Written by Sarah de Launey, edited by Nicola Simcock
“The nature of the communication will evince the necessity of secrecy; and we have promised Messrs. X. and Y. that their names shall, in no event, be made public”A secret message shrouded in mystery, this communication from Charles Pinckney in 1797 is the first recorded use of X outside mathematics.
From science to cinema, all over the world, X is used to represent the unknown. So ubiquitous, it’s easy to imagine that this has always been the case. At May’s SciBar Dr David Stewart, researcher in Pure Mathematics at Newcastle University, told the story of mathematical notation and explains how enigmatic, unassuming X emerged as champion of the unknown.
Like gin and tonic, mathematics and notation are a natural duo. However, just like gin and tonic, this is a relatively recent pairing. Early mathematics, recorded on Babylonian tablets dating back to 1700BC, describe calculations as a narrative without any specific mathematical symbols. These texts are conversational and at times dramatic, one speaker giving instructions to a second who then recounts the result to the reader. While a little time consuming, this engaging method was ideal for solving geometric problems or identifying patterns, such as predicting eclipses.
Drawing from Babylonian knowledge and traditions, the ancient Greeks expanded the field of mathematics and created many of the foundational texts we use today. Among the great scholars of this period, Pythagoras stands out as the epitome of a mad genius. He developed the Pythagorean theorem but refused to urinate towards the Sun; popularised the idea of mathematical proof but refused to pass a donkey in the street. Unlike the Babylonians and previous mathematicians, Pythagoras didn’t see mathematics as a simple means to solving practical problems, but as a discipline in its own right. He dedicated much of his time to exploring relationships between numbers ─ such as perfect and prime numbers ─ laying the foundations for modern number theory.
While the ancient Greeks may not have introduced our X, or in fact any consistent form of notation, they did develop many mathematical principles still used today. Euclid’s Elements of Geometry (the most successful textbook of all time) published over 2000 years ago, is still used in studies of geometry and number theory today.
At the turn of the common Era, it was not just the Greeks who had advanced their understanding of mathematics. Indian mathematicians and scholars, who were already well versed in Euclidean equations, made the most significant contribution to the discipline. Ever.
They invented nothing.
The discovery of zero was an incredible breakthrough. Its existence meant that calculations no longer needed lengthy explanation through text and paved the way for modern algebra, algorithms, and calculus. Almost 800 years after its first recorded use, the Persian mathematician Al-Khwarizmi brought the concept of zero, as well as minus numbers and the Arabic numerals, to Europe. With these new tools for abstraction, mathematics could finally be used to explore the unknown.
But what do we mean by the unknown? Donald Rumsfeld explains:
“As we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns—the ones we don’t know we don’t know.”
Rather than using common speech to describe these concepts, like the Babylonians, mathematics turned to symbolic notation as an efficient way of working with different types of unknowns.
With the arrival of the printing press in the 15th century there was an explosion of mathematical publications, each with their own system of notation. Luca Pacioli’s Summa de arithmetica, geometria, de proportioni et de proportionalita, for example, used Latin letters in alphabetical order as a way of identifying variables. Sadly, these notational proofs were considered too crude to accompany the main text and were relegated to the annex.
With no dominant form that stood out from the rest, the many notational systems coexisted to the end of the 16th century. In an attempt to expand and standardise notation, French mathematician Francois Viète tried to introduce a new system by which vowels would represent unknown unknowns and consonants as known unknowns. However, the absence of any regulatory body for mathematical notation meant that the preferential adoption of one system depended on the popularity of its author. Despite its importance for modern algebra, Viète’s book Opera Mathematica was not popular enough to catalyse standardisation.
Not until 1637 was X as we know it today first published, in René Descartes’ hugely popular La Géométrie (extract above). While according to the author the book “picked up where Viète left off”, Descartes opted for a different system of notation: X, Y and Z to represent unknown unknowns and A, B and C for known unknowns. But the question remains, why X? Could it have Christian connotations? As Descartes was famously devout. Could it be its form? Two arrow heads pointing to the answer. Or is there something intrinsically mysterious about X? Two infinitely long lines, meeting at one point, above the hidden treasure.
As Dr Stewart reminded us ─ this remains unknown.
A History of Mathematical Notations, Florian Cajori
Unknown Quantity: A Real and Imaginary History of Algebra Paperback – John Derbyshire
A History of Algebra From al-Khwārizmī to Emmy Noether – Waerden, Bartel L. van der