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An intellectual memoir, Angles of Reflection portrays a woman ...
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An intellectual memoir, Angles of Reflection portrays a woman deeply enmeshed in two male-dominated worlds--nineteenth century mathematics and twentieth century research academics--struggling to integrate the competing demands of family and career without sacrificing one to the other. As the strain of caring for her sick child mounts, Richards' view of De Morgan broadens to include his family in ways that both illuminate his work and force her to reevaluate her own career and relationships. In the process, she gives new meaning to the term "applied mathematics" by drawing life lessons out of De Morgan's logic, Newton's absolute space and time, and Leibnitz's relative visions of reality. She emerges from her ordeal profoundly altered, with a new appreciation of the ways that life, family, and work can inform and enrich one another.
Filled with insightful discussions of the debates among some of the finest mathematical minds of the 17th through 19th centuries, Angles of Reflection is both an intellectual journey through the history of mathematics and a gripping testament to maternal love.
About the Author:
Joan Richards is an Associate Professor of the History of Mathematics at Brown University. In addition to articles and reviews for scholarly publications, she is the author of Mathematical Visions: The Pursuit of Geometry in Victorian England.
I don't remember when I got the letter inviting me to spend a year at the Wissenschaftskolleg in Berlin. There certainly was no question that I would accept it. I had for the previous seven years poured myself into my family. My husband was engulfed with work at the Rhode Island Department of Education, I by the day-by-day demands of my job as a professor of the history of science at Brown University. The combination of raising two children and working in an institution that took undergraduate teaching seriously meant that I had long neglected "my work," my research and writing.
Not engaging it had its own set of negative consequences, however. Somehow I had managed to teach and finish my dissertation after Brady, my first child, was born. I finished my first book when Ned, my second child, was still in diapers. But my second book had not followed as expected. Raising two children had proved itself to be just too distracting to allow me to think through a whole new project. Though I scheduled a full workload around them—teaching, serving on committees, delivering papers—I could not muster the kind of total concentration required to write. Six years into my tenure, my department turned down my promotion. I could not refute their objection that I had not written a second book. What I could do was recognize that at nine and fourteen the boys were no longer as demanding as they had been before, and it was high time to concentrate on my research and writing.
So, when my sabbatical finally came due, I put all of my energies intofinding ways to extend with outside money the one semester I was granted. I had been successful and had a fellowship at the Dibner Institute in Cambridge, Massachusetts. The fellowship gave me a lovely office overlooking the Charles River in the midst of a group of people whose interests overlapped with mine. Working in the institute required that I commute, itself a privilege because it placed distance between my personal and my work life. Three or four mornings a week I left the house by seven-fifteen to catch the train to Boston, returning between five and five-thirty. In between my fellowship granted me time, space, and colleagues—a whole world for my work. It was wonderful.
The Wissenschaftskolleg offered yet more of the same—another year and another place in which to do my work. It required considerably more in terms of dislocation—the year would be in Berlin. I had never been to Germany for more than a week, my most recent contact with the language was a course, "German for Reading Knowledge," taken for my Ph.D. exams more than twenty years before. But I knew that for my work it didn't really matter where I was. A year in Germany would be a broadening adventure for the whole family. So, I am sure that I was thrilled when I got the letter offering me the fellowship and equally sure that I wrote back to accept with alacrity. But I do not remember now. At just about the time that it arrived my family was engulfed in a series of events that has blocked the specifics of the letter and my response entirely from my mind.
The scenes I do remember from that fall begin in a dark kitchen. It was about five-thirty in the evening on November 9 when I trundled up the back stairs, back from a long day in Cambridge. No lights were on, and I missed the conversation, music, arguments, and tromping around that the family could usually be counted on to provide. Bilbo, our black cocker spaniel, came hopefully into the kitchen but all else seemed still. Then I heard footsteps from Brady's far-away room. He was in a sequestered phase, which explained why the apartment seemed so deserted even though he was there.
As he came toward me the telephone rang, and Brady intercepted it. "Yes," he said. "She just came home. Yes, I'll get her."
"It's Dad," he said as he handed it to me. "He's in the hospital with Ned."
In the hospital with Ned? It was a strange concept. Ned was exuberantly healthy—perhaps he had fallen and broken a limb. I took the phone.
Rick sounded quite normal. "Ned and I need a ride home," he said. "Can you come pick us up?"
"Sure," I said, "but where are you?"
"I'm in a neurologist's office," he replied. "Ned had some kind of seizure at school. He's fine now though, and we are ready to come home. Brady can tell you about it, and I'll explain more when you come, but Dr. Gasparian's office is closing. Can you pick us up?" he repeated.
"Yes," I said but I had no idea where a neurologist's office might be. Rick gave me instructions and Brady and I went to the car.
As we drove the unfamiliar streets to the downtown Providence hospital complex I pumped my taciturn adolescent for information. He did not know a great deal but told what he could. "Ned had some kind of a fit in the lunchroom at school. I guess they couldn't reach you in Boston but they told Dad. He came to school and went in the ambulance with Ned. I think things got kind of screwed up at the hospital, but basically everything is fine now."
"Ned had a fit?" I asked. "What do you mean a fit? Did he have a temper tantrum?"
"No, Mom. You don't go to the hospital for a temper tantrum. I don't know what kind of fit—how should I know?"
It seemed a reasonable position. I peered through the rainy night, trying to follow Rick's directions through this unfamiliar part of town, finally locating the appropriate parking lot. Brady slumped his rapidly growing self in the front seat, knees on the dashboard. "Leave the keys. I'll listen to the radio," he said.
I left him there and followed Rick's directions to the third floor of a dreary office building where I found the nondescript offices of "Dr. Gasparian, Neurology." The secretary indicated a room in the back when I asked about my husband and child.
I found Rick and Ned in a small, white room. Ned flashed me his irresistible smile from a cross-legged perch on an examining table. He looked absolutely fine except for a large bruise half-hidden under the thick, light-brown hair on his forehead. Rick was sitting in one of two chairs on the other side of the room. He looked uncharacteristically disheveled: shirt dirty, pants crumbled, tie askew. His carefully trimmed beard was wilted. His usually neat gray hair bushed Einstein-like from his forehead. The eyes behind his horn-rimmed glasses were weary. A large poster of the brain, all brown-gray scrolls and little red labels, covered the wall behind him.
"What happened, Sweetie?" I sat in the other chair.
Rick leaned his shoulder tiredly against me. "Oh, Joan, I wish I knew. The school called me out of a meeting to say that Ned had had a seizure in the lunchroom. You had the car, so I got a cab and went as fast as I could. Ned was unconscious. We went in an ambulance to the hospital. Just about when we got there Ned threw up, all over everything. There were doctors everywhere. They tried to get Dr. Johnson, but he was out of the office. So they hooked Ned up to all sorts of machines, took blood, ordered a CAT scan, decided an EEG would be better and left us in a little room. That was about three and I guess the shifts changed. We just sat and watched TV. They didn't notice us again until it was five and we had missed the EEG. They sent us to Dr. Gasparian's anyway, so here we are. But his office is closed; he's not going to do it now."
"I banged my head on the table!" Ned said brightly. "See my bruise?"
When I had been briefed to this extent, we were interrupted by a slight, dark-complexioned man. There were introductions all around and then Dr. Gasparian asked me whether I could shed further light on Ned's condition. "Your son had a convulsive seizure at about noon today. Has anything like this ever happened before?"
My first reaction was "No," but as I turned it over in my mind I realized that might be too glib. Last spring Ned had once been sent home from school because something odd had happened. When he came in from recess, he said, the room went dark and he began to walk into furniture. The teacher told him to put his head on his desk and he did, but he got such a bad headache he could not concentrate. So, she sent him to the principal's office and they called me to fetch him. When I arrived, Ned had a splitting headache. Thinking that the problem might be that he had become dehydrated at recess, I gave him a glass of water. Then I supported him for the six-block walk home. When we got there I put him to bed in a darkened room, but he was already almost completely recovered. I could not keep him down for more than half an hour; the water had worked its magic.
The second incident was less noticeable. About a month before, when I had picked him up at the end of the school day, Ned had a terrible headache—he was shrunken into his body with great circles under his eyes. He said that he had had some trouble seeing in his last class, but it was not bad enough that anyone noticed. I treated him gently and he was totally normal by dinnertime.
I had told Rick about each of these episodes when it happened, but they were so minor that he had not registered them. They had been fleeting incidents, of no more moment than any of a myriad of other childhood ups and downs. I felt almost disloyal reconstructing them as seizures while the strange doctor gravely took notes and probed Ned for corroboration or clarification.
"We'll have to do an EEG as soon as possible," Dr. Gasparian said, and made an appointment for Friday afternoon. "I also need to know as much as possible about what happened today and on these other days. Ask the teachers or Ned's classmates what they remember, anything, and bring the information with you when you come."
I wanted to know more—about convulsive seizures, about an EEG, about the whole situation. How normal was a seizure? What caused them? Who had them? Why? What was an EEG? Would it hurt Ned? What would it show? How was Ned now? Did we need to do anything special? All of these questions and more filled my head, but Dr. Gasparian was not a man to chat. He showed us firmly to the door with a card showing Ned's EEG appointment on Friday, two days away.
In Dr. Gasparian's office Rick was calm, but on the car ride home it became clear that he had been deeply shaken. Nothing in the call that jerked him from a meeting at work had prepared him to find Ned unconscious. His own child had looked at him but not responded in any way. He had not grinned or even blinked when Rick spoke to him; he had not returned the pressure when Rick took his hand. Eyes open, he had lain blank and still on the couch in the nurse's office.
Rick was overwhelmed, but hardly more so than the school personnel who had summoned him. The story they told was simply weird. It seemed that when the children were dismissed for recess, there was a disturbance at the table where Ned had been sitting. When the lunchroom supervisor went to investigate she found that several children had dreamed up a new way to misbehave by taking refuge under the complex structure that combined table and chairs. She firmly told them to come out, which they all willingly did—all but Ned who ignored her and continued to wiggle on the floor. Irritated, she reached in to pull him out. It was only when his arm jerked uncontrollably in her grasp that she realized that she was dealing with more than a creatively naughty child. She then registered Ned's eyes rolling back into his head, his mouth alternately drooling and tensing, his head rhythmically pounding against the leg of the table. Horrified, she called for assistance. By the time another teacher arrived, Ned had stopped moving, but he was so tangled in the supports of the table and chairs that they could not pull him out. More teachers arrived: some shooed the crowd of entranced classmates onto the playground while others tried to figure out what to do with the limp child under the table. Ultimately they extricated Ned by lifting up the whole table-and-chair structure, thus allowing him to fall out onto the floor. They then carried him to the nurses' office where Rick found him.
When the ambulance arrived there was something to do besides stare at the comatose Ned. The emergency medical technicians briskly transferred Ned into their vehicle and made a place for Rick to sit beside him. Rick held Ned's wrist with its comforting pulse all of the way to the hospital; it was the only indication he had that Ned was not dead. He confessed he had never been so relieved as when Ned threw up all over him.
At home we ate a simple supper. Ned was completely exhausted and went to bed, Brady retreated to his room, and Rick fled to the paper in the living room. I went to the kitchen and the dishes. "What is happening to my little boy? What is the matter with Ned?" I asked the little cherub in a sunburst frame who hangs above my stove. From deep inside, the voice of my grandmother spoke. She was a western pioneer of a civilizing kind seldom represented in the movies. Fading strawberry-blond hair piled elegantly on her head, Grandmama knew how to ride sidesaddle, to make peach chutney, and to pronounce "Mama" properly, with the accent on the final syllable. "Joan," she said, "modulate your voice. I can't understand you when you shriek." The water flowled steady into the sink, the soap suds crept between my fingers, the world's order was unchanged. I modulated my inner voice and wondered what to do.
The phone rang. It was the pediatrician's office. Apparently Rick had an appointment with them scheduled after the EEG. Since there was no EEG he had thought he could let it go by. I relayed this to the nurse on the phone, she relayed it to the doctor and the word came back—we were supposed to go in anyway. They wanted to see Ned.
Rick and I woke and dressed our sleepy boy and drove him to the doctors' office. The nurse on the phone had indicated that our regular doctor, Dr. Johnson, was not there, but it did not matter to me much. The man seemed perfectly capable of treating my children's ear infections and staying on top of routine injections but I had never particularly liked him.
When we arrived we were shown into a small examining room. Rick and I sat side by side in the two straight-backed chairs against the short wall; Ned exposed the bottoms of his feet to us as he lay on the examining table, along the long one. A desk, with various neutral medical wares upon it, occupied the other long wall; there was a wheeled stool on which the doctor could presumably scoot from table to desk. A vapid poster about childhood occupied the facing short wall, which Rick and I contemplated while Ned snoozed.
After about ten minutes a tall, lanky man with sandy blond hair and a friendly manner arrived. "Hi," he said. "I'm Dr. Lyman." He looked at Ned, who was almost asleep, and grinned.
How are you, young man?" he asked. Ned grinned back. "That's quite an egg you've got there," he observed of Ned's bruise. "Does it hurt?"
"Only if you press on it," Ned replied, proudly fingering the lump.
Dr. Lyman began a standard checkup. He looked into Ned's eyes and his ears. He pulled up Ned's shirt to listen to his heart and lungs.
"Do you want these?" he asked of the electrodes still stuck to Ned's chest.
"Oh, yes!" Ned answered proudly. "My friends will want to see them."
"You look like the bionic man," Dr. Lyman said, and he cheerfully listened to Ned's chest around the strange attachments. He then moved into a number of simple neurological tests. He had Ned walk a straight line, checked his reflexes, and checked his peripheral vision. All seemed perfectly normal. "Well," he said finally, "you're certainly healthy! How do you feel?"
"Tired," Ned responded and lay comfortably back down on the examining table.
"He looks fine," Dr. Lyman said, turning to us. "How are you?"
The question was electrifying. Until it was asked, although Rick and I were deeply involved in the situation, we had had no acknowledged role at all. We were simply "the responsible adults," charged with being sure that directions were followed. Rick poured out his story. Dr. Lyman listened quietly, his upper body draped on the desk, his long torso and legs un-self-consciously moving a little on the wheeled three-legged stool. When Rick was done, Dr. Lyman sat up. "You have just had the worst experience that can ever happen to a parent," he said gravely. "There is nothing more frightening than seeing your child in seizure."
They looked at each other for a moment. Then the doctor broke the spell. "So, ... you've done it all!" he said with a broad smile. Rick visibly relaxed, and smiled back.
In the next fifteen minutes Dr. Lyman explained that what Ned had experienced, a period of about five minutes of convulsions followed by forty-five minutes of unconsciousness, followed by vomiting, exhaustion, and then his usual state was "normal" insofar as any seizure could be described as normal. The treatment for such seizures was simply commonsense—to clear the area of things like table legs so Ned would not hurt himself as he flailed about, and simply wait until the seizure was over. Unless convulsions last longer than five minutes, an episode of seizure might not even require a doctor to be called. As for the causes, most seizures are idiopathic—no specific cause is ever known, they are just part of an individual's makeup. Others have clear causes that can be found in the brain, but Dr. Lyman did not see any reason to put Ned into this category. We would, of course, follow through with more precise tests, but at the moment it was striking that Ned displayed none of the other neurological symptoms one would expect if there was a tumor or something like that.
"What about Ned now?" I asked. "Do we need to watch over him tonight? What do we do tomorrow? When is it safe to send him back to school?"
"What about Ned now?" the doctor echoed. "He looks fine to me! Are you fine, Ned?" The doctor grinned at the positive response. "He can certainly go back to school tomorrow. The good thing about a seizure is that when its over, it's over."
Rick and I left the building with Ned, so impressed by Dr. Lyman that we agreed to change pediatricians so Ned would be under his care. We took the privilege of frantic parents, and had negotiated the switch in doctors within the week.
Part II: Beim Wippen (On a See-Saw)
9. Verlezt (Injured)
10. Kontrolle (Checking)
11. Notfall (Emergency)
12. Der Fuchs (The Fox)
13. Villa Walther
14. Ein paar Tage (A Few Days)
15. Station H
16. Termin (Appointment)
17. Anruf (Telephone Call)
18. Weihnachten (Christmas)
Part III: Er mus bewegen (He must exercise/It must move)
19. Dritte Operation (Third Operation)
20. Zimmer Acht (Room Eight)
21. Der Katheter (The Catheter)
22. Ärzte (Doctors)
23. Camp Oh-Ha-Ha
24. Schiene (Splints)
25. Erwachsene (Adults, Grown-ups)
26. Niederlage (Failure)
29. Frühling (Spring)
Part IV: Waiting for Changes
30. America (Amerika)
31. A Quiet Business