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As Radius; Sine 90 d.
Is to the Hypothenufe NR 380.
So is the S'ne of the Angle NRK 61. 32.
Equal to the Height of the Object and the Hill together.
And fo is the Cofine of NRK, viz. RNK 28, 28. To the Logar fide RK, 179. 4
To which, if you add the measured diftance PR 212. their Sum will be 391.9, for the whole length PK. Then,
In the Triangle POK, you have given, PK 391.9 Foot, and the 4 OPK 22 2'5.by which you may find OK, by Cale 1ft, of Trigonometry BR, L, T. Thus, As Radius, Tang. 45 d.
Is to the S de PK 391. 9
So is the Tangent of the LOPK 22. 25'. To the Height of the Hill OK 160.19Foor. Which fubtracted from 335. the whole Height, there remains 174.81 Foct for the Altitude of the Object NO.
Prob. 1. How (mding upon an Obje of aknown Height,) to find the distance from thence, to fome other remote Object.
Fig. 6 Suppofe CA to be the fide of a Fort or Bulwark 22. 5 Foot high, and being upon the Platform at C, you fee a
Tree, or other Object at B, whose distance you would know from the Foot of the Wall at A.
The Lines AB and ACbeing drawn, and the he gth of the Wall 22. 5 Foot, fet from A to C, where by your Inftrument directed to B, you find the Degrees cut to be 7′1°. 25′, which Angle lay down; fo have you the Right-angled Triangle CAB, in which there is given, (1.) CA, the heighth of the Wall 22. 5 Foct (2.) the Angle obferved at C, 77: 25' Deg. by which you may find the ditance AB (ly Cafe 2. of Right Angled plane Triangles.) thus,
As Radius, Tangent 45 Deg.
To CA, the heighth of the Wall 22. 5 Foot.
So is the Tangent of ACB, the Angle olferved, 71.25 Deg.
To the diftance AB. 66. 28 Foot.
And if you would find the Length of the Line CB, you may (by Cafe 3 of Right Angle Triangle) thus,
As the Sine of the Angle obferved at C 71°.25' Deg.
Is to the diftance BA, 66. 28 Foot.
Fig. 68. Is Suppofed to be a piece of a River, and you meafuring along one fide of it, would as well know the Breadth of it, as alfo make a true Plot there
of, putting down what remarkable things are feen on the other fide.
Being at 1. the firft Station, caufe one of your Affiftants to go to the next bend of the River, as 2. and there fet up a mark for you; then fee what Angle from the Meridian 1.2 makes, which let be N. W. 6 deg. alfo feeing feveral marks on the other fide of the River, take their bea: ings, as the House A which ftands upon the Bank, and is a Mark for the Breadth of the River bears N.W. 52 degrees, the Wind-mill B up in the Land, bears N W. 40 deg the Tree C by the Water-fide, bears N. W 17 dig all this note down in your Field Book, and measure the distance from 1 to O2, 1 Chains, 20 Links. After this coming to 2, fee how the next bend of the River bears from you, viz. 3; which let be N. E. 15 deg. See alfo how the Houfe A there bears from you, viz. S. W. 20 de g. the Wind-Mill S. W. 50 deg the Tree, N. W. 77°. Alfo as you are going forward if you fee any thing more at this fecond Station, take the Bearing thereof, as a noted Caftle D upon the Land, bears NW. 28, as a Church E clofe by the River Brink NW,4°. measure the distance from 02, to 3. And placing your Inftru ment at 3, he Church bears troin you N W. 88 d Caftle up in the Land at D you cannot fe for the Church, therefore let it alone for D 4.
the next Station. But here you may fee forward a little Village F, the firft House thereof bears from you NW 32 deg.Measure the diftance from 3 to 4,and Planting your Inftrument in 4,the firft Houfe of the Vil lege F bears from you South Weft 32 d. and the Cafle D, which you could not fee at the third Station, S W. 24 Having put down all this in your Field-Bock, it will be as follows,
2 N. E, 15°. 18 Chains 20 Links.
(The end of the little Village S W32°,
To srotract this, draw the Line N. S toc a Meridian, and laying your Protiator upon it, the Center shereof to @ I; against NW. 6° make a Mark for the Line that goes to @ 2. Allo aguinst N. W. 17. make a Mark for the Tree, and against 40° and 52' for the Wind-Mill and Housi. Then froin
I, through these marks draw the Lines
A, B, C, 2. Secondly, Take froin your Scale 18 Chains and 20 Lin, and fer it off upon the Line ©2, which will reach to 2. There lay again the Center of your Protractor ; the Diameter thereof Parallel to the Line NS, and make marks as you see in the field. Book, againft N E 15° NW 77 SW 20°, SW 50, NW 28., NV 4°, and through these inasks draw Lines, the first Line directs to your third Station ; the second Line NW 77° directs you to the Tree Cupon the River Bank, for that Line cutting the Line Oi C, fhews you by the Intersection where the Tree food, and also the Breadth of the River. Also the Line SW 20° cuts the Line from the firft Station NW 52°, in the place where House A ftood upon the Bank of the River. If therefore you draw a Line from A to Cit will represent the farther Bank of the River, And so you may proceed on Plotting, according to the Notes in your Field-Book ; : and you will not only have a true Plot of the River, but also know how far the Wind