ca. 220 BC: The distance to the Sun The first mathematically-based attempt at determining the Sun-Earth distance is due to Aristarchus of Samos (ca. 310-230 BC). The procedure followed by Aristarchus is illustrated on the diagram below; form a triangle by connecting the Earth (E), Sun (S) and Moon (M). At the first or third Moon quarter, the triangle so described in a right-angle triangle (a=90 degrees). The angle b can be measured by an observer on Earth, which then allows the angle c to be computed (c=90-b when a=90 degrees). The ratio of the Earth-Moon segment (EM) to the Earth-Sun segment (ES) is by definition equal to sin(c) (in modern trigonometric parlance; Aristarchus expressed this differently).

While sound in theory, in practice Aristarchus's procedure is highly inaccurate in the Earth/Sun/Moon case; this is because EM is much smaller than ES, implying that b is very close to 90 degrees, so that c is in turn very small. This has the consequence that a small measurement error on translates in a large variation in the ratio EM/ES (again in modern parlance, a measurement error db is amplified by a factor 1/(sinc)^2, which is large when c is very small). Aristarchus mesured b=87 degrees, while the true value is in fact 89 degrees 50 minutes. This may seem a small error, but because of the large error amplification Aristarchus' value leads to EM/ES=19, instead of the true value EM/ES=397. Nonetheless, Aristarchus' calculation was the first to mathematically set the spatial scale of the cosmos.