It could be we all know what I'm trying to say with this example: that

something nontrivial can be going on when people see patterns in data and form hypotheses. and also that the mental process can turn out to have been logically in appropriate to the data, as in the case of Kepler, but no matter, he was prepared (by his culture and the mathematics of his time) to make the leap anyway.

http://en.wikipedia.org/wiki/Hemiola#EtymologyThe word hemiola comes from the Greek adjective ἡμιόλιος, hemiolios, meaning "containing one and a half," "half as much again," "in the ratio of one and a half to one (3:2), as in musical sounds."[1] The early Pythagorians, such as Hippasus and Philolaus, used this term in a music-theoretic context to mean a perfect fifth.[2]...

The words "hemiola" and "sesquialtera" both signify the ratio 3:2, and in music were first used to describe relations of pitch. Dividing the string of a monochord in this ratio produces the interval of a perfect fifth.

http://en.wikipedia.org/wiki/Hemiola#The_perfect_fifthHemiola is the ratio of the lengths of two strings, three-to-two (3:2), that together sound a perfect fifth.[16]

The justly tuned pitch ratio of a perfect fifth means that the upper note makes three vibrations in the same amount of time that the lower note makes two... The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune.

From high to low, the violin strings go E, A, D, G.

D is a fifth above G, A is a fifth above A, and so on. Each step is a

sesquialtera and that was a Medieval music term, a concept in Kepler's culture. Here is the kind of data he had in front of him to look at:

- Code: Select all
`Radii Years`

.72 .61

1.0 1.0

1.5 1.9

5.2 11.9

9.5 29.4

If you cube 9.5 and then take the square root, do you get 29.4? He recognized somehow that you do.

Let's paste 9.5^1.5 into google, and 5.2^1.5, and 1.5^1.5 ...pretty close.

.72^1.5 gives almost exactly .61

Some of the others are off by one in the tenths digit, like 29.3 instead of 29.4. Just roundoff.

Anyway take the number and

multiply it by its own square root like asking for another HALF slice of pie. A sesqui altera. Half another, please.

What other example of "hypothesis formation" can you think of that you might like to share?