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* OF HALF THE SUM OF THE THREE SIDES, AND THE DIFFERENCE BETWEEN THAT HALF SUM AND THE SIDE OPPOSITE THE ANGLE REQUIRED, SO IS THE SQUARE OF RADIUS TO THE SQUARE OF THE CO-SINE OF HALF THE ANGLE REQUIRED.
Those who make themselves well acquainted with TRIGONOMETRY, will find its application easy to many useful purposes, particularly to the mensuration of heights and distances. These are here omitted, because as this work is designed principally to teach the art of common FIELD-SURVEYING, it was thought improper to swell its size, and consequently increase its price, by inserting any thing not particularly connected with that art.
It is recommended to those who design to be surveyors, to study TRIGONOMETRY thoroughly; for though a common field may be measured without an acquaintance with that science, yet many cases will occur in practice, where a knowledge of it will be found very beneficial; particularly in dividing land, and ascertaining the boundaries of old surveys. Indeed no one who is ignorant of TRIGONOMETRY, can be an accomplished surveyor.
SURVEYING is the art of measuring, laying out, and dividing land.
The most common measure for land is the acre; which contains 160 square rods, poles, or perches; or 4 square roods, each containing 40 square rods.
The instrument most in use, for measuring the sides of fields, is Gunter's chain, which is in length 4 rods, or 66 feet; and is divided into 100 equal parts, called links, each containing 7 inches and 92 hundredths. Consequently, 1 square chain contains 16 square rods, and ten square chains make 1 acre.
In small fields, or where the land is uneven, as is the case with a great part of the land in New England, it is better to use a chain of only two rods in length, as the survey can be more accurately taken.*
PRELIMINARY PROBLEMS. PROBLEM I. To reduce two rod chains to four rod chains.
RULE. If the number of two rod chains be even, take half the number for four rod chains, and annex the links, if any; thus, 16 two rod chains and 37 links, make 8 four rod chains and 37 links.
But if the number of chains be odd, take half the greatest even number for chains, and for the remaining number add 50 to the link: Thus, 17 two rod chains and 42 links makes 8 four rod chains and 92 links.
PROBLEM II. To reduce two rod chains to rods and decimal parts.
* As there is no standard of long measure established by the General Government, the standard yard, which is kept in the State Treasurer's office, is adopted as a standard of long measure for Connecticut. Each chain of 2 rods, should be 11, and each of 4 rods, 22 yards, in
RULE. Multiply the chains by 2, and the links by 4, which will give hundredths of a rod; thus, 17 two rod chains and 21 links, make 34 rods and 84 hundredths; expressed thus, 34.84 rods.
If the links exceed 25, add one to the number of rods, and multiply the excess by 4: thus, 15 two rod chains and 38 links make 31.52 rods.
PROBLEM III. To reduce four rod chains to rods and decimal parts.
RULE. Multiply the chains, or chains and links, by 4; the product will be rods and hundredths: thus, 8 chains and 64 links, make 34.56 rods. Note. The reverse of this rule, that is, dividing by 4, will reduce
rods and decimals to chains and links: thus, 105.12 rods, make 26 chains and 28 links. PROBLEM IV. To reduce square rods to acres.
RULE. Divide the rods by 160, and the remainder by 40, if it exceeds that number, for roods or quarters of an acre: thus 746 square rods make 4 acres, 2 roods, and 26 rods.
PROBLEM V. To reduce square chains to acres.
RULE. Divide by 10; or, which is the same thing, cut off the right hand figure: thus, 1460 square chains make 146 acres; and 846 square chains make 84 acres and 6 tenths.
PROBLEM VI. To reduce square links to acres.
Rule. Divide by 100000; or, which is the same thing, cut off the 5 right hand figures : thus, 3845120 square links make 38 acres and 45120 decimals. Note. When the area of a field, by which is meant its superficial
contents, is expressed in square chains and links, the whole may be considered as square links, and the number of acres contained in the field, found as above. Then multiply the figures cut off by 4, and again cut off 5 figures, and you have the roods; multiply the figures last cut off by 40, and again cut off 5 figures, and you have
the rods. EXAMPLE. How many acres, roods, and rods, are there in 156 square chains and 3274 square links?
15)63274 square links.
21)23840 Answer. 15 acres 2 roods and 21 rods.
PROBLEMS for finding the area of right lined figures, and PROBLEM VII. To find the area of a square or rectani gle.
RULE. Multiply the length into the breadth; the product will be the area.
PROBLEM VIII. To find the area of a rhombus or rhomboid.
RULE. Drop a perpendicular from one of the angles to its opposite side, and multiply that side into the perpendicular; the product will be the area.
PROBLEM IX. To find the area of a triangle.
RULE 1. Drop a perpendicular from one of the angles to its opposite side, which may be called the base; then multiply the base by half the perpendicular, or the perpendicular by half the base; the product will be the area. Or, multiply the whole base by the whole perpendicular, and half the product will be the area.
RULE 2. If it be a right angled triangle, multiply one of the legs into half the other; the product will be the area. Or, multiply the two legs into each other, and half the product will be the area.
RULE 3. When the three sides of a triangle are known, the area may be found arithmetically, as follows:
Add together the three sides; from half their sum substract each side, noting down the remainders; multiply the half sum by one of those remainders, and that product by another remainder, and that product by the other remainder; the square root of the last product will be the area.*
EXAMPLE. Suppose a triangle whose three sides are 24, 20, and 18 chains. Demanded the area.
24+20+18=62, the sum of the three sides, the half of which is 31. From 31 substract 24, 20, and 18; the three remainders will be 7, 11, and 13.
31X7=217; 217x11=2387; 2387X13=31031, the square root of which is 176.1, or 17 acres, 2 roods, and 17 rods.
As the addition of logarithms is the same as the multiplication of their corresponding numbers; and as the number answering to one half of a logarithm will be the square root of the number corresponding to that logarithm : it follows, that if the logarithm of the half sum of the three sides, and
* Better expressed thus. From half the sum of the sides substract each side separately. Multiply the half sum and the several remainders together, and the square root of the product will be the
the logarithms of the three remainders, be added together,
0.845098 The second remainder, 11
1.041393 The third remainder, 13
The square of the area, 31030.083 nearly
RULE 4. When two sides of a triangle, and their contained angle, that is, the angle made by those sides, are given, the area may be found as follows:
Add together the logarithms of the two sides, and the logarithmic sine of the angle; from their sum substract the logarithm of radius, the remainder will be the logarithm of double the area.
EXAMPLE. Suppose a triangle, one of whose sides is 105 rods, and another 85, and the angle contained between them 28° 5'. Demanded the area. One side, 105
2.021189 The other side, 85
1.929419 Sine angle, 28° 5'
Note. Radius may be substracted by cancelling the left hand figure
of the index, or substracting 10, without the trouble of setting down the ciphers.
BY NATURAL SINES.
Multiply the two given sides into each other, and that duct by the natural sine of the given angle; the last product will be double the area of the triangle.
Nat. sine of the angle 28° 5', 0.47076.
105X85=8925, and 8925X0.4707634201, the double area of the triangle.