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Topic: Why trig?
Replies: 21   Last Post: Apr 3, 1997 4:17 PM

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Julio Gonzalez Cabillon

Posts: 1,353
Registered: 12/3/04
Re: Why trig?
Posted: Apr 2, 1997 9:57 PM
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On Wed, 2 Apr 1997, Ralph A. Raimi wrote:
>To carry back the formula for cos(A+B) to ancient times is, however,
>not easy. There is a proof of the equivalent attributed to Ptolemy,
>but it requires quite a bit of knowledge of Euclid to follow it.

Certainly, it is a bit troublesome to carry back that formula to ancient
times. Incidentally, was Ptolemy applying *plane* trigonometry?

>But even if a trig class doesn't get the full story, it is very good
>to tell as much of the chronological story as possible. On the other
>hand, historical trigonometry is not about periodic functions, but
>is confined to angles inscribed in circles somewhere, i.e. angles that
>can occur in triangles;

And not always about angles inscribed in circles somewhere...

>and the extension of the trig functions to larger
>domains is more recent. Here, history can be mischievous if one is not

Indeed. There is plenty on historical aspects of trigonometry in:

von Braunmuehl, Anton (1853-1908): "Vorlesungen ueber Geschichte
der Trigonometrie", [Lectures on the history of trigonometry],
2 volumes, B. G. Teubner, Leipzig, 1900 & 1903.

[The 'e' in *Braunmuehl* and the first 'e' in *ueber* are in lieu of the
German umlaut]

Almost a century later, this standard book on the subject is still
fascinating reading and "something to write home about".

>Old fashioned trig books would give a definition of sine as a
>ratio in a right triangle, hence for acute angles only, and then extend
>the definition to the other quadrants by a woefully long list of
>adjustments. From a strictly historical point of view that may be more
>or less the way it happened [...]

Sorry, but which evidence do you have to support your statement?

Now I'd like to add some recent exchanges from MAA math-history-list in
connection with trigonometry, threaded by Ken E. Pledger {KEP}, Randy K.
Schwartz {RKS}, and me {JGC}. Here goes:

KEP: There are some other interesting words connected with a segment.
Our word "arc" derives from the Latin "arcus" (bow), "chord" underlies
the modern English "cord" (string), and the line across the segment
perpendicular to the chord at its mid-point is traditionally called the
"sagitta" (Latin: arrow).

JGC: Amedeo Agostini in his article _Le funzioni circolari e le funzioni
iperboliche_ (Ulrico Hoepli, Milano, 1937) states:

E andata quasi completamente in disuso la considerazione della
funzione senv a = 1 - cos a, alla quale fu dato il nome di
*sagitta*, o *sinusversus* (donde il nome di *sinus totus* dato
al raggio della circonferenza). Accanto al *senoverso* si considero
anche il cosenoverso a = senv (pi/2 - a). Per l'arco supplementare
dell'arco 'a' si introdussero le funzioni analoghe col nome di
subsenoverso e subcosenoverso. Tra queste funzioni *verse* e il
seno e il coseno sussistono quindi le relazioni

senv a = 1 - cos a

cosv a = 1 - sen a

s.senv a = 1 + cos a

s.cosv a = 1 + sen a .

Having in mind its geometrical concept it is not difficult to imagine why
the *versed sine* was considered an arrow. Thus the Arabs had the term
"sahem" (arrow), which, later, was latinized as "sagitta". Interestingly
enough, this term was used by Leonardo of Pisa in his writings.

RKS: In a posting to this list on 15 Jan 1997, I wrote the following:

George Gheverghese Joseph, in his wonderful book *The Crest of the
Peacock: Non-European Roots of Mathematics* (Penguin Books, 1991) fills in
many details. See the sections on Indian Trigonometry (pp. 280-286) and Arab
Trigonometry (pp. 338-344).

The history of our word *sine*, in particular, makes for fascinating
reading. Because of the resemblance of a circular arc and its chord to an
archer*s bow and bowstring, respectively, Indian astronomers referred to
these geometric objects as *capa* (bow) and *samasta-jya* (bowstring). The
half-chord, in terms of whose length the Indians stated what we would refer
to as a sine, was called *jya-ardha*, eventually abbreviated to *jya*...

Julio Gonzalez Cabillon

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