Spring 1891
Triangles

Uncle Jakob found the twelve-year-old Albert sitting at the kitchen table, drawing triangles, again. Jacob was the youngest of five siblings.

Unlike Albert’s father, Hermann, Jakob had been able to pursue higher education and had qualified as an engineer. Hermann had not resented his brother’s opportunity, and together the two brothers built a successful company providing generators and electrical lighting to municipalities in southern Germany. Jakob oversaw the technical side while Hermann handled sales. Perhaps more essential to the partnership, Hermann was able to secure loans from his wife’s side of the family.

Jakob’s training and knowledge were regularly put to the test by Albert, who had an inexhaustible supply of questions. One day he was asking for details on how the generators worked. Then he had to know about the capacity of the wires that ran to the lighting fixtures. Most of all, Albert wanted to learn about light. But recently, geometry had become his focus.

“I see geometric shapes have captured your attention, nephew.”

Albert nodded, his bright eyes eager. “Somehow, triangles seem to blend nature and science. I even see geometric designs in the flowers in the garden.”

Jakob raised an eyebrow as he pulled up a chair across from his precocious nephew. “Hmm. Well, have you ever heard of Pythagoras?”

Albert reflected for a moment. “The name is familiar,” he said hesitantly. “But I don’t remember who he is.” The look on Albert’s face let Jakob know Albert wanted to know more.

Leaning forward, the learned man explained, “Pythagoras was a Greek mathematician who lived between 569 and 475 BC. He is sometimes called the first mathematician, meaning he was one of the first scientists on record to have made significant contributions to the field of mathematics.” As Albert nodded, Jakob continued. “He was more than just a mathematician, though; he studied and worked with religion and philosophy. What’s more, he was also a musician; he played the lyre.”

Albert’s hazel eyes danced with curiosity. “Now that’s a man I would like to know more about!”

Jakob smiled and beckoned Albert to follow him to a nearby bookcase. After searching for a moment, he pulled a small book from the shelf and handed it to the boy. “When I saw you drawing triangles the other day, I knew it would not be long before you would want to explore the mystery of Pythagoras and his theories.”

Albert grabbed the book and marched back to the table. He did not even notice Uncle Jakob had left, smiling and shaking his head. “Give that boy a book, and it is like tossing a sponge into a pail of water. He absorbs every drop of knowledge,” he muttered to himself walking out the door.

The house was quiet as a church as Albert lost himself in the book on Pythagoras. Warm summer winds blew the yellow cotton curtains, and they flapped through the open window over the kitchen sink. The young mathematician’s feet dangled from the wood-and-thatched chair at the rectangular table topped with butcher block. As he read, he began to realize that his uncle Jakob had given him his first real intellectual puzzle. Deep in thought, Albert was unaware that he had almost chewed through his pencil as he stared at the diagram of a right triangle. His eyebrows drawing closer and closer together as he read, Albert became determined to prove the Pythagorean theorem.

Losing himself in his contemplation, Albert absentmindedly began playing with his compass as he turned pages in the book. He would read a few paragraphs and then gaze at the compass face, letting his mind wander in speculation. There was no way for him to know that the energy of the compass took his mind beyond space and time. Albert was far away, and unaware of where he was as triangles of all shapes and sizes danced in his imagination. He was determined to meet this challenge and prove the theory. Albert did not tell anyone what he was working on.

By the second week of intense focus, Albert’s theories were swirling round and round in his head. Finally, on a Friday, wild with excitement, he sat filling a sheet of paper with cryptic drawings and numbers so furiously, the pencil lead broke. His hand twitching, he stared at the torn paper and broken pencil. He screamed, wadded up the paper with his shaking hands and threw it across the room. The budding scientist put his head down on the table and sobbed.

His mother, Pauline, rushed from the stove where she had been stirring the stew for the evening dinner. With a glance, she surmised what had happened. She knelt and put a comforting arm across her son’s shoulder. “Now, now, Albert. It’s okay.”

Albert turned and buried himself in his mother’s hug. “It’s not okay, Mama. There is a way to prove this theorem, but I can’t find it.” His face was still pinched with anger as he spoke.

Pauline thought for a moment, then brightened. “You need a break. Do something else.”

“Like what, Mama?”

Pauline shrugged. “I don’t know. Maybe play your violin. You know how music soothes you and clears your head.”

Gently tugging at the frowning boy, Pauline urged him from his seat. “Come, Albert. Invite Johann. The two of you could practice the Mozart Sonata for the recital at school next month.”

Albert didn’t want to see anyone. He was stuck. He was getting nowhere. His mother’s words reminded him how the family loved the recitals the two of them played during the holidays and how music lifted his spirits. And it was true that he did enjoy it when Johann joined in from time to time. Sighing in resignation, the young mathematician surrendered. “Oh, all right, I will go and get Johann.”

* * *

“Wow, did your pet goldfish die or something, Albert? You look terrible!” Johann shook his head in disapproval as his friend led him through the front door.

Despite himself, Albert had to smile at Johann’s cheerfulness. “Ah, I’m just stuck on a problem and don’t know how to get out of it.” Albert waved his arm as if to brush away his vexation. He was still hiding his mission and didn’t even want his good friend to know what he was pursuing. Albert ushered Johann into the parlor. “My mother thinks taking a break will help. We need to practice for the recital, anyway.”

Used to Albert’s moods, Johann nodded. “Okay, I can practice for an hour. My father needs me at the alehouse to help serve the evening meal.” He wiped his hands on his lederhosen and sat on the wooden piano bench, his legs stuffed under the piano, and shuffled the sheet music on the music stand. Albert had already memorized the piece, so he readied his violin as he stood next to Johann.

After fifteen minutes of stops and starts to refine their duet, the notes sparkled. The music’s sweetness began to seep into Albert’s troubled heart. He closed his eyes and, like fireworks, a burst of triangles within the notes flew in rhythm across his violin. His imagination blossomed and flowed with new ideas as Albert opened to new dimensions inside himself.

After another thirty minutes of playing, Albert had regained a sense of peace—as well as a new enthusiasm for his project. Albert began to fidget with his brow. He urged Johann to his feet and helped him on with his jacket. “It’s good, Johann, we’re ready for our recital,” Albert pronounced, propelling Johann to the door.

Attempting to straighten his jacket amid the hustle, Johann said, “Well, I guess we are ready.” Then Johann dug in his feet and turned to Albert, hiding a grin. “But are you sure you wouldn’t like to practice a few more times? I could stay a couple more minutes...”

“No, no, I am certain we are ready. Hurry up now, I don’t want you to be late for work,” Albert replied, almost slamming the door shut and utterly oblivious to the fact that Johann knew precisely what Albert was up to. On the porch, Johann smiled and shook his head as he turned to walk back to the alehouse. He had grown to love Albert and, truth be told, he was happy that his friend had regained his happiness.

With the breakthrough in awareness he had gained when he and Johann had been playing the Mozart piece, Albert became more confident as he worked over the next days. And with the confidence came serenity. The boy would awaken each morning with awareness of the music of the Pythagorean theorem dancing in his imagination. It was as if he was viewing the mathematics of it in its completeness from high above. And he knew he would find its temporarily elusive proof.