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William Shepherd.

Systematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) online

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determined from one or more triangles, which connect with
a shorter and temporary base assumed near the beach. A
boat then explores the othng j and at every rock, shallow, or
remarkable sounding, the bearings of the station staves are
noticed. These observations furnish so many triai^les, from

These staves will be described in the next article, IneUing.



LEVELLING. 531

which the situation of the several points are easily ascertained.
When a correct map of the coast can be procured, the
labour of executing a maritime survey is materially shortened.
From each notable point of the surface of the water, the
bearings of. two known objects on the land are taken, or the
intermediate angles subtended by three such objects are ob-
served. To facilitate the last construction, an instrument called
the Station- Pointer has been invented, consisting of three
brass rulers, which open and may be set at the given angles."

LEVELLING.

Levelling is the art of finding a line parallel to the hori-
zon at one or more stations, to determine the height or depth
of one place with respect to another, for the purposes of
laying out grounds, draining morasses, conducting water, &c.
Two or more places are on a true level when they are equally
distant from the centre of the earth. Of course one place is
higher than another, or out of the level with it, when it is farther
from the centre of the earth ; and a line equally distant from
that centre, in all its points, is called the true level. Hence,
since the earth is spherical, that line must be a curve, and
make a part of the earth's circumference, or at least one that
is parallel, or concentrical to it.

' The practice of levelling therefore consists : 1 . In finding
and marking two or more points that shall be in the circum-
ference of a circle, whose centre, and that of the earth, shall
coincide ; and 2. In comparing the points thus found with
other points, to ascertain the difference in their distances
from the earth's centre.

With respect to the theory of levelling, the following obser-
vations may be set down : A plumb line, hanging freely in the
air, points directly towards the centre of the earth ; and a line
drawn at right lines, crossing the direction of the plumb-line,
and touching the earth's surface, is a true level only in that
particular spot ; but if this line, which crosses the plumb, be

2 M G



532 MATIIF.MATICS.

continued for any considerable distance, it will rise above the
earth's surface ; and the apparent level will be above the true
one, because the earth is spherical, and this rising will be as
the square of the distance to which the said right line is pro-
duced, that is, as much as it is raised above the earth's surface
at one mile's distance, it will rise four times as much at the
distance of two miles, nine times at the distance of three
miles, and so on. This circumstance is, as we have already
observed, owing to the globular figure of thg earth, and the
rise is the JifFerence between the true and apparent levels,
the real curve of the earth being the true level, and the
tangent to it, tlie appareirt level. It appears tlrat the less
distance we take between any two stations, the truer will be
the operations in levelling; and as soon as the difference
between the true and apparent levels becomes perceptible,
it is necessary to make an allowance for it, even if the dis-
tance between two stations does not exceed a few chains in
length.

Levelling may be performed very expeditiously by the
assistance of a large theodolite, capable of measuring with
precision the vertical angle subtended by a remote object ;
the distance being known or calculated, and allowance made
for the effect of the earth's convexity, and the influence of
refraction. But the better method is to employ a spirit-level ^
accompanied by a pair of square staves, each of which is
composed of two parts that slide out into a rod of ten feet in
length, every foot being divided centesimally. A vane slide*
up and down upon each set of these staves, which, by brass
springs, will stand at any given height. These vanes are about
ten inches long, and four broad : the breadth is first divided
into three -wjual parts, the two extremes painted white, the
middle space divided again into tlucc equal parts, which are
less; the middle one of tlicin is also [>ainte(I white, and the
two other parts black ; and thus they are suited to all com-
mon distances. These vanes have each a brass wire across -a



LEVELLING. 533

small square hole in thf centre, which serves to point out the
height correctly, by coinciding with the horizontal wire of the
telescope of the level.

Levelling is distinguished into two kinds, the simple and
the compound ; the former, which rarely admits of applica-
tion, assigns the difference of altitude by a single observation ;
but the latter discovers it by means of a series of observations
carried along an irregular surface, the aggregate of the several
descents being deducted from that of the ascents. The staves
are therefore placed successively along the line of survey, at
propef intervals, according to the nature of the ground, and not
exceeding three or four hundred yards, the levelling instru-
ment being always^ planted nearly in the middle between them,
and directed backwards to the first staff, and then forwards to
the second. The difference' between the heights intercepted
by the back and the fore observation, must evidently give at
each station the quantity of ascent or descent; and the error
occasioned by the curvature of the globe mfty, in very short
distances be overlooked, as it will not amount at each station
to the hundredth part of a foot. The final result of a series
of operations, or the differences of altitude between the ex-
treme stations, is discovered by taking the measures of the
back and fore observations collectively, and the excess of the
latter above the former, indicates the entire quantity of
descent.

The following observations wJH, render the whole subject
easy and clear to any comprehension.

To find the distance between (he apparent and true level at
the distance of a mile. In the right-angled triangle ABC,
fig, 7, the side A C, the semi-diameter of the earth, is given,
suppose 3,982 miles, and the side AB=l milej to find the
hypothenuse C B. .t-""'^

CA2=158o6324
AB2- 1



CA^ + A B^^: 15856325= CB2



And / 158563'25=3982.0001253=CB.



534 MATHEMATICS.

Consequently the apparent level at the distance of one mile
from the observer's station is higher than the true level by
EB=:CB CE = .OOOI25 of a mile=S inches nearly.

The same tiling may be done another way. By Eucl. III.
36. EC + EBxEB=AB2; consequently, 2 EC + EB:
A B : : A B : : E,B. But E B in the first term is so small
in comparison of E C, that it may be neglected, therefore it

willbe2EC:AB:: AB:EB, and B=4^'=4S^

' 2EC 2EC

nearly ; that is, the ditterence between ihc true and apparent

level is equal to the square of the distance between the places,

divided by the diameter of the earth ; and since the diameter

of the earth is a given quantity, it is always proportional to

the square of the distance.

The diameter of the earth = 79t>4 miles, if A E be equal

A E ^

1 mile as before, then the difference of levels will be - -

' 2EE

- >. -=: 8 inches nearly. If AE=:2 miles, then the dif



2'



= 7964

ference of levels will be z^gr '-(w^u, ^^ inches, &c.

Hence, proportioning the excesses in altitude according to
the squares of the distances, the following Table is obtained,
shewing the height of the apparent, above the true level, fof
every 100 yards of distance on the one side, and for each mile
on the other.

TABLE I.



Dist.

or

AE.



Yards.

100

200

1000



Diff. of
L^or



Inches. Miles



Dist.

or

AE.



0.026
0.103
2..'>7 1



Diff. of

Level, or>

E R. I



Feet. In.

2
8
2 8



Dist.

or
AE.



Miles.

4
6
8



Diff. of

Level, or

B.



Feet. In.

10 7
23 11
42 6



Dist. Dift". of
or Level, or
AE. EB



Miles.

10

j 12
i 14



Feet. In.
66 4

97 7
130 1



LEVELLING. 535

As, however, levelling is usually performed by measures
consisting of chains instead of yards, the following Table is
calculated, shewing the difference of a true and apparent level,
in distances of from 5 to 100 chains.



TABLE n.



Dist.


Diff. lev.


i>t.


Difl. lev.


Dist.


Diff. lev.


Dist.


Diff. lev.


in




in




m




in




chains.


sub. in.


chains.


sub. in.


chains.


sub. in.


chains.


sub. in.


5


0.03


30


1.12


55


3.78


80


8.00


10


0.12


55


1.53


60


4.50


85


9.03


15


0.28


40


2.00


65


5.31


90


10.12


20


O.oO


45


2.53


70


6.12


95


J 1.28


25


0.78


50


3.12


75


7.03


100


12.50



Example. It is required to find if water can be brought
from a spring on the side of a hill at R, fig. 8, to the house
S, by means of pipes laid under ground.

Fix the theodolite x at any convenient distance from the spring,
suppose five chains, then enter on a paper, the difference
between the true and apparent level from the table No. II,
0.03 inches : let an assistant place a staff at a, the spring, and
another assistant, a staff at b towards S : the spirit level is to
be directed to a, and the assistant there is to place the vane
so as, when the staff is held perpendicularly, the hair in the
telescope may cut the middle of the vane, suppose at 6.5 inches,
which is to be entered in the paper, under the back-sights : let
the other assistant now take tlie fore-sight at the staff b, which
suppose to be 3 feet, 6.4 inches, which is to be entered under
the fore-sights : then take the measure of the distance of the
instrument to the staff b, suppose 15 chains, which enter in
column Dki. and ih the next column against it; is the dif^



536 MATHEMATICS.

ference between the tnie and apparent level taken from the
table, or 0.28 inches. Measure next to the second station
suppose 10 chains, which cuter, and against it write the
difference between the true and apparent level 0. 1 2 ; take the
back-sight to the staff b, viz. 1 foot, 2.7 inches, which is to
be entered as before, then take the fore-sight to the staff c, and
so on, whatever be the number of stations. When the obser-
vation is carried to S, and the level finished, the columns in
the paper, or field-book, are to be added up ; and from the
sum of the heights, under the back-sights, take the sum of
the differences of the level under the same, the remainder will
be the correct sum of the heights of the vane : the same is to
be done under the fore-sights, and the difference of these
corrected heights gives the diflference of the true level ; and
the sum of the distances under both distances will be the
whole distance from R to S, thus :



The first Assistant's Notes, orThe second Assistant's Notes,'



Sta-
tiot.

.1

2
3



Back-Sights.



Height V.
Feet. Id.



6.5
2.7
0.8



3 10.00
0.43



957



Dist. in
chains

5
10
15



30



Diff. lev
inches.

0.03
0.12
0.28



0.43



or Fore-Sights.



Sta-
tion.

1

2
3



Height V.
Feet. In.

3 6.4
2 6.9

4 6.5



7.8
9.57



6 8.23



Dbt. in
cliains

15
10
13



38
30



6S



Diff. lev.
iuches.

0.28
0.12
0.12



0.52



V



Here it is evident, that the sum of tn^. heights of the vanes
in the fore- sights exceeds those of the back-sights, by rather
more than 6 feet 8 inches, and so much is S below the tnie
level of R, and the whole distance fro'm R to S being 68



LEVELLING. 53?

chains, water may be readily brought from the spring on the
hill, to the house in the valley.

The table, No. I, will answer several useful purposes, as
follow,

1. To find the height of the apparent level above the true,
at any distance. If the given distance be contained in the
table, the correction of level is found on the same line with
it : thus at the distance of 1000 yards, the correction is 2.57,
or somewhat more than Q.\ inches^ but at the distance of 10
miles, it is 66 feet 4 ineh*- If the caa\.t distance be not
found ill the table, multiply the square of the distance in
yards by 2.57, and divide by KXXyj^, or cut off six places on
the right hand for decimak : the rest are inches ; or multiply
the square of the distance in miles by 66.4, and divide by
102=100.

2. To find the extent of the visible horizon, or how far
can be seen from any given height on a horizontal plane
at sea, as on the frozen ocean. Suppose the eye of an ob-
server be at the height of 6 feet above the surface of the
earth, he will see 3 miles every way. Hence it is plain, that a
man thus situated, will be able to see another person of the
same height at the distance of 6 miles. If the eye be
situated on a ship's mast 130 feet high, it will see an extent
of 14 miles every way. If a light house be 130 feet above
the level of the sea, it will be visible to an eye on the surface
at the distance of 14 miles ; but if the observer mount
the mast 97 feet, then the distance is increased 12 miles,
consequently the light-house will be visible at 26 miles
distant.

3. Suppose a spring be on one side of a hill, and a house
on an opposite hill, with a valley between them : and that the
spring seen froia tlip. house apppa^* ^y a levelling instrument
to be on a level with the foundation of the house, which is at
a mile distance from it : then is the spring 8 inches above the
true level of the house ; and this difference would be barely
sufficient for the water to be brought in pif^s from the



55S MATHEMATICS.

spring to the house, the pipes being laid all the way in the
ground.

4. If the height or distance exceed the limits of the table :
then if the distance be given, divide it by some aliquot part
till the quotient come within the distances in the table, and
take out the height answering to the quotient, and multiply it
by the square of the divisor for the height required. So if

40
the top of a hill be seen %t 40 miles distance, then = 10,

..-r^ 4

to %vhich 66} answers in the table : now 66^ x 4, o 6&i x 16,
gives 106 1^ feet for the height of the hill.

When tlie height is given, divide it by one of these square
numbers, 4, 9, 16, 25, 36, &c. till the quotient come
within the limits of the table, and multiply the quotient by the
square root of the divisor, that is, by 2, 3, 4, o, 6, &c. for
the distance sought. Thus when the top of Teneriffe, said to
be 15840 feet high, just comes into view at sea, divide
15840 by 225, or the square of 15, and the quotient is about
70 feet, to which 102^ miles will answer : and lOf x 15 154f^
miles for the distance of the mountain from the place of
observation.

DIALLING.

Dialling is the art of drawing dials on the surface of any
given body, whether plane or curved : it is founded on the
apparent motions of the heavenly bodies, or rather on the real
diurnal motion of the earth.

A dial is a plane, upon which lines are described in such a
manner, that the shadow of a wire, or of the upper edge of a
plate, called a btile, or gnomon, erected perpendicularly on the
plane of thi dial, may shew the time of the day.

The universaV principle on uliIcU dialling depends, is this :
if the whole earth a P c p, fig. 9, were transparent and
hollow, like a sphere of glass, and had its equator divided in
twenty-four parts by so many meridian semicircles, a, b, c, d,
&c. J one 6f which is the geographical meridian of any given



DIALLING. 559

place, as London, which may be supposed to be at the point
a; and if the hours of XII. Mere marked at the equator, both
upon that meridian and the opposite one, and all the rest of
the hours in order on the other meridians, tliose meridians
wouUi be hour-*circIes answering to the latitude of London :
then if this glass sphere bad an opaque axis P E ^j terminating
in the poles P and p, the shadow of the axis would fall upon
every particular meridian and hour, when the sun came to tlie
plane of the opposite meridian ; and would consequently shew
the time at London, a<l at all uilier places on the meridian of
London.

If this sphere were cut through the middle by the plane
ABCD, in the rational horizon of London, one half of the
axis EP would be above the plane, and the other half below
it ; and if straight lines were drawn from the centre of the
plane, to tliose points where the eircumference is cut by the
hour circles of the sphere, those lines would be hour lines of
a horizontal dial for London ; for the shadow of the axis would
fall upon each particular hour-line of the dial, when it fell
upon the like hour circle of the sphere.

If the plane, which cuts the sphere, be upright at AFGG,
fig. 10, touching the given place at F, and directly facing the
meridian of London, it will become the plane of an erect south
dial ; and if right lines be drawn from its centre E, to those
points of its circumference where the hour-circles of the sphere
cut it, these will be the hour-lines of a vertical or direct south
dial for London, to which the hours are to be set, as in the
figure ; and the lower half Ep of the axis will cast a shadow
on the hour of the day in this dial, at the same time that it
would fall upon the like hour-circle of the sphere, if the dial
plane were not in the way.

Without enicrJng into the miiiwii*^f incliniug and reclining
dials, &c., we may observe generally, that the plane of every
dial represents the plane of some great circle upon the earth,
and the gnomon the earth's axis, whether it be a small wire,
as in the figure just referred to, or the edge of a thin plate, as
in common horizontal dials.



540 MATHEMATICS.

Dialling may be performed by a common globe, which is t
be elevated to the latitude of the given place, and turned about
until any one of the twenty-four meridians cuts the horizon in
the north point, where the hour of XII is supposed to be
marked, the rest of the meridians will cut the horizon at the
respective distances of all the other hours from XII. Then
if these points of distance be marked on the horizon, and the
globe be taken out of the horizon, and a flat surface be put
into its place, even with the surface of the horizon, and if
straight li" *c Urawu nrom Uic bnr f the board to those
points of distance on the horizon whiph were cut by the iwe-
ty-four meridian semicircles, these lines will be tlie hour-lines
of a horizontal dial for that particular latitude, the edge of
whose gnomon must be in the very same situation that the
axis of the globe was, before it was taken out of the horizon :
that is,, the gnomon must make an angle with the plane of the
dial, equal to the latitude of the place for which the dial is
made.

For the practice of dialling either with the globe, or by the
principles of spherics, we must refer the reader to the works
now to be mentioned. Of these, for persons not conversant
in mathematics, the second volume of Brewster's edition of
Ferguson's Lectures will be reckoned quite sufficient. Ley-
bourn's Diallii^ Bion in his Use of Mathematical Instruments
the works of Leadbeater and Emerson may be consulted
with advantage. Mr. W. Jones has a work on Instrumental
Dialling, and Dr. Horsley has treated on the subject in his
mathematical tracts. Several ingenious constructions of dials
are to be found in Montucla's Recreations, as well as in
Ferguson's Lectures.

END OF VUL,. I.



J. >rCreery, Printer,
Bbek-Uone-Comrt, Ftet-Strect, Londou.



FoU.



MATHEMATrcSc




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Online LibraryWilliam ShepherdSystematic education: or Elementary instruction in the various departments of literature and science; with practical rules for studying each branch of useful knowledge (Volume 1) → online text (page 44 of 44)