८ VEL, Trigonometria Practica. PARTIII. WHEREIN The Doctrine of PLAIN and SPHERICAL Sciographia:}or{Dialling, By Calculation after a Navigation:} Or Sailing by { The Plain Mercators } Chart. Middle Latitude. By William Leybourn, Philomathemat. LONDON: Printed Anno Domini, M.DCC.IV. ANCILLA MATHEMATICA. VEL, Trigonometria Practica. SECTION I. OF GEOMETRY. N this Selfion of GEOMETRY, I shall treat only of fuch Practical Parts thereof, as the Doctrine of Plain (or Right-lined) Triangles, (both for their Illustration and Demonstration) becomes subservient: As, I. In ALTIMETRIA: By which the Height of any Objet (acceffible or inacceffible) may be obtained; As of Towers, Steeples, Trees, &c. II. In LONGIMETRIA: By which the Distance of one Objet from any Place, or of many Objets one from another, (whether approachable, or in-approachable) may be known, their true Pofitions laid down, and a Map made of them. III. In PLANOMETRIA: By which all Kinds of Superficies, (Regular or Irregular) as Plains, Land, &c. may be Measured. Fig. 1. Fig. II. T CHAP. I. Of ALTIMETRIA. I. Of an Altitude that is Accesible. Suppofe AC to be a Tower, Steeple, or other upright Objelt, and you standing at B, were required to tell the Height thereof. First, Measure the Distance from B, to the Foot of the Object A, which suppose to be 432.5 Feet. -Secondly, At B, (by a Quadrant, or other Graduated Instrument) look to the top of the Objeld at C, where we will suppose the Degrees found by the Inftrument to be 32.25 deg. -By this Obfervation, the Distance Measured, and the Object, you have a Right-angled Triangle constituted; in which, there is Given, (1.) A B the Distance Meafured 432.5 Foot. (2.) The Angle at B, observed by your Instrument, 32.25 deg. And, (3.) The Right Angle at A: To find the Leg. CA, which will be the Height of the Object: (By CASE II. of R, A, P, T,) thus: As Radius, Tangent 45 deg. Is to the Distance Measured A B, 432.5 Feet; And thus having found the Height of the Objelf to be 272.89 F. you may find the Length of the Visual Line (which is the Hypo. tenuse) CB, (by CASE V. of R, S, P, T.) For, As the Sine of the Angle Obferved, B, 32.25 deg. So is the Radius, Sine 90 deg. To the Length of the Visual Line CB, 511.39 Foot.. II. Of an Altitude Un-occeffible. Suppose DE to be an Object, as Steeple, Tower, or the like: And that you ftanding at G, were required to know the Height thereof; but (by reason of fome broad Moat, or other Impediment, you cannot come to measure from G to E. In this Cafe, - First measure from G, towards E, as far as conveniently you can, suppose to F, 95.25 F. -Then making Obfervation at G, you you find the Degrees of your Quadrant to be 52.50d. and Ob- Fig. II. Jerving at F, you find the Degrees to be 63.25. Now from this Distance Meafured, and the Two Obfervations by the Instrument made, the Altitude D E may be attained unto by Trigonometrical Calculation, thus: First, Upon a Sheet of Paper, draw a Line at Pleasure, as HK, upon which affume a Point for the Place of your first standing, as at G, and upon G, protract the Angle observed 52.50 d. drawing a Line through those Degrees at Liberty. Secondly, By help of a Scale, fer your Measured Distance 95.25 F. from G to F; and upon F, by a Scale of Chards, protract an Angle of 63.25 d. as you observed them to be; and through them draw another Line at Pleafure, which will cut the former Line drawn, in the Point D, which will represent the top of the Objett to be measured; and a Perpendicular let fall from D, upon the Ground-line Η K, will fall in the Point E, and fo will the Line D E reprefent the Objelt it felf. Thirdly, By these Lines thus drawn, you will have constituted Two Triangles; one DEF, Right-angled atE; and the other DFG, Obtufe-angled at F; by the resolving of which, the height of the Inacceffible Objet DE, will be found. For, 1. In the Oblique-angled Triangle DFG, you have given, the Side FG (which was the Measured Distance) 95.25 F. and the observed Angle DGF, 52.50 d. And the Angle obferved at F being 63.25 d. the Complement thereof to 180 d. viz. 116.75 d. is the Quantity of the Obtuse Angle DFG, so have you in the Oblique-angled Triangle DFG, Two Angles given; the Sum of which, viz. 169.25 d. taken from 180 d. there remains 10.75 d. for the Angle FDG: And now in the Triangle DFG you have given Two Angles FDG, and DGF, with the Side FG, oppofite to FDG; whereby, you may find the Side D'F, (by Ax. 1.) For, As the Sine of FDG, 10.75 d. Is to the Side F G, 95.25 Foot. So is the Sine of FGD, 52.50 d. To the Side D F, 405.14. 2. In the Right-angled Triangle DEF, (having found the Hy-potenuse as betore) you have given, (1.) The Obferved Angle at F, 63.25 d. (2.) The Hypotenuse, last found, DF 405.14 Foot; by which you may find DE, the Height of the Objet, (by Cafe IV. of R, 4, P, T,) thus; As |