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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?

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I am not completely convinced on the Greek connection. The Vatican manuscript (see image 1 at ibiblio.org/expo/vatican.exhibit/exhibit/d-mathematics/… ) has text and diagram labels seemingly both in minuscule, while the copy at Bodlein ( bodley.ox.ac.uk/imacat/img0023.jpg ) seems to have running text in script and diagram labels in majuscule. –  Willie Wong Jun 19 '10 at 19:13
@W.Wong: I admit I am uncertain (I am unschooled in this area); and those images (thanks for the links!) are not easy to interpret. A conflating factor is that, if the scribe who copied the manuscript was converting a purely majuscule document to one also employing minuscule, he might have substituted for diagram labels as well. Regardless, we know the originals were solely majuscule (because minuscule didn't exist), and later authors (Galileo, Descartes) consistently employed uppercase Latin letters in their diagrams. –  Joseph O'Rourke Jun 19 '10 at 20:38
I have this impression, I think from Descartes, that the Greeks published very few diagrams. –  Theo Johnson-Freyd Jun 19 '10 at 20:41
Thinking about where I choose lower- vs uppercase letters, I seem to use lowercase for small atomic things (points, functions, group elements) and uppercase for big space-like things (manifolds, spaces of functions, groups, categories). This seems natural as soon as you realize that mathematical objects are Things that can be Considered, which I suppose is rare at the high school level. –  Matt Noonan Jun 19 '10 at 22:28
@J. O'Rourke: precisely, my point was that I am not sure whether the "tradition" can be continuously traced back to the Greeks in the time of Euclid. Considering that very few manuscripts of Geometry survived into the medieval times, I find TonyK's interpretation below more likely: that someone somewhere introduced the notation (perhaps for legibility) that majuscule is used for labels. And his manuscript happened to be the lucky one that survived. If I come across any historians, I'll bring up the question. May make a nice PhD problem. –  Willie Wong Jun 20 '10 at 12:24
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6 Answers

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.

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@Victor: I must admit, you've shaken my confidence in my blithe assumption that lowercase rules now at advanced levels. It would take a careful inventory to assess accurately. Both Tristan Needham's Visual Complex Analysis and Marcel Berger's A Panoramic View of Riemannian Geometry consistently use lowercase for point labels in diagrams. A.D. Alexandrov's Convex Polyhedra solely uses uppercase (ca. 1950). I love your French origins theory! :-) –  Joseph O'Rourke Jun 20 '10 at 23:59
This is because in complex analysis and differential geometry a point is also a variable (in $\mathbb C$ or $\mathbb R^n$, respectively), and such variables are by convention lowercase. –  rem Apr 6 '13 at 10:46
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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.

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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.

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Here is one of the oldest fragments, from about 100 BC: math.ubc.ca/~cass/Euclid/papyrus/papyrus.html#lookingat –  Rhett Butler Apr 5 '13 at 20:14
Your post suggests you have experience rading math in Spanish. Have you considered contributing to the Spanish language mathoverflow/MSE proposal? area51.stackexchange.com/proposals/64529/… –  Brian Rushton Feb 1 at 3:07
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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

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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.

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Uh, I thought one of the points Joseph raised was that, at the time of Euclid, there were no such thing as minuscule script: when everything was written in majuscule, I doubt using them will make diagram labels stand out. Medieval scribes who copied manuscripts would likely have used the Greek uncial script, which is still "all-caps". Also, making the text easier to follow is probably one of the things furthest from the minds of those scribes: have you seen one of those illuminated Latin manuscripts? No punctuation or inter-word spacing, with liberal doses of abbreviations they had. –  Willie Wong Jun 19 '10 at 18:51
Now, your argument would be much more probable in the renaissance, except then the language of choice would most likely be Latin or French. –  Willie Wong Jun 19 '10 at 18:53
My point was that you don't have to go back to when lowercase didn't exist. You only have to go back to when books were hand-written, with no special typefaces to distinguish symbols from text. And this applies to Latin and French too. –  TonyK Jun 20 '10 at 0:01
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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.

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I have seen the same convention followed ie Vertices as capital Roman, sides as lower case Roman and angles as Greek letters. –  ARi Jul 3 '13 at 10:29
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