To find the distance of two inaccessible objects from each other.

EXAMPLE.

Wanting to know the distance between a house and a mill, which were separated from me by a river, I took arother station B at the distance of 300 yards from the first station A: now, from the first station A, the angle sub

tended

Draw the perpendiculars Ag, Bh. Then, by Eucl. II. 12,

And AB Bg :: rad. : 425 cosine of 64° 51' the CABO.

Ba-AB-Aa2

In like manner Ah=

=268;

2aA

And AB Ab :: rad. : :536 the cosine of 57° 35' the BAO; both the same as above.

tended by B and the mill was 58° 20′, and by the mill and the house 37°; from B, the angle subtended by A and the house was 53° 30', and by the house and the mill 45° 15′. What is the distance of the house and mill?

NOTE. Much after the manner of the last examples many curious and useful problems may be resolved; for if we can determine our distance from one remote object, we can do the same for any number of objects; or if the distance between two remote objects can be determined, those between any number of objects may be determined likewise. So, we may determine the angles and siles of fields, or of very large tracts of land, and that whether we be within them, or any where without them, whence the angles can be seen; hence also ships at sea may deter

mine their distances from known visible ports; and plans may be taken of countries, towns, harbours, fleets, fortifications, &c.*

PROBLEM

* Of many other methods and instruments used to find the altitudes and distances of objects some are here subjoined.

1. One very easy method is by a square, with a plummet AE suspended from one corner A, and the two sides. BC, DC, meeting in the opposite angle C, divided into 10, or 100, or 1000 equal parts; and two sights on the side AB.

B

D

It is evident, that in taking any altitude AB with the square, the plummet will always cut off from the square a triangle similar to that formed by the base line aF, the perpendicular FB, and hypotenuse Ba.

ca

E

D

If

PROBLEM XII.

To find the distance of the most remote point, that can be seen on the earth's surface, from the top of a mountain, and its diameter.

EXAMPLE

If the height AB of the mountain, called the peak of of Teneriffe, be 4 miles, and the angle ABC made by a plumb

If the angle BaF be equal to 45°, then the plumb line will pass through the opposite angle of the square, and the distance DA will be equal to the altitude BF: so if DA be 60, then BF will be 60 also.

If the angle be greater than 45°, as at the station C, then the part of the side cut off at, will be to the whole side ab, as aF, to FB so if ae be 6 divisions, of which ab is 10, and the dis tance CA be 36 feet; then 6; 10 :: 36: 60 feet itude BF.

the alt

If the angle be less than 45°, as at the station E, then the part is cut off the other decimated side, and be: ce :: EA : BF so if the parts cut off be 6, and the distance to feets, then 10 6: 100: 60 feet

BF.