
Re: Why trig?
Posted:
Apr 2, 1997 9:57 PM


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 >careful.
Indeed. There is plenty on historical aspects of trigonometry in:
von Braunmuehl, Anton (18531908): "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 mathhistorylist 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 midpoint 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: NonEuropean Roots of Mathematics* (Penguin Books, 1991) fills in many details. See the sections on Indian Trigonometry (pp. 280286) and Arab Trigonometry (pp. 338344).
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 *samastajya* (bowstring). The halfchord, in terms of whose length the Indians stated what we would refer to as a sine, was called *jyaardha*, eventually abbreviated to *jya*...
Julio Gonzalez Cabillon

