"Some forty years ago Prof. Dee already had the reputation of being the greatest wizard of his time. He was approached by a man with an extraordinary request for assistance. This man was a layman but like many Englishmen of that time he was fascinated by such magical knowledge as had filtered down into books accessible to laypeople. He had in his possession something that was possibly unique, a container full of dust from another world. Of course by 'another world' I don't mean another planet. Any such place could be reached if one travelled long enough. I mean another world that co-exists with ours and cannot be reached by ordinary physical means. The man claimed that the dust was handed down from the Atlantean civilization. This is almost surely nonsensical, but I have no better knowledge of the dust's origin.

"Upon doing research Prof. Dee discovered that there was a process whereby the dust could be incorporated into rings that would transport the wearer into other worlds. He fashioned the rings for this amateur magician. Prof. Dee naively gave up the rings with no further efforts to follow events, despite having misgivings about the man's trustworthiness and wisdom. The man had given Dee a false name and address, for one thing. But Dee wanted to deal with him fairly, without trickery. We never heard from him again. For all we know, he could have used the rings, gone to another dimension, and never come back. Without evidence of the existence of more dust we thought we were at a dead end. Not surprisingly, we forgot about the incident and our thoughts turned to other matters.

"It was some years later that Prof. Dee met a man who claimed to be one of the great figures in magical history, no one less than Nicholas Flamel."

"The discoverer of the Philosopher's Stone?" exclaimed Mrs. Macready. "But he must have been over four hundred years old!"

"Indeed. One of the properties of the Stone is to extend life indefinitely. Flamel passed on the secret of the Philosopher's Stone to Prof. Dee. I believe he did so slyly, counting on Dee's lack of expertise in alchemy. You remember, of course, that the Stone was created through alchemy, of which Flamel was the undisputed master. The secret would be protected and preserved but Flamel wouldn't have to worry about the proliferation of Stones, or Stones falling into the wrong hands, because it was unlikely Dee, or anyone else, could master the techniques of its creation. The Professor confided to me, after many attempts, his failure to follow Flamel's footsteps.

"A lesser man might have put Flamel's secret away, a hidden treasure, but we were much younger men in those days and we felt we had all the time in the world to attack the big problems of magical theory. Prof. Dee had the brilliant thought that if all correct magical systems were equivalent, as the Grand Unification Theory says, then it should be possible to replicate any result of alchemy through the manipulation of symbols alone. Thus, Dee set himself the possibly even more difficult task of finding a system of spells that would create the Philosopher's Stone."

Ludovico noticed that Denis was looking perplexed. "Not to worry. These strands will tie together." He continued with his story. "Dee tried for many years to produce the Philosopher's Stone but always fell short. He created spells that ran on to more than a hundred pages, spells of mind-boggling complexity, yet there was always something missing in them. Now, a lesser magician might have given up, or worked away fruitlessly for the rest of his life, but Dee was no ordinary magician. He remembered the Theory of Incompleteness of the medieval scholars. Briefly stated, it is that the system of symbols as mapped out in Lavoisier's Tree of Symbols, so familiar to generations of young witches and wizards, is incomplete. Because our system of magic is incomplete there are spells that are impossible to write. Dee thought this was the explanation for his failure to create the Philosopher's Stone. Systems of magic are different in different worlds, although there could be considerable overlap. Therefore, there are magical symbols and connections between symbols that exist in other worlds but not in ours. This was a theory that the Ministry has long suppressed. It is whispered that the books in which it was written were systematically sought out by agents of the Ministry and destroyed. Today, we do not know how far the ancient scholars went in their speculations; we only have second-hand accounts."

"There's a point I'm confused about," objected Mrs. Macready. "Suppose magic from other worlds doesn't work properly in ours, and our magic isn't effective in other worlds. Then the missing parts, they can't be—well, effective."

"Yes, you're quite right. They don't have the status of the known part of the Tree; they're ghostlike, is the term I have used. They can be completely ignored for all the everyday magic that we do. Yet for spells of great complexity and subtlety they are necessary. The consequences may be few but they are very significant, indeed. Think of all those 'unbreakable' security spells used to protect our greatest valuables, or to lock away our criminals. Are they really unbreakable, or are there overlooked flaws?

"Dee knew, of course, that the theory was heretical. It undermined centuries of Ministry teaching. He did nothing to communicate his speculations. Only I and a select company of theoretically-minded wizards ever heard about it. Yet the ideas leaked out beyond our circle and spread rapidly if furtively. Soon, Dee was hearing that he was a champion of Incompleteness. The Ministry was in a quandary. To take actions to suppress the theory was to draw attention to it. Eventually there was no choice. This is the world that our younger generation has grown up in, one in which it is forbidden to discuss, study, write about Incompleteness or the other world problem."

Denis spoke up. "I can't believe all this fuss over a bit of theory. Surely it can't be so dangerous to the Ministry."

"I'm afraid I could cite many historical precedents in your own culture for violence, bloodshed, imprisonment and so forth over 'a bit of theory', as you put it. I suppose it's a universal human tendency to attach emotional values to their ideas, however abstract. The Ministry has always identified with the Tree of Symbols. It is the symbol they rally around, the way an army rallies around their nation's flag. They have always stood behind the idea that there is only one system of magic and it is complete and perfect. I could explain further but I will trust Mrs. Macready to satisfy your curiosity, or I will be speaking until dawn."

Mrs. Macready felt the tingle of an old memory run through her. She had read brief mentions of the Incompletists in the books she studied in the distant, dusty recesses of the school library. One day she decided to bring the matter up in her History of Magic class, but the teacher would not answer her questions. "Go to Prof. Dee," she said, "and see if he will tell you about them." Fine, she thought, and immediately stood up and walked out of the classroom, to the astonishment of the teacher and the other pupils.

The empty stone corridors of Gladhearts during class time had never seemed emptier, Mrs. Macready recalled. She found herself hurrying along, her heels clicking on the stones and echoing. Up two flights of wide wooden stairs with banisters polished by centuries of children, and it was a short corridor to the entrance of Prof. Dee's office, in one of Gladhearts' towers. The entrance at that time was like an intricate wooden puzzle, with pieces that folded out of the way to permit passage, once the password was uttered. All the children knew the password. It was 'licorice'.

Prof. Dee offered her a pickle from a small dish in his hand. "How may I help you?"

"Could you explain the Theory of Incompleteness to me? Prof. Austin, my History of Magic teacher won't."

"Ah, young lady, there's a reason we don't want to teach the Theory of Incompleteness. The theory is a speculation. If you want to make progress and do some really useful magic your education must be based on what is tried and true. Naturally, your mind is curious. This is a wonderful quality, but I advise you to concentrate on knowing the Tree thoroughly. It will unlock the universe to you."

It was a disappointing answer but she had accepted it at the time and did not raise the issue again. She remembered how she and the other students had idolized Prof. Dee. His example had fired her with ambition to become a scholar of magic. He never spoke condescendingly to the students. He was always honest and helpful. But she realized now how unsatisfactory and frustrating his answer was for the pursuit of genuine scholarship.

Ludovico's voice brought her back to the present. "Now, Mrs. Macready, you understand why I have made you an audience to this? I need your help. I am an old man and there is only so much I can accomplish by myself."

"What about the Professor himself?" she asked, but she could guess what the answer would be.

"Prof. Dee never had the nerve to be a revolutionary. He long ago turned his back on research in controversial areas. He is often called upon as a consultant to the Ministry. I suppose they pay him well for his time. He is honored and respected. There is even talk of him becoming Minister of Magic. Naturally, he does not communicate with an outcast and fugitive like myself."

Mrs. Macready asked, "But why me? I can't believe it was because I was a student of Prof. Dee's all those years ago, when I was a child."

"He often spoke of you as one of his most promising students. Did you know that? He was profoundly disappointed when you dropped out of magic altogether. He was delighted, of course, when he heard that you had joined the Bureau, but even then he thought you had too much talent for, may I say, routine enforcement work."

It gave Mrs. Macready a glow after all those years to hear that Dee had thought so highly of her. She really had had no idea. However she had a natural tendency to resist being persuaded. "Was that intended as flattery?"

"Every word was the truth. I know you are sworn to uphold the decrees of the Ministry, but you do not need to abandon your duties. We have common purpose. We need to find the rings that Dee fashioned. Only you, with the resources of the Bureau behind you, can do it. I have never known where to begin on a trail that is nearly forty years cold. Whoever obtains the rings can personally enter other worlds, not only proving their existence but gaining the opportunity to develop a more complete system of magic than what we now possess. That person stands to gain great power.

"Well, I have planted the seed of thought in your mind. I hope you will give it long and serious consideration."

"Why do you suppose I won't arrest you now? You've told me enough to have you put away for unlawful research and public communication."

"You and I are birds of a feather, as they say. You have a higher calling than your obligations to work. I am convinced that you would not try to arrest me."

Ludovico was right; she did not feel any inclination to arrest him. For some reason, her instinct was to let matters be. Was this because she saw him as Prof. Dee's former research partner? Perhaps she understood that he, personally, was no threat to the Ministry, and had no interest at all in affairs of power. Then there was the mystery of the rings and of the wolves and the gateway between worlds. Perhaps they would need to be awkward allies after all.

He finally turned around and confronted her. "But if you choose to, I will not go without a struggle. I think you would be well advised to seek reinforcements." For the first time there was an undercurrent of menace to his speech. Mrs. Macready felt a surge of adrenaline. Her heart jumped. Momentarily she felt as if she was challenged and she itched to reach for her wand; but she let the moment pass and did nothing.

Ludovico continued. "I must ask you and your friend to leave now. Don't think that I would be so obliging as to wait here for you to bring back your colleagues from the Bureau. I bid you adieu." Mrs. Macready suspected that the old wizard had some tricks up his sleeve. After all, she reasoned, he had spent many years as a fugitive. He would not give up his freedom easily and he had the power and the cunning to evade the Bureau's best efforts. She bid Ludovico goodnight and led Denis out the door.

Mrs. Macready told Denis that there was one item of business she had to attend to before she went home for the night. "I have to wipe our your memory," she said.

"No, no! You must be joking! After all that I've heard. You want to deprive me of my story?"

Before Denis could object further she tapped him lightly on the temple with her wand. He immediately looked confused and disoriented. She held onto him lightly by the arm and steered him to the sidewalk where she hailed a cab. In his dazed state he barely noticed when she removed the notebook from his pocket.