An Exquisite Mathematical Font

Question

EDIT: The extension to this question, Replicating an Exquisite Font, is a means to the same end with the construction of such a font being the primary goal.

I've recently been wondering: "where on Earth can I find that old-timey LaTeX font with the really nice integrals?" I've seen a paper or two on differential equations using it and plenty of old math text use it. For reference, here is an example image from A Source Book in Mathematics by Smith, David Eugene (1860-1944).

also from another book, The Evanston colloquium: lectures on mathematics delivered from Aug. 28 to Sept. 9, 1893 before members of the Congress of Mathematics held in connection with the World's Fair in Chicago at Northwestern University, Evanston, Ill.

which is a more compressed albeit exactly similar font.

If someone would tell me the name of this font, I would be greatly indebted to them; either this, or reference a modern version. Essentially, whatever LaTeX platform is consistent with this font is the platform I will use.

IMPORTANT EDIT

I am also happy to see any customized stylizations and creative solutions to mimic the font style(s) in the images.

Answer 1

That book was published by Dover Books which was a legendary English metal foundry and publishing house (today only a shadow of itself, so can be said of the US business side).

Sooo... Your chances of finding a digital version of such font are slim to none. The font used is a transitional typeface, with a heavy body to compensate for paper and ink quality. I guess the closest thing you could get available in the TeX Archive would be some of the free baskervilles and their matching math fonts, but they will be spindlier.

Answer 2

I propose two solutions after about two hours I have arrived at this conclusion: one more complicated and one very fast. For my memories the \pi greek character is similar to Mathematical Pi 1 which is not a free font. After there many problems....to build a book that has such symbols it would take a century! :-) You would need to define each character individually and as pointed out by @egreg there are symbolic aesthetic issues. The book is old and those fonts don't exactly exist but there are modern ones that are similar or nearly so.

Personally this output I not like very much but it is a possibility: I have tried after two hours differents combination and I have build this MWE:

I have used partially an old question of @egreg redefining the symbol sum and product. Changing sum and product symbols There is the presence of \usepackage[lite]{mtpro2} that you can download here.

\documentclass[12pt]{article}
\usepackage[lite]{mtpro2}
\usepackage{mathspec} 
\defaultfontfeatures{Mapping=tex-text}
\setmainfont{Old Standard}
\setmathsfont(Latin)[Uppercase=Italic,Lowercase=Italic,Scale=0.95]{Old Standard}
\setmathsfont(Greek)[Uppercase=Regular,Lowercase=Regular,Scale=0.85]{Baskervaldx}
\DeclareSymbolFont{mathptmxlargesymbols}{OMX}{ztmcm}{m}{n}
\DeclareMathSymbol{\upsumop}{\mathop}{mathptmxlargesymbols}{"50}
\DeclareMathSymbol{\upprodop}{\mathop}{mathptmxlargesymbols}{"51}
\begin{document}
Old text from an ancient book it is
[s=\frac{\sin n\pi}{\pi}\int_{0}^{x}\frac{\phi(a)da}{(x-a)^{1-n}}]
[ \upsumop_{i=1}^n f_i(x)\upprodop_{j=1}^n g_j(x), \sqrt{z^2+b^2}]
\end{document}

If you found the best character you can substitute the mathspec fonts into the dots.

\setmainfont{.....}
\setmathsfont(Latin)[Uppercase=Italic,Lowercase=Italic,Scale=0.95]{.....}
\setmathsfont(Greek)[Uppercase=Regular,Lowercase=Regular,Scale=0.85]{.....}

The 2nd MWE is this:

\documentclass[12pt]{article}
\usepackage{mathspec}
\defaultfontfeatures{Mapping=tex-text}
\setallmainfonts{Old Standard}
\DeclareSymbolFont{mathptmxlargesymbols}{OMX}{ztmcm}{m}{n}
\DeclareMathSymbol{\upsumop}{\mathop}{mathptmxlargesymbols}{"50}
\DeclareMathSymbol{\upprodop}{\mathop}{mathptmxlargesymbols}{"51}
\begin{document}
Old text from an ancient book it is
[s=\frac{\sin n\pi}{\pi}\int_{0}^{x}\frac{\phi(a)da}{(x-a)^{1-n}}]
[ \upsumop_{i=1}^n f_i(x)\upprodop_{j=1}^n g_j(x), \sqrt{z^2+b^2}]
\end{document}

that is more simple but you have not the same characters of the book.

I aesthetically prefer the last solution. Both the MWEs are compilable with the engine XeLaTeX.

Addendum: On the suggest of the user @palopezv I add a MWE with Times where it is possible to compile with the pdfLaTeX or XeLaTeX.

\documentclass[12pt]{article}
\usepackage{newtxtext}
\usepackage[lite]{mtpro2}
\usepackage{nicefrac}
\usepackage{parskip}
\DeclareSymbolFont{mathptmxlargesymbols}{OMX}{ztmcm}{m}{n}
\DeclareMathSymbol{\upsumop}{\mathop}{mathptmxlargesymbols}{"50}
\DeclareMathSymbol{\upprodop}{\mathop}{mathptmxlargesymbols}{"51}
\begin{document}
An introduction for an ancient textbook using the font Times from newtxtext:
[s=\frac{\sin n\pi}{\pi}\int_{0}^{x}\frac{\phi(a)da}{(x-a)^{1-n}}]
[ \upsumop_{i=1}^n f_i(x)\upprodop_{j=1}^n g_j(x), \sqrt{z^2+b^2}]
After we can have using the nicefrac command:
[s=\frac{\upsumop_{i=1}^n a_i}{\upprodop_{j=1}^n b_j}\left[\frac{\Gamma(\mu+1)}{\Gamma(\nicefrac{1}{2})}\right]]
\end{document}

Answer 3

Here is a possible approach without the need of creating your own font:

1. Draw this integral symbol with Inkscape or Adobe Illustrator, extract its path in SVG format
2. In LaTeX, use the TikZ package to draw the same path

In the following example, The \MyIntTop command corresponds to the top half of the integral drawn with Inkscape. I applied rotation and translation to get the bottom half of the integral sign based on symmetry. Of course, you can draw the entire integral sign as SVG.

Then, the important command is \my_customize_math_operator:cnnnnnn, which declares a custom math operator with parameters to fine-tune horizontal and vertical positioning of subscript and superscript. You can play with the magnification scale and superscript/subscript placements to find your desirable settings.

A major problem of this approach is that it does not support style selection via \mathchoice. If you are using LuaTeX, then this problem can be bypassed by using \mathstyle. If you have to use other TeX compilers, then I think there is no other approach than manually specifying different inline/display style operators.

The following example only works with LuaLaTeX

\documentclass{article}
\usepackage{fontspec}
\usepackage{unicode-math}
\usepackage{tikz}
\usepackage{adjustbox}
\usepackage{expl3}
\setmainfont{TeX Gyre Schola}
\setmathfont{TeX Gyre Schola Math}
\usetikzlibrary{svg.path}
% top half of the symbol drawn with Inkscape
\newcommand{\MyIntTop}{%
\begin{tikzpicture}[yscale=-1]
\filldraw[fill=black] svg "m 25.357197,131.54532 c 0,0 0.100226,-7.95149 0.868626,-9.87696 0.7684,-1.92547 2.405424,-4.27882 4.510172,-4.1362 2.104749,0.14263 2.271791,1.8185 2.004523,2.49598 -0.267269,0.67749 -2.238384,-0.004 -1.937705,-0.71313 0.200452,-1.00994 -0.935444,-1.06522 -1.570208,-0.53037 -0.634765,0.53486 -1.102486,0.85128 -1.503391,2.52715 -0.400904,1.67587 -0.501131,10.29135 -0.501131,10.29135 z";
\end{tikzpicture}%
}
% build the entire symbol based on symmetry
\newcommand{\MyIntCombined}{%
    \adjustbox{rotate=180, lap=0.8mm, raise=0.1mm}{\MyIntTop}\MyIntTop%
}
\newcommand{\MyIntAdj}[1]{
    \adjustbox{#1}{\MyIntCombined}
}
\ExplSyntaxOn
% customize superscript and subscript positioning
% #1: math symbol new command name
% #2: math symbol base command 
% #3: superscript vshift
% #4: superscript hshift
% #5: subscript vshift
% #6: subscript hshift
% #7: subscript/superscript style
\cs_set:Npn \my_customize_math_operator:cnnnnnn #1#2#3#4#5#6#7 {
\tl_new:c {l_#1_sub_tl}
\tl_new:c {l_#1_super_tl}
\tl_new:c {l_#1_math_style_tl}
\bool_new:c {l_#1_finish_bool}

% declare the command
\cs_set_protected:cpn {#1} {
    \tl_clear:c {l_#1_sub_tl}
    \tl_clear:c {l_#1_super_tl}

    % check subscript
    \peek_catcode:NTF \c_math_subscript_token {
        \use:c {#1_sub:Nn}
    } {
        \peek_catcode:NTF \c_math_superscript_token {
            \use:c {#1_super:Nn}
        } {
            \use:c {#1_make_op:}
        }
    }
}


\cs_set_protected:cpn {#1_sub:Nn} ##1##2 {
    \tl_set:cn {l_#1_sub_tl} {##2}
    %\tl_show:c {l_#1_sub_tl}
    % check for superscript afterwards
    \peek_catcode:NTF \c_math_superscript_token {
        \use:c {#1_sub_super:Nn}
    } {
        \use:c {#1_make_op:}
    }
}

% superscript after subscript
\cs_set:cpn {#1_sub_super:Nn} ##1##2 {
    \tl_set:cn {l_#1_super_tl} {##2}
    %\tl_show:c {l_#1_super_tl}
    \use:c {#1_make_op:}
}

\cs_set_protected:cpn {#1_super:Nn} ##1##2 {
    \tl_set:cn {l_#1_super_tl} {##2}
    %\tl_show:c {l_#1_super_tl}
    % check for subscript afterwards
    \peek_catcode:NTF \c_math_subscript_token {
        \use:c {#1_super_sub:Nn}
    } {
        \use:c {#1_make_op:}
    }
}

% subscript after superscript
\cs_set_protected:cpn {#1_super_sub:Nn} ##1##2 {
    \tl_set:cn {l_#1_sub_tl} {##2}
    %\tl_show:c {l_#1_sub_tl}
    \use:c {#1_make_op:}
}

% retreive current math style
% only works in LuaTeX

%    \cs_set:Npn \my_save_math_style: {
%        \iow_term:x {math~style~is~\mathstyle}
%        \int_case:nn {\mathstyle} {
%            {0} {\tl_set:cn {l_#1_math_style_tl} {\displaystyle}}
%            {1} {\tl_set:cn {l_#1_math_style_tl} {\crampeddisplaystyle}}
%            {2} {\tl_set:cn {l_#1_math_style_tl} {\textstyle}}
%            {3} {\tl_set:cn {l_#1_math_style_tl} {\crampedtextstyle}}
%            {4} {\tl_set:cn {l_#1_math_style_tl} {\scriptstyle}}
%            {5} {\tl_set:cn {l_#1_math_style_tl} {\crampedscriptstyle}}
%            {6} {\tl_set:cn {l_#1_math_style_tl} {\scriptscriptstyle}}
%            {7} {\tl_set:cn {l_#1_math_style_tl} {\crampedscriptscriptstyle}}
%        }
%        \tl_show:c {l_#1_math_style_tl}
%    }
% make the operator
\cs_set_protected:cpn {#1_make_op:} {
    #2
    \tl_if_empty:cF {l_#1_sub_tl} {
        \c_math_subscript_token {
            \mkern#6\relax
            \adjustbox{raise=#5}{$#7 \tl_use:c {l_#1_sub_tl}$}
        }
    }
    \tl_if_empty:cF {l_#1_super_tl} {
        \c_math_superscript_token {
            \mkern#4\relax
            \adjustbox{raise=#3}{$#7 \tl_use:c {l_#1_super_tl}$}
        }
    }
}


}
% declare script style
\my_customize_math_operator:cnnnnnn {myintscript} {\MyIntAdj{scale={0.4}{0.4},raise=1mm}} {-1pt} {0mu}  {0.5pt} {-2mu} {\scriptstyle}
% declare display style
\my_customize_math_operator:cnnnnnn {myintdisplay} {\MyIntAdj{scale={0.7}{0.7},raise=1mm}} {-3.5pt} {1.5mu} {1pt} {-10mu} {\scriptstyle}
\int_new:N \l_myint_style_int
% finally, declare our operator
\newcommand{\myint}{
    \int_case:nnF {\mathstyle} {
        {0} {\myintdisplay}
    } {\myintscript}
}
\ExplSyntaxOff
\begin{document}
$\int_a^b \myint_{abc}^{bef} \myint^{abc}_{def} \myint \myint_a \myint^b$
[\int_a^b \myint_{abc}^{bef} \myint^{abc}_{def} \myint \myint_a \myint^b]
This equation gives
[
\myint^x_0 \frac{ds}{\sqrt{a-x}}
]
\end{document}

Answer 4

Disclaimer: This is probably a terrible idea.

Define a new command \exint (exquisite integral), which is just the ordinary integral, but sheared left 30% and thickened 50% using transition matrix [[1.5 -.3] [0 1]]. Spacing is then adjusted (sort of) with a phantom integral. An invisible rule sets the height for limits of integration (if any).

To get the limits of integration closer to the example, the upper limit should have a thin space (\,) and the lower limit should have a negative thin space (\!).

Here is the code:

\documentclass{article}
\usepackage{mathtools} % for \mathrlap
\newcommand{\exint}{\pdfliteral{ q 1.5 0 -.3 1 0 0 cm}\mathrlap{\int}\pdfliteral{ Q}\phantom{\int}\rule[-1.85ex]{0pt}{4.65ex}}
\begin{document}
[
s=\frac{\sin n\pi}{\pi}\exint_{!0}^{,x}\frac{\phi(a)da}{(x-a)^{1-n}}\
]
\end{document}

The idea for the shearing matrix comes from this answer by David Carlisle