On Finding the Way. 297
knows the direction to within eight points. Similarly, he is sure to twelve points, if his limits, on either hand, are E.N.E. and W.N. W. respectively.
C requires no further explanation.
Now, if a man can answer all three questions, A, B, to within eight points of the compass, and C, he is four and a half times as well off as if he could only answer A ; as will be seen by the following considerations. A knowledge of B in addition to A, is of only one-third the use that it would be if C also were known.
1. Let P (fig. 1) be the point where the traveller finds himself at fault, and let P n be a distance within which the path certainly lies ; then the circle, E D F, somewhere cuts the path, and the traveller starting from P must first go to D, and then make the entire circuit, D E H F D, before he has exhausted his search. This distance of P D+ D E H F D= P D + 6 P D nearly, = 7 P D altogether, which gives the length of road that the man must be prepared to travel over who can answer no other than the question A. Of course, P D may cut the path, but I am speaking of the extreme distance which the lost man may have to travel.
2. Supposing that question B can be answered as well as
M Fig, 1.
question A, and that the direction of the line of road lies