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Reading about the history of mathematical typesetting before TeX, I found an interesting article from Ivan Niven titled: "A SIMPLE PROOF THAT π IS IRRATIONAL" which is dated back to 1946 when there was no TeX (and probably no computer at all). The typesetting was great, and I was wondering if this is an originally typeset paper from 1946, or a new remake?

I have created a remake with plain TeX and also LaTeX, and the similarities between the plain TeX output and the original is astonishing! I am eager to know more about the similarities between the fonts and typesetting of the original document, and the output of plain TeX remake of this article. Compare these two for yourself:


A 1946 article from Ivan Niven

Fig. 3: Original document


Remaking the original article with Plain TeX

Fig. 4: Remade with plain TeX


If this is from the old days, how they could achieve such a nice output?

The article is published in:

Bulletin of the American Mathematical Society (Bull. Amer. Math. Soc.), Volume 53, Number 6 (1947), page 509, and it was first became available in Project Euclid in July 4th, 2007. In the dedicated page for BAMS on Project Euclid we read:

The digitization and unrestricted availablity of the backfile of the Bulletin of the American Mathematical Society (1891-1991) is made possible with the generous support of the Gordon and Betty Moore Foundation, the Mathematical Sciences Research Institute, and the American Mathematical Society.

Compare the whole page of two documents. They are nearly identical! Even the hyphenation is mostly the same. They have slight differences in math spacing:


This is the code I used to remake the paper:

\magnification=\magstep1
\baselineskip=12pt
\hsize=5.0truein
\vsize=8.7truein
\font\footsc=cmcsc10 at 10truept
\font\footbf=cmbx10 at 10truept
\font\footrm=cmr10 at 10truept
\font\bigrm=cmr12 at 14pt
\font\smallbf=cmbx10 at 8truept
\parindent=0.15in
\pageno=509    

\centerline{\bigrm\bf A SIMPLE PROOF THAT $\pi$ IS IRRATIONAL}
\smallskip\smallskip

\centerline{\smallbf IVAN NIVEN}

\smallskip\smallskip

Let $\pi = a/b$, the quotient of positive integers. We define the
polynomials
$$\displayindent=0.3in\displaywidth=1.3in f(x)={x^n(a-bx)^n \over n!},$$
$$\displayindent=0.3in\displaywidth=3.3in F(x) = f(x) - f^{(2)}(x)+f^{(4)}(x)-\ldots+(-1)^nf^{(2n)}(x),$$
the positive integer $n$ being specified later. Since $n!f(x)$ has integral
coefficients and terms in $x$ of degree not less than $n$, $f(x)$ and its
derivatives $f^{(j)}(x)$ have integral values for $x=0$; also for $x=\pi=a/b$,
since $f(x)=f(a/b-x)$. By elementary calculus we have
$$\displayindent=0.01in\displaywidth=4.0in{d \over dx}\{F'(x) \sin x - F(x) \cos x\} = F''(x) \sin x + F(x) \sin x = f(x) \sin x$$
\noindent and
$$\int^\pi_0 f(x) \sin xdx = [F'(x) \sin x - F(x) \cos x]^\pi_0 = F(\pi) + F(0).\leqno(1)$$
Now $F(\pi)+F(0)$ is an {\it integer}, since $f^{(j)}(\pi)$ and $f^{(j)}(0)$ are integers. But
for $0<x<\pi$,
$$0 < f(x) \sin x < {\pi^n a^n \over n!},$$
so that the integral in (1) is {\it positive}, {\it but arbitrarily small} for $n$
sufficiently large. Thus (1) is false, and so is our assumption that $\pi$ is
rational.
\smallskip
{\footsc Purdue University}

\kern +10pt
\hrule width 1.0in
\kern +10pt

{\footrm Received by the editors November 26, 1946, and, in revised form, December 20, 1946.}

\bye
  • 5
    Monotype machine? – egreg Jul 12 '15 at 22:17
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    If I highlight the contents of the pdf file and copy-and-paste it to a plain-text file, it's evident that the original wasn't just scanned but fully OCR'd. It's also evident that the file is based on a scan rather than re-typeset via some variant of TeX. For instance, the greek character "\pi is rendered in plain-text in several different ways, e.g, TT and 7T. In addition, other glyphs set in math italics also don't show up correctly in the plain-text file. This wouldn't happen if the pdf file were based on the output of a tex file, as opposed to an OCR'd version of a scan. – Mico Jul 12 '15 at 22:53
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    see this paper of Knuths where he explains that the design of cm fontsand tex came about partly to get back to that era of AMS publication quality math.lsa.umich.edu/~millerpd/docs/501_Winter13/Knuth79.pdf – David Carlisle Jul 12 '15 at 22:59
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    TeXbook preface: By preparing a manuscript in TeX format, you will be telling a computer exactly how the manuscript is to be transformed into pages whose typographic quality is comparable to that of the world’s finest printers;... – touhami Jul 13 '15 at 0:08
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    in 1947, the ams bulletin was typeset by the george banta printing company, menasha, wisconsin. they were one of a handful of compositors specializing in technical composition of very high quality. they would have used monotype machines, that being the only system in existence at that time that was capable of such work with relatively little additional handwork. – barbara beeton Jul 13 '15 at 15:47
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in 1947, the ams bulletin was typeset by the george banta printing company, menasha, wisconsin. they were one of a handful of compositors specializing in technical composition of very high quality. they would have used monotype machines, that being the only system in existence at that time that was capable of such work with relatively little additional handwork.

the cited article, available from project euclid, is a scan from the original. although the font is very similar to computer modern (in fact, it may very well be the font on which computer modern is based -- monotype modern 8a), there are characteristics that will positively identify it as not tex output.

compare this integral from the scan

integral from monotype page

with this one produced by tex

integral from tex output

both the shape and the spacing are different.

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