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H, through two marks having a considerable angular distance, and turning the vernier plate 180°, H continues to pass through the same points.

In

A

H

H

Fig. 814.

The vertical wire o will be in a plane perpendicular to the horizontal axis A, when, moving the vertical circle backward and forward, a distant and well-defined mark continues accurately on v. order to know whether the horizontal wire, h, be in its proper position, or if the line of collimation, H, when the vertical circle indicates zero, be perpendicular to the vertical axis, A., it is only necessary to reverse the telescope, and that the wire h is in a plane parallel to the azimuth circle, will be determined by a backward and forward motion of the vernier plate. When these adjustments shall have been perfected, by often repeating them one after another in a different order, whether the levels are parallel to the plane of the azimuth circle will be known by leveling this circle, making the vertical circle indicate zero, if its telescope have a level attached to it, revolving the vernier plate and seeing if the bubbles continue in the middle. Whether the level which may be attached to the telescope be perpendicular to the axis A, will be known by bringing A, over two of the leveling screws, and then, by aid of these screws flinging A out of level, or by revolving the telescope in its Ys, if it be capable of such a motion.

If, on any occasion, it be desired to make the vertical circle coincide with the greatest possible accuracy with a vertical plane, we may suspend a plumb line before the telescope and observe when the line of collimation traces this line.

The Variation of the Magnetic Needle may be conveniently determined with the theodolite by the process of equal altitudes.

Let the magnetic bearing of the sun before noon at a determinate altitude be e°, and at the same altitude after noon f°, and suppose x = variation; then will the angular distances of the sun from the true meridian be, before noon, e±x, afternoon, ƒx, but these distances are equal,

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The equal altitudes may be taken at any corresponding hours before and after noon, and in any season of the year, but the most favorable time is about the 21st of June, when the sun, being near the summer solstice, will change his declination but a few seconds

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between the observations; and, if these determinations be made several successive days, both before and after the 21st, the errors, lying in opposite directions, will tend to balance each other; thus, suppose the values of x

for June to be

then

18, 19, 20, 21, 22, 23, 24,

X1, X2, X3, X4, X5. X6. Xq;
x1+x2+x3+ X1 + Xs + Xs + X1
7

The observations may be made several times during the same day, but the best hours will be when the sun is in a position nearly

east or west.

The Art of Leveling consists in finding the difference of elevation between two places: and is not only necessary in the construction of railroads, aqueducts, and canals, but is useful for laying the foundations of edifices, and for other like purposes. The operation may be readily executed by aid of the theodolite, or, more conveniently, with the Leveling Instrument made expressly for the purpose, and consisting of a telescope, level, and tripod, being in all respects similar to the theodolite, except in not possessing the graduated circles. The instrument is to be firmly planted midway between two stations, situated at a convenient distance, and its telescope made to revolve accurately in the horizontal plane, when the depressions of the first and sec

ond stations are to be noted by sighting at a graduated staff held vertically on these two points in succession. In like manner the second station is to be compared

B

Fig. 82.

with a third, the third with a fourth, and so on, to the last.

For an example, suppose it required to find the elevation of the fountain, A, above a dwelling house at B, from the following notes, in which the back altitudes are marked and the forward

ones

2 +1-25

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Through a given point to draw a line perpendicular to a given

line.

Join the given point, P, and any convenient point, Q, of the given line; with the middle, O, of the line PQ as a centre, describe the semicircumference PP'Q, cutting the given line in P'; the line drawn through P, P', will be the perpendicular required. Why?

Fig. 83.

If the given point be P' in the given line, set one foot in P' and the other in any convenient point, O, out of the line, describe a circle and draw the diameter QOP; PP' will be the perpendicular required.

Other methods of drawing perpendiculars may be employed, as indicated in the figures, but the "Right-angle" is preferable in practice.

$ + 4

Fig. 833.

Fig. 834.

Fig. 832.

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X

Through a given point to draw a line parallel to a given line. Set one foot of the dividers in the given point, P, and with any convenient centre describe the circumference PBAX, cutting the given line in A and B; take the chord PB and apply it from A upon the circumference at X; PX will be the parallel required. (Why?) Do the same thing with the Rightangle and Straightedge.

Fig. 84.

PROBLEM III.

From a definite point in a given line to make an angle equal to a given angle.

Around the given angle, A, with any convenient radius, AB, describe the arc BC; around the given point, P, with the same radius, describe the arc QR, Q being a point in the given line; apply the chord BC in QR; QPR will be the required angle.

B

P

R

Fig. 85.

PROBLEM IV.

To construct a triangle, having given its three sides. Draw an indefinite line, KL, in the required position, and apply one of the given sides, C, from K to L; with the other sides, A, B, as radii, describe around the centres, K, L, arcs intersecting in Q; KQL will be the triangle required.

K

Fig. 86.

Scholium I. This problem enables us to plot a field when it is surveyed by the chain, that is, when its diagonals are known, either by actual measurement or by computation. The student will find it a profitable exercise to plot the pentagon given in the preceding section under the chain.

Scholium II. The preceding Graphical Problems give us the six following Problems of Construction, whereby any geometrical problem, solved algebraically, may be executed in a geometri

cal way.

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Lay off b from one extremity of a towards the other, as from the right hand towards the left in the figure; the remainder will be = x.

We observe that if b exceed a, x will be drawn to the left instead of to the right (180), and that x will be minus instead of plus in the equation x = a — b.

Fig. 88.

>], ༩ ;

Fig. 882.

PROBLEM VII.

To construct the square root of the sum of the squares of two lines,

x = (a2+b2).

Make a and b the sides of a right angled triangle;

will be the hypothenuse.

PROBLEM VIII.

Fig. 89.

To construct the square root of the difference of the squares of two lines.

x = (a2 — b2)‡.

On the greater line, a, describe a semicircle, and from the extremity of a lay off the chord b; x will be the chord joining the extremities of a and b.

a

Fig. 90.

PROBLEM IX.

To construct a fourth proportional,

bc

abc: x, or ax = bc, or x =

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