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Re: Finding longitude in the 12th century
From: Lu Abel
Date: 2012 Sep 1, 10:49 -0700
Alex, my question was a bit rhetorical but thanks for the history -- which proves my thesis that it would have been almost impossible to produce a sight reduction method such as HO214 in the 12th century even if the basics of trig were available.  Forget comptometers to ease calculations, imagine doing any large-scale calculations using Roman numerals rather than decimals that only came to Europe in the 15th or 16th century!  I looked up "decimal numbers" in Wikipedia and it contains an almost useless history of them, citing obscure civilizations that might have used them three millenia ago, but not giving a whit of history on exactly how and why they displaced Roman numerals in Europe.  But further Google searches did point out that decimals migrated from Arabia to Europe in the 15th or 16th century.   Come to think of it, I remember decimals sometimes being called "Arabic numerals"

So back to the original question -- could someone have determined their latitude in the 12th century?  The answer seems to be a strong "no" -- at least with respect to any subsequent technique such as lunar distances or the equivalent for star/planet movement that rely on spherical trigonometry -- the theory may have been known, but practical use of that theory was impossible.

From: "eremenko@math.purdue.edu" <eremenko@math.purdue.edu>
To: NavList@fer3.com
Sent: Saturday, September 1, 2012 5:49 AM
Subject: [NavList] Re: Finding longitude in the 12th century

I can tell the story briefly.

> Ah, but then where did those log trig tables come from?

Tables of trigonometric functions were first developed by Greek
astronomers in the Roman Empire. Probably the oldest set which survived
to our time in in Ptolemy (II century AD). These are tables of chords,
but they are equivalent to the tables of sines, because of the very simple
relation between the sine and the chord.
They were computed by hand, using certain relations from trigonometry.
For example, geometry tells us that sin 30d=1/2, then there are theorems
which from the sine of an angle give you the sine of 1/2 of the angle,
and from sines and cosines of two angles give you sine of their sum.

Computing these tables with many digits was an enormous labor, especially if
you take into account that they had no our decimal system.
(Just try to add or multiply large Roman numerals, and you will see:-)
They used Greek numerals, which seem to be even less convenient.

There was a question whether people in medieval time really interested
in calculations. Not much. But some such people always existed: astronomers
(and astrologers).

Next great breakthrough was the introduction of decimal system and
logarithms....

However the main idea of logarithms existed before that. The idea is
that logarithms transform multiplication into addition.
Addition is relatively easy, while multilication is hard.
To see this, try to multiply b y hand two 6-digit numbers.

So the main property of log is that log(xy)=log(x)+log(y).
However the trig functions have similar properties.
Just slightly more complicated. Indeed, for example
cos(x)cos(y)=(1/2)(cos(x+y)+cos(x-y)).
Suppose you want to multiply A and B. Using the trig tables find
x and y such that cos(x)=A, cos(y)=B.
Compute x+y and x-y. Then use the table to find their cos.
Then the formula above gives you the product.
So you only do additions, subtractions, division by 2, and look to the
tables. This method is called prostapheresis.

So you can use trig tables instead of log tables, to multiply.

Logarithms and the decimal system were introduced almost simultaneously,
Decimal system was promoted by Simon Stevin in the end of XVI century,
and logarithms by John Napier (with important contribution of
Henri Briggs, who advised to combine logs with decimal system, thus
introducing convenient decimal logs) and Joost Burgi in early XVII.
(So Columbus had no logarithms and no decimal system).

Logs were computed by the following
formula, which is essentially the definition:
log(x)=area under the hyperbola y=1/t from t=1 to t=x. This area can be
computed by various means, for example, approximating the region
under the curve by many rectangles.

I think it was Kepler who said that invention of logarithms increased
10 times the life span of astronomers:-) The invention happened during his
life time, and I suppose it made possible his great discoveries
which were based on enormous amount of calculations.

Napier and Burgi comuted log tables with enormous number of digits.
They also computed logs of sines. Since then everyone can solve spherical
triangles quickly.

Alex.

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