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Yahoo
19-05-2025
- General
- Yahoo
How the Passionate Male Friendship Died
One of my favorite monuments looks like it belongs to a married couple. Draped in marble flowers and guarded by fat cherubs, it features two stone portraits joined by a knotted cloth, and script that describes an intimate bond: a 'beautiful and unbroken marriage of souls and a companionship undivided during thirty-six complete years.' But this memorial has nothing to do with a husband and a wife. It commemorates a friendship—one between two men, Sir John Finch and Sir Thomas Baines, Renaissance-era doctors who traveled, worked, and lived together in the 1600s and were buried side by side at Christ's College in Cambridge. 'They who while living had mingled their interests, fortunes, counsels, nay rather souls,' reads their tombstone, 'might in the same manner, in death, at last mingle their sacred ashes.' This exuberant expression of love is a far cry from popular depictions of male friendship today, which tend to portray men as struggling with vulnerability, or reticent to form bonds unless prompted by the women in their lives. Of course, Finch and Baines could have been a couple; they lived in a culture that criminalized homosexuality, in which 'friendship' might have served as a guise. But a platonic relationship as effusive as theirs also wouldn't have been out of the ordinary: In their time, across Europe, passionate bonds between men were publicly celebrated, and women were typically viewed as lacking the emotional depth required for true friendship. Many historians have a ritual they use to shake themselves out of their complacent 21st-century perspectives. When my friend Jo sits down at her desk to write about 19th-century America, she tries to remember that back then, pigs used to wander the streets of New York. While writing my latest book, on the history of friendship, going to Finch and Baines's monument was my trick. In an age when women's friendships are widely idealized—think Sex and the City's foursome and pink glitter T-shirts that read Best Friends Forever—Finch and Baines's memorial helped remind me that the cult of female friendship hasn't always been the norm, and that the way friendship is viewed now may not be how it's viewed forever. [Read: The agony of texting with men] I'm a historian of emotion: I study how cultural narratives act on people as individuals—and how changing social factors play out in our hearts and homes. Scholars in my field often talk about the concept of 'emotional communities' to understand how the behavior connected to a particular feeling can change across time and place. An emotional community shares expectations about which emotions should be felt—which are shown, which are hidden, how each should be expressed. Such rules are enforced through institutions such as schools and courts, and via literature, art, and the rituals of family life. As a society's emotional rules change, so do the ways individuals expect to feel—including in friendship. Psychologists have shown that, far from following a universal template, friendship has 'styles' that differ subtly from place to place. According to the psychologist Roger Baumgarte, some cultures, such as that of the United States, seem to favor a more 'independent' style of friendship, in which friends are highly respectful of one another's autonomy and might become uncomfortable if they sense that someone is overstepping a boundary. In other cultures, such as those of Cuba or China, friends are expected to 'intervene' more in a friend's life and might feel snubbed if help is not given. Although this kind of research can quickly give way to reductive stereotypes, it does illuminate how diverse people's expectations about friendship can be. These expectations can also change across time. Finch and Baines's 17th-century emotional community, for instance, was shaped by highly romantic ideas about male friendship. The two men lived toward the end of a period of astonishing intellectual transformation in Europe, when artists, politicians, scientists, and philosophers had rediscovered, via the scholars of the Islamic Golden Age, great works of classical antiquity and wanted to make their ideals their own. One such work, Aristotle's Nicomachean Ethics, written in the fourth century B.C.E., divided friendship into three tiers. The bottom two tiers were populated by ordinary kinds of friends, in what he called friendships of utility and friendships of pleasure (the only kinds that women were supposedly capable of). Friendships of utility, Aristotle wrote, were 'commercially minded,' based on mutual help and quid pro quo. Friendships of pleasure were bonds formed through diversion and entertainment: You might gravitate toward a friend because they make you laugh, or you might sit with them at a game because they support your favorite athlete. But the third tier, which Aristotle called 'perfect' friendship, was something else—a bond between two men 'alike in virtue,' who saw each other as a 'second self.' It was, as later philosophers explained, as if 'one soul dwelled in two bodies.' This is how Baines and Finch saw themselves. They strove to be 'perfect' friends, and by all accounts, they seem to have succeeded. [Read: Why friendship is like art] Another Renaissance figure who believed himself to be a 'perfect' friend was the French philosopher Michel de Montaigne, whose essay 'On Friendship' continues to influence writing on the subject today. In his bond with Étienne de La Boétie, a fellow lawyer and author, de Montaigne believed that he had found the kind of ideal friendship that only a few men in a generation could possibly hope to achieve—and his essay is a soaring depiction of their transcendent connection. In one of the most quoted lines in the history of friendship, he writes: 'If you press me to say why I loved him, I feel that it cannot be expressed except by replying: 'Because it was him: because it was me.'' Like other intellectuals of the time, de Montaigne did not consider women capable of this ideal. Women's brains could not endure this 'clasp of a knot so lasting, and so tightly drawn,' he wrote. His claim was based on the era's misogynistic medical theories, which held that women's brains were colder and weaker than men's. Women were willows, explained the scientist and poet Margaret Cavendish in 1655, liable to bend in the smallest draft—not sturdy oaks. In this way, most women were declared too flighty, capricious, and stupid for the commitment that true friendship required. They were also thought to be too easily distracted by romantic relationships with men. 'The reason why most women are so little affected by friendship,' wrote the 17th-century French nobleman François de La Rochefoucauld, 'is that it tastes insipid when they have felt love.' Women's friendships from this era were not generally memorialized in stone or glorified in reams of soaring poetry. In the archives, their stories appear as fragments compared with the large, easily accessible corpus on male friendship. But certainly they existed. In my research, I found evidence of women's bonds in scraps and shards—glimpses of women grieving over their deceased friends or helping one another through illnesses, evidence of friends forming households and raising children together, running joint businesses, defending others in court. Men knew the power of these alliances. Friendships gave women agency in a world not designed for them. It is perhaps no wonder that their friendships tended to be dismissed. [Read: What thirty years of female friendship looks like] Within 100 years of Finch and Baines's burial, the rules of their emotional community began to shift. The second part of the 18th century was a period of revolutionary fever and social reform across Europe and America. As abolitionist, women's-rights, and anti-poverty campaigns gathered momentum, so did discussions about emotion and sympathy (the old word for empathy). Poets, artists, and philosophers in this new Age of Sensibility began to speak in reverent tones of the supposed heightened sensitivity of 'the female mind.' They helped advance the idea that women enjoyed some special skill in friendship—and were capable of deeply emotional connections with the poor, the disenfranchised, and one another. By the 19th century, though, this belief in women's empathy had hardened into a new ideal of middle-class Victorian femininity: the 'angel of the house.' The consummate woman was supposed to be tender, loving, and eternally supportive; devoted friendships were seen as proof of her compassionate nature. Girlhood bonds were considered practice for the affection and sacrifice necessary for future roles as wives and mothers. Often, girls were primed with stories of highly romantic female friendships: One popular collection of nursery tales depicts two friends, Beatrice and Alice, who 'loved each other dearly' and 'with their arms about each other would sit under the deep shadow of the trees listening for the cuckoo's notes.' In Victorian novels, the girls who enjoy intimate childhood bonds—think Jane Eyre and Helen Burns—grow up to win the prize of marriage and motherhood, and 'difficult,' solitary, awkward girls, such as Lucy Snowe in Villette, marry unhappily, if at all. Romantic friendships between men didn't fall completely out of favor during this time. Photographs from the 1850s to the early 1900s show male friends holding hands or draping their arms around each other; sentimental letters between men in this period also abound. But by the second half of the 19th century, a new narrative about men's friendships was on the rise. Some began to depict male friendships as blundering and superficial. And as gay culture became more visible, and European sexologists stoked fears of 'sexual inversion,' a growing self-consciousness around male intimacy emerged. In 1863, the English feminist campaigner Frances Power Cobbe published an essay, 'Celibacy v. Marriage,' in the widely read Fraser's Magazine, in which she reiterated what had by then become a familiar story about male and female friendships. Whereas women friends enjoyed 'one of the purest of pleasures and the most unselfish of all affections,' Cobbe wrote, to men, friendship was little more than forming an 'acquaintance at a club.' Americans still, to an extent, live among the ghosts of these Victorian forebears, holding women to high standards of intimacy and portraying male bonds as clumsy and inept. We also live in an age of social fragmentation, in which experts, worried about loneliness and isolation, are puzzling over how to bring people together. To foster more connections, we'll need to reexamine our emotional rules—which ones are worth preserving and which ones we might be better off without. As a historian, I can tell you this: If we want to reimagine the terms of friendship, we can. *Lead image credit sources: Fitzwilliam Museum / Bridgeman Images; Harris Brisbane Dick Fund / The Met; Royal Institute of British Architects / The Met Article originally published at The Atlantic
Atlantic
19-05-2025
- General
- Atlantic
When Men Weren't Afraid to Love Their Friends
One of my favorite monuments looks like it belongs to a married couple. Draped in marble flowers and guarded by fat cherubs, it features two stone portraits joined by a knotted cloth, and script that describes an intimate bond: a 'beautiful and unbroken marriage of souls and a companionship undivided during thirty-six complete years.' But this memorial has nothing to do with a husband and a wife. It commemorates a friendship—one between two men, Sir John Finch and Sir Thomas Baines, Renaissance-era doctors who traveled, worked, and lived together in the 1600s and were buried side by side at Christ's College in Cambridge. 'They who while living had mingled their interests, fortunes, counsels, nay rather souls,' reads their tombstone, 'might in the same manner, in death, at last mingle their sacred ashes.' This exuberant expression of love is a far cry from popular depictions of male friendship today, which tend to portray men as struggling with vulnerability, or reticent to form bonds unless prompted by the women in their lives. Of course, Finch and Baines could have been a couple; they lived in a culture that criminalized homosexuality, in which 'friendship' might have served as a guise. But a platonic relationship as effusive as theirs also wouldn't have been out of the ordinary: In their time, across Europe, passionate bonds between men were publicly celebrated, and women were typically viewed as lacking the emotional depth required for true friendship. Many historians have a ritual they use to shake themselves out of their complacent 21st-century perspectives. When my friend Jo sits down at her desk to write about 19th-century America, she tries to remember that back then, pigs used to wander the streets of New York. While writing my latest book, on the history of friendship, going to Finch and Baines's monument was my trick. In an age when women's friendships are widely idealized—think Sex and the City 's foursome and pink glitter T-shirts that read Best Friends Forever—Finch and Baines's memorial helped remind me that the cult of female friendship hasn't always been the norm, and that the way friendship is viewed now may not be how it's viewed forever. I'm a historian of emotion: I study how cultural narratives act on people as individuals—and how changing social factors play out in our hearts and homes. Scholars in my field often talk about the concept of 'emotional communities' to understand how the behavior connected to a particular feeling can change across time and place. An emotional community shares expectations about which emotions should be felt—which are shown, which are hidden, how each should be expressed. Such rules are enforced through institutions such as schools and courts, and via literature, art, and the rituals of family life. As a society's emotional rules change, so do the ways individuals expect to feel—including in friendship. Psychologists have shown that, far from following a universal template, friendship has 'styles' that differ subtly from place to place. According to the psychologist Roger Baumgarte, some cultures, such as that of the United States, seem to favor a more 'independent' style of friendship, in which friends are highly respectful of one another's autonomy and might become uncomfortable if they sense that someone is overstepping a boundary. In other cultures, such as those of Cuba or China, friends are expected to 'intervene' more in a friend's life and might feel snubbed if help is not given. Although this kind of research can quickly give way to reductive stereotypes, it does illuminate how diverse people's expectations about friendship can be. These expectations can also change across time. Finch and Baines's 17th-century emotional community, for instance, was shaped by highly romantic ideas about male friendship. The two men lived toward the end of a period of astonishing intellectual transformation in Europe, when artists, politicians, scientists, and philosophers had rediscovered, via the scholars of the Islamic Golden Age, great works of classical antiquity and wanted to make their ideals their own. One such work, Aristotle's Nicomachean Ethics, written in the fourth century B.C.E., divided friendship into three tiers. The bottom two tiers were populated by ordinary kinds of friends, in what he called friendships of utility and friendships of pleasure (the only kinds that women were supposedly capable of). Friendships of utility, Aristotle wrote, were 'commercially minded,' based on mutual help and quid pro quo. Friendships of pleasure were bonds formed through diversion and entertainment: You might gravitate toward a friend because they make you laugh, or you might sit with them at a game because they support your favorite athlete. But the third tier, which Aristotle called 'perfect' friendship, was something else—a bond between two men 'alike in virtue,' who saw each other as a 'second self.' It was, as later philosophers explained, as if 'one soul dwelled in two bodies.' This is how Baines and Finch saw themselves. They strove to be 'perfect' friends, and by all accounts, they seem to have succeeded. Another Renaissance figure who believed himself to be a 'perfect' friend was the French philosopher Michel de Montaigne, whose essay 'On Friendship' continues to influence writing on the subject today. In his bond with Étienne de La Boétie, a fellow lawyer and author, de Montaigne believed that he had found the kind of ideal friendship that only a few men in a generation could possibly hope to achieve—and his essay is a soaring depiction of their transcendent connection. In one of the most quoted lines in the history of friendship, he writes: 'If you press me to say why I loved him, I feel that it cannot be expressed except by replying: 'Because it was him: because it was me.'' Like other intellectuals of the time, de Montaigne did not consider women capable of this ideal. Women's brains could not endure this 'clasp of a knot so lasting, and so tightly drawn,' he wrote. His claim was based on the era's misogynistic medical theories, which held that women's brains were colder and weaker than men's. Women were willows, explained the scientist and poet Margaret Cavendish in 1655, liable to bend in the smallest draft—not sturdy oaks. In this way, most women were declared too flighty, capricious, and stupid for the commitment that true friendship required. They were also thought to be too easily distracted by romantic relationships with men. 'The reason why most women are so little affected by friendship,' wrote the 17th-century French nobleman François de La Rochefoucauld, 'is that it tastes insipid when they have felt love.' Women's friendships from this era were not generally memorialized in stone or glorified in reams of soaring poetry. In the archives, their stories appear as fragments compared with the large, easily accessible corpus on male friendship. But certainly they existed. In my research, I found evidence of women's bonds in scraps and shards—glimpses of women grieving over their deceased friends or helping one another through illnesses, evidence of friends forming households and raising children together, running joint businesses, defending others in court. Men knew the power of these alliances. Friendships gave women agency in a world not designed for them. It is perhaps no wonder that their friendships tended to be dismissed. Within 100 years of Finch and Baines's burial, the rules of their emotional community began to shift. The second part of the 18th century was a period of revolutionary fever and social reform across Europe and America. As abolitionist, women's-rights, and anti-poverty campaigns gathered momentum, so did discussions about emotion and sympathy (the old word for empathy). Poets, artists, and philosophers in this new Age of Sensibility began to speak in reverent tones of the supposed heightened sensitivity of 'the female mind.' They helped advance the idea that women enjoyed some special skill in friendship—and were capable of deeply emotional connections with the poor, the disenfranchised, and one another. By the 19th century, though, this belief in women's empathy had hardened into a new ideal of middle-class Victorian femininity: the 'angel of the house.' The consummate woman was supposed to be tender, loving, and eternally supportive; devoted friendships were seen as proof of her compassionate nature. Girlhood bonds were considered practice for the affection and sacrifice necessary for future roles as wives and mothers. Often, girls were primed with stories of highly romantic female friendships: One popular collection of nursery tales depicts two friends, Beatrice and Alice, who 'loved each other dearly' and 'with their arms about each other would sit under the deep shadow of the trees listening for the cuckoo's notes.' In Victorian novels, the girls who enjoy intimate childhood bonds—think Jane Eyre and Helen Burns—grow up to win the prize of marriage and motherhood, and 'difficult,' solitary, awkward girls, such as Lucy Snowe in Villette, marry unhappily, if at all. Romantic friendships between men didn't fall completely out of favor during this time. Photographs from the 1850s to the early 1900s show male friends holding hands or draping their arms around each other; sentimental letters between men in this period also abound. But by the second half of the 19th century, a new narrative about men's friendships was on the rise. Some began to depict male friendships as blundering and superficial. And as gay culture became more visible, and European sexologists stoked fears of 'sexual inversion,' a growing self-consciousness around male intimacy emerged. In 1863, the English feminist campaigner Frances Power Cobbe published an essay, 'Celibacy v. Marriage,' in the widely read Fraser's Magazine, in which she reiterated what had by then become a familiar story about male and female friendships. Whereas women friends enjoyed 'one of the purest of pleasures and the most unselfish of all affections,' Cobbe wrote, to men, friendship was little more than forming an 'acquaintance at a club.' Americans still, to an extent, live among the ghosts of these Victorian forebears, holding women to high standards of intimacy and portraying male bonds as clumsy and inept. We also live in an age of social fragmentation, in which experts, worried about loneliness and isolation, are puzzling over how to bring people together. To foster more connections, we'll need to reexamine our emotional rules—which ones are worth preserving and which ones we might be better off without. As a historian, I can tell you this: If we want to reimagine the terms of friendship, we can.
Yahoo
05-05-2025
- Politics
- Yahoo
Conclave: Inside the Sistine Chapel
Faith is an integral part of millions of Americans' daily lives. 'One Nation Under God' is committed to reflecting America's voices, values and communities, covering stories of faith and religion from the latest headlines to stories of hope. . (NewsNation) — 'Extra omnes.' With these words in Latin, which mean 'everybody out,' the highest-ranking officials of the Catholic Church will be locked inside the Sistine Chapel for a secret vote to elect a new pope. After Pope Francis died April 21 at age 88, cardinals from all over the world gathered in St. Peter's Square for his funeral. Kings and queens, presidents and prime ministers and hundreds of thousands of others attended as well. Gambling on the next pope is an older tradition than you might think Now, after nine days of mourning and complex preparations, including installing a chimney where the cardinals' ballots will be burned, the secret conclave will begin May 7. After a Mass to begin the historic meeting, the cardinals will enter, and a massive Renaissance-era door will be shut. The Church leaders will vote once on the first afternoon, and four times a day — twice in the morning and twice in the afternoon — on the following days. Two-thirds are required to elect a new pope. If the smoke coming from the chimney is black, it signals to the world outside that a pope has not been elected, but when the smoke is white, 'habemus papam' (Latin for 'We have a pope'). The backdrop of the conclave is the Sistine Chapel, and the site could not be more dramatic. Decorated with frescoes by Michelangelo and other renowned Renaissance artists, it is named after Pope Sixtus IV della Rovere, a patron of the arts who oversaw the construction of the main chapel in the 15th century. Pope Julius II commissioned Michelangelo to paint the ceiling with scenes from Genesis, and he later returned to paint 'The Last Judgment' on one of the walls. JD Vance defends Trump's AI pope photo 'The Last Judgment' depicts Christ's second coming and the final judgment. The dead rise up to heaven or descend to hell as they are judged by Christ. Several prominent saints such as St. Peter, John the Baptist, St. Catherine and St. Bartholomew are depicted. Perhaps most shocking are the souls that are damned. Michelangelo painted some naked, others ensnared in snakes, still others as ghouls. Those images could influence the cardinals who will be choosing the next leader of the Catholic Church. Copyright 2025 Nexstar Media, Inc. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.
Emirates Woman
01-05-2025
- Entertainment
- Emirates Woman
Chanel embraces the allure of La Dolce Vita in its Cruise 2026 spectacle at Lake Como
Fashion by Aminath Ifasa 4 hours ago The Chanel Cruise show is always a masterclass in elegance, but this season, the French maison elevated its artistry to new heights, transporting guests to the timeless allure of Italy's La Dolce Vita. Against the breathtaking backdrop of Lake Como's iconic Villa d'Este—a 16th-century jewel frequented by royalty and Hollywood legends—Chanel unveiled its Cruise 2026 collection to an audience of luminaries, including Keira Knightley and Margaret Qualley. A celebration of effortless sophistication, the collection drew inspiration from the villa's enchanting gardens, where magnolias, wisteria, and camellias bloom in abundance. Delicate lacework mirrored the floral splendor, while the designs embodied the villa's legacy of Renaissance-era glamour and relaxed refinement. From intricately embellished tweed sets to fluid silk slips and effortlessly chic co-ords, the lineup redefined vacation dressing with a signature Chanel twist. Each piece exuded a sense of luxurious ease, blending nonchalance with high-octane allure—a fitting tribute to the storied location. As one of the final collections crafted by Chanel's in-house team before Matthieu Blazy's anticipated debut, the show was a dazzling testament to the house's enduring legacy of craftsmanship and innovation. Here, we revisit the standout moments from an unforgettable evening where fashion met fantasy. Check out our favorite Chanel Cruise 2026 looks here Images: Supplied & Feature Image: Supplied
Scientific American
28-04-2025
- Science
- Scientific American
This Cutting-Edge Encryption Originates in Renaissance Art and Math
The portly, balding sculptor-turned-architect must have drawn a few curious gazes as he set up a complicated painting apparatus in the corner of a Renaissance-era piazza. He planted his instrument, which involved an easel, a mirror and a wire framework, near the then unfinished cathedral of Florence in Italy—a cathedral whose monumental dome he would soon design. His name was Filippo Brunelleschi, and he was using the apparatus to create a painting of the baptistry near the cathedral. This demonstration of his recently discovered laws of perspective is said to have occurred sometime between 1415 and 1420, if his biographers are correct. The use of the laws of perspective amazed bystanders, altered the course of Western art for more than 450 years and, more recently, led to mathematical discoveries that enable elliptic curve cryptography. This is the security scheme that underpins Bitcoin and other cryptocurrencies and has become a fast-growing encryption method on other Internet platforms as well. But how did Renaissance art lead to the mathematics that govern modern cryptography? The tale spans six centuries and two continents and touches on infinity itself. Its characters include a French prisoner of war and two mathematicians struck down in their prime—one by illness and the other by a duelist's pistol. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today. Merging Perspective and Geometry The first steps in the path from Brunelleschi to Bitcoin involved connecting the visual geometry within the rules for perspective to Euclidean geometry, the orderly realm of lines and points that we're taught in school. French mathematician Girard Desargues, who researched the geometry of perspective in the 17th century, was the first contributor. His findings, however, were couched in rather obscure language and struggled to find an audience. His key contributions were included in a book that had a print run of 50 copies, small even for that era, and many of those copies were eventually bought back by the publisher and destroyed. During Desargues's lifetime, only fellow French mathematician Blaise Pascal became an ardent disciple of his work. Pascal contributed his own theorem to the study of what became known as 'projective geometry.' Despite Desargues's obscurity, he made a revolutionary breakthrough by adding the concept of points and lines at infinity to Euclidean geometry. By including those points, projective geometry could be merged with Euclidean geometry in a way that was consistent for both systems. In Desargues's system, every pair of lines meets at exactly one point, with no special exceptions for parallel lines. Furthermore, parabolas and hyperbolas are equivalent to ellipses, with the addition of one or two points at infinity, respectively. These insights, though valuable, would languish in obscurity for more than 100 years. When they reappeared, it was not because Desargues's work was rediscovered. Rather, a different French mathematician, Gaspard Monge, began to work on the same questions and derived similar results. A Mathematician at War The most comprehensive work on projective geometry in this era, however, came in the 19th century from French engineer and mathematician Jean-Victor Poncelet, under rather trying circumstances. Poncelet attended France's prestigious École Polytechnique, graduating in 1810. He then joined France's corps of military engineers as a lieutenant and was ordered to what is now Belarus to support Napoleon's invasion of Russia in 1812. He and his fellow troops overran a burned out and abandoned Moscow in September of that year, and when the Russians refused to sue for peace after losing the city, Poncelet was with Napoleon when the army left Moscow and began the return to France. Poncelet remained with the French army right up to the Battle of Krasnoye in Russia, where he was separated from his unit and possibly left for dead. After the battle, he was scooped up by the Russian army and marched to Saratov, Russia, more than 700 miles from Krasnoye and more than 2,000 miles from his home in Metz, France. Although Poncelet was not confined to a prison, he was 'deprived of books and comforts of all sorts,' according to an English translation of his introduction to his first book on projective geometry. As a coping mechanism, he decided that he would try to redevelop all the math he had learned up to that point. He could not carry out this plan, however, saying that he was 'distressed above all by the misfortune of my country and my own lot.' Instead he essentially expanded on Monge's work and recreated Desargues's work independently. In hindsight, it is perhaps not surprising that a prisoner of war thousands of miles from home and unsure of when, or even if, he would be repatriated would focus his efforts on understanding points at infinity—a distance that might have seemed quite intelligible to someone in Poncelet's situation. After the war that had been sparked by that invasion ended, Poncelet returned to France and his two-volume work on projective geometry, published in 1822, was far more well-received and widely read than Desargues's work. Integrals and Curves At around the same time that Poncelet was finishing his book on projective geometry, Norwegian mathematician Niels Henrik Abel was studying elliptic integrals. These integrals are rather difficult expressions that started off as parts of an attempt to measure the circumference of an ellipse. Abel discovered that there are certain circumstances where the inverse of these elliptic integrals—what are called elliptic curves—could be used instead. The curves, it turned out, are much easier to work with. Further research into elliptic curves would be left to others, however; Abel died from tuberculosis at age 26 in 1829, mere months after publishing an important paper on the subject. In the early 1830s French mathematician Évariste Galois laid the groundwork for a new field of mathematics. Galois would die tragically but also stubbornly in a duel at age 20, but before his death he laid out the principles of group theory, in which mathematical objects and operations that follow certain rules constitute a group. The French had managed to unite projective geometry with Euclidean geometry, but it would fall to a German mathematician, August Möbius (of Möbius strip fame) to figure out how to merge projective geometry with the Cartesian coordinate system familiar to algebra students as a means of graphing equations. The system he developed, which uses what are called homogeneous coordinates, play a pivotal role in elliptic curve cryptography. Several decades later, in 1901, another French mathematician, Henri Poincaré, realized that points with rational coordinates—that is, points with coordinates that can be represented as fractions on the graph of an elliptic curve—composed a group. What Poincaré realized is that if you defined an operation (typically called 'addition') that took two rational points on the graph of the curve and yielded a third, the result was alwaysanother rational point on the curve. This process onlyworked if you used the homogeneous coordinates discovered by Möbius that include a point at infinity, however. Importantly, elliptic curve groups turned out to be Abelian, which meant that the order in which those addition operations were performed didn't matter. This is where matters stood until the mid-1980s, when Victor S. Miller, then a researcher at IBM, and Neal Koblitz of the University of Washington independently realized that you could build a public-private key cryptographic system based off elliptic curve groups. Encryption Keys Public-private key encryption, which is how almost all traffic on the Internet is secured, relies on two encryption keys. The first key, a private one, is not shared with anyone; it is kept securely on the sender's device. The second key, the public one, is composed from the private key, and this key is sent 'in the clear,' meaning that anyone can intercept it and read it. Importantly, both keys are required to decrypt the message being sent. In elliptic curve cryptography, each party agrees on a certain curve, and then each performs a random number of addition operations that start from the same point on the same curve. Each party then sends a number corresponding to the point they've arrived at to the other. These are the public keys. The other party then performs the same addition operations they used the first time on the new number they received. Because elliptic curve groups are commutative, meaning that it doesn't matter in what order addition is carried out, both parties will arrive at a number corresponding to the same final point on the curve, and this is the number that will be used to encrypt and decrypt the data. Elliptic curve cryptography is a relative latecomer to the encryption game. The first suite of tools did not appear until 2004, far too late to become a standard for the Web but early enough to adopted by the inventors of Bitcoin, which launched in 2009. Its status as the de facto standard for cryptocurrencies made people more familiar with it and more comfortable implementing it, although it still lags behind RSA encryption, the standard method in use today, by a wide margin. Yet elliptic curve cryptography has distinct advantages over RSA cryptography: it provides stronger security per bit and is faster than RSA. An elliptic curve cryptographic key of just 256 bits is roughly as secure as a 3,072-bit RSA key and considerably more secure than the 2,048-bit keys that are commonly used. These shorter keys allow for faster page rendering for Web traffic, and there's less processor load on the server side. Principles from elliptic curve cryptography are being used to try to develop cryptographic systems that are more quantum-resistant. If trends continue, the mathematics behind the vanishing point discovered by Renaissance artists 600 years ago may turn out to be a fundamental part of Internet encryption in the future.
