Four hundred years ago, in 1609, Galileo heard about the newly invented telescope. He succeeded in making one for himself, and he immediately used it to look at the night sky. The things he discovered were astonishing: moons around Jupiter, the phases of Venus, and the stars of the Milky Way.

When he examined the moon, he saw craters and plains, which he depicted in the drawing below:

Notice that there are a few bright spots of light on the dark side, near the midline. Galileo realized that these were the tops of mountains – they were high enough that their tops caught some sunlight when the surrounding terrain was in shadow.

Almost anyone else would have stopped with that observation, but Galileo thought of a way to use mathematics to actually measure how high these mountains were.

To see how he did it, consider the diagram below:

Here we are looking at the half-moon, with the mountain on the horizon and its top being illuminated by the Sun. Galileo measured the distance of one of the bright spots (a mountaintop) to the midline as being 1/20 of the radius of the moon. Notice that a right triangle is formed, with its hypotenuse being R+h. By the Pythagorean theorem, (R + h)2 = R2 + (R/20)2

The radius of the Moon, R, was already known, so the only unknown in the equation is h.

To solve for h, first, expand the squared term on the left side:

R² + 2Rh + h² = R² + R²/400

Collect terms on the left :

h²+ 2Rh – R²/400 = 0

This is a quadratic formula in h, so use the quadratic formula, with a = 1, b = 2R, and c = – R²/400

or

The approximate radius of the moon had been known since antiquity. The value Galileo used, in modern units, was 3220 kilometers, and the above formula then gives h = 4.0km.

Galileo knew this number was not very precise, but the fact that it is about the same as for high mountains on earth provided some evidence that the earth is not completely different from the moon.

Tags: Math

This entry was posted on Thursday, May 28th, 2009 at 5:41 am and is filed under Thrilling Math. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.