How medieval astronomers made trig tables | Hacker News Reader

How medieval astronomers made trig tables

johndcook.com

119 pointsColinWright41 comments

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.

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.

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

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

The "Classical" world is as limited as "the middle ages" (perhaps even moreso); it mostly deals with the Mediterranean civilisations - and Iran is a long way from there.

According to my AP World textbook, at least, classical refers to a lot of geographically diverse civilizations, from the Byzantines to the Song Chinese to the Aztecs

What's wrong with 14th century then?

Christianism. The man probably thought he lived around the year 800.

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.

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

What do you suggest as the proper (for the purposes of the title, western-known) name of the period when al-Kāshī lived then? He obviously hasn't lived in the western-named and western-bound "Islamic Golden Age", being born a century later. Note that the goal is to convey the time period, not the geographical location.

Edit: see also how Wikipedia names that period of Persian history (my next answer for links): "Middle Ages."

then? It is not by the process of elimination that we'd get the precise era in which Al-Kāshi lived.

If either of us is to tolerate a certain degree of imprecision (for me that medieval concerns not only Europe and for you that the Islamic Golden Age is not a fixed time interval) then we'll come to a understanding, other than that, I think there is not a proper name of that period.

Edit: Note that the goal is to convey the time period, not the geographical location. When it comes to history those two are inter-dependent you cannot make a random choice of words.

It's not a random choice, it's a well established practice, even for non-European regions.

al-Kāshī was born in Persia in 1380:

http://en.wikipedia.org/wiki/Jamsh%C4%ABd_al-K%C4%81sh%C4%AB

That period in Persian history is in Wikipedia named "Middle Ages (652–1501)"

http://en.wikipedia.org/wiki/Iran#Middle_Ages_.28652.E2.80.9...

That name ("medieval", meaning simply "middle ages") is also used to denote a period in the history of Japan:

http://en.wikipedia.org/wiki/History_of_Japan#Medieval_Japan...

"The medieval or "feudal" period of Japanese history, dominated by the powerful regional families (daimyō) and the military rule of warlords (shōgun), stretched from 1185 to 1573/1600"

I might stand corrected, BUT:

"The Middle Ages was a period in Western history spanning the time from the 5th to the 16th century"[0]

and

"Middle Ages, the European historical period from the 5th to the 15th century (476-1453). By analogy, the term is also used to refer to periods in nations outside of Europe having similarities in social and military development"[0]

Iran, even during that period of Islamic rule , is far from similar to Europe on both the social and military level.

0.http://en.wikipedia.org/wiki/Middle_Ages_(disambiguation)

On the very page you link at the moment ( http://en.wikipedia.org/wiki/Middle_Ages_(disambiguation) ) it is also explicitly stated that the term is used for other regions too:

"Medieval or Mediaeval (the adjectival form of "Middle Ages") may refer to: Middle Ages, the European historical period from the 5th to the 15th century (476-1453).

By analogy, the term is also used to refer to periods in nations outside of Europe having similarities in social and military development."

So using it for other regions is common. One of these regions is actually Iran. I already gave link that the term is actually used in history of Persia (today known as Iran, the page in Wikipedia is named "Iran," http://en.wikipedia.org/wiki/Iran#Middle_Ages_.28652.E2.80.9... ) for the period 652–1501.

So using it for other regions is common Given that these regions are "having similarities in social and military development".

I was suggesting that the use of the term may not have been accurate given the fact Iran was different from Europe.

Nonetheless, Al-Kāshī(1380-1429) lived under Timurid dynasty (1370–1507)[0] succeeding Iran's "Middle Ages" making him thus, not an "Iranian medieval" astronomer.

0.http://en.wikipedia.org/wiki/History_of_Iran#Timurid_dynasty...

I'm glad that you finally accepted that there is a period known as "Middle Ages" in Iran.

However the Timur certainly didn't start new ages, his dynasty is just the last Mongolian one, and all Mongolian rulers are medieval:

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

"Genghis Khan invaded Transoxiana in 1219 during his conquest of Khwarezm. Before his death in 1227, he assigned the lands of Western Central Asia to his second son Chagatai, and this region became known as the Chagatai Khanate. In 1369, Timur, of the Barlas tribe, became the effective ruler while continuing the ceremonial authority of Chagatai Khan's dynasty, and made Samarkand the capital of his future empire."

There's really no reason to make the division currently present on that page:

1 Prehistory

2 Classical Antiquity

3 Medieval Iran

4 Timurid dynasty (1370–1507) <=== ? ? ? Why, How ? ?

5 Early modern era (1502–1925)

It's obviously an error that somebody managed to plant on that Wikipedia page. Another page is better, where medieval period at least naturally includes the last Mongolian dynasty and the early modern era starts with Safavids (1501), as "Encyclopaedia Iranica" also claims:

http://www.iranicaonline.org/articles/safavids

"The period of the Safavids, the dynasty that took control of Persia in the early 16th century, is often considered the beginning of modern Persian history"

However note also that other authors see Islamic Middle Ages lasting "down until the seventeenth century" and "it may be argued that certain continuities existed in Islamic civilization down until the advent of modern secular and national ideologies in the nineteenth century CE." (Medieval Islamic Civilization: An Encyclopedia: http://www.amazon.com/Medieval-Islamic-Civilization-Encyclop...)

Well, you have certainly proven that one must take anything on Wikipedia with a grain of salt. Kudos.

However the point that medieval gives the false impression that Al-Kāshī was a European astronomer still stands but unlike what you suggested before; changing the title, I hope this changed the impression.

See all my previous answers: it is absolutely normal to use "medieval" for historical periods outside of Europe, even for Japan. I also qouted printed encyclopedia "Medieval Islamic Civilization: An Encyclopedia" and "Encyclopaedia Iranica."

How a 15thC Iranian mathematician did trig.

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.

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.

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

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

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.

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.

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

Do you really think you'll get six digit accuracy with that?

Depends how long your string is.

Also depends on a ruler marked with 6 digits of accuracy.

Making it longer will not fix the problem. The longer the string, the more it stretches. Replace it with a rod. The longer the rod, the more it sags.

No, that's not too obvious. As has already been pointed out, you're going to need a circle the size of a building if you want to get more than 2-3 digits of precision (assuming that you can make your measurements with millimeter accuracy). How are you going to verify that your inscribed circle has no serious imperfections? Try making a building-sized circle on the ground using string -- it will be off of true by much more than a millimeter, I assure you.

At one point doing some field experiments I needed to measure out a large distance very precisely, and found that using construction measuring tape (the fiber glass kind from Home Depot) has problems because of stretch. And really any kind of material will have enough elasticity to cause problems for long measurements. I ended up making a tensioner to ensure I always took the measurement at about the same force. Getting high precision gets complicated and expensive real fast!

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?

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?

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

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

>> Then sin 1° is a fixed point of f.

Why ?

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?

Yes.

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

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