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Fig. L.

6. Two Sides, and an Angle oppofite to the Greater of them Given; To find the third Side.

Reckon the Difference of the given Sides from the Pole, one way; and the Sum of the fame, the other way. Count alfo, the given Angle upon the Equinoctial: And extend the ftrait Line (or lay a Ruler) cutting the Meridian, as in the laft; for now it will cut but once; and fo the Third Side will admit only of one fingle Answer.

The End of the Second Part.



Any are the Ways by which Plain and Spherical Triangles may be both Geometrically and Inftrumentally performed: As by Scales of Natural Sines, Tangents, Secants, and Equal Parts, by Protraction: Alfo, by Scales of Artificial Numbers, Sines, Tangents, and Verjed Sines, as Mr. Gunter long time fince contrived them, to be ufed with Compaffes: And I have now lately contrived an Inftrument, which I call TRISSOTETRAS, which printed on a large Sheet of Paper, and pafted upon a Board, all Triangles, both Plain and Spherical, may be Refolved by Infpection, without Pen, Compaffes, opening Joints, or other Moveable; fave only the Extenfion of a Thread, or thin Streight Ruler, upon the Inftrument, the Description and Ufe whereof I may hereafter publish by it felf. But, the beft and most abfolute Way of Refolving Triangles; and to what Ufes foever they be applied, is by the Canons of Artificial Sines, Tangents and Logarithms; both Decimal and Sexaginary: And fuch a Canon was intended to be joined to this Book, at this time; but must be referred till farther Opportunity, which may be fhortly.



Trigonometria Practica.



The Doctrine of PLAIN and SPHERICAL TRIANGLES is applied to Practice: In

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The Plain



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Trigonometria Practica.





N this Section of GEOMETRY, I fhall treat only of fuch Practical Parts thereof, as the Doctrine of Plain (or Right-lined) Triangles, (both for their Illuftration and Demonftration) becomes fubfervient: As,

I. In ALTIMETRIA: By which the Height of any Object (acceffible or inacceffible) may be obtained; As of Towers, Steeples, Trees, &c.

II. In LONGIMETRIA: By which the Distance of one Object from any Place, or of many Objects 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 Meafured.

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Fig. 1.

Fig. II.




I. Of an Altitude that is Acceffible.

AC to be a Tower, Steeple, or other upright Object, and you ftanding at B, were required to tell the Height thereof. First, Measure the Distance from B, to the Foot of the Object A, which fuppofe to be 432.5 Feet, -Secondly, At B, (by a Quadrant, or other Graduated Inftrument) look to the top of the Object at C, where we will fuppofe 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 Meafu red 432.5 Foot. (2.) The Angle at B, obferved by your Inftrument, 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 Meafured A B, 432.5 Feet;

So is the Tan. of the Angle obferved by Inftrument at B, 32.25 d.
To the Height of the Object C A, 272.89 Feet.

And thus having found the Height of the Object to be 272.89 F. you may find the Length of the Vifual Line (which is the Hypotenufe) CB, (by CASE V. of R, S, P, T.) For,

As the Sine of the Angle Obferved, B, 32.25 deg.
Is to the Altitude of the Object, 272.89 Foot;

So is the Radius, Sine 90 deg.

To the Length of the Vijual Line CB, 511.39 Foot..

II. Of an Altitude Un-acceffible.

Suppofe 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 reafon of fome broad Moat, or other Impediment, you cannot come to meafure from G to E. In this Cafe, Firft measure from G, towards E, as far as conveniently you can, fuppofe to F, 95.25 F. Then making Obfervation at G,



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