What follows is my speculation, based on what I know about the state of mathematics in the time of Descartes:Since Descartes himself is often credited with uniting geometry and algebra via the introduction of coordinate systems, it must be true that Descartes worked at a time when geometry and algebra were regarded as separate disciplines. This means that "rectification" in geometry did NOT involve assigning a NUMERICAL value to the length of a curve. The pure geometrical approach to this problem would be to actually CONSTRUCT a straight line that was guaranteed by geometrical proof to have the exact same length as the given curve.
If this is in fact what was meant in that era by "rectification" (and I'm pretty sure this is correct), then Descartes is correct. You cannot construct a straight line that is the exact same length as a given arbitrary circular arc, using only ruler and compass construction.
This leads to the next point: how do we know that this construction is impossible? The word "necessarily" in your quotation is a comment from the author of the article. It would NOT have been known in Descartes' day that the attempt to rectify curves was doomed to failure. The word "necessarily" is justified, because Galois theory shows that the construction is impossible. However, Galois theory was not developed until much later, and in fact applied algebra to the problem, so it could not have come into play until after Descartes had connected geometry to algebra. It thus lay far in Descartes' future.
The basic approach of Galois theory to this problem is to use Descartes' coordinate geometry to assign numerical coordinates to locations and then to show that the ruler and compass can only construct points that have coordinates that are within a certain class of numbers. Then it is shown that the number pi does not belong to this class of numbers, and hence cannot be constructed. Therefore, it is impossible to rectify a semicircle of radius 1, so there cannot be a universal method of rectification using ruler and compass. Incidentally, the square root of pi and the cube root of 2 is also not in the proper class of numbers, and so Galois theory also shows that squaring the circle and duplicating the cube (two classical geometry problems) are impossible.
As far as I know, there is no proof purely WITHIN geometry that shows these constructions to be impossible; only the algebraic approach via Galois theory works. If anyone knows of a purely geometric proof, I would be interested in a reference, but I do not think one exists.
It is also possible, I suppose, that the author of the article is referring to the fact that there is no algebraic formula for the length of a segment of an ellipse or hyperbola. Whether this makes these curves "unrectifiable" is a matter of is more a matter of taste than fact: once you step outside of pure geometry, the question naturally arises as to what exactly you will consider to be a "solution."
Instead of a construction, you now have a computation, but what properties must the computation have? The integration itself provides a universal procedure for "rectifying" a curve in the sense of providing a numerical value for its length, but the procedure is one of approaching the value as a limit. Does a well-defined procedure that gives the answer to any desired accuracy, but is not a finite procedure, count as a solution? In that case, all smooth curves can be rectified.
On the other hand, if you require a formula or a finite procedure, what terms are you allowed to use in your formula? If you allow only basic functions, you can produce at most polynomials, but the solution to this problem will not be a polynomial. If you allow more "advanced" functions, such as sine or cosine, more curves become rectifiable, but these functions themselves are defined analytically as infinite series, so the infinity of the process has merely been hidden within the functions.
In fact, the integral for the length of an ellipse is a function which cannot be expressed in terms of any of the standard "elementary functions." This does not mean that the integral has no answer, only that the answer has no name. You could always derive an infinite series for this function and then name it'something suchs "ell" for ellipse. This makes it no more or less acceptable than familiar functions like sine or cosine. So in this case, the author of the article may have meant simply that these functions do not have a common name.
Hope this helps!
--Stuart Anderson
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