Sophie Germain and Fermat’s Last Theorem

Sophie Germain-1776-1831-grangerOne doesn’t often meet professional mathematicians so I am reposting this piece (which I originally wrote for the OSU history of science blog) in honor of Matt Klassen, the head of the mathematics department at DigiPen Institute of Technology. I met Matt, who also happens to be a fine guitarist, at David Russell’s masterful guitar concert in Seattle last weekend. Dr. Klassen has presented on “Non-Associative Loops on Fermat Curves of Odd Degree,” which reminded me of the French mathematician, Sophie Germain (1776-1831):

On May 20, 2014, the OSU Department of Mathematics sponsored a history lecture by Dr. David Pengelley, of New Mexico State University. Dr. Pengelley presented an animated lecture on some of the work of Sophie Germain. Dr. Pengelley’s interest in Germain was sparked by his use of primary historical sources in his teaching of mathematics. This led him to the National Library of France (Bibliothèque nationale de France) where he found a store of Germain’s original manuscripts which had not been studied in over two hundred years. Revisiting Germain’s work as a mathematician, Dr. Pengelley found that Germain had developed a sophisticated plan for proving Fermat’s Last Theorem, making significant contributions to number theory. Until recently, her work was known only via a footnote in another mathematician’s treatise (Legendre, Essai sur la Théorie des Nombres, 1823). In an age when women were usually not well-educated and when they were explicitly excluded from scientific academies, Germain’s substantial achievements were indeed remarkable.

Sophie Germain was only thirteen when the French Revolution broke out, forcing her to spend most of her time indoors. During that period, she turned to her father’s library. Fascinated by books on mathematics, she taught herself against her parent’s wishes (Pengelley relates that at one point they even took away her clothes and candles to prevent her from studying at night!). Germain’s father was a silk merchant so it was not through his mentorship that she developed her abilities but rather through her own effort and perseverance. At one point, Germain took on the identity of a student at the École Polytechnique who had died (Antoine-August LeBlanc). When the professor discovered that it was really a woman who was submitting such fine work under LeBlanc’s name, he was astonished. Germain eventually corresponded with Johann Carl Friedrich Gauss (1777-1855) in Göttingen, one of the most celebrated mathematicians of the time. Pengelley recounts that upon receiving a letter from Germain, Gauss praised the way she contributed to the “charms of this sublime science,” as giving him great joy.

Pengelley gave a cogent and fairly detailed explanation of the theorem by Pierre de Fermat (c.1601-1665) that Germain was hoping to prove. Basically, the theorem states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer n greater than two. At the time that Germain was working on the problem, it was known that the theorem could be proven to hold for some numbers but much work remained before the theorem could be proven conclusively. Germain’s letters and manuscripts demonstrate that she had a good handle on the problem and that she had made considerable progress toward a solution. Pengelley found that she had made a mistake in one of her proofs but peering closer found scribbled in the margins, “voyez errata”—Germain’s own admission that she saw she had made an error!

Germain did win a prize from the French Academy of Sciences for her work on elasticity and she eventually was able to attend the Society’s meetings, but she was never made a member nor was any of her work published. Her manuscripts were taken by Guillaume Libri, described by Pengelley as a “mathematician, historian, bibliophile, thief, and friend of Sophie Germain.” Because Libri ended up with her manuscripts, they were preserved and eventually made available for Pengelley’s research. Finding a proof for Fermat’s theorem has been a problem that has long attracted the attention of mathematicians. In 1995, the mathematician Andrew Wiles, with help from Richard Taylor, finally solved the Fermat Theorem. It had been one of the most famous problems in mathematics and Sophie Germain’s efforts made an important contribution to the discovery of a proof. Dr. Pengelley’s work is of interest to historians not only for the way he used primary sources to teach mathematical concepts but he also revived interest in an under-appreciated figure, Sophie Germain, whose achievements deserve to be more widely celebrated.


Do we still need professors?

Books on my shelves

The job market for PhDs, especially in the Humanities, is not good. My advisor told me it was the worst she’d ever seen…and that was several years ago when I first begin my PhD studies. My own professional association, the History of Science Society, has made a concerted effort to identify jobs outside of academia for graduates and, importantly, to promote those positions as “not failures.” Further, a number of books and articles have advocated for alternative careers  for PhDs (a recent example is discussed here) while others rightly bemoan the state of the business-model university.

However, we should not lose sight of the larger picture. Income inequity for everyone has grown enormously over the past four decades. The financial crash of 2008 has further battered our economy. Job prospects are not just bad for academics; they are discouraging for everyone. So the push to value other careers – which could translate to other low paying, unstable jobs as an alternative to the low pay and tenuous employment status of adjuncts – seems misguided. My sense is that most PhDs would be happy to use their skills in other settings if they were compensated at a decent rate and with reasonable job security.

Carla lecturing at a museum cropped

But we still need good professors. In the stories that followed the defeat of the Harvard debating team by the Bard College-trained debate team from the Eastern New York prison, the expertise of the professors who worked with them did not receive nearly as much press as the fact that the Bard students in prison had to study with restricted access to materials and no use of the internet. Indeed, the work ethic of the students and their intellectual aptitude deserves to be celebrated. Nevertheless, it is also worth recognizing the key role that professors played in preparing these students to think clearly and argue effectively.

It is a real loss when a retiring professor is not replaced in favor of doubling the class size or employing less expensive (i.e. “contingent”) faculty. This has become an accepted scenario in the current climate of higher education. It is not just jobs that are at stake. Employment shapes our daily lives in important ways: as a source of income and sense of achievement and community. The loss of tenured positions, which in secondary education has meant tenable positions, is a real loss for new PhDs and the work force at large.

But we should also keep in sight what is lost in the bigger picture when that not-replaced professor is not there. The future world with its increasingly-global-scale complexity needs people who are thoughtful, energized, and well-educated. Professors challenge, support, and prepare those kinds of people. Education is not a collection of facts but an engagement of mind and body, a traversing of what is known and not-yet-understood. Professors serve as skilled guides and seasoned fellow travelers.

The current dearth of decent jobs in academia should not make us lose sight of what universities have achieved nor of what role they could play in the future. Equal opportunities in education have been in many ways America’s great promise. We live in the best of times and the worst of times – a time when our universities still are among the most excellent institutions of higher learning in the world, and a time when those institutions are under the most virulent attack.

The recent success of the Bard college prison debate team has made visible persistent racism and a U.S. prison industry that has grown exponentially over the same decades that have seen the sharp decline in academic positions. The United States with only 5% of the world’s population incarcerates nearly 25% of the world’s prisoners. Instead of funding one of the world’s largest prison systems, we could do a better job of funding higher education. The success of the Bard college debate team could be a catalyst for more deeply appreciating what professors do and for not giving up on the professorial vocation without a fight.

Well-made Things

Under the labelsThe replacement jar for my now-apparently-vintage Hamilton Beach blender arrived this past weekend from eBay. A blank label under the inventory label, carefully obscured the data on origins which proclaimed that the replacement jar was “Quality Constructed in China.” It came complete with two tiny insects squished and preserved on the rubber gasket, a lid of dubious quality plastic and a rotating blade of creaky construction.

This is not a complaint (the glass jar will work fine) so much as a comment on how much has changed since eBay first appeared as a venue for selling and buying gently-used genuinely second-hand things. Much of the clothing and household items now sold at thrift stores such as Goodwill are of such low quality that they probably should not have been made in the first place. My sense is that it is much more difficult for students, artists, and young people starting out or older ones living on a limited pension to stretch their dollars in a reasonable way. Being able to buy well-made things – even if they are second-hand – imparts a certain dignity to one’s daily life.

I do love things that truly are quality material objects, such as the handsome table and bench set that my father-in-law made for us when we were living in our first home – a 1930s craftsman bungalow with a thirty-inch wide and seven-foot-long counter made of old-growth Douglas-fir in the kitchen and French doors looking out on an enclosed back porch across the entire width of the house. We were not able to take our lovely Highland Drive home with us on our many adventures since then, but the bench and table set – with its Douglas-fir inlay back and maple trim – has come with us and we’ve found the perfect niche for it in nearly every house to which we’ve moved.

Bench set

I’m sometimes working at this very table as I am writing my version of The Great American Dissertation. You can find out more about my research on things material and ephemeral in eighteenth-century France here.

A Patron Queen of Sciences in the Seventeenth-Century


In my readings in the history of science during the seventeenth century, I have come across the name of Queen Christina (1626-1689) repeatedly, but only lately have I begun to pay attention. I have yet to find a comprehensive list of the many early modern figures in science Queen Christina of Sweden employed, encouraged, patronized, prodded to teach her, or with whom she corresponded, but it would be a long and fascinating roster. Descartes, Galileo, and Bernini would top such a list but one could include many others, such as Count Luigi Fernando Marsili, the author of a 1725 volume on physical oceanography (a copy of which was recently acquired by the Special Collections and Archives Research Center at Oregon State University).

Several books and even a movie (starring Greta Garbo) have detailed the remarkable life of Queen Christina – her assumption of the Swedish throne at age six upon the death of her father, Gustav Adolf (killed while defending Protestantism in the Thirty Year’s War), and her abdication of the throne some twenty years later (to convert to Catholicism and move to Rome!). While in Rome, she established one of the most powerful courts in Europe even as a ruler without a country.

My interest in Queen Christina (she was technically crowned King) was sparked again by reading a provocative dissertation on Queen Christina’s physical presence. In her dissertation, “Figuring a Queen: Queen Christina of Sweden and Embodiment of Sovereignty,” Camilla Kandare argues that Queen Christina used the movement of her body to gather power. Unlike many of Christina’s biographers, Kandare claims that during the seventeenth-century, it was not how Queen Christina looked that mattered but rather how she carried herself. While some contemporaries commented on her aging female body, her messy hair, or her habit of wearing men’s shoes, it was the way she moved that gained her access to the highest echelons of religious and political life in Rome. The way Queen Christina processed, rode, strolled, danced, curtsied, and kissed, all served to bolster her prestige and power, even as an older woman.

Early Music Vancouver recently presented a concert of music from the court of Queen Christina, along with a performance of dances with which she would have been familiar. On July 26, 2015, a band of top-notch musicians and a dancer took the stage in the Roy Barnett Recital Hall at the University of British Columbia’s School of Music. The music and dance selections followed the outline of Christina’s life from “Stockholm, the Court of the Lion of the North,” through her “Abdication and journey to Rome,” to her visit to Versailles, and back to Rome.

Soprano Ellen Hargis used snippets of Christina’s diaries, letters, and commentaries of the day to paint episodes from Christina’s life. For example, to portray Christina’s education, Hargis read from the young Christina’s letter to her Latin teacher in which she promised “to speak only Latin…even though I have failed at that in the past.” The music of Baldassare Donato (c.1530-1603), Luigi Rossi (c.1597), Antonio Cesti (1623-1669), Jean-Baptiste Lully (1632-1687), and Giacomo Carissimi (1605-1674) and others was played with nuance and spirit by Ellen Hargis (soprano), Paul Luchkow (violin), Christopher Bagan (harpsichord), and Lucas Harris (theorbo & baroque guitar). Marie-Nathalie Lacoursiére, a baroque dancer from Montreal gave splendid and convincing performances of dances reconstructed from period notation complete with lavish costuming. In 1657, Christina had visited the court of Louis XIV where she would have seen the Folias danced using choreographies like those employed by Lacoursiére to splendid effect. For decades, Christina was an intimate friend of Cardinal Azzollino (1623-1689) who attended her at her death bed. Lacoursiére’s funerary-inspired final dance evoked the sorrow of the cardinal through a pensive and beautiful tribute, also danced with a repertoire of period steps.

Stalking the Elusive Tanoak

tanoak acorns

Two weekends ago, I went on an urban hike through the Washington Park Arboretum in Seattle across highway 520 from the University of Washington. We went in search of a tanoak, a tree I’d been reading about in Frederica Bowcutt’s new book, The Tanoak. We found a barred owl on a low branch and a Douglas-fir that had been spectacularly struck by lightening in March, but the tanoak proved elusive. Our friends had to go on to other engagements but we stuck it out and, following directions from the visitor’s center, wound our way to the south end of the park to the “Pacific Connections” area where we eventually found a scrubby tree loaded with acorns – once a major food source for native Americans in California and southwestern Oregon.

In her book, The Tanoak Tree: An Environmental History of A Pacific Coast Hardwood recently published by University of Washington Press, botanist Frederica Bowcutt draws on two decades of research to demonstrate the importance of this underappreciated hardwood tree. She not only offers the kinds of details that make history fascinating (the tannins in acorns are effective as a cough suppressant, for example, who knew?!), but also addresses larger questions such as how western science and native ecological knowledge might work together to save promote human health and landscapes that are sustainable, beautiful, and productive.

The book is generously illustrated with photos and figures, from Bowcutt’s own 1995 shot of tanoak chips hauled away on a truck to be sold for $125/ton to an image showcasing the clean lines of a handsome rocker made from tanoak to a mid-twentieth century ad grimly urging factory owners to mechanize so they “don’t have to pay the man who isn’t there.” On page 25, the nutritional content of acorns is compared to wheat and barley. Forest managers, loggers, European peasants, Native American protesters, and cooks all make an appearance as the author takes the reader through chapters covering different parts of the tanoak and perspectives on how it has been used/viewed by humans: aesthetics, acorns, bark, weed, hardwood, plague, landscapes, and conservation.

In The Tanoak, one learns of the complicated effects on workers when a local renewable resource (bark from tanoak) becomes obsolete as it is replaced by a non-renewable resource (chromium salts). One also finds that different purposes for leather informed the tanning processes and the quality of leather produced. Native Americans used animal brains and other animal parts to process skins to make soft leather chamois; Spanish cattlemen made a heavy-duty grade of leather for their saddles, first using tanoak bark and then chemicals.

Bowcutt explores in depth the complicated history of how tanoaks have been used. Scattered stands of trees, such as the pattern favored by Tanoaks, are more difficult and costly to harvest for timber. Furthermore, Tanoak was not as suitable for railroad ties – the leading use of wood in the late nineteenth century – as eastern hardwoods and redwood. By the mid-twentieth century, Tanoaks were treated with herbicides to favor the cultivation of quicker growing trees such as Douglas-fir or Redwood which were used for timber in the post-war building boom (with less expensive lumber, by 2013 house sizes more than doubled from the average of 983 square feet in 1950).

Over the course of two hundred years, the Tanoak went from being an important source of food for native populations to becoming an important commodity in the early twentieth century leather trade to being considered a “trash tree” that competed with the more commercially lucrative Redwood and Douglas Fir. The author demonstrates the caloric and nutritive value of acorns as tested in recent scientific research against the continued bias against supposedly “wild” indigenous foods. In fact, native peoples did not simply gather acorns at random but cultivated the best varieties and stands through selective thinning of trees and low intensity burns. Tanoak acorns fed native populations in California and southern Oregon for thousands of years before the settlement of western North America by Europeans.

By the end of this book, the reader is convinced of the significance of this little-known tree. In this densely packed history of Notholithocarpus densiflorus, Bowcutt restores the Tanoak to a central place in the history of human/nature interactions. She offers realistic interventions to address the Tanoak’s decline in the face of wasteful forestry practices, and now, as it faces danger from an imported pathogen, Phytophthora ramorum, the cause of sudden oak death. Western scientific research, Native American management practices, better government regulation of nursey trade (the origin of the current sudden oak death epidemic), and a deeper understanding of the Tanoak’s varied history will go far in holding out hope for preserving this useful and beautiful tree. Of special interest to environmental historians and forestry managers, this book is also rich in insights for the general reader.


“I was talking with a fisherman in a bar in Anchorage…” This was the beginning of the lecture I described in the last post by Mari K. Reeves, ecologist with the U.S. Fish and Wildlife Service in Alaska.

globe bluemarblewestIn Reeves’ talk, the fisherman with whom she had been talking had a surprising analogy for “Mother Nature.” He argued that nature is not a mother, but rather a baby. Babies are resilient, he pointed out, but they need a great deal of care. While they may heal from some injuries quickly, seemingly minor trauma can have long-term consequences that may not be immediately apparent. Connecting the fisherman’s insight with her One Health perspective, Reeves showed a modified graph based on C. S. Holling’s 1973 visualization demonstrating a threshold for resilience that takes into account the accumulation of stressors. While an animal, human, or ecosystem may be able to recover from an insult readily if it is already in a relatively healthy state, recovery is more difficult when stressors accumulate.

Multiple stressors and low- level stressors can have significant impacts, but are very difficult to study. To demonstrate the effects of small, non-toxic doses of a contaminant, Reeves showed a video of an experiment in which one could see that when an uncontaminated (control) fish smells a predator, it immediately sinks to the bottom of the tank to hide. In contrast, fish exposed to minute amounts of copper seemed fine, but were not able to smell the predator signal the same way that the uncontaminated fish did. When the predator smell was released into the tank, these exposed fish did not sink to the bottom and hide, and not smelling danger, they quickly became prey.

What does this have to do with Reeve’s study of frogs? When studying stressors affecting the health of frogs, it can be difficult to pinpoint relatively small changes in their environment. However, as for the fish in the experiment above, even minute stressors such as barely measurable levels of pesticides or heavy metals can have significant impacts on frog populations, especially when combined with other stressors. As advocates of the One Health perspective have argued, this is true for other creatures as well, including humans.

The One Health Seminar Series has been organized by veterinarian/ PhD-candidate Rhea Hanselmann ( ).

Frog Blog!

Alaska Wood FrogAlaska Ecologist Mari K. Reeves imparted an infectious enthusiasm for the wonder of frogs when she spoke on “Amphibian Abnormalities and their Environmental Linkages: Observations and Musings after a Decade of Research” at Oregon State University last month (4/3/2015), as part of the One Health Seminar Series that veterinarian/ PhD-candidate Rhea Hanselmann has been promoting on campus ( ).

One Health is “the concept that the health of humans, animals, and the environment are directly linked, and that the condition of one can affect the health of the others.” It is a global movement endorsed by the World Health Organization (WHO) and many other national and international organizations seeking to unite healthcare efforts for more comprehensive prevention and treatment of disease. Given that 70% of new and recurring human diseases are related to animal carriers or environmental stressors, it makes sense to address well-being in this more holistic way. Hanselmann, who is researching the effects of environmental change on the incidence of Hanta-virus infections, is working with Luiz Bermudez in the Department of Biomedical Sciences to develop One Health courses. They hope that OSU will begin to offer integrated classes and eventually a One Health credential program for undergraduate students.

Frogs have historically been a favorite creature for biological studies, because so much of their development is visible as they change from egg to tadpole to frog. Because they then migrate to totally different habitats, they are also exposed to a wider variety of environmental hazards than most other species. When a frog population starts declining rapidly, or includes many frogs with deformed or extra limbs as was noticed in the Midwest several years ago, how can one tell what is going on? Pollutants, parasites, predators, climate change – any of these could be driving the demise of a frog population. In a large-scale study published by Reeves and her colleagues in Ecological Monographs (August 2010), the researchers found that the most important factor in frog survival was where they lived. The quality of their habitat proved to be a mitigating factor for other stressors, contributing to better health and survival rates. Again, the corollary lesson for humans seems clear: protecting our habitat is the best way to protect our health.

Unfortunately for frogs, clean habitat is hard to come by. A 2013 study by the U.S. Environmental Protection Agency showed that over half of U.S. streams and rivers are in poor condition. Reeves ended her talk with a call for help in cleaning up our waterways. Her descriptions of frogs (which included listening to a recording of them croaking!) inspired a deep sense of delight and wonder in these amazing creatures. Furthermore, in helping frogs, we very likely will be helping ourselves in ways we don’t yet understand.