Why LaTeX uses upright uppercase Greek letters by default in math mode?

Question

Why upper-case Greek letters are typeset in upright font, i.e., \mathrm? While at the same time, some upper-case Greek letters those are also Latin Letters ,e.g., A for $\Alpha$, B for $\Beta$, are typeset in \mathnormal font, which leads to inconsistency? Can anyone give some explanations?

From the "Short math guide for LaTeX" by AMS, I've found:

For obscure historical reasons, the default font for lowercase Greek letters in math formulas is italic while the default font for capital Greek letters is upright/roman.

So what are the historical reasons?

Answer 1

Computer Modern's greek (bottom) font is (loosely) based on Monotype's 155M Greek font (top):

The Monotype 155M font itself is related to the Porson greek font, which was one of the most used typeface for greek in english speaking countries. One of the characteristic of this font is that it has upright capitals but slanted lowercase. This might seem surprising today, but the first italic fonts all had upright capitals (see for example the image of the Aldine Press article on wikipedia)

Notice that in France, the traditional font for greek was Didot's, which has both upright capitals and lowercase, which is why using upright lowercase greek was standard in France.

You can see versions of the Didot and Porson fonts on the Greek Font Society website.

Answer 2

The situation with uppercase Greek letters is in fact better than for lowercase Greek letters: the latter are provided only in the slanted shape. Upright uppercase Greek letters which look like their Latin counterpart can be typeset in Plain with {\rm A}, etc... and the other uppercase Greek letters are by default upright but can easily be typeset in slanted shape with {\mit\Gamma}, as the math italic font does contain all the Greek alphabet (well not all, monotonic only and some variant letters are missing; let's say: contains a sufficient cut of the Greek letters for scientific use) in slanted shape and the slot positions for the uppercase Greek letters is the same both in the roman and math italic Knuth fonts.

In LaTeX you can do \mathnormal{\Gamma} to get a slanted \Gamma.

Knuth says (TeXBook, page 434):

It’s conventional to use unslanted letters for uppercase Greek, and slanted letters for lowercase Greek;

But had he decided to also provide a font with upright lowercase Greek, then the 11 uppercase Greek letters not Latin like could have been moved a common font with lowercase and not strangely put in the first 11 slots of the OT1 encoding.

Ultimately I think the problem is to have decided that Latin letters in math were most of the time to be slanted. Another logic could have prevailed to provide, alongside the Computer Modern Math Italic font a Computer Modern Math Upright font, separate from the Computer Modern Roman font. Then we would have had from the start the choice to typeset math upright, or slanted, or a mixture.

In the decades following Knuth's initial software with its accompanying fonts, hundreds or even thousands of fonts have been proposed for text, and a handful (or only slightly more) for math. So, although the situation of text fonts was very active, with in particular the need of font encodings of the European languages, the situation with math fonts remained stagnant for decades, only to be changed in recent times by the emergence of the OpenMath.

Answer 3

As an extension of the reply provided by Philippe Goutet, let me point you to the Fourier font. Simply by calling

\usepackage[upright]{fourier}

as substitute to

\usepackage[sloped]{fourier}

you may activate the traditional French convention, i.e. both lower and uppercase Greek characters upright all across the document. Perhaps there are other fonts with a similar switch.