Dawn approached. Levin's notebook was nearly full. Finn snored softly against a statue's base. Mira, quiet and tired, watched Kael.
Kael stretched, his muscles aching from sitting so long. The long night of explanations had been invigorating, but now, a familiar caution settled in. He had shown too much.
He looked at Levin, who was carefully reviewing his new notes, a smudge of ink still prominent on his cheek.
"Levin," Kael began, his voice low.
Levin looked up, his hazel eyes alert, the earlier frustration gone.
"What I showed you," Kael said, "about the math. It's… unusual. For someone like me, from a village."
Levin closed his notebook slowly. "Unusual is an understatement, Kael. Master Elms, with all his academy training, has never made these concepts so… accessible."
"I need it to remain unusual, for now," Kael stated. "I don't want word of it spreading. Not yet."
A flicker of understanding, or perhaps calculation, crossed Levin's face. "You fear unwanted attention? From whom?"
"Attention from anyone who might ask questions I'm not ready to answer," Kael replied. "My… abilities… they are best kept quiet for a while."
Levin considered this. He was a noble; he understood the value of discretion and the danger of revealing too much too soon. "And what do you propose?"
"An agreement," Kael said. "I can continue to help you with your mathematics. Help you understand it, not just memorize it. You clearly need to catch up, and you said yourself it's important for… Aetheric understanding."
Levin's interest was piqued. "Continue? To what extent?"
"Through what you called 'secondary math' – equations, geometry, the basics of life science as it pertains to measurable phenomena. Perhaps a little beyond, into the foundational concepts that underpin the more advanced theories you mentioned, like calculus. Enough to give you a solid understanding to build upon."
Levin's eyes widened slightly. This was a far more extensive offer than he'd dared hope for. "And in exchange for this… extensive tutelage?"
"Information," Kael said. "You live in a different world than I do. Tell me about the capital. About the Royal Academy. About the great noble houses, the politics. About how 'Aetheric manipulation' truly works, beyond just what is commonly known. And," Kael added, thinking of Eldham, "perhaps, when the time is right, your… influence… might be helpful for some projects I have in mind back in my village. Practical things. A market, a way to manage local currency, improving our tools."
Levin listened intently. The offer was compelling. Kael's teaching in one night had illuminated more than months with Master Elms. The information Kael sought was common knowledge to him, if not always to commoners. And the idea of having some discreet influence on "practical things" in a village… it had a certain appeal, a way to apply his station that wasn't just about courtly manners or dry academics. It sounded like something real.
"The projects you mention," Levin asked, "what are they?"
Kael briefly outlined the supermarket, the concept of local banknotes, and the need for better tools and processes, much like he'd explained to Mira and Finn, but framed it as improving village efficiency and trade. He watched Levin's face, gauging his reaction.
Levin tapped his quill against his chin. "A centralized market with its own currency… ambitious for a village. Potentially lucrative if managed well. And a source of local stability." He looked at Kael. "I find these… 'practical things'… intriguing. Very well, Kael. I agree to your terms. Your discretion for mine. Your mathematical instruction for my knowledge of the world, and… we can discuss how I might discreetly assist your village endeavors when the time comes. No one in my household, or any visitors, will know of your true abilities from me. You will simply be… a surprisingly knowledgeable village boy I encountered."
"And I will not speak of who you are, or what we discuss, to anyone outside of my trusted friends here," Kael said, glancing at Mira and a still-sleeping Finn.
"Agreed," Levin said, a faint smile touching his lips. He extended a hand. It was a formal gesture, one Kael hadn't expected. Kael hesitated for a moment, then took it. The handshake was brief, firm. A pact was made.
With the deal struck, a different kind of energy filled Kael. He had a new ally, a source of information, and a clear path to legitimizing his knowledge in the future. He looked over at Mira, who had been listening intently to the exchange.
"Mira," Kael said. "Finn, wake him up. If Levin is going to learn math properly, it starts from the beginning. And you two are going to learn with him."
Finn grumbled as Mira shook him awake. "Huh? Learn what? More number headaches?"
"Mathematics," Kael stated. "From the very basics. It's important. For everything."
Levin watched, a mixture of amusement and curiosity on his face, as Kael prepared to teach his village friends. This was going to be an unusual vacation.
Kael found a relatively smooth patch of flagstone near the bench. He picked up his twig again. "Alright," he began, looking at Mira and Finn. Levin settled back on the bench, observing. "Before we talk about adding or subtracting, we need to understand what numbers are."
He drew a single short line in the dust. "This is one. One pebble, one apple, one Finn." Finn grinned. "It represents a single thing." He drew another. "This is two." He continued up to five. "We group things to count them easily. Our hands have five fingers each. Two hands, ten fingers. That's why we often count in groups of ten. It's called a base-ten system."
He gathered a small handful of pebbles from the edge of the gravel path. He laid out nine pebbles. "This is nine. If I add one more," he placed another pebble, "I have ten." He then arranged those ten pebbles into a neat little pile. "We can call this pile 'one ten.' If I have another pebble, I have 'one ten and one one,' which we write as eleven."
Kael spent the next hour on this, patiently explaining place value – units, tens, hundreds – using pebbles, then showing how to write the numbers. Mira, who handled market goods and had a good sense of quantities, picked it up quickly. Finn, more accustomed to tangible objects, struggled a bit with the abstract representation until Kael related it to counting nails or stacks of firewood.
"Okay, so if I have three piles of ten pebbles, and seven loose pebbles, that's thirty-seven?" Mira asked, arranging her own set.
"Exactly!" Kael said. "The 'three' tells you how many tens, the 'seven' tells you how many ones."
Levin interjected from the bench, "Master Elms spent a week on numeration. He used an abacus and chants. Your pebble method is… surprisingly clear for such a foundational concept."
Kael smiled faintly. "Next, addition." He set out two small groups of pebbles. "If Finn has three apples," he gestured to three pebbles, "and Mira gives him two more apples," he added two pebbles to Finn's group, "how many apples does Finn have now?"
"Five," Finn said immediately, then looked surprised at how obvious it was when laid out.
"Right. Addition is just combining groups. 3 + 2 = 5." Kael wrote the symbols in the dust. He went through several examples, gradually increasing the numbers, showing them how to "carry over" when a sum in the units column went over nine, explaining that ten units become one ten, which moves to the tens column.
Subtraction was next. "If Mira has eight coins," Kael laid out eight pebbles, "and she pays three coins for a loaf of bread," he removed three pebbles, "how many does she have left?"
"Five," Mira said, her eyes following the pebbles.
"Correct. 8 - 3 = 5. Subtraction is taking away, or finding the difference." He explained "borrowing" for more complex subtraction problems, again using the pebbles to demonstrate how one ten could be broken back down into ten units if needed.
Levin, despite knowing these operations, found himself listening intently to Kael's clear, logical breakdown. Kael wasn't just stating rules; he was demonstrating the why behind them, building from tangible reality to abstract symbols.
Multiplication was introduced as repeated addition. "If Kael gives three sweets to each of you," Kael said, looking at Mira, Finn, and then including Levin with a nod, "how many sweets did he give out in total? That's 3 sweets, and he did it 3 times. So, 3 + 3 + 3. Or, we can write it as 3 x 3." He arranged pebbles in three rows of three. "Nine sweets."
He showed them how to build multiplication tables through this repeated addition and by visualizing arrays. Mira quickly saw how this could help her calculate the total price of multiple identical items. Finn related it to patterns in weaving or laying out bricks.
Division was "sharing equally" or "how many groups." "If I have twelve cookies," Kael drew twelve circles, "and I want to share them equally between the three of you, how many does each person get?" He had them physically group the circles. "Each person gets four. So, 12 ÷ 3 = 4." He also explained it as repeated subtraction – how many times can you take 3 away from 12?
By mid-morning, the sun was high and warm. Finn was actually looking less bewildered and more engaged, especially when Kael used examples involving lengths of wood or numbers of tools. Mira was already mentally applying the concepts to her market dealings. Levin had occasionally interjected with a more formal term or a question about how these simple concepts related to the algebraic equations he'd been struggling with, and Kael would briefly bridge the gap, showing how understanding that "x" was just an unknown quantity of "pebbles" made solving for it much more intuitive.
"Alright," Kael said, noticing Finn starting to fidget. "That's enough numbers for a bit. My head's starting to hurt too."
A genuine laugh escaped Levin. "After a night of advanced geometry and a morning of… pebble arithmetic, I believe I understand the sentiment." He stood up, stretching. "The head gardener mentioned there's a small stream at the far end of the estate, with shallow pools. He says the water is cool. Perhaps a… less academic pursuit is in order before the midday meal?"
Mira and Finn looked at Kael, their eyes lighting up.
Kael considered. He had achieved his primary objective with Levin. And his friends deserved a break. He, too, felt the pull of simple, childish enjoyment, a feeling that was becoming rarer. "A stream sounds good," Kael agreed.
As they walked, Levin, surprisingly, took the lead, pointing out different fruit trees and flowering bushes, his earlier formality softening slightly in the less structured environment. He even attempted a joke about a oddly shaped cloud, which fell flat but earned him a polite smile from Mira.
Reaching the stream, a clear, babbling brook shaded by willows, Finn was the first to kick off his worn boots and wade in, letting out a whoop as the cool water splashed his legs. Mira, more reserved, contented herself with dipping her feet in from a smooth rock. Levin watched them, a hesitant, almost wistful look on his face.
Kael, seeing Levin's hesitation, picked up a flat, smooth stone. With a practiced flick of his wrist, he sent it skipping across the water's surface – one, two, three, four times. "Your turn," he said to Levin, offering him another stone.
Levin looked surprised, then a faint, shy smile appeared. He took the stone. His first attempt sank immediately with a plunk. Finn let out a good-natured laugh. Levin's ears turned red, but then he tried again, and this time, the stone skipped twice. A small, genuine grin spread across his face.
For a little while, the weight of their respective worlds – Kael's fading past and uncertain future, Levin's noble pressures, Mira's burgeoning responsibilities, Finn's toil – seemed to lift. They were just children, skipping stones by a sun-dappled stream.
Later, as they walked back towards the main house for the promised midday meal (Levin had discreetly arranged for extra portions to be prepared, citing "unexpected but welcome young visitors to the estate staff"), Levin fell into step beside Kael.
"About your village projects, Kael," Levin said, his tone more casual now. "The centralized market and the local banknotes… it's an interesting model for stimulating an isolated economy. My father, the King Regent, often discusses the challenges of improving trade and prosperity in the more remote baronies." He paused. "There's a merchant guild in the capital, the Golden Quill Consortium. They have… considerable influence and resources. They are always looking for new, secure trade routes and stable local markets. If your Eldham market could demonstrate consistent management and reliable local currency exchange, backed by tangible goods as you described… it might attract their attention. Or, at least, the attention of one of their less conservative factors."
Kael listened, his mind immediately processing the implications. "And how would one… attract such attention?"
Levin smiled faintly. "Perhaps a well-drafted proposal, outlining the current successes and future potential, delivered through… an appropriate channel. I may know of someone who occasionally advises a junior member of the Consortium's board. No promises, of course. But interesting ideas, backed by solid groundwork, tend to find their way."
Kael nodded slowly. "Thank you, Levin. That is… very useful information."
The offer was subtle, but clear. Levin was holding up his end of the bargain, not just with information about the world, but with a potential, tangible path for Kael's ambitions. The sun felt warmer now, the future a little less daunting.