Oblique-angled Triangle A MO: In which there is given, (1.) The Side MO 130.00 Fathom. (2.) The Angle АМО, 104.50 Deg. And, (3.) The Angle AOM, 37.82 Deg. And having Two Angles given, you have the third also given, viz. MAO, 37.68 Deg. whereby you may find the other Two Sides MA and O A (by Axiom II.) thus: As the Sine of MA O, 37.68 Deg. Is to the Side M O, 130.00 Fathom: So is the Sine of A OM, 37.82 Deg. To the Side MA, 130,41 Fathom, the Distance of the Ship at And fo is the Sine of A MO, 104.50 (or 75.50 Deg.) To the Side O A, 205.91 Fathom, the Distance of the Ship at Again, Observing 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 those Angles, you shall have another Oblique-angled Triangle MBO: In which you will have given as in the former, (1.) The Angle BMO, 65.50 Deg. (2.) BO M 68.00 Deg. And consequently MBO, 46.50 Deg. together with the measured Distance MO 130 Fathom, by which you may find the other Two Sides M B and OB, 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 BMO, 65.50 Degrees, To the Side B O, 163.08 Fathom: And fo is the Sine of the Angle BOM 68.00 Degrees, To the Side B M, 166.18 Fathom. Lastly, Observing 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 MC be 164.39 Fathom, and OC 45.62 Fathom. Bb2 And Fig. VIII. Fig. IX. 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 First, For the Distance A B. In the Oblique Triangle ABM, there is given, the Sides A M 13041 Fathom, and BM 168.18 Fathom, and the Angle contained by them, AMB 39.00 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 BM, 296.59 Fathom, Then say, (By Axiom II.) As the Sine of the Angle A BM, 51.69 Degrees, So is the Sine of the Angle A MB, 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 Distances of many Places upon the Land, one from another be obtained, by making of ObServation, by a Theodolite, Semi Circle, (or other. Graduated Instrument) from Two Places, from whence all the other may be. feen: An Example whereof I shall 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 shall leave to the Ingenuity of the Practitioner. Wherefore, Let ABCDE be feveral Places, as Churches in a Town or City, or fuch like Objeds. 1. Make Choice of Two such Places, from either of which, you may fee all the Places whose Distances you require; which Places let be F and G, distant from each other 1000 Foot, more or less. 2. Set 2. Set up your Instrument 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 Instrument to the second Place G, direct the Sights which are upon the Diameter thereof, back towards the First Place at F, where fix it: Then, turning the Index about, direct it to the several Places, A, B, C, D and F, as before; noting the Degrees cut by the Index. Which we will suppose to be fuch as are noted in this Table. And the Distance from F to G 1000. From these Obfervations, to make a Plot or Map of the Situation 1. Upon a Sheet of Paper, draw a Right Line, as FG, to'con- 2.. Place the Center of a Protraffer 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 several Objets at A, B, C, &c. when you Obferved at F, and through those Points draw Right Lines, at Pleasure; as FA, FB, FC, FD, FE. 3. Lay the Center of the Protractor to the Point G, and the Diameter thereof upon the Line FG, and there holding it fast, make Marks against those Degrees of the Protrador, as the Index did cut upon the Instrument, when you made Obfervation at G; and through those Points, and the Point G, draw Lines; as GA, GB, GC, GD, and G E, croffing the former Lines (drawn from F,) in the respective Points, A, B, C, D and E. Which Points will lye upon your Paper, in the same Position as the Places you took notice of, were fituate on the Ground on which they ftood: And being thus laid down, if you take with your Com paffes Fig. IX. passes the Distance between any Two of them, and measure it upon the fame Scale you laid down the Line FGby, it will give you the Distance between those Two Places. But, their Distances may be more exact and accurately attained unto by Trigonometrical Calculation. For, in every Triangle, as in FGA, FGB, F G C, F G D, and FGE, there is given, (1.) The Angle A FG, observed at F. (2.) The Angle AGF, observed at G; and, (3.) The Side FG, (the Stationary Distance) included between them, to find the other Sides, AF and AG. The Practice whereof I commit to the Ingenuity of the Practitioner. CHAP. III. OF PLANOMETRIA. OR, Of the Mensuration of Plain Superficial Figures. I. Of the Geometrical Square A BCD, whose Side A B is 27.32 Foot. By Logarithms. II. Of the Parallelogram (or Long Square) EFGH, whose Length EF is 27.25 Yards, and Breadth EG, 6.29 Yards. |