Handbook of Typography for the Mathematical Sciences

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Lots of nicely or at least acceptably typeset books and papers have used the italic forms. Cohen-Tannoudji's Mecanique Quantique Hermann uses the upright forms. The Greek pi used for the constant seems to get no special treatment: usually, in math, lowercase Greek letters are italic and uppercase ones are upright. Concerning the choice of italic versus upright in subscripts and superscripts: the "rules" for those are simply the usual ones.

On the other hand, when a letter, word or word fragment in a subscript represents some "label" instead of a variable, it is convenient to write it in upright roman: for instance, T crit or T c for a critical temperature, x min , x max , P Jones. As John Walker was kind enough to mention my kinetics book thank you! I cannot resist asking which of the various John Walkers he is: the famous one who won the Nobel Prize for Chemistry in and whom I used to know slightly when we were students in the s, the parasitologist at Bristol, a biochemist at Canterbury, New Zealand, or another John Walker altogether?

Athel Cornish-Bowden asks, "which of the various John Walkers he is". I was thinking the same question. Based on the texts cited in John Walker's messages, I expect that one of Athel's guesses is correct.

Another candidate is the John Walker who founded Autodesk and co-wrote AutoCAD, a program that definitely influenced the graphic display of information. Will the real John Walker please stand up? I hope this is not too far off topic. It seems that there can be a certain anonymity in being a highly accomplished John Walker. In answer to your questions: I would like to win a Nobel Prize unlikely , I enjoyed Athel Cornish-Bowdens textbook highly recommended for anyone learning biochemical kinetics , I liked the Incredibles, I have used biochemistry but I washed my hands afterwards, I have heard of parasitology no first hand experience fortunately , and I have struggled with AutoCad for several hours to produce what I thought was a nice clear drawing only to find that my five year old son could outdo me in a few minutes with paper and crayons.

Adjustments, in response to your advice, have been posted in the proof above. Thank you so much everyone for your thoughtful and helpful contributions, as well as for your patience with my stumblings around on this. It would be useful now if an authoritative mathematical expert could provide our readers with a few fundamental links on mathematical typography links to a few major style sheets, for example that show the best practices note the plural of experts; this would help our readers interested in the general topic.

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With best regards to all the contributors to this thread, ET -- Edward Tufte. To be certain, I'm no authoritative expert on mathematical notation, but the subject interests me as I do have occasion to typeset math notation and find a goodly measure of aesthetic pleasure in the look of the equations as well. I've followed this thread with keen interest; it's one of the best on the board.

I did a fair amount of looking around for information on this subject, especially on conventions for setting "e" natural log , and found that most sources have set "e" in italic type, or at least obliquely. However, the U. To my eye, this makes good sense, as the roman e in a good typeface has the dignity befitting one of the fundamental mathematical constants; the italic e just looks more, well, variable.

I'm still looking for who decided to set variables in italic. The answer may be in the one source that comes up most often and which I haven't looked at, Florian Cajori's "A History of Mathematical Notation" of This might not show up well in my HTML, but in print the difference really catches the eye, and there appear to be three different spacings: full space, no space, thin space the thin space only appears because the superscript characters are not kerned to the punctuation.

I ask because I'm a bit suspicious of whether the NIST writer elected to set e rather than e simply because e had already been committed to the elementry charge. I found very few references to any of these in chemical or biological science books. Looking beyond the sciences, Burchfield's New Fowler's Modern English Usage says nothing about the base of natural logarithms but does discuss the mute e.

I had come across this discussion by accident, when doing a web search for information about the typesetting of mathematics. When I saw the question come up of whether or not e should be an italic in equations, I did not immediately remember the answer. Using the Roman form of the e did seem like a good idea for reducing confusion. Taking a look at my copies of Whittaker and Watson, Abramowitz and Stegun, and another mathematical book taken more or less at random, I found that all three used the italic form of the e to represent the base of the natural logarithms.

However, these days, I think that a bold soul who would wish to experiment with the other usage would escape censure. I do recall that occasionally, operators like cos or sin in equations will appear in boldface as well; this is rare, but if this is done, should e be in boldface as well? My first inclination would be to shudder in horror and say "No", but, on the other hand, if one were using a variable named e in the same equation as e standing for 2. And, thus, in copy handed to a typesetter, particularly as the italics in equations are not normally indicated each one by underlining, it had not been considered worth-while to indicate e as not being italicized as a standard practise.

As to the question of whether or not Bembo is a suitable medium for mathematical typography, I might note that I have a foreign-language book on some aspect of mathematics not immediately accessible at the moment; I think it may have been in Hungarian which is set in a font resembling Linotype's Antique No. At one time, the question "can you typeset mathematics in Today, thanks to the computer, one has more-or-less complete freedom in this regard.

I am quite sure that I have seen the older style of number used within equations in older works, but I do agree that this would strike the modern reader as bizarre and as an affectation. There is also the danger that the numeral 1 could be confused with a small capital I. None the less, I applaud your efforts to make use of other typefaces, generally recognized as beautiful, in the typesetting of mathematics, than those most commonly used.

John Savard -- John Savard email. Italicizing of "e"; and what space of models to use Should e be italic or roman when it denotes the base of natural logarithms? I say it should be italic. The argument given above is that it represents a function, namely the exponential function. It doesn't. The exponential function has a name: "exp".

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When you use that name, you don't use italics, just as you don't for "sin" or "cos". But in e x , it's just the name of a particular number, different from x only in being constant rather than variable. There is no mathematical tradition of setting constants in roman rather than italic type.

There is one of using Greek letters; but they are almost always italic Greek letters, in so far as that terminology makes any sense. I'm at work at present, and almost all my real mathematics books are at home. But I've just checked the following, and they all use an italic e for the base of natural logarithms:. That's quite a range of books and publishers, and so far as one can tell from superficial appearance they aren't all using the same typesetting software or the same style guide.

I think it's reasonable to conclude that italicizing e is the usual choice. Another stylistic quibble, though here I'm on more controversial territory: You will very seldom see a real mathematician write "log" with a subscripted "e" italic or otherwise. Usual practice is simply to write "log" for natural logarithms. In school textbooks and other contexts long infected by base logarithms, "ln" is lamentably common. Subscripted "e" is very rare.

Introduction

And a mathematical point, for a change! ET protests at the use of quintic polynomials to describe the variation of price according to age, citing the enormous size of 80 5 to point out the absurdity. But the enormous size of 80 5 is neither here nor there. Taking logarithms of the artist's age seems like a peculiar thing to do. Sure, it means that the gradient of your curve is an elasticity. But the only reason why economists are interested in elasticities is that changes in many quantities notably prices are best considered as fractions of the quantities themselves.

But this doesn't seem to be true of ages; at least, I see no reason why it should be. Galenson's polynomial fitting is certainly pretty arbitrary. It may be a daft thing to do; I haven't read his book, and the "dequantification" ET mentions is good reason to be suspicious. But it's not obviously insane; if you have a bunch of data points which you think may be error-contaminated samples from an unknown smooth curve and no convincing underlying theory yet , fitting a lowish-degree polynomial is a decent enough thing to do. A 5th-order polynomial doesn't seem like an unreasonable choice here.

What I have grave doubts about is the assumption that the curve should be smooth, which ET makes too when suggesting fitting a linear model to the log prices. Artists may go through fairly clearly delineated "periods"; one of the clearest and most famous examples is Picasso, represented by the right-hand graph above. If there's a meaningful price-versus-time curve, why should it be smooth or even continuous? Lest I be misunderstood, let me add that I think it's preposterous for such a book as I understand Galenson's to be to be so light on quantitative price data.

Gareth McCaughan's comments are very thoughtful and interesting. Yes, there are 6 terms.

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I forgot the intercept term with t to the zero power. Its meaning is mystical, however, providing an estimate of the ln price of the artist's non-existent paintings at the artist's instant of birth. Depending upon what space we're in, the especially mystical ln 0 may arise by forcing 0,0. My concern with quintic age are problems with units of measurement and interpretation.

Other than expressing a bend, the regression coefficients have no interpretative substantive meaning. What do we learn by calculating that "a million year change in the quintic age of an artist corresponds to beta sub 5 unit change in ln price of the artist's paintings"?! Better that summaries, such as regression coefficients, have quantitative interpretations. With regard to the lower-power terms, sometimes I almost understand what a square year is, but certainly not a cubic year, or quartic year.

Thompson Andrei D. Polyanin Andrei D. Polyanin Frank E. Jones Christopher M. Aslam Chaudhry Ronald J. Tallarida Lance D. Chambers Lance D. Chambers James S.

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