232-108 BCE

103,049

Here’s where this number comes from: Take a sequence of symbols, (a b c d e f g h i j), say. Construct as many groups as you want by sticking parentheses around any two or more symbols or groups. For example, ((a b) c d (e (f g)) h i j). Or (a (b (c (d e f g) h) i) j). Or (a (b (c d e)) f g (h i j)). There are 103,049 ways of doing this with ten symbols, so 103,049 is the tenth Schröder number, named after the man who published this result in 1870. But it turns out that the same number is given in Plutarch – attributed to Hipparchus (190-120 BCE) – as the number of “affirmative compound propositions” that can be made from ten simple propositions. It is only in 1994 that somebody connected the dots, and realized that Schröder numbers had been discovered 2000 years before Schröder.

This is just one example of the very high level of mathematical, scientific, and technical accomplishment attained within the Hellenistic world — the world of Greek culture after Alexander. Lucio Russo calls Hellenistic science The Forgotten Revolution. A couple more examples from his book:

Everybody knows that Aristotle – and thus “the Greeks” – thought that heavy objects fall faster than light ones. Supposedly it took Galileo to prove him wrong. But in fact there is a clear statement in Lucretius (*De Rerum Natura* II:225-239) that objects of different weight fall at the same speed, unless air resistance kicks in; Russo argues that circumstantial evidence points to Hipparchus as the source.

Russo also argues that Hellenistic thinkers understood that gravity could account for the spherical shapes of the earth and planets, and that the balance between gravity and linear velocity could account for circular orbits. He shows that some strange passages in Vitruvius and Pliny about the sun making planets go around by shooting out triangular rays make sense if you assume the authors were looking at, but not understanding, vector diagrams of successive straight line motions bent into a circle by a centripetal thrust.

Russo argues that scientific progress largely came to an end by 150 BCE, and the Roman period saw an actual decline in scientific understanding. Later writers like Ptolemy and Galen, often taken to represent the height of Classical learning, were derivative, and didn’t really understand their predecessors: a stark reminder that a civilization may avoid collapse, and even maintain a decent level of prosperity, but regress intellectually.

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