Suppose for example, it reads 84 degrees. Repeating the operation at B for the measure of the angle C B A, suppose it to be 95 degrees. Measure the distance from A to B and let it be 80 yards.
Now here is the statement of our problem:
How to resolve a triangle of which the base is known to be 10 yards, and two of its angles. Well, we have said above that the sum of the three angles is always the same, equal to 180 degrees, having on one side 84, and on the other 95, that makes together 84 by 95, equal to 179 degrees. The difference between this number and 180 is 1 degree, therefore the angle ABC measures one degree.
We know that an angle of one degree corresponds to a distance of 57 yards. Multiply the base of our triangle by 57 yards and you obtain a distance of the church from the points A and B, 10 by 57, equal to 570 yards. Nothing is more simple than this.
The smaller the measured angle the further off the object will be. As seen in our figure, the upright lines, m o, m’ o’, m, o,, do not vary, but according to their distances from point C, they form various angles, ac, a’c’, a,c,, becoming smaller and smaller.
A graphometer is not always to be had. When approximate distances only are required, the following contrivance may be used. Trace on a cardboard of large size a semi-circumference which one divides first into 180 equal parts, then each of these is divided again in 2, 3, 4 divisions, etc., according to the size given to the circumference, which constitutes a large protractor.
To measure an angle place the cardboard upright in an horizontal position, supporting it by the center of the semi-circumference by means of a screw fixed on a stick. Then proceed as stated above.
From a pin stuck in the center mark the spot where the visual ray passes, go to A and to B, and you get approximately the desired result.