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While contemplating the existence of math, I came across an interesting problem: Why are variables often lowercased?

There may not be a reason, but if there is, I would like to find out. Maybe it's because they sometimes take up less space or some other slightly useful answer, or maybe it has to do with something else. Unfortunately, I'm not exactly sure anyone can answer this definitively, as Viete is dead.

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    $\begingroup$ We often use lowercase vs uppercase when dealing with the whole vs parts. For example a set is usually denoted by an uppercase letter while elements of that set are denoted by lowercase letters. $\endgroup$ Commented Oct 6, 2011 at 17:20
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    $\begingroup$ @Unreasonable Sin: Except for when you have proper classes involved (e.g. NBG set theory), where sets are lowercase and classes are uppercase. :-) $\endgroup$ Commented Oct 6, 2011 at 17:24
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    $\begingroup$ @FUZxxl this is true especially for the "field" $\mathbb{N}$ ^^ $\endgroup$ Commented Oct 6, 2011 at 18:03
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    $\begingroup$ Viete started it all, and used upper case vowels. A few decades later, Descartes used lower case letters late in the alphabet. The Cartesian conventions won. $\endgroup$ Commented Oct 6, 2011 at 18:15
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    $\begingroup$ Mary Somerville was 19th century, a few centuries too late, at least if you are thinking of mathematics. It was Viete (with an accent grave on the first e), latinized name Vieta, though there can be a (very thin) case made for a few precursors. Viete used upper case vowels for variables, upper case consonants for unspecified parameters. This was just before 1600. $\endgroup$ Commented Oct 7, 2011 at 0:25

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The obvious place to look up something like this is Cajori's book A History of Mathematical Notations (in 2 volumes), volume 1 of which is freely available on the internet (URL below). Both volumes are currently available as Dover reprints.

http://www.archive.org/details/historyofmathema031756mbp

In searching the on-line .pdf file for "small letter" and "capital letter" (these two phrases worked best for me), I found that styles differed in the generation or two before Newton and Leibniz, with uppercase letters used by Francis Vieta (1590s) and lowercase letters used by Thomas Harriot (1631) and Descartes (1637). There may be a convention that began with Leibniz in which numerical variables are lowercase and geometrical variables (e.g. for points) are uppercase, google "Leibnizian procedure", but I'm not very sure about this. In any event, even if Leibniz began such a convention, I'm sure there still would have been a lot of individual variation in the late 1600s to late 1700s. However, regardless of who is responsible for these algebraic and geometric notation conventions (lowercase for algebra, uppercase for geometry), I believe they were widely used in the literature beginning at least by the late 1700s (this being based on my own observations of many hundreds of 19th century journal volumes and books I've looked through in the past 30+ years).

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Pierre de Fermat was still using uppercase letters, mostly vowels like $A$ and $E$, for variables. For example his method of adequality exploits an infinitesimal quantity denoted $E$ that has been much written about; see e.g., this article.

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Small letters are easy and fast while writing compare to the uppercase letters.This makes the mathematics faster.

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