After a hearty breakfast of porridge and custard with Farmer Koto and his family, Mati and Sata bought everything they could squeeze into the rucksacks, from carrots to hard crackers, dried onions to porridge oats.
Ilika sat down with the farmer and made a list, and the total came to nearly eight small silver pieces. Since the list didn’t include all the valuable information the farmer had given, Ilika felt good about handing the man a great silver piece.
The farmer and the teacher shook hands, and Ilika went out to see if his students were ready.
After making sure the provisions were all packed, and tying Misa’s blankets into a neat bundle with rope, Buna stood up and looked around. She saw Boro and Sata helping each other shoulder their heavy packs, and smiled.
When everyone was ready, with sun hats on their heads, all eleven travelers waved farewell to the family and made their way out to the road.
Ilika stopped and opened his shoulder bag. “This will do nicely.”
“But Ilika!” Kibi challenged with a puzzled look, “we’re still right in front of the farmhouse where we ate breakfast!”
“Today we have a wonderful opportunity that would be hard to find anywhere else. Are you all ready to do some serious trigonometry?”
Toli spoke with slight exasperation. “We’ve already measured about a hundred trees using tangent!”
“You told us about sine and cosine,” Rini added calmly, “but we haven’t
tried them.”
“Trees are easy,” Ilika responded. “You can walk right up to them, and they form a convenient right angle with the ground. Today we’ll get more serious. See that snow-covered mountain, the pointy one that looks tallest?”
Mati turned in the saddle and shaded her eyes from the morning sun.
“Yeah.”
Everyone else gazed toward the northeast. The jagged white peak, wreathed in thin clouds, stood out starkly against a deep blue sky.
“I want to know how tall it is, and how far away,” Ilika continued.
Miko spread his arms. “Very tall, and very far!”
“I want to know exactly, in feet.”
Several students shrugged.
Toli frowned with worry. “I bet this’ll take more than the tangent of forty-five degrees.”
“Yeah, a bit more,” Ilika admitted with a smile. “First we need a note-taker. Um . . . Rini.” He handed paper and pencil to the quiet lad, then pulled the little hand-held device out of his bag and flipped open the cover.
“The best name for this tool, in your language, is knowledge processor.”
Buna quickly got close so she could see.
Ilika pressed some tiny buttons. “Compasses in your kingdom are too big to carry with us, so we’ll make do with a simulation.”
“Is that magic, like your bracelet?” Buna asked, grinning as she looked at the screen.
“Same kind of . . . magic.”
Everyone gathered around. The little screen appeared to contain a compass rose and needle.
“Buna, this road runs fairly straight across the meadow. Take a bearing, please.”
Buna received the knowledge processor with trembling hands. She had never touched Ilika’s bracelet, nor imagined she ever would. Now she was holding, and was supposed to use, an even larger magical item, one with a constantly changing magical picture. “I’m . . . not sure . . . how.”
“Face the direction the road goes, hold it out in front of you so you can move your eyes up and down to see the road and the compass. The thick lines
are five degrees, the thin lines are one degree.”
“Um . . . seven . . . eight . . . ten . . . no, nine . . . eight! . . . no, seven . . .
damn!”
“That’s okay. Just take some time with it, look way up the road, and see where the needle is most of the time.”
“Most of the time . . . where are you little needle? Eight!” Buna declared.
“Got that, Rini?” Ilika asked.
“Got
it!”
Ilika stepped close to Tera. “Mati, compass bearing to the mountain, please.”
Mati received the little device, then nudged Tera slightly so they faced the mountain. “Um . . . let me see . . . no . . . yes . . . no . . . most of the time . . .
thirty-one.”
“Neti, the angle between the road and the mountain?”
Neti stared at the two numbers on Rini’s paper.
Toli was about to explode with the answer, but Ilika, Boro, Miko, and Rini all flashed him stern looks.
“Twenty . . . three?” Neti dared to say with a very worried expression.
“Right!” Toli burst out.
“I did it!” Neti almost screamed, bouncing up and down. “For the first time, I did it in my head!”
Ilika smiled. “You sure did! Now, Rini, draw a right triangle. We are at one of the acute angles, call it A1. The right angle is ahead of us, up the road somewhere. The mountain is at the other acute angle. Yes, that looks good.”
Everyone gathered around to see what Rini had drawn.
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“Now it’s time to move on. Toli, I want you to use that nice, even stride of yours and measure the distance from here to . . . somewhere up the road, we don’t know where yet. If you forget your count, you’ll have to walk all the way back here and start over.”
“I won’t forget!” he squeaked.
“Sata, your turn to take the compass. You need to find the place up the road where the mountain has changed its angle by one degree. It will be about half a mile, I think.”
Sata received the device from Mati and started walking up the road.
Misa grabbed her bundle of blankets. It had been cold last night, and just looking at the white mountains made her shiver.
Farmer Koto stood up from pretending to weed the front garden when the travelers finally took to the road. He had heard the rumors of sorcerers who could cast spells to put whole towns to sleep, and start or put out fires at will.
These were strange folks who spoke of unnatural things like trigomancy and tangometry, but they didn’t seem to be doing any harm.
As an afterthought, just to be sure it was still solid and real, he tossed the great silver piece up and caught it. Then he turned his mind to how to get his precocious middle daughter married off before she ran away.
Sata stopped about every hundred yards, pointed the compass at the mountain, then shook her head.
Toli strode along behind, counting out loud.
After several more stops, Sata announced that her reading was getting close, but she wasn’t quite happy with it yet. She marched on.
The others walked behind. Misa came last, dancing or hopping over clumps of grass with her new moccasins.
Sata stopped and took one more compass bearing. “That mountain is at thirty-two degrees, or I’m a billy goat.”
Boro grinned. “You’re much cuter than a billy goat.”
Sata
blushed.
Ilika took off his rucksack. “We’ll be here awhile, so get comfortable. Toli, how far?”
Toli brought his strides right up to Sata’s position. “One thousand and four!”
“I have determined,” Ilika announced, “that a Toli-stride is about three point two feet long. Kibi, how many feet between the two points, rounded to the nearest foot?”
She received paper and pencil and went to work. Everyone else sat near Rini to look at his drawing, or near Kibi to watch her do the multiplication.
After a few minutes, she handed the answer to Rini.
“We are at A2, Rini, and the line segment we just measured is B. From here to the right angle is C, and the far side of the triangle is D. Neti, how many degrees in angle A2?”
Neti grinned with happiness as she picked out the important numbers and worked out the difference in her head. “Twenty-four!”
Ilika looked in his shoulder bag and found a sheet with many numbers arranged in neat columns. “Miko, we need the tangents of twenty-three and twenty-four degrees.”
Rini made notes as Miko found the numbers. Neti watched, then smiled at the boy she loved.
NEBADOR BookTwo: Journey 234
“Boro, what is the relationship of the tangent function to the sides of a right triangle?”
“Wow . . . err . . . um . . .” Boro began as he lay back in the grass, scrunched his face and searched his memory, “the tangent of the . . . um . . . angle is equal to the . . . far side . . . divided by the . . . um . . . near side.”
“Perfect,” Ilika complimented.
“Whew!”
“Toli, the tangent of angle A2 equals . . .”
“D over C.”
Ilika nodded. “Rini, the tangent of angle A1?”
Rini looked at his drawing, then got another piece of paper and wrote down Toli’s answer and his own.
“Okay,” Ilika announced, looking around. “Everyone take a good look at these two equations. We need to solve them.”
Sata frowned. “I see a huge problem. Both have two unknown variables.
No matter which one we solve for, there will always be another one, so we can never do a calculation!”
All the others looked at Ilika with blank faces. Misa chewed on a blade of grass.
“If we had only one equation,” Ilika went on, “that would indeed make it unsolvable. But since we have two equations . . .”
Rini’s mouth suddenly opened.
“Do you see it, Rini?”
NEBADOR BookTwo: Journey 235
“I’m not sure, but when you look at something from two different directions, you can sometimes see a lot more about it.”
“Yes. Since we have two equations, we can solve both for the same variable, then drop out that variable and put them together.”
“I think we should solve for D,” Rini suggested. “It looks easier.”
Toli agreed, and did his work first. Then Rini did his part and combined the results.
“Hurray!” Buna cheered. “Only one variable!”
“Now we have to solve for C,” Ilika continued, “and it’s going to take some work. Everyone ready?”
They
nodded.
“Buna, would you please distribute the right side?”
“Me?”
“I think you could use the practice.”
Buna sat down with paper and pencil. After several scratch-outs, she started turning red. “I don’t remember how,” she whimpered in a broken voice.
“You just . . .” Toli started to say, but Boro stuck an elbow in his ribs so hard he doubled over with pain.
Buna stared at the paper for several more minutes. Suddenly she
NEBADOR BookTwo: Journey 236
screamed, “I CAN’T!” then jumped to her feet and stomped off into the grass, sobbing deeply.
Everyone remained silent, and Toli looked at the ground. Kibi and Misa went to sit with their friend. The rest stretched out in the grass to take a break.
After wiping her tears on her sleeves, Buna glanced back toward her teacher every minute or two. Other students also looked at Ilika off and on, expecting him to give Buna a hint or ask someone else.
Kibi knew him better than that, so she stretched out in the grass to watch the clouds go by.
About a quarter hour later, Buna wandered back with Kibi and Misa at her side. “I guess . . . I’m ready to, you know, try it again, if that’s okay,” she said in a timid voice.
“It sure is,” Ilika said softly. “Maybe . . . there’s something you need to ask for that will help.”
Everyone gathered near. The other girls gave their friend reassuring words and touches.
Buna thought for a long moment. “Would you . . . show me the formula again?”
Buna took the paper after Ilika wrote the formula, then looked at the problem. Toli handed her a pencil without daring to open his mouth. Buna carefully worked out the distribution, digit by digit, symbol by symbol, while everyone watched in silence.
A shepherd on a hillside about a mile away heard the cheering and
clapping, and looked toward the meadow, but had no idea what the happy occasion might be. He glanced around at his sheep, then went back to whittling designs into his walking stick.
Ilika could see a sparkle in Buna’s eyes that hadn’t been there earlier.
“Now we can do that multiplication. Who needs to practice? Mati!”
With help from Sata, Mati lowered herself to the ground. Buna handed her paper and pencil, and stayed close to watch.
Mati was slow and careful at arithmetic, but always remembered the steps.
She worked her way through the difficult multiplication as Buna watched with wide eyes. At the very end, with a nod from Ilika, Mati rounded the answer.
Ilika smiled. “Now we need to divide both sides by C. Would you take care of that, Kibi?”
“Sure,” she agreed, got another sheet, and quickly finished.
“You can keep going,” Ilika prompted, “distribute that C on the right side, please.”
After a deep breath and a glance at the formula, Kibi went to work. Again Buna watched closely.
“And you have some simplifying you can do,” Ilika nudged.
Kibi frowned. “I . . . don’t see it,” she said with a hint of irritation in her voice that Ilika heard clearly.
“You have two places where you are multiplying by one.”
Kibi closed her eyes for a long moment, took a deep breath, then opened them. C divided by C nearly jumped out at her, twice.
NEBADOR BookTwo: Journey 238
Everyone
clapped.
“Now I think . . . Neti should subtract that tangent of twenty-three degrees from both sides.”
Neti grinned. “I’m the subtraction queen today!” She looked at the numbers for a long moment as her smile slowly faded. “Ilika . . . is it okay if I don’t do this one in my head?”
“Yes, Neti,” he said, chuckling. “I don’t think I could!”
Neti breathed a sigh of relief, took the sheet of paper Miko handed her, and began to work through the difficult arithmetic problem.
Buna was right beside Neti for the entire process, watching every step, sometimes holding her breath for a moment, sometimes smiling.
“Here’s a challenging step for Rini. Get rid of that number on the right
NEBADOR BookTwo: Journey 239
side.”
Rini made quick work of his teacher’s request.
“You fell into a trap,” Ilika pointed out.
Rini looked hard at his work, then laughed. “Oh, donkey turds!”
“What?” Mati probed with obvious interest.
“I accidentally inverted C.” Rini corrected his mistake. “Now I should invert both sides, right Ilika?”
Ilika nodded. Everyone watched Rini do the division.
Rini read the answer, the largest number he had ever used, then flopped backwards into the grass.
After running around in the meadow for a few minutes to unwind, everyone settled down to a snack of bread, goat cheese, and plums. Boro sat silently with his brow furrowed. Ilika waited until he found his voice.
“I was just wondering . . .” Boro finally began, “what good is figuring out things like this?”
“On a ship, if that mountain’s in our way, our lives might depend on knowing how far away it is.”
“But it takes so long to figure it out!”
“We’re doing it by hand. On my ship, you would have instruments and tools to do the work for you, giving you the answer in seconds. But to
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intelligently guide your tools, you have to understand the process. If you don’t, then the numbers become the masters, and you become the slaves.”
Boro looked very thoughtful as he ate his plum.
Toli looked at Rini’s drawing. “The road would have to go straight for ten miles to get to the right angle.”
“Which it doesn’t,” Buna pointed out sharply.
Ilika stepped in. “The right angle is somewhere far up in the mountains, somewhere we probably couldn’t find. That’s why we had to work with two different angles.”
“Can we figure out the slanty line now?” Rini asked with sparkling eyes.
Ilika smiled. “Yes, we’ve done the hard part, now finding the hypotenuse will be easy. Since we know the near side, we’ll use the cosine function. I’ll go through the steps while all of you watch.”
They quickly closed the food sacks and crowded around.
“I look up the cosine of twenty-four degrees, and put in the value of C, the near side. H is unknown. Next I divide both sides, eliminate the identity, invert, and I’ve solved for H. Everyone with me?”
Some of them took a moment to catch up.
Ilika looked around. “I think . . . Sata needs some division practice.”
She got pencil and paper and worked through the problem. “Wow. That’s big.”
“That’s the answer to my first question,” Ilika announced. “Since a mile in your kingdom is about six thousand feet, that mountain is twelve miles away, plus a stone’s throw.”
After everyone took a break to throw rocks toward the mountain, none of which made it, Ilika sat down with Rini’s drawing again. All the students gathered around. Misa continued picking little flowers.
“Now we need another angle measurement, angle A3. It’s a vertical angle, so we use an instrument called an inclinometer. It’s simpler than a compass, and just compares the angle to the gravity of the planet.” He touched some keys on his knowledge processor and the display changed. “Miko and Neti, would you take a reading to the mountain top?”
Miko received the knowledge processor, which now had a circular scale and a line going from the center to the outside. He quickly discovered that the line always pointed downward. “So . . . I should use it sideways?”
“Yes,” Ilika answered. “Sight along one edge while Neti reads it. Then switch and see if you get the same number.”
While Miko sighted to the top of the mountain, holding the little device as still as he could, Neti peered at the screen. “Seven . . . no, eight degrees.”
Miko read eight or nine degrees, so they settled on eight.
“Kibi,” Ilika said with a smile, “you’re in charge of finding the elevation.
You don’t have to do all the work yourself, but you’re in charge.”
After showing her teeth and snarling slightly at Ilika, Kibi sat down by Rini and his sheet of numbers. Rini soon finished drawing another triangle to represent the elevation problem.
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“Rini, we’re at A3, right?” Kibi asked.
“Yep.”
“That makes the bottom line of this triangle the same as H, right Toli?”
“Exactly.”
“Sata, review for me the tangent function.”
“The far side divided by the near side.”
Kibi got a piece of paper and started writing a formula. “Boro, check me.”
“Looks
right.”
“Buna, tangent of eight degrees, please.”
She shuffled through the papers, then read the number to Kibi.
Kibi looked at her equation in which the unknown was still part of a division problem, and took a leap. “Toli, Sata, Rini, check this step for me.”
They all nodded.
“Whew. Any volunteers for the big bad multiplication?”
“I’ll do it!” Toli blurted out.
No one else made a sound. He worked his way through the four partial answers, finally summing them and inserting the decimal point.
“Thank you, Toli. I’ll round that to ten thousand one hundred and thirty-five . . . feet.”
The next ten minutes were filled with dancing, whooping and hollering, hand shakes, and pats on the back. Ilika was very proud of all his students.
Each had, in his or her own way, wrestled dragons to help solve their first serious trigonometry problem.
Misa peeked out from under her new sun hat. “Can we go now?”
Deep Learning Notes
What harm could possibly come of letting Farmer Koto overhear them using big words he didn’t know?
The calculations in this chapter are not very precise because of the rough bearing measurements made with a hand-held compass. A ship’s compass, even in the days of wooden sailing ships, would be much more precise. The variance between magnetic direction (to the magnetic poles, as measured by the compass) and true direction (to the axial poles) is not a factor because the calculations only use the differences in the bearings. Also, the length of a
“Toli-stride” cannot be measured perfectly.
Even though it took a long time, what was gained by waiting for Buna to get over her feelings of failure when trying to apply the distributive property?
Hint: it showed most clearly when others were doing calculations after that.
Ilika agreed with Boro that the process was too difficult to use “on the spot”
during navigation, and so the ship’s tools would do most of the work, most of the time. It is not humanly possible to retain a skill that is rarely used, but it is possible to retain a “sense of” the skill, so it can be refreshed quickly when
needed, and so the work of calculating tools can be intelligently guided and checked.
One of the most important reasons to have some personal experience with mathematics is to watch for “unreasonable” answers from our tools. If I press
“pi” on my calculator and get 3.14159265358, I may not notice (or care) that 3.14159265359 would be more accurate. If I get 7.3, I better stop and figure out why. If I had no experience with the real “pi” and just trusted the calculator, my results would be garbage, of course.
By putting Kibi “in charge” of calculating the elevation, what was Ilika having her practice?
