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II. To take a Diftance, (Acceffible or In-acceffible) at Two Sta
Fig. VII. Being in a Field at E, there is a Wind mill (or other Object) in another Field at F, (feparated by a River or other Impediment) whofe Distance is required.
In any other Part of the Field, remote from E, caufe a Mark to be fet up, as at D; and measure the Distance between E and D,. which let be 115.00 Foot. Then by your Inftrument at E, find the Quantity of the Angle FE D, which fuppofe to, be 106.50 Deg. And going from E to D, make Obfervation of the AngleFDE, which fuppofe to be 57.10 Deg.
By thefe Two Angles, and the Menfured Diftance, you have conftituted an Oblique Triangle DE F, in which there is given, (1.) The Meafured Distance E D, 11500 Foot. (2.) The Two Angles at D and E, 106.50 Deg. and 57.10 Deg. And, (3.) The Angle at F, (the Complements of the other Two to 180 Deg.) 16.40 Deg. Whereby you may find the Sides E F and DF, (by Axiom. II.) thus:
As the Sine of the Angle at F, 16.40 Deg.
Is to the Meafured Distance DE, 115.00 Foot;
To the Distance E F, 341.98 Foot.
And fo is the Angle at E, 105.50 Deg. (cr 73.50.)
III. If there were Three (or more) Ships on the Sea, and you be
Fig. VIII. L Shoar at M, are required to tell how far those Ships are from
ET the Three Ships be A, B, C, and you being upon the
you, and alfo, how far from each other.
First, Being at M, make choice of fome other Place upon the Shoar, at fome confiderable Diftance, as at 0 130.00 Fathom.
Secondly, Being at M, cbferve the Quantity of the Angle AMO, which we will fuppofe to contain 104.50 Deg. and then remo ving to O, and obferving the Angle MO A, you find it to be 37.82 Deg, through which Degrees, Lines being drawn from M and O, will crofs each other in A, which is the Place of the firft Ship, which, with the Line of Diftar.ce M O, will form the
Oblique-angled Triangle A MO: In which there is given, (1.) The Side M O 130.00 Fathom. (2.) The Angle A M O, 104.50 Deg. And, (3) The Angle A OM, 37.82 Deg. And having Two Angles given, you have the third alfo given, viz. MA O, 37.68 Deg. whereby you may find the other Two Sides M A and OA (by Axiom II.) thus:
As the Sine of MA O, 37.68 Deg.
Is to the Side M O, 130.00 Fathom:
To the Side M A, 130,41 Fathom, the Diftance of the Ship at
And fo is the Sine of A M O, 104.50 (or 75.50 Deg.)
To the Side O A, 205.91 Fathom, the Distance of the Ship at
Obferving from M and O, to B, you found the Angle BMO Fig. VIII. to contain 65.50 Deg. and MOB 68.00 Deg. wherefore, if upon M and O, you lay down thofe Angles, you fhall have another Oblique-angled Triangle M BO: In which you will have given as in the former, (1.) The Angle BMO, 65.50 Deg. (2.) BO M 68.00 Deg. And confequently M BO, 46.50 Deg. together with the measured Distance MO 130 Fathom, by which you may find the other Two Sides M B and O B, by Axiom II. as in the former. For,
As the Sine of MBO, 46.50 Degrees,
Is to MO, 130.00 Fathom:
So is to the Sine of B MO, 65.50 Degrees,
To the Side B O, 163.08 Fathom :
And fo is the Sine of the Angle B OM 68.00 Degrees,
To the Side B M, 166.18 Fathom.
Obferving again from M and O, to C, you find the Angle CMO to be 11.75 Deg. and MOC 132.75 Deg. the which Angles laid down, you have a third Oblique-angled Triangle CMO, wherein there is given, as before, all the Three Angles, and One Side, MO, whereby you may find the other Two, MC and OC, (by Axiom II.) as in the former.
So will M C be 164.39 Fathom, and OC 45.62 Fathom.
And thus have you the Distances of all the Ships, from the Two places, M and O, on the Shoar.
Now for their Distances one from another:
And Firft, For the Distance A B.
In the Oblique Triangle A B M, there is given, the Sides A M 13041 Fathom, and BM 168.18 Fathom, and the Angle contained by them, A M B 39.co Deg. whereby the third Side AB may be found, (By Cafe II. of O, A, P, T,) thus:
As the Sum of the Sides, A M and B M, 296.59 Fathom,
Then fay, (By Axiom H.)
As the Sine of the Angle A BM, 51.69 Degrees,
So is the Sine of the Angle A M B, 39.00 Degrees,
And in the fame manner may the Distance from B to C, and from C to A, be found.
According to this Method may the Diffances of many Places upon the Land, one from another be obtained, by making of Obfervation, by a.Theodolite, Semi Circle, (or other. Graduated Inftrument) from Two Places, from whence all the other may be. feen: An Example whereof I fhall give, with the manner of making the Obfervations; and protracting of them, whereby the Triangles to be refolved for the Performance will be confpicuous: And for the refolving of them. (it being altogether the fame with that foregoing) I fhall leave to the Ingenuity of the Practitioner. Wherefore,
Let A B C D E be feveral Places, as Churches in a Town or City, or fuch like Objeds.
1. Make Choice of Two fuch Places, from either of which, you may fee all the Places whofe Diffances you require; which Places let be F and G, diftant from each other 1000 Foot, more
2. Set up your Inftrument at F. and direct the Sights on Fig. IX. the Diameter thereof, to the other Place at G. and there fix it: Which done, direct the Sights to the feveral Places, A, B, C, D and E, noting what Degrees of the Inftrument are cut by the Index at every Obfervation.
3. Then removing your Inftrument to the fecond Place G, direct the Sights which are upon the Diameter thereof, back towards the Firft Place at F, where fix it: Then, turning the Index about, direct it to the feveral Places, A, B, C, D and F, as before; noting the Degrees cut by the Index. Which we will fuppofe to be fuch as are noted in this Table.
And the Distance from F to G 1000.
From thefe Obfervations, to make a Plot or Map of the Situation of the feveral Places.
1. Upon a Sheet of Paper, draw a Right Line, as FG, to contain 1000 Foot, of any Scale, which reprefents the Line of the Diftance of the Two Places, where you Obferved at F and G.
2.. Place the Center of a Protracter upon F, and the Diameter thereof upon the Line FG, and there holding it fast, make Marks against the feveral Degrees that were cut by the Index, when the Sights were directed to the feveral Objects at A, B, C, &c. when you Obferved at F, and through thofe Points draw Right Lines, at Pleasure, as FA, FB, FC, FD, F E.
3. Lay the Center of the Protractor to the Point G, and the Diameter thereof upon the Line F G, and there holding it fast, make Marks against thofe Degrees of the Protractor, as the Index did cut upon the Inftrument, when you made Obfervation at G; and through thofe Points, and the Point G, draw Lines; as GA, GR, GC, G D, and G E, croffing the former Lines (drawn from F,) in the refpective Points, A, B, C, D and E. Which Points will lye upon your Paper, in the fame Pofition as the Places you took notice of, were fituate on the Ground on which they food: And being thus laid down, if you take with your Com
Fig. IX. paffes the Distance between any Two of them, and measure it upon the fame Scale you laid down the Line F G by, it will give you the Distance between thofe Two Places.
But, their Distances may be more exact and accurately attained unto by Trigonometrical Calculation.
For, in every Triangle, as in F GA, F G B, F G C, FG D, and FG E, there is given, (1.) The Angle A FG, obferved at F. (2.) The Angle A GF, obferved at G; and, (3.) The Side FG, (the Stationary Distance) included between them, to find the other Sides, A F and A G. The Practice whereof I commit to the Ingenuity of the Practitioner.
CHA P. III.
Of the Menfuration of Plain Superficial Figures.
I. Of the Geometrical Square A B C D, whofe Side A B is
AS IF to 27.32 F :: 27.32 F : to 746. 38 F.
Logar. of 27.32
Logar. of 746.38
The Superficial Content of the Square in Feet.
II. Of the Parallelogram (or Long Square) EF G H, whofe Length
EF is 27.25 Tards, and Breadth EG, 6.29 Yards.
As I 27.25 6.29: 17.03
Logar. of 27.25