Aristarchus claimed that when the sun casts its rays on the moon to form a half moon at night, a right triangle is formed between the Earth, Sun, and Moon. Therefore, the distance from the sun to the moon, the Moon to the Earth, and the Earth to the Sun forms a right triangle. From this, he attempted to measure the angle between the Moon, Earth, and Sun (<MES ) and found it to be 87 degrees. We are not exactly sure how he got this, but we do know that he was off by 2.85 degrees making it the biggest flaw in his whole argument.
Using 87 degrees for <MES, he then used a form of trigonometry (ratios in right triangles) to figure out the ratio of the Earth to the moon compared to the Earth to the sun (EM/ES).
We, however, can figure this out by taking the cosine of 87 degrees and approximating it as 1 to 19 (1/19).
We do this because it is easier to work with fractions than large decimals. Although, changing the angle to the correct 89.85 degrees, the actual ratio is 1 to 382 (1/382).
You get this figure by adding the 2.85 degrees back to Aristarchus' 87 degrees and taking the cosine of 89.85 degrees and approximating it to 1 to 382.
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by Ryan, Teresa, and Tabitha