In 1965 my late friend and colleague John Stallings wrote "How not to prove the Poincaré conjecture." This work appeared in the Proceedings of the Wisconsin Topology Seminar and is still available on John's Berkeley website. It begins with the declaration: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now no one has known about it." It continues with the exposition of the main ideas relating the conjecture to statements in Algebra, and is certainly what Stephen Miller had in mind when he wrote, in the AMS Notices, after John's death, "His 1960s papers on the 3-dimensional Poincaré Conjecture are both brilliant and hilarious at the same time."

In 1445 Nicholas of Cusa wrote *De Geometricis
Transmutationibus* (On Geometric Transmutations); my account is
based on a recent translation into French of all of Nicholas' mathematical
works: *Nicolas de Cues, Les Écrits
mathématiques* by Jean-Marie Nicolle (Honoré Champion,
Paris, 2007). This was the first of Cusa's writings in which he
addressed the problem of *squaring the circle*. Literally,
squaring the circle means devising the straightedge-and-compass construction
of a square whose area equals that of a given circle. This
means a construction relating a segment of length
1 (the radius of the circle)
to a segment of length $\sqrt{\pi}$ (the side of the square).
Nicholas's plan was start from
an equilateral triangle and construct an *isoperimetric* circle;
this is
the content of the First Premise in *De Geometricis Transmutationibus*.
If the triangle had perimeter 1, the circle would have diameter $1/{\pi}$.
Then the composition of
two more standard straight-edge and compass constructions
could start from that diameter and generate first a segment
of length $\sqrt{1/\pi}$, and from that one
a segment of the reciprocal length $\sqrt{\pi}$.

Nicholas of Cusa (1401-1464) was one of the leading intellectual figures in early 15th-century Europe. He is often described as a transitional figure between the Middle Ages and the Renaissance, and in fact he was personally involved in one of the great events that mark that transition: Pope Eugene IV sent him to Constantinople in 1437 as part of a delegation to negociate the participation of the Eastern Orthodox hierarchy in the Council of Florence. They came, with an entourage of distinguished Greek scholars who stayed, and lectured, in Florence; contributing to the surge of interest in humanistic learning which led to the new age.

Nicholas' principal occupations were ecclesiastical politics and
administration
(he was named Cardinal in 1449) and, relatedly,
theology/philosophy. Those were a tumultuous times for the Church;
Nicholas was at the center both of bitter jurisdictional
controversies and of intense disputation about the exact
wording of dogmatic texts, where the placement of a comma could
assume cosmic importance. In those days philosophy, theology and
natural science were closely linked: the physical structure
of the universe had deep theological implications. Nicholas'
energetic and erudite mind, in *a priori* meditation,
led him to scientific insights that turned out to be prophetic.
For example, he understood that the earth, the sun and the moon were objects moving through space; and he rejected
the idea that all orbits had to be circular or even
that the universe had a center (*De Docta Ignorantia*, Book II).
Here he was a predecessor of
Kepler (who referred to him as "divine") and of Giordano Bruno.

Nicholas' interest in mathematics seems to have been its
status as an impregnable logical system. He believed that by
testing his philosophical theories in mathematics he could
produce convincing evidence of their validity. He outlines the
parallelism between geometry and theology in *De Circuli Quadratura*,
dated July 1450. "Transport yourself by assimilation from these mathematics
to theology. ...
Just as the circle is perfection in a figure,
since any perfection of figures is worked into it, its surface
contains all the surface of all figures and has nothing in common
with all the other figures, but is absolutely one and simple in
itself; likewise absolute eternity is the form of all forms ...
having nothing in common with any other form. And whatever the figure
of the circle therein may be, since it has neither beginning nor end,
it has resemblance with eternity ... . ... Likewise, if a triangle wanted to triangulate the circle, or a square to square it and so forth for the other
polygons, thus also intellectual nature wants to understand [God]."

One of the thought schemata Nicholas devised for use in theology was the
"concidence of opposites." Here is how he applied the principle
to the proof of his First Premise.
The construction involves a parameter, namely the position of
the point $e$ on the line $cb$. Nicholas observes that when
$e$ is at the midpoint $f$ the length of the segment $ah$
is smaller than the desired radius, and that when $e$ is at $b$
the length is larger. He applies the principle: *ubi est dare
magis et minis, quod ibi sit dare aequale* (where one can give
a greater and a lesser, one can also give an equal; essentially
the intermediate value theorem) and
concludes correctly that for some intermediate position
$x$ the length $ah$ must be exactly equal to that radius,
"and that is the point $e$ equidistant between $b$ and $f$."
The last statement made with no justification.

The construction is in fact plausible:
suppose the sides of the triangle have length 1.
Then $ef = \frac{1}{4}$; similarity of triangle
$abf$ with a half-equilateral triangle, and the Pythagorean
theorem, yield $af = \frac{1}{2\sqrt{3}}$;
so $ae = \sqrt{\frac{7}{48}}$,
and $ah = (5/4)\sqrt{\frac{7}{48}}$; the First Premise
states that $2\pi\cdot ah = 3$, which implies
$\pi = \frac{6}{5}\sqrt{\frac{48}{7}} = 3.1423...$ . This value, which Nicholas
could have calculated but never mentions explicitly, was within the
bounds $[\frac{223}{71}, \frac{22}{7}] = [3.14084... , 3.14285...]$ established by
Archimedes. Therefore, until better approximations to
$\pi$ were available, there was no way to prove Nicholas's
construction wrong, even though there were obvious gaps
in his proof.

Nicholas circulated copies of his work among his friends, who included Paolo Toscanelli (1397-1482), a Florentine astronomer and physician. He had been a classmate of Nicholas', and they remained good friends for life. Toscanelli wrote back with objections. To us, now, it is clear that there was no way the argument could be repaired. Nicholas' solution was to devise a different, and considerably more complicated, construction.

Nicholas would have done better to stay with his first construction. The new one was reprinted and minutely analyzed by Regiomontanus (Johannes Müller, 1436-1476) who showed that the implied value for $\pi$ was outside the Archimedean bounds (Nicolle calculates it as 3.154); this is part of a 60-odd page appendix to hisNicholas of Cusa was attacking a problem dating back to the ancient Greeks. The solution would have made him famous forever, and might even have helped bolster his side in theological disputations. No one knew at the time that squaring the circle is impossible: the proof requires calculus, which was 200 years away; and even then it was not discovered until 1882.

John Stallings was also attacking a famous problem: 50 years old, a very long time in modern mathematics. In this case the problem was not impossible, but the methods that led to its solution lay far in the future. Richard Hamilton's introduction of the Ricci flow, which led to Gregory Perelman's ultimate victory, came out in 1982, some 17 years after Stallings wrote "How not to prove the Poincaré Conjecture." But Stallings discovered his error by himself, before publishing, whereas Nicholas seems to have believed until the end that he had squared the circle, but perhaps had not been able to find the right argument to substantiate his claim.

Here is how Stallings ends his story: "... I was unable to find flaws in my 'proof' for quite a while, even though the error is very obvious. It was a psychological problem, a blindness, an excitement, an inhibition of reasoning by an underlying fear of being wrong. Techniques leading to the abandonment of such inhibitions should be cultivated by every honest mathematician."

We will see that any length occurring
in a compass and straight-edge construction starting from length one
must be an *algebraic number*, i.e. it must be a root of a polynomial
with integer coefficients. Considerably more intricate is the proof
that $\pi$, and therefore $\sqrt{\pi}$, is *transcendental,* i.e.
not algebraic. Some
references are given here.

- To see why this works in general, note first that if
points $P$ and $P'$ have their coordinates in
**S**, then by the Pythagorean theorem their distance $PP'$ = $r$ must also belong to**S**. So the circle of radius $PP'$ about $P$, say, has the equation $(x-p_1)^2 + (y - p_2)^2 = r^2$. Another circle constructed from two points with coordinates in**S**will have a similar equation, say $(x-q_1)^2 + (y - q_2)^2 = s^2$. All these coefficients lie in**S**. The coordinates of the intersection points of the two circles (if they intersect) will be the pairs $(x,y)$ satisfying both equations. From the first equation we can write $y = \pm\sqrt{r^2 -(x - p_1)^2} + p_2$. Substituting this value in the second equation yields a polynomial equation in $x$; it looks like it might have degree 4, but the higher powers cancel and it is a*quadratic*equation withcoefficients in**S**. The quadratic formula involves arithmetic and a square root, so the solutions it produces will again belong to**S**. For intersections of a circle and a line no cancellation is needed; the equation is quadratic; and for the intersection of two lines it is linear.

Menso Folkerts, "Regiomontanus'
role in the transmission and transformation of Greek mathematics,"
in *Tradition, transmission, transformation: proceedings of
two Conferences on Pre-Modern Science held at the University of
Oklahoma,* ed. by F. Jamil Ragep and Sally P. Ragep. With Steven Livesey.
Leiden; New York; Köln; Brill 1996.