Uppercase Point Labels in High-School Diagrams: from Euclid?

Question

I wonder if the convention of labeling points in geometric diagrams with uppercase symbols ultimately derives from Greek mathematics, which was originally written in "majuscule" (uppercase) Greek script (in contrast to the "minuscule" script that was introduced much later (9th century?)). Certainly Euclid and Archimedes used only uppercase, and all of Descartes diagrams in La Geometrie (1637) follow the same convention.

It seems that middle- and high-school textbooks continue to use uppercase labels (is this only in the U.S.?), but college texts do not follow this as rigidly. This was brought home to me when I wrote a chapter for high-school teachers and the editors changed all my lowercase vertex labels to uppercase. I much prefer lowercase for point labels, although I do not quite know why I have this preference. (Maybe because uppercase seems like SHOUTING?) But when writing for an audience accustomed to a particular convention, it seems prudent to follow that convention.

My questions are: (1) Is the Greek majuscule script the origin of the uppercase diagram-labeling convention? (2) In so far as I am correct that the uppercase convention is followed up to high school but dissolves at more advanced levels, why does it persist to one level but dissolve beyond?

Answer 1

On the one hand:

1. Ratdolt's 1482 editio princeps (albeit of a Latin translation of an Arabic translation) labels points with lower case letters.

2. Herwegen's 1533 editio princeps of the original Greek text also labels points with lower case letters.

Moreover, for instance, as late as 1565, a corrected edition of Commandino's Latin translation of Archimedes still labels points with lower case letters. One might well argue that this continues the manuscript tradition, as exemplified by the Codex Vaticanus.

On the other hand, just a few years later:

1. Billingsley's 1570 English translation labels points rather with upper case letters.

2. Commandino's 1572 Latin translation labels points with upper case letters.

3. Clavius's 1574 Latin translation labels points with upper case letters.

So, writing admittedly as a complete layman with regards to the textual transmission of Greek Mathematics, I would suggest that the custom of labelling points with upper case letters might well be an innovation of mid-late 16th century publishing, and that it in any event demonstrably has nothing to do with the earliest printed editions and the manuscript traditions they would have derived from.

Answer 2

I don't want to speculate on the historical origins, but I am surprised by your point (2): as far as I am aware, the use of upper case latin characters for points is completely standard in geometry regardless of the sophistication level in English language books. I've just had a quick look at Coxeter's Introduction to Geometry and Coxeter and Greitzer, and that's certainly their convention. I am less certain of how international this convention is: for example, all Russian texts that I can remember follow it, but, for example, Marcel Berger's geometry doesn't: it may please you that his points are set in lower case; however, he denotes lines and, more generally, sets by $A,B, H,K$ (sorry for shouting). At the same time, at least the English edition of Michele Audin's Geometry uses upper case characters for points. This contradicts my informed guess that lower case use has French origins, unless the translation was adapted to conform to English language use. On the other hand, authors of college textbooks do not always follow the best practices.

Answer 3

From Wikipedia: "In classical Greek, as in classical Latin, only upper-case letters existed. The lower-case Greek letters were developed much later by medieval scribes to permit a faster, more convenient cursive writing style with the use of ink and quill."

On the other hand (Wikipedia again) "the oldest surviving Latin translation of the Elements is a 12th century work by Adelard, which translates to Latin from the Arabic."

In other words, there is no clear connection between Euclid using uppercase (the only script he knew), and us using it too (what symbols did the old arab scholars use?). There is no point either in looking at "old" books like Coxeter's. The convention is surely older!

I will hazard the guess that the convention is quite old, say 1700-1800, and that it started with some random edition of Euclid that became slightly more popular than others. Which one, I do not know, but it is improbable that it is the first English translation by Billingsley (very famous, with 3D pop-out models of solids, but English was not the most influential language at the time), or Oliver Byrne's [color coded edition](http://en.wikipedia.org/wiki/Oliver_Byrne_(mathematician)#Byrne.27s_Euclid) (which is beautiful, but became well known only recently, AND does not use labels :). See also this page...

BTW, in Spanish points are also represented with capital letters.

Answer 4

As mentioned by the OP in the comments, Reviel Netz is an academic who thinks about these kinds of questions. For example, he has a great chapter in "The Archimedes Codex" (by Netz and his coauthor William Noel) on the topic of how the Greeks labelled their figures. He explains how the philological method - comparing and contrasting different manuscripts to reconstruct a common origin manuscript - can be applied to the various figures in the extant codices of the Archimedean tradition. He applies the philological method to construct what the figures drawn on the beaches of Syracuse in the third century BCE might have looked like.

However, Netz also notes the innovations of the medieval scribes. He states (p. 111):

Medieval scribes, at this point, took some crucial steps in the invention of more effective typesetting interfaces. The use of several cases - an uppercase alongside a lower case - is of great value. It allows us to separate the letters referring to the diagram (which remains in upper case) from the rest (which is now in lower case).

Thus, Netz appears to argue that the real innovation is not so much writing in majuscule for the figures, but in writing a mix of majuscule and miniscule at the same time. As pointed out by others, this is a medieval construct.

(Netz states in passing that the ancient Chinese used different colors to identify the components of their figures.)

Answer 5

If you don't have italic or bold typefaces available -- for instance, if you are writing by hand, in Greek -- then you really have to use uppercase for diagram labels, to make the text easier to follow.

Answer 6

This only answers the minor question (put in parentheses) assuming a very late origin of lower case letters: According to David Side: "The Library of the Villa Dei Papiri at Herculaneum" Roman lower case letters have been found in manuscripts written before 79 AD.

And an expert historian even has evidence for Greek lower case letters some hundred years back: "Lowercase letters, also called ‘small', 'minuscule letters' in contrast to the majuscule, or uppercase, capital letters, which were the earliest form of writing in ancient alphabets, date back to the 3rd. century BC in Greece where in fact the lowercase letters were used to write faster on papyrus and parchment." http://en.allexperts.com/q/Ancient-Classical-History-2715/lower-case-letters.htm

Answer 7

The convention of labeling points in geometric diagrams with uppercase symbols derives (at least also) from Greek mathematics. I cannot judge about the question whether this is the "ultimate" reason. But if we look into Euclid's elements, we find many diagrams where all points and also all lengths (magnitudes) are labeled with Greek uppercase symbols. It is said that Euler introduced the convention to label points of triangles with uppecase Latin letters and sides with lower case Latin letters and angles with lower case Greek letters. Probably the middle- and high-school textbooks, as many other elements of mathematics, adhere to the custom invented by Euler. But when we have a closer look into his books, first of all his INTRODUCTIO IN ANALYSIN INFINITORUM, we find that he used also lower case symbols to label points. So we should drop the word "ultimately" from the original question.