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of ATC. The side AC is the semi-diameter of the earth, and the hypothenuse CT is equal to the same semi-diameter added to BT the height of the eye. Then
AC: R:: TC Sec. ACT-ATH the depression."
100. Artificial Horizon.-Hadley's Quadrant is particularly adapted to measuring altitudes at sea. But it may be made to answer the same purpose on land, by means of what is called an artificial horizon. This is the level surface of some fluid which can be kept perfectly smooth. Water will answer, if it can be protected from the action of the wind, by a covering of thin glass or tale which will not sensibly change the direction of the rays of light. But quicksilver, Barbadoes tar, or clear molasses, will not be so liable to be disturbed by the wind. A small vessel containing one of these substances, is placed in such a situation that the object whose altitude is to be taken may be reflected from the surface. As this surface is in the plane of the horizon, and as the angles of incidence and reflection are equal, (Art. 91.) the image seen in the fluid must appear as far below the horizon, as the object is above. The distance of the two will, therefore, be double the altitude of the latter. This distance may be measured with the quadrant, by turning the index so as to bring the image formed by the instrument to coincide with that formed by the artificial horizon.
101. The Sextant is a more perfect instrument than the quadrant, though constructed upon the same principle. Its arc is the sixth part of a circle, and is graduated to 120 degrees. In the place of the sight vane, there is a small telescope for viewing the image. There is also a magnifying glass, for reading off the degrees and minutes. It is commonly made with more exactness than the quadrant, and is better fitted for nice observations, particularly for determining longitude, by the angular distances of the heavenly bodies.
A still more accurate instrument for the purpose is the Circle of Reflection. For a description of this, see Borda on the Circle of Reflection, Rees's Cyclopedia, and Bowditch's Practical Navigator.
* See note I, and Table II.
SURVEYING A FIELD BY MEASURING ROUND IT.
ART. 105. THE most common method of surveying a field is to measure the length of each of the sides, and the angles which they make with the meridian. The lines are usually measured with a chain, and the angles with a compass.
106. The Compass.-The essential parts of a Surveyor's Compass are a graduated circle, a magnetic needle, and sight holes for taking the direction of any object. There are frequently added a spirit level, a small telescope, and other appendages. The instrument is called a Theodolite, Circumferentor, &c. according to the particular construction, and the uses to which it is applied.
For measuring the angles which the sides of a field make with each other, a graduated circle with sights would be sufficient. But a needle is commonly used for determining the position of the several lines with respect to the meridian. This is important in running boundaries, drawing deeds, &c. It is true, the needle does not often point directly north or south. But allowance may be made for the variation, when this has been determined by observation. See Sec. V.
107. The Chain.-The Surveyor's or Gunter's chain is four rods long, and is divided into 100 links. Sometimes a half chain is used, containing 50 links. A rod, pole, or perch, is 16 feet. Hence
108. The measuring unit for the area of a field is the acre, which contains 160 square rods. If then the contents in square rods be divided by 160, the quotient will be the number of acres. But it is commonly most convenient to make the computation for the area in square chains or links, which are decimals of an acre. For a square chain =4×4 16 square rods, which is the tenth part of an acre. And a square link=0×180=10800 of a square chain =100'000 of an acre. Or thus,
625 links, or 272 feet = 1 square rod,
1 chain or 16 rods,
- 1 rood or 40 rods,
= 1 acre or 160 rods.
be cut off, the remaining if five places be cut off, the Thus the square of 16.32
109. The contents, then, being calculated in chains and links; if four places of decimals figures will be square chains; or remaining figures will be acres. chains, or 1632 links, is 2663424 square links, or 266.3424 square chains, or 26.63424 acres. If the contents be considered as square chains and decimals, removing the decimal point one place to the left will give the acres.
110. In surveying a piece of land, and calculating its contents, it is necessary, in all common cases, to suppose it to be reduced to a horizontal level. If a hill or any uneven piece of ground, is bought and sold; the quantity is computed, not from the irregular surface, but from the level base on which the whole may be considered as resting. In running the lines, therefore, it is necessary to reduce them to a level. Unless this is done, a correct plan of the survey can never be exhibited on paper.
If a line be measured upon an ascent which is a regular plane, though oblique to the horizon; the length of the corresponding level base may be found, by taking the angle of elevation.
Let AB (Fig. 30.) be parallel to the horizon, BC perpendicular to AB, and AC a line measured on the side of a hill. Then, the angle of elevation at A being taken with a quadrant, (Art. 4.)
R: Cos. A AC: AB, that is,
As radius, to the cosine of the angle of elevation;
So is the oblique line measured, to the corresponding horizontal base.
If the chain, instead of being carried parallel to the surface of the ground, be kept constantly parallel to the horizon; the line thus measured will be the base line required. The line AB (Fig. 30.) is evidently equal to the sum of the parallel lines ab, cd, and eC.
PLOTTING A SURVEY.
111. When the sides of a field are measured, and their bearings taken, it is easy to lay down a plan of it on paper. A north and south line is drawn, and with a line of chords, a protractor, or a sector, an angle is laid off, equal to the angle which the first side of the field makes with the meridian, and the length of the side is taken from a scale of equal parts, (Trig. 156-161.) Through the extremity of this, a second meridian is drawn parallel to the first, and another side is laid down; from the end of this, a third side, &c. till the plan is completed. Or the plot may be constructed in the same manner as a traverse in navigation. (Art. 76.) If the field is correctly surveyed and plotted, it is evident the extremity of the last side must coincide with the beginning of the first.
Draw a plan of a field, from the following courses and distances, as noted in the field book;
Let A (Fig. 31.) be the first corner of the field. Thro' A, draw the merid. NS, make BAN=78°, and AB-2.46 Thro' B, draw N'S' par. to NS, make S'BC-16°, and BC=3.54 Thro' C, draw N"S" par. to NS, make DCN"=83°, &CD=2.72
112. To avoid the inconvenience of drawing parallel lines, the sides of a field may be laid down from the angles which they make with each other, instead of the angles which they make with the meridian. The position of the line BC (Fig. 31.) is determined by the angle ABC, as well as by the angle S'BC. When the several courses are given, the angles which
any two contiguous sides make with each other, may be known by the following rules.
1. If one course is North and the other South, one East and the other West; subtract the less from the greater.
2. If one is North and the other South, but both East or West; add them together.
3. If both are North or South, but one East and the other West; subtract their sum from 180 degrees.
4. If both are North or South, and both East or West; add together 90 degrees, the less course, and the complement of the greater.
The reason of these rules will be evident by applying them to the preceding example. (Fig. 31.)
The first course is BAN, which is equal to ABS'. (Euc. 29. 1.) If from this the second course CBS' be subtracted, there will remain the angle ABC.
If the second course CBS', or its equal BCN", be added to the third course DCN"; the sum will be the angle BCD. The sum of the angles CDS, NDE, and CDE, is 180 degrees. (Euc. 13. 1.) If then the two first be subtracted from 180 degrees, the remainder will be the angle CDE.
Lastly, let EP be perpendicular to NS. Then the sum of the angles DES, PES, and AEP the complement of AEN, is equal to the angle DEA.
We have then the angle ABC=62°,
With these angles, the field may be plotted without drawing parallels, as in Trig. 173.
FINDING THE CONTENTS OF A FIELD.
113. There are in common use two methods of finding the contents of a piece of land, one by dividing the plot into triangles, the other by calculating the departure and difference of latitude for each of the sides.
When a survey is plotted, the whole figure may be divided into triangles, by drawing diagonals from the different angles. The lengths of the diagonals, and of the perpendiculars on the bases of the triangles, may be measured on the same scale of equal parts from which the sides of the field were laid down. The area of each of the triangles is equal to