40 The New York Review
is not (“It was neither a boat, nor an
aeroplane... ”). “Probabilist” works
in the theme of chance. First read-
ing aloud and then disentangling the
mathematically sophisticated “Permu-
tations by Groups of 2, 3, 4 and 5 Let-
ters” (“Ed on to ay rd wa... ” becomes
“One day toward... ”) is a delight, even
if you have to work a little. As a whole,
Queneau said in a published conversa-
tion with the French painter and poet
Georges Ribemont- Dessaignes, he
hoped that the collection might act as
a “rust remover of literature” without
“boring the reader too much.”
In 99 Variations on a Proof Ording ex-
plains that, after discovering Queneau’s
Exercises in Style, he “wanted to see
what effect constrained writing strate-
gies would have on a mathematical nar-
rative—a proof.” Mirroring Queneau,
each of Ording’s variations is born of
a simple fact, but a mathematical one,
expressed in this cubic equation:
Let x be real. If x^3 –6x^2 +11x–6
=2x– 2 , then x=1 or x=4.Admittedly, there is a bit to unpack
here. For starters, this equation involv-
ing interacting cubes, squares, and the
like is just an example of “solving for
x,” where you discover that if you plug
in 4 for x on both sides you get the same
number (6), and if you plug in 1 for x
on both sides you also get the same
number (this time it is 0).^5 But—and
more interesting—these are the only
two “real” numbers (i.e., numbers with
decimal expansions or numbers on the
familiar “number line”) for which this
is true. This part of the proof reaches
back to the earliest recorded arithme-
tic exercises, in which a “proof” was an
actual demonstration of a calculation.
Ording’s choice of theorem runs the
risk of being a deterrent to readers
with “little or no predisposition to the
subject matter,” whom he hopes to at-
tract. It introduces a heavy dependence
on notation—x’s and y’s make regular
appearances among the hundred vari-
ations—and there is a fair amount of
moving them about through the use
of algebra. Legend has it that Stephen
Hawking’s editor on A Brief History of
Ti me warned him that every equation
included in the book would reduce his
audience by half. He chose to include
just one: E =mc^2. By that arithmetic,
this little aside and the paragraph above
may have cost us fifteen sixteenths of
our readers. Ording’s book would suf-
fer an even more dramatic loss.
But Ording’s assertion seems a clear
nod to Queneau. As a simple (even “bor-
ing”) statement that two combinations
of numbers work out to be the same, it
serves as a mathematical doppelgänger
for Queneau’s bumptious commuter.
That it too could inspire so many vari-
ations and accompanying reflections
deserves applause, even from the non-
predisposed reader.^6 There is much that
can be learned—and plenty of rust to be
removed—for both the uninitiated (but
open- minded) and the sophisticated.
Early mathematical arguments and
styles are well represented in the book,
ranging from an imagined Babylonian
justification (chapter 16, “Ancient”)
to a proof using Euclidean geometry
(chapter 52, “Antiquity”). I’d challenge
anyone to check the work in the glyphs
that make up the former. Chapter 88,
“Dialogue,” is a back- and- forth between
a master and a disciple—a form of rea-
soning through a problem that Ording
posits is “a likely candidate for the oldest
style of presenting a mathematical argu-ment.” Some variation of it still goes on
during office hours all around the world.
There are two medieval efforts,
each of which serves as an example
for the utility of a simplified—even if
abstract—notation that cleans up the
flowery and somewhat laborious text.
English is really not well suited to alge-
bra. The first, chapter 34, is formatted
to look like an illuminated manuscript
and restates the theorem as follows:Suppose that the intensity of a
quality is as the cube of its exten-
sion and 9 times that less 6 times
its square. It will be demonstrated
that when this quality achieves an
intensity of 4, its extension is 1 or 4.In the accompanying commentary Or-
ding credits the proof as following the
method of Leonardo of Pisa (aka Fibo-
nacci) and the scholastic philosopher
Nicole Oresme, with connections to an
ancient Chinese text, The Nine Chap-
ters on the Mathematical Art.
The second medieval- style proof,
chapter 70, continues the high languagebut in a normal font, opening with “In
the name of God, gracious and merci-
ful!” and closing with “It is now time that
we should conclude this demonstration
with gratitude to God and praising all
of His prophets.” Sandwiched between
is a clever geometric proof that includes
circles, conics, and intersecting lines. We
learn from Ording’s commentary that
this kind of proof could be found in the
eleventh- century Tre a tise on Demon-
stration of Problems of Algebra, a mas-
terpiece by the Persian mathematician
Omar Khayyam, who is now surely bet-
ter known for his poems in the Rubaiyat.
Khayyam’s Algebra was the first writ-
ten attempt to address all the ways in
which cubic equations can arise and
then be solved via clever geometric con-
structions. With no x’s and y’s present
it may seem to the modern reader to be
ironically titled, but the word “algebra”
is derived from the Arabic al- jabr, which
refers to the process of simplification
(e.g., “cancellation”) of mathematical
relations. Its origin is usually attributed
to the ninth- century Persian mathema-tician al- Khwarizmi, whose name lives
on in the word “algorithm,” which re-
fers to any recipe of instructions that
underlies a computer program and, for
some, is poetry of a different sort.
Fast forward several hundred years
to Renaissance Italy, where solving
equations was a form of dueling, with
mathematicians challenging one an-
other to contests for money and honor.
Ording memorializes this in chapter
43, “Screenplay,” a script in which a
character announces:I hereby declare that on this the
tenth of August in the year fifteen
hundred and forty- eight of our
Lord in our fair city of Milan, the
visitor Niccolò Fontana Tartaglia
of Brescia challenges Girolamo
Cardano, represented here by
Ludovico Ferrari Esquire, to a duel
of the mind. The dishonored shall
pay the winner two hundred scudi.All of the characters mentioned here
are real historical figures. Out of such
challenges grew formulaic approaches
to the solution of the cubic equation
you get if you move the pieces around
in Ording’s original theorem.
Such a formula—analogous to the fa-
mous quadratic formula—is among the
discoveries to be found in Ars Magna,
Girolamo Cardano’s sixteenth- century
magnum opus. Elsewhere Ording re-
produces a two- page spread from Ars
Magna (chapter 7, “Found”) with the
exact cubic relation that is the focus of
the book. Ording claims to have been
“stunned” to find it there, but one can’t
help but wonder if that isn’t where his
project started. (There are an infinite
number of cubic equations with real
solutions to investigate.) References
to Cardano are sprinkled throughout,
suggesting that he’s something of a
hero to Ording. Khayyam also receives
a handful of mentions, but generally—
as Ording points out in chapter 71,
“Blog”—early Eastern contributions
to mathematics continue to be less well
known to Westerners than their later
Western counterparts.The order of the variations in Ording’s
book confuses the timeline of discov-
ery, even if it might reflect the time-
lessness of mathematical truths. But
99 Variations on a Proof is not really
meant to be a history lesson, even if it
is by necessity partly that. One aim is to
display the variety of possible proofs,
the different ways in which we might
convince one another that there are
only two solutions, 1 and 4, to this lit-
tle problem. To this end, various proofs
exhibit particular styles of logical argu-
mentation, including at least two that
involve indirect argument.
Chapter 13, “Reductio ad Absur-
dum,” bears the title of a form of proof
in which one assumes the conclusion
to be false—that there is another solu-
tion different from 1 and 4—and with
that additional hypothesis, as Ording
explains, concludes that “some third
statement that was already known to
be true—an axiom or proven propo-
sition—is false,” in this case that 0 =1.
This is a roundabout way to show that
the stated conclusion must therefore
be true—a feat of logical magic called
a “proof by contradiction” that makes
use of the famous “law of the excluded
middle”: either there is a third solution
or there isn’t, and no other option (“theA ‘wordless’ visual proof of the solutions to the cubic equation x^3 –6x^2 +11x–6=2x–2;
from Philip Ording’s 99 Variations on a ProofPhilip Ording(^5) In case your algebraic skills need to
be dusted off: plugging in 4 for x in
x^3 –6x^2 +11x–6 gives 64 – 96 + 44 – 6, or
6; as does plugging in 4 for x in 2x–2
(which gives 8 – 2); and analogously
for plugging in 1 in both expressions
(1 – 6 + 1 1 – 6 = 0 and 2 – 2 = 0).
(^6) For those who read French, I must rec-
ommend Ludmila Duchêne and Agnès
Leblanc’s Rationnel mon Q: 65 exer-
ci ces de styles (Paris: Hermann, 2010),
which uses the fact (theorem) that the
square root of two is an irrational num-
ber as its core (it can’t be written as a
fraction, a fact that blew the minds of
the ancient Greeks). They don’t succeed
in producing ninety- nine variations, but
the sixty- five they do achieve are won-
derful. If you happily read mathematics,
I also recommend John McCleary’s Ex-
ercises in (Mathematical) Style: Stories
of Binomial Coefficients (Mathemat-
ical Association of America, 2017) for
ninety- nine often but not always very
technical takes inspired by the numbers
that some might know as the entries in
“Pascal’s triangle.”