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 makeshift 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 dashing the climber against the rocks. A short stick is useful, but not necessary. There are three cautions to be borne in mind.

1. As you go down, test every stone carefully. If the movement of the rope displaces any one of them, after you have been let down below it, it is nearly sure to fall upon your head, because you will be vertically beneath it. Some climbers use a kind of helmet as a shield against these very dangerous accidents. 2. Take care that the rope does not become jammed in a cleft, or you will be helplessly suspended in mid-air. Keep the rope pretty tight when you are clambering about the ledges: else, if you slip, the jerk may break the rope, or cause an overpowering strain upon the men who are holding it above.

Turf and solid rock are much the best substances for the rope to run over. In the Faroes, they tar the ropes excessively; they are absolutely polished with tar. Good ropes are highly valued. In St. Kilda, leather ropes are used: they last a lifetime, and are a dowry for a daughter. A new rope spins terribly.

Leaping Poles.--In France they practise a way of crossing a deep brook by the help of a rope pa.s.sed round an overhanging branch of a tree growing by its side. They take a run and swing themselves across, pendulum fashion. It is the principle of the leaping-pole, reversed.

The art of climbing difficult places.--Always face difficult places; if you slip, let your first effort be to turn upon your stomach, for in every other position you are helpless. A mountaineer, when he meets with a formidable obstacle, does not hold on the rock by means of his feet and his hands only, but he clings to it like a caterpillar, with every part of his body that can come simultaneously into contact with its roughened surface.

Snow Mountains.--Precautions.--The real dangers of the high Alps may be reduced to three:--1. Yielding of snow-bridges over crevices. 2. Slipping on slopes of ice. 3. The fall of ice, or rocks, from above. Absolute security from the first is obtainable by tying the party together at intervals to a rope. If there be only two in company, they should be tied together at eight or ten paces apart. Against the second danger, the rope is usually effective, though frightful accidents have occurred by the fall of one man, dragging along with him the whole chain of his companions. Against the third danger there is no resource but circ.u.mspection. Ice falls chiefly in the heat of the day; it is from limestone cliffs that the falling rocks are nearly always detached. When climbing ice of the most moderate slope, nailed boots are an absolute necessity; and for steep slopes of ice, the ice-axe (described below) is equally essential.

Alpine Outfit consists of ropes, ice-axe or alpenstock (there must be at least one ice-axe in the party), nailed boots, coloured spectacles, veil or else a linen mask, m.u.f.fettees, and gaiters.

I give the following extracts from the Report of a Committee appointed by the Alpine Club in 1864, on Ropes, Axes, and Alpenstocks:--

Ropes.--We have endeavoured to ascertain what ropes will best stand the sharp jerk which would be caused by a man falling suddenly into a creva.s.se, or down an ice-slope: and on this subject we lay before the Club the result of nearly a hundred experiments, made with various kinds of rope purchased of the best London makers. We considered that the least weight with which it was practically useful to test ropes, was twelve stone, as representing the average weight of a light man with his whole Alpine equipment. In the preliminary experiments, therefore, all ropes were rejected which did not support the strain produced by twelve stone falling five feet. Under this trial, all those plaited ropes which are generally supposed to be so strong, and many most carefully-made twisted ropes, gave way in such a manner as was very startling to some of our number, who had been in the habit of using these treacherous cords with perfect and most unfounded confidence. Only four ropes pa.s.sed successfully through this trial; these were all made by Messrs.

Buckingham and Sons, of 33, Broad-street, Bloomsbury, and can be procured only of them. We confined our further experiments to these ropes, one of which failed under severer tests, while the remaining three, made respectively of Manilla hemp, Italian hemp, and flax, proved so nearly equal in strength that it may fairly be doubted which is on the whole to be preferred. Each of these three ropes will bear twelve stone falling ten feet, and fourteen stone falling eight feet; and it may be useful to say that the strain upon a rope loaded with a weight of fourteen stone, and suddenly checked after a fall of eight feet, is nearly equal to that which is caused by a dead weight of two tons. None of these ropes, however, will bear a weight of fourteen stone falling ten feet; and the result of our experiments is, that no rope can be made, whether of hemp, flax, or silk, which is strong enough to bear that strain, and yet light enough to be portable. We believe that these ropes, which weigh about three-quarters of an ounce to the foot, are the heaviest which can be conveniently carried about in the Alps. We append a statement of the respective merits of the three kinds, all of which are now made by Messrs. Buckingham, expressly for the Club, and marked by a red worsted thread twisted in the strands:--

No. 1. MANILLA HEMP. Weight of 20 yards, 48 oz. Advantages--Is softer and more pliable than 2. Is more elastic than 2 and 3. When wet, is far more pleasant to handle than 2 and 3. Disadvantages--Has a tendency to wear and fray at a knot.

No. 2. ITALIAN HEMP. Weight of 20 yards, 43 oz. Advantages--Is less bulky than 1 and 3. Is harder, and will probably wear best, being least likely to cut against rocks. Disadvantages--Is much more still and difficult to untie than 1 and 3. When wet, is very disagreeable to handle, and is apt to kink.

No. 3. FLAX. Weight of 20 yards, 44 oz. Advantages--When dry, is softer, more pliable, and easier to handle than 1 and 2, and will probably wear better than 1. Disadvantages--When wet, becomes decidedly somewhat weaker, and is nearly as disagreeable to handle as 2.

Knots.--There can be no doubt that every knot in a rope weakens its power of resisting a sudden jerking strain. How great a loss of strength results from a knot we cannot undertake to estimate, but that the loss is a very serious one the following statement will show: these ropes which we report will resist the strain of fourteen stone falling eight feet, will not resist it if there is a knot in any one of them; or even if the knots used in attaching them to the point of support, or to the weights, be roughly or carelessly made. The rope in these cases breaks at the knot, for two reasons; partly because of the folds, as they cross in the knot, are strained suddenly across each other, and one of them is cut through; and partly because the rope is so sharply bent that the outer side of each fold in the knot is much more stretched than the inner side, so that the strain comes almost entirely upon one side only of each fold.

For the first reason, we found it necessary to put a pad of some kind inside the knot--leather, linen, or a little tow or waste rope will do.

For the second reason we preferred knots in which the folds are least sharply bent round each other; that is, in which the curves are large. We therefore conclude that--1st. No knot, which is not absolutely necessary, ought to be allowed to remain on the rope: 2nd. The tighter and harder a knot becomes, the worse it is: 3rd. The more loose and open a knot is made, the better it is:--and we append diagrams of those knots which we found by experiment weaken the rope least. For Alpine ropes, only three sorts of knots are ever required, and we suggest one of each kind:--No. 1 is for the purpose of joining two ends. No. 2 is for the purpose of making a loop at one end. No. 3 is for the purpose of making a loop in the middle when the ends are fastened. No. 4 is a knot, of which we give a diagram in order that no one may imitate it. It is one of those which most weaken the rope. The only one which seemed to be equally injurious is the common single knot, of which no diagram is necessary. As the topes which we have recommended are very liable to become untwisted, unless the loose ends are secured, we advise travellers, in order to avoid knots, to have the ends of every piece of rope bound with waxed twine. It should also be known that it is very unsafe to join two pieces of rope by looping one end through the other, so that when the jerk comes, they will be strained across each other as two links of a chain are strained across each other. Unless a pad of some kind divides the loops, one will cut the other through.

[Four diagrams of knots on this page].

Axes.--The axes made in England for the purpose of being taken out to Switzerland, may be divided into two cla.s.ses, namely: travellers' axes, intended to be used for chipping a few occasional steps, for enlarging and clearing out those imperfectly made, and for holding on to a snow-slope,--and guides' axes, which are the heavier implements required for making long staircases in hard blue ice. We have had three models prepared, of which diagrams are appended; the first two represent the lighter axe, or what we have termed the travellers' axe; and the third, the heavier instrument required for guides' work. Diagram No. 1 represents a light axe or pick, of a kind somewhat similar to that recommended by Mr. Stephen, in a paper published a short time ago in the 'Journal.' It has, in the first place, the great advantage of lightness and handiness, while its single blade, to some extent, combines the step-cutting qualities possessed by the two cutters of the ordinary double-headed axe, though the latter instrument is on the whole decidedly superior. The small hammer-headed axe, though the latter instrument is on the whole decidedly superior. The small hammer-head at the back is added in order to balance the pick, and in some degree to improve the hold when the axe-head comes to be used as a crutch handle. This form, it should be understood, we recommend on account of its lightness and of its convenient shape. Diagram No. 2 represents a travellers' axe, slightly heavier than the first; and as this is the shape which appears to us the best adapted for mountain work of all kinds, we desire shortly to state our reasons for recommending it to members of the Club.

[Fig 1 and Fig 2, shapes of axes, are on this page.].

In the first place it is absolutely necessary that one of the cutters should be made in the form of a pick, as this is by far the best instrument for hacking into hard ice, and is also extremely convenient for holding on to a snow-slope, or hooking into crannies, or on to ledges of rock.

For the other cutter we recommend an adze-shaped blade, and we are convinced that this is the form which will be found most generally useful, as being best suited for all the varieties of step-cutting. The hatchet-shaped blade used by the Chamouni guides is no doubt a better implement for making a staircase diagonally up a slope, but on the other hand it is exceedingly difficult to cut steps downwards with a blade set on in this manner; and as mountaineers rarely come down the way by which they went up, if they can help it, it is obvious that this objection to the Chamouni form of axe is conclusive.

We recommend that the edge of the blade should be angular instead of circular, although the latter shape is more common, because it is clear that the angular edge cuts into frozen snow more quickly and easily.

The curve, which is the same in all the axes, approaches to coincidence with the curve described by the axe in making the stroke. A curve is, in our opinion, desirable, in order to bring the point more nearly opposite the centre of percussion, and to make the head more useful for holding on to rocks or a slope.

The axe shown in diagram No. 2, though slightly heavier than No. 1, is not of sufficient weight or strength for cutting a series of steps in hard ice. To those gentlemen, therefore, who do not object to carrying weight, but who desire to have an axe fit for any kind of work, we recommend No. 3. As this is exactly similar in shape to No. 2, differing from it only in size, we have not thought it necessary to give a separate diagram of No. 3.

As to the mode of fastening, which is the same in all three axes, we should have felt some diffidence in giving an opinion had we not been fortunate enough to obtain the advice of an experienced metal-worker, by whom we were strongly recommended to adopt the fastening shown in the diagrams, as being the method generally considered best in the trade for attaching the heads of hatchets, or large hammers likely to be subjected to very violent strains. It will be seen that the axe-head and fastening are forged in one solid piece, the fastening consisting of two strong braces or straps of steel, which are pressed into the wood about one-eighth of an inch, and are secured by two rivets, pa.s.sed through the wood and clenched on each side. The braces are put at the side, instead of in front of and behind the axe, because by this means, the strain which falls on the axe acts against the whole breadth of the steel fastenings, and not against their thickness merely.

We believe that this is the firmest method of fastening which can be adopted, and that so long as the wood is sound, it is scarcely possible for the head of the axe to get loose or to come off; and it has the further advantage of strengthening the wood instead of weakening it, and of distributing the strain produced by step-cutting over a large bearing.

It should be added that these axe-heads and fastenings ought to be made entirely of steel.