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**1. Why did Aristarchus make his measurements of the Sun's distance at the time of the half moon?**

Aristarchus watched for the phase of the Moon when it was exactly half full, with the Sun still visible in the sky. Then the sunlight must be falling on the Moon at right angles to his line of sight. This meant that the lines between Earth and the Moon, between Earth and the Sun, and between the Moon and the Sun form a right triangle.

A rule of trigonometry states that, if you know all the angles in a right triangle plus the length of any one of its sides, you can calculate the length of any other side. Aristarchus knew the distance from the Earth to the Moon. At this time of the half Moon he also knew one of the angles, 90°. All he had to do was measure the second angle between the line of sight to the Moon and the line of sight to the Sun. Then the third angle, a very small one, is 180° minus the sum of the first two angles ( the sum of the angles in any triangle = 180°)

Measuring the angle between the lines of sight to the Moon and Sun is difficult to do without a modern transit. For one thing, both the Sun and Moon are not points, but are relatively big. Aristarchus had to sight on their centers (or either edge) and measure the angle between - quite large, almost a right angle itself. By modern day standards his measurement was very crude. He measured 87° while the true value was 89.8°. He figured the Sun to be about 20 times the Moon's distance, when in fact it is about 400 times as distant. So although his method wand ingenious, his measurements were crude. Perhaps Aritarchus found it difficult to believe the Sun was so far away, and her erred on the nearer side. We don't know.

Today we know the Sun to be an average of 150,000,00 kilometers away. It is somewhat closer to the earth in December (147,000,000 km), and somewhat farther in june (152,000,000 km).

**2. Well these are relevant to me but perhaps not to get the answer, I was wondering how he knew the distance from the moon was it the moon trick where it was 1/110 which is 1 diameter of the moon divided by 110 moons in itself?**

As well I was wondering about the half moon when the sun is in the sky? What is meant by this?

As well I was wondering about the half moon when the sun is in the sky? What is meant by this?

**3. To get a right angle**