298 Art of Travel.
certainly within the points of the compass, P s and P R. Draw the circumscribing parallelogram, G L H E m, whose sides are respectively parallel to P s and P R. Join L M. By the conditions of this problem, the path must somewhere cut the circle E D r ; and since L m cuts L H, which is a tangent to it, it is clear it must cut every path-such as a a, parallel to L H, or to P R-that cuts the circle. Similarly, the same line, L m, must cut every path parallel to P s, such as b b. Now, if L M cuts every path that is parallel to either of the extreme directions, P R or P s, it is obvious that it must also cut every path that is parallel to an intermediate direction, such as c c, but
P PH PD
L-cos HPL- Cos J RPS;
the consequence of which is that P L exceeds P D by onesixth, one-half as much again, or twice as much again, according as R P s = 60°, 90°, 120°, or 140.°
The traveller who can only answer the questions A and B, but not C, must be prepared to travel from P to L, and back again through P to m, a distance equal to 3 P L. If, however, he can answer the question C, he knows at once whether to travel towards L or towards m, and he has no return journey to fear. At the worst, he has simply to travel the distance P L.
The probable distance, as distinguished from the utmost possible distance that a man may have to travel in the three cases, can be calculated mathematically. It would be out of place here to give the working of the little problem, but I append the rough numerical results in a table.
| |
Extreme length of Road it may
be necessary to travel. |
Probable length of Road it will
be necessary to travel. |
|
Knows A, alone
Knows A; and B to within |
7 times the "least distance." |
2 times the "least distance." |
|
8 points. Not C. |
44 11 |
2} |
|
Knows A; and B to within
12 points. Not C. |
7¢ 11 |
24 |
|
Knows A and C; and B to
within 8 points .. |
14 |
4 |
|
Knows A and C; and B to
within 12 points.. |
24 11 |
l4 |
|
Knows A and C; and B to
within 13 points |
3f 11 |
l4 |
* The words "least distance," mean the least distance that the traveller can specify with absolute confidence, as that within which the path, &c., he wishes to
regain, is situated.
).F_
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