at the opposite end, making a dot at 107 degrees from the line, and draw a line from the left hand end of the base line through this dot.  If we extend these lines until they intersect, we will have the required triangle, and can measure the two sides, which will be found to be about 12 inches and 8 inches long, and the third angle will measure just 26 degrees.  It doesn’t make any difference on what scale we draw the triangle, whether it be miles, yards, feet, inches or fractions of an inch, the proportions will be the same.  If the base line had been 6 half-inches, or 3 inches long, and the same angles were used, the other two lines would measure 12 half-inches, or six inches, and 8 half-inches, or 4 inches.  If the base line were 6 quarter-inches long, the sides would be 3 inches and 2 inches long.

“Now, for example, I am going to measure the distance to that tree over there.  Get out your chain and measure off a straight line 10 feet long.  Now, I’ll set the surveying instrument with the plumb-bob right over the end of this line, and sight through the two sight holes until I bring the, two vertical hairs in line with each other and the tree.  Look at the compass needle.  It points to the 173 degree mark on the cardboard ring.  Now, Bill, you hold the rod at the other end of our base line while I swing this instrument around and sight it.  There, the needle points to 92 degrees, and subtracting this from 173 the difference, 81 degrees, is the angle at the right end of our base line.  We’ll do the same thing at the other end of our line.  See, the compass needle points to 189 degrees, and now sighting to the pole at the other end of the line we find that the needle points to 268.  The difference, 79 degrees, is therefore the size of the angle at the left end of our base line.  Now we will draw this out on paper, as we did our first triangle, using quarter-inches to represent feet.  Our base line was 10 feet long, and we will therefore draw a line 10 quarter-inches, or 2-1/2 inches long, on our drawing board.  On this line we will construct the triangle, using the angles 81 and 79 degrees.  There, that’s how our triangle looks, and the right hand side measures 7-1/4 inches, while the left hand side measures 7-5/16 inches.  That is, 29 quarter-inches for one side and 29-1/4 quarter inches for the other.  As each quarter-inch represents a foot, you will find that the tree is about 29 feet from the right end of our base line and 29 feet 3 inches from the left hand end.  Of course, our instrument is not perfect, neither is our drawing; but if you measure it off with the chain you will see that I am not very far from correct.”

[Illustration:  Fig. 84.  Determining the Distance to the Tree.]

Mapping the Island.