Disclaimer: Mathematically speaking, my ownership of any rights to the Harry Potter franchise is 0%.

A/N: Thanks to aplusbex for pointing out that there may be some confusion about what maths Arithmancy covers. Basically, N.E.W.T.s are equivalent to A-levels, which do include calculus.

Thanks to everyone else who suggested ideas about arithmancy and spellcrafting. I have some of my own as well, but I've made notes on all of them.

Chapter 3

Breakfast the next morning was served Scottish style with black pudding, lorne sausage, and tattie scones alongside the bacon, eggs, and toast. The floating candles from the feast the night before were gone, and the Hall was brightly lit by the sunlight streaming through the windows and the blue sky on the enchanted ceiling above. Overhead, a whole parliament of owls came winging its way into the Hall, delivering letters and the occasional parcel to students and professors alike. Hermione hadn't quite believed (or wanted to believe) that owls were the main component of the post system in the magical world, but she couldn't deny it now.

Hermione sat near Percy again. The other Weasleys were scattered around a few seats away. Percy fielded a few questions from the other first years until Professor McGonagall approached him.

"Mr. Weasley, the first years' class schedules," she told him.

"Thank you, Professor," he said, rising to pass them out.

McGonagall then touched Hermione on the shoulder and told her, "Miss Granger, your first class will be Transfiguration with me in Classroom 1B on the first floor. Please come early. I will need to discuss your schedule with you personally."

"Yes, ma'am," she said, wondering if there had been a problem with the scheduling. She hoped there wouldn't be anything to disrupt her plans.

She was snapped out of her thoughts by one of the second years across from her whispering, "There look, next to the other Weasley."

"Wearing the glasses?" the boy's friend said. They were obviously talking about Harry Potter.

"Yeah, that's him."

"I can't see a scar from here. Did you see it?"

"Yeah, like a lightning bolt, just like everyone says."

"That's not very polite, you know," Hermione interrupted them. They stared at her in surprise. "Talking about him behind his back like that."

"It's Harry Potter," the first boy said, as if that made it alright.

"So? He's just here to learn, like the rest of us."

The second boy rolled his eyes. "You must be muggle-born."

"And what's wrong with that?"

"Nothing!" the first boy said quickly. "It's just that you wouldn't know how important Harry Potter is to the rest of us."

"Well, he doesn't know it either. I met him on the train yesterday, and he said he was muggle-raised—"

But there was a clatter as the two boys dropped their silverware and leaned toward her over the table, wide-eyed. "You met him?" the second boy said. "What's he like?"

She groaned to herself. "You know, you could just talk to him like a normal person." She finished her breakfast quickly after that, trying to avoid being asked any more such questions, and trudged up to her dormitory for her Transfiguration book. She considered taking the whole stack, but she decided she'd never be able to haul them back up the stairs that afternoon. Most of those books were pretty big.

She made it to Classroom 1B at a quarter to nine, easily the first student in the room, despite the difficulty of finding her way. There was only one fully enclosed connection between the residential and academic wings of the Castle, and it was down on the ground floor. Professor McGonagall was waiting for her when she arrived.

"Miss Granger. Good," McGonagall said. "I have your class schedule here. There was only one small difficulty. The third-year Arithmancy class conflicts with the first year Gryffindor-Ravenclaw History of Magic class. Now, I was able to remedy this by placing you in the Hufflepuff-Slytherin section, if that's alright with you."

"Oh—of, course, Professor," Hermione said quickly once she realised she'd been asked a question.

"Good. I've placed your name on Professor Binns's roster for that section. However, as Professor Binns's memory for anything that's happened since 1954 is questionable on the best of days, it is possible that he may fail to recognise you or call on you. Please see me if you have any problems with him."

Well, that was reassuring. "Yes, ma'am."

"Here is your schedule Miss Granger. I'll also write you a pass for Professor Quirrell's class so you have time to get your other books after this."

"Thank you, ma'am." Hermione took her seat and looked over the slip of parchment. She had Defence, Herbology, and History today—she would need her gloves as well as her books—and Charms and Arithmancy tomorrow. She balked when she saw Thursday: Astronomy ended at two in the morning, and she had to be up in time for Charms at ten-thirty. Then Potions only met on Friday for some reason, though the word on Professor Snape didn't exactly have her looking forward to that one.

The other students began to file in, her fellow Gryffindors and the Hufflepuffs. Hannah and Susan waved to her as they took their seats. Ron and Harry were the last to arrive, running in and just barely beating the first bell.

Then Professor McGonagall stood and addressed the class: "Excellent. Five points to Gryffindor and five points to Hufflepuff for everyone getting to their first class on time." She must have a very good memory to rely on a head count and skip calling the roll after having only seen most of them at the Sorting, Hermione thought.

"Transfiguration is some of the most complex and dangerous magic you will learn at Hogwarts," she said. "Anyone messing around in my class will leave and not come back. You have been warned.

"It is extremely important that you understand the fundamental rules governing transfiguration because it is easier here than in most branches of magic for something to go wrong. I want you all to write this clearly at the top of your notes and memorise it." She wrote in large letters on the blackboard: Some transfiguration is permanent, and some is not.

"Some transfiguration is permanent, and some is not," she repeated aloud.

Hermione nodded to herself as she wrote the words at the top of her notes. This had been explained in graphic detail in the first chapter of A Beginner's Guide to Transfiguration, but it was serious enough to require a strong reminder.

"For example," McGonagall said, "this is not a permanent transfiguration." She waved her wand over her desk, and the desk changed into a pig. The class cheered and some of them laughed, but McGonagall held up her hand. They instantly fell silent again. "This is not a permanent transfiguration," she repeated. "Left alone, it would change back into a desk within a few hours. Nor is it, in strictest terms, a pig, but only a magical construct that looks and behaves like a pig—and that only as well as my own knowledge of pigs allows. For any transfiguration, permanent or not, the result is only as good as the image in your mind."

She changed the pig-construct back to a desk and picked up the chalk again, writing a number one under the first line. "Now, under this line, you will write the most important safety rules of transfiguration. First, transfigured food is not edible. Say that with me, please…Transfigured food is not edible. You are not attempt to transfigure any food in this class or outside of it. We will not be covering that topic.

"In most cases, it will be obvious that transfigured food is not edible from the taste and smell, which is why this is not an even more dire rule than it is already. But transfigured food may look right; it may smell right; and if it is done extremely well, it may even taste right, but it is not edible. It won't kill you unless it was transfigured from something poisonous, but it will make you quite ill, and we don't want to have any of that.

"Second," she continued, writing the next line on the blackboard, "transfigured clothing will change back at the worst possible time." This prompted some giggles from the class, but McGonagall remained stern. "It may sound funny now, but you don't want to have to worry about your robes untransfiguring themselves when you're outdoors in the middle of winter. You are not to attempt to mend or modify any clothing using transfiguration in this class unless you are specifically instructed to do so.

"Third, transfiguring money is illegal and will rarely fool anyone. I shouldn't even have to tell you that one. Most people can't make transfigured gold or silver last long enough to fool anyone anyway, but there always seems to be one person in Professor Dumbledore's Alchemy class who doesn't get the message. It is illegal, and the goblins in particular do not take kindly to it." Some members of the class shuddered, having seen the goblins in action at Gringotts.

"Fourth, and most importantly, human transfiguration should never be attempted below N.E.W.T. level. Again, human transfiguration should never be attempted below N.E.W.T. level. Transfiguring any living subject is more difficult than normal, and human transfiguration is particularly dangerous when it goes wrong, although it is usually reversible. You will not attempt to transfigure any living subject unless specifically instructed, and you will not attempt human transfiguration on yourself or anyone else, even if instructed. Do I make myself clear?"

"Yes, Professor." To their credit, every student answered. Minerva McGonagall's presence was that commanding, especially when she was talking about things that were usually reversible.

"We will revise these rules several times over the course of the year, and we will study in depth the consequences should they be broken. I expect all of you to keep them most scrupulously. I have not had any students permanently injured by transfiguration during my tenure at Hogwarts, and I do not intend to break that streak.

"Today, then, we will begin with the free transfiguration of a matchstick into a needle. This is not the easiest transfiguration, even with its small scale, because any kind of metal is difficult, but I believe in laying the groundwork early for the more conceptually challenging forms of transfiguration later on…"

Professor McGonagall spent most of the period explaining in great detail exactly how transfiguration worked on a fundamental magical level, the silent incantation for free transfiguration and how it related to the actual transformation, and the mental concepts of changing form and substance that were required to be held in mind to make it work. Most of the class seemed disappointed and impatient to get to the actual wandwork, but Hermione found it fascinating. It seemed to mesh together bits of Arithmancy, a dash of muggle chemistry, a surprisingly large dose of Platonic philosophy, and practical magical instruction. Granted, most of it was in Magical Theory, but she thought Professor McGonagall did a better job of explaining it.

With about half an hour left in the class, she finally handed out the matches. Free transfiguration was indeed very difficult, not like the simple charms Hermione had tried. She worked very hard, applying the silent incantation while focusing on the mental forms, just as McGonagall had said. By the end of class, she was dismayed to see she was only halfway there; she had succeeded in transfiguring her match into a metal toothpick, but not a needle. But she swelled with pride when McGonagall actually smiled at her and showed her results to the class. It was only then that she noticed that the other nineteen students all still had nothing but matches.

Hermione rushed down to the ground floor to get back to the West Wing and then up the fourteen flights of stairs to her dorm to grab the rest of her books and got to Defence class only a few minutes late. Her legs felt like lead by the end of it, though, and she was panting like she'd just run about a mile. She could tell it was going to be a long year. At this rate, she was worried might need to see Madam Pomfrey for muscle strain before the week was out, if there was even anything that could be done for that. She could see it was wearing on some of her classmates, too.

Meanwhile, Defence class itself meant sitting in a sickeningly thick odour of garlic while listening to Professor Quirrell stutter his way through basic principles of jinxes. Hermione had a very hard time giving him his due respect as a professor, since it looked like he wouldn't be able to defend himself from so much as a swarm of pixies if he had to.

After lunch was Herbology. Hermione had been dreading Herbology more than any of her other classes since she was decidedly not the outdoorsy type. But she soon found there were advantages to a more hands-on class. Professor Sprout believed in taking fewer notes, since One Thousand Magical Herbs and Fungi was about as good a reference as anyone could hope for, and instead focusing on the practical side of growing and working with plants. First year would mainly teach identification of common plants and the basic principles of gardening, primarily with non-magical plants—or at least plants that were known to muggles—since an awful lot of them tended to find their way into potions. Hermione quickly learnt that being able to get some fresh air and work with her hands gave her a chance to relax and partially disengage her overactive, analytical mind. It wasn't what she was used to, but she enjoyed it much more than she expected.

History of Magic, on the other hand, was no better than Defence. Professor Binns successfully called her name once on the role and then called her "Miss Grant" every other time he addressed her, presumably after someone he had taught in 1954. She'd have to keep an eye out to make sure her grades got in correctly. He began lecturing at the very beginning—the earliest known evidence of magic: stone circles unearthed in Turkey that were estimated to be ten thousand years old, twice as old as Stonehenge. Hermione had enjoyed A History of Magic, but Professor Binns was so mind-numbingly dull that she could barely stay awake. The Hufflepuffs in the class could barely stay awake, too, and the Slytherins were too hypnotised to make any trouble, for which she was grateful. She was a little disappointed in the castle ghosts in general. It seemed like only Sir Nicholas and the Fat Friar were the only ones who were really talkative.

She spent the rest of the afternoon checking out the library, partly for its own sake and partly so she wouldn't have to go all the way back up to Gryffindor Tower. The library was certainly impressive, and why not? Since there didn't seem to be any public libraries in the magical world, and the largest collection of scholars was right here, why shouldn't Hogwarts have the most books, too? But even so, there were thousands of them! She did some quick figuring and decided that the Hogwarts library must have a majority of all the books on magic ever written in Britain, and there was quite a selection from Canada, the United States, and Australia, as well a substantial foreign language section. She was pleased to see there were also extra copies of all the textbooks for all seven years.

Hermione ate diner mostly in silence that night. Percy Weasley was already engrossed in his O.W.L. classes. He boasted about taking twelve O.W.L.s this year, which surprised Hermione since she thought one could only take ten courses. Percy was suspiciously evasive when she asked him how he could schedule that. Fred and George also said hello to her, but she was trying to keep her distance from them, having heard of their reputation by now. Her fellow first years made small talk about their classes and complained about how much homework they had already. That was a lot like her secondary school all over again. She'd have to try to find some studious Ravenclaws to hang out with if she wanted to converse with people more on her level.

In the evening, she sat in the Common Room and read up for the next day for a while before going to bed. She tried to talk to her roommates some, but Sally-Anne was already falling asleep, Lily was busy doing her hair, and Parvati and Lavender, easily the chattiest girls in the room, were far more preoccupied with things like clothes, gossip, and Quidditch than schoolwork, so she gave up on that pretty quickly.

"Parvati Patil?"

"Present."

"Padma Patil?"

"Present."

"Sally-Anne Perks?"

"Present."

"Harry Pot—EEK!"

Professor Flitwick dropped the class roster and swung his arms in little circles as he lost his balance and toppled off the pile of books he was standing on with a thud. A few people laughed, but Hermione, among others, was concerned until she saw his hands appear on top of the desk and he climbed back to his feet.

"Ah, terribly sorry about that, class," he squeaked. "Now, then, Mr. Potter?"

"Uh, present, sir," said the shy, dark-haired boy.

Professor Flitwick was obviously part goblin with his wrinkly bald head and bushy white beard. And at about three-foot-six, he was small even by goblin standards, but he was definitely very knowledgeable about his subject—the rumour was he had been a world-class duelling champion back in the day and had earned a Doctor of Sorcery in Charms.

Much like Professor McGonagall, he began with a long lecture on the theory of Charms. Transfiguration may have been difficult and dangerous, but Charms wasn't exactly easy either. You had to say exactly the right words with the right inflection and rhythm, with the correct wand movement and a clear picture of the spell in your mind. The slightest slip of the tongue could produce disastrous and unpredictable results, although the magic became more forgiving with experience.

Unlike Professor McGonagall, however, Professor Flitwick believed in starting with the very simplest spells and elements of wand handling. When it came time to start the practical lesson, he told the class to get out their wands and simply wave them to produce uncontrolled sparks. That was a lot of fun. Everyone's sparks were different colours, and sparks were flying all over the classroom. Some were only one colour, while some were more than one, like Harry's red and gold, and Morag MacDougal produced an entire rainbow. Neville Longbottom seemed to need to use a lot of effort to produce any sparks. He didn't get very many, and the ones he did were a horribly clashing purple and orange.

Professor Flitwick then taught them the incantation to force their wands to produce white sparks, Argentious. This, Hermione deduced, was to help learn how to control their raw magical energy to cast the Lumos charm, which was the first spell taught in The Standard Book of Spells (Grade 1). Hermione's sparks were pretty close to white to start with, so the charm came easily to her. Professor Flitwick praised her work to the class and said that her natural colour showed a high degree of magical control.

After lunch, though, came the class she had most been waiting for: Arithmancy. She wasn't intimidated by going to a class with students two years older than her; she had sat in on the trigonometry class for a while last year, although the pace had wound up being too slow for her. But she did turn some heads as she walked confidently into the classroom.

Professor Vector was already seated at her desk, and a few students from all four houses sat around the room. A tall and, if she did say so, handsome boy in Hufflepuff robes sitting near the door turned toward her as she entered. "Can I help you?" he asked. "What classroom are you looking for?"

"This one," Hermione answered. The boy's eyebrows shot up, as did some others.

"Miss Granger will be joining our class this year, Mr…Diggory, is it," Vector explained, checking her roster for his name.

"What?" spat a black-haired boy in Slytherin robes. "She can't be in this class, she's just a m—firstie."

"Mr. Montague, I have interviewed Miss Granger personally, and I am confident that she can handle the material," Vector defended her. "And you will show appropriate respect."

Montague settled in, grumbling, and Hermione smiled a little. With only Professors Snape and Vector and Madam Hooch from Slytherin among the staff, that house really didn't get as much internal control as it needed. Professor Vector, it seemed, made sure to do her part.

Hermione sat down next to Alicia Spinnet, the reserve Quidditch chaser she had met at breakfast. About a minute later, a Ravenclaw wearing his hair long came in and sat on her other side. He looked past her and gave Alicia a questioning look.

"Hermione Granger," Alicia whispered. "She tested in. Hermione, this is Roger Davies. He's on the Ravenclaw team." They nodded in acknowledgement to each other just before Professor Vector stood to call the role. The class was nearly half Ravenclaws, with only three Gryffindors besides herself. With a few notable exceptions, most Gryffindors didn't seem to be the analytic type.

"Welcome to Arithmancy class," she told them. "Arithmancy is perhaps the most analytical and challenging branch of magic. It is not for those who are looking for an easy O. But it is also useful in ways that other forms of magic cannot hope to match. Most modern spellcrafting and even some potions innovations are done by arithmancy. It forms the foundation for curse-breaking, magic reversal, spell detection, and spell analysis. It is an integral part of alchemy and advanced astronomy, and an excellent aid to improve rune-based magic and wards. It is also used for statistical prognostication, which, in my humble opinion, is far more reliable than any other form of divination because it can be backed up with hard numbers, although I'm sure Professor Trelawney would disagree." Everyone but Hermione laughed.

"We will mainly focus on arithmantic prognostication this year, since that requires the least complex mathematics, but we will also cover the mathematical and magical foundations of spell analysis and spellcrafting. These things can be quite tricky and must be done with great care, so I expect a full and focused effort from each of you. In particular, there is to be no fooling around with untested spells. Doing that is far more likely to land you in the hospital wing than anything else." Everyone nodded. Professor Vector's reputation regarding fooling around was similar to Professor McGonagall's.

"I would like to begin the class with a short quiz," Professor Vector continued. She began to hand out pieces of parchment. "You will not be graded on it, and, indeed, most of you will probably not be able to finish it. It is merely to ascertain your level of mathematical instruction. I do this because students often come into this class with very different backgrounds in the subject. In fact, I've noticed that maths is the one area in which muggle schooling consistently outstrips our own," she added, looking pointedly at the Montague boy who had called Hermione out. "Please begin. You have ten minutes."

Hermione looked down at the parchment and saw that the quiz, unsurprisingly, was just arithmetic and basic algebra. She could do this in her sleep—literally. She'd had dreams about more complex maths than this. Even checking her answers, she was the first to put her quill down after less than five minutes. There were some snickers from people behind her who thought the little firstie had given up, but Alicia and Roger weren't laughing. They could see that she'd at least written something.

Professor Vector collected the quizzes and gave them a quick glance over, clearly just seeing how far people had got on them. She paused over one of them and scanned it from top to bottom, checking it against an answer sheet. Then she smiled at Hermione knowingly before continuing on, prompting some surprised and confused looks from the rest of the class.

"Good. It looks like you all have a solid foundation in arithmetic, so we can advance straight into numerology and probability. Later, we'll get into algebra and geometry. The O.W.L. exam in Arithmancy requires proficiency in these topics as components of basic spellcrafting and spell reversal. In fact, the maths portion of the exam is remarkably similar to muggle O-levels, or whatever they're called now. Should you choose to continue at N.E.W.T. level, we will study more advanced spellcrafting, an introduction to curse-breaking, and the equivalent of muggle A-level maths, that is, trigonometry and calculus.

"For a first look today, we will study the magical properties of the number seven and the ways in which sevens appear more often in magic than you would otherwise expect…"

Hermione found the first lecture not to be very interesting, but sitting through a couple weeks of tossing numbers around would be worth it when they got to the actual magic. She quickly concluded that Professor Vector was more subtle than any of her other professors in the way she interacted with her students, but they seemed to be on the same wavelength. She never mentioned her score on the quiz, but Hermione was certain that if it had been anything less than perfect, she would have received a slightly sadder smile.

"No, really, Cedric, she wrote the answers in like two minutes," she heard Roger Davies say to the Diggory boy on their way out. "Wait, there she is. Hey, Granger, when did you learn to do maths like that?" he asked.

Hermione blushed slightly as she tried to think back to her chaotic lesson schedules. "Um…that maths in particular? It must have been…two and a half years ago."

Cedric, Roger, and Alicia, who was standing nearby, were gobsmacked. Cedric recovered first: "I can see why Professor Vector let you in the class…Miss Granger, would you like to join our study group? You'll probably need some help with the magic, since there will be third year spells, and you could help us with the maths."

Hermione was tempted to point out that she wasn't going to do their homework for them, but Roger and Cedric, at least, seemed pretty bright about that sort of thing from the way they'd answered Professor Vector's questions in the lecture. "Yes, I'd like that. Thank you," she said.

"Great. We were planning on meeting in the library on Mondays and Wednesdays after classes end—oh, but when do your Flying Lessons start?"

"I don't know. They haven't mentioned them." Hermione was not looking forward to that class.

"They'll be Thursdays for Gryffindors, unless they've changed it," Alicia said.

"Well, that's fine then," Cedric said. "So, tomorrow afternoon?"

"Sure. I'll be there."

"Hey, Hermione," Parvati said when they made it up to their dorm that night. "Why weren't you in History class today?"

"Oh, I'm taking History with the Hufflepuffs and Slytherins. I have Arithmancy Tuesday and Thursday."

"Arithmancy? I didn't know you could take that in first year."

"Well, I met Professor Vector this summer and tested into third year."

"Wow, that's great. I was a little worried. I didn't think you'd be one to skip class."

"Ooh, did you meet Cedric Diggory? I think he's in that class," Lavender interrupted.

"Yeah…he asked me to join his study group."

Parvati and Lavender both squealed loudly.

"Oh my God, what's he like?"

"Is he really cute?"

"Uh—yeah, I guess. But he's really nice, too, and he seemed pretty smart."

"Hermione, that's awesome," Lavender gushed. "Do you know if he's single?"

"I don't know! We only talked about maths!" Hermione said quickly.

"Padma said she thinks he's got his eye on Cho Chang in second year," Parvati replied.

"Ooh, tell me everything!"

The two of them soon devolved into a conversation about who was allegedly dating whom. Hermione rolled her eyes and went back to her books. Some girls, she thought.