A PRACTICAL USE OF TRIGONOMETRY.
It was Binney Gibbs who had come up the river from Fort Yuma several
days before, with General Elting, to meet the second division, and guide
them to "The Needles," the point at which the line was to cross the
Colorado. The other divisions, which had followed the Gila route, and
crossed the Colorado at Fort Yuma, where the desert was narrower, had
reached the Pacific ere this, and gone on to San Francisco. The hardest
task of all, that of running a line over the desert where it was two
hundred and fifty miles wide, had been reserved for Mr. Hobart's men,
who had proved themselves so capable of enduring and overcoming
hardships.
Binney had waited impatiently in camp until the transit-party reached
it, expecting to see Glen ride in at its head with the front flag. Then
he had borrowed a horse, and set forth to find the boy whom he had once
considered his rival, but whom he now regarded as one of his best
friends.
After the first exchange of greetings, they stood and looked at each
other curiously. Glen's hair hung on his shoulders, and the braid that
bound the brim of his sombrero was worn to a picturesque fringe,
matching that of his buckskin shirt. He was broader and browner than
ever; and though his face was still smooth and boyish, these last three
months had stamped it with a look of resolute energy that Binney noticed
at once.
He, too, was brown, though not nearly so tanned as Glen, in spite of the
burning suns of the Gila Valley; for his work had kept him under cover
as much as Glen's had kept him in the open air. As General Elting's
secretary, Binney had spent most of his time in the ambulance, that,
fitted up with writing-desk and table, was the chief-engineer's
field-office, or in temporary offices established in tents or houses
wherever they had halted for more than a day at a time. He had evidently
met with barbers along the comparatively well-travelled Gila; while, as
compared with Glen's picturesquely ragged costume, his was that of
respectable civilization. Although he, too, was the picture of health,
his frame lacked the breadth and fulness of Glen's, and it was evident
at a glance that, in the matter of physical strength, he was even more
greatly the other's inferior than when they left Brimfield.
Glen could not help noting this with a feeling of secret satisfaction;
but, as they rode towards camp together, and Binney described his
winter's experiences, Glen began to regard him with vastly increased
respect. He thought he had studied hard, and done well to master the
mysteries of adjusting and running a level, perfecting himself as a
rodman, and learning to plot profile; but his knowledge appeared
insignificant as compared with that which Binney had picked up and
stored away. Not only had he learned to speak Spanish fluently, but he
had become enough of a geologist to talk understandingly of coal-seams
and ore-beds. He had the whole history of the country through which he
had passed, from the date of its Spanish discovery, at his tongue's end.
He spoke familiarly of the notable men to whom, at General Elting's
dictation, he had written letters, and altogether he appeared to be a
self-possessed, well-informed young man of the world.
Poor Glen was beginning to feel very boyish and quite abashed in the
presence of so much wisdom, and to wonder if he had not been wasting his
opportunities on this trip as he had those of school. His thoughts were
inclining towards a decidedly unpleasant turn, when they were suddenly
set right again by Binney, who exclaimed, "But, I say, old man, what a
fine thing you fellows have done this winter! The general declares that
you have made one of the most notable surveys on record; and it's a
thing every one of you ought to be proud of. You should have heard him
congratulate Mr. Hobart. He asked at once about you, too, and wants to
see you as soon as you get in. He seems to take a great interest in you,
and has spoken of you several times. I expect, if you choose to keep on
in this business, you can always be sure of a job through him. He seems
to think it queer that you should be a year older than I am; but I told
him it was certainly so, because I knew just when your birthday came."
Glen was on the point of saying that, if Binney knew that, it was more
than he did, but something thing kept him silent. He hated to
acknowledge that he knew nothing of his real birthday, nor how old he
really was, but he wondered if he could truly be a year older than this
wise young secretary.
At this point the conversation was interrupted by their arrival at camp,
and by General Elting stepping from his tent to give Glen a hearty
handshake as he exclaimed,
"My dear boy, I am delighted and thankful to see you again. I tried to
persuade our friend Mr. Hobart, when I last saw him at Santa Fe, that,
in spite of your performance on that railroad ride you and I took
together last summer, you were too young to make the trip I had laid out
for him. He said he didn't know anything about your age, but that you
were certainly strong and plucky enough for the trip. I made him
promise, though, to try and induce you to go back from Isletta; but he
doesn't seem to have succeeded."
"No, sir," laughed Glen, "and I'm awfully glad he didn't, for it's been
the most glorious kind of a trip, and I have enjoyed every minute of
it."
"I am glad, too, now that it is all over; but I must tell you that, if I
had not been assured that you were a whole year older than my young
secretary here, I should have insisted on your going back, for I
considered it too hard and dangerous a trip for a boy so young as I had
supposed you to be until then."
Here was another good reason why Glen was glad he had remained silent on
the subject of his birthday.
"Now what do you think of running a line across the desert ahead of us?"
continued the chief-engineer; "are you as anxious to undertake that as
you were to cross Arizona?"
"Yes, indeed, I am, sir," replied Glen, earnestly. "I am anxious to go
wherever the second division goes; and if anybody can get a line across
that desert, I know we can."
"I believe you can," said the chief, smiling at the boy's enthusiasm,
"and I am going along to see how you do it."
The Colorado was so broad, deep, and swift that Glen wondered how they
were going to measure across it, and had a vague idea that it could be
done by stretching a long rope from bank to bank. He asked "Billy"
Brackett; and when the leveller answered, "By triangulation, of course,"
Glen showed, by his puzzled expression, that he was as much in the dark
as ever.
"You have studied geometry and trigonometry, haven't you?" asked the
leveller.
Glen was obliged to confess that, as he had not been able to see the use
of those studies, he had not paid much attention to them.
"Well, then, perhaps you'll have a better opinion of old Euclid when you
see the practical use we'll put him to to-morrow," laughed "Billy"
Brackett.
Glen did see, the next day, and wondered at the simplicity of the
operation. The front flag was sent across the river in a boat, and on
the opposite side he drove a stake. While he was thus engaged, a line a
quarter of a mile long was measured on the bank where the rest of the
party still remained, and a stake was driven at each end of it. The
transit was set up over one of these stakes, and its telescope was
pointed first at the other and then at the one across the river, by
which means the angle where it stood was taken. It was then set over the
stake at the other end of the measured line, and that angle was also
taken. Then Mr. Hobart drew, on a leaf of his transit-book, a triangle,
of which the base represented the line measured between the two stakes
on his side of the river, and one side represented the distance across
the river that he wished to find. He thus had one side and two angles of
a triangle given to find one of the other two sides, and he solved the
problem as easily as any boy or girl of the trigonometry-class can whose
time in school has not been wasted as Glen Eddy's was.
It was a simple operation, and one easily performed, but it involved a
knowledge of the four fundamental rules of arithmetic, of proportion, or
the rule of three, of geometry, of trigonometry, and of how to use a
surveyor's transit; all of which, except the last, are included in the
regular course of studies of every boy and girl in America who receives
a common-school education.
Glen had also been sent across the river, where he held his rod so high
up on the bank that the cross hair in the telescope of the level cut
just one tenth of an inch above its bottom. Then, when "Billy" Brackett
came over, and went on beyond Glen, he set the level up so high on the
bank that, through it, he could just see the top of the rod, extended to
its extreme length. So they climbed slowly up out of the Colorado
Valley, and began to traverse the dreary country that lay between it and
the Sierra Nevada.
For the first hundred miles or so they got along very well, so far as
water was concerned, though the mules and horses speedily began to grow
thin and weak for want of food. The patches of grass were very few and
far between, and the rations of corn exceedingly small; for in that
country corn was worth its weight in gold, and scarce at that.