How medieval astronomers made trig tables

http://www.johndcook.com/blog/2014/11/23/compute-trig-tables/

by ColinWright • 12 years ago

119 41 12 years ago

exDM69• 12 years ago

Very interesting, makes me want to order the spherical trigonometry book that was the source for this text. Spherical trigonometry is sometimes very non-intuitive.

I recently took a few courses of university level astronomy. And as has been the tradition at least from the times of Ptolemy's Almagest in the 2nd century, we started with spherical trigonometry.

When doing exercises, we'd draw triangles on oranges we got from the cafeteria. That's way better than trying to draw spheres on a flat piece of paper.

sxp• 12 years ago

Wouldn't it be easier to calculate the values by drawing a giant circle on the ground and measuring the values? You can mark out the angle on the circle and then measure the X & Y values of different points on the arc.

pokpokpok• 12 years ago

accurate measurement tools weren't so easy to come by as paper and time

wging• 12 years ago

You'd also probably need to do some fairly precise calculations in order to manufacture accurate measurement tools. Chicken-and-egg sort of problem?

ColinWright2• 12 years ago

It's tough to get 6 digits of accuracy that way. Think, if the circle is 10 metres across, 1 mm is still 0.0001, just 4 digits.

Trying to do it physically is orders of magnitude less accurate.

breadbox2• 12 years ago

You might think that if you've never tried it. But how do you get a giant circle on the ground in the first place? I mean, sure that giant circle you've drawn there looks circular, but how do you know that it is? Normally you'd measure it and verify it, but the whole point is that you're trying to use this circle to determine what those verified measurements ought to be in the first place. Even a hairsbreadth misshapen circle is going to give you wrong numbers. No, there's no comparison to finding a precise mathematical formula and exploiting it.

willvarfar• 12 years ago

http://www.instructables.com/id/How-to-draw-a-circle-with-st... or is that too obvious?

gradi3nt• 12 years ago

Very interesting. Life was seriously tough before series expansions were invented!

KhalilK6• 12 years ago

Meta note:

Al-Kāshī was not a "medieval astronomer", the Middle Ages are related to European history and Al-Kāshī lived in Iran during the Islamic Golden Age.

m-photonic• 12 years ago

But "ancient" astronomers would have been even less accurate, and headlines need words that people can instantly recognize.

michael_fine2• 12 years ago

Yeah, post-classical would have been the more accurate term, thought I'm not sure it would've had the same comprehensibility.

nichtich• 12 years ago

What's wrong with 14th century then?

acqq6• 12 years ago

Meta meta note: "The metaphor of a golden age begins to be applied in 19th-century literature about Islamic history, in the context of the western cultural fashion of Orientalism."

http://en.wikipedia.org/wiki/Islamic_Golden_Age

It seems nobody named any period of Islamic history "golden age" before (apparently Muslim scholars after 12th century had a different value system). Anyway, the same western cultural circles also use "middle ages" to denote approximately the time period between 5th and 15th century.

Moreover, Islamic Golden Age refers to "the 9th to 12th centuries, and (is) sometimes extended to include parts of the 8th and 13th centuries" but Al-Kāshī was born in 1380, died in 1429 (born in 14th, died in 15th century!).

So "medieval" matches the time when Al-Kāshī lived, "Islamic Golden Age" (note, the term also invented by the western intellectuals for their own purposes) simply doesn't.

KhalilK6• 12 years ago

My whole point is that medieval gives the false impression that he was a European astronomer.

analog31• 12 years ago

Thanks! I love reading about how people solved problems without our contemporary tools.

srean• 12 years ago

You may find the work of Kerala mathematicians on Taylor series of sin, cos, and arctan quite intriguing

http://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and_...

They happen to be at the center of a minor controversy and semantic games of what exactly is Calculus because they were doing these series approximations routinely about a couple of centuries before the European recognition of the Taylor series or calculus for that matter.

thewarrior2• 12 years ago

> Then sin 1° is a fixed point of f.

Why ?

CarolineW• 12 years ago

The fact that it's a fixed point comes from the triplke angle formula just a few lines above. Substitute and simplify, and you'll see that f applied to sin(1 deg) returns sin(1 deg).

As to why it conveges when iterated, starting from a value that's close enough, that's a different question. Is that what you actually intended to ask?

thewarrior2• 12 years ago

Yes.

raverbashing• 12 years ago

numpy? really?

"For deg2rad"? REALLY?

But to be fair:

>> math.sin(2 * math.pi/360.)

0.01745240643728351

>> numpy.sin(2 * numpy.pi/360.)

0.017452406437283512

VikingCoder• 12 years ago

Okay, I'll voice my desire...

My understanding of this comes from Carl Sagan's original Cosmos miniseries...

Kepler and Brahe were reluctant allies. Brahe had the observational data Kepler needed to try to understand the motions of the planets...

...but how the hell did Kepler do it?

I've gone through Differential Equations, I've done Linear Algebra quite a bit, I'm good at computer graphics... I had to derive the equivalent of Bresenham's Ellipse drawing algorithm for a test... And I have no idea how it's even remotely possible for Kepler to have taken astronomical observational data, and derived the motions of the planets from that data.

Has anyone ever done a "For Dummies!" write up of how this was possible?

plikan13• 12 years ago

I didn't read the book, but from leafing through Stephen Hawking's "On the shoulder of giants" I got the impression this book tries to reconstruct how Copernicus, Kepler etc... acquired their understanding of the universe, based on their original texts. So maybe that's the book you want?

How medieval astronomers made trig tables

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