Translation from the Latin of "Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem" (1763). The passage Euler is referring to is the "Excerpta" in part 6, p. 6 of Descartes' 1701 "Opuscula posthuma". Before reading this paper I had not heard of the "quadratrix" before, and I recommend learning a bit about it before reading this. I found Thomas Heath, "A history of Greek mathematics", vol. I, chapter VII to be helpful, in particular pp. 226-230. The quadratrix is a "mechanical curve" that can be used to rectify the circle. The usual problem of squaring the circle is to construct a square with the same area (or perimeter) as a given circle, in a finite number of steps using compass and straightedge. Descartes worked in the reverse direction: from a given square he constructed the radius of a circle with the same perimeter, but in an infinite number of steps. In this paper Euler reconstructs Descartes' argument and develops some consequences of it. Euler finds that \[ \sum_{n=0}^\infty \frac{1}{2^n} \tan \frac{1}{2^n}\phi = \frac{1}{\phi} - 2\cot 2\phi. \] Integrating this yields \[ \prod_{n=1}^\infty \sec \frac{1}{2^n} \phi = \frac{2\phi}{\sin 2\phi}. \] I'd like to thank Davide Crippa from the University of Paris 7 for some helpful back and forth about this paper. One of the only citations to this paper that I have found is in Pietro Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence, 1792, pp. xxi--xxiii. The full text of it is available on Google Books.
A saving this discovery from oblivion, also publishes many singular formulas and series pertaining to measuring the circle, by which geometric approximations of this kind can be applied to a greater extent and further ones can be found. For instance, he has demonstrated that with q denoting the length of a quadrant of a circle whose radius is = 1, q = sec. 1 2 q • sec. 1 4 q • sec. 1 8 q • sec. 1 16 q • sec. 1 32 q • etc., from which one can conclude the following rather neat and elegant construction With the quadrant AOB established, the normal BC to the radius OB intersects the line OC bisecting the angle AOB at C. Then, the normal CD at C to this OC intersects the line OD bisecting the angle AOC at D. Similarly, DE, the line normal to this OD, intersects the line OE bisecting the angle AOD at E. Again then, EF , the line normal to OE, intersects the line OF bisecting the angle AOE at F , and so on. One continues in this manner until finally the radius OA is reached; this construction finally stops at the point Z. Then having done this, the line OZ will be precisely equal to the length of the quadrant Bcdef gA. 4 As well, one can easily derive many other constructions of this kind from the formulas of the Author. It will be helpful to note that the points B, C, D, E, F are found on a curve such that, by putting any angle AOD = φ and the line OD = v, 4 Translator: Euler in fact does not explain why OZ = Bcdef gA later in the paper. The following explanation is from Ed Sandifer. OBC is a right angle so sec π 4 = OC/OB, hence OC = sec π 4 . As well, sec π 8 = OD/OC, hence OD = sec π 4 • sec π 8 , etc. Then using the formula we get OZ = Bcdef gA.
it turns out that v = q sin. φ φ .5 Then, taking any ratio between the angle φ and the right angle, whose measure is the arc q, let φ = m n q; it will be v = n m sin. m n q and the line v can thus be assigned geometrically. Further, taking the angle φ as continually decreasing, it will come finally to a vanishing angle, for which sin. φ φ = 1, and then the line v is clearly equal to the quadrant q. One can make any number of additional similar formulas.
In Excerpts from the Manuscripts of Descartes,6 a certain geometric construction which quickly approaches the true measure of the circle is briefly described. This construction, which either Descartes himself had found, or which had been communicated by someone else, especially at that time indicates brilliantly the insightful character of its discoverer. Those who later handled this same argument, as far as I know at least, have not made mention of this extraordinary construction, so that it is in danger that it disappear altogether into oblivion. This demonstration, which is given with nothing added to it, can be supplied without difficulty; truly not only the elegance of this fertile construction merits more study, but the notable conclusions that can be derived from it would altogether by themselves be worthy of attention. This most beautiful construction is proposed thus in the words of Descartes himself: Fig. 1 I find nothing more suitable for the quadrature of the circle than this: if to a given square bf is adjoined a rectangle cg contained by the lines ac and bc which is equal to a fourth part of the square bf : likewise a rectangle dh is made from the lines da, dc, equal to a fourth part of the preceding; and in the same way a rectangle ei, and further infinitely many others on to x: and this line ax will be the diameter of a circle whose circumference is equal to the perimeter of the square bf .
The strength of this method therefore consists in that by continually adjoining rectangles of this type, cg, dh, ei, etc., whose top right angles fall on the extended diagonal of the square, finally leading to the point x at which ends the diameter ax of a circle whose circumference is equal to the perimeter of the square bf , or four times the line ab.
Since each of these rectangles is equal to a fourth part of the preceding one, as is already observed by Descartes himself, it is clear that the sum of all these rectangles will be equal to a third part of the square bf ; indeed this is clear continued to infinity is = 1 3 . Descartes further indicates the rule on which this construction rests; he begins namely with regular polygons of 8, 16, 32, 64, etc. sides, whose perimeters are mutually equal to the perimeter of the square bf . Now as ab is the diameter of a circle inscribed in this square, it is thus affirmed that ac is the diameter of a circle inscribed in an octagon, and indeed that ad and ae are the diameters of circles inscribed in a 16gon and 32gon respectively, and so on. Thus one sees that ax is the diameter of a circle inscribed in a polygon of infinitely many sides, whose circumference will therefore be equal to the perimeter of the square.
For clarity, I shall elaborate the demonstration of this construction. I observe that what are spoken of presently as the diameters of circles can equally well
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