“Zeno?” Amethyst asked with a cocked eyebrow. “Who’s Zeno?”
“Zeno is one of my old research partners, my first in fact,” Evenstar said as he set the envelope in his hoof onto the table. He stared at the unopened letter and let out a sigh. “I haven’t heard from him in years… Amethyst? Would you care to do the honors?”
A purple aura surrounded the note as Amethyst lifted it from the desk and brought the letter opener to its edge.
“Careful now…” Evenstar said, his gaze fixed on the note.
I’d never seen him like this. It was as though the letter that floated before him held more importance than anything else in the lab. I distinctly remember the entire room becoming dead silent as Amethyst gently sliced through the envelope’s edge. She wasn’t incredibly worried, as she had performed the task thousands of times before, but Evenstar still leaned ever closer as she brought the blade through the paper.
The moment it was open, Evenstar snatched it from her grasp with his hoof and withdrew the letter inside. It wasn’t particularly notable, aside from the ornate signature that adorned the bottom of the page. In fact, as Evenstar’s eyes scanned through the lines, I could make that there wasn’t much on the page. For such a short discourse, Evenstar spent quite a while reading it through dozens of times. I remember watching his face, trying to get any hint as to what the letter said. At first he seemed, happy, then concerned, and shortly after, extremely excited.
“So?” Amethyst asked, breaking the silence with a quiet voice.
“Hm?” Evenstar looked over to Amethyst, his eyes absently staring at her for a brief moment. “Oh! Right,” he exclaimed as he began to regain his composure. He looked to the page once more and began to read.
“Evenstar, I have a paradox that’s been bothering me for the past few months. I figured it would be best to bother you with it as well for old time’s sake.
“If a pony and a tortoise race, the pony can never win. This is due to the fact that the pony must first reach the starting position of the tortoise, and by that time, the tortoise will have moved ahead farther, continuing the cycle.”
The room fell into an uneasy silence. Personally, I was awaiting an answer to the question. I knew it was fundamentally against how the world worked, but for some reason, I was stumped at the math behind it. Evenstar, however, had a very different look about him. He sat with a confident smirk on his face, like he was hiding a secret from the two of us.
“Well?” Amethyst started, “what’s the solution?”
“I don’t know,” he said with a confident smirk, “but I think, between the three of us, we can find one.” He turned and looked over the room. His aura of confidence seemed to build with every second as he stepped over to a particularly cluttered table, placed his foreleg onto its surface, and in one fluid motion, flung its contents to the ground.
I nearly cried out as I watched countless instruments fall to the floor, but a purple aura stopped their fall.
“You haven’t done that in a while,” Amethyst said with a smirk as she placed the items in different locations across the library. “A bit of advanced warning would have been nice.”
“My apologies, Amethyst,” Evenstar muttered absently as he stepped to another table and repeated the maneuver. “But we’ll need a lot of space to deal with this question.”
“What?” I asked, trying to insert myself into the confusion.
“Don’t you see, Starswirl?” Evenstar said as he cleared a third table. “Ponies are beginning to research the infinitely small.”
“So?”
“Starswirl…” Evenstar turned to me with a face of pure determination. “Infinitesimals are the central unit of the Analog equations. They’re fundamental to the decades of research I’ve performed.” He turned to Amethyst. “Note: glass, water, protractor, compass, several rulers, mop, salt, and a towel.”
Amethyst quickly jotted the list onto a small notebook before trotting through the shelves gathering equipment.
“Starswirl,” he said, pulling my attention from Amethyst, “I need to you come here and help me with something.”
“Yes?”
“Come here and stand next to me.”
I stepped over to Evenstar’s side and watched as he took a few strides forward. “Alright, when I say go, you walk forward at a canter. I’ll walk normally.”
“I’ve got the items,” Amethyst said as she emerged from the final row of shelves.
“Excellent work. Put them on one of the cleared tables, we’ll need them soon.” Evenstar turned his attention toward me. “Ready?”
I nodded.
“3… 2… 1… go.” We both started off the track in one of the most interesting races I’d ever taken part in. I watched as I quickly overtook my mentor as he mentally noted every detail about the experiment. It was the first time I had ever seen him in his element, discovering something new about the world. We repeated the test, this time with our positions switched, and he began to place rulers down at varying intervals along the course.
If you had told me, just two months prior, that I would have the time of my life having a race with Evenstar, I would have laughed at you. In fact, now that I think about it, I’m chuckling to myself. Regardless, it was the first time I had been exposed to the rigors of the scientific process, and this was just the beginning.
We repeated that simple experiment for hours, all the while making detailed notes on our starting positions, where we passed, and what time we did so. It was all so enjoyable for me, like I was doing something revolutionary, even if it was just being a pawn for Evenstar. Amethyst participated by taking detailed notes and guiding the experiments to perfect repetition.
“Interesting…” Evenstar muttered as he scanned the data over again; his eyes straining over the darkness. “We’ve repeated the experiment hundreds of times, but we’re still not seeing anything against the obvious answer…”
Amethyst looked to one of the three chalkboards she has littered with math. “Everything works out, it’s exactly as we expect it.”
“So then why is the math still not backing it up?” Evenstar wondered aloud as he pointed at the graph in the middle of the chalkboard.
Amethyst had plotted the question on the board, and the curve it drew was a rather interesting one. It started off at an initial height of 1, but as it stretched to the right, it dropped by half each time. Near the end of the board, she found that the two lines seemed to merge, but for some reason, the math proved that they didn’t.
“Why…” Amethyst whispered as she scrutinized the lines again. “They never meet.”
I paused for a moment as I tried to get a better understanding of the graph. “Why…” I started as my mind grew lost in the math. “I don’t even understand this anymore.” I’m not sure why I said that, but Evenstar spoke before I had a chance to internalize any feelings of shame.
“The vertical axis represents the distance between you and me. The horizontal axis describes… time.” Evenstar caught himself on that final word. I would come to recognize that hesitation as a spark in his mind. “It doesn’t… It represents an interval.” His voice was so quiet I could barely hear the words he uttered, but his furrowed brown and searching eyes shouted that he had stumbled across a stroke of brilliance.
“Is something wrong?” Amethyst asked as she looked at the board once more.
“The graph…” he said confidently. “The graph is wrong.”
“But, I’m sure those values are—“
“Not the values, Amethyst. The criteria,” he said as he pointed to an empty section of the board. “Draw another graph, but this time…”
Amethyst looked to him with a cocked eyebrow as a small piece of chalk floated beside her. “This time… what?”
“The horizontal axis… it needs to be…” His brow furrowed in thought. “There’s something wrong with it.”
I looked to the board once more. By now, the graph was incredibly confusing, and I was beginning to forget the basic gist of the problem. “What…” I started, stirring the silence, “What exactly are we trying to do?” The words rolled off my tongue as though I had a mind to give up on the entire project.
“We’re trying to get this graph to accurately display our findings,” Evenstar said again, “but the graph’s horizontal axis… it’s… It’s only showing intervals!” He declared as he turned to Amethyst. “Amethyst! I need a new graph. This time, label the bottom axis as Time rather than x. Draw any diagonal line, and fit the data to it!”
“What?” I asked, confused by his sudden change in demeanor.
“The graph. The one that slopes off toward infinity, it’s wrong. That would be true if we slowed down as we walked, but we obviously didn’t. Now… I’m going to fit the data to a graph our velocity. We retained the same speed, which is why this graph doesn’t move.”
“Alright,” Amethyst said as she plotted the first set of data. “Now what?”
“Go ahead and draw another line in accordance with my speed.” As Amethyst drew the second line’s plot points and joined them together, they crossed. Never before in my life had I see a stallion so excited about crossing lines.
“What’s the big deal?” I asked as Evenstar gleefully cantered to the board.
“The big deal…” he started as his eyes darted across the board, is that we just used infinity against itself. “Look here.” He tapped on the intersection. “Using the old method, this never reached the value of the intersection, it was always just a small fraction of a unit off. Now, with time dropping to infinity as well, we’ve effectively removed that hindrance… We went to infinity.”
As though he had been captured by his own statement, Evenstar let his hoof slide down the board as he fell squarely on his rump. It was funny to me, and I believe I heard Amethyst snicker as well, but there was a beauty to his actions. He had, within a matter of hours, surpassed a huge threshold that had sat in his way for a long time. Infinitesimals like these were fundamental to the Analog equations he had devoted his life’s work to. Seeing them fall together so beautifully, watching them cancel themselves out so perfectly, well it was something that was lost on me at the time.