> At 10:40 PM 02/05/1999, Gordon Fisher wrote:
>
> > ... Hans Hahn in an article on non-archimedean systems of quantities
> > notes that Bettazzi handled some questions of this kind in his *Teoria
> > delle grandezza* (1890), and Hahn also cites Veronese's *Fondamenti ...".
> > Hahn expresses an opinion that the study of such systems goes back to
> > Paul du Bois-Reymond (work published between 1870 and 1882), and Otto
> > Stolz (appeared from 1879 to 1896).To which I added the comment that
> > there was some study of such systems much earlier, e.g. involving
> > so-called horn angles in ancient Greece.
>
> "We know - Heath remarks - that there were controversies in ancient
> times about the nature of the 'angle of contact' (the 'angle' formed,
> at the point of contact, between an arc of a circle and the tangent to
> it, which angle was called by the special name _hornlike_, keratoeidh/s),
> and the 'angle' complementary to it (the 'angle of a semicircle'). The
> question was whether the 'hornlike angle' was a magnitude comparable
> with the rectilineal angle, i.e. whether by being multiplied a sufficient
> number of times it could be made to exceed a given rectilineal angle".
>
> In his "Elements", Euclid handled almost exclusively rectilinear angles.
> None the less, a single reference to angles with curved sides indeed
> appears in Book III, proposition 16.
>
> The straight line drawn at right angles to the diameter of a
> circle from its extremity will fall outside the circle, and
> into the space between the straight line and the circumference
> another straight line cannot be interposed; further the angle
> of the semicircle is greater, and the remaining angle less
> than any acute rectilinear angle.
>
> "The controversies doubtless arose long before his [Euclid's] time, and
> such a question as the nature of the contact of a circle with its tangent
> -remarks Heath- would probably have a fascination for Democritus, who,
> as we shall see, broached other questions involving infinitesimals."
>
> Whatever the case, it seems that such a crucial question as the nature of
> the horn angle had not a fascination for Euclid.
>
> Julio
>
Euclid defines an angle between any two lines, not necessarily straight
ones, then defines rectilineal angle. But non-rectineal angles are
mentioned only once in the 13 books, in that Proposition III 16. Why
bother to introduce a notion that is mentioned once and never again? I
think Euclid wanted to "kill off" that notion. Hornlike angles were used
before Euclid, see the Aristotelian proof of the equality of the angles
in an isoceles triangle. What Euclid shows in III 16 is that hornlike
angles are not magnitudes that have a ratio to rectilineal angles, in the
sense of book V, though he does not say it explicitly. It follows that
hornlike angles are not legitimate objects to apply in proofs. E.g. the
Aristotelian proof above has to be replaced by an allowed one. I think
that the proof of I 5 was found by replacing the angles between a
straight line (the base of the triangle) and a semicircle by a
rectilineal angle which is placed in more or less the same situation
where the hornlike angle used to be. Of course an argument has to be
devised to show the equality of the two angles on both sides of the base.
Avinoam Mann