The Winners Write the Textbook

 Guest Author: Dan Walls

It is said that the winners write history. While usually this is reserved for the perspectives in history textbooks and other writings, it also finds true in the evolution of mathematical history as well. Beg to differ? Ask Recorde. He has Leibniz and his winning calculus notation to thank.

Daniel Walls_925508_assignsubmission_file_Sh to ch to x DW

From “sh” to “chi” to X Images from

Find x. Well, we have found it several times as mathematicians, used it several times in problems, and assumed it as the universal unknown. The unknowns span from just a mathematics variable and into popular culture, spanning the X-files, the X-factor, and Project X. But where did we get x? In a TEDx (again, why x?) talk, Terry Moore presents a brief explanation of what he has found is the reason we use x, and the reason is perhaps more comical than expected.

Moore explained that he undertook learning Arabic to better understand the history of mathematics. He explains that the word “shalan” translates to “something,” as in something unknown or something arbitrary. He goes on to explain that when the Europeans, mainly the Spanish, came to translate the Arabic mathematical findings, they were presented with a problem: the Spanish did not have a sound for “sh.” Therefore, they picked a hard “ch”/”ck” sound, as in the Greek “Chi”, symbolized as “X”, which, when translated to Latin became “x.” Moore jokes at the end, “Why is it that X is the unknown? X is the unknown because you can’t say ‘sh’ in Spanish.”

Having watched this video, I was inspired to ask, “What other notation have we taken for granted?” I then went and thought about why we use symbols. While yes, mathematics could be done with entirely words and documented arguments, the use of symbols stems from our use of language. Terrence W. Deacon explains in her book “The Symbolic Species,” that we, as humans, have developed symbols that “don’t just represent things in the world, they also represent each other. Because symbols do not directly refer to things in the world, but indirectly refer to them by virtue of referring to other symbols, they are implicitly combinatorial entities whose referential powers are derived by virtue of occupying determinate positions in an organized system of other symbols” (99). The use of symbols (variables) to describe other symbols (words, or other variables) has become a part of our general nature.

Besides from x, a symbol perhaps most used by mathematicians is the equal sign, =. What does this mean and how did it evolve? Until about 400 years ago, there were numerous symbols that meant equal. At first, there was no sign for equal, but rather it expressed rhetorically with such words as aequales, faciunt, or gleich, taking form in a variety of languages. At one point, just the abbreviation aeq was used. As a variety of other mathematical notations had formed, so did a multitude of ways to write an equals sign. Buteo used [ to show the function of equality, and Diophantus used two parallel lines, ||, to show that two quantities were equal. Descartes suggested that Daniel Walls_925508_assignsubmission_file_Descartes Equal DW be used to signify equality, and, for a while, he had begun to gain some popularity for this notation—as well as developing a widely, used coordinate system. (Cajori 297-301)

Up to the 17th century, the = symbol had a plethora of meanings, including parallel lines, difference, or even plus or minus. In Berlinghoff and Gouvea’s Math through the Ages, it is noted that the symbol is suggested by Recorde in the 15th century, but not adopted for a couple hundred years later until Leibniz preferred = to other symbols in his calculus notation, which had proven to be more successful than Newton’s. Just think, if we had used Newton’s calculus notation, we would still be using Daniel Walls_925508_assignsubmission_file_Descartes Equal DW instead of =. (100)

So Leibniz wins! And therefore so does Recorde and the traditionally accepted sign, =, is now used in all of the textbooks.

Berlinghoff, William P.Gouvêa, Fernando Q.Math Through The Ages: A Gentle History For Teachers And Others. Farmington, ME : Oxton House Publishers ; 2004. Print.
Cajori, Florian. A History Of Mathematical Notations. New York : Dover Publications, 1993. Print.
Deacon, Terrence William. The Symbolic Species: The Co-evolution Of Language And The Brain. New York : W.W. Norton, 1997. Print.
Moore, Terry. “Why is ‘x’ the unknown?” TED. Feb 2012. Accessed 28 March 2016.

About Matthew Simonson

I am a second-year Network Science doctoral student at Northeastern University in Boston. I model homophily and time-varying dynamics on social networks.
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