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.
The Tables will be found on the next page.
[Tables I and II appear on p 34].
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.")
[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 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.
[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 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.
[Small drawing illustrating 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 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.