1/22<h2>Dating the manuscript </h2> The brown calf leather and wood binding of this manuscript date from the 15th century.

2/22Small blind-stamp decorations (lilies, finial, and quatrefoil) decorate the front, side and back cover.

3/22<h2>Signs of time</h2> Traces of iron clasps from the original binding are clearly visible. They caused damages which a later restoration of the binding attempted to correct. Finally remains of nails in the front and back cover are found. Continue with <a href="https://alfa.obspm.fr/heritage/touching?screen=gP8AyLbXDiz8xK5YADWG" rel="noopener noreferrer" target="_blank">Touching manuscript path</a>

4/22On these images of the bottom and top sides of the codex, it is possible to grasp the complex structure of the codex as composed of quires of paper of different sizes, origins, and quality. This reflects the context in which the manuscript was assembled: during the 15th century, a student in the art faculty in Paris, Conrad Heingarter, collected here his notes on mathematical astronomy, mathematics, and other disciplines taught at the art faculty in Paris. He did not care much about the material uniformity of the codex thus created. There are many manuscripts of this kind in the corpus of Alfonsine astronomy.

5/22<h2>Binding the knowledge together</h2> The codex is composed of several quires many of which show traces of independent circulation before being bound together at the end of the 15th century. This page is a very good example of this phenomenon. The paper of the page is marked by several traces of uses and we see the title “Philosophia” marked in the middle of the page, as if the folded folio on which this page is marked was meant to be a paper folder for the quire before it was bound in the codex attested today. Continue learning about <a href="https://alfa.obspm.fr/heritage/typical?screen=VgwEKvQLABO2z3Fv2u5g" rel="noopener noreferrer" target="_blank" style="color: rgb(0, 0, 0);">typical manuscripts.</a>

6/22 The page layout in this manuscript is very diverse. Starting from this pages, the four following folios of the same quire we can see cases of layout in one, two, three, or four columns sometimes with breaks in the middle of the page. This lack of stability in the page organization is a consequence of the content of the copied text but also of the production process of this manuscript as the accumulated notes of Conrad Heingarter in the art faculty while he was a student at Paris University.

7/22<h2>Rubrication </h2> Here we can see what manuscript specialists call “rubrication” from the latin “ruber” for red. Once the initial text has been copied in black or brown ink a second set of marks in red is added in general to enhance the readability of the text. These marks often&nbsp;include capital letters and different kinds of punctuation signs like virgula or pied de mouche.

8/22In this case the rubrications were probably made by Conrad Heingarter himself. As he does not systematically rubricate all his manuscripts, we know that the rubricated parts especially attracted his attention. The naive illustration used in the capital letter is an indication of Heingarter's young age when he made the rubrication. A few other instance of such naive illustration can be found in the manuscript.

9/22<h2>Guess the science</h2> The second leaf (fol. 2r-v) of Heingarter’s student manuscript contains a set of allegorical poems on different arts and sciences&nbsp;- “de omnibus artibus”. First, the scribe states why one should learn such disciplines as Grammar, Logic, Rhetoric, Music, Arithmetic, Geometry, and Astronomy. The second column of fol. 2r seems to be composed of riddles related to the listed fields of study and represented through a certain allegory, like “a student with a feather, slate-pencil, and chalk” for Grammar. [or at least that’s how I understood it] Fol. 2v contains riddle-verses on the following arts and sciences: Metaphysics, [Monastica?], Alchemy, Perspective, Mathematics, Astrology, Pyromancy, Hydromancy, Geomancy, Theriomancy (or Zoomancy), Physics, Medicine, Ethics, Canonical and state law [I suppose that the title “keiser” refers to the law of the king/state, based on the text of the verse], Economics, and Politics. Each verse personifies a certain science, explaining the ways in which it is useful.

10/22<h2>Differentiating the works in a manuscript </h2> Manuscripts from the late medieval period in Europe often contain more than one work. They are actually small libraries with different texts. The boundaries between different texts is not always marked, but when it is the copist often relies on an “Explicit” to do so.

11/22<h2>Importance of a "colophon" </h2> These small texts, placed at the end of a work may contain different types of information such as a “title” of the finished work, the name of the copyist, the place and date of copy. “Explixit” or “Colophon”, as they are also named, are very precious for the historians. In this particular case, only the first two elements are provided and allow us to identify Conrad Heingarter as the copyist of this work on proportion.

12/22<h2>A hand-drawn ruler </h2> This little ruler is inserted on the top of a page in a treatise addressing computations with fractions of different kinds. It is possible that it represents some sort of computation tool used by medieval astronomers and mathematicians. In any case, it is also useful and operational as a diagram. Continue learning about <a href="https://alfa.obspm.fr/heritage/seeing?screen=iWUWuJ01AI1qzbyRwyFV" rel="noopener noreferrer" target="_blank">the visual dimension of a manuscript.</a>

13/22﻿The ruler puts in relation sexagesimal and natural fractions. One sees for instance that two thirds correspond to 40, one half to 30, one fourth to 15 and so on. This type of correspondence can be very handy for interpolation in order to get a quick estimation of the result by mental computations.

14/22This way to compute approximate interpolations is described in some astronomical text as “interpolation by denomintation”. This last word, very important in medieval mathematics, is found on the right of the rulers.

15/22<h2>The use of fractions </h2> One specificity of computation in mathematical astronomy, at least until the end of the 15th century, is that they make heavy use of fractions in general and sexagesimal fractions in particular. Fractions were thus a specific topic in medieval mathematics with different sub-branches and dedicated treatises that astronomers needed to master. This page is from one of those treatises on fractions and we can see that Conrad Heingarter studied it thoroughly while being a student at the art faculty.

16/22One topic about fractions was to classify them in different types. Different classifications were in circulation, often transmitted from Arabs sources. One of them is summarized in this little table.

17/22The principle of this classification is to consider fractions of fractions at many different levels, like illustrated in this small diagram where an initial division by 3 is composed with two successive division by four. Continue learning about <a href="https://alfa.obspm.fr/scientific/disciplines?screen=cDjyLZ0BhhXnjArUIzli" rel="noopener noreferrer" target="_blank" style="color: rgb(0, 0, 0);">mathematical tools.</a>

18/22<h2>Planetary theory as geometric visualization</h2><h2> </h2>Among the different texts a late medieval astronomer needs to master, the planetary theory texts are essential. They provide a geometrical definition for most of the quantities that need to be computed with tables, they also allow to visualize the circular motions which are used to decompose celestial phenomena. Finally, they can act as a support to geometrical reasoning of different kinds.

19/22These texts generally rely on diagrams named “figura” or “theorica”. Here we see one of them related to Mercury. While many aspects of this diagram are shared with hundreds of others found in different manuscripts it is nonetheless quite original because the Mercury epicycle is represented with different sizes. We do not know of any Latin medieval planetary theory where the epicycle could have varying sizes (this kind of idea can be found, for the sun, in al-Andalus; it was also systematically explored in India). The fact that this, so far as we know, strictly graphical innovation, occurs for Mercury is interesting. Indeed the “theorica” of Mercury has a device to change the apparent size of its epicycle.

20/22<h2>An unconventional approach to Parisian Alfonsine Tables</h2> This is a page from a work giving a very peculiar version of the Parisian Alfonsine Tables of which only 4 manuscripts are known. The most striking feature of this version of the Parisian Alfonsine tables is its compactness. Instead of being given every degree, the planetary equations, or corrections, are given every five degrees. In addition to this, only the main quantities are provided and not the auxiliary ones allowing to perform the necessary interpolation. In other words, these tables can be of very litle use for computation! It is quite like an instrument with no or imprecise scales that can not be used for measurement. This shows that tables were not always computational tools, but also a means to display mathematical relations between astronomical quantities complementary to that of diagrams and text.&nbsp;

21/22<h2>Nicole Oresme and the proportion diagrams</h2> This series of very impressive geometrical and proportion diagrams begins on the previous page. They are a part of the “Algorismus proportionum” by Nicole Oresme, a treatise dedicated to the computation of proportions. This, with knowledge of fractions, was part of the fundamental mathematical knowledge a late medieval astronomer had to master in order to practice his art.

22/22<h2>Find the center?</h2> In these two diagrams we see Conrad Heingarter learning the geometrical construction allowing one to find the center of a given circle. These are part of a text named “inveninendum centrum circuli”. It is one of the multiple mathematical skill a young astronomer needed to learn.

This is a mid-15th-century manuscript, probably of Parisian provenance.It is interesting firstly for being the student manuscript of Conrad Heingarter, a famous astronomer and astrologer of the time, who served at the court of John II, Duke of Bourbon. The manuscript seems to have been composed during Heingarter's study years at Paris University.