An Interactive, Multi-layer Edition Of Jacob Bernoulli's Scientific Notebook Meditationes As Part Of Bernoulli-Euler Online

Introduction
The

Meditationes
is the scientific notebook of the mathematician Jacob I Bernoulli (1654–1705), a member of the Bernoulli dynasty from Basel. The notebook consists of 367 pages; more than 90% of the 287 entries deal with issues in mathematics and physics. Parts of the
Meditationes have been published in six volumes over the past decades according to rather varying standards.

Our project will provide a complete edition of the manuscript for the first time, providing facsimiles, transcriptions (from diplomatic to normalized texts), translations, and comments. The edition is part of

Bernoulli-Euler Online
(BEOL), a platform for early modern mathematical texts. BEOL makes texts available to historians of science and will provide tools for working with the resources available on the platform.

Transcription of the Meditationes

Regions and normalized text
The transcription of the
Meditationes is based on the digital facsimiles of the manuscript. For each entry – a Meditatio –, regions are defined on the manuscript page and transcribed individually. Regions are categorized as title regions containing the entry’s number, text regions (mainly philological text), mathematical notation regions (mainly equations), and mathematical figure regions. Regions can be connected by cross references.

Figure 1: Part of Meditatio 165

Figure 1 shows a part of Meditatio 165. Two regions are shown (marked in green). The region at the bottom of the page (M165-07-T) contains a cross reference sign “#” (marked in red) that reoccurs in the region in the page’s margin (M165-08-TN). This means that the text in the margin belongs to a specific place in “M165-07-T”. When presenting the transcriptions, we offer a multi-layer graphical user interface (GUI). The diplomatic layer shows the single regions aligned with their transcriptions. When specifying a region in the GUI, the associated transcription is shown to the user. The next layer shows a text that is easier to read with the margin region directly integrated into the text of region “M165-07-T”. The normalized text reflects the Meditatio’s logical structure: it is a sequence of paragraphs that can span regions and pages. The example below shows the normalized text that combines the transcription of “M165-07-T” and “M165-08-TN” (indicated by curly braces):
[...] hoc est, { quia

d
d
x

.

d
x

:
:

d

y

3

.

I
n
t
.

d

y

3

, erit [...] proportional‹es› ‹?› }

d
d
x

.

d
x

:
:

d

y

3

.

I
n
t
.

d

y

3

& quia etiam [...]

Regions are geometrically defined and shown as overlays. The regions can be converted to IIIF image URLs. The International Image Interoperability Framework (IIIF) defines a URL-based syntax for accessing images, allowing for interoperability among image repositories. (Appleby et al.) With IIIF, regions can be easily viewed as image files, defining their format, resolution, and even rotation. Rotation is useful when text has been added vertically in a margin, as in “M165-08-TN”, which is easier to read when it is rotated 270 degrees clockwise.

Philological and mathematical text
The transcription of the Meditationes is based on two markup systems: XML for the philological aspects of the text, and LaTeX for mathematical notation. LaTeX is the input format preferred by the specialists working on the project, and can be automatically converted into MathML if necessary. To represent both the mathematical and philological aspects of the text, LaTeX mathematical notation is embedded in the XML-based transcription.

The XML transcriptions can be transformed to TEI/XML by means of XSL transformations.
In some cases, the surrounding context has to be taken into account when rendering the LaTeX, e.g., if text is underlined or struck through. This can be achieved by dynamic macro insertion to avoid redundancy in the transcription source files.

Figure 2: Title Text of Meditatio 54

Figure 2 shows the title text of Meditatio 54. In the manuscript, philological text and mathematical notation are not differentiated: both kinds of text are underlined. The excerpt below shows a part of the transcription source. When transforming the XML to HTML using an XSL transformation, macros are inserted dynamically if needed. In case of “abc”, the macro “\underline{abc}” is inserted dynamically.
<hi rend="underline">In <choice><abbr>ang<am>.</am></abbr><expan>ang<ex>ulo</ex></expan></choice> <formula notation="tex">abc</formula>. ductâ obliquè <lb/>
...
</hi>
Philological text can be converted to HTML for rendering in a web browser. The

MathJax
library is used to render mathematical notation. MathJax generates HTML and CSS from LaTeX or MathML, and can be instructed to render mathematical notation as underlined or struck through.

When using LaTeX macros directly in the source, custom macros are used that are replaced when transforming the XML source code to HTML. This way, the usage of MathJax-specific macros can be avoided, making it possible to use other rendering environments, such as a LaTeX-based typesetting system for print output. We have found

Pandoc
to be a useful tool for changing the implementation of a macro. The XML transcription files are preprocessed: for each formula, the LaTeX is passed to Pandoc, which replaces custom macros with the implementation to be used in the production environment.

https://pandoc.org/MANUAL.html#latex-macros

Translations
For each Meditatio, translations in English, German, and French are available. Translations mainly refer to the normalized text, and can be aligned with the regions by their IDs. To avoid redundancy, complex formulae are referenced rather than re-encoded. In some cases, additional macros are needed for language support.
<formula notation="tex" id="M151-09-M_1">
\left.
\begin{array}{lcr}
& & BE \\
\language{Trigono}{Dreieck}{triangle}{triangle} & & \frac{4}{3}a \\
\language{Quadrato}{Quadrat}{square}{carré} & & a \\
\language{Pentagono}{Fünfeck}{pentagon}{pentagone} & & \frac{4}{5}a \\
\language{Hexagono}{Sechseck}{hexagon}{hexagone} & & \frac{2}{3}a \\
\language{Heptagono}{Siebeneck}{heptagon}{heptagone} & & \frac{4}{7}a \\
\end{array} \
\right\}
\language{ erit }{abschneidet, so wird }{, one has }{, on aura } CE \language{ vel }{ oder }{ or }{ ou } Ce (b)\ =
\left\{
\begin{array}{r}
\frac{1}{3}a \\
0 \\
\frac{1}{5}a \\
\frac{1}{3}a \\
\frac{4}{7}a \\
\end{array}
\right\}
</formula>
The excerpt above shows LaTeX code for a formula referenced from the translation file. Depending on the target language, the macro “\language{...}{...}{...}{...}” has to be replaced with the correct argument. In case of the English translations, the first argument is taken into account, in case of German the second and so forth. This can be accomplished by using Pandoc to preprocess the XML.

Integration into Bernoulli-Euler Online
The transcriptions, facsimiles, translations, and comments of the
Meditationes will be imported into
Bernoulli-Euler Online (BEOL), a web-based platform for studying early modern mathematics. The platform already contains correspondence (currently about 2000 letters) relating to members of the Bernoulli dynasty and Leonhard Euler. The material in BEOL has been imported from existing editions, both print and digital. For BEOL, a comprehensive index of persons and bibliographical items had to be created. This index makes it possible to search for persons and bibliographical items in a global manner, overcoming the previous divisions into different parts of the edition. (Schweizer et al., 2017; Alassi et al., 2018) BEOL will serve as a single entry point for historians of science interested in the works of members of the Bernoulli dynasty and Leonhard Euler.

The BEOL platform is based on
Knora, a generic framework for storing, sharing, and working on primary sources and data in the humanities. All operations (reading, creating, changing of data) are performed through calls to the Knora API over HTTP. This API will enable clients to integrate the material available on BEOL into other platforms and vice versa. BEOL contributes to the development of Knora. Project-specific requirements are generalized so they can be implemented as generic components. These components can then be used by projects other than BEOL.

Bibliography

Alassi, Sepideh, Tobias Schweizer, Martin Mattmüller, Lukas Rosenthaler, and Helmut Harbrecht (2018). ‘A Digital Edition of Leonhard Euler’s Correspondence with Christian Goldbach’. In
Digital Humanities 2018 Conference Abstracts. https://dh2018.adho.org/a-digital-edition-of-leonhard-eulers-correspondence-with-christian-goldbach/.

Appleby, Michael, Tom Crane, Robert Sanderson, Jon Stroop, and Simeon Warner ‘IIIF Image Api 2.1.1’. https://iiif.io/api/image/2.1/.

Schweizer, Tobias, Sepideh Alassi, Martin Mattmüller, Lukas Rosenthaler, and Helmut Harbrecht (2017). ‘Integrating Historical Scientific Texts into the Bernoulli-Euler Online Platform’. In
Digital Humanities 2017 Conference Abstracts, 585–87.
https://dh2017.adho.org/abstracts/147/147.pdf.