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and

74857 : 1410 : .45087 : 8.494=Tb, and

18.005-8494=bA=9.511 then measure off 9.511 from b to A.

Fig. 90 is drawn to a scale of 8 chains to 1 inch.

Another case of inaccessible distance or obstruction is shown at Fig. 91, where after ranging from a to A the curve runs on to D and L, after passing for some distance through water. -This is by no means an unusual case, where a line runs along the bank of an estuary or along the sea-shore. From the plan measure carefully the distance A D, and make A D some round number, as in the figure where it is made equal to 21 chains, the radius bcing 30 chains.

Divide the length of the arc by the radius, and by means of the quotient, from the table of the lengths of arcs, find the angle subtended by the arc A D.

21 ) 30 ( 7000. Length of arc •69813 corresponds to 40° 0020

7' sufficiently near. •70013 Then the angle subtended by the arc=40° 7' ;

40° 7'

20° 3' 30" = angle of deflection at D; 2 90°—20° 3' 30" = 69° 56' 30" = DA O or 0 AD. Now take out the tabular versine of the angle subtended by the arc AD= 40° 7', and also the sine of the same angle.

Nat. Versine 40° 7' = 23526 x 30= 7.058=A V;
Nat. Sine 40° 7' = 64434 x 30=19:330=V D.

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Supposing that as yet we have been setting out by tangents, &c. and offsets, then the tangent At is already fixed on the ground; with the box or pocket sextant set out A V at right angles to A t, and make A V=7.058; and at V set out V D at right angles to A V, and make V D=19-330.

Now at A make the angle DAO=69° 56' 30", when the visual line will intersect the point D measured off from V.

Το carry on the course of the curve from D to L. The three angles of a triangle being equal to 180°, and O V D being one of them, observe that 90° — 40° 7' = 49° 53' = the angle V ĐO; by means of this angle VDO, and then at D, set off T D at right

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147

angles to DO; the curve may then be continued on to L by tangents, &c. and offsets as before.

If setting out by theodolite, then at A fix A t, by the method already shown, reverse the telescope and make the angle TAD = 20° 3' 30", and also make A V at right angles to A t, and from V, set out V D at right angles to A V; if D is sighted off from V by an assistant at the same time that the angle TAD is being set off, the visual intersection will be at D, and V D will measure as above 19:33. To carry on the curve from D, plant the theodolite at that point ; fix the vernier at 21 times the tangential angle for 1 chain, there being 21 chains from A to D, and move round to zero; then reverse the telescope which fixes DT.

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very analogous case of inaccessible distance, but requiring different treatment, occurs in Fig. 92, when the curve A K crosses a river at b and c. The point b we may set out by any of the methods above explained, but not so the point c, for we cannot measure from b to c; neither can we set out the versine A V, nor the sine V C.

By the means already explained, after measuring Ac on the plan, we obtain the angle at the centre, half of which is equal to the angle of deflection. Having the angle at the centre, we obtain the length of the tangent, which set off as at AT. Take out the secant and subtract the radius, which gives T d. To obtain the angle dTA, subtract the angle at the centre from 180°, and divide the remainder by 2, which gives the angle Od A, O being the centre and left out in the figure to save space; 90° — the angle od A = the angle TdA; and 180° — TDA+DAT= A d T, which last angle set out at T. Now at d make the angle Ad V=to the angle of deflection, and at A set out the angle of deflection TA c, and the intersection at e will give the point on the curve which was required.

Having the angle at the centre, we may set out the sine of the angle, as at A C, and the versine at C, at right angles to A C, will give the point d.

A perfect understanding of the above very simple, though perhaps very tedious remarks, will enable us to find out another way to extricate ourselves from some of the difficulties which often arise, more particularly in uncultivated districts, where we may be at times, to a certain extent, feeling our way, and where time is precious.

In Fig. 93, let the curves and the tangents be altogether in the midst of a large wood, through which we have reached our

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way as far as A; required to set out the curve A L, with a radius of 20 chains; the direction of the tangent A T, however short it may be, is fixed by our previous work. With the theodolite at A, and the telescope on A T, pick out some line A D, on which there is a clearing, or where it is most easy to make one. Measure the angle T AD, and subtract it from 90°, which leaves the angle DAO=ADO; twice either of these subtracted from 180°, gives the angle at the centre subtended by the chord or arc A D.

Thus, let TAD=12° 15'; 90° -12° 15'=77° 45'=DA O or A D 0 ;

180°—(77° 45' x 2)=24° 30'=the angle at the centre. Observe that twice the sine of half the angle at the centre is equal to the chord of that angle; the sine of 12° 15'= 21218;

21218 x 2 x 20 (radius)=8:487. Then 8·487, set out in the direction A D, will be a point on the curve.

To find the chainage of this on the arc itself, take out from the Table, giving the lengths of circular arcs to radius 1, the tabular numbers to 24° 30', and multiply them added together by the radius, which will give the length of the arc, thus, Tab. No. 24° = '4188790

0-30' = 87266

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•4276056 × 20 = 8.552 = length of arc,

Х which added to the chainage at A, will give that at D.

Removing the instrument, with the verniers clamped to the point D, bringing the instrument to bear bodily on A, and there clamp it; then setting the verniers to 360° and 180°, will give the direction of the tangent TTS, when we may repeat the operation for a longer arc, as D L, or a shorter one, as d, as circumstances may make convenient.

CHAPTER XI.

The Variation of the Magnetic Needle.--Rules to find the

Variation.

To find the variation of the needle. Having found a convenient spot, set up a prismatic compass, when the sun's lower edge or limb is a semidiameter above the horizon,t take the bearing of its centre from the north or south, whichever is nearer ; the bearing subtracted from 90° will be the sun's magnetic amplitude, from the east or west by the needle. Now to find the

I sun's true amplitude for that day.

As the cosine of the latitude is to radius, so is the sine of the sun's declination|| at setting or rising to the sine of his amplitude from the west or east, which will be north or south, as the sun's declination is north or south, and the distance in

* That is the variation of the maguetic north, as shown by the needle, from the true north ; this variation is different at different parts of the world and there is also a diurnal variation.

† At this time, although apparently elevated, on account of the refraction of the atmosphere, as explained in the article “ Sextant,” the centre is in fact about on the horizon.

# The magnetic amplitude is an arc of the horizon, contained between the sun or star at its rising or setting, and the magnetical east or west point of the horizon, indicated by the magnetical compass, or the azimuth compass ;or it is the difference of the rising or setting of the sun, from the east or west points of the compass. The true amplitude is an arc of the horizon, intercepted between the true east or west points, and the centre of the sun or a star at its rising or setting; the amplitude is therefore of two kinds, eastern or western; these are also northern or southern, accordingly as they fall in the north or south quarters of the horizon. The complement of the amplitude, or what it wants of 90°, or the angular distance the point of rising or setting from the north or south point, is called the AziMUTH. Azimuths, called also vertical circles, are great circles of the sphere intersecting each other in the zenith and nadir, and cutting the horizon at right angles.

§ The latitude of a place on the terrestrial globe is its angular distance from the equator. It is measured on the meridian, being that part of it which is intercepted between the zenith of the place and the equator. If the place is situated north of the equinoctial line, it is said to have north latitude ; if on the other side it is said to have sonth latitude.

|| The necessary tables for declination and corrections will be found in the Nautical Almanuc.

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degrees and minutes between the true east or west, and the magnetic, is the variation of the needle. To prevent mistakes, sketch the horizontal circle to represent the visible horizon, and on it the several data or angles, when it will easily be seen how the variation is found, whether by addition or subtraction, and on which side of the north it lies.

Suppose the variation sought at sunset; draw by hand the circle N WS E, Fig. 94, to represent the horizon ; the centre C will be the observer's station ; through this draw N S and W E; then on the north or south of W, accordingly as the sun sets north or south of the true west, draw Co, representing the position of the sun at its setting; and another line C w, for the magnetic west, either north or south of O, as observed to be; by means of these it will be easily seen whether the magnetic amplitude and true amplitude are to be added or subtracted, to give the variation. In our diagram w o, is the magnetic amplitude, and W O the true amplitude; the angle W o must therefore be subtracted from w O, to give Ww, the distance of one from the other; and n, the magnetic north, 90 degrees from w, must be westward of Ń, the true north. Observe also, that in the above rule the fourth proportional "found is the sine of the amplitude, and therefore opposite to this sine in the tables we shall find the angle or amplitude; and this will be of the same name as the declination ; that is north, when the declination is north, and south, if the declination is south.

A second method, more simple, though it takes more time, is to observe the azimuth of the sun at equal altitudes on different sides of the meridian, which will be at equal hours before and after noon ; the mean reading will give the true meridian.

An approximation may be determined by setting up a rod vertically on a level piece of ground, having first, with the length of the rod as radius, and from the exact spot where it is to be set up as centre, described an arc of a circle, and noting where this arc is intersected by the shadow of the rod a few hours before noon, and then again at the same time in the afternoon; a line through the middle point on the arc between the intersections, and the foot of the rod will be the meridian.

Whichever of these two ways is adopted, the difference between the meridian thus found, and the magnetic, will be the variation, which should be shown on all maps of whatever description.

When the meridian has been thus determined, it should be permanently staked out, unless fixed objects can be found to answer the same purpose, and this more particularly on exten

sive surveys.

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