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Find the Bearing and Distance of CI by Right An. GLED TRIGONOMETRY, Case IV. as follows : As CK, the Southing of CI, 109
2.03743 : Radius
10.00000 :: KI, the Westing of CI, 71.7
Note. In this way the Course and Distance may be found from
one Angle of a Field to another. Having found the Line CI divide 3470, the num. ber of Rods to be contained in the Triangle ICN, by one half the Line CI, viz. 65, the Quotient will be the length of the Perpendicular PN, viz. 53.4.
Now, by the Bearings of CI and CD it appears that they form an Angle of 60° 20'; wherefore in the Triangle CPN are given the side PN 53.4 and the Angle at C 60° 20', to find the Hypothenuse CN.
As Sine PCN 60° 20 9.93898
: Hyp. CN 61.5
Thus the dividing Line must go from I to a Point on the Line CD, which is 61.5 Rods from C. The Bearing and Distance of this Line may be found by the directions given above for finding the Bearing and Distance of the Line CI. Or, they may be found by Ob. lique Trigonometry Case III.
Another method of finding the Distance CN. Having ascertained the Latitude and Departure of the Line CI, set them down in a Traverse Table ; find the Latitude and Departure of the Line CD, and place them in the Table; the Difference between the Northing of the Line IC and the Southing of the Line CD will be the Southing of the Line ĎI. viz. 6.6; and the Sum of the Eastings of those Lines, as they are both Easterly, will be the Westing of the Line DI, viz. 123.9. Proceed to calculate the Area of the Triangle ICD, which will be found to be 6522 Rods, nearest. Note. As in this Triangle two Sides and their contained Angle
are given, the Area may be found by Prob. IX. Rule 4. Page 39. Having found the Area of this Triangle, proceed to find CN according to PROB. II. Page 71, as follows :
As the Area of the Triangle ; Is to CD the Base ; So is the quantity to be contained in the Triangle ICN; To CN its proportion of the Base.
As 6522 : 115 : : 3470 : 61.2
A third method of finding the Distance CN. To the Logarithm of double the Area to be contained within the Triangle ICN add Radius; from this Sum subtract the Logarithmic Sine of the Angle at C; and from the Remainder subtract the Logarithm of the Side IC; the last Remainder will be the Logarithm of the Side CN.
The double Area of the Triangle ICN is 6940; the Angle at C is 60° 20'; the Side IC is 130. Double Area 6940
Sine ICN 60° 20'
Side IC 130
Note. Radius may be added by placing a Unit before the Index
of the Logarithm for the double Area, without the trouble of setting down the Cyphers.
By Natural Sines. Divide the double Area by the Natural Sine of the given Angle, and that Quotient by the given Side ; the last Quotient will be the Side CN.
Nat, Sine of the Angle at C 60° 20' 0.86892
From the above the following general Rule may
To find the Side of a Triangle when the Area is given, with one of the Sides and the Angle contained between the given Side and the Side Required.
To the Logarithm of double the Area add Radius; from this Sum subtract the Logarithmic Sine of the given Angle, and from the Remainder subtract the Logarithm of the given Side; the last Remainder will be the Logarithm of the Side required.
Or, By Natural Sines : Divide the double Area by the Nat. Sine of the given Angle, and that Quotient by the given Side ; the last Quotient will be the Side re. quired.
Other methods of surveying Fields are taught by some authors on this Subject. The preceding, however, will be found most useful in actual practice, Other instruments besides those mentioned in this Book are also sometimes used; such as the Plain Table, Semicircle, Perambulator, Theodolite, &c. But of these instruments very little use is made in NewEngland; and they are not often to be met with. For general practice none will be found more useful than a common Chain, and a Compass upon Rittenhouse's construction. A Surveyor should also provide himself with an Offset Staff, ten Links in length, and accurately divided into Links. This should be made of firm, hard wood, and will be found very convenient in taking Offsets, and also in measuring the Chain; which should be often done, as from a variety of causes a Chain is liable to become inaccurate.
It will be observed that in this work there are no descriptions of Mathematical and Surveying instruments. The Compiler omitted such descriptions from a belief that nothing which can be written on the subject will enable a person to understand them without an actual inspection of the instruments themselves, and some instruction from those acquainted with them.
The general principles here taught may be applied to the surveying of Townships, Roads, Rivers, Har. bors, &c.
Of the Variation of the COMPASS and ATTRACTION of the Needle. "HE Variation of the Compass is the number of Degrees that
the Magnetic Needle points from the true North, either East or West. This differs in different places, and in the same place at different times. It is, at present, in Connecticut, a few degrees to the Westward. That is, the Needle points to the Westward of North and is gradually approaching the true North.
The following method of ascertaining the Variation, by the North Star, has been adopted by many Surveyors, as the most eligible to be practised on Land. It was communicated to the Compiler by Moses WARREN, jun. Esq. of Lyme, an experienced Surveyor, with permission to publish it.
The Star, commonly called the North Star, is not directly North, but revolves round the Pole in a small Circle, once in 24 hours. It cannot therefore be due North but twice in that period ; and that is within a very few minutes of the time when a Star, called Alioth, in the Constellation of Ursa Major, or the Great Bear, is directly over or under it. There is also another Star nearly in an opposite direction from the Pole, called Gamma, in the Constellation of Cassiopeia. When these three Stars are vertical the North Star is very near the Meridian ; and when they are horizontal, it is at its greatest Elongation, that is, at its greatest distance east or West of the Pole, and on the same side as the Star in Cassiopeia. The Variation may be calculated when the Star is on the Meridian, or when at its greatest Elongation ; more accurately, however, at the latter period, because its motion being then nearly vertical for some time gives the observer a better opportunity to complete his observation.*
* The following Figure exhibits a view of the relative situation of these