MEASUREMENTS.
Distance.--To measure the Length of a Journey by Time.--The pace of a caravan across average country is 2 1/2 statute, or 2 geographical, miles per hour, as measured with compasses from point to point, and not following the sinuosities of each day's course; but in making this estimate, every minute lost in stoppages by the way is supposed to be subtracted from the whole time spent on the road. A careful traveller will be surprised at the accuracy of the geographical results, obtainable by noting the time he has employed in actual travel. Experience shows that 10 English miles per day, measured along the road--or, what is much the same thing, 7 geographical miles, measured with a pair of compasses from point to point--is, taking one day with another, and including all stoppages of every kind, whatever be their cause,--very fast travelling for a caravan. In estimating the probable duration of a journey in an unknown country, or in arranging an outfit for an exploring expedition, not more than half that speed should be reckoned upon. Indeed, it would be creditable to an explorer to have conducted the same caravan for a distance of 1000 geographical miles, across a rude country, in six months. These data have, of course, no reference to a journey which may be accomplished by a single great effort, nor to one where the watering-places and pasturages are well known; but apply to an exploration of considerable length, in which a traveller must feel his way, and where he must use great caution not to exhaust his cattle, lest some unexpected call for exertion should arise, which they might prove unequal to meet. Persons who have never travelled--and very many of those who have, from neglecting to analyse their own performances--entertain very erroneous views on these matters.
Rate of Movement to measure.--a. When the length of pace etc., is known before beginning, to observe.--A man or a horse walking at the rate of one mile per hour, takes 10 paces in some ascertainable number of seconds, dependent upon the length of his step. If the length of his step be 30 inches, he will occupy 17 seconds in making 10 paces. Conversely, if the same person counts his paces for 17 seconds, and finds that he has taken 10 in that time, he will know that he is walking at the rate of exactly 1 mile per hour. If he had taken 40 paces in the same period, he would know that his rate had been 4 miles per hour; if 35 paces, that it had been 3.5, or 3 1/2 miles per hour. Thus it will be easily intelligible, that if a man knows the number of seconds appropriate to the length of his pace, he can learn the rate at which he is walking, by counting his paces during that number of seconds and by dividing the number of his paces so obtained, by 10. In short the number of his paces during the period in question, gives his rate per hour, in miles and decimals of a mile, to one place of decimals. I am indebted to Mr. Archibald Smith for this very ingenious notion, which I have worked into the following Tables. In Table I., I give the appropriate number of seconds corresponding to paces of various lengths. I find, however, that the pace of neither man nor horse is constant in length during all rates of walking; consequently, where precision is sought, it is better to use this Table on a method of approximation. That is to say, the traveller should find his approximate rate by using the number of seconds appropriate to his estimated speed. Then, knowing the length of pace due to that approximate rate, he will proceed afresh by adopting a revised number of seconds, and will obtain a result much nearer to the truth than the first. Table I. could of course be employed for finding the rate of a carriage, when the circumference of one of its wheels was known; but it is troublesome to make such a measurement. I therefore have calculated Table II., in terms of the radius of the wheel. The formulae by which the two Tables have been calculated are, m=l x 0.5682 for Table I., and m=r x 3.570 for Table II., where m is the appropriate number of seconds; l is the length of the pace, or circumference of the wheel; and r is the radius of the wheel.
b. When the length of Pace is unknown till after observation.--In this case, the following plan gives the rate of travel per hour, with the smallest amount of arithmetic.
For statute miles per hour--Observe the number of paces (n) taken in 5.7 seconds: let i be the number of inches (to be subsequently determined at leisure) in a single pace; then ni/100 is the rate per hour.
For geographical miles per hour--The number of seconds to be employed is 5. This formula is therefore very simple, and it is a useful one. (A statute mile is 1760 yards, and a geographical mile is 2025 yards.)
For finding the rate in statute miles per hour in a carriage--Observe the number of revolutions (n) made by the wheel in 18 seconds: let d be the number of inches in the diameter of the wheel; then n d/200 is the rate per hour.
The above method is convenient for measuring the rate at which an animal gallops. After counting its paces it may be through a telescope, during the prescribed number of seconds, you walk to the track, and measure the length of its pace. If you have no measuring tape, stride in yards alongside its track, to find the number of yards that are covered by 36 of its paces. This is, of course, identical with the number of inches in one of its paces.
Convenient Equivalents.--The rate of 1 mile per hour, is the equivalent to each of the rates in the following list:--
Yards. Feet. Inches. 29.333, or 88.000, or 1056.000, in one minute or 0.488, or 1.466, or 17.600, in one second
Measurement of Length.--Actual measurement with the rudest makeshift, is far preferable to an unassisted guess, especially to an unpractised eye.
Natural Units of Length.--A man should ascertain his height; height of his eye above ground; ditto, when kneeling: his fathom; his cubit; his average pace; the span, from ball of thumb to tip of one of his fingers; the length of the foot; the width of two, three, or four fingers; and the distance between his eyes. In all probability, some one of these is an even and a useful number of feet or inches, which he will always be able to recollect, and refer to as a unit of measurement. The distance between the eyes is instantly determined, and, I believe, never varies, while measurements of stature, and certainly those of girth of limb, become very different when a man is exhausted by long travel and bad diet. It is therefore particularly useful for measuring small objects. To find it, hold a stick at arm's-length, at right angles to the line of sight; then, looking past its end to a distant object, shut first one eye and then the other, until you have satisfied yourself of the exact point on the stick that covers the distant object as seen by the one eye, when the end of the stick exactly covers the same object, as seen by the other eye. A stone's throw is a good standard of reference for greater distances. Cricketers estimate distance by the length between wickets. Pacing yards should be practised. It is well to dot or burn with the lens of your opera-glass a scale of inches on the gun-stock and pocket-knife.
Velocity of Sound.--Sound flies at 380 yards or about 1000 feet in a second, speaking in round numbers: it is easy to measure rough distances by the flash 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 11 1/4 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 11 1/4 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 11 1/4 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 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.
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 sextant 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-glass.--Either dry sand or water may be used in an hour-glass; if water be used, the aperture through which it runs must, of course, be smaller.