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Now we have, in the right-angled Triangle DBE, given the Hypothenufe D B 53.7 oppofite the right Angle at E, and the Angle at D 52°. 10' oppofite the required Side BE; therefore,

As the Radius,

Is to the Hypothenufe B D, 53.7,
So is the Sine of the Angle at D, 52°. 10',

To its oppofite Side BE, 42.41,

10.

1.729986 9.897516

1.627502

Thus having found the Side B E, the Height of the Object above the Center of the Quadrant at the Time of Obfervation, if to that is added 4.9 Feet, the Height of the Quadrant, it gives B A 47.31 for the Height of the Object above the Plain c A.

Prob. 35. To measure the Object AB above the Objet BC.

At any convenient Distance, as at D, with a Quadrant take the Angle A D C, which fuppofe 50°. 56'; take the Angle BDC 38°. 30'; and then measure the Distance from your Station at D to the Base of the Object at C, which suppose 48.6 Feet.

A

B

C

Conftruction. Draw the BafeD Line DC at Pleafure; at D, by Prob. 8, make an Angle A D C of 50°. 56', and draw A D; at the fame Point D, make an Angle BDC of 38°. 30", and draw DB; fet off, by the diagonal Scale, on the Bafe-Line 48.6 from D to C; at C erect a Perpendicular, which, being drawn to meet DA and D B, completes the Figure; for A B is the Height of the Object above BC.

Now, in the Triangle D A C, right-angled at C, there are given the Angle CDA 50°. 56′, the Angle at A, being the Complement of that to 90°, 39°.4', and the Side DC 48.6, to find the Perpendicular AC.

As

As the Sine of the Angle at A, 39°.4',

Is to the Side D.C, 48.6,

Co. Ar. 6.200509

1.686636 9.890093

1.777234

So is the Sine of the Angle at D, 50°. 56',

To the Perpendicular A C, 59.87,

And, in the Triangle DB C, there are given the Angle CDB 38°. 30', the Angle DBC 51°. 30, found by fubtracting 38° 30' from 90°, and the Bafe DC 48.6, to find the Perpendicular CB:

As the Sine of the Angle at B, 51°. 30′,
Is to the Side D C, 48.6,

So is the Sine of the Angle at D, 38°.30′,

To the Perpendicular B C, 38.65,

Now from 59.83 the Height of AC,
Take 38.65 the Height of BC.

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(required.

21.18 Height of A B above B C, as was

This Object being acceffible, that is, which could be come at, fo as to measure the Distance of the Station from the Foot of the Object, one Station is fufficient: But, if any Obstacle lies in the Way, fo as to make it inacceffible, it is neceffary to have two Stations, as in Prob. 34; and the Center of the Inftrument fhould be kept at the fame Height, while each of the Angles is measured at each Station.

Prob. 36. To measure the Distance between two Objects, A and B, when you can measure from the Station to one of the Objects.

C

Take a Station at a convenient Distance from the Object, as C, and obferve the Angle C 70°56′ : If either of the Objects is acceffible, meafure the Distance from your Station to the acceffible Object, as fuppofe CB 49.7 Feet; if you cannot measure to the other Object A in a right Line, go to it, and take the Angle at A, which here will be 50°. 10, and then the Angle at B, which is the Supplement of the other two to 180°, will be 58°.54'.

Α.

B

Conftruction.

Conftruction. Draw the right Line C B, on which lay from the diagonal Scale 49.7 from C to B; at C, by Prob. 8, make an Angle of 70°. 56', and draw CA; at B make an Angle of 58°.54', and draw B A, meeting CA in A, which conftructs the Triangle CA B, and A B is the Distance required.

In the Triangle AC B, there are given the three Angles, and the Side C B 49.7, to find the Side A B, or the Distance of the two Objects: Whence,

As the Sine of the Angle at A, 50°. 10', Co. Ar. 0.114689 Is to the Side C B, 49.7,

So is the Sine of the Angle at C, 70°. 56′,

1

To the Side A B, 61.17, the Distance of the
Objects A and B,

1.696356 9.975496

€1.786541

Another Example. Suppose I am at C, and can measure from C to each of the Objects A and B, to find the Distance between the Objects themselves.

With the Inftrument measure the Angle A CB 59°. 30'; and then measure the Distance CB 49.2 Feet, and the Diftance CA 51.7 Feet.

C

Conftruction. Draw the right Line C B, on which lay from C to B 49.2; at C, by Prob. 8, make an Angle of 59°. 30', and draw CA, on which fet off 51.7 from C to A; then draw A B, meeting C B in B, and the Triangle ABC is conftructed, A B being the Distance required.

A

B

In the Triangle AC B, there are given the Side A C, the Side BC, and the included Angle at C, to find the Bafe A B; which may be done by firft finding the other Angle, by Prob. 28, as follows:

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Now,

As the Sum of the given Sides, 100.9,

Is to their Difference, 2.5,

Co. Ar. 7.996109
C.397940

So is the Tangent of the Half-Sum of the unknown Angles, 60°. 15',

} 10.242948

To the Tangent of Half their Difference, 2°. 29', 8.636997

To 60°. 15' the Half-Sum of A and B.
Add 2. 29 the Half-Difference.

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As the Sine of the Angle at A, 57°.46', Co. Ar. 0.072690
Is to the oppofite Side B C, 49.2,

So is the Sine of the Angle at C, 59°. 30′,

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1.691965
9.935220

To the Side A B, 50.14, the Diffance of the 1.699875
Objects A and B,

Prob. 36. To measure the Distance between an Object at A from any Place, as C, where, by Reafon of fome Obstacle,

it cannot be come at.

In any Place on the fame Plain with C place a visible Mark, as B, and measure the Angle A CB 88°. 20; meafure the Distance BC, which fuppofe 76.35; then with the Inftrument take the Angle ABC 36°. 10'.

B

MA

Conftruction. Draw the Bafe-Line B C, on which lay 76.35 from C to B; at C, by Prob 8, make an Angle of 88°20', and draw CA; at B, by the fame Problem, make an Angle of 36°. 10', and draw B A, meeting CA in A, which conftructs the Triangle BCA, and AC is the Distance fought.

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2

Now, in the Triangle B C A, there is given

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Whence, all the three Angles and the Distance BC being known, the Side A C, or the Distance of the Obje& A from C, may be found thus:

As the Sine of the Angle at A, 55°. 30', Co. Ar. 0.084006 Is to the Side B C, 76.35,

So is the Sine of the Angle at B, 36°. 10',

To the Side A C, 25.85,

1.557507

9.770952

1.412465

S

Of SURVEYING.

URVEYING, or Measuring of Land, confifts in taking the Dimenfions of a Piece of Ground, as a Field, laying it down in a Draught or Map, and finding the Content in Acres, Roods, &c.

The firft Part of these, that is, taking the Dimenfions, confifts of two Parts; firft, taking the Angles, and then measuring the Diftances: To perform which feveral Inftruments are neceffary, as a Plain-Table, Theodolite, Semicircle, Compafs, and a Chain, alfo a Protractor for plotting.

The Plain-Table is a Board, about fifteen Inches broad, and twelve long, encompaffed with a Frame, jointed at the Corners From a Brafs Center in the Middle are projected three hundred and fixty Degrees of a Circle, each Degree being halved, and numbered at every tenth Degree, both Ways, to fave Subtraction: On the contrary Side, a Semicircle is ufually projected, divided and numbered in the fame Manner, to one hundred and eighty Degrees, both Ways. To the Table is ufually fitted a Compafs, to fet it North or South,

and

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