Then the manufacture of new arrows began, the boys working earnestly now, because they were interested.
After awhile Sam took out his map and began pricking the course upon it.
"I say, Sam," said Bob Sharp, "how do you do that ?" "How do I do what?
Prick the map ?" "No, I mean how do you know where we are and which way we go ?" "That's just what I want to know," said Sid Russell.
"And me, too," chimed in Billy Bunker and Jake Elliott.
"Well, come here, all of you," replied Sam, "and I'll show you.

We started there, at camp Jackson,--you see, don't you, where the Coosa and the Tallapoosa rivers come together and we are going down there," pointing to a spot on the map, "to the sea, or rather to the Bay near Pensacola." "Are we! Good! I never saw the sea," said Sid Russell, speaking faster than any of the boys had ever heard him speak before.
"Yes, that is the place we're going to, and presently I'll tell you what we're going for; but one thing at a time.

You see the course is a little west of south, nearly but not quite southwest.

The distance, in an air line is about a hundred and twenty-five miles: that is to say Pensacola is about a hundred and ten miles further south than camp Jackson, and about fifty miles further west." "That would be a hundred and sixty miles then," said Billy Bowlegs.
"Yes," replied Sam, "it would if we went due south and then due west, taking the base and perpendicular of a right angled triangle, instead of its hypothenuse." "Whew, what's all them words I wonder," exclaimed Billy.
"Well, I'll try to show you what I mean," said Sam, taking a stick and drawing in the sand a figure like this: [Illustration] "There," said Sam, "that's a right angled triangle, but you may call it a thingimajig if you like; it doesn't matter about the name.
Suppose we start at the top to go to the left hand lower corner; don't you see that it would be further to go straight down to the right hand lower corner and then across to the left hand lower corner, than to go straight from the top to the left hand lower corner." "Certainly," replied Billy, "it's just like going cat a cornered across a field." "Well," said Sam, pointing with his finger, "if I were to draw a triangle here on the map beginning at camp Jackson and running due south to the line of Pensacola, and then due west to Pensacola itself, with a third line running 'cat a cornered' as you say, from camp Jackson straight to Pensacola, the line due south would be about a hundred and ten miles long and the one due west about fifty miles long, while the 'cat a cornered' line would be about a hundred and twenty five miles long." "How do you find out that last,--the cat a cornered line's length ?" asked Tom.
"I can't explain that to you," said Sam, "because you haven't studied geometry." "Oh well, tell us anyhow, if we don't understand it," said Sid Russell, who sat with his mouth open.
"Sid wants to find out how to tell how far it is from his head to his heels, without having to make the trip when he's tired," said Bob Sharp, who was always poking fun at Sid's long legs.
"Well," said Sam smiling, "I know the length of that line because I know that the square described on the hypothenuse of a right angled triangle is equal to the sum of the squares described on the other two sides." "Whew! it fairly takes the breath out of a fellow to hear you rattle that off," replied Sid.
"Come," resumed Sam, "we aren't getting on with what we undertook.

Now look and listen.