# Book: Real Analysis (Boman and Rogers)

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

Thumbnail: Real number line with some constants such as (pi). (Public Domain; User:Phrood).

## How We Got From There To Here: A Story of Real Analysis

The irony of this section is that it exists to tell you that this book was not written for you it was written for your students. After all, we don't need to teach you about Real Analysis. You already understand the subject. The purpose of this text is to help your students make sense of the formal definitions, theorems, and proofs that they will encounter in your course. We do this by immersing the student in the story of how what is usually called Calculus evolved into modern Real Analysis. Our hope and intention is that this will help the student to appreciate why their intuitive understanding of topics encountered in calculus needs to be replaced by the formalism of Real Analysis.

The traditional approach to this topic (what we might call the “logical” story of Real Analysis), starts with a rigorous development of the real number system and uses this to provide rigorous definitions of limits, continuity, derivatives and integrals, and convergence of series typically in that order. This is a perfectly legitimate story of Real Analysis and, logically, it makes the most sense. Indeed, this is a view of the subject that every mathematician-in-training should eventually attain. However, it is our contention that your students will appreciate the subject more, and hopefully retain it better, if they see how the subject developed from the intuitive notions of Leibniz, Newton and others in the seventeenth and eighteenth centuries to the more modern approach developed in the nineteenth and twentieth centuries. After all, they are coming at it from a viewpoint very similar to that of the mathematicians of the seventeenth and eighteenth centuries. Our goal is to bring them into the nineteenth and early twentieth centuries, mathematically speaking.

We hasten to add that this is not a history of analysis book. It is an introductory textbook on Real Analysis which uses the historical context of the subject to frame the concepts and to show why mathematicians felt the need to develop rigorous, non-intuitive definitions to replace their intuitive notions.

You will notice that most of the problems are embedded in the chapters, rather than lumped together at the end of each chapter. This is done to provide a context for the problems which, for the most part, are presented on an as-needed basis.

Thus the proofs of nearly all of the theorems appear as problems in the text. Of course, it would be very unfair to ask most students at this level to prove, say, the Bolzano-Weierstrass Theorem without some sort of guidance. So in each case we provide an outline of the proof and the subsequent problem will be to use the outline to develop a formal proof. Proof outlines will become less detailed as the students progress. We have found that this approach helps students develop their proof writing skills.

We state in the text, and we encourage you to emphasize to your students, that often they will use the results of problems as tools in subsequent problems. Trained mathematicians do this naturally, but it is our experience that this is still somewhat foreign to students who are used to simply “getting the problem done and forgetting about it.” For quick reference, the page numbers of problems are listed in the table of contents.

The problems range from the fairly straightforward to the more challenging. Some of them require the use of a computer algebra system (for example, to plot partial sums of a power series). These tend to occur earlier in the book where we encourage the students to use technology to explore the wonders of series. A number of these problems can be done on a sufficiently advanced graphing calculator or even on Wolfram Alpha, so you should assure your students that they do not need to be super programmers to do this. Of course, this is up to you.

A testing strategy we have used successfully is to assign more time consuming problems as collected homework and to assign other problems as possible test questions. Students could then be given some subset of these (verbatim) as an in-class test. Not only does this make test creation more straightforward, but it allows the opportunity to ask questions that could not reasonably be asked otherwise in a timed setting. Our experience is that this does not make the tests “too easy,” and there are worse things than having students study by working together on possible test questions beforehand. If you are shocked by the idea of giving students all of the possible test questions ahead of time, think of how much (re)learning you did studying the list of possible questions you knew would be asked on a qualifying exam.

In the end, use this book as you see fit. We believe your students will find it readable, as it is intended to be, and we are confident that it will help them to make sense out of the rigorous, non-intuitive definitions and theorems of Real Analysis and help them to develop their proof-writing skills.

If you have suggestions for improvement, comments or criticisms of our text please contact us at the email addresses below. We appreciate any feedback you can give us on this.

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## The Real and the Complex: A History of Analysis in the 19th Century

The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

The late Carl B. Boyer, author of the well-known The History of Mathematics, and the professor from whom, as an undergraduate, I first studied this subject, writes in the third edition of his book (revised after his death by Uta Merzbach) that

The nineteenth century deserves to be known as the Golden Age of mathematics. The additions to the subject during these one hundred years far outweigh the total combined productivity of all preceding ages. The century was also one of the most revolutionary in the history of mathematics. The introduction to the mathematician&rsquos repertoire of concepts such as non-Euclidean geometries, n-dimensional spaces, noncommutative algebras, infinite processes and nonquantitative structures all contributed to a radical transformation that changed the appearance, as well as the definitions, of mathematics.

As the above quote makes clear, advances took place in many different areas of mathematics during this period. Jeremy Gray, the author of the book now under review, has previously written Worlds out of Nothing, which chronicles the history of geometry during the nineteenth century, and now, in what can be described as a companion piece to that book, has turned his attention to 19th century developments in analysis.

The text is organized roughly chronologically, and actually begins slightly before the dawn of the nineteenth century with the work of Lagrange, who tried to firm up calculus by reducing it to algebraic manipulation of power series. (See A Historian Looks Back: The Calculus as Algebra and Selected Writings by Judith Grabiner, and its review in this column.)

Other mathematicians whose work at about this time helped provoke future developments include Fourier and Legendre, who worked on trigonometric series and elliptic integrals, respectively. These difficult mathematical ideas forced people to think hard about the underpinnings of analysis. (David Bressoud, for example, begins his A Radical Approach to Real Analysis with a discussion of the &ldquocrisis&rdquo caused by Fourier&rsquos work: &ldquoThe edifice of calculus was shaken to its foundations. [W]hile most scientists realized that something had happened, it would take fifty years before the full impact of the event was understood.&rdquo) Chapters 2 and 3 of the text discuss Fourier and Legendre as a way of setting the stage for future developments.

There follow three chapters on Cauchy. Chapters 4 and 5 discuss his work on real analysis and chapter 6 talks about his contributions to complex analysis, which in the early 19th century was not even yet recognized as a subject in its own right. After this, there are three chapters discussing the early history of elliptic functions and elliptic integrals, concentrating on the work of Abel, Jacobi and Gauss. We then return to Cauchy and his work on complex function theory in the middle third of the century, and chapter 11 then explains how elliptic function and complex function theory were brought together.

These chapters constitute roughly a third of the book. The second third (chapters 13 through 20) of the text begins with potential theory and moves from there to the further development of complex function theory during the mid-19th century. Riemann&rsquos geometric approach to the subject is discussed in some depth over several chapters, and is compared and contrasted with the more algebraic approach of Weierstrass.

Finally, chapters 22 through 29 address developments in the rigorization of analysis in the latter part of the century. There are chapters here on uniform convergence of series of functions, the fundamental theorem of calculus, and the construction of the real numbers by Dedekind and Cantor. Chapters in this group also discuss Lebesgue theory, Cantor&rsquos set theory and the beginnings of topology. These final chapters &ldquoare offered as pointers to some of the ways analysis was to develop in the 20th century, and to branch into new domains, abstract set theory and topology.&rdquo

The author discusses not only mathematics but also personalities. Many of the chapters in the book are organized around people, and good biographies of them, complete with photographs, appear throughout. The lives of these mathematicians are placed in historical context, and the author does a good job of conveying the fact that mathematical advances are often the product of false starts and mistaken ideas.

This book, like Worlds out of Nothing, is part of the Springer Undergraduate Mathematics Series and, we are told, derives from a series of lectures given by the author to (British) senior undergraduate students. Consistent with these undergraduate origins, the author has employed a number of useful pedagogical devices. The three thirds of the book that are discussed above are set off from one another by two chapters, 12 and 21, that are called &ldquorevisions&rdquo and which pause and reflect on what has come before I confess that I cheated a bit and read these chapters first, figuring that I might as well know in advance what to look out for.

There is also a final (very short) chapter 30, titled &ldquoAssesment&rdquo, in which Gray describes his class&rsquos end-of-course essay assignment and reproduces the advice on essay writing that he gave his students. The earlier chapter 21 also touched on assessment issues. Analogous chapters on writing (12, 21 and 31) appeared in Worlds.

In addition to these chapters, Gray has, in an Appendix, included translations (all but one done by himself) of portions of important papers, including works by Fourier, Dirichlet, Riemann, and Schwarz.

Notwithstanding these nice features, I doubt this book would prove very successful as a text for undergraduates on this side of the Pond. For one thing, there is the obvious difference between a British undergraduate mathematics education and an American one, and, in addition, the prerequisites for reading this book seem a bit daunting. Unlike other history-minded analysis books such as Bressoud&rsquos aforementioned Radical Approach and How We Got From There to Here: A Story of Real Analysis by Rogers and Boman, a substantial background in analysis seems to be assumed here. (This is presumably because these two books are not really intended as histories of analysis, but instead as introductory courses in analysis from a historical perspective.) In addition, although billed as a text, Gray&rsquos book is curiously devoid of exercises some appear sporadically throughout the text, but there are very few of them (less than 20 throughout the entire book).

Although I don&rsquot anticipate using this book as the text for a course, I view it as an extremely valuable addition to my library and a book that I am sure I will consult frequently in the future: this is a text that should serve well as a desk reference for a faculty member teaching either analysis (real or complex) or the history of mathematics. Certainly, no good university library should be without it. We are also told in the preface that two other volumes on 19th century mathematics history (on algebra and differential equations, respectively) are planned, and I eagerly await their publication.

## A Story of Real Analysis

The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis.

This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context.

However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.

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## Open College Books

In many parts of the world, academic books are not adequately available, and almost everywhere they are very expensive. Fortunately, the Internet contains many open-access texts created by university professors, nonprofit groups, or governments, with useful information in the sciences, the humanities, and medicine. These books can be downloaded for free by anyone and most of them can be printed out for free. I have compiled a list of some of these sources. If you need a book but can't find it here, try the excellent Open Textbook Library of the University of Minnesota's Center for Open Education. Also try FreeTechBooks and Rice University's OpenStax. . I have also included some links to related educational resources such as educational videos and online courses, as long as they are free of charge. There are also some lists to subject collections of open-access articles at Cold Spring Harbor's Perspectives in Medicine.

My name is John J. Dziak and my day job is as an associate research professor in statistics at the Edna Bennett Pierce Prevention Research Center at the College of Health and Human Development of the Pennsylvania State University. I am maintaining this site as a volunteer -- as a caveat, I do not have any expertise in most of these subjects.

## ‘The New World’: Terrence Malick’s Magic Portrayal of America’s Original Sin

After a much-discussed hiatus that lasted for two decades, Terrence Malick returned to filmmaking with the highly praised anti-war epic called The Thin Red Line, and as the critics and film lovers around the globe competed to express their acclaim for the powerful cinematic poem that proved Malick was far from getting out of touch, many wondered what the director’s next step might be. Surprisingly enough, Malick reassigned his attention to a script he had finished in the late seventies and kept it in a drawer until 2004. It was his take on the well-known classic story of Pocahontas and the English pilgrim’s “discovery” of America a story nobody thought of touching after Disney’s 1995 animated version. But Malick felt he had something new to say. The New World, of course, centers on Pocahontas and the well-charted historical events that unfolded after the British reached the American shores and established the colony of Jamestown in 1607. As Malick portrays “first contact” as a consequential, far-reaching event that changed the land and its people forever, the name of the film gains more depth than one might perceive at first glance.

When the English established Jamestown and started exploring the completely foreign world they chose to transform into their new home, both the Native American and European societies were faced with unfamiliar cultures that shook their notions of the world and broadened their horizons. Just as Pocahontas and her tribe were alien, puzzling and exciting to the newcomers, so did John Smith and John Rolfe, two distinguished settlers, present a shocking novelty to “the Naturals,” as the settlers addressed the local tribes. The New World is definitely not the historical adventure film, an emotionally captivating romance with thrilling elements of war that a viewer with no previous Malick experience might expect. This is a beautiful, carefully staged, thoughtful rendering of the first collision of cultures we all know ended in tragedy, a depiction of a specific sensitive period of American history that was at the same time painful and unfair just as it was necessary and stimulating. Malick knows what the United States were built on, and he doesn’t shy away from dealing with it.

The production of The New World started in July 2004, and principal photography was done with by mid-November. The film experienced its wide-release premiere at the beginning of 2006, only to be met with shyly positive reviews from the critics. It was a much more critically polarizing endeavor that the universally greeted The Thin Red Line its reception might be more closely compared to that of Days of Heaven back in 1978. But similarly to Days of Heaven, Malick’s fourth feature film just needed time, as its stature grew steadily in the years that followed. A lot of distinguished and widely respected critics loved it from the get-go. Roger Ebert, for example, gave his full support immediately. Six years after its release, John Patterson of The Guardian published his analysis of the film calling it a misunderstood masterpiece. “This decade hasn’t been up to much, movie-wise, but I am more than ever convinced that when every other scrap of celluloid from 2000-2009 has crumbled to dust, one film will remain, like some Ozymandias-like remnant of transient vanished glory in the desert. And that film is The New World, Terrence Malick’s American foundation myth, which arrived just as the decade reached its dismal halfway point,” Patterson wrote. “It is a bottomless movie, almost unspeakably beautiful and formally harmonious. (…) It is both ancient and modern, cinema at its purest and most organic, its simplest and most refined.”

Malick wanted the film to feel as authentic as possible, and as a famous perfectionist, you realize how much preparation and attention to detail he demanded. Casting was logically a big part of this process. As a fan of Disney’s Pocahontas, Malick decided to hire Christian Bale and Irene Bedard, both of whom had a role in the animated feature. Choosing for the role of Pocahontas was a far more complicated matter. Q’orianka Kilcher was selected after a laboring process: a perfect choice, it seems, but also a choice that proves to what degree the filmmaker strived for authenticity. Kilcher was only 14 at the time of the filming, as close to Pocahontas real age of 10-12 in the time of these historical events as possible. Hiring a lot of Native American actors and actresses wasn’t enough—Malick hired people to teach the actors the rarely spoken Algonquin language. Blair Rudes, a professor of linguistics at UNC-Charlotte, was brought in to reconstruct a form of the extinct Powhatan language specifically for the needs of the story. Some reports suggest Malick didn’t allow Kilcher and Farrell to interact in any way before their first scene together so as to capture the most authentic reaction possible from both of them. Naturally, the choice of filming locations and the meticulous design of costumes and props were also heavily influenced by the filmmaker’s desire to present as accurate a depiction of the period and circumstances as possible.

Malick didn’t cut the film and then add the music: he had an already completed score by James Horner before he even started editing the picture. This meant a lot of Horner’s music ended up discarded, which caused a rift between the two artists. Horner wasn’t all that thrilled that only a couple of his fragments made it to the final cut, calling his collaboration with Malick “the most disappointing experience he’s ever had with a man,” saying he “never felt so let down by a filmmaker in his life.”

Shot by the great cinematographer Emmanuel Lubezki (three-time Academy Award winner for Gravity, Birdman and The Revenant, a film influenced by The New World), with the predominant use of natural light, edited by Richard Chew, Hank Corwin, Saar Klein and Mark Yoshikawa, based on Malick’s own screenplay written three decades before the film’s premiere, The New World is a visually stunning, stylish portrayal of the very beginnings of the American civilization. With Q’orianka Kilcher, Colin Farrell, Christian Bale and Christopher Plummer delivering great performances, The New World is one of the most impressive films of the beginning of the 21st century.

Screenwriter must-read: Terrence Malick’s screenplay for The New World [PDF]. (NOTE: For educational and research purposes only). The DVD/Blu-ray of the film is available from the Criterion Collection and other online retailers. Absolutely our highest recommendation.

The following is an excerpt from the book ‘The Cinema of Terrence Malick: Poetic Visions of America,’ written by Mark Cousins, ‘Praising The New World.’

### PRAISING ‘THE NEW WORLD’ by Mark Cousins

We cannot separate our reading of a film, our sense of where it fits in culture and aesthetics, from the raw circumstances in which we saw it. I watched The New World, a film that flopped around the world despite headlining one of America’s most famous movie stars, on a rainy Tuesday afternoon in the Vue cinema in Edinburgh. Usually a Colin Farrell movie would be booked into all the multiplexes simultaneously but this one was hard to find. It played in one suburb in Glasgow and on just a handful of other screens in Scotland, as if it had the plague and needed to be quarantined. For a prestige picture, it was booked with what seemed like reluctance, minimally, resentfully.

The film made me cry. I began to do so about thirty minutes after it started and continued, on and off, in waves, just like Yang Kuei-Mei sitting on the park bench in the last shot of Tsai Ming-liang’s Vive l’amour. By the end of The New World, it seemed to me, I had experienced something like a Bach’s Mass in B Minor or a poem by Percy Bysshe Shelley. It was about rapture and the end of rapture. It showed me seeing. It made me sensible. That week, I was to begin teaching at the Screen Academy in Edinburgh—Scotland’s new film school. After seeing The New World, I realised that the best, most inspiring first class would be to watch it on the big screen. The Screen Academy negotiated a discount rate, and my students went.

None responded as I did. None cried, none thought that they’d seen a cinema masterclass, none seemed compelled to find words to describe how it went beyond cinema. Thus began The New World’s descent for me, its relegation, my growing suspicion that I went too far to meet it that rainy Tuesday morning, that I mingled too many of my ideas with its ideas, as if it was a lover for whom I had fallen, to whom I had not listened, in whom I had seen my own selfish sense of what lovers are.

The New World continued to become stale for me when I read the first edition of book you now have in your hands, whose subtitle is Poetic Visions of America. As I read Mottram’s essay, then Patterson’s, then Campbell’s, McGettigan’s, Orr’s and McCann’s, I realised that The New World was each of the things that the films they were writing about was. It is certainly Emersonian, as Badlands is, in that it is an essay about human subjectivity and its response to nature. It is also, surely, Heideggerian, predicated on the idea that deep down in the well of ourselves is Being, something rendered inaudible by the tinny sound of life modernising. It also uses commentary, as Badlands, Days of Heaven and The Thin Red Line do, to fetch water from that well and bring to the surface traces of deep thought and, as a result, render that surface numinous.

Like the other films discussed in this book, The New World is about having paradise and watching it slip away, the slippage caused by the very same stuff that allows us to feel the bliss of paradise in the first place. As in Badlands, The New World uses doorways and gateways to contend that the built world frames human beings in a different way to the bowers of the natural world, and causes us to re-see them because of this framing. Also, like some of the other movies conjured on these pages, The New World seems as if it could have been written by Herman Melville. Like Private Witt in The Thin Red Line, it asks the question ‘This great evil, where does it come from? How did it steal into the world? What root did it grow from? Who’s doin’ this? What’s killin’ us, robbin us of life and light?’ And, like the other films described here, its characters are often in a dream state, a condition that has ‘gone upwards’ to the movie itself, imparting to it their reverie.

This realisation that The New World is like the other films described deepened my deflation. That the fourth is like the other three is disappointing to me but then I am in and of a culture that values innovation and iconoclasm over continuity and confirmation. The New World does not value iconoclasm. It praises rather than destroys. Its glorious beauty is not fuelled by the desire to move forward into the future of ideas, but backward, as it were, or downwards into thoughts and wisdoms that are no longer apparent. This backwardness disappoints me. The film that made me cry began, on reading this book, to set off a train of thought that started with its over-familiarity, moved through its thematic and philosophical repetitions, on to question why another film had been made about Edenic America, and concluded with a bang, and some irritation, with another question: Why not produce a movie about something pressing on our modern age, like Iraq?

This point led to other thoughts: America has been irritating of late. Its centripetal self-regard has been showing too much and not been sufficiently outweighed by its openness, liberty of mind and optimism, so to see another film about its discovery by Europe, that politico-mythic next step that for so long was thought of (by Europeans) as a first step, seems like more navel-gazing. Who cares? Get over yourself. We’ve gotten over you. We no longer flock to see your horse operas, your primal law-makers, the stetsoned first-footers of what you consider to have been that splendid nineteenth-century party—America.

It is clear from its first scenes that The New World doesn’t talk to America in this lippy, impatient, twenty-first century, teenage way. The film isn’t ‘over’ the country. It elides the centuries since the ‘founding’ fathers as if they—the centuries—were a regrettable detour that, desirably, could be redressed. It is still calling those days ‘new.’ This last point, my attempt to be honest about the apparent conservatism of The New World conversely fails to be as depressing as it should be. The film shows that its director’s worldview and philosophical assumptions haven’t budged, but immobility can mean passionately, obsessively correct as well as blinkered. Once Gaugin found his style and situations, he strayed little because they took from the air around him, and made flesh, some of the discourses of his times. Maybe the writer-director of The New World has been doing the same thing all these years, precipitating timeless insights, just as Gore Vidal has been lobbing word bombs at his home country from the very same point since the 1950s.

With this thought—the longevity of the filmmaker’s rightness—the deflation starts to reverse. The stature of The New World starts to grow again for me. It is time, today, also a Tuesday but sunny, to watch it once more.

The first shot, of the rippling surface of a pool, is scored to the ululation, twitter and squawk of a gigantic aviary. A Darwinian world, shoals of fish or eels, seem to slip and swim beneath the pool’s surface, breaking and convulsing it. To stare at water in cinema is to expect a Spielbergian shark or contemplate a Tarkovskian void but here, in the first moments of this film, we are looking at a gene pool, the origins of life. A voice, that of Pocahontas, says ‘Come spirit… we rise out of the soul of you.’ Then we are at her feet as she reaches to the heavens, hands apart, beckoning the spirit to descend. Then we are under the water, as if we have become the spirit she addresses, and are looking up at Algonquin Indians in Virginia in 1607, staring and pointing out to sea. They have spotted something. Deep down in the sound mix, softening and underscoring the aviary, are distant, gathering horns. Then violas join them in heralding what the Algonquins have seen, then violins rush to the scene, above the rest, and we cut to seventeenth-century boats drifting gracefully towards the land, the strain in their holds heard in the creak of massive beams and the click of rigging. Then we are in the black bowels of one of the boats, in whose gloom we see two white eyes of a dog, its wet hair hanging over its face as if it has been for a swim. Then we get closer and see that the dog is a movie star, Colin Farrell, who peers out into the light.

Then, six and a half minutes in to this gathering opera (the music is from Wagner’s Das Reingold), we cut to a field of long grass which, given that it’s this director, is photographed as if it is the sea. Wind rushes across it in waves, as it does in the trees at beginning of Tarkovsky’s Mirror and the forest shots in David Lynch films. The field is alive, a corner of a slumbering planet. The Panavision lens makes the distant edge of the field, the horizon, curve as if it is a huge quadrant of the earth. Then Colin Farrell, playing John Smith, is walking through the field. The sun is behind him—this filmmaker usually shoots into the sun, giving his actors haloes, making their faces penumbral. Tribesmen arrive. With great gentleness, Smith reaches out his hand and they touch it and almost nibble it as if they are chickens eating corn or, as Smith says in close-miked voice-over, ‘like a herd of curious deer.’ When we hear his thoughts, they sound like he is praying to his European god. Interweave his prayers with Pocahontas’s to her earth mother god and you get a theological duet, a song to two sirens.

At 12 minutes 40 seconds, in a sea-field, Smith sees Pocahontas. They look at each other. The light is to his right and behind him (of course), but his body is clocked at 45 degrees so the sun glances off his chest. We cut to her and, magically, impossibly, the light is to her right too. Sixteen minutes later she touches his lips. Smith says these people have ‘no guile, no jealousy, no sense of possession,’ but how could he possibly tell? All he has is what he sees and feels in these moments, on that first day, a day that will change the world.

Two hours later, I am crying again, not as much as before because I am watching on DVD this time, on a screen too small to overwhelm. As I watch I get stuck at how beautiful Q’Orianka Kilcher, who plays Pocahontas, is, and Farrell too, and become suspicious of their casting because, in our time, the iniquities of the fashion and advertising industries have made us rightly uneasy about worlds where everyone is beautiful. I free myself to enjoy the film again by the thought that it, The New World, is explicitly asserting the beauty of this moment in Virginia in 1607, and that these beautiful actors are what John Orr calls ‘the expressive externalisation of private emotions’—feelings of peace and harmony—that, themselves, are beautiful. Then I get stuck again, by Kilcher’s age. She is fifteen. This filmmaker does fifteen well. Sissy Spacek was 24 when Badlands was released but seems fifteen in the film. The question in The Thin Red Line, ‘Love, where does it come from, this flame in us?,’ is the question a 15-year-old would ask. This director keeps asking it, as if he himself is stuck at that age.

As I watch I think and scribble, mostly superlatives about people seeing and feeling and how the film depicts this and how it is mostly about seeing and feeling, their immediacy, their necessity and their limits. Again and again as I scribble, I write a name, David Hume. When I saw The Thin Red Line I also thought of the historian-philosopher Hume, who published A Treatise on Human Nature in 1740, Enquiries Concerning Human Understanding in 1748, Enquiries Concerning the Principles of Morals in 1751 and whose Dialogues Concerning Natural Religion was published post-humously, in 1779. The day after I saw The Thin Red Line I went to Hume’s grave in Edinburgh, plucked a weed, taped it to a bit a paper and mailed it to the filmmaker’s agent, with a note saying what it was and thanking him—the filmmaker—for what I considered to be a masterpiece.

A month later, as I left a theatre, I was met by my partner who said that I should hurry home because I would be getting a phone call at nine o’clock. I did so and, on the button of nine, the phone went and a high-pitched voice said ‘Hi, it’s Terry Malick here.’ He asked if the weed was really from Hume’s grave. He asked what the grave is like (a wide, squat Georgian tower). He thanked me again and again for the weed. He said he had read Hume as a student and admired him. I remember that in trying to explain why I sent the weed, I mentioned George Steiner’s book Real Presences. We talked some more. Malick said he would call again, and hung up. He did call again. This time I asked if he’d agree to be interviewed for my television series Scene by Scene. He said that TV makes him nervous and that he is a bad speaker and, certainly, even in this least pressured of circumstances, a phone call with a cinephile, he was hesitant and diffident. I felt bad for asking, then, presumptuously, suggested a book-length interview, like the Truffaut-Hitchcock one or Faber & Faber’s ‘Filmmaker on Filmmaker’ series. Again, he said that he wasn’t sure, that he just liked talking about philosophy. Malick called a third time, then we said our goodbyes.

The connection, of course, made me alert to the Humeanism of Malick’s work. Hume wrote that his mother was ‘wake minded’ and so is Malick and so are his films. Following in the footsteps of John Locke’s Empiricism, Hume argued that passage beyond our sense perception is impossible. We have only these perceptions—he called them impressions—with which to construct our sense of the outside world. These impressions, argued Hume, imprint themselves on our mind and, when repeated, form ideas. An idea is an aggregate of confirmed impressions—that, for example, the sun in the morning will feel warm on my face. We do not build our understanding of the outside world logically (Hume criticised the concept of cause and effect), but by impressions accumulating into ideas. Each, said Hume, recalls the other. There is a flow between the two.

The lack of metaphysics in this empiricism, its non-essentialism sounds, at first, very un-Malick. That human beings are receivers of information, that there is nothing mysterious or soul-like inside the black box of themselves, seems to undermine the Heideggerean rapture that we see in Malick’s films, their attempts to biopsy the human soul. But this is perhaps to misunderstand Hume and Malick. The mystery in Malick’s work lies in his ability to use the medium of film to show that it is the process of receiving impressions of the world that is transcendent. He may in addition posit theories of self, but looking at Malick through the lens of Hume shows that the pre-cognitive experience of engaging sensually with the world is where at least some of the wonder lies.

So, in The New World, we see impressions turning into ideas. We see Smith feeling and hearing this new world, observing patterns of tenderness and openness in the Algonquins, patterns that, Hume would say, aggregate into ideas. They coalesce and take root. Through observing him, touching him and listening to him, Pocahontas’s sense of what sort of man Smith is grows magically before us, as green shoots in the jungle do in time-lapse scenes in natural history programmes. And, over time, because Malick’s films are cyclical and repetitive rather than forwardly directed, we see ideas becoming complex ideas—another Humean term. This tree is beautiful. The one I saw yesterday is beautiful. The one from the day before was too. We grow accustomed to this certainty, this beauty-hit, and come to expect it. It enters our nature, this assumption of constancy, of guaranteed rapture.

If Malick is in some sense stuck, and if in some sense I get stuck watching his films and reading about them, perhaps this is why. The constancy in his career, the ongoing praise he lavishes on the mid-moment in human experience when we are drinking in, with ‘beaded bubbles winking at the brim, and purple-stained mouth’, perhaps derives from that sense that because life has been beautiful so far, it should always be so. And in each of his films he gets knocked for six, bowled over by the un-expected, the post-Edenic, the scepticism at the very least, that is required in order to brace oneself for the tragedy of life.

In The New World, Malick gets knocked over yet again. Each time he goes away from our screens, he seems to unlearn what life has taught him and start again with new characters like Pocahontas who are too young to have learnt, or from a place that doesn’t know or doesn’t need to know. They are wide open, these characters, Humean sponges drinking in experience, glorying in the aviary that is life.

Whether we need more Malick films is an open question. I hate the thought that there might be no more but would, to be honest, love him to drop both Hume and Heidegger next time, and venture out into a new world. That’s not to say that I don’t adore the game of skittles he’s played these last three decades, but isn’t it time for another game? This one came courtesy of Time-Warner, just as the Vatican funded the paintings of Caravaggio in Santa Maria del Popolo.

Making The New World, a documentary shot during the production of the film in 2004, directed and edited by Austin Jack Lynch.

Colin Farrell shares his candid observations on Malick’s directing style and the inspiration he drew from the natural landscape.

Conquest of Paradise is the second chapter of the Directors Series’ examination into the films and career of director Terrence Malick, covering his pair of experimental historical epics: The Thin Red Line (1998) and The New World (2005).

### EMMANUEL “CHIVO” LUBEZKI

“When Terry Malick and I talked about shooting the life of Che Guevara last year, we wrote down a set of rules—our dogma—to follow,” says cinematographer Emmanuel Lubezki. “The idea was to capture the reality of this man’s life with natural light, no cranes, no big rigs. It would be subjective—from the man’s point-of-view.”

“A little while later, Terry called to say ‘we’re not doing Che but doing a story based on the foundation of this country,’” Lubezki adds. “The code of belief, the dogma, would be the same.”

“I was naturally interested in this new project. No light. Handheld. And, one of the most important tools, the Steadicam [manned by Jörg Widmer and James McConkey]. We, in the camera department, would think of ourselves as the 5 o’clock news team, capturing the reality in front of us. It would just happen to be the reality of a different century.”

Their dogma was natural back light. “We chose back light, not because it is pretty but because it helps the evenness of the light,” Lubezki explains. “Terry shoots out of sequence. A close-up in October might be used on a scene later in December somewhere else. By shooting back light, keeping softer light on the faces of the actors, and keeping the sky white [Terry doesn’t like blue skies], we were able to serve the story and keep the continuity.”

The team was always looking for locations that would be versatile, not only to serve the story but also to serve Malick’s need for moving story elements from place to place. “We would shoot inside the forest and trees when the light was toppy and use the canopy of trees as a big silk,” Lubezki explains. “When the light disappeared, we would go to the fields, using the direction of light that would serve us best. Sometimes, Terry would shoot a scene in the forest and then re-shoot it in the fields, deciding later which played best.”

To capture the earthy look as naturally as possible, Lubezki chose Kodak 5218 for most of the picture. It gave him the flexibility to work at a deeper stop. “We shot everything anamorphic with a depth-of-field between f/16 and f/11,” he says. “By shooting at that depth-of-field, we could really make the audience feel as if they were in this world with these characters.”

“Sometimes, lack of depth-of-field becomes a barrier and we wanted to take that barrier away. To help the lenses as much as possible, Panavision created a new lens for us to use on the XL cameras. They combined the E-series lens that is heavy with the C-series lens. This allowed us to have close focus, keeping with our ‘dogma’ of being in the action and solve the anamorphic contradiction between resolution [which it gives us] and depth-of-field [which is not always possible].”

“I wanted to use only the 35mm and 40mm most of the time,” he adds. “We added the 50mm for telephoto, when we needed to get close to the actors, but still give them some room to move.”

Following the philosophy and ideas of photojournalists, Lubezki and team were constantly thinking of themselves as still photographers, moving fast with a camera hanging around their necks, creating images of life and reality in the places where they landed.

“Terry plans everything in his head,” says Lubezki, “but he is always looking for those moments that are unplanned—those happy accidents that breathe reality into a story. For him, the behavior of people, nature, even where the wind blows are all moments to capture. These are ‘happy accidents’ that can’t happen on the stage because you are always restricted to walls or props. And, to artificial light. Artificial light is simple. It is a specific color temperature and feel. But, natural light is complex and sometimes chaotic. A bounce from the floor or a reflection from the sky can do so much.”

“Terry’s desire to free the actor also freed us from the artificial. The many elements and feelings that the natural environment and light evoked contributed to our desire to capture this story in a different way.” (Source: from article by Pauline Rogers on the International Cinematographers Guild website)

Emmanuel “Chivo” Lubezki is arguably the best cinematographer working in Hollywood today. Thanks to Jorge Luengo Ruiz for this breathtaking supercut.

The director of Nomadland, Chloé Zhao, talks about what she has learned from Terrence Malick’s epic reimagining of the tale of John Smith and Pocahontas, including the power of rooting historical drama in subjective experience. Read more at the Criterion’s Current.

Here are several photos taken behind-the-scenes during production of Terrence Malick’s The New World. Photographed by Merie Weismiller Wallace © New Line Cinema, Sunflower Productions, Sarah Green Film, First Foot Films, The Virginia Company LLC. Intended for editorial use only. All material for educational and noncommercial purposes only.

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