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N this section of GEOMETRY, I shall treat only of such Practical Parts thereof, as the Doctrine of Plain

(or Right-lined) Triangles, (both for their Illustration and Demonstration) becomes fubfervient: As, I. In ALTIMETRIA: By which the Height ofany Objet

(accessible or inaccessible) may be obtained ; As of Towers,

Steeples, Trees, &c. II. In LONGIMETRIA: By which the Distance of one

Obje&t from any Place, or of many Objects one from another, (whether approachable, or in-approachable) may be known,

their true Positions laid down, and a Map inade of them. III. In PLAN OM ETRIA: By which all kinds of Su

perficies, (Regular or Irregular) as Plains, Land, &c. may be Measured.

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1. Of an. Altitude that is Accelăbič. : Fig. 1. . I ,

and you standing at B, were required to tell the Height thereof. First, Measure the Distance from B, to the Foot of the Objekt A, which suppose to be 432.5 Feet, --Secondly, At B, (by a Quadrant, or other Graduated Inftrument) look to the top of the Obje&t át C, where we will suppose the Degrees found by the Instrument to be 32.25 deg. --By this Observation, the Distance Measured, and the Objell, you have a Right-angled Triangle constituted ; in which, there is Given, (1.) A B the Distance Measured 432.5 Foot. (2.) The Angle at B, observed by your Instrument, 32.25 deg. And, 13.) The Right Angle at 8: To find the Leg. CÃ, which will be the Height of the Obje&t: (By CASE II. of R, A, P, T,) thus: As Radius,: Tangent 45 deg.

Is to the Distance Measured A B; 432.5 Feet; So is the Tan. of the Angle observed by Instrument at B, 32.25 d. 1 To the Height of the Objeff C A, 272.89 Feer.

And thus having found the Height of the Ohjeł to be 272.89 F.
you may find the Length of the Visual Line (which is the Hypo.
ienuse) C B, (by CÀ SE V. of R, S, P, T.) For,
As the Sine of the Angle Observed, 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..

IL Of an Akitude Un-occessible.
Fig. II. Suppose D E to be an Ohjelt, as Steeple, Tower, or the like:

And that you standing at G, were required to know the Height thereof; but (by reason of some broad Moat, or other Impediment, you cannot come to measure from G to E. In this case, - First measure from G, towards E, as far as conveniently you can, suppose to F, 95.25 F. --Then making Observation at G,


you find the Degrees of your Quadrant to be $2.50 d. and Ob- Fig. II. jerving at F, you find the Degrees to be 63.25. Now from this Distance Mexfured, and the Two Observations by the Instrument made, the Attitude 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 assume a Point for the Place of your first standing,

a 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, set your Measured Distance 95.25 F. from Ġ to F; and upon F, by a Scale of Chards, protract an Angle of 63.25.d. as you oblerved 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 Object to be measured ; and a Perpendicular let fall from D, upon the Ground-line H K, will fall in the Point E, and so will the Line DE represent the Objelt it felf.

Thirdly, By these Lines thus drawn, you will have constituted Two Triangles; one D E F, Righi-angled at E; and the other DF G, Obiuse-angled at F; by the resolving of which, the height of the Inaccessible Object DE, will be found. For,

1. In the Oblique-angled Triangle D F G, you have given, the Side FG (which was the Measured Diftance) 95.25 F. and the observed Angle D G F, 52.50 d. --And the Angle observed at F being 63.25 d. the Complement thereof to 180 d. viz. 116.75 d. is the Quantity of the Obtufe Angle D F G, fo have you in the Oblique-angled Triangle D F G, Two Angles given; the Sum of which, viz. 169.25 d. taken from 180 d. there remairs 10.75 d. for the Angle FDG: And now in the Triangle D. F G you have given Two Angles F D G, and D G F, with the Side F G, oppofite to FDG; whereby, you may find the Side DF, (by Ax, I.) For, As the Sine of FDG, 10.75 d.

. Is to the Side FG; 95.25 Foot: So is the Sine of F GD, 52.50 d.

To the Side D F, 405.14.

2. In the Right-angled Triangle D E F, (having Yound the Hypotenuse as betore) you have given, (1.) The Observed Angle at F, 63.25 d. (2.) The Hypotenuse, last found, DF 405.14 Foor; by which you may find D'E, the Height of the Objeft, (hy Cafe IV.

. of R, A, P, T,) thus;


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Fig. II. As the Radius, Sine 90 d.

Is to the Hypotenuse DF, 405.14 F.
So is the Siné of DF E, (the Angle observed at F) 63.25 d.
To the Leg. D E, 367.77 Foor: Which is the Height of the

And now (if you please) you may find the Visual Line GD,
(by Cafe V. of R, A, P, T,) thus:
As the Sine of the Angle at G, 52.50 d.

Is to the height of the Obječi DE, 361.77 F.
So is Radius, Sine god.
To the Length of the Visual Line D G 456.00 F.

And also, the Distance EF, (by Case II. of R, A, P, T,) thus :
As the Sine of the Angle observed at F. 63.25 d.

Is to the Height of the Object, DE, 361.77 F.
So is the Co-line of the Angle at F, 63.75 d.
To the Distance E F, 182.35 Foot.
Unto which, if you add the Distance F G, 95.25 Foot, their

Sum will be 277.6 Foot. And that is the whole Distance

from G to E, the Foot of the Object. III. Of the Altitude of an Obje&t standing upon a Hill, Un-accessible.

Suppose M O to be such an Object; and you standing at L, were required to tell the Height thereof.

Firft, Úpon Paper, or the like, draw a Right-line at Pleasure, as Q R ; and therein, assume any Point, at Pleasure, for the Place of your standing, as L; where, with your Instrument dire&ted to the top of the Objeft, you find the Degrees cut, to be 40:52 d. and directed to the bottom of the Object at O, the Degrees'cut, to be 22.25 d. Wherefore upon L, protract an Angle of 40.52 d. and draw a Line L, at Pleasure: And also b, an Angle of 22.25 d. and draw the Line L c at Pleafure.

Secondly, Go forwards, in a Right-line towards the Object, some considerable Distance, as to N, 212.5 Foot; and there, by your Instrument directed to the top of the Obje&tat M, you find the Degrees cut, to be 61.82 d. through which draw a Line at Pleasure, as Na, crossing the Line L b in the Point M, which is the top of the Object: From whence, a Perpendicular let fall upon the


Fig. III.

Ground.line QR, as MP, that Line shall be equal to the Ai Fig. II. tude, of the Obje&, and the Hill together.

Now, by the Interfe&tions of these Four Visual Lines, L M, LG, N M, and N O, there are constituted Four Right-lined Triangles, viz. L MP, and NOP, both Right-angled at P: And LMN, and N MO, Oblique-angled : By the resolving of which, from the Distance Measured, LN, and the several Angles observed, at L and N, the required Altitude my he obtained. For,

1. In the Oblique-angled Triangle' L M N, there is given,
(1.) The Angle M L Ñ, 49.52 d. (2.) The Angle L N M,
118.18 d. (it being the Complement of the Angle O NP 61.82 d.
to 180 d.) (3.) The Side Measured LN, 212 5 Foot: And
having the Angles at L and N, the Sum of them 158.70 d. tuken
from 180 d. there will remain 21.30 d. for the Angle L M N.
From which Things given, the Two other Sides, L M and MN,
may be found, by Axiom II. thus :
As the Sine of L M N, 21.30 d.

Is to the Side L N, 212.5 Foot;
So is the Sine of M L N, 40.52 d.
To the Side M N, 380.08 Foot:
And fo is the Sine of 61.82 d. (the Complement of MNL, ,

to 1 80 d.)
To the Side L M. 515.66
In the Right-angled Triangle NMP, there is given, (1.) The
Hypotenuse M N, 380.08 Foot. (2.) The Angle M NP, 61.82d.
whereby you may find NP, (by Case IV. of R, A, P, T,) thus:
As the Radius, Sine sod.

Is to the Hypotenuse M N, 380.08 Foot;
So is the Sine of the Angle M NP, 61.82 d.
To the Side MP, 335.03 Foot:

Which is the Height of the Obje& and the Hill together.
And so is the Co-sine of MN P, viz. NMP, 28.18 d.

To to Leg. or Side NP, 179.49 Foot.
To which, if you add the measured Distance L N, 212.05, their

Sum will be 391.54, for the whole length L P. Then;

3. In the Triangle LOP, (Right-angledar P) you have given, (1.) The Side L P, 391.54 Foot. (2.) The Angle OL P, 22.25 d.



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