the

eye, back away

from the tree until the top

of the tree is just visible.

(c) Have the distance measured from your eye to the ground and from your toe to the foot of the tree.

Δ A ABC A ADE. Why?

AD DE. Why?

=

(d) What must be added to DE to find the height of the tree?

8. Use your triangles to find the heights of various. objects in your vicinity.

E. THE USE OF THE QUADRANT AND SEXTANT

I. Drawings from Bettinus

1. In order to measure angles accurately, a surveyor has a very complicated and very expensive instrument, called a transit. In primitive times a much simpler instrument, a sextant or a quadrant, was used.

2. (a) The instrument was called a quadrant if it was a quarter of a circle, and a sextant if it was a sixth of a circle.

(b) An improved and more complex sextant is still used on shipboard for determining latitude.

(c) In its simplest form, that used by Thales, a quadrant is a frame holding a 90° arc, with a moving arm. It is used for measuring angles.

Figures A and B show two different styles of quadrants.

3. To find the width of a river, with a tape and quadrant.

E

D

(a) Let AB and CD be the banks of the river.

(b) Locate some object as A on the opposite bank. (c) Let one pupil stand at C, directly opposite this landmark.

(d) Let another pupil walk in the direction of the line AC to some point E.

(e) With a quadrant, sight an angle of 60°.

(f) What kind of a A is AFE?

(g) What lines are equal?

(h) Measure EF and EC to find AC. How?

4. (a) The drawings shown are taken from an old book by Bettinus, printed early in the seventeenth

century. They illustrate the early use of the quadrant.

(b) The drawing from Bettinus on page 153 shows the use of a quadrant in finding the depth of a well.

Which line measures the unknown depth?

(c) Find two similar triangles in the figure.

(d) Measure three lines and use in a proportion to find the depth of the well.

(e) Assume reasonable values for these lines and compute the depth.

5. By mounting a quadrant or protractor on a frame a very good substitute for a transit may be made. This will be more serviceable if one is mounted in a horizontal position and another in a vertical position.

Any pupil who is clever with his hands can make such a substitute for a transit.

6. (a) Suppose A is a hostile camp, a ship at sea, an

island, or other inaccessible spot.

(b) A gunner at B wants to know the distance AB.
He makes out a line BC 1 AB, and a line
CD 1 BC. These lines may be any length.
(c) From D, he sights to A and notes the intersection
with BC at E.

(d) By measuring BE, EC, and CD, he can find the
distance AB. How and why?

7. (a) The distance may be found in another way.