Note. If in the right line ABa you measure Ba equal AB, and the line CBc be produced until the angle a be equal to the angle A ; the distances Bc, ac, will be equal to BC, AC

2. Wanting

2. Wanting to know the breadth of a river, I measured 100 yards in a right line close by one side of it ; and at each end of this line I found the angles, subtended by the other end and a tree close by the other side of the river, to be 53 and 79° 12'. What is the perpendicular breadth?

°

2

3. Two ships of war, intending to cannonade a fort, are, by the shallowness of the water, kept so far from it, that they suspect their guns cannot reach it ; in order therefore to measure the distance, they separate from each other half a mile, or 880 yards; then each ship observes the angle, which the other and the fort subtend, and they are found to be 85° 15' and 83° 45'. What is the distance between each ship and the fort?

4. Suppose I want to know the breadth of a river, or my distance from an inaccessible object O, and that I have no instrument for taking angles, but only a chain or cord for measuring distances; and suppose, that from each of the two stations A, B, which are 500 yards asunder, I measure in a direct line from the object O 100 yards, viz. Aa and Bb each equal to 100 yards, and that the diagonal Ab is by measure 550 yards, and the diagonal aB 560. What then is the distance of the object from each of the stations A and B ?

Let fall the perpendiculars AP, BQ. Then, in the triangle ABa

Ba : BA+Aa :: BA-Aa : BP-Pa,

That is, 560 : 600 :: 400 : 4281=BP-Pa,

Its half 2141
Half sum

280

BP 4945
aP 65
a

And

Then

Aa 100 : aP 655 :: rad. : s. LOAP 41° 5'
AB B 500 : BP494 :: rad. : $._BAP 31 20

Leaves BAO 57.35
Again, in the triangle ABb,
Ab550 : AB--B1600 :: AB-B3.100 : ACQB436

The half diff. 21877
Half sum

275 AQ 4937 Q?

Then

Bb 100 : 560 :: rad. : s. __BQ 34° 37'
BA 500 : AQ4937i rad. :s ZABQ 80 32

The sum 115 9
Taken from

180

* Or the angles ABO and BAO may be otherwisé - found

thus :

Draw

VOL. II.

AA