The Art of Travel; Or, Shifts and Contrivances Available in Wild Countries. Galton Francis
of a gun and its report; for even a storm of wind only makes 4 per cent. difference, one way or the other, in the velocity of sound.
Measurement of Angles.--Rude Measurements.--I find that a capital substitute for a very rude sextant is afforded by the outstretched hand and arm. The span between the middle finger and the thumb subtends an angle of about 15 degrees, and that between the forefinger and the thumb an angle of 11¼ degrees, or one point of the compass. Just as a person may learn to walk yards accurately, so may he learn to span out these angular distances accurately; and the horizon, however broken it may be, is always before his eyes to check him. Thus, if he begins from a tree, or even from a book on his shelves and spans all round until he comes to the tree or book again, he should make twenty-four of the larger spans and thirty-two of the lesser ones. These two angles of 15 degrees and 11¼ degrees are particularly important. The sun travels through 15 degrees in each hour; and therefore, by "spanning" along its course, as estimated, from the place where it would stand at noon (aided in this by the compass), the hour before or after noon, and, similarly after sunrise or before sunset, can be instantly reckoned. Again, the angles 30 degrees, 45 degrees, 60 degrees, and 90 degrees, all of them simple multiples of 15 degrees, are by far the most useful ones in taking rough measurements of heights and distances, because of the simple relations between the sides of right-angled triangles, one of whose other angles are 30 degrees, 45 degrees, or 60 degrees; and also because 60 degrees is the value of an angle of an equilateral triangle. As regards 11¼ degrees, or one point of the compass, it is perfectly out of the question to trust to bearings taken by the unaided eye, or to steer a steady course by simply watching a star or landmark, when this happens to be much to the right or the left of it. Now, nothing is easier than to span out the bearing from time to time.
Right-angles to lay out.--A triangle whose sides are as 3, 4, and 5, must be a right-angled one, since 5 x 5 = 3 x 3 + 4 x 4; therefore we can find a right-angle very simply by means of a measuring-tape. We take a length of twelve feet, yards, fathoms, or whatever it may be, and peg its two ends, side by side, to the ground. Peg No. 2 is driven in at the third division, and peg No. 3 is held at the seventh division of the cord, which is stretched out till it becomes taut; then the peg is driven in. These three pegs will form the corners of a right-angled triangle; peg No. 2 being situated at the right-angle.
Proximate Arcs.-- 1 degree subtends, at a distance of 1 statute mile, 90 feet. 1' subtends, at a distance of 1 statute mile, 18 inches. 1' subtends at a distance of 100 yards, 1 inch. 1" of latitude on the earth's surface is 100 feet. 30' is subtended by the diameter of either the sun or the moon.
Angles measured by their Chords.--The number of degrees contained by any given angle, may be ascertained without a protractor or other angular instrument, by means of a Table of Chords. So, also, may any required angle be protracted on paper, through the same simple means. In the first instance, draw a circle on paper with its centre at the apex of the angle and with a radius of 1000, next measure the distance between the points where the circle is cut by the two lines that enclose the angle. Lastly look for that distance (which is the chord of the angle) in the annexed table, where the corresponding number of degrees will be found, where the corresponding number of degrees will be found. If it be desired to protract a given angle, the same operation is to be performed in a converse sense. I need hardly mention that the chord of an angle is the same thing as twice the sine of half that angle; but as tables of natural sines are not now-a-days commonly to be met with, I have thought it well worth while to give a Table of Chords. When a traveller, who is unprovided with regular instruments, wishes to triangulate, or when having taken some bearings but having no protractor, he wishes to lay them down upon his map, this little table will prove of very great service to him. (See "Measurement of distances to inaccessible places.")
Triangulation.--Measurement of distance to an inaccessible place.--By similar triangles.--To show how the breadth of a river may be measured without instruments, without any table, and without crossing it, I have taken the following useful problem from the French 'Manuel du Genie.' Those usually given by English writers for the same purpose are, strangely enough, unsatisfactory, for they require the measurement of an angle. This plan requires pacing only. To measure A G, produce it for any distance, as to D; from D, in any convenient direction, take any equal distances, D C, c d; produce B C to b, making c B--C B; join d b, and produce it to a, that is to say, to the point where A C produced intersects it; then the triangles to the left of C, are similar to those on the right of C, and therefore a b is equal to A B. The points D C, etc., may be marked by bushes planted in the ground, or by men standing.
The disadvantages of this plan are its complexity, and the usual difficulty of finding a sufficient space of level ground, for its execution. The method given in the following paragraph is incomparably more facile and generally applicable.
Triangulation by measurement of Chords.--Colonel Everest, the late Surveyor-General of India, pointed out (Journ. Roy. Geograph. Soc. 1860, p. 122) the advantage to travellers, unprovided with angular instruments, of measure the chords of the angles they wish to determine. He showed that a person who desired to make a rude measurement of the angle C A B, in the figure (p. 40), has simply to pace for any convenient length from A towards C, reaching, we will say, the point a' and then to pace an equal distance from A towards B, reaching the point a ae. Then it remains for him to pace the distance a' a" which is the chord of the angle A to the radius A a'. Knowing this, he can ascertain the value of the angle C A B by reference to a proper table. In the same way the angle C B A can be ascertained. Lastly, by pacing the distance A B, to serve as a base, all the necessary data will have been obtained for determining the lines A C and B C. The problem can be worked out, either by calculation or by protraction. I have made numerous measurements in this way, and find the practical error to be within five per cent.
Table for rude triangulation by Chords.--It occurred to me that the plan described in the foregoing paragraph might be exceedingly simplified by a table, such as that which I annex in which different values of a' a" are given for a radius of 10, and in which the calculations are made for a base = 100. The units in which A a', A a", and B b', Bb", are to be measured are intended to be paces, though, of course, any other units would do. The units in which the base is measured may be feet, yards, minutes, or hours' journey, or whatever else is convenient. Any multiple or divisor of 100 may be used for the base, if the tabular number be similarly multiplied. Therefore a traveller may ascertain the breadth of a river, or that of a valley, or the distance of any object on either side of his line of march, by taking not more than some sixty additional paces, and by making a single reference to my table. Particular care must be taken to walk in a straight line from A to B, by sighting some more distant object in a line with B. It will otherwise surprise most people, on looking back at their track, to see how curved it has been and how far their b' B is from being in the right direction.
Measurement of Time.--Sun Dial.--Plant a stake firmly in the ground in a level open space, and get ready a piece of string, a tent-peg, and a bit of stick a foot long. When the stars begin to appear, and before it is dark, go to the stake, lie down on the ground, and plant the stick, so adjusting it that its top and the point where the string is tied to the stake shall be in a line with the Polar Star, or rather with the Pole (see below); then get up, stretch the string so as just to touch the top of the stick, and stake it down with the tent-peg. Kneel down again, to see that all is right, and in the morning draw out the dial-lines; the string being the gnomon. The true North Pole is distant about 1½ degree, or three suns' (or moons') diameters from the Polar Star, and it lies between the Polar Star and the pointers of the Great Bear, or, more truly, between it and [Greek letter] Urs ae Majoris.
The