Levin gestured towards the stone bench under the gazebo, the invitation still hanging awkwardly in the air. "If you are… indeed merely lost," he said, his tone a fraction less hostile but still laced with suspicion, "then rest. Perhaps one of the groundskeepers can direct you once they make their rounds." He made no move to sit himself, instead standing stiffly, his recovered notebook clutched to his chest like a shield.
Kael exchanged a quick glance with Mira and Finn. Mira gave a tiny, almost imperceptible shrug that said, 'Your call, but this is weird.' Finn just looked hungry, his eyes still occasionally flicking towards the apple trees in the orchard.
"Thank you," Kael said, stepping towards the bench. "We've been walking for hours." He sat, deliberately choosing a spot that gave him a clear view of Levin and the notebook. Mira and Finn followed, perching on the edge of the bench, their unease palpable in the way they held themselves.
The silence stretched, broken only by the distant chirping of crickets and the rustle of leaves in the evening breeze. Levin continued to stand, eyeing them as if they were peculiar insects that had wandered into his terrarium.
"So," Kael began, breaking the quiet, his voice casual. "You said your tutor focuses on memorizing methods. Does he not explain the… the logic behind them? Why a certain formula works, or how it connects to other mathematical ideas?"
Levin frowned, clearly taken aback by the directness of the question from a boy who looked like he'd just been wrestling with a particularly stubborn patch of dirt. "Master Elms is a highly regarded scholar from the Royal Academy. He teaches the prescribed curriculum. Understanding the 'why' is… a pursuit for advanced theorists, I am told. For students, mastery of application is paramount."
"But isn't application easier if you understand the foundation?" Kael pressed gently. "Like with your multiplication of 98 by 23. Knowing that 98 is (100 - 2) isn't just a 'trick'; it's using the distributive property of multiplication over subtraction: a(b-c) = ab - ac. That's a fundamental principle."
Levin blinked. "Distri-what property?" He opened his notebook again, his earlier defensiveness momentarily forgotten in the face of Kael's calm, almost matter-of-fact delivery of a term he'd clearly never encountered. He found the problem: 98 x 23. His own work involved the standard vertical multiplication, with multiple lines for partial products.
"It means you can distribute the multiplication across the parts of the subtracted number," Kael explained, leaning forward slightly. He picked up a fallen twig and smoothed a patch of gravel with his foot. "Think of it visually. If you have a rectangle that's 23 units long and 100 units wide, its area is 2300. If you then cut off a smaller rectangle that's 2 units wide from that 100-unit width, you're removing an area of 2 times 23, which is 46. What's left is a rectangle 98 units wide and 23 units long. So, 2300 minus 46. It's the same area." He sketched a quick diagram in the gravel with the twig.
Mira and Finn watched, fascinated. They'd seen Kael use similar visual explanations for laying out market stalls or calculating timber needs, but applying it to pure numbers was new.
Levin stared at the crude diagram, then at Kael. A flicker of something – resentment, perhaps, or disbelief – crossed his face. "You make it sound… simple. Trivial, even." He straightened up, a touch of aristocratic pride returning. "Perhaps your 'tricks' work for basic arithmetic, village boy. But the mathematics required for true scholarship, for understanding the Aetheric Harmonics or Celestial Mechanics, is far more complex. It cannot be reduced to… twig drawings."
A challenge. Kael recognized it instantly. Levin, despite his initial astonishment, couldn't quite accept that a commoner, a child no less, could possess a deeper grasp of mathematics than he, a noble presumably receiving the best education available. His pride was stung.
"Perhaps," Kael said evenly. "You mentioned you were reviewing. What other concepts are you working on? Maybe I've encountered them in my… readings."
Levin's eyes narrowed. This was it, a chance to reassert his intellectual standing, to put this strangely knowledgeable urchin in his place. "Very well. Since you seem so confident in your 'patterns.'" He flipped a few pages in his notebook. "Consider this: solving for an unknown variable in a linear equation. For example," he tapped a neatly written line, "if 7x + 12 = 47, what is x?"
Kael didn't hesitate. "Five."
Levin's jaw tightened. "You… you didn't even write anything down."
"You don't always need to," Kael said. "The goal is to isolate x. So, you want to get rid of the '+ 12' on the left side. To do that, you subtract 12 from both sides of the equation to keep it balanced. 47 minus 12 is 35. So now you have 7x = 35. Then, to get x by itself, you divide both sides by 7. 35 divided by 7 is 5. So, x = 5." He spoke clearly, methodically, laying out each logical step.
Finn scratched his head. "What's x?"
"It's just a symbol, Finn," Kael explained patiently. "It represents a number we don't know yet, and we're trying to find it. Think of it like a mystery box. The equation is a set of clues to tell you what's inside the box."
Mira, ever practical, asked, "But why do you need to find what's in the box if you don't know what it is in the first place?"
"That's a good question, Mira," Kael said, turning to her. "Imagine you're a merchant. You know you sold 7 identical items, and after adding the 12 silver pieces you already had in your pouch, your total is 47 silver pieces. The equation '7x + 12 = 47' helps you figure out the price, 'x', of each individual item you sold."
Levin watched this exchange, a complex mix of emotions playing on his face. He was annoyed that Kael had solved it so quickly, yet also intrigued by Kael's ability to explain it in such simple terms to his clearly uneducated companions. Master Elms would have launched into abstract algebraic theory.
"A simplistic analogy," Levin sniffed, though his conviction was wavering. "Let us try something that requires a modicum of actual calculation. The sum of the angles in a quadrilateral."
"Three hundred and sixty degrees," Kael replied instantly.
Levin flinched as if struck. "How… how could you possibly know that without measurement or construction?"
"Any quadrilateral can be divided into two triangles by drawing a diagonal line from one vertex to an opposite one," Kael said, picking up his twig again and sketching a four-sided shape, then bisecting it. "And the sum of the angles in any triangle is always one hundred and eighty degrees. Since a quadrilateral is made of two such triangles, its internal angles sum to two times one hundred and eighty, which is three hundred and sixty."
He looked at Levin. "Do you know why the angles in a triangle sum to 180 degrees?"
Levin looked down at his notebook, then back at Kael, a hint of defiance in his eyes. "It is an axiom. A foundational truth stated by the First Geometers."
"It can be proven," Kael said. "If you draw a line parallel to one side of the triangle, passing through the opposite vertex..." He began to sketch again, explaining alternate interior angles and angles on a straight line.
As Kael spoke, his voice took on a different quality. The childish hesitancy was gone, replaced by the focused passion of a teacher absorbed in his subject. He wasn't showing off; he was genuinely explaining, laying out the logical progression of thought, making the abstract concrete.
Finn was lost, but he watched Kael with a sort of dumbfounded admiration. Mira, however, was leaning forward, her brow furrowed in concentration, actually trying to follow Kael's explanation of parallel lines and angles. She might not grasp all the terms, but she recognized the clear, structured thinking that Kael applied to everything, from market logistics to now, apparently, the shapes of things.
Levin listened, his initial intent to stump Kael slowly morphing into something else. He was still a proud noble, irked by this village boy's effortless superiority in a subject that was a constant source of his own academic torment. But Kael wasn't just giving answers; he was unveiling a way of thinking about mathematics that was entirely new to Levin. Master Elms presented math as a set of rigid rules to be memorized and applied. Kael presented it as a landscape of interconnected ideas, explorable and understandable.
"Why do you learn all this?" Kael asked Levin directly, pausing his geometry explanation. "You clearly find it… challenging. And you said yourself that 'mastery of application is paramount.' Application to what, exactly? What is the ultimate purpose of you learning these equations and geometric principles?"
Levin looked momentarily flustered by the personal question. "It is… it is expected," he said, a little stiffly. "A nobleman must be educated. Mathematics trains the mind, hones logical thinking. And… and it is said that the deepest understanding of Aetheric manipulation, the true power of the mages, is rooted in the highest forms of mathematics. Only those with a profound grasp of numbers and their relationships can hope to truly command the fundamental forces." He said the last part almost reverently, echoing a sentiment he had clearly heard many times.
"So, math is the language of magic?" Mira blurted out, then clapped a hand over her mouth, as if surprised by her own interjection.
Levin looked at her, then at Kael. "In a manner of speaking. The common hedge wizard might rely on rote incantations and instinct. But the Archons, the Grand Mages… they converse in equations. Their 'spells' are theorems brought to life." He didn't elaborate further, as if this was a truth too profound for uninitiated ears.
Kael nodded slowly. This confirmed his suspicions, the whispers he'd gathered from his System during the battle. "So, the better one understands mathematics, the better one might, theoretically, understand and perhaps even control these Aetheric forces?"
"Theoretically," Levin conceded, a touch of his earlier hauteur returning. "But theory and practice are leagues apart. One might understand the geometry of a perfect sword thrust, but that does not make one a swordsman." He seemed to regain some confidence from this analogy. "You may have a knack for these… elementary calculations, village boy. But the higher mathematics, the Aetheric Calculus, the N-Dimensional Geometries… those are beyond your ken."
"Perhaps," Kael said, unfazed. "But one must walk before one can run. Even calculus is built upon the foundations of algebra and geometry. If the foundation is shaky, the tower will inevitably be unstable." He met Levin's gaze. "You seem to have the intellect, and clearly the motivation, given what you've said about its connection to Aetheric manipulation. What if your difficulty isn't a lack of aptitude, but perhaps… a lack of interest because the methods you've been taught obscure the beauty and logic of it all?"
Levin was silent for a long moment, considering Kael's words. The sun had now fully set, and the garden was bathed in the soft, silvery light of the rising moon. The scent of night-jasmine was stronger now, almost intoxicating.
"Beauty?" Levin finally said, the word tasting foreign on his tongue when applied to mathematics. "I have found only frustration."
"Then perhaps," Kael offered, a ghost of a smile playing on his lips, "you've been looking at it from the wrong angle." He picked up the twig again. "Let's go back to that quadrilateral. You know it's 360 degrees. What if I told you there's a general formula for the sum of interior angles of any polygon, no matter how many sides it has?"
Levin leaned forward, despite himself. "A general formula?"
"Yes," Kael said. "(n-2) times 180 degrees, where 'n' is the number of sides." He then proceeded to explain why that formula worked, breaking polygons down into constituent triangles emanating from a single vertex.
Finn, by now, had subtly retrieved a slightly bruised apple from his pocket and was munching on it quietly, his initial awe at the numbers game giving way to a patient endurance. Mira, however, was still surprisingly engaged, asking clarifying questions. "So, for a five-sided shape, a pentagon, it would be (5-2) times 180? Three times 180… 540 degrees?"
"Exactly, Mira!" Kael said, a genuine smile this time. He turned back to Levin. "See? It's not about memorizing a different rule for every shape. It's about understanding a single, underlying principle."
Levin slowly sat down on the edge of the stone bench opposite Kael, his notebook still in his lap but now open to a fresh page. He picked up his quill, dipped it in the small inkpot on his discarded lap desk, and looked at Kael. "Alright, village boy. Kael. Explain it again. From the beginning. And this time," a note of challenge, but also a sliver of genuine curiosity, entered his voice, "don't leave anything out."
Kael felt a quiet sense of satisfaction. This wasn't just about showing off his knowledge. It was about planting a seed, about awakening a different kind of understanding. He glanced at Mira and Finn. They were out of their depth with the specifics, but they were here, part of this strange, moonlit classroom.
"Alright," Kael said. "Let's start with the very basics. What is a number?"
And so, under the silent gaze of the stars, surrounded by the fragrant blooms of a noble's garden, Kael Dray began to teach. He explained number systems, place values, the properties of prime numbers, the logic of fractions and decimals, the fundamentals of algebraic manipulation, and the visual elegance of geometric proofs. He didn't just give rules; he dissected them, showed their origins, their connections, their inherent rationality.
He taught not just Levin, but also, in their own way, Mira and Finn. He adapted his explanations, using analogies they could understand – market trades for positive and negative numbers, land division for fractions, building frameworks for geometric shapes.
Hours passed. Finn eventually dozed off, his head resting against a surprisingly soft patch of moss at the base of a statue. Mira, however, remained attentive, her quick mind grasping the practical applications even if the abstract theory sometimes eluded her. She was particularly fascinated by Kael's explanation of ratios and proportions, immediately seeing how it could be applied to mixing dyes or scaling recipes.
Levin was a revelation. Once Kael bypassed the rote methods and tapped into the 'why,' Levin's mind, trained for Aetheric sensitivity and perhaps possessing an inherent talent for complex systems, began to absorb the concepts at an astonishing rate. His frustration melted away, replaced by an almost feverish intensity. He asked sharp, insightful questions, challenged Kael's assertions (forcing Kael to further refine his explanations), and filled page after page of his notebook with newfound understanding.
The initial animosity between the noble boy and the village urchin had dissolved, replaced by the shared absorption of intellectual discovery. Kael, for these few precious hours, wasn't a fading echo or a lost child; he was a teacher, reveling in the joy of imparting knowledge. And Levin, for perhaps the first time, was experiencing mathematics not as a dreaded chore, but as a fascinating, logical, and powerful system.
The first pale light of dawn was beginning to streak the eastern sky when Levin finally leaned back, his quill set aside, a look of weary but profound satisfaction on his face.
"I… I understand," he said, looking at the pages filled with his own neat, now confident script. "It all… connects." He looked at Kael, and for the first time, there was no trace of condescension, only a grudging, burgeoning respect. "You are no ordinary village boy, Kael."
Kael merely smiled. "And you're a faster learner than you give yourself credit for, Levin."
The fragile bridge had solidified. It was no longer just about mathematics; it was about a shared journey into understanding. And as the new day began to illuminate the grand mansion and its hidden garden, new possibilities, and new complications, undoubtedly lay ahead.