Part 4 (1/2)
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.”)
[Table of Chords to Radius of 1000].
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 s.p.a.ce 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.
[Contains Table for Rough Triangulation without the usual instruments, and without Calculation”].
Measurement of Time.--Sun Dial.--Plant a stake firmly in the ground in a level open s.p.a.ce, 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 1 1/2 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.
[Small drawing ill.u.s.trating these directions in above text].
The one essential point of dial-making is to set the gnomon truly, because it ensures that the shadows shall fall in the same direction at the same hours all the year round. To ascertain where to mark the hour-lines on the ground, or wall, on which the shadow of the gnomon falls, the simplest plan is to use a watch, or whatever makes.h.i.+ft means of reckoning time be at hand. Calculations are troublesome, unless the plate is quite level, or vertical, and exactly facing south or north, or else in the plane of the Equinox.
The figure represents the well-known equinoctial sun-dial. It can easily be cast in lead. The spike points towards the elevated pole, and the rim of the disc is divided into 24 equal parts for the hours.
Pendulum.--A Traveller, when the last of his watches breaks down, has no need to be disheartened from going on with his longitudinal observations, especially if he observes occulations and eclipses. The object of a watch is to tell the number of seconds that elapse between the instant of occulation, eclipse, etc., and the instant, a minute or two later, when the s.e.xtant observation for time is made. All that a watch actually does is to beat seconds, and to record the number of beats. Now, a string and stone, swung as a pendulum, will beat time; and a native who is taught to throw a pebble into a bag at each beat, will record it; and, for operations that do not occupy much time, he will be as good as a watch.
The rate of the pendulum may be determined by taking two sets of observations, with three or four minutes' interval between them; and, if the distance from the point of suspension to the centre of the stone be thirty-nine inches, and if the string be thin and the stone very heavy, it will beat seconds very nearly indeed. The observations upon which the longitude of the East African lakes depended, after Captain Speke's first journey to them, were lunars, timed with a string and a stone, in default of a watch.
Hour-gla.s.s.--Either dry sand or water may be used in an hour-gla.s.s; if water be used, the aperture through which it runs must, of course, be smaller.
CLIMBING AND MOUNTAINEERING.
Climbing.--Climbing trees.--Colonel Jackson, in his book, 'How to Observe,' gives the following directions for climbing palms and other trees that have very rough barks:--”Take a strip of linen, or two towels or strong handkerchiefs tied together, and form a loop at each end, for the feet to pa.s.s tightly into without going through; or, for want of such material, make a rope of gra.s.s or straw in the same way. The length should embrace a little more than half of the diameter of the trunk to be climbed. Now, being at the foot of the tree, fix the feet well into the loops, and opening the legs a little, embrace the tree as high up as you can. Raise your legs, and pressing the cord against the tree with your feet, stand, as it were, in your stirrups, and raise your body and arms higher; hold fast again by the arms, open the legs, and raise them a stage higher, and so on to the top. The descent is effected in the same way, reversing, of course, the order of the movements. The ruggedness of the bark, and the weight of the body pressing diagonally across the trunk of the tree, prevent the rope from slipping. Anything, provided it be strong enough, is better than a round rope, which does not hold so fast.”
A loop or hoop embracing the body of the climber and the tree, is a helpful addition. Large nails carried in a bag slung round the waist, to be driven into the bare trunk of the tree, will facilitate its ascent.
Gimlets may be used for the same purpose. High walls can be climbed by help of this description; a weight attached to one end of a rope, being first thrown over the wall, and the climber a.s.sisting himself by holding on to the other end. Trees of soft wood are climbed by cutting notches two feet apart on alternate sides. Also by driving in bamboo pegs, sloping alternately to left or to right; these pegs correspond to the ”rungs” of a ladder.
Ladders.--A notched pole or a knotted rope makes a ladder. We hear of people who have tied sheets together to let themselves down high walls, when making an escape. The best way of making a long rope from sheets, is to cut them into strips of about six inches broad, and with these to twist a two-stranded rope, or else to plait a three-stranded one.
Descending cliffs with ropes is an art which naturalists and others have occasion to practise. It has been reduced to a system by the inhabitants of some rocky coasts in the Northern seas, where innumerable sea-birds go for the breeding season, and whose ledges and crevices are crammed with nests full of large eggs, about the end of May and the beginning of June.
They are no despicable prize to a hungry native. I am indebted to a most devoted rock-climber, the late Mr. Woolley, for the following facts. It appears that the whole population are rock-climbers, in the following places:--St. Kilda, in the Hebrides; Foula Island, in Shetland; the Faroe Islands generally; and in the Westmarver Islands off Iceland. Flamborough Head used to be a famous place for this accomplishment, but the birds have become far less numerous; they have been destroyed very wantonly with shot.
In descending a cliff, two ropes are used; one a supply well-made, many-stranded, inch rope (see ”Ropes”), to which the climber is attached, and by which he is let down; the other is a much thinner cord, left to dangle over the cliff, and made fast to some stone or stake above. The use of the second rope is for the climber to haul upon, when he wishes to be pulled up. By resting a large part of his weight upon it, he makes the task of pulling him up much more easy. He can also convey signals by jerking it. A usual rock-climbing arrangement is shown in the sketch. One man with a post behind him, as in fig. 1, or two men, as in fig. 2 are entrusted with the letting down of a comrade to the depth of 100 or even 150 feet. They pa.s.s the rope either under their thighs or along their sides, as shown in the figures. The climber is attached to the rope, as shown in fig. 2. The band on which he sits is of worsted. A beginner ought to be attached far more securely to the rope.
[Fig 1 and Fig 2 appear on p 45].
(I have tried several plans, and find that which is shown in Fig. 1 to be thoroughly comfortable and secure. A stick forms the seat' at either end of it is a short stirrup; garters secure the stirrup leathers to the knees; there is a belt under the arms.)
It is convenient, but not necessary, to have a well-greased leather sheath, a tube of eighteen inches in length, through which the rope runs, as shown in both figures. It lies over the edges of the cliff, and the friction of the rock keeps it steadily in its place.
It is nervous work going over the edge of a cliff for the first time; however, the sensation does not include giddiness. Once in the air, and when confidence is acquired, the occupation is very exhilarating. The power of locomotion is marvellous: a slight push with the foot, or a thrust with a stick, will swing the climber twenty feet to a side. Few rocks are so precipitous but that a climber can generally make some use of his hands and feet; enough to cling to the rock when he wishes, and to clamber about its face. The wind is seldom a gale above, but the air will be comparatively quiet upon the face; and therefore there is no danger of a chance gush das.h.i.+ng the climber against the rocks. A short stick is useful, but not necessary. There are three cautions to be borne in mind.