptible to being bought off by bribes of stuffed animals from the zoo gift shop.

Now Scott is thirteen, and the other day he picked up a copy of American Zoo lying around the house, and he read the whole thing. He asked if we could go back to the zoo one last time, just to double-check my findings in the book against the real thing. Sign me up; I think we’ll go on Father’s Day.  

David Grazian is associate professor of sociology at the University of Pennsylvania. He is the author of American Zoo: A Sociological Safari, Blue Chicago: The Search for Authenticity in Urban Blues ClubsOn the Make: The Hustle of Urban Nightlife, and Mix It Up: Popular Culture, Mass Media, and Society.

What is Calculus?

By Oscar Fernandez

This is the first of three short articles exploring calculus. This article briefly explores its origins. The second and third articles explore its substance and impact, respectively. They will be published in the coming weeks.

What is calculus? If you were watching Jeopardy on May 31, 2019 you were treated to one whimsical answer: “developed by 2 17th century thinkers & rivals, it’s used to calculate rates of change & to torment high school students.” Funny, Jeopardy. While that answer isn’t totally accurate, what I do like about it is its structure—history, substance, and impact. This is a tried-and-true powerful framework for understanding new concepts that marries context with content. In this three-part series on calculus I’ll give you a short introduction to calculus’ history, substance, and impact to provide you with a more fulfilling answer to the question “what is calculus?” First up: a short tour of the origins of calculus.

Three Big Problems That Drove the Development of Calculus

By the mid-1600s, scientists and mathematicians had spent millennia trying to solve what I’ll call the three Big Problems in mathematics: the instantaneous speed problem, the tangent line problem, and the area problem. The figure below illustrates these.

(Reprinted, with permission, from Calculus Simplified (Princeton University Press))

The instantaneous speed problem (a) popped up in many places, most notably in connection with Isaac Newton’s studies of gravity. You see, gravity continuously accelerates a falling object, changing its velocity from instant to instant. To fully understand gravity, then, requires an understanding of instantaneous velocity. This didn’t exist before calculus. The tangent line problem (b) arose mainly as a mathematical curiosity. The ancient Greeks knew how to calculate tangent lines to circles, but until calculus no one knew how to do that for other curves. The area problem (c) popped up in a variety of places. Ancient Egyptian tax collectors, for example, needed to know how to calculate the area of irregular shapes to accurately tax landowners. Many hundreds of years later, the ancient Greeks found formulas for the areas of certain shapes (e.g., circles) but no one knew how to find the area of any shape.

From understanding gravity to calculating taxes to mathematical curiosities, the three Big Problems illustrate the broad origins of calculus. And for millennia they remained unsolved. What made them so hard was that they could not be solved with pre-calculus mathematics. For example, you’ve been taught that you need two points to calculate the slope of a line. But in the tangent line problem you’re only given one point (point P in (b)). How can one possibly calculate the slope of a line with just one point?! Similarly, we think of speed as “change in distance divided by change in time” (as in “the car zoomed by at 80 miles per hour”). That’s a problem for the instantaneous speed problem (a), because there’s zero change in time during an instant, making the denominator of “change in distance divided by change in time” zero. We can’t divide by zero, so again we’re stuck.

The Two Geniuses That Figured Everything Out

It wasn’t until the mid-1600s that real progress on solving the three Big Problems was made. One thing the Jeopardy answer above got right was the allusion to the two 17th century thinkers credited with making the most progress: Isaac Newton and Gottfried Leibniz. You probably know a few things about Newton—you may have heard about Newton’s Three Laws of Motion, which forms the foundation of much of physics—but you’ve likely heard little if at all about Leibniz. That’s because, in short, Newton used the eventual power and influence he gained after making his many discoveries and advances public to discredit Leibniz’s role in the development of calculus. (Read more about the feud here.) Yet each of these great thinkers made important contributions to calculus. Their frameworks and approaches were very different, yet each provides tremendous insight into the mathematical foundations of calculus and how calculus works.

In the next post in this series we’ll dive into those foundations. We will discuss the ultimate foundation of calculus—limits—and the two pillars erected on that foundation—derivatives and integrals—that altogether constitute the mansion of calculus. And we will discover an amazing fact: all three of the Big Problems can be solved using THE SAME approach. As is true with so many thorny problems, we will see that all that was required was a change in perspective.

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided

World Oceans Day: Safeguarding the Oceans for Future Generations

Photo credit: @ORCA_web on Twitter

The ocean covers more than two thirds of the surface of our planet, and is the mysterious home of countless species. The seemingly endless sea, both evocative and intimidating, can strike us as eternal, unchanging, infinite.

However, despite appearances, the marine environment is fragile and vulnerable, and in the 21st Century faces a wider array of threats than at any other point in human history. Growth in population, industrialisation and technological advancement has given humanity the ability to delve deeper and farther into the ocean, and as a result our impact on the ocean has been profoundly damaging.

ORCA believe that everyone who cares about the marine environment have a responsibility to play their part to make sure that these critical habitats are protected and the marine life that calls it home are safe. We do this by training volunteer “citizen scientists” to monitor whales, dolphins and porpoises, using them as indicator species to give us a picture of the overall health of the ocean. We know that if whales and dolphins are thriving, the ecosystem is strong and that other species are doing well.

By doing this we can understand the impact of the many threats that marine species are facing in the 21st century, and identify trends in the places we can find marine mammals and the populations in different parts of UK and European waters.

This includes well known issues such as marine plastic – thousands of tons of plastic waste that finds its way into the ocean and is ingested by a host of different species. This can fill the stomach of animals that mistake it for food, causing them to starve and eventually die, with some specimens found with huge quantities of plastic waste in their stomach.

It also includes less know threats, such as bycatch. This is when marine animals are tangled in fishing gear, which can cause horrific injuries and even death. Hundreds of thousands of whales, dolphins and porpoises are victim to this each year, both smaller animals who are caught and drowned in nets and larger whales which are entangled in lines from crab and lobster pots.

Monitoring whales and dolphins is even allowing ORCA to work to address some of the least understood issues facing marine mammals, including ship strike. Large whales globally are regularly involved in collisions with ships of all types and sizes, but the problem is so poorly understood we don’t truly understand even the scale of the challenge facing us. ORCA are analysing large whale behaviour to better understand how they react to ships, as well as analysing data to identify more high risk areas. We hope to be able to give ships better advice on how to minimise the risk of collisions.

We’re delighted to have the opportunity to highlight some of these threats in the upcoming Europe’s Sea Mammals – not only have we been able to include the data we use to try and mitigate some of these threats, but we have even been able to profile some of the challenges facing whales and dolphins to raise awareness, and hopefully inspire people to make small changes in their own lives.

We can all do our part to keep the ocean safe and we have a responsibility to be custodians of the sea. To find out some of the ways you can get involved, be sure to pick up Europe’s Sea Mammals, as well as visiting www.orcaweb.org.uk to find out how you can be an ocean hero.

Europe’s Sea Mammals Including the Azores, Madeira, the Canary Islands and Cape Verde
A field guide to the whales, dolphins, porpoises and seals
By Robert Still, Hugh Harrop, Tim Stenton, and Luis Dias

This cutting-edge photographic identification guide to Europe’s sea mammals—the only such guide of its kind—covers the 39 species of whales, dolphins and porpoises and 9 species of seals found in the region, which spans the eastern Atlantic from Iceland to Macaronesia, and the Mediterranean, Caspian and Baltic seas. Written and illustrated by a team of professional tour guides with extensive experience presenting the region’s sea mammals, the guide features more than 180 color photographs, maps and graphics, highlights key identification features and includes information on the range, ecology, behaviour and conservation status of each species. Produced with the marine conservation charity ORCA, the book presents mapping data from a decade of surveys, which shows both current distribution and changes over time.

Europe’s Sea Mammals is an essential companion for whale watchers and anyone else who is interested in this enigmatic group of mammals.

  • The only photographic guide dedicated to this popular whale-watching region
  • Features more than 180 color photos, maps and graphics
  • Highlights key identification features and provides essential information on the range, ecology, behaviour and conservation status of each species

Insect of the Week: Bees in the Forest

Adapted from page 23 of The Lives of Bees:

Crowd of forager bees recruited to help exploit a square of comb filled with sugar syrup, at the start of a hunt for a wild colony’s home.

How abundant are wild colonies of honey bees? Building on the 1978 study of the density of honey bee colonies living within the Arnot Forest, other biologists have investigated this matter at various sites in North America, Europe, and Australia. The first of these additional studies was led by Roger A. Morse, the entomology professor at Cornell University who generously let me start working in his honey bee laboratory when I was still a high school student back in 1969. He and a team of seven graduate students conducted their study in the spring of 1990, in the small port city of Oswego, on Lake Ontario in northern New York State. Their investigation was triggered by the discovery of a colony of Africanized honey bees—a hybrid between European subspecies and the African subspecies A. m. scutellata —nesting in a shipment of pipes from Brazil. The presence of these exotic honey bees raised concerns that Africanized bees, and the fearsome ectoparasitic mite (Varroa destructor) that these bees could carry, might have been introduced to North America, so attempts were made to locate all the honey bee colonies living near the port so they could be checked for Africanized bees and Varroa mites. Newspaper and radio advertisements were run offering a $35 reward for information on honey bee colonies living in the semicircular area within 1.6 kilometers (1 mile) of the port. Eleven wild colonies living in trees and buildings, and one managed colony residing in a backyard beehive, were found. This work revealed that in this small city, the density of the wild colonies was 2.7 colonies per square kilometer (7 colonies per square mile), much higher than what Kirk and I had found in the woods of the Arnot Forest. Fortunately, no Africanized honey bees or Varroa destructor mites were found.

A still higher density of wild colonies was found in a remarkable study conducted by a team of biologists led by M. Alice Pinto at Texas A&M University in 1991–2001. This group worked in the Welder Wildlife Refuge, a 31.2 square- kilometer (12.2 square- mile) nature preserve in southern Texas. Their aim was to track the “Africanization” of a population of wild honey bees living in the southern United States, and they did so by sampling the colonies living in this wildlife refuge before, during, and after the arrival of Africanized honey bees from Mexico. Africanized honey bees are derived from a founder population of an African subspecies, A. m. scutellata, that was introduced to Brazil from South Africa in 1956. The purpose of this introduction was to crossbreed a tropical- evolved African subspecies with several temperate- evolved European subspecies already in Brazil to create a honey bee well suited to tropical conditions. However, several colonies of A. m. scutellata escaped from the quarantine apiary, thrived in the Brazilian climate, and spawned strong populations of wild colonies of this subspecies throughout the American tropics.

The Lives of Bees: The Untold Story of the Honey Bee in the Wild
By Tom Seeley

Humans have kept honey bees in hives for millennia, yet only in recent decades have biologists begun to investigate how these industrious insects live in the wild. The Lives of Bees is Thomas Seeley’s captivating story of what scientists are learning about the behavior, social life, and survival strategies of honey bees living outside the beekeeper’s hive—and how wild honey bees may hold the key to reversing the alarming die-off of the planet’s managed honey bee populations.

Seeley, a world authority on honey bees, sheds light on why wild honey bees are still thriving while those living in managed colonies are in crisis. Drawing on the latest science as well as insights from his own pioneering fieldwork, he describes in extraordinary detail how honey bees live in nature and shows how this differs significantly from their lives under the management of beekeepers. Seeley presents an entirely new approach to beekeeping—Darwinian Beekeeping—which enables honey bees to use the toolkit of survival skills their species has acquired over the past thirty million years, and to evolve solutions to the new challenges they face today. He shows beekeepers how to use the principles of natural selection to guide their practices, and he offers a new vision of how beekeeping can better align with the natural habits of honey bees.

Engagingly written and deeply personal, The Lives of Bees reveals how we can become better custodians of honey bees and make use of their resources in ways that enrich their lives as well as our own.

Artemis Leontis on Eva Palmer Sikelianos: A Life in Ruins

This is the first biography to tell the fascinating story of Eva Palmer Sikelianos (1874–1952), an American actor, director, composer, and weaver best known for reviving the Delphic Festivals. Yet, as Artemis Leontis reveals, Palmer’s most spectacular performance was her daily revival of ancient Greek life. For almost half a century, dressed in handmade Greek tunics and sandals, she sought to make modern life freer and more beautiful through a creative engagement with the ancients. Along the way, she crossed paths with other modern artists such as Natalie Clifford Barney, Renée Vivien, Isadora Duncan, Susan Glaspell, George Cram Cook, Richard Strauss, Dimitri Mitropoulos, Nikos Kazantzakis, George Seferis, Henry Miller, Paul Robeson, and Ted Shawn.

 Who is Eva Palmer Sikelianos? Please introduce her to us.

Creative, brilliant, and stunning with floor-length red hair, she was an American actor and director who loved women and ancient Greece and forged an artistic alliance with her Greek poet husband Angelos Sikelianos that shaped twentieth-century Greek culture.

The life of Eva Palmer Sikelianos (1874–1952) reads like a novel. She was born into a wealthy New York family. Her father, Courtlandt Palmer, a freethinker trained at Columbia Law School, was from the Palmers of Stonington, Connecticut and her mother, Catherine, a musician, from the Amory family of Boston. Home-schooled among artistic luminaries and business scions, she boarded at Miss Porter’s School in Farmington, Connecticut after her father’s sudden death in 1888. She studied Greek and Latin at Bryn Mawr College.

Throughout her life, Eva was a non-conformist. She seduced Natalie Clifford Barney in Bar Harbor (by Barney’s recollection) and followed her to Paris in the early 1900s. They aimed to create a woman-centered utopia reviving the spirit of Sappho’s Lesbos. Willful anachronism was part of their artistic practice. Eva, immersed in theatricality and Greek sources, devised the hairstyles, dress, music, gestures, scenery, and props for Barney’s performances that played with gender ambiguity.

She pursued another Greek-inspired utopia in Greece when her tempestuous relationship with Barney became unbearable. Dressed in sandals and a tunic she wove for one of Barney’s plays, she traveled to Greece with Raymond Duncan (brother of dancer Isadora) and his wife Penelope in 1906. She met Penelope’s brother Angelos Sikelianos. They married and settled in Greece the next year, giving birth to a son, Glafkos, in 1909. For 25 years, she promoted endangered musical practices, weaving, and handicrafts as forms of resistance to economic domination by the West. With Sikelianos as the public face of the events, she produced and directed two festivals of drama and games in 1927 and 1930 in the archaeological site of Delphi, spending all her money. The Delphic Festivals—expressions of an international modernist Hellenism grounded in the Greek present— used modern Greek expressive means to signal their Greece’s transhistorical survival. They helped popularize the performance of Greek drama in ancient sites in Greece and across the Mediterranean. 

Now deeply in debt, Eva returned to the U.S. in 1933 to raise cash. She directed Greek plays and wove costumes to keep a roof over her head. The struggle to survive broke her health but not her spirit. By the mid-1940s, British and American interference in Greece’s internal affairs had radicalized her politically. She wrote over a thousand letters to politicians and newspaper in protest of American imperialism. The House Un-American Activities Committee listed her name four times on suspicion of campaigning to “disarm and defeat the United States.”

Though denied a visa to return to Greece in 1951, she did eventually return in the spring of 1952 to attend the third Delphic revival, a small festival of drama and part of the “Return to Greece” postwar tourist campaign supported by the American Marshall Plan. There she witnessed her vision for a Greek revival deployed by Greek and American officials to transform Greece into a tourist destination, something she strongly opposed. She suffered a stroke on site and lies buried in the village cemetery of Delphi.

Although best known for reviving the Delphic Festivals, her most spectacular performance was her personal unscripted daily revival of the ancient life. For almost half a century, she performed Greekness in handwoven tunics and sandals in defiance of fashion norms, working to make modern life freer, more authentic, and better.

Why did you write about Eva Palmer Sikelianos? How do you value her achievement today?

 She is an overlooked person of the 20th century, airbrushed from depictions of her era’s cultural attainments. I felt that it was time for someone actually to delve into the sources of her life to piece together her story.

Careful digging yielded enormous returns. I read the Greek translation of 163 of Eva’s previously unpublished letters to Barney published in 1995 and discovered over 600 more personal letters in a library in Athens, including 182 exchanged with Barney. Thus, I learned that her connections to Natalie Barney’s turn-of-the-century lesbian salon ran deeper than anyone anticipated. Studying her official archive, I saw that she was also a bigger player in the Delphic Festivals than previously acknowledged; Angelos Sikelianos deliberately covered over her seminal work. There were more, important unexplored episodes. Her search for alternative, tonalities linked her to Richard Strauss, Dimitri Mitropoulos, Konstantinos Psachos, an expert in Byzantine music, and Khorshed Naoroji, the classically trained pianist from Bombay who sought Eva’s help to recover Hindi music. Through Nairoji, her costuming for the Prometheus Bound became interwoven with Gandhi’s advocacy of khadi in India’s decolonization movement. She was also a missing figure in the genealogy of modern dance from Isadora Duncan to Ted Shawn and Martha Graham. Her legacy shaped Greece’s post-War tourist development.

What do we see when we connect these pieces? Her slight, endlessly determined figure, finding its place as a crucial link in a series of artistic and socio-political endeavors, helps clarify the relationships of tesserae in the mosaic of cultural modernism. With her in the picture, modernism’s mantra to “make it new” (Ezra Pound), which placed ultimate value on novelty, becomes “make it ancient”: a determined effort to renew the modern world by capturing the latent energy in ancient sources as creative ground for expression. This was something many modernists actually aspired to.

Moreover, her presence connects the early twentieth-century search for new artistic forms with a queer temporal sensibility in women’s classical learning. Through Eva Palmer’s performances of the Greeks, we see the anachronistic temporality of other modernist projects which moved not progressively forward toward fulfillment, but backward, into the holes of history, to recover a past that never was in order to suggest an as yet unimagined future.

Sometimes a book needs to be written because a person’s strange life suddenly makes sense. I’ve known about Eva Sikelianos for as long as I can remember. For many years, those Greek tunics and the Delphic Festival pressed unproductively against my brain. I did not have the right frame to process them. In the early 2000s, when I read the 163 letters to Barney, in which they imagine new gender and social roles by going back in time, the anachronism of her untimely aspects suddenly made sense.

The idea that new bodily exercises recycling older aspects, performed habitually with self-awareness, may alter the field of possibilities of a restrictive inherited world is quite current. Many people today believe that identities are not innate to individuals but performed through the vocabularies, gestures, and materials recognized by society. This counter-Enlightenment position was tested by a people in the aestheticist movement of Eva Palmer’s youth. Given a new spin in the late twentieth century by Michel Foucault and some of his interpreters, it became ascendant in the twenty-first. Millions of people today work at being themselves by becoming otherwise. They change their names; cross dress to challenge gender binaries; transition to live and present in alignment with another gender identity; tattoo their bodies; draw on ancient traditions and distant sources to transform their consciousness. They aspire, somewhere, somehow, to change the social order through their alteration of the field of possibilities defining themselves.

My hope is that a textured interpretation of the life of Eva Palmer Sikelianos, who, for the first half of the twentieth century, sustained a conscious practice of living differently while absorbing the shocks and heartbreaks of noncomformity, might serve as a meditation on our times.

You mention several artists with whom Eva Palmer Sikelianos’s crossed paths. Who are some of the other famous people in her story?

Her archival materials cover a lot of ground. We find in them the old upper-class families across the world: the Barneys, Beechers, Roosevelts, Vanderbilts, and Andrew Carnegie in the U.S.; Antonis Benakis and Eleftherios Venizelos in Greece; Rabidranath Tagore and the Naorojis in India. This is a sampling.

In the performing arts, I’ve mentioned Barney’s turn-of-the-century lesbian salon. Sara Bernhardt, Colette, Marguerite Moreno, and Isadora and Raymond Duncan are some of the people she worked with that first decade. Mrs. Patrick Campbell offered her acting work on the condition that she cut her ties with Barney, but she wouldn’t. George Cram Cook, Susan Glaspell, photographer Nelly Sougioutzoglou-Seraidare, filmmaker Dimitris Gaziadis, artist Yannis Tsarouchis, Koula Pratsika, an important figure in Greek dance history, and Aliki Diplarakou Lady Russell, who won the Miss Europe contest in 1930, were part of the Delphi scene. In the U.S., Eugene O’Neill supervised her work on the Federal Theater Project. Artist Katherine Dreier introduced her to Ted Shawn. She corresponded with Paul Robeson and W.E.B. DuBois on politics and asked Robeson to play Prometheus in a performance she never staged in Delphi. She was the teacher, friend, and possibly lover of Mary Hambidge, weaver and founder of the Hambidge Center craft community in Georgia, who gave her shelter in the 1930s and 1940s. Greek folklorist Angeliki Hatzimihali was also a dear friend.

She had many connections in the music world. Soprano Emma Calve and harpsichordist Wanda Landowska were possibly her lovers. Natalie Curtis Berlin, a pianist who collected Native American music, was a family friend. She collaborated with Psachos, Greece’s first professor of Byzantine music, and Greek musicologist Simon Karas. Dimitris Mitropoulos stayed in her house near Corinth in 1924 composing music for Angelos’s poetry, and Richard Strauss and his architect Michael Rosenhauer visited her in Delphi two years later while planning to build a music hall on Philopappos Hill in Athens.

Her literary connections begin with Oscar Wilde and continue with Barney, Lucie Delarue-Mardrus, Collette, and Renée Vivien. Playwright Constant Lounsbery and novelist Elsa Barker were longtime friends. In Greece she knew the most prominent writers of the twentieth century: Sikelianos, Kostis Palamas, Nikos Kazantzakis, Kostas Karyotakis, and George Seferis. She corresponded with translators Kimon Frier and Rae Dalven. With Henry Miller she nominated Sikelianos for the Nobel Prize.

She knew all the heads of archaeological schools and many classicists and archaeologists in addition to artists working in the margins of the discipline: Joan Jeffery Vanderopol, Georg von Pscehke, and Alison Frantz. Archaeologist Theodore Leslie Shear was her correspondent on Cold War political matters in the 1940s. Bryn Mawr College presidents, professors, and alumni were her acquaintances, lovers, or good friends, including president M. Carey Thomas, Lucy Donnelly, Virginia Yardley, and Edith Hamilton, popularizer of Greek literature and myths with whom she corresponded about Greek tragedy for over a decade.

So many people, so much material! How did you deal with this embarrassment of riches?

There really is a lot. I visited over 15 archives and libraries in 12 cities in 3 countries on 2 continents.

The archival abundance was especially challenging because much of it was underprocessed. No biographical account existed except the autobiographical episodes in Eva Palmer Sikelianos’s book, Upward Panic. There was not even a reliable timeline of her life to build on. Moreover, access and the organization and contents of archives kept shifting. In one archive, for example, I was given access, then denied it, then given it again to a collection of very personal letters that had been hidden for years. There I found the letters I mentioned earlier. I could not have written a book without those letters. Barney’s own literary executor François Chapon didn’t know they existed. Later I learned that there were 10 more boxes of uncatalogued materials from a later period in Eva’s life.

Every biographer should have such problems!

Besides the challenges of reading and intepreting source materials, selecting was the bigger challenge. There is more material than can possibly fit into a book. To deal with this overabundance, I wrote a detailed chronology to give order to people and events. This is where I map out the historical coordinates of Eva Palmer Sikelianos’s life from birth to death, her burial, and the handling of her remains. I also connect her to many of the people with whom she interacted.

The book I chose to write takes another direction. It brings focus to her life’s work—her body of work and her work shaping her life —as it shifts from one medium to another: from lesbian performances and weaving, patronage of Greek music, staging ancient drama to writing, translating, and political activism in the last phase of her life. I follow the continuities and discontinuities to link Eva Palmer, the stage manager and actor who made Sappho a model of emulation for twentieth-century lesbian identity, with Eva Sikelianos, the director of the Delphic revivals and unconventional dresser who made her life a Greek revival.

Did the source materials present other challenges?

The intimate sources of a person’s life always raise ethical questions in addition to the practical and legal ones of access and permission. On the legal side, who owns the rights and when, where, and how to track that person or entity down. What happens when they are not found? The ethical questions are more layered. We can ask the executors of a deceased person’s work for permission. But what about the wishes of people invested in the person’s legacy? I’m not talking just about people who are now dead. I believe the wishes of anyone living today who has a stake in Eva Palmer Sikelianos’s legacy actually matters. Yet the wishes are many and conflicting.

I struggled long and hard with the question of how to handle the intimate materials. It was not a problem of permission to reveal details of Eva Palmer/Sikelianos’s sexual life. Eleni Sikelianos, Eva’s great grand-daughter and an important American poet, who is her literary executor, wrote in 2005 about her “lesbian theater director great-grandmother.” People heard her grandfather Glafkos express his firm wish that Eva’s sexual orientation should be known and respected. She gave me permission to use every source and to write what I think ought to be said.

Initially, I intended to avoid writing about her intimate relations. I considered them irrelevant to her creative processes. I was trying to get at Eva’s ideas as they expressed themselves publicly in the media of her life and staged performances. As I dug deeper into the sources, I read the many letters exchanged between Eva and Barney and dozens of other women who corresponded intimately with Eva. Some love letters mixed creativity, passion, pain, and despair with deep cultural knowledge about Greek sources and efforts to recreate it. There were letters from Eva’s brother and mother too. Some were especially harsh, belittling her life choices. Her mother wished her to reveal herself yet would not accept her choices.

As I read the archival material, particularly the most personal letters, I sensed just how arbitrary and absurd yet resolutely stigmatizing the social rejection of same-sex orientation has been. At the same time, I observed that Eva’s encounter with the world as a lesbian was inextricably intertwined with her efforts to make herself free and anachronistically ancient. This was true when she cultivated a lesbian life in the company of Barney in the early 1900s and even more true when she “went Greek” after she married Angelos Sikelianos. I chose to disclose those points of intimacy that spoke to the creative process.

So the great loves of her life were…

Natalie Clifford Barney and Greece. Angelos Sikelianos is folded into Greece, and many other women are folded into both great loves. Eva’s correspondence from 1906 to 1907 suggests that she had trouble imagining herself close to Angelos when they first met. She was actually trying to find new footing away from Barney. Yet, in writing Upward Panic thirty years later, Eva made Angelos out to be the love of her life. She grew to love him, I think, but in an expansive way that embraced his dreams and artistic work under the big umbrella of Greece. In her words, she loved “his country, his people, his language, and most of all his dreams.”

She also loved his sister Penelope, and this love is a driving force in my narrative. I guess you will have to read the book to learn how.

You call her “Eva.” How much intimacy do you feel for her?

I use the first name, “Eva,” for a practical reason. Many people in her story share the last names Palmer and Sikelianos. I generally refer to her siblings also by their first names, Robert, May, and Courtlandt, and to Angelos Sikelianos and Penelope Sikelianos Duncan as Angelos and Penelope. The Duncans are Isadora and Raymond, whereas Natalie Clifford Barney is Barney. The reason is not familiarity but to keep my subjects clear, since many people share the last name. Additionally, this spares me the trouble of specifying the maiden, married, or maiden and married, unless there is a point in using these, as in this and the next paragraphs.

As a general principle, I wanted my feelings for Eva Palmer Sikelianos to be irrelevant to the project. This is not to say that I had none or that they did not interfere with the research. I spent many years digging deep into biographical sources. They moved me. There were times when they caught my breath. When I opened the inaccessible dossiers containing over 600 letters for the first time my jaw dropped. Each dossier contained several folders of letters spilling out in loosely bundled stacks, tied with purple, pink, baby blue, and flowered satin ribbons from the time of their receipt. I found traces of tears and mud. The letters absorbed me so utterly that I could do think about anything else. I barely cared about putting dinner on the table. Yet while I was researching and writing, I was also teaching, advising, socializing, and caring for people in my intimate circle. Sometimes I would actually forget “Eva” or be very frustrated that her life always occupied me.

Researching a human subject inspires a certain kind of identification, even attachment, that is different for me at least from other subjects. Eva expressed love, pain, desolation, despair, bias, weakness, doubts, and extreme certainty. I did not always sympathize with her ideas or find her work extraordinary. Since she was a historical subject, I didn’t control her story, as I would have if she were a fictional subject. I compensated by wanting to protect her when things got bad, or to cover for her when she was wrong or just insufferable.

These were some of the feelings I struggled with while writing the book. They may become confused with my using her given name. But my calling her “Eva” is not meant to signal familiarity. I am now as far from knowing “Eva” as I was when I began. But it does solve a practical problem.

What happened in the way you think as a result of writing this biography? Did it change you?

A lot. It changed my thinking about LGBQT culture, as I describe above. This was life changing.

It also changed my attitude to life writing. When I started writing a book about Eva Palmer Sikelianos, a surprising number of colleagues questioned my decision. “You’re not writing a biography,” one person I respect a great deal stated decisively. Several others took time to enumerate why they dislike biographies. Actually, I wasn’t intending to write a biography. Initially, this was going to be a book of essays about Eva Palmer Sikelianos’s work reviving Greek cultural heritage in five media.

But I could not avoid the life-historical research. Classical models offered precedents for personal constructions of dress, behavior, and identity in addition to cultural products. My interest in the former grew. While ideas of Greece take monumental forms in many of their neoclassical manifestations, they are quite liquid in their passage through life. I came to ask: What new shapes do Greek textual and material fragments take when they inhabit people’s daily life? What becomes of both the ancient ruins and the modern person? How did Eva incorporate them in her daily activities? How did they script her life? How did her intense investment in finding the latent life in ruins change over time to become increasingly an art of life?

I now consider the book’s biographical mode to be vital to its contribution. In working to recover the story of a woman whose rich and diverse work in Greece was fitted to the procrustean bed of patriarchal, nationalist, and heteronormative discourses, I honor several decades of biographical writing that studies the gaps in classical scholarship for clues of women invested personally in the study of Greece and other ancient worlds, who had nontraditional careers, lived their eccentric lives mainly in obscurity, and somewhere, somehow, tried to change the social order through their alteration of the field of possibilities defining not just knowledge but themselves.

Why did you feature “ruins” in your title? She was such a creative person!

She was indeed a creative person but also a broken one working in a country associated with ruins on multiple levels. The title plays on the multiple meanings of “ruins”: as an inspiration for creative work, a signifier of financial crisis and decay, and a central feature of Greece—both literal and metaphorical—in modern times.

Eva’s creativity had its source in ruins. She saw the latent grandeur of the Greeks everywhere, and, over the course of half a century, she embraced fragments of the past, giving them new life. This was her attitude before she came to Greece, when she and Barney were imagining an alternative social order for women reconstructed from the gaps in Sappho’s fragments. It became a worldview grounded in materials and practices after she moved to Greece. She saw fragments of the past everywhere: in archaeological sites, museum objects, folk practices, and the modern Greek language. She sought them out and became increasingly involved with them, conversing with people who excavated artifacts for a living or who collected folk survivals, using their discoveries to make new things. Ancient sites and things were a canvas for her creative interventions.

Concurrently, her life trajectory moved from abundance toward ruination. To produce the first Delphic Festival in 1927, she took out loans against the value of two houses. She never recovered the expenses. For the next six years, she experienced the humiliation of avoiding her creditors, whom she could not repay. She left Greece to escape them and also Angelos, who kept asking her for more money. For the next two decades, she depended on the generosity of friends in the U.S. to board her in exchange for her weaving.

With World War II and the Nazi devastation of Greece during the three-year occupation, Eva’s own fall into ruins and Greece’s intersected. From her humble living quarters in the U.U., she read about women and children who were tortured, forests denuded, and villages plundered or totally destroyed. People she loved disappeared. Reading about an economy in ruins and sensing the people’s hardships, she had to rethink the relations between ruins and her advocacy for Greece.

Ruins are a central metaphor for Greece in its modern history, denoting a rift with the past which the present must work endlessly to repair. During the decade when I was researching and writing this book, Greece experienced a huge government debt crisis leading to economic contraction and coinciding with the arrival of hundreds of thousands of asylum seekers and refugees, a true humanitarian crisis. Poverty spread, bringing hunger and illness. Neighborhoods were abandoned and fell into ruins, where homeless people now stay. Today’s Greece evokes the Greece of Eva. Meanwhile, imagery from classical ruins has appeared repeatedly in foreign media representations to symbolize the sorry state of the Greek economy. It works to flatten and ridicule current reality by identifying it with—what else—the rift with the glorious past of Greece.

I think that Eva Palmer Sikelianos: A Life in Ruins tells a good riches to rags story. It should also generate reflections on the representations and functions of ruins today.

Artemis Leontis is professor of modern Greek and chair of the Department of Classical Studies at the University of Michigan. She is the author of Topographies of Hellenism and the coeditor of “What These Ithakas Mean…”: Readings in Cavafy, among other books. She lives in Ann Arbor, Michigan.

Should We Celebrate Mother’s Day Every Week?

Doepke, Zilibotti, Love, Money, and Parenting book coverThe modern Mother’s Day holiday in the United States was first celebrated in 1908 in a time of strictly separated gender roles. While some single women were working for pay, married women were usually out of the labor force. Indeed, so called “marriage bars” formally prohibited the employment of married women in many occupations. With much of paid work reserved for married men, married women had to shoulder most of the burden of unpaid work, including caring for children, on their own.

No wonder, then, that there was a perceived need at a time for an occasion to specifically celebrate mothers. When mothers were working without pay and little other formal or informal recognition, a dedicated holiday provided at least an occasional opportunity to honor mothers’ profound contributions to their families and society at large.

But gender roles have evolved a long way since 1908. Bars against women’s employment in the labor market were gradually lifted, and after World War II many married women, including mothers, joined the labor force. Today, a large majority of mothers combines raising children with working for pay.

Conversely, fathers have become more involved in childcare. Until the 1970s, men’s participation in child-rearing was minimal, but today fathers take an increasingly active role in caring for their children. Fathers now spend considerably more time with their children and are less likely to be found in the bar or on the golf course compared to earlier generations.

These changes might suggest that today, there is less need for a mothers-only holiday. In the interest of gender equality, might it be time to abandon the gender-specific celebrations of Mother’s Day and Father’s Day in favor of the single, inclusive “Parents’ Day”?

A closer look at the numbers suggests that, in fact, we need just the opposite – namely a Mother’s Day every week.

While differences in gender roles have become smaller over time, women continue to do a lot more childcare than men, not just in the United States, but across all economically advanced economies. Data from the OECD (a club of mostly rich countries) shows that women still do on average two-thirds of unpaid work in the economy, of which childcare is a major component. Gender equality in this dimension is almost in reach in Sweden, where women do 56 percent of unpaid work, compared to 63 percent in the United States. East Asian countries have the longest way to go: in Japan and Korea women still do more than 80 percent of unpaid work.

While some of these differences reflect that men spend more time working for pay, that is only part of the story. Women do the majority of housework and childcare even among couples where both spouses are working full time. As a result, women end up with less free time: across all OECD countries, women enjoy less leisure than men do.

In the United States, the change in gender roles actually has slowed in recent decades. Women’s labor force participation rose quickly from the 1970s to the 1990s but has stalled since, and is now lower than in many European countries.

The nature of motherhood has also been affected by a transformation in the nature of parenting in recent decades. We describe in our book Love, Money, and Parenting: How Economics Explains the Way We Raise Our Kids how sharply rising inequality has raised the stakes in parenting starting in the 1980s. While in the 1960s and 1970s obtaining a high-school degree came with the expectation of a secure future as members of the middle class, after decades of stagnation in median earnings in the economy by now only a college degree can provide the same level of security.

American parents responded to this changed environment by adopting more time-intensive parenting styles geared at helping their children succeed in a harsh economic climate. Typical couples now spend twice as much time on caring for their children than what was typical in the 1970s. Activities aimed at supporting children’s educational achievement, such as helping them with homework, rose the fastest.

This trend towards intensive parenting has contributed to a persistent gap in the parenting engagement of mothers and fathers. As parenting became more intense, fathers’ contribution went from very little to substantial. But in absolute terms, mothers increased their time spent on parenting even faster. As a result, mothers now have a full five hours less of leisure time per week compared to the 1970s.

Given these numbers, there are good reasons to use this Mother’s Day not just for thanking mothers for everything they do for their children and their families, but also to consider what can be done for the long-run trend toward more gender equality to resume. American women already get more education and are substantially more likely to graduate from college than men. But women will not be able to make full use of these skills in the labor market and have equal career opportunities until fathers carry a fair share of the load of parenting.

What would help, therefore, is a Mother’s Day every week –  reshuffling one-seventh of mothers’ weekly childcare duties to fathers would still fall well short of equality, but it would be a good start toward closing the gap.

Matthias Doepke is professor of economics at Northwestern University. Fabrizio Zilibotti is the Tuntex Professor of International and Development Economics at Yale University. They are the coauthors of Love, Money, and Parenting.

Rachel McCleary and Robert Barro on The Wealth of Religions: The Political Economy of Believing and Belonging

mcclearyWhich countries grow faster economically—those with strong beliefs in heaven and hell or those with weak beliefs in them? Does religious participation matter? Why do some countries experience secularization while others are religiously vibrant? In The Wealth of Religions, Rachel McCleary and Robert Barro draw on their long record of pioneering research to examine these and many other aspects of the economics of religion. Places with firm beliefs in heaven and hell measured relative to the time spent in religious activities tend to be more productive and experience faster growth. Going further, there are two directions of causation: religiosity influences economic performance and economic development affects religiosity. Dimensions of economic development—such as urbanization, education, health, and fertility—matter too, interacting differently with religiosity. State regulation and subsidization of religion also play a role. Timely and incisive, The Wealth of Religions provides fresh insights into the vital interplay between religion, markets, and economic development.

How did you come to write the book?

Robert is an economist and Rachel is a moral philosopher. In thinking about religion, we took as our starting point the work of Adam Smith, the founder of economics, who believed that moral values and organized religion were key forces in political economy and society. Nevertheless, social scientists—particularly economists and political scientists—have tended to underestimate the importance of religion, particularly the role of beliefs and values. We think that Adam Smith was right. Beliefs and religiosity are central determinants of which societies prosper and which deteriorate.

What does your book bring to the conversation on the economics of religion that hasn’t been discussed before?

Another contribution to the study of religion is bringing together the ideas of Adam Smith with those of the German sociologist Max Weber. Religious beliefs and values motivate people to behave in certain ways. This view, as we discuss in our book, is integral to forms of Protestantism with its emphasis on unmediated, individual responsibility for one’s salvation. We bring a quantitative approach to the relationship between beliefs, values, and economic behavior. In so doing, we examine the role of religious beliefs across world religions and countries. Our research has an international perspective with a focus on believing and belonging in the major religions of the world.

We focus on the role of religious beliefs and belonging to organized religions in the economic, political, and social development of nations and individuals. We are filling an important gap in the literature on religion by providing an international perspective. Much of the work in the sociology of religion is focused on local or regional patterns of religiosity. The sociology of religion has a strong focus on the United States, centering research around assumptions about religious patterns and organizations in the United States. In our research, we apply economic analysis to world religions and across countries.

How does religion fit into the story of developing nations? Does religious fervor help or hinder efforts to increase economic development?

To better understand the relationship between religion and economic growth, we need to look at a two-way causation. Religiosity has a two-way interaction with political economy. With religion viewed as the dependent variable, a central question is how economic development and political institutions affect religious participation and beliefs. There is a clear overall pattern whereby economic development associates with decreasing religiosity. However, there is no evidence that greater education diminishes religious beliefs.

Looking at the other direction of causation with religion as the independent variable, we study the effects of religion on economic, social, and political behavior. A key issue is how religiousness affects individual traits such as diligence, honesty, thrift, and integrity, thereby influencing productivity and economic performance. Another channel involves religion’s effects on literacy and education (human capital) more broadly. For example, there is evidence that Protestantism is more favorable than Catholicism as an influence on education and work ethic.

We find that social capital and cultural aspects of religion—communal services, rituals, religious schools—are significant mainly to the extent that they influence beliefs and, hence, behavior. For given beliefs, more time spent on communal activities would tend to be an economic drag for the believer as well as the entire community. Moreover, the costs of formal religion include the time spent by adherents and religious officials on religious activities. In addition, time and money are expended on buildings, sacred objects, and so on. Our general view, based on empirical evidence, is that believing relative to belonging (attending) is the main channel through which religion matters for economic and other social outcomes.

Can religion help to explain why some nations develop faster than others?

We found evidence that economic growth was stimulated when religious beliefs were high compared to religious participation. This pattern applied, for example, to Japan and parts of Western Europe. An overall expansion of religiousness—greater beliefs accompanied by the typically associated attendance at formal religious services—was not strongly related to growth. Religiously sponsored laws and regulations hindered economic growth in some places, notably in Muslim countries, which typically did not have favorable institutions with respect to corporations, credit markets and insurance, and inheritance.

How did the conflict between Protestantism and the Catholic Church affect economic development in early modern Europe? Do we still see the impact of that today?

As Max Weber argued, the rise of Protestantism beginning with the Reformation in the 1500s enhanced work ethic and the accumulation of human capital and, thereby, contributed to the industrial revolution. We found evidence that this mechanism still operated in Western Europe in the modern era.

Competition increases the quality of services provided by different religions. The introduction of Protestantism into Western Europe challenged the monopolistic status of the Roman Catholic Church, pressuring that organization to respond in two ways. First, by lowering the nature and pricing of religious goods, the Catholic Church sought to retain believers. Second, the Catholic Church promoted those aspects of its theology that distinguished it from other religions.

We discuss in our book how the beatification of saints is a unique mechanism of the Catholic Church. With the rise of Evangelical faiths, religious competition became particularly strong in Latin America, vernacularly referred to as “The Catholic continent,” where Catholicism had enjoyed a monopoly since the region was colonized by Spain in the 1400s. Today, in regions of the world where competition with types of Protestantism is increasing, the beatification of local saints revives religious fervor and deters adherents from converting to types of Protestantism.

Is religious fervor impacted by fluctuations in the economy? If so, how?

There is evidence that adverse economic shocks and natural disasters tend to increase the demand for religion. This pattern has been observed, for example, for earthquakes in Italy, flood-related declines in agricultural harvests in Egypt, declines in incomes during the Asian Financial crisis, and adverse effects from a poorly designed land reform in Indonesia. In the other direction, increased economic development—particularly movements away from agriculture and toward urbanization—tend to lower the demand for religion. However, it is wrong to conclude that sustained economic growth causes religion to disappear.

What do you hope readers will take away from reading this book?

We hope our readers will appreciate the possibilities of interdisciplinary research on a variety of religion topics. The application of economic ideas to religion broadens our understanding of ways in which beliefs and practices influence individual and group behavior.

We find that social capital and cultural aspects of religion—communal services, rituals, religious schools—are significant mainly to the extent that they influence beliefs and, hence, behavior. For given beliefs, more time spent on communal activities tend to be an economic drag for the believer as well as the entire community. The costs of formal religion include the time spent by adherents and religious officials on religious activities and the time and money expended on buildings, sacred objects, and so on. Our general view, based on empirical evidence, is that believing relative to belonging (attending) is the main channel through which religion matters for economic and other social outcomes.

Rachel M. McCleary is lecturer in the Department of Economics at Harvard University. Her books include The Oxford Handbook of the Economics of ReligionRobert J. Barro is the Paul M. Warburg Professor of Economics at Harvard. His books include Education Matters: Global Schooling Gains from the 19th to the 21st Century and Economic Growth. They both live in Massachusetts.

Daniel Kennefick on No Shadow of a Doubt

In 1919, British scientists led extraordinary expeditions to Brazil and Africa to test Albert Einstein’s revolutionary new theory of general relativity in what became the century’s most celebrated scientific experiment. The result ushered in a new era and made Einstein a global celebrity by confirming his dramatic prediction that the path of light rays would be bent by gravity. Today, Einstein’s theory is scientific fact. Yet the effort to “weigh light” by measuring the gravitational deflection of starlight during the May 29, 1919, solar eclipse has become clouded by myth and skepticism. In No Shadow of a Doubt, Daniel Kennefick provides definitive answers by offering the most comprehensive and authoritative account of how expedition scientists overcame war, bad weather, and equipment problems to make the experiment a triumphant success.

What compelled you to write this book?

The story of the 1919 eclipse is one of the most dramatic and significant in the history of science, and one that I’ve always found fascinating. What compelled me to research it closely was my puzzlement about the criticisms of Eddington which I heard repeated more and more, especially while working on volume 9 the Collected Papers of Albert Einstein, which covered Einstein’s life during the year 1919. I found the complaints about Eddington’s supposed bias in favor of Einstein unconvincing, especially the claim that Eddington’s pacifism was responsible for his desire to prove Einstein right. I thought that it was time someone looked closely at the actual data analysis decisions, using original documents preserved in the archives. I decided to write the book because I found the complete story of the eclipse which I put together to be fascinating and the centenary seemed like a perfect occasion to tell that story. I also felt that there was a danger that important work on the 1919 eclipse was being overlooked. As part of my research I learned that a re-analysis of the photographic plates taken in 1919 was conducted in 1978 by English astronomers at the Royal Greenwich Observatory using modern plate-measuring equipment and computers. They completely vindicated the work of the original team, and yet their re-analysis had gone totally unrecognized and unread. It was even misrepresented in the one book which did allude to it, Stephen Hawking’s A Brief History of Time. So I felt it was important to restore some balance to the story of what happened in 1919.

You say that the 1919 solar eclipse is perhaps one of the most important eclipses in history, but there are critics who contend that Arthur Eddington placed too much emphasis on the eclipse proving Einstein’s theory of relativity. Why do you think that’s a weak counter-argument?

The problem here is that the modern critics distort the story by their focus on just one participant, the famous astrophysicist Eddington. Incidentally, he was known to his family by his middle name Stanley; he never went by Arthur. Since Eddington was only involved in this one test of general relativity, it is easy to make it seem that there has been too much emphasis on the 1919 eclipse test. But Eddington himself never regarded confirmation of the theory as depending upon this one test. It’s just that modern commentary rarely talks about anything beyond Eddington’s role, which doesn’t even tell the complete story of this one test. There were two expeditions in 1919, and Eddington was only involved in one of them. The other one, organized by the Royal Observatory, Greenwich to Sobral in Brazil, obtained the most important data.

Having said all that, there is a sense in which the 1919 test was of very special importance. There were only three tests of Einstein’s new theory of gravity that were possible to do a century ago. One of these—the explanation of the perihelion shift of Mercury—was impressive, but since Einstein knew the result his theory had to “predict” it didn’t count as a prediction in the usual sense. The other test was the solar redshift measurements, but this confirms only the principle of equivalence and is not strictly speaking a test of general relativity as such. The prediction that light is deflected when it passes through the gravitational field of the Sun was a test of the complete theory that Einstein could not know the answer to beforehand. The 1919 expeditions were the first time that this observation had ever been successfully made. The agreement achieved was very dramatic and the fact that the experiment could not be repeated until the next suitable eclipse, in 1922, added even more drama to the occasion. So the truth is that the 1919 expedition was a special occasion in the history of science.

Can you talk a bit about the circumstances surrounding the Principe and Brazil expeditions that made this experiment so significant?

There were three circumstances that made this eclipse extraordinary. The first is that the eclipse took place on a day, May 29th, when the Sun is in the star field of the Hyades cluster. This is the closest star cluster to the Earth and there is no other place on the ecliptic (the Sun’s path through the sky) with so many bright stars so close together. Thus, an eclipse taking place on that day is perfectly suited to performing this experiment. Such an eclipse will next occur in 2310, so the expedition planners realized that it was especially important to try the experiment in 1919. Unfortunately, as late as November 1918, it looked unlikely that ships could be found to carry the teams to their preferred stations on the island of Principe and in northeastern Brazil. The reason for the suspension of shipping was World War I which fortunately ended abruptly later that same month. Had the war lasted any longer, it is unlikely that the expeditions could have departed. Even as it was a civil war broke out in Portugal, a key stop on their route, before their departure, and Eddington had no idea which ship would take him to Principe when he left England in March 1919.

This second circumstance, that of a war torn world, very nearly scuppered the planning for the expeditions, but undoubtedly helped make the team so famous when they returned successfully. The triumph of science over the tribulations of history really caught the public imagination. Certainly an aspect of this public response was that the expedition was mounted from England in order to test, and confirm, the theory of a German scientist, Albert Einstein, so it had an additional aura of reconciliation about it, at a time when postwar feelings were very bitter.

A third favorable circumstance was the relevant expertise of the expeditions’ personnel, especially the director of the Greenwich Observatory, Frank Watson Dyson. Einstein’s prediction was that the presence of the Sun near stars would cause tiny shifts in their positions, because the Sun’s gravity would deflect the starlight on its way to the Earth. Dyson and Eddington, but especially Dyson, were experts in this kind of differential astrometry, the measurement of small shifts in star positions. They had spent years (decades, in Dyson’s case) measuring the proper motion and the parallax of stars, which depends on the measurement of similar small star shifts. Thus by good fortune this special opportunity to test Einstein’s opportunity was undertaken by the ideal team who were able to overcome all obstacles, including bad weather and difficulties with instrumentation.

Will we ever see a solar eclipse quite like this in our lifetime?

No, we won’t. Obviously an eclipse with this special star field won’t occur again for nearly two centuries. But in addition, the advance of technology means that there are few important scientific tasks which require an eclipse. Radio telescopes do not require a solar eclipse to test Einstein’s light deflection prediction. These instruments can do the test far more accurately than can be done with optical telescopes at an eclipse. But in another sense, replicating the drama of 1919 is open to anyone. Experiments at the recent 2017 eclipse have shown that a modern amateur astronomer can do the experiment alone to an accuracy better than what was achievable in 1919. Another total solar eclipse will cross America in 2024 and we can hope that other enthusiasts will study the eclipse then. If enough people do the experiment and are able to pool their data, they could achieve a result far more accurate than any ever achieved by professional astronomers at an eclipse. We are living at a moment in history in which the means to do this experiment are within the reach of many people.

What do you hope that readers will take away from this book?

What first made me skeptical of Eddington’s modern critics was their claim that the expedition’s work was influenced by Eddington’s bias in favor of General Relativity as well as his militant pacifism. I found these arguments unpersuasive because I knew that Eddington’s views were highly unusual. Other astronomers of the period were highly skeptical of, or even hostile towards, general relativity. War resisters like Eddington (and Einstein) were a despised minority during World War I. It didn’t make sense that this man could have single-handedly persuaded everyone involved to share his peculiar biases. Sure enough, careful reading of the documents in the archives, including letters and data analysis notes, made it clear that the decisions which were being criticized today weren’t even taken by Eddington but by others in the expedition, especially Dyson. Both Eddington and Dyson made it clear in their letters that Dyson was skeptical of Einstein’s theory to begin with. As I puzzled through Dyson’s notes, I began to unravel the reasoning behind his decision, and I found that it made a lot more sense than did the arguments of some of the modern critics. Furthermore, his reasoning is completely vindicated by the results of the 1978 re-analysis. But I also came to realize that Dyson’s decision depended heavily on input from his assistant, Charles Davidson, and that the success of the expedition was made possible by the multinational Astrographic project, which Dyson worked on and which two of the telescope lenses they used were constructed for. I realized I needed to learn about the man who made those lenses, a fellow Irishman called Howard Grubb, and about the institutional framework which was used to organize the expeditions at a very difficult time. The minutes of the meetings of that Joint Permanent Eclipse Committee and the letters written home to his mother and sister by Eddington made the expeditions come alive for me, and I wanted to share that with other readers. I hope they come away, as I did, with a conviction that the history of science cannot be told fully without understanding the role of all the scientists involved, rather than just one or two famous names. Part of the charm of the story is the different characters who contributed to doing something extraordinarily challenging under impossibly difficult circumstances.

Daniel Kennefick is associate professor of physics at the University of Arkansas, Fayetteville. He is the author of Traveling at the Speed of Thought: Einstein and the Quest for Gravitational Waves and a coauthor of An Einstein Encyclopedia (both Princeton).