On the Sizes and Distances (of the Sun and Moon) (Ancient Greek: Περὶ μεγεθῶν καὶ ἀποστημάτων [ἡλίου καὶ σελήνης], romanized: Perì megethôn kaì apostēmátōn [hēlíou kaì selḗnēs]) is widely accepted as the only extant work written by Aristarchus of Samos, an ancient Greek astronomer who lived circa 310–230 BCE. This work calculates the sizes of the Sun and Moon, as well as their distances from the Earth in terms of Earth's radius. The book was possibly preserved by students of Pappus of Alexandria's course in mathematics, although the evidence for this is sparse. The editio princeps was published by John Wallis in 1688, using several medieval manuscripts compiled by Sir Henry Savile. The earliest Latin translation was made by Giorgio Valla in 1488. There is also a 1572 Latin translation and commentary by Frederico Commandino.
Symbols The work's method relied on several observations:
The apparent size of the Sun and the Moon in the sky. The size of the Earth's shadow in relation to the Moon during a lunar eclipse The angle between the Sun and Moon during a half moon is 90°. The rest of the article details a reconstruction of Aristarchus' method and results. The reconstruction uses the following variables:
Half Moon Aristarchus began with the premise that, during a half moon, the moon forms a right triangle with the Sun and Earth. By observing the angle between the Sun and Moon, φ, the ratio of the distances to the Sun and Moon could be deduced using a form of trigonometry.
From the diagram and trigonometry, we can calculate that
S L = 1 cos φ = sec φ . {\displaystyle {\frac {S}{L}}={\frac {1}{\cos \varphi }}=\sec \varphi .}
The diagram is greatly exaggerated, because in reality, S = 390 L, and φ is extremely close to 90°. Aristarchus determined φ to be a thirtieth of a quadrant (in modern terms, 3°) less than a right angle: in current terminology, 87°. Trigonometric functions had not yet been invented, but using geometrical analysis in the style of Euclid, Aristarchus determined that
18 < S L < 20. {\displaystyle 18<{\frac {S}{L}}<20.}
In other words, the distance to the Sun was somewhere between 18 and 20 times greater than the distance to the Moon. This value (or values close to it) was accepted by astronomers for the next two thousand years, until the invention of the telescope permitted a more precise estimate of solar parallax. Aristarchus also reasoned that as the angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18–20 times larger.
Lunar eclipse Aristarchus then used another construction based on a lunar eclipse:
By similarity of the triangles, D L = t t − d {\displaystyle {\frac {D}{L}}={\frac {t}{t-d}}\quad } and D S = t s − t . {\displaystyle \quad {\frac {D}{S}}={\frac {t}{s-t}}.}
Dividing these two equations and using the observation that the Sun and Moon appear the same size to people on Earth, s / S = ℓ / L {\displaystyle s/S=\ell /L} , yields
ℓ s = t − d s − t ⟹ s − t s = t − d ℓ ⟹ t ℓ + t s = 1 + d ℓ . {\displaystyle {\frac {\ell }{s}}={\frac {t-d}{s-t}}\ \ \implies \ \ {\frac {s-t}{s}}={\frac {t-d}{\ell }}\ \ \implies \ \ {\frac {t}{\ell }}+{\frac {t}{s}}=1+{\frac {d}{\ell }}.}
The rightmost equation can either be solved for ℓ / t {\displaystyle \ell /t} or s / t {\displaystyle s/t}
… excerpt ends here. Continue reading the full article.
