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The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162. Free version here. Thanks to Aaron Meyerowitz's answer to question 72792 for the reference.], angles might be measured either by the area of a sector of unit radius having the angle or by the arc length of such a sector. If the former convention is adopted then it can be proven using a completely unexceptionable Euclidean argument that $\lim_{x\to 0} \sin(x)/x = 1$.

Also, whichever convention is adopted (or so it seems to me), using completely unexceptionable Euclidean arguments, it is possible to prove the angle addition formulas for sine and cosine. Using these two ideas, it is straightforward to find the derivatives of sine and cosine, and from there one can derive an algorithm for computing digits of sine and cosine (and for computing $\pi$) using the relatively sophisticated mean-value version of Taylor's theorem.

The equivalence of the two definitions of sine (or of angle measurement) apparently depends on something like Archimedes' postulate: "If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D." (Again, thanks to Aaron Meyerowitz.) Of course, it is just this postulate that Archimedes needed to prove that the area of a circle is equal to the area of a triangle with base the circumference of the circle and height the radius. And something like it is surely necessary to derive any algorithm for computing digits of $\pi$. (Except, and this confuses me a bit, it seems that if we used the area definition of angle, we could derive an algorithm for computing sine without depending on this postulate, and from there we could get an algorithm for computing digits of $\pi$ since $\sin(\pi)=0$.)

I am looking in general for elucidation of the conceptual connections between the ideas I have so far discussed and of their background. But here are two more specific questions.

  • First, in what sense is a postulate like Archimedes' needed in the foundations of geometry? (I wonder, in particular, if in a purely formal development we might get by without it, but we would somehow be left without assurance that what we had axiomatized was really geometry.) Also, are more intuitive alternatives to Archimedes' postulate?
  • Second, what is really needed to get an algorithm for computing digits of sine? Does it really require such complicated technology as Taylor series? It seems like if one uses the area definition of angle, one might be able to give an algorithm using unexceptionable Euclidean techniques and without so much as invoking the notion of limit.
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    $\begingroup$ I don't understand your claim that something like Archimedes' postulate is necessary to derive an algorithm for computing the digits of $\pi$. I can define $\sin(x)$ as the unique solution of the differential equation $y'' = -y$ satisfying $y(0) = 0, y'(0) = 1$ and derive everything else (angle addition, the geometric interpretation) from this, then define $\pi$ as the smallest positive real number such that $\sin(\pi) = 1$ and show that the area of the unit circle is $\pi$. As far as I can tell none of this requires anything like Archimedes' postulate. $\endgroup$ – Qiaochu Yuan Jul 14 '12 at 21:05
  • $\begingroup$ Sorry, that should say $\sin (\pi) = 0$. $\endgroup$ – Qiaochu Yuan Jul 14 '12 at 21:06
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    $\begingroup$ Well, you could also "define" $\pi$ by as the result of some algorithm that outputs its digits. I am supposing that $\pi$ is defined as the ration of the circumference of a circle to its diameter. However, as suggested by my parenthetical comment at the ends of the second paragraph of my question, things might turn out differently if you defined $\pi$ as the area of a circle of unit radius. $\endgroup$ – Marian Jul 14 '12 at 21:20
  • $\begingroup$ By the way, the definition in terms of area seems conceptually more fraught since it presupposes commensurability between areas and linear distances instead of just arc lengths and linear distances. It is only relatively recently in the history of mathematics that people thought it made sense to compare areas to distances. $\endgroup$ – Marian Jul 14 '12 at 21:26
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    $\begingroup$ Yes, you are right, my second comment was a bit overblown. But it seems to me that what you are now saying is quite different from what you were saying in your original comment, so I am confused by the phrase "all I am claiming is that". $\endgroup$ – Marian Jul 15 '12 at 0:46
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To answer your question 1: it really depends on which set of axioms you consider 'canonical'. If you look at it from the viewpoint of the first-year courses in calculus and vector analysis and the like, you build up the Euclidean plane as $\mathbb{R}^2$, and you define curve length as some kind of integral, prove its independence of parametrization, etc. Within this framework, one will never need to use an axiom such as that of Archimedes, because you simply "define" curve length, and the question whether this definition is actually reasonable, i.e. corresponds to all manner of intuitive ideas about arc-lengths, is usually left to the imagination of the student.

Likewise, in the framework of Euclidean geometry, you need to have some kind of definition of arc-length. That this is put in the form of an axiom, and not simply given as a definition, signals the fact that Archimedes intends to justify his definition. He apparently finds it reasonable, and perhaps somehow consistent with everyday geometric intuition, that, when one has two curves with identical endpoints, which are both convex "in the same direction", and the first of which lies wholly or partly outside the second, and nowhere inside it, the first curve has greater length than the second. Perhaps he thinks of the first curve as a rubber band, that has been stretched out starting from the same position as the second curve. (I don't know whether this makes the axiom more intuitive, but I've tried to imagine his reasoning and this is what it feels like for me.)

Anyway, the point that I am trying to get across here is, you need to do something. Arc-length is not a given once you just start from the usual Euclidean axioms. Think about it: straight line segments can be compared, owing to the isotropy of Euclidean space (which incidentally is an unstated axiom in Euclid's Elements), which is how we can talk about the length of a line segment -- or at least about length relative to some sort of reference line segment. On the other hand, how would you compare a curved line with a straight line? You need to come up with a definition, at least, but an axiom, which is supposed to appeal to intuitive geometrical reasoning, is even better.

[By the way, this story does not hold for surface areas, since areas bounded by straight lines and those bounded by curved lines can be compared! Here Euclid can appeal to his "common notion" that the whole is greater than the part -- which by the way shows he clearly hadn't heard of open intervals, but that is another matter. For example, a circle is smaller than a circumscribed polygon (this is how Euclid phrases it, but he's referring to their areas), and greater than an inscribed polygon. Using this kind of reasoning, he shows that circles are to each other as the squares on their diameters, which shows that, in this case at least, arguments mixing lengths and areas are more straightforward than ones mixing lengths of straight and curved lines.]

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There are geometries such as the Dehn plane with infinitesimals in which the Archimedes postulate does not hold. So the postulate is not needed to do a bit of geometry. See the classical book of Artin "geometric algebra" for an exposition of the relations between an axiomatic approach of plane geometry and the algebraic approach via coordinates over a general (not necessarily archimedean) field.

For your second point, the sine and cosine can be computed using the duplication formulas, e.g. $$ \sin(\theta) = 2 \sin\Bigl({\theta\over 2}\Bigr) \cos\Bigl({\theta\over 2}\Bigr), \quad \cos(\theta) = 2\cos^2\Bigl({\theta\over 2}\Bigr) -1 $$ instead of the Taylor expansion. Just compute recursively $\sin({\theta \over 2^k})$ and $\cos({\theta \over 2^k})$ starting with the approximation $\cos({\theta \over 2^n}) \simeq 1$ and $\sin({\theta \over 2^n}) \simeq {\theta \over 2^n}$ for $n$ large.

This is more or less what Archimedes did to compute $\pi$ using the tangent instead. This is also a standard method to compute the Jacobi elliptic functions such as $sn$ or $cn$ that appears in many numerical libraries. Amusingly, you can compute the sine by trisecting the angle, a fact which certainly would have raised the interest of Ancient Greece mathematicians, just by using the formula $$-4 \sin^3\Bigl({\theta\over 3}\Bigr) + 3 \sin\Bigl({\theta\over 3}\Bigr) = \sin(\theta).$$

So for example, to compute $\sin(1)$, we can start with $n=5$ and $x = 1/243$ and iterate the polynomial $x\mapsto -4x^3+3 x$ five times to get $\sin(1)\simeq 0.841472...$, that's five correct decimals.

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