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PROBLEM. 173. To lay out a given quantity of land in a rectangular form, huring one of the sides given.
Divide the given area, reduced to square chains or square rods, by the given side of the required rectangle; the quotient is the other side.
Example 1.- To lay off 2 10 acres in a rectangular form, one of the sides being given, equal to 80 rods.
2410A=2100 square chains=38 100 square rods. 8.0)38 10.0( 180 rods :—the required side of the rectangle.
174. A great number of similar problems might be proposed. The solution of them does not properly belong to surveying, but combines the application of geometrical and algebraical principles; it is tracing the lines only that belongs to surveying. The manner of tracing lines, having been already explained, it seems unnecessary to add the numerous examples usually given under this head of the subject.
OF THE VARIATION OF THE COMPASS. 175. The line, indicated by the magnetic needle when it is allowed to move freely about the point of support, has been named the magnetic meridian (140); and is, in general, a different line from the true meridian, which is determined by a plane passing through the place and the axis of the earth.
176. The angle which the magnetic meridian makes with the true meridian at any place on the surface of the earth, is called the variation of the needle at that place, and is east or west, according as the north end of the needle lies on the east or west side of the true meridian.
177. The variation is different at different places, and even at the same place it does not remain constant for any length of time. The variation is ascertained by comparing the magnetic, with the true meridian.
178. The best practical method of determining the true meridian of a place is by observing the north star. If this star were precisely at the point in which the axis of the earth,
the vertical plane passed through it and the place, with the surface of the earth, would be the true meridian. But, the star being at a distance from the pole, equal to 1° 35' 41", it performs a revolution about the pole in a circle, the polar distance of which is 1° 35' 41": the time of revolution being 23 h. and 56 min.
To the eye of an observer, this star is continually in motion, and is due north but twice in 23 h. 56 min., and is then said to be on the meridian. Now, when it departs from the meridian, it apparently moves east or west, for 5 h. and 59 min., and then returns to the meridian again. When at its greatest distance from the meridian, east or west, it is said to be at its greatest eastern or western elongation.
The following tables show the times of its greatest eastern and western elongations.
The eastern elongations are put down from the first of April to the first of October; and the western from the first of October to the first of April; the time is computed from 12 at noon. The western elongations in the first case, and the eastern in the second, occurring in the day time, cannot be used. Some of those put down, are also invisible, occurring before daylight is gone in the evening, or after daylight in the morning. In such case, if it be necessary to determine the meridian at that particular season of the year, let 5 h. and 59 min. be added to, or subtracted from, the time of greatest eastern or western elongation, and the observation be made at night, when the star is on the meridian.
The following table exhibits the angle which the meridian plane makes with the vertical plane drawn to the pole-star, when at its greatest eastern or western elongation : such angle is called the azimuth. The mean angle only is put down, being calculated for the first of July of each year.
Lat. 32° Lat. 34 Lat. 36o Lat. 38° Lat. 40° | Lat. 42° | Lat. 44
Azimuth Azimuth Azimuth Azimuth Azimuth Azimuth Azimuth
Take a board, of about one foot square, paste white paper
1° 53 1° 55 | 1° 59 2° 2 2° 5 2° 2° 13
2°03' | 2° 4' 2° 8
23 | 2° 6'
1° 52} | 1° 55 1° 58 2° 13' | 2° 5
1° 54' 1° 57' 21' 2° 43' 2° 9 1° 49 1° 51} | 1° 54' 1° 571 | 2° 0 | 2° 4
telescope of the theodolite. Let this board be so fixed to a vertical staff, as to slide up and down freely : and let a small piece of board, about three inches square, be nailed to the lower edge of it.
About twenty-five minutes before the time of the greatest eastern or western elongation of the pole-star, as shown by the tables of elongations, let the theodolite be placed at a convenient point and levelled. Let the board be placed about one foot in front of the theodolite, a lamp or candle placed on the shelf at its lower edge; and let the board be slipped up or down until the pole-star can be seen through the hole. The light reflected from the paper will show the cross hairs in the telescope of the theodolite.
Then, let the vertical spider's line be brought exactly upon the pole-star, and, if it is an eastern elongation that is to be observed, and the star has not yet reached it, it will move from the line towards the east, and the reverse when the elongation is west.
At the time the star attains its greatest elongation, it will appear to coincide with the vertical spider's line for some time, and then leave it, in the direction contrary to its former motion.
As the star moves towards the point of greatest elongation, the telescope must be continually directed, by means of the tangent screw of the vernier plate ; and when the star has attained its greatest elongation, great care should be taken that the instrument be not afterward moved.
Now, if it be not convenient to leave the instrument in its place until daylight, let a staff, with a candle or small lamp upon its upper extremity, be arranged at thirty or forty rods from the theodolite, and in the same vertical plane with the axis of the telescope. This is easily effected, by revolving the vertical limb about its horizontal axis without moving the vernier plate, and aligning the staff to coincide with the vertical hair. Then mark the point directly under the theodolite; the line passing through this point and the staff makes an angle with the true meridian equal to the azimuth of the pole