# (8b) Parallax

Index

5c. Coordinates

6. The Calendar

6a. Jewish Calendar

7.Precession

8. The Round Earth

8a. The Horizon

8b. Parallax

8c. Moon dist. (1)

8d. Moon dist. (2)

9a. Earth orbits Sun?

9b. The Planets

9c. Copernicus
to Galileo

10. Kepler's Laws

10a. Scale of Solar Sys.

11a. Ellipses
and First Law

## "Pre-Trigonometry"

Section M-7 describes the basic problem of trigonometry (drawing on the left): finding the distance to some far-away point C, given the directions at which C appears from the two ends of a measured baseline AB. This problem becomes somewhat simpler if:
1. The baseline is perpendicular to the line from its middle to the object, so that the triangle ABC is symmetric. We will denote its side by r:

AC = BC = r

2. The length c of the baseline AB is much less than r. That means that the angle α between AC and BC is small; that angle is known as the parallax of C, as viewed from AB.

3. We do not ask for great accuracy, but are satisfied with an approximate value of the distance--say, within 1%.
The method presented here was already used by the ancient Greeks more than 2000 years ago. They knew that the length of a circle of radius r was 2πr, where π (a modern notation, not one of the Greeks, even though π is part of their alphabet) stands for a number a little larger than 3, approximately

π = 3.14159...

(The Greek mathematician Archimedes derived π to about 4-figure accuracy, though he expressed it differently, since decimal fractions appeared in Europe only some 1000 years later.)

 Draw a circle around the point C, with radius r, passing through A and B (drawing above). Since the angle α is so small, the length of the straight-line "baseline" b (drawing on the right; distance AB renamed) is not much different from the arc of the circle passing A and B. Let us assume the two are the same (that is the approximation made here). The length of a circular arc is proportional to the angle it covers, and since
b covers an angle    α
2π r covers an angle 360°

we get

2π r = (360°/ α) b

and dividing by 2π

r = (360°/2 π α) b

Therefore, if we know b, we can deduce r. For instance, if we know that α = 5.73°, 2 π α = 36° and we get

r = 10 b

## Estimating distance outdoors

Here is a method useful to hikers and scouts. Suppose you want to estimate the distance to some distant landmark--e.g. a building, tree or water tower.

The drawing shows a schematic view of the situation from above (not to scale). To estimate the distance to the landmark A, you do the following:

 Stretch your arm forward and extend your thumb, so that your thumbnail faces your eyes. Close one eye (A') and move your thumb so that, looking with your open eye (B'), you see your thumbnail covering the landmark A.   Then open the eye you had closed (A') and close the one (B') with which you looked before, without moving your thumb. It will now appear that your thumbnail has moved: it is no longer in front of landmark A, but in front of some other point at the same distance, marked as B in the drawing.   Estimate the true distance AB, by comparing it to the estimated heights of trees, widths of buildings, distances between power-line poles, lengths of cars etc. The distance to the landmark is 10 times the distance AB.   Why does this work? Because even though people vary in size, the proportions of the average human body are fairly constant, and for most people, the angle between the lines from the eyes (A',B') to the outstretched thumb is about 6°, close enough to the value 5.73° for which the ratio 1:10 was found in an earlier part of this section.   That angle is the parallax of your thumb, viewed from your eyes. The triangle A'B'C has the same proportions as the much larger triangle ABC, and therefore, if the distance B'C to the thumb is 10 times the distance A'B' between the eyes, the distance AC to the far landmark is also 10 times the distance AB.