by any swordsman. Before, she had been facing humiliation. What Diane de Tremontaine faced now was total destruction.
The gentleman asked one question, and without hesitation, the duchess gave him the only answer possible.
And so it was with a light heart and a light step that Benjamin Hawke left Tremontaine House and walked in the direction of his own, whistling again the air that Ixkaab Balam had overheard earlier that day in Riverside.
But the Duchess Tremontaine, her hands trembling, sat frozen in a brocade chair that—along with every other part of a life that she cherished—she might very soon see for the last time.
• • •
Doctor Volney was talking about triangles, which were Micah’s fourth-favorite regular shape, after octagons, hexagons, and triskaidecagons. Of course, she had a separate list for irregular polygons; you’d think that would go without saying, but then one day she had been drawing with a stick in the dirt out in front of the house and Aunt Judith had asked her whether she liked the regular septagon better than the concave hexagon (of course, Aunt Judith hadn’t known their names then, and neither had Micah) and had been absolutely unable to understand why it was a ridiculous question, so there you go.
“And so we see,” said Volney, “that the ratio of the length of each side to the sine of its opposite angle is the same as the ratio of the length of each other side to the sine of its opposite angle. Which means what, in terms of our earlier discussion? Milner?”
“Given the derivations we’ve just been through, sir, the ratio of the length of any given side of the triangle to the sine of its opposite angle is twice the radius of the circumscribed circle.”
“Good for you.” Micah looked at Rafe on the bench beside her, about to grin, and then realized his mouth was still scrunched up and his eyebrows drawn together, just like when he’d walked into the room, and that meant he was upset about something.
She would ask him later what it was. Because right now she was just too happy to think about it. Of all the lectures Rafe had taken her to, Volney’s delighted her the most, because it was Volney who talked most about the kinds of things she had always spent all her time thinking about. Sine. Cosine. Tangent. The very words tasted delicious.
“It should be clear, then, that, if we know the radius of the circumscribed circle, all we need to learn the length of any side is the angle opposite.”
Circumscribed circle. This, this was why she was willing to put up with people who got upset with her. Amos and Judith and Seth never talked about circumscribed circles.
“Given that,” Volney continued, “suppose a right triangle with an additional angle of sixty degrees. Suppose further that the radius of the triangle’s circumscribed circle is seven and one part in two. Now, lay the triangle we have hypothesized on a sphere. What then is the length of the side of the triangle opposite the other angle?”
Silence as students around the room scratched on their slates. Micah caught sight of what Rafe was writing and was surprised to see that he was on the wrong track. She grunted softly and, when he looked up, gestured to his calculations and shook her head, at which point he stopped writing, his hand hovering in the air with the pen.
Finally a voice from somewhere in the room called out, “Three and three parts in four.” Micah sighed: this was exactly the same error Rafe had been making.
“Just right, Pearson.”
Micah’s brows knit in consternation. Why was Doctor Volney teaching them something wrong?
“The trick here is in recognizing—since we started with a right triangle, a known radius, and an included angle—that the hypotenuse of a circumscribed right triangle is always equal to the circle’s radius.”
Micah began breathing hard. She pulled urgently on Rafe’s sleeve, but she knew from the expression on his face—confusion was an easy one to recognize—that she was on her own. She felt an immense pressure somewhere deep within her. She knew this feeling well and she hated it fiercely, but she could never seem to control it. The pressure built and built and built. She had to do something. The pressure was almost crushing.
“Now, if we assume that—”
“No!” She knew it was the wrong thing to do, but she couldn’t help herself. “That’s wrong!”
And now everyone in the room was looking at her. She wanted to make herself tiny, or run away, or disappear into thin air. The horrible pressure was now joined by a hideous embarrassment. Her breath came even faster and she began rocking back and forth very quickly, her eyes wide with fright.
Rafe put his arms around her and squeezed hard. “Want to do angles?” he said, and she nodded, grateful. “Tell me the angle in an equilateral triangle,” he whispered in her ear.
“Sixty,” she said at once, deeply grateful to have something else to focus on.
“In an equilateral pentagon.”
“One hundred eight.”
“Square.”
“Ninety.”
“Breathe, breathe slowly. Hexagon.”
“One hundred twenty.” She saw the shapes as he named them and, though her breath was still heavy and her hands gripped the bench no less tightly, at least she was able to stop rocking.
“What, if I may be so bold to ask,” said Doctor Volney, “is your name?” She knew the answer but somehow couldn’t speak to say it.
“Octagon.”
“One hundred thirty-five.”
“Breathe. Decagon.”
“One hundred forty-four.” Micah remembered telling Rafe that this helped her. She also remembered that if she breathed deeply it helped her calm down, so she inhaled. Exhaled. Inhaled.
“Did I fail to speak clearly?” Doctor Volney again. She needed to answer him. “What is your name, boy?”
She could breathe again, now, so she could speak. She shrugged Rafe off and he stopped whispering. She could do this. “Micah, sir. Micah Heslop.”
“And, Master Heslop, would you be so good as to honor us by explaining yourself?”
That one was easy, too.
“Your conclusion. It’s wrong.” She was glad he had asked her. Immediately a small pulse of relief diminished her urge to rock.
You could have heard a feather fall. “Oh? How so?”
But she had to continue; speaking made it better. “How can you not see it?” She couldn’t keep the frustration out of her voice. “The numbers you’re talking about are right if the triangle is on a flat surface. But if you’re on a sphere, the surface changes and the number of angles in any triangle has to add up to greater than one hundred eighty. You can’t lay a flat triangle perfectly on a sphere without breaking it somewhere, and then it’s not a triangle anymore. The question doesn’t even make sense.”
There. She was still agitated, but her breathing,
