A Question of Numbers

by Andrea Penrose

  “The theorem . . . what the devil is Napoleon’s Theorem?” muttered Arianna. She bit her lip—and then shot up from her chair and rushed to fetch the valise containing the books she had brought with her from London. Among them were a handful of copies of The Ladies’ Diary, an annual almanac that featured practical information like calendars, moon phases, and recipes, along with puzzles and riddles to test the intellect.

  It also had a long tradition—going back to 1713, when Henry Brighton became the editor—of featuring some of the most difficult mathematical problems of the day. Each year, the Ladies’ Diary then published the proofs submitted by readers to the previous year’s challenges.

  Grabbing up the issues, she began to thumb through them. She seemed to remember . . .

  Yes—there it was on page 47! An Answer to Question VII, provided by Mrs. W. Rutherford—many of the readers were women who possessed extraordinary skills in mathematics—based on a proof first suggested by N. Bonaparte. Arianna began reading Mrs. Rutherford’s explanation . . .

  Take any shape triangle . . .

  Spotting a pencil and ruler beneath an open notebook, Arianna grabbed them and a fresh sheet of paper. She copied the original triangle shown on the map.

  After re-checking that the measurements were a perfect match, she resumed reading.

  And using each side of that triangle as a base, draw three equilateral triangles pointing outward . . . She set to work drawing in the new triangles using dashed lines.

  When that was done, Arianna moved on to working out the next set of measurements.

  The lines which connect the centers of those three new triangles will then create a new equilateral triangle . . .

  She quickly marked in the new equilateral triangle using dotted lines and then leaned back to study the final result.

  “Oh, you clever devil,” she murmured, “But fortunately for us, a little too clever for your own good.”

  Chapter 24

  “Napoleon’s Theorem!” announced Arianna, slapping the map and her diagram down side by side upon the breakfast room table.

  Grentham finished sealing a note he had just written to Wellington and handed it to one of the footmen. “I’d much rather that you present me with Napoleon’s arse,” he responded, before instructing the fellow to wait for the duke’s reply.

  She gave a tight-lipped smile. “You may wish to swallow your sarcasm, sir, when the map’s triangle helps hand you his head on a platter.”

  That got his attention. “A triangle! How the devil is a triangle drawn atop an obscure part of the countryside going to do that?”

  “Ah.” Her smile became more pronounced. “That’s the beauty of mathematics.”

  Paloma raised a quizzical brow. “Mathematics?”

  “My wife is a wizard with numbers,” said Saybrook. “It’s quite magical what she can get them to reveal.”

  The explanation only caused Paloma to appear more mystified. “But how does Napoleon have anything to do with a triangle”

  “When I was in Elba, the emperor and I had several private meetings, and in one of them we discussed mathematics,” answered Arianna. “You see, he’s gifted in the subject—indeed he’s quite proud of the fact that he proved a mathematical theorem, which is now named after him.”

  “But how—” began Constantina.

  “Yes, yes—allow me to explain.” Arianna set the map and her diagram of the triangles side by side. “I won’t go through the actual mathematics, but Napoleon’s Theorem is all about triangles.” She tapped at the outline of the long, narrow triangle drawn in ink on the map. “One begins by choosing any sort of triangle—like the one here. One then creates an equilateral triangle from each of the three sides of the original triangle.”

  Arianna pointed to her diagram, where she had copied the exact shape of the original triangle from the map, and slowly traced a finger over the three new triangles she had added with dashed lines. “The theorem posits that the center point of these three equilateral triangles creates a new triangle that’s also an equilateral triangle,” she said, a note of excitement creeping into her voice as she indicated the triangle drawn in dotted lines atop the others. “And that the exact center of that newest equilateral triangle will also be the exact center of the original triangle.”

  Picking up a pencil, Arianna drew an ‘x’ to indicate the spot.

  “Forgive me,” murmured Saybrook, “but I fail to see how that’s important—”

  “Oh, but it is!” she replied. “Now again, I won’t explain the mathematics. But when I say the center of the original triangle, I mean its center of gravity. In other words, if you made it out of wood, attached a string to that exact point and suspended it in the air, it would hang perfectly balanced.”

  Arianna showed what she meant by holding out her hand exactly parallel to the floor. “The key is that it’s only by using Napoleon’s Theorem that we can determine that center point of the original triangle.”

  She turned back to the documents. “And voila!” she announced, “By transferring the measurements from the diagram I was able to determine where on the map that point falls.” Her fingers drummed a triumphant tap-tap on the paper. “X marks the spot!”

  Saybrook let out a low whistle.

  “As you see, the center of the triangle drawn on the map falls on the road to Nivelles, just past the Château d’Hougoumont.”

  Arianna drew a deep breath. “And I also found some tiny writing at the bottom of the map. It’s a message addressed to Vecchio, explaining that he must deliver this to NB, as a precaution in case the other two couriers don’t reach him.”

  “Napoleon Bonaparte,” murmured the earl.

  She nodded. “The main text reads, To find your secret weapon, look for a tall oak, whose trunk is scarred by lightning. So, I’m quite certain that at the point I’ve identified using Napoleon’s Theorem, we’ll find a tall oak tree scarred by lightning. And in it is hidden a secret weapon that Napoleon means to use in the upcoming battle with Wellington.

  “A secret weapon? Carried by some shadowy phantom?” Grentham looked skeptical. “We haven’t the time or resources to chase after specters.”

  “It’s not as mad as it sounds,” said Saybrook. “Remember the affair with Reynard.” A previous mission for the minister had involved a terrifying invention that was tiny but capable of great destruction. “The triangle is a very clever code, and only a few close confidantes of Napoleon would know it’s the key to deciphering the message.”

  “Very well, we’ll have to consider the possibility and how to deal with it,” conceded Grentham. “But it’s not our immediate concern. Wellington has said he doesn’t expect Napoleon to start advancing until the end of the month.”

  Arianna refolded the map, reluctantly agreeing he was right. “What have you decided about Pierson?”

  “I’ve asked for a meeting with the duke as soon as possible in order to request a detachment of his most skilled soldiers.” The minister made a face. “He won’t like it. Given how few battle-tested men he has, they are worth their weight in gold.”

  “And if he says no?” asked Saybrook.

  A flutter of silence. “Your friend Leete is under the command of the Prince of Orange,” responded Grentham. “How willing would he be to accept a signed and sealed document from the Prince Regent asking for absolute obedience to any demand made by the bearer of said document?”

  “I think,” replied the earl, “that Leete could be convinced of where his duty lies.”

  “That may not be necessary,” said Paloma. “I have an idea that might have an even better chance of success. I know of some French Royalists who fought with the partisans in Spain against Napoleon’s forces. They’re experienced at clandestine missions . . . and unlike gentlemanly officers, they have no compunction about doing what needs to be done to achieve an objective.”

  Grentham gave a pointed look around the table.

  The earl voiced no objections. Arianna, too, remai
ned silent. War demanded difficult choices. But friendship was a bond that didn’t break easily.

  “I’ve been thinking,” added Paloma. “It was only by chance that Vecchio discovered the ruffians were holding Lord Grentham, and thus decided to accompany them to deliver the prize. Which means the two captives must know the protocol for approaching the hideaway, as they were meant to be bringing Pierson’s daughter—things like the password or signal to allow them inside ”

  “You’re right,” said Arianna. “I fear I’ve let Vecchio’s presence cloud my thinking. He’s been at the heart of the mystery since the beginning.”

  “But no more,” interjected Saybrook. “It’s a good plan, assuming the Royalists are willing.”

  “They will agree out of principle. However . . .” Paloma arched a cynical look at the minister. “ . . . the promise of money will buy an assurance that their loyalty won’t waver.”

  Grentham nodded. “Whatever it costs.”

  “Then I had better be off and begin making the arrangements.” She rose. “In the meantime, I suggest you question the prisoners.”

  “Saybrook and I will see to that,” said Grentham. “I’ll soften up their resistance by threatening to cut out their livers with a dull knife, leaving the honorable earl as a far more pleasant alternative.”

  Paloma had no sooner left the room than their footman returned with Wellington’s reply for the minister.

  The seal yielded with an audible crack as Grentham hurriedly unfolded the missive. It appeared to be a lengthy response, for it took him several long moments to read it.

  “Hmmph.” He passed it to the earl without further comment.

  Saybrook, too, spent some time studying its contents before looking up. “The duke decided to take a calculated risk regarding last night’s assassination attempt on Andronovich.” A wry grimace. “No surprise there, as he’s never been faint of heart.”

  “And?” asked Arianna, recalling Grunwald’s last gasping breaths. She hoped his warnings hadn’t been in vain.

  “And so he allowed the assassin to enter the soirée without challenge. It wasn’t until the man pulled a pistol that the British officers shadowing him leapt into action.”

  “Risky indeed,” she remarked. “I take it Andronovich survived?”

  “The chandelier suffered a mortal wound, but the Russian escaped with naught but a few bruises from being knocked to the floor and covered by one of the rescuers.”

  The earl allowed a small smile. “Andronovich was suitably grateful, for to save himself from execution, the would-be assassin confessed as to who sent him. So, the question of an alliance was settled within seconds. After a quick conference, the heads of the Prussian and Austrian delegations agreed with Andronovich to put aside all differences, and the Prussian announced to the crowd that Blücher would fight against the French, if Napoleon dared to cross the border.”

  “Yet another triumph for Wellington,” murmured Constantina. “Thank Heaven—the man seems invincible.”

  “Let us hope so.” Saybrook’s expression, however, indicated he was not yet ready to celebrate victory. “But I daresay his most difficult battle remains to be fought.”

  Grentham’s grunt echoed the sentiment. “Come, we need to go interrogate our prisoners.” Catching Arianna’s frown, he added, “Have no fear, Lady Saybrook. Like you and your husband, I’m of the opinion that torture isn’t terribly useful. There are far better ways of extracting the truth.”

  “And I shall go make myself useful by keeping an eye on the children,” murmured the dowager. “An endeavor, I might add, which is also not for the faint of heart.”

  “I doubt they will dare disobey orders and risk a roasting by the Dragon’s fire,” murmured Grentham as he passed her chair.

  Constantina waggled her cane at him. “Don’t be impertinent to your elders.”

  “I would hope I’m wise enough not to provoke a person who’s holding a lethal weapon mere inches from my . . . nose.”

  “So far, we’ve been extraordinarily lucky in having come through the dangers relatively unscathed,” observed Saybrook as he rose. Like the others, he was moving a little stiffly. “Would that our luck will hold for a little while longer.”

  The room turned quiet. Birds were twittering in the back garden, the cheerful notes rising above the low buzzing of the bees flitting through the flowers. A gentle breeze ruffled through the ivy framing the mullioned windows, setting the glossy leaves to dancing against the glass.

  Arianna had returned to poring over the map, intent on making sure she hadn’t missed anything. But a flutter of sunlight slipped through the greenery and darted across the paper, causing her to look up. Enclosed within the high walls of the weathered stone, the garden was a little world unto itself, she reflected, one suffused with an aura of peaceful harmony. An illusion, of course—spiders were weaving their deathtraps, birds were pecking at worms, insects were devouring the plants.

  And yet, at that moment it looked so achingly, impossibly beautiful.

  Sophia let out a sigh and shifted in her chair. She, too, was staring out at the garden and seemed caught up in her own unsettled mood.

  “Who was it that said life is nasty, brutish and short?”

  Arianna lifted her shoulders. “Not someone with whom I care to become acquainted.” Another flutter of light touched the map. “My own thoughts are dark enough as it is.”

  “Hobbes—it was Thomas Hobbes,” said Sophia. “A political philosopher from several centuries ago, if I remember correctly.” Her expression turned troubled. “I suppose those who are familiar with the inner workings of governments and quests for power must be irredeemably cynical as they watch the constant struggle between good and evil.”

  “Cynicism and pragmatism are not always the same thing,” said Arianna carefully. She had an inkling of what had provoked such musings.

  The comment seemed to give Sophia pause for thought. Rising from her chair, her friend went to the window and pressed a palm against the glass, as if seeking warmth from sunshine.

  “Do you think I’m naïve?” blurted out Sophia. “Or merely a bloody lackwit?”

  Love makes fools of us all—especially when we least expect it. Arianna felt a wry smile tug at the corners of her mouth. “It matters naught what I think. Only you can unravel the complexities of your heart.”

  “But that’s just the point—I only seem to be tying myself in knots.”

  “If emotions were easy, life would indeed be simpler.” She reflected back on the twists and turns of her own arduous journey through some very dark paths. “But it would be far less nuanced and rewarding. Struggle is painful, but I think we gain a better elemental understanding of ourselves and what we ultimately value.”

  Sophia didn’t answer, and Arianna didn’t wish to press. Her friend’s past experience with men had given her good reason to be wary and mistrustful. She had inner demons that she must battle on her own terms.

  A fact that Arianna knew all too well. Holding back a sympathetic sigh, she turned back to puzzling over the map and what the ‘secret weapon’ might be—and who might be the messenger. Conundrums, conundrums. Perhaps, she thought wryly, the bees flitting from flower to flower in cheerful oblivion to the dangers lurking all around were to be envied . . .

  “Might I ask one more question?” Sophia turned away from the window, her face now wreathed in shadows.

  Arianna set down her pencil. “Of course.”

  “No doubt it’s even more foolish than the last one. But nonetheless . . .”

  “No question is foolish. Save for the one that asks why we bother to plumb the depths of our feelings,” she responded. “You’re wise to examine your heart very carefully.”

  “I—I don’t think I shall ever be as wise as you are,” replied her friend.

  She chuffed a laugh. “Wisdom is not something you ever fully possess. You may think you have grabbed hold of it, but poof—like quicksilver, it simply slips away through your fing
ers.” A smile. “So ask your question, and I’ll answer it as best I can.”

  “How . . . how did you and the earl come to have such a strong bond of trust?”

  “Slowly, and with a good deal of stumbles,” replied Arianna truthfully. “We’re both very private people, who don’t find it easy to share our innermost feelings. To do so makes you vulnerable. And being vulnerable is very frightening. You must, at some point, simply make a leap of faith. And only you will know when that moment comes.”

  Sophia nodded, setting off a flitting of shadows around her face. “I think I understand what you’re saying.” The curl of her lashes hid her eyes. “I know you don’t like Grentham—”

  “Like and dislike are far too simplistic terms. The minister and I have had our differences in the past,” she interrupted. “Quite severe ones, in fact. But I think we’ve come to understand and respect each other. More than that, I’ve come to believe he’s a very honorable man in his own way.” After a hesitation, she added, “And though he keeps it well hidden, he has a dry sense of humor, which says a lot about a man.”

  The words seemed to surprise Sophia, but then a tiny twitch softened the tension around her mouth. “I sense he’s equally shocked to find himself thinking of you and Saybrook as . . . friends.”

  “As I said, life’s journey is full of unexpected and unimagined twists and turns.”