I am very glad to see this post, as I have wondered about this myself. I think I had a relatively early dawning of "typographical awareness" and I have yet to encounter a math or science text that uses text figures (proportional oldstyle, or even oldstyle at all). I have occasionally typeset texts with small amounts of math and used them, but I'm not sure what I would do for an extensively mathematic text. As much as I would like to use oldstyle figures, I feel that only titling figures are accepted in these technical areas, in a way similar to how only one form of the theta is recognized as a mathematical symbol, or only one form of the lowercase g is used in IPA transcriptions. However, I don't know for sure why this is, and indeed, I'm very curious. My best guess is that mathematicians, scientists, etc. may see it as a legibility problem for certain characters, but I think overall equations and charts look both more attractive and more easily scannable/readable when set in OSF. Anyway, it's a very interesting topic. I look forward to more responses.

Josh

I'm involved in scientific research as my "day job," and have been intrigued by this question before, as well.

I once read a book by Werner Heisenberg (reprinted by Dover: http://www.amazon.com/Physical-Principles-Quantum-Theory/dp/0486601137/r...) that used text figures in the equations. It was mostly legible, with the exception of the 0 and the 1. Specifically, the 0 looked too much like a lc /o/ (which is occasionally used to, for example, indicate an element of solid angle -- anyone who tells you /o/ is never used as a variable or operator is lying) and the 1 of the typeface used looked too much like some variant of a small caps /I/, which would make someone question if that is how they choose to indicate the imaginary unit at first glance -- not a good thing to be confused about!

My personal thoughts are to avoid text figures in equations. First, most of the target audience for such material is not used to looking at text figures in type. As a result, it distracts from the content when the figures are different than what the reader looks at day in and day out. Second, some text figures suffer from the ambiguities noted with the Heisenberg book above, though that can be fixed through appropriate selection of the typeface. Finally, once one delves into science mathematics beyond, say, the high school level there are few numbers in most equations -- equations at this level are mostly combinations of variables and operators, both upper and lowercase (and often with uc and lc greek and sans/serif/blackletter in one equation!). When a number appears, there is no reason why it should blend in with the lowercase. If anything, it should stand out as a pure number in a sea of variables, and make the reader wonder what quirk of calculus made it appear there.

Refer to the attached images for a comparison (the moment of rotational inertia of a solid, uniform sphere of mass m and radius R). I personally find the version with lining figures much easier on the eyes -- especially combinations like "4π," and the fractions.

I think the biggest aesthetic problem with text figures in this situation is their varied vertical alignment: for instance, see the 4/3 fraction in the denominator of the fifth large fraction. The fraction bar appears to be closer to the 4 than the 3 — which, of course, it is. If the numerator and denominator of the small fraction had several more digits, the effect might disappear, but it's very noticeable with only the two digits. Of course, a fraction like f/e, for instance, gives you the same problem. I recall a math text that adjusted the numerator and denominator baselines to avoid this issue:

An interesting solution, but it made equations with several fractions look inconsistent and haphazard.

In the useful examples provided by SciTechEngMath, the oldstyle figures don't help the legibility of fractions and superscripts, certainly, but I definitely like the look of them in a line of text. For instance, I actually think 4π looks very nice with OSF as does any nonfractional term with a combination of digits and variables. OSF next to italic lc letters or lc Greek letters is, I think, very attractive. I recall that Bringhurst's chapter on page proportions had a bit of math (also, like the examples above, in Minion) and used OSF — but this was all inline, I think, and didn't include anything as complicated as the above. I thought of it, though, because of the frequent use of π and φ (he uses the letter, I think, not the math symbol ϕ), which look quite nice with OSF.

Josh

IMHO, the lining option is far better than the oldstyle for scientific data. BTW, I think even in common text OsF is not always the better choice, although there is a solid general opinion favouring them.

These figures are more distinctive and elegant, but become distracting if the text has lots of numbers. The argument about x-height of OsF to match lc is valid. But when you have a text/language with many ascenders, numbers with descenders may become a bit strange or appear misaligned. This may not be an issue for English, where |y| is very common and increases the descender frequence. But for Italian or Portuguese, languages with much less descender usage, this may cause OsF 3, 4, 5, 7 and 9 not to fit so well to the text.

Besides this, sequences of figures with ascenders (like 886), in a passage without other numbers, may give the idea of a lining standard (and even to be set in bigger type size) for the average reader.

Just for literature I have no doubt to take OsF as a primary choice.

Between disciplines from which I handled some amout of material, I never saw OsF figures used for Mathematics or Biology. But I saw some books about Philosophy, History or Law using OsF. For these, OsF just work if few numbers were present. About chess notation, I tried to do this when editing a small chess magazine, but had to give it up. There is so much amount of lining characters in chess notation and this makes OsF appear very small and shifted towards bottom.

An interesting possibility is to have figures in an intermediary size. Let's say, a lining which matches uc height, an oldstyle which matches lc x-height and a third set using lining numbers with height mid-sized between x- and X-height.

"...and a third set using lining numbers with height mid-sized between x- and X-height."

As in the typeface Bell, for instance. I'm sure there are other examples.

I'm adopting this in my font project, but don't remember other examples. In some fonts with larger and complete small caps, to use SC figures achieves partially this effect (as in Greta and Guardian). Tasman, still not released, uses old style with ¾ of the lining height, another possibility to explore.

Oldstyle figures are OK everywhere except in formulae, equations, and fractions.
They are even used by some for superior figures as reference numbers.
And some even use them in all-cap and all-small cap settings.
IMO, chess moves are fine with oldstyle figures, in fact they improve the appearance of text.
But only if you're playing with Staunton pieces, the Man Ray design requires modernist sans serif type with lining figures.

Everybody is quite used to OSF these days, due to Georgia's pervasiveness.
This issue has been discussed at Typophile with regard to legal documents.

Prior to 1800, all typography, including the first log tables, used oldstyle figures, because that was all there was.

Josh:

I see what you mean about changing the height of the baseline. That would be very effective when there is only one or at the most two fractions on a given line, but would turn quite hairy quite fast.

(Please forgive the following confusion of symbols -- the preview on my browser showed both forms of phi becoming one and the same, even though they show up correctly in the typing window!)

I am curious about your comments regarding the use of ϕ versus φ (two stroke versus single stroke lc "phi"). I have seen both used in various contexts in both papers and books. The one I see the most is the two stroke form (ϕ), which is almost invariably the one selected for the azimuthal coordinate of a spherical coordinate system and is also frequently used for the electric potential in certain literature. When I see the one stroke form (φ) it has usually been in the context of quantum mechanics to represent a wave function, and is sometimes mixed in expressions with the two stroke version when they wish to use spherical coordinates (i.e., the two stroke form represents the azimuthal coordinate while the one stroke represents the wave function, so you may have some perverse form like φ(ϕ)). I was unaware of there being much of a convention regarding this.

Could you please share what you know about the choice between the two forms?

Thanks!

There are two places in Unicode where the distinction occurs.

Basic Greek:
http://www.unicode.org/charts/PDF/U0370.pdf
03C6
03D5
This enables the distinction as either between an archaic "scripty" form (open top, one-stroke -- BTW you got that backward) and the more modern closed form, OR the "legacy" use of a few Greek letters (e.g. also theta) for scientific/math purposes.

Here are the strictly mathematical characters, which extend to whole alphabets:
http://www.unicode.org/charts/PDF/U1D400.pdf

AFAIK, there are more fonts which have the two "basic" encodings than those which are specialist with whole "math greek" alphabets.
For instance, I have included the basic variants in my Modern Suite fonts, which have no special math features, as primarily stylistic options -- although they may be used for math as the Unicode values are correct.

SciTechEngMath:

I'm afraid that, not being a mathematician or scientist, I can't comment on whether there's any technical distinction between the forms of phi, other than I have read in typography texts that φ is preferred in Greek text, while ϕ is preferred as a mathematic symbol. I have heard the same about the two forms of theta (ϑ in text vs θ as a math symbol). However, in texts that contain only small amounts of math, I have seen the lc Greek forms used, probably because they simply look nicer.

Josh

Thank you both for sharing, Nick and Josh!

Nick, in response to your comment regarding me getting the two backward: it appears as though several fonts on my system have the two reversed or, as is the case with how Typophile is presently displaying for me, identical. Inspection of the page element with the trusty UnicodeChecker.app shows that the page has character U+03C6 when I mention the single stroke form, and U+03D5 when I mention the two stroke form. Experimentation shows that Lucida Grande (among others) has the correct forms in the correct spots, whereas Adobe Garamond Premier Pro (among others, including all of the Japanese faces I have installed) seems to have the two forms reversed. Fascinating, but completely unrelated to "old style figures in mathish places." I apologize for so rudely sidetracking this thread.

Chess moves are fine with oldstyle figures [...] But only if you're playing with Staunton pieces.

As most games are played online these days, your criteria would make MICR a proper choice for these games. But a web-hinted one, of course. :-)

Everybody is quite used to OSF these days, due to Georgia's pervasiveness.

I should had pointed my observation is based on local usage. Brazil has a very poor typographic culture and things like small caps, OsF and ligatures are widely unknown. So, OsF are still seen as a stylish, fancy design instead of an usual option for numbers – even in Georgian times. Maybe time will improve this.

I do remember that in one British book - I think it was "Teach Yourself Calculus" - there were log tables in the back which were set using oldstyle figures rather than ranging figures.

So, while for reasons outlined above, oldstyle figures are generally inappropriate for complex equations (in addition, 0 and 1 could easily be confused with small capital O and I, again because of lack of context) in older works, where oldstyle figures were the norm, they would be used where feasible.

The Doctrine of Fluxions, by James Hodgson, and published in 1736, and available on Google Books, is set in Caslon, and although this textbook on calculus has equations in it, it uses oldstyle figures in them; back when Caslon was what people used when they set type, that's what they used for equations too, and they managed.

On the other hand, when it is merely tabular matter instead of equations, oldstyle figures are not a problem, which is why they've been used even in recent works. For an old example, Astronomia Accurata, or the Royal Astronomer and Navigator, by Robert Heath, from 1760, also on Google Books, might be commended.

As an example of the early use of ranging figures in mathematics, Mathematical Treatises, by the Reverend John West, dating from 1838 might be noted - it is set in Bell, apparently.

On the other hand, Cours d'Analyse de l'École Polytechnique, by Jean-Marie-Constant Duhamel, dating from 1841, although set in Bodoni, uses oldstyle figures throughout.

These are just a few random examples, of course.

My son Julian is working on a PhD in Geophysics and I just saw a proof from the prerelease version of his paper, soon to be published in "The Bulletin of the Seismological Society of America". I can't say that I fully grasp the meaning of the text but just looking at the type, I see no use of oldstyle figures but much use of italics. I'll have to ask him when all of these distinctions occur over the holidays.

It is the convention in mathematics that the variables used in formulas are printed in italics. It makes them stand out more from numerals, distinguishes them from the conventional characters used to stand for functions, and fits better with Greek lowercase letters also used for variables (from, of course, Porson Greek).

From here, "The Monotype 4-line System of Setting Mathematics" and "Three Typefaces for Mathematics" might be of interest.

And Donald Knuth's famous paper in which he introduced TEX to the world, also being available for free download, would be worth a look as an introduction to the subject.

The log tables my mother used at school in the 1930s had old style figures. Printed in 1920s.
I posted an image a few years back at Typophile, but can't find the thread.

Thank you for posting those links, quadibloc!

For those interested in some of the nitty-gritty of the rules that have been imposed upon mathematical typesetting, this is an interesting summary (with the later half giving technical details of possible TeX implementations):
http://www.tug.org/TUGboat/Articles/tb18-1/tb54becc.pdf

Tragically, the article is obsolete as ISO 31 has been superseded by ISO 80000 as of 2009. But, the guidelines given for typesetting formulas are still reasonable guidelines. I have been very pleased to see that more and more places are following the advice concerning the use of roman type for the imaginary unit, the differential and in non-variable subscripts. European publishing houses (Springer-Verlag and Cambridge University Press come to mind in my recent memory) seem to do a better job (on average) of sticking to these rules than their American counterparts, though. Like anything in the design of a text, these small details make a big difference in the speed of comprehension when encountering a new equation.