Re: History of trigonometry

Narasimham wrote:
Ross A. Finlayson wrote:
Nick wrote:
"Ken Pledger" <ken.pledger@xxxxxxxxxxxxx> wrote in message
news:ken.pledger-E6DD48.11304703012007@xxxxxxxxxxxxxxxxxxxxx
In article
<2386070.1166882761600.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
zeros <nimzeros@xxxxxxxxx> wrote:

....
When did someone define the sine of an angle greater than 90 degrees?

I think this may be happened in 17c. But I can't find any record about my
question....


It's an interesting question, which most popular historians of
mathematics scarcely mention.

In fact, http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html says
that:

"He made large bounds forward in the study of modern analytic geometry and
trigonometry where he was the first to consider sin, cos etc. as functions
rather than as chords as Ptolemy had done."

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Trigonometric_functions.html#76
says that:

"The first actual appearance of the sine of an angle appears in the work of
the Hindus."

"Chapters of Copernicus's book giving all the trigonometry relevant to
astronomy was published in 1542 by Rheticus. Rheticus also produced
substantial tables of sines and cosines which were published after his
death. In 1533 Regiomontanus's work De triangulis omnimodis was published.
This contained work on planar and spherical trigonometry originally done
much earlier in about 1464. The book is particularly strong on the sine and
its inverse."

See reference for more.

Nick
---
Where that is so, is there a similar notion to trigonometry for
unfolding regular n-gons, n-gonometry? How about a 3-D or n-D analog
for pyramidometry and n-hedrometry? I figure these are quite well
explained somewhere. Where can I learn more about these things and
their directions of development?

Basically my question here is: what's 4-gonometry.

May be the tetrahedronometry.It should include dihedrals and trihedral
"solid" angles.Unfortunately, this 3D counterpart did not develop so
well or used so much like plane trigonometry.It could have involved
quaternions.

Why is trigonometry called trigonometry?

Very clear from its Sanskrit roots: Tria (three,sounds are similar in
English,German and French),Kona (gonio is angle),Matra (size,measure
or unit).These three currently used Indian words are common parlance.

Narasimham


Hello Narasimhan,

Thank you, yes, the word roots maintain their generally understood
meaning. I didn't know Sanskrit was that close to Greek, only ever
having heard of Sanskrit as being a dead language, i.e., no longer
spoken, with a fixed vocabulary and grammar, like Latin or ancient
Greek, although somebody added "windsurfing" to Latin some years ago, I
thought it was more insular.

I research those terms, or rather, "google" them, besides a reference
to Lexell with some description of a generalized polygonometry in the
late 1700's blind Euler era, they appear to be more about the spherical
"trigonometry", I think that has to do with great circles and so forth,
and don't know, I'm wondering more about extension of planar
goniometry. The polygonometry is referenced with regards to the
precession of orbits and so on, I've read a decent textbook on using
quaternions for orbital computations. I can't recall it, not having
had the background, but it was pretty good. I read some of your posts
and am impressed, I wish I was a better differentialist. I'm average.
I have here a copy of a Dover reprint of the "Theoretical Kinematics".
There's quite some discussion about trigonometry these days, otherwise
as usual it seems there is much about foundations.

I wrote a little program last night to graph them, these evolving
coordinate systems, I'll try and get an applet together to illustrate
this "n-gonometry."

It might be interesting to consider how totally fascinating geometry
can be. While that may be so, I'm sure for some it's remarkably
boring. Ha ha ha. Consider for example, the interior angle of a
polygon is 2 pi radians, except a triangle's interior angle is 1pi
radians. Consider the exterior angle, it's (2 pi - 2 pi / n) * n,
where n is the number of sides. So, for example, for a polygon of 100
sides, its exterior angle is 198 pi. The exterior angle or sum of
exterior angles of a polygon of n sides is (2n-2) pi radians. The
radian is the arc length subtended by the angle, of the unit circle.
So, the 180 degree angle is pi, radians, that's the length of the
perimeter of the circle swept through from the beginning to end of that
angle. Fascinating.

The circle, it's 360 degrees or 400 gradients. Nobody uses gradients
anymore, and that's hyperbole.

So, is the exterior angle of the circle 2 pi? That would be
non-archimedean, as it were. In smooth infinitesimal analysis, the
circle has infinitely many sides, it is the infinitely sided regular
polygon, yet with no corners, is it zero?

I'll try and have some applet to illustrate the drawing of these
periodic "n-gonometric" functions and perhaps some analytical results
of them shortly. Good day.

Ross

.

Relevant Pages

  • Re: History of trigonometry
    ... except a triangle's interior angle is 1pi ... So, for example, for a polygon of 100 ... its exterior angle is 198 pi. ... radian is the arc length subtended by the angle, of the unit circle. ...
    (sci.math)
  • Re: History of trigonometry
    ... except a triangle's interior angle is 1pi ... So, for example, for a polygon of 100 ... its exterior angle is 198 pi. ... radian is the arc length subtended by the angle, of the unit circle. ...
    (sci.math)
  • Re: Sine code for ANSI C
    ... >> compute the sine of small arguments quickly, but if it has to take ... > angle is greater than pi. ... All the library writer knows is that there's little *precision* ... small arguments it radians are easy to compute. ...
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  • Re: Calculating Pi
    ... when you take the sine of a small angle, ... This method leads naturally to ... then sin x is very nearly 0. ...
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  • Re: History of trigonometry
    ... "The first actual appearance of the sine of an angle appears in the work ... Now as the chord ... I would observe that the fact that I use a reference doesn't ...
    (sci.math)