Hazō's office was a sanctuary, a small space adorned with books and scrolls that lined the shelves. The air was still, and the only source of light came from a flickering candle that cast dancing shadows on the walls. In this intimate setting, Hazō, a prodigious teenager with a deep understanding of Sealing, embarked on his arduous journey that would ultimately lead him to discover Calculus from first principles.
Seated at his cluttered desk, Hazō stared at the blank parchment before him. His mind was a whirlwind of ideas, each fragment begging to be explored. He closed his eyes, taking a deep breath, and allowed his thoughts to settle. In the depths of his mind, he recalled the patterns and motions he had witnessed throughout his training in the art of Sealing.
The teachings of Sealing emphasized the understanding of spatial relationships, the manipulation of energies, and the harmony of movement. As Hazō immersed himself in the intricacies of this mystical practice, he gradually recognized the hidden connections between Sealing and mathematics. It was through this revelation that Hazō found the inspiration to embark on his quest to uncover Calculus.
Opening his eyes, Hazō's focus sharpened. With his quill in hand, he began to sketch simple shapes, each representing an object in motion. His strokes were deliberate, capturing the arcs and curves that defined their paths. As he examined his sketches, he pondered the essence of motion and change.
Hazō knew that in order to capture the concept of change mathematically, he needed to understand how an object's position varied with time. With this in mind, he started by considering two positions of an object, one at an initial time and another at a slightly later time. He marked these positions on his sketch, denoting them as "P1" and "P2."
As he observed the two positions, Hazō instinctively recognized that the change in position was determined by the difference between P2 and P1. In his mind, he visualized this difference as a vector, an arrow connecting the two points on his sketch. He labeled this vector as "ΔP," signifying the change in position.
The next step in Hazō's chain of logic was to understand how the change in position, ΔP, related to the change in time, which he denoted as Δt. Deep in thought, he began to analyze the concept of velocity—the rate at which an object's position changed with time.
In his mind's eye, Hazō envisioned an object moving from P1 to P2. He imagined dividing the vector ΔP by the corresponding change in time, Δt. This division, he reasoned, would yield a new vector—one that represented the object's average velocity between the two points.
Hazō labeled this new vector as "v," signifying velocity. His intuition told him that as the change in time approached zero, the average velocity vector would converge to a particular direction and magnitude. This observation sparked his realization that he was inching closer to the concept of instantaneous velocity, the very essence of Calculus.
With a surge of excitement, Hazō continued his exploration. He contemplated the idea of shrinking the time interval, making Δt infinitesimally small. As he did so, the object's motion became more refined, its path bending and curving with greater precision. In his calculations, Hazō let Δt approach zero, symbolized as "dt," and discovered that he had reached the threshold of Calculus.
The final link in Hazō's chain of logic involved recognizing that the instantaneous velocity, represented by the vector v, was the derivative of the object's position vector with respect to time. The realization washed over him, filling him with
a profound sense of achievement. Hazō's heart raced as he grasped the magnitude of his discovery. He had successfully derived the concept of the derivative, a fundamental principle of Calculus, from his own reasoning and observations.
Excitement fueled his determination as he continued to explore the intricacies of change. Hazō now shifted his focus to the integral, another vital aspect of Calculus. With his quill poised, he delved into the depths of his knowledge, seeking to understand how to capture the cumulative effect of change over a given interval.
Drawing upon the concepts he had previously uncovered, Hazō devised a new line of reasoning. He imagined an object moving along a curved path, its position changing continuously. To comprehend the cumulative effect of its changing position, he contemplated the notion of adding up an infinite number of infinitesimal changes.
With each infinitesimal change in position, symbolized as "dP," Hazō recognized that he could create a new vector representing these infinitesimal changes. He denoted this vector as "ds," signifying the infinitesimal arc length along the object's path.
As he worked through his calculations, Hazō realized that the cumulative sum of these infinitesimal changes, symbolized by the integral, captured the total change in position over the entire interval. This revelation sparked a profound understanding within him—an understanding that he had discovered the basis of integration.
With a sense of awe and wonder, Hazō marveled at the interconnectedness of these newfound principles. He had unlocked the hidden treasures of Calculus, derived from the depths of his own reasoning and guided by his knowledge of Sealing. The mathematical language he had developed, a fusion of Sealing symbols and conventional notation, bore witness to the convergence of ancient wisdom and modern understanding.
Days turned into weeks, and weeks into months, as Hazō immersed himself further into the intricacies of Calculus. He refined his notation, crafted proofs, and expanded his understanding of its applications. His office walls became a mosaic of diagrams, equations, and sketches—a testament to his relentless pursuit of knowledge.
Throughout his journey, Hazō couldn't help but feel the presence of Akane, his beloved girlfriend who had tragically passed away. He often found solace in her memory, drawing strength from the echoes of her laughter and the warmth of her love. In a heartfelt tribute to her, Hazō named his foundational discoveries in Calculus after her—Akane's Derivative and Akane's Integral—a perpetual reminder of the inspiration she had brought into his life.
As Hazō's reputation spread, scholars and mathematicians marveled at the genius of this young prodigy. His breakthroughs in Calculus from first principles revolutionized the field, opening new horizons of knowledge and understanding. Hazō's legacy endured, not only as a mathematical pioneer but as a symbol of perseverance, resilience, and the power of the human spirit. He may have been the first to discover these revolutionary relationships (at least, the first to discover and then
share them becuase who knows what the Thinker clans were sitting on), but he would not be the last to use them.
In the quiet solitude of his office, Hazō continued to push the boundaries of his knowledge, forever driven by the desire to unravel the mysteries of the universe. And as the flickering candlelight illuminated his face, casting dancing shadows on the walls, Hazō knew that he had embarked on a lifelong journey—one that would shape not only his own destiny but the course of mathematical exploration for generations to come.
And then the Dragon came and ate the entire world, because he was too busy escaping his grief through rigorous math to stop the Great Seal from collapsing.
THE END