As many of you may know, sometime around the 14/15th centuries an Indian mathematician by the name of Madhava of Sangamagrama derived the Maclaurin series for sine and cosine for the first time in recorded history, predating Newton by 250 years, give or take. On a unit circle, his formulas simplify to $$\sin(s)= s-\frac{s^3}{3!}+\frac{s^5}{5!}+\ldots$$ $$\cos(s)= 1-\frac{s^2}{2!}+\frac{s^4}{4!}+\ldots$$ where $s$ is the length of some arc of the unit circle. So put yourselves into a medieval mathematician's shoes for a minute. You have some basic premises:

$P1$. The "sine" and "cosine" are literally just geometric lengths inscribed inside a circle

$P2$. $s$ is also a geometric length

$P3$. $s$ has an upper bound of $2\pi$; it probably doesn't make sense to have an arc-length of a shape be longer than the shape's circumference

$P4$. Lengths cannot be negative

Now, let $T(s)$ be a good-enough Taylor Polynomial of Sine's Maclaurin Series, situated on the unit circle. Notice the following:

  1. $T(\frac{3\pi}{2})\approx -1$. Plug some more nearby values in and it's not terribly difficult to notice that $T$'s output on $(\pi,2\pi)$ is a "negative mirror" of $T$'s output on $(0,\pi)$. But since we accepted $P1$, this output contradicts $P4$.
  2. $T$'s output on $(-\infty,0)$ is well-defined, and replicates the behavior we see on $(0,2\pi)$. But $P2$ and $P4$ suggest that $T$ should NOT be defined on this interval. The fact that it is is a little spooky...
  3. Similarly, $T$'s output on $(2\pi,\infty)$ is also well-defined and replicates the behavior we see on $(0,2\pi)$. But $P3$ suggests that it should be undefined.

(Similar weirdness happens with Cosine's series)

So it seems that not only is $T$ surprisingly well-defined for completely invalid inputs (2,3), it's also nonsensically defined for half the inputs that are valid (1). So I'm curious: were these contradictions noticed in Madhava's time? And if so, is there any historical consensus/evidence/speculation on how they were reconciled?

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    $\begingroup$ As you can see even from modern expositions (e.g. Bressoud), Kerala mathematicians had a computational scheme derived geometrically for acute angles. It was used to increase accuracy of astronomical calculations, and was quite labor intensive. So they did not preoccupy themselves with what happens when one tries to put some other values through that scheme. This sort of playing around algebraic mentality only emerged in 17-18th century when computational methods were better developed. $\endgroup$
    – Conifold
    Aug 13, 2023 at 6:23


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