"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. Here is the line we would follow if we could go straight from Camp Jackson to Pensacola. If we could follow it, I would only have to guess how many miles we march each day, and mark it down on the map. But we can't go straight, because of swamps and creeks and canebrakes, so I must keep looking at my compass to find out what direction we do go; then I mark on the map the route we have followed each day, and the distance, and each night's camp gives me a new starting point." "Yes, but Sam," said Tom, suddenly thinking of something. "Well, what is it, Tom?" "Suppose you guess wrong as to the distance travelled each day?" "Well, suppose I do; I can't miss it very far." "No, but it gives you a wrong starting-point for the next day, and two or three mistakes would throw you clear out." "Yes, but I make corrections constantly. You see, I have changed the place of last night's camp a little on the map." "How do you make corrections?"