William E

William E. Lopez

Approx 2,366 words

Billy and the Wizards

W. E. Lopez

“I don’t think they’ll even come close,” Billy told the two older boys. They were dressed in Boy Scout khaki while he wore Cub Scout blue.

“Well, if we don’t do something, we won’t have more than a guess to submit!” Nelson replied. As an over achiever, Billy had been skipped two grades by his teachers. He was younger than Nelson and Bobby, but since all three were members of the school computer club, a natural friendship had formed.

“Mr. Bain said the patrol who comes up with the most correct answer to the distance from here to that bridge, will not have to pull camp cleanup or wash dishes after meals. The Pumas and the Black Widows are pacing off the distance now, while we Wizards sit on our fat behinds.” Bobby was not anxious to pull camp cleanup, and hated to give up at anything. “If Nellie hadn’t sprained his ankle getting out of the pickup truck, we’d have a fair chance.”

“I didn’t do it on purpose,” Nelson complained. “It’s no fun, believe me.”

The Scoutmaster had brought the nine boys out to the desert, not far from Indio, to lay off a map-reading course for other scouts who were testing for merit badges next weekend. When Billy heard about the overnight campout, he begged to come along and Mr. Bain allowed it.

“Look, fellas,” he said to the older boys. “It’s gotta be three miles, at least, from here to that bridge. And the terrain is pretty rugged, with lots of gullies and dry washes. I’ll bet we can survey the distance and be at least as accurate as the other guys.”

“Survey, with what, genius? Mr. Bain said we could only use the stuff we normally carry on a hike. You didn’t smuggle along a global positioning system or something did you?”

“Of course not,” the younger boy replied. “But we can use that climbing rope we always carry for emergencies. Anybody know how long it is?”

“Exactly fifty yards,” Bobby said. “I was with Jimmy Bain and his dad when they bought it.”

“But, what good is a climbing rope gonna do us?” Nelson asked.

“We’ll use it as a compass,” said Billy.

“Dumkopff!” said Bobby in annoyance. “A compass has to be magnetic so it can point north. You can’t magnetize a rope.”

“Not that kind of compass,” Billy answered. “The kind you draw circles with. The Egyptians built cities and laid out the pyramids using knotted ropes for measuring. The Romans built roads all over Europe, long before the magnetic compass was invented.”

“They sure did,” agreed Nelson. “I know we’re at least as smart as they were, but how are we going to do it?”

“We have to determine the distance from that bridge to this benchmark,” Billy said, pointing to a small concrete block in the ground with a brass plaque on top. Government surveyors placed the benchmarks at locations all over the country to aid in map making. In fact, Mr. Bain had a U.S. Forestry map which included the benchmark and the bridge, and which he would use to determine the winning answer.

“Bobby, run and fetch the climbing rope while Nellie and I organize our data.”

“What data?” Bobby said. “We don’t know anything yet.”

“We know plenty,” Billy told him. “All we have to do is get our ducks in a row, then we can work on the problem.” Bobby didn’t seem to understand, but he ran to bring back the climbing rope. When he returned, Billy set about explaining what he wanted them to do.

“Bobby, walk twenty paces that way,” he pointed in the direction opposite from the bridge, “then stand so you can sight over the benchmark to the bridge.” As Bobby did that, Billy picked up a tent peg. When Bobby signaled he was in position, Billy took the rope and held it taught while using the tent peg to draw a line in the dirt.

“Okay, Bobby. Now walk twenty paces toward the bridge so I can draw another arc.” Billy drew another line in the dirt.

“Oh, I see,” said Nelson. “I remember Mrs. Kennison teaching that to us in sixth grade. Now we connect the intersection of the two arcs, and we have a line perp-endicular to our base line. But what good will that do?”

Billy uses the rope to draw arcs of equal radii at points equidistant from the benchmark. Connecting a line between the intersections of the arcs establishes a line perpendicular to the line of sight from the benchmark to the bridge

“We need the perpendicular to lay out a triangle. Surveying is based mainly on the relationships of triangles,” Billy said. “Now, we lay the rope end over end, being very careful as we do so, until we have measured 500 yards out in that direction where the ground is very smooth and level.”

“500 yards!” Bobby said. “I thought this was going to be easy?”

“Would you rather walk three miles over to the bridge, counting paces all the way, and then walk back?” Nelson asked. “Give Billy a chance to show what he can do.”

The boys measured 500 yards using the rope. When they were finished, Billy sketched in the dirt for them. “Visualize this,” he said. “Our original line of sight is the unknown distance we have to measure. We’re at this point, Angle B, and Angle C is back there at the benchmark, 500 yards away.”

“Right, Einstein, but until we know the value of Angle B, we can’t begin to calculate the distance. How are we gonna measure B?”

“Simple,” Billy said. “We walk the dog. Rather, you and I do, Nellie has that bum ankle.”

“What dog? Are you crazy or something?” Bobby asked.

“I’ll stand here while you walk the angle from the line to the benchmark, round to our new line of sight to the bridge. We all paced off the distance between the goal lines at school just last week preparing for this trip, so all we have to do is count paces, you first, and then me. I’d like to have Nelson pace it also so we could get better data, but with his bad ankle I don’t think he should. Nellie can take charge of the pencil and paper and keep a record.”

To be as accurate as possible, both boys paced the distance in each direction. Then they converted their paces to yards and had Nelson compute the average.

“I make it 73.9581 yards,” Nelson said.

“We’ll just have to hope it will be accurate enough,” said Bobby.

“Okay, Bobby. The length of the rope is fifty yards, so the circumference of the full circle is 314.1592 yards. How many degrees did we measure?” They watched while Bobby worked with paper and pencil, rubbing out his first answer and recomputing, then doing it a third time just to be sure.

“I make it 84.7497 degrees,” Bobby said. “We know that the angle at the benchmark is 90 degrees, so the remaining angle at the bridge must be 5.2502 degrees. I see how you worked that, and it was pretty slick, I’ll admit. But, what do we do now?”

“We know that the angle subtended at the bridge equals 500 yards at a given radius, we just don’t know what the radius is. Any suggestions?” asked Billy.

Bobby thought a moment then began writing in his notebook again. In a moment he began explaining. “I divide 5.2502 into 360 degrees to find out how many times this arc will go into the entire circumference, and I get an answer of 68.5688 times. Then I multiply by 500 yards to get the circumference of the entire circle, which is 34,284.4082 yards.”

“You’re right as rain,” Nelson said. “I’m working the same arithmetic just to keep a check on you.”

“I don’t know whether to be glad I’m right, or annoyed that you are checking on me,” Bobby said.

Billy and Bobby use the rope to “walk the dog.” While one boy holds the rope at B, the other paces off the distance between the two lines of sight to A and C. Their average is 73.9581 yards. Knowing the length of the arc, they can calculate the angle of arc and then determine the value of angle A.

From that point, it’s only a matter of determining the length of the radius of a circle, which has the distance from A to C.

The distance from the bridge to the benchmark is 5,456.533 yards.

“A little checking just serves to keep our work accurate,” Billy said.

“We find the radius using the old C = 2 Pi times r. Substituting 34,284.4082 for C, dividing by Pi, we get a diameter of 10,913.0660 yards. Half of that is the radius or, distance to the bridge, of 5,456.5330 yards. Everyone check on that?”

“Close enough,” Bobby agreed. “I differ from you by about two yards. 5,454.6447 yards is my answer.”

“Lots of decimal points floating around,” Billy said as he did his own work. “Why don’t we split the difference and call it 5,554 yards?”

“Suit’s me,” Nelson and Bobby agreed.

The boys had enjoyed the mental gymnastics of working out a solution using their brains. They went back to the encampment area where they soon erected their tents and were relaxing on their sleeping bags when the Pumas and Black Widows came dragging back in the late afternoon. They didn’t need to ask to see that the boys were hot and exhausted.

“Too bad for you Wizards,” one of the other boys called. It was Jimmy Bain, sometimes called Jimmy the Jock by the boys in the Wizard Patrol if they were feeling generous, Jimmy the Jerk when they weren’t. “We had a good hike and you guys get to wash the dishes.”

“We had a nice rest while you ape-men were exercising your legs,” Nelson said.

“And we’ll see who is doing the dishes,” Bobby jeered.

When all the boys had returned to the camp, Mr. Bain called them together at the hood of his pickup. He took a clean sheet of paper and asked the patrol leaders for their best estimates of the distance to the bridge.

“The Black Widows say it’s 5,508 yards to the bridge,” Jimmy Bain proudly announced. His dad wrote down the figure where all could see.

“You’re off by a mile,” jeered Ronnie from the Puma Patrol. The correct distance is 5,431 yards.” He heavily emphasized the word correct.

“Too bad for you Wizards,” the Scoutmaster said. “It’s a shame Nelson sprained his ankle, but fair is fair….”

“Wait a minute, Mr. Bain,” Nelson said. “We have an answer too, and we say the correct distance is 5,455 yards to the bridge.”

“But how could you come up with that answer?” the Scoutmaster asked. “You didn’t even leave camp. You spent the afternoon down there on the flats just walking around.”

“We were surveying, Mr. Bain,” Billy said proudly.

“Surveying? And how did you do that?” Mr. Bain asked.

“Nothing to it,” Bobby said. “All we had to do was organize the ducks and then walk the dog.”

The rest of the boys’ chuckled while Bobby blushed. “Aww, let Billy explain it. He understands it better than I do.”

Billy drew a couple sketches in the dirt as he explained how the Wizards had arrived at their answer. “It really isn’t such a big deal,” he admitted. “Eratosthenes used a similar method two thousand years ago when he estimated the circumference of the earth using reasoning alone.”

“Okay, okay, I give up,” said the Scoutmaster. “Let me measure the distance on my map and we’ll see who came up with the best answer.” Pulling his forestry map from the truck, he unfolded it and used the same sheet of paper with the answers on it as a straight edge. He laid the edge across the symbols for the bridge and the benchmark, and marked the paper with two tick marks. Then he moved his paper down to the bottom of the map where the map scale was graphically represented, and aligned the tick marks along the scale. He was very careful as he did so, because he knew the boys were each hoping their answer would be the correct one.

“I make it out to be 5,462 yards, fellas.” He made a note of that and compared it with the answers each patrol leader had given.

“Widows, you seem to be long by 46 yards, while the Pumas appear to be short by 31 yards. Still, pretty good measuring considering what you had to work with.” He scratched his head as he stared at the paper before him. “What I can’t figure out is how the Wizard Patrol could get so much closer, only 7 yards different than my answer. You boys sure you don’t have a map of your own in your kit?”

“No, sir,” Nelson assured him. “We simply used reasoning and geometry. With real surveying instruments, we should have gotten much closer, but we’ll stick with the answer we came up with.”

“Hmmm, your answer could be better than mine. Even the width of a pencil lead makes a difference on a map of this scale. You said Errol somebody did the same thing about two thousand years ago? I guess I must have slept through that class, or something.

“All the same, you boys are the winners, and I’ll definitely appreciate your assistance laying out the compass course in the morning. As for you Pumas and Widows, the Pumas will do camp clean up tonight, prepare the meal and wash up afterwards. The Widows will have the duty in the morning, and I want this campsite to be as pristine when we leave as it was before we arrived. You understand me?”

The Wizards did not need to gloat over their satisfaction at having one the competition. The disappointment in the eyes of the other Boy Scouts was satisfaction enough.