Disclaimer: JK Rowling owns Harry Potter, and no one owns mathematics.

A/N: All maths in this story is real and correct to the best of my knowledge. Hermione's seemingly superhuman maths skills are based on real child prodigies. All arithmancy in this story besides the tiny amount mentioned in the books is stuff I made up. This story covers Years 1-4. The sequel, Lady Archimedes is in progress and will cover years 5-7.

I've seen a number of stories where some character, usually Harry, mysteriously has superhumanly good skills with Ancient Runes and warding, but I've yet to see anything comparable for Arithmancy and spellcrafting, even though this idea is as rich, if not more so with possibilities. So I decided to try it myself, changing Hermione from a bookworm to a maths whiz and giving her skills equal to some of the most gifted real child prodigies in the world, so that she can explore the true potential of Arithmancy. Yes, there will be maths, but my intent is that you won't need to understand the maths to enjoy the story. The important part is all the advanced spellcrafting Hermione will eventually be doing, and trust me, it will be epic.

A quick note on the characterisation: many reviewers have complained that Hermione is too weak, a pushover, too submissive to bullies, etc. in this story. In answer, I say that this was deliberate, but it changes dramatically over her first three years, most especially in third year. Unlike Harry, Hermione comes from an easy life. She has a loving family and is financially well-off. And that means from a narrative standpoint, she needs to go through trials and tribulations to match Harry's (many of which would be plausible behind the scenes in canon) to forge her into the strong, confident, powerful witch we see by the end of this story. So if you have a problem with how she is acting, I urge you to stick it out to the end of third year. I think you'll like the results.

Fair warning: this story includes some Harry/Ginny towards the end and Hermione/George in the sequel. Personally, I'd prefer not to spoil the pairings at the beginning, but that seems to be the etiquette here.

Quick Reference

First Year: Chapters 1-22

Second Year: Chapters 23-38

Third Year: Chapters 39-59

Fourth Year: Chapters 60-84

Years 5-7: in the sequel, Lady Archimedes

Chapter 1

Hermione Granger had always been a very bright child. No, she wasn't go-to-university-at-age-ten material like those kids she saw on the news every so often, but she had skipped a year in primary with no trouble, although she had a September birthday anyway, so it hadn't been that big an adjustment.

Ah, but maths, that was where she excelled. Oh, she loved books and all kinds of learning, else she could never have skipped a year, but numbers were her first love. She lived and breathed them from the time she first learnt to count, and her obsession had only grown with the passing years.

Multiplying large numbers in her head was trivial. Anyone could learn that if only they would bother to take the time. No, being able to do anything that a four-function calculator could do, and often as fast, only made her a curiosity, not a prodigy. But that was just the start. As the years went by, private tutors began pulling her aside each day during maths lessons to teach her long division, then probability, then algebra, then trigonometry. She knew by the time she turned six that she was on a different plane entirely from her year-mates when it came to numbers, and for all the trouble it caused for her, a part of her enjoyed the stares she got when people saw her happily working problems in GCSE and A-level maths while all the other children were learning their fractions.

Actually, this past year hadn't been so bad. Since she had skipped Year Four, she had advanced to secondary school a year early, and the kids there were at least a little more dedicated to their schoolwork than in primary school, where she had never really fit in and spent more time reading and doing sums than playing tag or, heaven forbid, dodge-ball. She'd even found a couple of girls at secondary school who shared her love of epic fantasy and science fiction, although she had to go all the way up to the sixth-formers to find anyone who could keep up with her in maths. Still, after some rough years in primary school, things were really looking up.

Then, Hermione's world was turned upside-down when a woman who was dressed like she had stepped out of a 1930s film showed up at the Grangers' door. Their hotel room door.

"Mr. and Mrs. Granger?"

"How do you do? My name is Minerva McGonagall." She offered her hand to shake. "Is your daughter here with you?"

Dan only lightly shook her hand. "Yes, she is. Is there a problem?"

"Not at all, Mr. Granger. You see, I represent an exclusive school in Scotland for gifted children, and we would like to extend an invitation to a Miss Hermione Granger to attend."

"Really?" Minerva caught a glimpse of a head of bushy brown hair as the girl in question leapt from her seat and ran to the door, only to be blocked by her parents.

"Now hold on there, Hermione," her father said, then to their visitor, "So you tracked us down on our holiday out of the country just to invite her?"

Minerva had been wondering about that herself. She was still a bit queasy from the international portkey she'd had to take to get to Italy, when she could surely have waited a week and caught them in Britain. Still, procedure was procedure. This was the hardest part, though: convincing them to let her come in and that she wasn't a—what had that Dame Finch-Fletchley called her? "A wandering lunatic," she believed it was.

"My apologies for interrupting your holiday," she continued, "but it is our standard practice to contact all of our scholarship recipients in the final week of July, regardless of where they are staying at the time. Here is my card." She handed over a muggle business card that said "Hogwarts School. Minerva McGonagall: Deputy Headmistress" and showed a muggle post address while in public. "If it is inconvenient, I can come back another time."

"No, no," Dan said quickly. The card at least looked somewhat verifiable. If it was true, they might as well listen to her, and if there was any funny business, it was best to get it out of the way quickly. "Please do come in." He turned around and subtly motioned for Emma and Hermione to stay back from the strange woman and offered her a seat. He sat in between them and McGonagall—and by the room phone—just in case the strange woman tried anything.

"Ms. McGonagall, my name is Daniel Granger, this is my wife, Emma, and you seem to know our daughter, Hermione," he said. "I'm sure you can appreciate how unusual this seems to us, but I suppose we might as well hear you out."

"I appreciate that, Mr. Granger."

Emma examined the business card. "I don't think I've ever heard of a…Hogwarts School, Ms. McGonagall," she said. "Do you have any actual documentation with you?"

"I do, Mrs. Granger," Minerva answered. "You would not have heard of it before because the full name of the school is Hogwarts School of Witchcraft and Wizardry."

Dan's and Emma's mouths hung open, but Hermione spoke up for the first time: "Witchcraft and wizardry? Do you mean like magic—real magic?"

"Yes, Miss Granger, magic. I am a witch, and our spells have told us that you are as well."

Hermione's eye grew wide at the revelation, but her father's narrowed. "Ms. McGonagall, I think you've said enough."

"Please, Mr. and Mrs. Granger, I know this sounds difficult to believe, but if you'll permit me to give you a demonstration…"

The strange woman pulled a small stick from the sleeve of her dress.

"A magic wand?" Dan said sceptically. His sense that this was a prank or a scam rose even higher. He knew enough about supposed psychics and paranormal practitioners to know the usual tricks.

"Quite so. Perhaps a simple Levitation Charm?"

No doubt to be performed with an invisible thread, Dan thought. But there was one thing that didn't add up. Just where was this McGonagall woman planning to go with this. She couldn't fake an entire school. If it was a school for stage magicians or something like that, why try to recruit Hermione, who had never shown any interest in the art? And if it was a prank, who and why? And if it wasn't either of those…well, Daniel Granger was nothing if not curious. Hermione hadn't got it all from her mother, after all. In the unlikely event that McGonagall could do anything besides make a playing card spin in midair or turn her wand into a bouquet of flowers, it really would be worth hearing. "Alright, then, let's see it," he said.

McGonagall waved her stick and muttered something in what sounded like Dog Latin, and then the coffee table in middle of them rose into the air.

"Holy…!" Dan fell right out of his chair. That was a lot more than a playing card. Emma paled at the sight, and Hermione gasped in surprise. He ran his hands all around the table, over and under, looking for wires or invisible supports. "But how…?"

"As I said, Mr. Granger, magic." The table spun around twice in the air and then settled back down to the floor.

"Wow…" Hermione said.

"Magic…" Emma stammered. "Magic…So…Hogwarts School of…Witchcraft and Wizardry, you said?" Emma stammered.

"That's right," McGonagall answered. "I am Deputy Headmistress of the school as well as Professor of Transfiguration."

"Spells to change one thing into another. For example…" She touched her wand to the coffee table and muttered another incantation. The table turned into a large tortoise.

Dan very nearly fell off his chair again. That was definitely no trick. He pinched himself. Not dreaming, either. Hermione actually applauded the sight.

Once McGonagall restored the coffee table, Emma said, "So you're saying Hermione can do those things, too?"

"She will be able to—with appropriate training, of course. That was a very advanced spell."

"But how could you possibly know that she's a…a witch?" The word sounded insulting, but it was hardly the strangest part about this whole thing.

"We have very ancient spells in operation at Hogwarts that automatically detect all magical births within the shores of Britain, Mrs. Granger, but surely you have noticed yourselves…unusual things happening around your daughter—what we call 'accidental magic'."

Dan and Emma looked at each other, and they both instantly knew the answer. It certainly explained a few things, including one particularly nasty temper tantrum in which a whole room full of books had come flying off their shelves. They just nodded, but Dan started up again: "Well, if that's the case, why wait until this school of yours starts—I assume it starts at eleven or twelve? Why not contact us earlier?"

That was another issue that always seemed to come up. McGonagall tried to explain it gently and hoped the conversation wouldn't devolve into a political argument: "Hogwarts accepts all magical children who have reached the age of eleven and makes its first contact with them the summer before their first year because children rarely have much ability to control their magic before age eleven. Because of this, for children with no magical relatives, then, there is very little reason for them to have contact with the magical world. I admit there are drawbacks to our system, but as you can probably guess, the magical world values its secrecy, so we choose not to make contact until later."

This seemed to placate the parents for the moment. But as she watched her parents digest this information, Hermione was just getting started. She held out her hand towards McGonagall and said, "Please, ma'am, may I try a spell?"

McGonagall chuckled at the girl. That particular question was surprisingly rare, even from muggle-borns. "No, I'm afraid not, Miss Granger," she said. "A wand must be specifically suited to your own innate magic, or you will find it very difficult to use, especially at your age. You will need to purchase your own wand before the term begins. However, I can give you your official Hogwarts acceptance letter." She pulled an envelope out of her handbag and levitated it over to the girl.

Hermione was almost entranced by the old-fashioned envelope. It was made of parchment, she noted, and addressed with flowing script in emerald-green ink. She took it in hand gingerly and read the front aloud:

Miss H. Granger

The Smaller Bedroom

Suite 405

Hotel San Zulian

Venice, Italy

Dan shot to his feet. "How did you know what room our daughter was sleeping in?"

Minerva sighed. This was happening more and more every year. Perhaps it was time to change the addressing spells. "I did not know, Mr. Granger," she said. "The letters are addressed automatically."

"So you can just automatically find anyone wherever they are?"

"For the most part, yes. There are ways of concealing one's movements when they are called for, but that's hardly the concern of a student. In any case, tracking charms like that are strictly regulated."

Grumbling, but mollified for the moment, Dan sat back down. Hermione broke the ornate wax seal and slid the letter out of the envelope. She read:

HOGWARTS SCHOOL of WITCHCRAFT and WIZARDRY

(Order of Merlin, First Class, Grand Sorc., Chf. Warlock,

Supreme Mugwump, International Confed. of Wizards)

Dear Miss Granger,

We are pleased to inform you that you have been accepted at Hogwarts School of Witchcraft and Wizardry. Please find enclosed a list of all necessary books and equipment.

Term begins on 1 September. There will be an orientation for incoming students of non-magical parents held at 9:00 AM on 27 July at Platform 10, King's Cross Station, London, at which time you will be able to submit your enrolment forms.

Yours sincerely,

Minerva McGonagall

"That is the beginning of the orientation, Mrs. Granger. The itinerary will include a tour of important magical sites in London, as well as time to purchase school supplies." McGonagall dug into her handbag again, raising eyebrows when her hand dove in deeper than the size of the bag, and pulled out a large brochure. "I have some basic literature on Hogwarts here for you: classes offered, extracurricular activities, profiles of the professors, and so forth."

"Do witches actually wear pointed hats?" Hermione exclaimed as she read the lists of school supplies.

"I admit they are falling out of favour, but they are still part of the uniform, yes."

"Um, we may have a problem," Dan said as he reread Hermione's acceptance letter for himself. "We were planning on flying home on the 27th. Is it possible to do the orientation later?"

McGonagall blinked and adjusted her spectacles. It had been a while since something like this had happened. She considered her schedule and said, "If you like, I can meet with you personally on the following weekend. Or, if you prefer, I can arrange to have your flight changed to a day earlier at no cost to yourselves."

"You can ar…arrange…? What do you mean? It's pretty hard to change flights at the last minute."

With a barely-discernible hmph, the witch answered, "Well, I suppose it might be considered bending the rules a bit by muggle—that is non-magical standards, but I assure you that we can do it quite easily—and above board."

Emma wore a resigned look: "Well, then…I suppose we won't have any trouble going back a day early. And I doubt we'll be able to keep Hermione from trying to get a hold of these magical textbooks for another week."

Mr. Granger smirked at that remark, and a gleam flittered through Hermione's eyes. McGonagall couldn't help but think, Ravenclaw for sure, this one.

But her thoughts were interrupted when the girl spoke up again: "Excuse me, ma'am, but none of these look like books for non-magical classes at all."

"Hmm, she's right," Emma said, thumbing through the brochure again. "There's no English classes, no—what did you call it? Muggle history? And not any kind of sciences or maths that I've heard of, unless it's this Muggle Studies thing."

"You've just got to have maths classes," Hermione insisted.

"Yes, just for starters," Emma agreed, although Hermione was sure to need maths instruction far beyond what any secondary school could provide. "How is Hermione supposed to get a well-rounded education with classes so heavily biased toward magic?" She wondered to herself how she had got to the point of stringing those words together.

McGonagall held up a hand to stem the tide of questions. This was a common question, though not one to which she had an ideal answer: "Now, then, it's not quite as bad as it may look just from the course listings. Most of our classes have an essay component that is graded on language as well as content, so English is certainly not neglected. For maths, we offer an elective Arithmancy class beginning in third year that covers much of the same maths as muggle secondary school. I will admit that our curriculum is light on muggle sciences and history. We must cater to the needs of our students, and nearly all of our students, including muggle-borns—those from non-magical families—choose to live in our world. Our curriculum is designed to help our students succeed there." She held up her hand again for the obvious next question. "You can of course, hire tutors to cover the remaining subjects, and magical education is only compulsory up through the fifth year. Very rarely, we do have students leave after their fifth year to take an early apprenticeship or prepare for a muggle university. If you like, I can put you in contact with some former students who have done so."

Both Granger parents leaned back, digesting this information, though Hermione was still looking over the brochure and frowning.

"Well, I suppose all that does make a kind of sense," Emma said. "At least she would keep her options open. What do you think, Dan?"

Dan stared at the ceiling. "I think I'm having the strangest dream of my life, and I'd like to wake up, now, please," he mumbled. "But on the off chance this is real, I can't see why it would hurt to go home early and go to the orientation. Hermione, what's wrong?" he added when he saw his daughter's face.

"You don't have any maths the first two years?" she said. "Do you think I could I test into the Arithmancy class, Professor?"

"Wha…? Test into…?" McGonagall sputtered. "Well, that would be highly irregular. And I'm afraid that few students could do well in that class at your age, even clearly gifted ones like yourself."

"Oh, you haven't seen our Hermione around numbers, yet, Professor McGonagall," Emma defended her daughter.

"I could show you what I've been working on," the girl said, and before McGonagall could protest, she had disappeared in a flurry of brown hair and soon returned with a large book that she opened to a spot a little ways in and presented to her. "This is what Mr. Andrews has been teaching me this summer. I haven't got that far yet, though. I've only got up through differentiation of rational functions."

McGonagall's eyes narrowed in confusion, then slowly grew to the size of saucers as she began to thumb through the book. It was a muggle textbook, yes, but this was the kind of maths used in N.E.W.T.-level Arithmancy, and in much more detail. McGonagall had forgotten most of it, and she wondered if even Professor Vector knew all of it. "My goodness, you can really do this kind of work?"

"Of course." Hermione started to explain how to do one of the problems, but McGonagall cut her off.

"That's quite alright, Miss Granger, I believe you. Irregular it may be, but with maths skills like these, you could teach the non-magical part of the class. I'll ask Professor Vector if she is willing to interview you for a possible Arithmancy placement before you come to Hogwarts."

Hermione giggled at the name Vector. "Thank you, Professor."

With that crisis settled, McGonagall answered a few more of the family's questions about the school and gave them a brief overview of the magical world. She was sure they would need time to fully understand everything that had happened, just like all the muggle-born families, but she confirmed that they would be at the orientation and promised to contact them through the hotel with their new travel arrangements.

After she left the hotel room, she waited until she was out of earshot before allowing herself a heavy sigh and wondering why it was always the most studious children who caused her the most trouble on these visits.

Minerva McGonagall returned to Hogwarts after a long two days of visiting muggle-born families. As always, she was glad to be out of that muggle dress and back in proper witch's robes. But it was too late that night to bother reporting in that night, so she didn't head up to the Headmaster's office until after breakfast the next morning.

As usual, she didn't have to knock on Albus's door before he called out a hearty "Ah, do come in, Minerva." She entered the office and sat down among the many twittering contraptions (she had long suspected that most of them were completely useless, but Albus simply ignored any such comments about them).

"Sherbet lemon?" the Headmaster asked before popping one in his mouth himself.

"No, thank you."

"So, the visits with the muggle-born students took longer than usual I see."

"Yes, Albus. One of them turned out to be on holiday in Italy. I had to register an international portkey and rearrange a muggle aeroplane schedule to deal with her. Muggles travel so much these days, I'm beginning to think we should change our orientation procedures."

"Hmm, perhaps a consideration for next year. No other troubles, then?"

"No more than usual, although I'll need to talk to Septima about that one as well."

Albus's bushy eyebrows rose at that. "Septima? For a first-year student? Why would her involvement be needed?"

Minerva allowed herself a small smile. "Because if Miss Granger is as good as I think she is, Septima will want her for an apprentice before she's through with her."

Albus stroked his beard. "How intriguing," he mused. "Good arithmancers are hard to find."

"Indeed. So no problems here, then, Albus?"

"Only a spot of difficulty in contacting Harry Potter." At that moment, one of the devices on the walls chimed six times. Albus rose to inspect it. "Oh dear, it appears that all six letters I sent to Mr. Potter this morning just triggered as lost."

"Six!"

"Yes, this is the third day in a row. I'll have to arrange another post to him for tomorrow."

"Albus, if six letters couldn't get through to the boy today, I can't see how sending more will help. If I had to guess, I'd wager those awful relatives of his are keeping them from him."

"Now, now, Minerva, I left specific instructions with them…"

"Specific instructions my foot, Albus," Minerva cut him off. "I told you how awful those muggles were ten years ago. Perhaps I should visit the boy in person."

"No, Minerva, you're doing quite enough this week with the muggle-borns…I think that if Mr. Potter does not read his letter before his birthday, I'll send Hagrid to deliver it," Albus said with that characteristic twinkle in his eye. (Minerva had long ago decided that must be some kind of spell, but she couldn't for the life of her figure out what it was.) "I'm sure he'll be happy to see the boy again."

Minerva thought about those prim and proper muggles' likely reaction to the half-giant barging in on them and smiled in spite of herself. "Well, I suppose Hagrid is up to the task," she said.

The last month of summer was a whirlwind for Hermione. First, there was the orientation at King's Cross, where she met Sally-Anne Perks, Sophie Roper, Justin Finch-Fletchley, Terry Boot, Kevin Entwhistle, and the last-minute addition, Dean Thomas. Then, there was the wonder of Diagon Alley, buying her robes, her supplies, her wand (at her insistence, Mr. Ollivander had let her try a few simple spells in his shop), and of course, her textbooks.

All of the books were fascinating, of course. She was surprised to find that several of the textbooks covered multiple years, making them even more useful. She bought the Arithmancy textbooks all the way up through seventh year, and even in the first one, she was amazed at how something as smooth and organic as magic could be broken down mathematically. She devoured the history books, too, trying to learn as much as she could about her new world. Her respect for Albus Dumbledore shot up several notches when she learnt that he single-handedly defeated Hitler's dark wizard ally in World War II, but she was a little unnerved when Modern Magical History described a terrorist who sounded like a comic book supervillain who was defeated by a boy named Harry Potter only a decade ago. There had been a civil war in Magical Britain back then—against the muggle-borns. She was certainly glad that she didn't have to deal with that now.

In the meantime, Professor Vector had been nice enough to arrange a visit to her house to meet her a week after the orientation. At the appointed time, Emma Granger opened the front door to find a middle-aged woman with long black hair who was dressed in a flowing burgundy robe and a matching pointed hat, much like the strange dress of many of the people they had seen in Diagon Alley. Emma looked the woman up and down once before saying, "You must be Professor Vector."

Vector turned up her nose slightly and she answered haughtily, "And you must be Mrs. Granger. How do you do? I'm here for the interview with your daughter."

Emma felt vaguely annoyed by the woman's demeanour, but quickly invited her in and started some tea.

"Mr. Granger?" Vector shook Dan's hand.

"How do you do?"

"And you must be Hermione."

"Pleased to meet you, Professor Vector. Thank you for coming," the girl said politely.

Vector shook her hand stiffly. The child didn't look like much—all teeth and curls, not that appearances mattered. She was quite polite, though, and plainly excited, but to be honest, Vector wasn't expecting much. She was only doing this as a favour to Minerva. Despite what her colleague said, she found it hard to believe that an eleven-year-old could qualify that highly in Arithmancy.

They retired to the living room, where the family sat down, and Vector cautiously took a chair, eyeing the electric lights and the switched-off television curiously. Unlike far too many of her fellow Slytherins, she had no quarrel with muggles, and she recognised how often muggle-borns outperformed their peers academically (there was something to be said for muggle primary school), but she didn't think she would ever get over the culture shock each time she entered their world.

"Thank you for the tea," she told Emma. "Now, Miss Granger, I, of course, am Septima Vector, Professor of Arithmancy at Hogwarts. Professor McGonagall informed me that you are interested in testing into my third year Arithmancy class as a first year."

Hermione looked nervous as Vector gave her a stern look, but she tried to answer calmly. "Yes, ma'am. It sounds terribly interesting—and useful because it deals with spellcrafting, according to Numerology and Grammatica. And I really wouldn't want to go two years without a maths course. I want to keep it fresh in my memory."

Well, she was certainly dedicated, Vector thought, especially to have started Numerology and Grammatica already. "You should know that I have never before considered early placement for my classes," she said. "Arithmancy is a very rigorous subject—in my opinion, the most rigorous taught at Hogwarts, and I expect a full effort from all of my students, regardless of their age." Hermione started frowning. "However, Professor McGonagall informed me that your mathematical prowess is the finest she has ever seen from a first year, and she insisted that I take a look. So if you could show me what kind of maths you have been taught, it would give me an idea of your possible placement."

"Of course, ma'am. I've got my calculus book right here." Hermione jumped up and grabbed the thick textbook from one of the stacks on the side table. She didn't see Professor Vector twitch in surprise at the word "calculus". She opened the book to the right section and said, "My lesson this week is differentiation of compound functions. In principle, it's a very simple application of the Chain Rule. You just treat the inner function as a variable when taking the derivative of the outer function, then multiply it by the derivative of the inner function. Of course, with more complex functions, it can be very complicated—"

"Miss Granger," Vector cut off the enthusiastic child, "may I see that book?"

"Of course, ma'am." She handed it over, and Vector looked over the open pages. The description of the Chain Rule was correct, of course, and as she flipped to the previous pages, she was amazed to see the elements of calculus explained in such detail. The N.E.W.T.-level Arithmancy book was smaller than this one, and the maths parts only took up half of it. And the child certainly acted like she understood it, which would be astounding if true. Vector needed to see this for herself.

"Miss Granger, if I gave you an equation based on this material, would you be able to solve it for me?"

"Yes ma'am," she nodded emphatically.

"Very well, do you have any parchment?"

Emma rolled her eyes. The fact that the magical world seemed to be stuck in the nineteenth century had not escaped her. "No, but we have a pen and paper right here."

"Of course." Vector took the unfamiliar muggle writing implements. Holding a pen should have been the same as holding a quill, but it still felt a little awkward. She began writing a formula for the Granger girl to differentiate. Just to be sure, she made it a fiendishly complicated formula, one that seventh-years would struggle with, which required her to apply the Chain Rule twice, and on a rational function at that. A good student would try and probably get a mostly-correct answer. A fake would be forced to give up at once. "Mm-hmm. Very good," she said, handing the paper over. "Perhaps this one, then?"

Hermione paled when she saw the complex formula, but she set her face with a determined expression and got to work, leaning over the coffee table as she began figuring. Even watching it upside-down, Vector could see that the girl was serious, much to her surprise. She was definitely doing real algebra and what looked like real calculus. It took ten minutes of figuring, including checking her work twice and handing it back with a nervous look, but she finished it.

Vector looked over the paper. In neatly-written letters, Hermione had shown her work in great detail. The professor worked through each step herself, growing more and more excited as she found no mistakes. As she reached the end, she felt faint as her haughty, sceptical facade crumbled, and she was forced to conclude that the answer was correct. "Miss Granger," she said, "do you realise that the majority of my seventh-year students could not solve this equation correctly as quickly as you have just done?"

Hermione smiled nervously, unsure what to say, but her father jumped in and said, "That's our Hermione. She's been doing secondary-level maths for years, now."

"I should certainly say so," Vector said, any hint of superiority in her voice gone. "That is without a doubt the most extraordinary display of mathematical prowess I have ever seen from a first-year student." Hermione's smile grew broader. "Of course, arithmancy is more than just calculus. For example, have you learnt multiplication of matrices, Miss Granger?"

"Yes, ma'am. That was in my Algebra II class."

Vector didn't particularly know was "Algebra II" meant, but she took another sheet of paper and wrote down two grids of nine numbers side by side. "I see. So perhaps you could multiply these matrices together, then?"

"Yes ma'am." Hermione took the paper and started figuring.

Vector hadn't thought she could be more impressed with this child after the calculus display, but she was wrong. She knew full well that the problem she had given her required forty-five arithmetic operations, and as she watched, Hermione worked it out, entirely in her head, in forty-five seconds. It actually took Vector longer to check the answer than it had taken her to compute it.

"I don't know how you did that so fast, Miss Granger, but you are again correct…Well, then, I assume you have also learnt geometry and trigonometry?"

"Yes, ma'am."

"Excellent." Vector drew up a complex geometric figure for which some of the distances and angles had to be solved. It required the use of both trigonometry and the geometry of circles, but Hermione made short work of it, giving the exact answers in terms of square roots.

"Well, Miss Granger, I think your mathematical skills are above reproach." Vector was conversational now, although she was not about to compromise on quality. "However, if you place in my class, you will need to be adequately versed in the theory of magic as well. So, can you tell me why the wand motion is important when casting a charm?"

It was an easy question from Magical Theory, which Hermione had, of course, already read, and she repeated the book's explanation almost word-for-word: "The movement of the wand aligns the ambient magical energies with those embodied by the charm's intended effects. This alignment of energies makes the charm much easier to cast, especially for beginning magic users."

"Mm-hmm, very good…Something a little more difficult, then. What are the principle magical properties of the number seven?" That was straight out of chapter one of Numerology and Grammatica.

"Well, seven is the most magically powerful number. More spells include an arithmantic factor of seven than any other number, and in magical fields that include instances of sevenfold symmetry, the resonant energies often cause the magical effects to be more powerful and more stable. Numerologically, arranging objects in groups of seven can—"

"That's enough, Miss Granger, thank you. It's clear that you have read the course books very thoroughly. The important thing is that you can apply what you have learnt. For example, what is the geometric structure that describes the magical fields of the Lumos Charm?" That was one of the homework questions for chapter three.

"A sphere, ma'am."

"And why is that?"

"The Lumos Charm produces light by confining the uncontrolled magical energies that produce sparks around the tip of the wand. The most efficient shape of the confining field is a sphere."

Now that was the kind of magical intuition Vector was looking for. And she surprised herself that she really was hoping to find it. Here was one of the most extraordinary minds she had ever seen, and she really wanted to see what the girl could do. She asked a few more questions along these lines and then decided to see just how far the she had got in her studies. "Alright, one last question," she said. "What is the arithmantic difference between a jinx and a hex?"

Hermione paled, and she looked down at her feet. "I…I don't know, ma'am. I thought that jinxes were spells that were just irritating, and hexes were spells that were actually harmful…"

"That's quite alright, Miss Granger, this is actually a fifth-year topic. The answer is that while the definitions you will learn in Defence Class are roughly correct, jinxes are described using algebraic equations, while hexes, which are more powerful spells, are described using transcendental equations."

Hermione's analytical mind started spinning at the implications of this, but she filed them away for future reference as the professor smiled and continued speaking.

"I must say, Miss Granger, in my twenty years of teaching, I have never seen a child with a greater aptitude for arithmancy at your age. I will inform Professor McGonagall to add third-year Arithmancy to your schedule when you arrive at Hogwarts."

"Yes!" Hermione leapt to her feet and nearly tripped when she narrowly prevented herself from hugging the professor. Instead, she managed to restrain herself to shaking the witch's hand vigorously. "Thank you! Thank you, so much, Professor Vector. I won't let you down."

"No, Miss Granger, I'm sure you won't." Vector looked back to the girl's parents. Her father was beaming with pride, and her mother looked more than a little smug. She probably deserved that, she admitted. She gave the family a brief outline of the full five-year curriculum and answered a few more questions about Hogwarts and the wizarding world in general. She noted that Emma was quick to ask about career prospects, but of course she could say they were quite diverse for a skilled arithmancer.

As a bit of a courtesy (and a suddenly renewed curiosity), Vector asked a few questions about the muggle world, and in particular, what they used their advanced maths for. While she only understood about half of their answers, that half was impressive. The applications to muggle sciences were amazing, like that mission of sending people to the Moon that Professor Sinistra always raved about. Equally impressive was Hermione's knowledge of these endeavours, even if they weren't up to her level in pure maths.

When she finally took her leave, Vector shook her head and thought, That girl's going to have my job by the time she graduates.